Keywords: Energy cascade, unsteady turbulence, inhomogeneous turbulence, direct numerical simulation

Three-dimensional turbulence is characterised by a multi-scale mechanism in which the energy injected by external forcing and/or boundary conditions is transferred to smaller scales, where it is dissipated into heat by molecular viscosity. Although this energy cascade is a commonly accepted conceptual picture of turbulence, providing the basis for statistical theory, its precise understanding is yet missing. This thesis consists of three parts and investigates the turbulent kinetic energy cascade from time, space, and scale-point of view.

The first part discusses quasi-cyclic temporal behaviour observed in turbulence driven by a steady forcing. To construct a minimal three-equation model reproducing quasi-cyclic behaviour, lower-Reynolds-number periodic flow is decomposed into scales representing the external forcing, energetic modes, and energy dissipation. Furthermore, the similarity between this low-dimensional model and turbulence is assessed using phase-space orbits and bifurcation analysis.

The second part discusses the physical-space locality of nonlinear interactions in turbulence. More specifically, the relation between the spatial locality and the multi-scale nature of nonlinear interactions is discussed using a system in which the nonlinear term of the Navier-Stokes equations is restricted in a space-local sense. The results show that the nonlinear interactions restricted to a space-local domain of radius R can sustain the energy cascade for wavenumbers 1/R.

The third part discusses nonequilibrium scaling of turbulence induced by large-scale fluctuations. First, large-scale inhomogeneity is considered as a perturbation around the k -5/3 wavenumber scaling of the energy spectrum, allowing us to derive a nonequilibrium scaling correction proportional to k -7/3 . This nonequilibrium scaling, decaying faster than the equilibrium scaling as a function of the wavenumber, is confirmed numerically. Furthermore, by applying the same methodology for large-scale inhomogeneity and unsteadiness, a unified description of the scaling law between the fluctuations of the normalised energy dissipation rate and the Reynolds number is proposed.

Résumé

La turbulence tridimensionnelle est caractérisée par un mécanisme multi-échelle dans lequel l'énergie injectée par un forçage externe et/ou des conditions aux limites est transférée à des échelles plus petites jusqu'à ce qu'elle soit dissipée en chaleur par la viscosité moléculaire. Bien que cette cascade d'énergie soit une image conceptuelle communément acceptée, fournissant la base de la théorie statistique de la turbulence, sa compréhension précise fait encore défaut. Cette thèse se compose de trois parties et étudie la cascade d'énergie cinétique turbulente du point de vue du temps, de l'espace et de l'échelle.

La première partie traite du comportement temporel quasi-cyclique observé dans un écoulement turbulent maintenu par un forçage constant. Pour construire un modèle minimal à trois équations reproduisant un comportement quasi-cyclique, l'écoulement à faible nombre de Reynolds est décomposé en échelles représentant le forçage externe, les modes énergétiques et la dissipation d'énergie. De plus, la similitude entre ce modèle à basse dimension et la turbulence est évaluée à l'aide d'orbites dans l'espace des phases et d'une analyse de bifurcation.

La deuxième partie discute la localité dans l'espace physique des interactions non linéaires dans la turbulence. Plus précisément, la relation entre la localité spatiale et la nature multi-échelle des interactions non linéaires est examinée à l'aide d'un système dans lequel le terme non linéaire des équations de Navier-Stokes est restreint dans l'espace. Les résultats montrent que les interactions non linéaires restreintes à un domaine spatial local de rayon R peuvent entretenir la cascade d'énergie à des nombres d'onde 1/R. La troisième partie discute le comportement hors équilibre de la turbulence induite par des fluctuations à grande échelle. Tout d'abord, l'inhomogénéité à grande échelle est considérée comme une perturbation autour de du spectre d'énergie d'équilibre (proportionnel à k -5/3 ), ce qui nous permet de dériver une correction proportionnelle à k -7/3 . Cette correction, qui décroît plus rapidement que le spectre d'équilibre (en fonction du nombre d'ondes), est confirmée numériquement. De plus, en appliquant la même méthodologie pour l'inhomogénéité et l'instabilité, une description unifiée de la relation entre les fluctuations du taux de dissipation d'énergie normalisé et le nombre de Reynolds sont proposés.

Mots clés: Cascade d'énergie, turbulence instationnaire, turbulence inhomogène, simulation numérique directe

i 要旨 3次元の乱流は「外力や境界条件によって大スケールへ注入されたエネルギがより小さなスケールへ輸 送され,最後は分子粘性によって熱として散逸する」というマルチスケール機構によって特徴づけら れる.この「エネルギカスケード」は乱流の統計理論に基礎づけを与える広く受けいれられた概念的 描像であるものの,その精緻な理解はいまだ得られていない.本学位論文は三部からなり,乱流の運 動エネルギカスケードを時間・空間・そしてスケールの観点から議論する. 第一部では,定常な外力で駆動される乱流にみられる時間準周期的なふるまいを議論する. Reynolds数 の低い周期流を外力・エネルギ・エネルギ散逸率を代表するスケールに分割することで,準周期変動 を再現するミニマルな3方程式モデルを構築する.また,相空間での軌道や分岐解析を通じて,この低 自由度のモデルと乱流の類似性を評価する. 第二部では,乱流の非線形相互作用の物理空間における局所性を議論する. 具体的には,Navier- Stokes方程式の非線形項を空間局所に制限した系を通じて,非線形相互作用の空間局所性とマルチスケ ール性の関係を調べる.その結果,半径Rの空間局所領域で定義される非線形相互作用が波数 1/Rで のエネルギカスケードを維持しうることを明らかにした. 第三部では,大スケール変動が励起する乱流の非平衡スケーリングを議論する.まず,大スケールの 非一様性をエネルギスペクトルのk -5/3 波数スケーリング周りの摂動として考慮することで,k -7/3 に 比例する非平衡スケーリング修正を導出する.この平衡スケーリングより波数の関数としてはやく減 衰する非平衡スケーリングは数値的に確認された.さらに,同様の手法を大スケールの非一様性と非 定常性に対して適用することで,無次元化されたエネルギ散逸率とReynolds数の変動の間に統一的に 成立するスケーリング則を提案する. キーワード:エネルギカスケード,非定常乱流,非一様乱流,直接数値計算
On the evening of October 4, 2020, I was waiting for the flight to France at the Kansai International Airport. Being in the middle of the COVID-19 pandemic, I was alone in the pretty much empty international departures concourse. The flight to the Paris Charles de Gaulle Airport was almost vacant, too; I could lie down to rest using three seats next to me. I still remember that half-exciting and half-anxious mixed feeling on that day about spending three years in a country I had never been to. This is how my PhD has started. After three years, I am now completing my PhD thesis. First and foremost, I express my sincerest gratitude to my two supervisors. Wouter J. T. Bos guided me all the way through my PhD. The works compiled in this thesis have been conducted in a double degree program (cotutelle) between École Centrale de Lyon and Osaka University. Even though the COVID-19 pandemic and the corresponding very strict immigration restriction policy of Japan prohibited me from going to Japan, I have been a member of the Fluid Mechanics Group (Goto lab) in Osaka during my PhD, too. Susumu Goto has been my supervisor for six years since I joined the lab in my final year of undergraduate and he taught me how to be a researcher through his enthusiasm for research. I spend my daily life with cheeful lab members at Laboratoire de Mécanique des Fluides et d'Acoustique (LMFA). I thank Tong Wu, Wesley Agoua, Henri Lam, Bruce (Xi Yuan) Yin, and Smiron Varghese for daily discussion, conversation, and occasional hang out. I also thank the past and present members of the Goto lab. In particular, I acknowledge Kiwamu Yoshii. Although we shared the same office at Osaka University for less than half a year, we share enthusiasm on many topics. During the three years, I have been interacting with many inspiring researchers. First, I appreciate my collaborators. I owe Tomohiro Tanogami a lot of things. He introduced me to the field of information thermodynamics and how turbulence and its energy cascade can be described in such a way. With Yohann Duguet, we planned to extend our study of the minimal model of quasi-cyclic behaviour from a dynamical system's point of view. I began a new project with Adrián Lozano-Durán and Alberto Vela-Martín, which is still in its very preliminary stage and is not included in this thesis. I look forward to finding new turbulence properties from an information-theoretic point of view. Next, I thank the researchers in the lab: Benjamin Miquel, Michio Otsuki, and Yutaro Motoori. In particular, I thank Masanobu Inubushi for his critical comments on my research and advice on my career. Finally, I thank my thesis committee members: Fujihiro Hamba, Genta Kawahara, and Kazuyasu Sugiyama, for Osaka-side and Laurent Chevillard, Sergio Chibbaro, Bérengère Dubrulle, Yohann Duguet, and Alain Pumir for Lyonside. Their comment greatly helped me to revise and improve my thesis. I have been fortunate to have financial support from several scholarship foundations during my study. I sincerely thank the Fujii International Scholarship Foundation for their support during the final year of the bachelor, Yoshida Scholarship Foundation for the master, and Takenaka Scholarship Foundation for the overseas PhD. Without their support, my career would have been completely different. I also appreciate opportunities to interact with other scholarship students, in which I can make invaluable friends. Here, I acknowledge the city of Lyon and Écully, where I spend three years. Although it is true that the PhD is often highly stressful, the occasional free afternoon in this beautiful city offers great refreshment. In particular, I love walking around Vieux Lyon (Old Lyon) and the promenade along the two rivers: le Rhone and la Saône. I have many more great memories in Lyon; the view of the city from Fourviére, a variety of activities offered in Parc de la Tête-d'Or (Park of the Golden Head), cuisine in bouchon, splendid Fête des Lumiéres (Festival of Lights) in December, and so on. I will miss Lyon a lot and definitely come back in the near future. Finally, I thank my family for their continuous support, which leads me to who I am today.

List of Frequently Used Symbols

Re

Reynolds number (defined on page 7)

A

Velocity gradient tensor (defined on page 8)

f Forcing field = (f x , f y , f z ) (defined on page 6) k Fourier mode = (k x , k y , k z ); also p, q (defined on page 9)

N Nonlinear term = (N x , N y , N z ) (defined on page 67) S Rate-of-strain tensor (defined on page 8)

u Velocity field = (u x , u y , u z ) (defined on page 6)

x Physical space coordinate = (x, y, z) (defined on page 6)

C

Normalised energy dissipation rate (defined on page 16)

Mathematical operators

• ≶ Band-pass filter (defined on page 65)

• < Low-pass filter (defined on page 12)

• > High-pass filter (defined on page 12)

• ens Ensemble average (defined on page 85) xii

List of Frequently Used Symbols

• Ω Spatial average (defined on page 7)

• phase Temporal phase average (defined on page 38)

• t Temporal average (defined on page 9)

Superscripts

• * Complex conjugate (defined on page 10)

• L Defined by the integral length (defined on page 107)

• L Spatially local component (defined on page 62)

• NL Spatially nonlocal component (defined on page 70)

• k f Defined by the forcing length (defined on page 103)

Subscripts

• η Kolmogorov length related quantity (defined on page 13)

• ω Enstrophy related quantity (defined on page 19)

• f Forcing length related quantity (defined on page 13)

• X X related quantity (defined on page 43)

• Y Y related quantity (defined on page 43)

• Z Z related quantity (defined on page 47)

Overview of the Thesis

This thesis investigates turbulence by focusing on one of the most important concepts in turbulence research: the energy cascade. In three-dimensional turbulence, it represents an overall tendency for the energy to transfer from large to small scales. Despite its wide acceptance as a key ingredient of the statistical description of turbulence, the precise mechanism of the energy cascade has been at the centre of debate in the community for over a century. This thesis consists of three parts, investigating quasicyclic temporal behaviour, spatial locality, and nonequilibrium scaling of turbulence, respectively. In other words, we investigate turbulence and energy cascade from three different perspectives: time, space, and scale.

Chapter 1 is a general introduction. We first introduce the concept of energy cascade with the role of inviscid invariants and focus on its scale-local nature ( § 1.1). Next, we review how this idea led to the equilibrium and nonequilibrium theory of turbulence ( § 1.2). We discuss the Kolmogorov-Onsager theory of small-scale universality of turbulence, emphasising the assumption of equilibrium in the small scales.

We also discuss the invalidity of such an assumption and the nonequilibrium nature of turbulence. Then, we further investigate a few candidates for the physical mechanism of the energy cascade ( § 1.3), which has been an open question in the turbulence community to this day.

Part I: Quasi-cyclic temporal fluctuations of turbulence and its minimal model

The main topic of Part I is the origin of quasi-cyclic temporal fluctuations observed in turbulence. Turbulence is often referred to as a typical example of chaos, which indeed captures the unpredictable nature of turbulence. An interesting feature in certain types of turbulence is the observation of quasi-cyclic fluctuations in random time signals.

In Chap. 2, we review these robust quasi-cyclic temporal fluctuations, discussing the concept of periodic orbits and highlighting an example of a closed system called von Kármán flow ( § 2.1). We also briefly review reduced-order modelling of developed and transitional turbulence ( § 2.2). In Chap. 3, we first observe and quantify the quasi-cyclic fluctuations of a flow in a box with periodic boundary conditions maintained by a steady forcing ( § 3.1). We then extract a minimal three-equation model from a low-Reynolds number periodic flow by regrouping energetic modes at different scales ( § 3.2).

The minimal model qualitatively reproduces quasi-cyclic behaviour, and we further conduct a bifurcation analysis to discuss the similarity between the steady-chaotic or periodic-chaotic transition in the minimal model and laminar-turbulent transition in real turbulence ( § 3.3).

Part II: Space-local Navier-Stokes turbulence

Turbulence generally contains structures of different sizes. The energy cascade picture can be understood as the energy exchange between structures of similar scales, known as the scale locality of turbulence. However, the scale-space representation of turbulence does not provide its physical-space description. The main topic of Part II is the other locality of turbulence, in physical space. In Chap. [START_REF] Araki | Inertial range scaling of inhomogeneous turbulence[END_REF], we first review studies on the physical-space locality of turbulence ( § 4.1). We then introduce the space-local velocity field, defined by the neighbouring vorticity field via a truncated Biot-Savart law ( § 4.2). We then conduct a post-process analysis on a turbulence dataset to compute the turbulence spectra with the space-local velocity and evaluate how spatially local turbulence is ( § 4.3). In Chap. 5, we examine the physical-space locality of turbulence by employing the space-local velocity field in situ in numerical simulation. We define a new system called the space-local Navier-Stokes equations by

Overview of the Thesis restricting the nonlinear term in space-local sense ( § 5.1). The direct numerical simulation of the spacelocal system exhibits qualitatively different behaviour compared to the post-process analysis reported in § 4.3 ( § 5.2). In particular, we focus on two scaling regimes in Fourier space. We observe that the E(k) ∝ k -5/3 scaling of the energy spectrum, corresponding to energy cascade, survives under the space-local constraint in scales smaller than the filter size ( § 5.3). This result illustrates the physical space-locality of the turbulent energy cascade, which should not be confused with the well-known scale locality. It is shown that enstrophy production is suppressed in scales larger than the filter size, and an alternative scaling, representing a conservative enstrophy cascade, emerges ( § 5.4).

Part III: Nonequilibrium scaling in inhomogeneous or unsteady turbulence

The well-known E(k) ∝ k -5/3 scaling of the energy spectrum is based on the equilibrium description of turbulence. However, turbulence is a nonequilibrium phenomenon, particularly in its large scales, as reviewed in § 1.2. In Part III, we focus on nonequilibrium scaling of turbulence due to large-scale modulation.

In Chap. 6, we discuss a correction to the E(k) ∝ k -5/3 scaling due to the large-scale inhomogeneity.

We first review past theories on scaling corrections due to large-scale unsteadiness and anisotropy ( § 6.1). Then, we theoretically examine the effect of weak statistical inhomogeneity as a perturbation to the equilibrium scaling ( § 6.2). To do so, we decompose E(k) = E 0 (k) + E 1 (k), where the subscript "0" and "1" indicates the equilibrium and nonequiilbrium contributions, respectively. Our examination yields E 1 (k) ∝ k -7/3 , which is assessed by direct numerical simulation of turbulence with a single inhomogeneous direction, known as Kolmogorov flow ( § 6.3). In Chap. 7, we discuss nonequilbrium scaling of the normalised energy dissipation rate C against the Reynolds number Re λ . We first review the scaling of the parametric plot of C (x, t) as a function of Re λ (x, t), for the case of both spatial inhomogeneity and temporal unsteadiness ( § 7.1). Then, we conduct a similar perturbation analysis, as done for the energy spectrum in Chap. 6, for C in terms of both inhomogeneous and unsteady fluctuations ( § 7.2). These theoretical findings yield the same scaling exponents and are numerically examined ( § 7.3). We further investigate the choice of the macroscopic length scale and how it changes the scaling ( § 7.4). We then linearise the newly obtained scaling expressions to obtain relations which can be compared to observations ( § 7.5). We also discuss the influence of the large-scale energy distribution on the C scaling ( § 7.6).

The final Chap. 8 provides a summary as well as some perspectives for the three parts of the thesis. The construction of the thesis is summarised in Fig 1.

Introduction: the Energy Cascade

This chapter is devoted to reviewing important concepts and past research on the energy cascade, representing the transfer of energy from large to small scales in three-dimensional turbulence. In § 1.1, we discuss energy transfer in turbulent flow. In § 1.2, we review Kolmogorov's famous 1941 theory on the small-scale universality of turbulence by emphasising the local equilibrium hypothesis. In this sense, Kolmogorov's 1941 theory describes the equilibrium state of turbulence. Note that it is not equivalent to the concept in statistical mechanics. We also discuss the nonequilibrium aspect of turbulence by focusing on the violation of the local equilibrium hypothesis. In § 1.3, we examine the physical mechanism of the energy cascade. Although it provides the phenomenological picture of turbulence, its precise mechanism has been a long-standing open question to this day. An important subject that we will not discuss in this thesis is the interaction of the energy cascade with solid walls. For discussions of that topic, we refer to [START_REF] Jiménez | Cascades in wall-bounded turbulence[END_REF] and [START_REF] Motoori | Generation mechanism of a hierarchy of vortices in a turbulent boundary layer[END_REF].

Energy cascade in three-dimensional turbulence

A one hundred years ago, Lewis F. Richardson, who dreamed about weather prediction 1 , described the motion of whirls in cumulus clouds in a parody of Jonathan Swift's poem 2 as [START_REF] Richardson | Weather Prediction by Numerical Process[END_REF], CH. 4/8/0, p. 66):

We realize thus that: big whirls have little whirls that feed on their velocity, and little whirls have lesser whirls and so on to viscosity -in the molecular sense. This poem has, thanks to its brevity and rich insight on the physical phenomena, been cited numerous times in textbooks, monographs, and research articles over a century3 .

1 He wrote in the preface of his book (Richardson 1922, p. vii):

Perhaps some day in the dim future it will be possible to advance the computations faster than the weather advances and at a cost less than the saving to mankind due to the information gained. But that is a dream.

See [START_REF] Lynch | Richardson's forecast: the dream and the fantasy[END_REF] for his projects, perception of his book, and the dawn of weather forecasting.

2 However, Richardson did not explicitly mention this in his book. The poem reads (De Morgan 1872, p. 377):

Great fleas have little fleas upon their backs to bite 'em, And little fleas have lesser fleas, and so ad infinitum.

And the great fleas themselves, in turn, have greater fleas to go on; While these again have greater still, and greater still, and so on.

Great whirls have little whirls That feed on their velocity And little whirls have lesser whirls and so on to viscosity Note that the terms "that" and "their" may refer to different things here. The authors also proposed that the original poem "might be modified as follows:"

Great whirls gobble smaller whirls And feed on their velocity; but Where great whirls grind, they also slow, And little whirls begin to grow -stretching out with high vorticity We remark here that the concept of whirls of various scales was captured before Richardson. Figure 1.1(a) is one of the most famous paintings of turbulence drawn by Leonardo da Vinci, who introduced the term "turbolenza". He depicted multi-scale eddies in water pouring into a pool, and their size even exhibits a power-law scaling (Marusic and Broomhall 2021, Figure 13). See [START_REF] Marusic | Leonardo da Vinci and fluid mechanics[END_REF] for a review of da Vinci's works in fluid mechanics. Figure 1.1(b) is another example of turbulence in a popular painting: Vincent van Gogh's "The Starry Night". His iconic brushwork fills the sky with swirling patterns, which were later found to agree with the turbulence theory [START_REF] Aragón | Turbulent luminance in impassioned van Gogh paintings[END_REF][START_REF] Beattie | Is the Starry Night turbulent?[END_REF].

In this section, we review the notion of an energy cascade in three-dimensional turbulence. First, in § 1.1.1, we define the inviscid invariants of the Navier-Stokes equations and the cascade of energy. In § 1.1.2, we discuss the scale-space description of the energy cascade, emphasising its locality.

Inviscid invariants and their cascades

We begin our discussion with the Navier-Stokes equations

∂u ∂t + (u • ∇)u = -∇p + ν∇ 2 u + f , (1.1)
defined in the three-dimensional domain x = (x, y, z) ∈ Ω with time t. For simplicity, we restrict ourselves to a triply periodic domain of size 2π: 0 ≤ x i ≤ 2π with x i ∈ {x, y, z}. Throughout this thesis, we consider turbulence far away from walls. Here, u(x, t) = (u x , u y , u z ) denotes the three-dimensional velocity field satisfying the incompressibility condition,

∇ • u = 0. (1.2)
A scalar field p(x, t) denotes the pressure (divided by density ρ) and f (x, t) = (f x , f y , f z ) is the forcing vector field driving the flow. The kinematic viscosity is denoted by ν. The curl of the velocity ω = ∇ × u

Up to the limit of viscosity. -

In Tamai (2016), Richardson's poem is cited in the one-line form but with slightly different wording:

Big whorls have little whorls Which feed on their velocity, And little whorls have lesser whorls And so on to viscosity. and Swift's poem is introduced in the one-line form, again with a slightly different choice of words:

Great fleas have little fleas Upon their backs to bite'em, And little fleas have lesser fleas And so ad infinitum.

1.1 Energy cascade in three-dimensional turbulence is called the vorticity field, and its governing equation reads

∂ω ∂t + (u • ∇)ω = (ω • ∇)u + ν∇ 2 ω + ∇ × f . (1.3)
We observe several differences between the vorticity and Navier-Stokes equations. First, the pressure gradient term disappears since ∇ × ∇p = 0.

(1.4)

Second, we have two nonlinear terms emerging from the advection term (u • ∇)u; The vortex advection (u • ∇)ω and the vortex stretching (ω • ∇)u terms, respectively. By non-dimensionalising the variables in (1.1), one obtains a single nondimensional quantity characterising the flow, called the Reynolds number

Re ≡ U L ν . (1.5)
Here, U and L denote appropriately defined characteristic velocity and length scales, respectively 4 . For large values of Re (small values of ν), the nonlinear term (u • ∇)u dominates the flow compared to the viscous term ν∇ 2 u and the flow becomes turbulent.

Turbulent flow can be found in every aspect of our daily life. When we fully open the tap, turbulent water flows into the sink to make larger noise and splash than the half-opened tap. When boarding an aeroplane, the captain sometimes makes a warning announcement to fasten your seatbelt to be safe under vibration caused by turbulent air motion. On a schematic level, turbulence is described as the complex motion of a fluid. However, despite its common appearance, providing a precise definition of turbulence is difficult. Rather, we can describe turbulence with its characteristics: multi-scale nonlinear interactions and their self-similarity, statistical universality, and anomalous scaling, to name a few. Given its importance in such a wide range of applications, turbulence has been studied extensively using theory, experiments, and simulations. Numerical simulations of turbulence, such as Direct Numerical Simulation (DNS) numerically solving the discretised Navier-Stokes equations (1.1), require extremely large computational resources. The history of the largest DNS in the last two decades can be found in [START_REF] Aoyama | Statistics of energy transfer in high-resolution direct numerical simulation of turbulence in a periodic box[END_REF], [START_REF] Ishihara | Study of high-Reynolds number isotropic turbulence by direct numerical simulation[END_REF], [START_REF] Yeung | Dissipation, enstrophy and pressure statistics in turbulence simulations at high Reynolds numbers[END_REF], [START_REF] Ishihara | Thin shear layers in high Reynolds number turbulence-DNS results[END_REF], [START_REF] Yeung | Extreme events in computational turbulence[END_REF], [START_REF] Yeung | Effects of finite spatial and temporal resolution in direct numerical simulations of incompressible isotropic turbulence[END_REF]Buaria, Pumir, Bodenschatz, and[START_REF] Buaria | Extreme velocity gradients in turbulent flows[END_REF]. Figure 1.2 visualises the fine-scale structures in one of the largest DNS of turbulence to this day [START_REF] Buaria | Extreme velocity gradients in turbulent flows[END_REF].

In the following, we focus on the invariants of the Navier-Stokes equations (1.1) to introduce the concept of a cascade. Here, an invariant means a global quantity which does not evolve in time. When we consider the inviscid (ν = 0) and force-free (f = 0) system, we find two global invariants: the total kinetic energy and helicity:

K ≡ 1 2 u • u Ω and H ≡ 1 2 ω • u Ω , (1.6)
respectively. Here, the bracket with subscript Ω denotes the spatial average

• Ω ≡ 1 (2π) 3 ˆΩ(•) dx . (1.7)
Once we go back to the viscous (ν = 0) and forced (f = 0) case, we have a governing equation for the kinetic energy dK dt = P -, (1.8) where the Right Hand Side (RHS) consists of the energy injection rate and the energy dissipation rate: respectively. Here, we define the symmetric/antisymmetric decomposition of the velocity gradient tensor:

P ≡ f • u Ω and ≡ 2ν S 2 Ω , (1.9)
A = S + Ω, ↔ ∂u i ∂x j Aij = 1 2
∂u i ∂x j + ∂u j ∂x i Sij + 1 2 ∂u i ∂x j - ∂u j ∂x i Ωij .
(1.10)

The symmetric part S is often called rate-of-strain tensor and the antisymmetric part Ω rate-of-rotation tensor. Note that the energy dissipation rate in (1.9) is often expressed in alternative forms. For example, by using the velocity gradient tensor, .11) This quantity is called pseudo-energy dissipation rate and is not strictly equivalent to the original definition (1.9). See Appendix A for details. Another common definition is by the square of the vorticity vector: .12) This is an estimation of the energy dissipation rate and equivalent to (1.9) when the flow is statistically homogeneous. See Appendix B for details. The helicity equation can be obtained by taking the inner product of (1.1) with ω.

pseudo = ν A 2 Ω . ( 1
homogeneous = ν ω 2 Ω . ( 1
One important remark on (1.9) is that the energy dissipation rate is a positive definite quantity, whereas the energy injection rate can be positive or negative. Now, we consider the so-called statistically steady flow, in which the time-average of (1.8) is zero:

dK dt t = P t -t = 0.
(1.13)

Energy cascade in three-dimensional turbulence

Here, the bracket with subscript t denotes the time average .14) Under this assumption, there is a balance between the energy injection and dissipation rates in a timeaverage sense. Since > 0, this balance implies that P t is positive, which does not necessarily hold instantaneously.

• t ≡ lim T →∞ 1 T ˆT 0 (•) dt . ( 1
Based on the injection-dissipation balance (1.13), we consider the scales corresponding to P and . The energy injection rate P is defined by an inner product of the forcing f and the velocity field u; more energy is injected when the two vector fields are aligned. The energy dissipation rate is defined by the square of the strain-rate tensor S ij , and thus it takes larger values when there is a strong velocity gradient.

In this thesis, we consider a case where the energy injection takes place at the largest scales of the flow (forcing scale; energy injection scale), and the energy dissipation occurs at the smallest scales (viscous scale; dissipation scale). However, it is important to note that this large-scale injection and small-scale dissipation picture, often implicitly assumed, is not trivial. For example, Vela-Martín and Jiménez (2021, p. 13) wrote that "the energy dissipation is a consequence, rather than a cause, of the energy cascade." Since two terms on the RHS of (1.13) represent different scales in turbulence and must be balanced under the assumption of statistical stationarity, there needs to be a transfer of energy from large to small scales. This energy transfer is caused by the nonlinear term of the Navier-Stokes equations. Since its spatial average is zero, the nonlinear term represents energy redistribution. In other words, it does not produce or eliminate energy. The question is how it redistributes energy among different scales. Is there a direct energy transfer from the forcing scale to the dissipation scale? Or are there more complicated interactions, including the intermediate scales between them? Turbulence research has found that the latter is the case; there is a multi-step mechanism to transfer energy from large to small scales. This leads to the concept of energy cascade5 . In the next subsection, we will review this concept from scale space.

Scale-local nature of the energy cascade

In this subsection, we review the concept and nature of energy cascade in scale space. First, we define the Fourier transform of the velocity field

u(k) = 1 (2π) 3 ˆu(x) exp(-ik • x) dx , (1.15)
and associated inverse transform .16) respectively. Note that we do not employ annotations to express the Fourier-space quantities. Instead, we explicitly denote the Fourier mode k as an argument in the following. The Fourier transform of the Navier-Stokes equations (1.1) yields 1.17) with the incompressibility condition

u(x) = ˆu(k) exp(ik • x) dk , ( 1
∂u i (k) ∂t = -ik m P ij (k) ˆδ(k -p -q)u j (p)u m (q) dp dq -νk 2 u i (k) + f i (k), (
ik • u(k) = ik j u j (k) = 0. (1.18)
For the full derivation process, see, for example, Pope (2000, § 6.4.2) and Lesieur (2008, § 5.3). We also assume solenoidality for the forcing term k j f j = 0. To express the nonlinear term (including the pressure gradient), we use the projection

P ij (k) = δ ij - k i k j k 2 (1.19)
and Kronecker's delta function δ ij .

The nonlinear term of (1.17) in the integral form has a constraint k = p + q, known as triad interaction.

It denotes that the energy transfer in Fourier space is possible only between the three modes (k, p, q) when they satisfy this condition. By computing a dot product of (1.17) with u * i (k), where • * denotes the complex conjugate, one obtains an equation for energy at mode k,

∂E(k) ∂t = -ik m P ij (k) ˆδ(k -p -q)u * i (k)u j (p)u m (q) dp dq T (k) -2νk 2 E(k) (k) + f i (k)u * i (k), F (k)
( 1.20) where

E(k) = u(k) 2 /2 = u i (k)u * i (k)/2.
The RHS of (1.20) consists of: (i) nonlinear energy transfer T at k with all the combination of Fourier modes (k, p, q) satisfying the triad interaction condition k = p + q, (ii) energy dissipation rate at k, and (iii) energy injection rate F at k. Figure 1.3(a) shows the schematic of energy transfer in Fourier space with triad interactions. The integrand of T (k) in (1.20) denotes the energy transfer from the giver mode p to the receiver mode k with the aid of the mediator mode q, which corresponds to the advection velocity. Next, we average (1.20) over a shell of radius k = |k| in Fourier space to obtain

∂E(k) ∂t = F (k) + T (k) -2νk 2 E(k). (1.21)
Here, the energy spectrum

E(k) = ˆE(k) dΩ k (1.22)
represents energy density at wavenumber k. The notation ´dΩ k denotes the integral over spherical shells of radius k. The RHS terms

F (k) ≡ ˆF (k) dΩ k and T (k) ≡ ˆT (k) dΩ k (1.23)
are called energy injection and energy transfer spectra, respectively. The last term 2νk 2 E(k) is the energy dissipation spectrum.

A question here is the scale locality of the energy transfer T (k) in three-dimensional turbulence. In other words, how does an individual triangle consisting of three Fourier modes (k, p, q) contribute to T (k)?

Equilibrium and nonequilibrium aspects of turbulence

A pioneering theoretical investigation was done by Kraichnan (1971b). Figure 1.3(b) shows the first numerical evidence that the energy transfer between wavenumbers k and p6 shows A valley and a ridge seen along the diagonal k = p in this plot reflects the local nature of the energy transfer. [START_REF] Ohkitani | Triad interactions in a forced turbulence[END_REF] investigated the triad interactions in more detail by categorising them into six groups according to the energy gain/loss of the largest, intermediate, and smallest wavenumbers. They found strong interactions in nonlocal nonlinear interactions with large differences in scale. However, the largest scale does not contribute to the energy cascade in the triad, as two comparable scales dominate it. [START_REF] Waleffe | The nature of triad interactions in homogeneous turbulence[END_REF] investigated triad interactions with the helical decomposition of the velocity field to discuss the stability and the energy transfer. Eyink (2005) decomposed the velocity in scale and space to investigate various cascades, including energy and helicity in three-dimensional turbulence and inverse enstrophy and direct energy cascade in two-dimensional configurations. He established a sufficient condition of the locality of the energy cascade. [START_REF] Domaradzki | An analysis of the energy transfer and the locality of nonlinear interactions in turbulence[END_REF] found significant cancellation of the local transfer through nonlocal interactions in the global energy transfer with smooth and sharp filters. [START_REF] Biferale | Inverse energy cascade in three-dimensional isotropic turbulence[END_REF] uncovered an inverse energy cascade in three-dimensional turbulence by considering only a subset of the nonlinear triad interactions. [START_REF] Cardesa | The turbulent cascade in five dimensions[END_REF] traced the space-time position of a given structure, defined by isocounters of intense bandpass-filtered energy, at four different scales. The intersection ratio of vortices of different scales reveals a signature of energy cascade; the eddies of a given scale emerge inside larger eddies with twice the size, while they give rise to eddies with half the size before dying out. This observation with a characteristic scale ratio of two supports the scale-local nature of the energy cascade.

Equilibrium and nonequilibrium aspects of turbulence

Although turbulence is a typical example of strongly out-of-equilibrium phenomena, earlier theoretical works were directed to the "equilibrium" aspect of turbulence. In this section, we review both "equilibrium" and "nonequilibrium" aspects of turbulence. Here, we remark that the definition of these terms does not coincide with statistical mechanics. In the following, we omit the quotation for readability. In § 1.2.1, we first introduce an assumption on the equilibrium in the small scales of turbulence, which is called the local equilibrium hypothesis. Then, in § 1.2.2, we review the famous Kolmogorov's 1941 theory on universal small-scale turbulence statistics. This theory sets the paradigm of current turbulence research. In § 1.2.3, we investigate the nonequilibrium aspect of turbulence associated with the breakdown of Taylor's dissipation law, which can be understood as an extension of the local equilibrium hypothesis. In § 1.2.4, we apply a similar discussion for two-dimensional turbulence to highlight the qualitative difference between two-and three-dimensional turbulence.

Local equilibrium hypothesis

We begin our discussion by recasting (1.21): 1.26) which is the governing equation of the energy spectrum. Here, we explicitly denote the time dependency. Note that the RHS consists of the energy injection F (k, t), energy transfer T (k, t), and the energy dissipation 2νk 2 E(k, t) at scale k and at time t, respectively. in Fig. 1.3(a). It is because the nonlinear term in [START_REF] Domaradzki | Local energy transfer and nonlocal interactions in homogeneous, isotropic turbulence[END_REF] is defined in a symmetric form: .24) where

∂E(k, t) ∂t = F (k, t) + T (k, t) -2νk 2 E(k, t), (
- i 2 P nlm (k) ˆd3 p u l (p, t)um(k -p, t) (1
P nlm (k) = km δ nl - knk l k 2 + k l δnm - knkm k 2 .
(1.25) where the kinetic energy 1.28) the power input .29) and the energy dissipation rate

K(t) ≡ ˆ∞ 0 E(k, t) dk , (
P (t) ≡ ˆ∞ 0 F (k, t) dk , ( 1
(t) ≡ 2ν ˆ∞ 0 k 2 E(k, t) dk . (1.30)
Note that the integral of the nonlinear transfer term is zero: .31) Refer to § 1.1.1 for the inviscid invariants of the Navier-Stokes equations.

ˆ∞ 0 T (k, t) dk = 0. ( 1
We now consider decomposing the integral in (1.27) into low-and high-wavenumber contributions. This leads us to the set of equations for large-and small-scale energy,

∂K < ∂t (k, t) = P < (k, t) -Π(k, t) -< (k, t) and ∂K > ∂t (k, t) = Π(k, t) -> (k, t).
(1.32) (1.33) Here, the superscripts • < and • > denote large-scale (low-wavenumber) and small-scale (high-wavenumber) quantities, respectively. Each term is defined by decomposing the integral of (1.28)-(1.30), for example,

K < (k, t) ≡ ˆk 0 E(p, t) dp and K > (k, t) ≡ ˆ∞ k E(p, t) dp , < (k, t) ≡ 2ν ˆk 0 p 2 E(p, t) dp and > (k, t) ≡ 2ν ˆ∞ k p 2 E(p, t) dp .
(1.34) (1.35) For the power input term, we restrict F (k, t) to appear only at the low-wavenumber region. It is an assumption which we employ for the rest of this thesis. For the integral of the energy transfer function, we take advantage of (1.31) to define the energy flux .36) so that the sum of (1.32) and (1.33) reduces to (1.27).

Π(k, t) ≡ - ˆk 0 T (k, t) dk = ˆ∞ k T (k, t) dk , ( 1
We now focus on the small-scale equation (1.33). Under the assumption that the temporal evolution of the small-scale energy K > (k, t) is slow, namely, .38) This is the formal expression of the Local Equilibrium Hypothesis (LEH) (Goto and Vassilicos 2016a), and is depicted in Fig. 1.4. The LEH expresses the instantaneous balance at scale k between the energy 1.2 Equilibrium and nonequilibrium aspects of turbulence flux across that scale and the energy dissipation rate in the smaller scales. In turbulence research, it represents the equilibrium aspect of turbulence. Again, we remark that the usage of this term is different from statistical mechanics.

∂K > ∂t (k, t) > (k, t), (1.37) it follows that Π(k, t) ≈ > (k, t). ( 1
log k Energy budget Π(k) (k) Π(k, t) ≈ > (k, t) (t) ∝ U (t) 3 /L(t)

Kolmogorov-Onsager theory of the small-scale universality

The current paradigm of turbulence research relies on the work of Kolmogorov (1941a). In this subsection, we review Kolmogorov's two similarity hypotheses to derive Kolmogorov's universal energy spectrum 1.39) by emphasising the importance of the LEH (1.38) discussed in the last subsection ( § 1.2.1). First, we introduce the local homogeneous and isotropy hypotheses according to Kolmogorov (1941a).

E(k) = C K 2/3 k -5/3 for k f k k η , (
The local homogeneous hypothesis states that the statistics of velocity difference between two points in the four-dimensional space (x 1 , x 2 , x 3 , t) depend on the difference between two points and not their position. The local isotropy hypothesis states that under the local homogeneous hypothesis, the statistics are invariant regarding rotations and reflections of the original coordinate. Turbulent flow satisfying these hypotheses is called Homogeneous and Isotropic Turbulence (HIT).

For the local isotropy hypothesis, Kolmogorov (1941a) stated:

[...] it seems very likely that in the case of an arbitrary turbulent flow with sufficiently large Reynolds number7 Re = LU/ν (1.40) in sufficiently small regions G of the four-dimensional space (x 1 , x 2 , x 3 , t) not lying close to the boundaries of the flow or other singularities of it, the hypothesis of local isotropy holds with high accuracy.

Here, by "small regions" are meant those whose linear dimensions and time scales are small relative to L and T = U/L, (1.41) respectively.

Chapter 1 : Introduction: the Energy Cascade Furthermore, Kolmogorov claimed that the local isotropy hypothesis cannot be proved rigorously. In the footnote spanning two-thirds of a page, there is the following statement:

By virtue of the chaotic mechanism of transfer of motion from lower-order fluctuations to higher-order ones, it is natural to assume that, within the limits of space regions which are small as compared to l (1) (= l, where l is the Prandtl mixing length), small-scale higher-order fluctuations are subject to an approximately spatially isotropic statistical regime. Within short time intervals this regime can naturally be regarded as being stationary, even when the flow as a whole is non-stationary. Thus, Kolmogorov (1941a) claimed the idea of local equilibrium in the local isotropy hypothesis. If we translate the above discussion to Fourier space, we obtain the LEH (1.38). Now, we introduce two similarity hypotheses:

The first similarity hypothesis states that for very large (but finite) Reynolds number, the small-scale (k k f ) statistics of turbulence can be uniquely determined by the parameters ν and . By defining the dissipation length scale .42) we can express the statistics with and η. This allows us to express the energy spectrum as

η ≡ ν 3 1/4 , ( 1
E(k) = 2/3 k -5/3 F (kη) for k f k, (1.43)
where F denotes a universal nondimensional function of a nondimensional argument.

The second similarity hypothesis further states that the small-scale statistics, this time larger than the viscous length scale k k η = η -1 , can be uniquely determined by the single parameter . The energy spectrum can be expressed as 1.44) where C K is a universal constant of order unity called the Kolmogorov constant. Figure 1.5 shows the energy spectrum of turbulence in various flow configurations, including theory, numerical simulation, and experiment to support the robustness of the similarity hypotheses. Remarkably, after the normalisation, all spectra collapse for k f k (the first similarity hypothesis). In the intermediate range of k f k k η , there is a universal k -5/3 scaling described by (1.44) (the second similarity hypothesis). Kolmogorov's theory on small-scale universality has set the paradigm of turbulence research. Many textbooks [START_REF] Tennekes | A First Course in Turbulence[END_REF]Frisch 1995, § 5-7;Pope 2000, § 6.1;Tsinober 2009, § 5;Lesieur 2008, § 6;Davidson 2015, § 5), lecture notes [START_REF] George | Lectures in Turbulence for the 21st Century[END_REF], § 14, Eyink 2007-2008), and review articles [START_REF] Hunt | Kolmogorov's contributions to the physical and geometrical understanding of small-scale turbulence and recent developments[END_REF]Lumley 1992;[START_REF] Nelkin | Universality and scaling in fully developed turbulence[END_REF][START_REF] Sreenivasan | The phenomenology of small-scale turbulence[END_REF][START_REF] Hunt | Developments in turbulence research: a review based on the 1999 Programme of the Isaac Newton Institute[END_REF]Dubrulle 2019) discuss this theory in more detail. We note some historical remarks to conclude this subsection. In the 1941 works of Kolmogorov (Kolmogorov 1941a;Kolmogorov 1941b;Kolmogorov 1941c), the analysis was conducted in real space. The Fourier-space expression (1.44) was first derived by Obukhov (Obukhov 1941a;Obukhov 1941b). Around the same time, several other researchers, including Onsager [START_REF] Onsager | The distribution of energy in turbulence[END_REF]Onsager 1949), Heisenberg [START_REF] Heisenberg | Zur statistischen Theorie der Turbulenz[END_REF], and Weizsäcker [START_REF] Weizsäcker | Das spektrum der turbulenz bei grossen Reynoldsschen zahlen[END_REF], reached the same result independently. To highlight the contribution by Onsager, we name this subsection as "Kolmogorov-Onsager theory of the small-scale universality". For more detail on the historical aspects, see Frisch (1995, § 6.5) and Eyink and Sreenivasan (2006, § 4-5) as well as § 1.3.1.

E(k) = C K 2/3 k -5/3 for k f k k η , (

Taylor's dissipation law and its violation

In this subsection, we discuss the extension of the LEH and its validity. The LEH serves as a keystone for a more applicational aspect of turbulence, along with the theory of small-scale universality. To see this, we first assume that the LEH is valid at the largest wavenumber of the flow, k → L(t) -1 , where L(t) is the time-dependent macroscopic length scale. Then, (1.38) becomes Adapted from Sreenivasan (1984, Figure 1).

Π(L -1 , t) ≈ (t), (1.45)
Chapter 1 : Introduction: the Energy Cascade since > (k, t) → (t) holds in this limit. By modelling the largest-scale energy flux as .46) where U (t) denotes the macroscopic velocity, we obtain .47) This relation is called Taylor's dissipation law (Taylor 1935) 8 . By introducing the scaling coefficient C to rewrite (1.47) as

Π(L -1 , t) ∝ U (t) 3 L(t) , ( 1
(t) ∝ U (t) 3 L(t) . ( 1
(t) = C U (t) 3 L(t) , (1.48)
Taylor's dissipation law allows us to measure the microscopic energy dissipation rate by the macroscopic quantities. Here, we call C the normalised energy dissipation rate. Figure 1.6 shows the Reynolds number dependency of C evaluated from (1.48) in a statistical sense [START_REF] Sreenivasan | On the scaling of the turbulence energy dissipation rate[END_REF]. It suggests that C tends to be independent of Re λ : .49) This constancy of C at higher Reynolds numbers has been reported in both DNS [START_REF] Kurien | Scaling structure of the velocity statistics in atmospheric boundary layers[END_REF][START_REF] Kaneda | Energy dissipation rate and energy spectrum in high resolution direct numerical simulations of turbulence in a periodic box[END_REF] and experiment [START_REF] Cadot | Energy injection in closed turbulent flows: Stirring through boundary layers versus inertial stirring[END_REF]). Thus, this scaling has been employed in numerous research papers and industrial applications through turbulence models (Frisch 1995, § 7.2). Refer to Lumley (1992) and [START_REF] Vassilicos | Dissipation in turbulent flows[END_REF] for the review on C . However, a naive implementation of Taylor's dissipation law (1.47) is problematic, as does the assumption and modelling procedures based on it. Indeed, the statistical relation (1.49) is valid and gives a good estimation of the average energy dissipation rate. However, it is not necessarily the case for an instantaneous balance (1.48), as there can be an imbalance between large-and small-scale dynamics associated with the nonequilibrium nature of turbulence. For a review of equilibrium and nonequilibrium dissipation scaling, see [START_REF] Vassilicos | Dissipation in turbulent flows[END_REF] and references therein. In Chap. 7 of this thesis, we will investigate the nonequilibrium scaling of C for inhomogeneous or unsteady flows. The invalidity of LEH has been investigated in various flow configurations, including wall-bounded flows [START_REF] Jiménez | Cascades in wall-bounded turbulence[END_REF], decaying turbulence under rotation [START_REF] Valente | Spectral imbalance in the inertial range dynamics of decaying rotating turbulence[END_REF], wake of side-byside square cylinder [START_REF] Zhou | Energy transfer in turbulent flows behind two side-by-side square cylinders[END_REF]. George (2014) claimed that the LEH is "not satisfied in general for non-stationary and inhomogeneous flows", despite the fact that the hypothesis has been a basis for the Kolmogorov's 1941 theory of turbulence. [START_REF] Silva | On the local equilibrium of the subgrid scales: The velocity and scalar fields[END_REF] investigated the LEH in the context of subgrid-scale modelling. Recently, in [START_REF] Kaneda | Velocity gradient statistics in turbulent shear flow: an extension of Kolmogorov's local equilibrium theory[END_REF], the LEH is extended to shear flow with several hypotheses to discuss the influence of mean shear on the dissipation-scale statistics. In Fig. 1.7, the temporal imbalance between the loss rate of energy at scale k and dissipated energy in scales smaller than k (Goto and Vassilicos 2016a). It is clear that by decreasing k (or focusing on larger scales), there is a larger imbalance between these two quantities. The same authors extended the analysis in decaying turbulence (Goto and Vassilicos 2016b). [START_REF] Bos | Spectral imbalance and the normalized dissipation rate of turbulence[END_REF] and [START_REF] Fang | An EDQNM study of the dissipation rate in isotropic nonequilibrium turbulence[END_REF] investigated the influence of spectral imbalance on the normalised energy dissipation rate. [START_REF] Mccomb | Taylor's (1935) dissipation surrogate reinterpreted[END_REF] claimed that Taylor's dissipation law (1.47) should be interpreted as estimating the peak inertial range energy flux rather than the energy dissipation rate. In [START_REF] Valente | Origin of the imbalance between energy cascade and dissipation in turbulence[END_REF], the systematic imbalance between the energy flux to the fine scale and its dissipation was investigated. In [START_REF] Rubinstein | Equilibrium" and "non-equilibrium" turbulence[END_REF], the concept of "equilibrium" was investigated, and the conclusion was similar to [START_REF] Mccomb | Taylor's (1935) dissipation surrogate reinterpreted[END_REF]; the energy dissipation rate cannot be equivalent to the large-scale energy flux. In Vela-Martín and Jiménez (2021), the "out-of-equilibrium" energy flux was investigated. Recently, Steiros has been working on a generalization of the turbulence theory for out-of-equilibrium effects and cascades far from initial conditions (Steiros 2022a;Steiros 2022b).

C = L U 3 = const. as Re λ . ( 1
One of the pioneering works on the nonequilibrium balance of energy budget in turbulence was Yoshizawa (1994). This study investigated the nonequilibrium property of the inertial-range statistics of energy with 1.2 Equilibrium and nonequilibrium aspects of turbulence

log k log E(k) ∝ k -5/3 ∝ k -3 ln k/k f -1/3 k f k ηω

Energy accumulation

Inverse energy cascade

Forward enstrophy cascade the Lagrangian derivative of the energy transfer. A k -7/3 scaling correction gives the deviation from the equilibrium scaling. This correction due to unsteadiness was further investigated in [START_REF] Rubinstein | Time-dependent isotropic turbulence[END_REF][START_REF] Woodruff | Multiple-scale perturbation analysis of slowly evolving turbulence[END_REF][START_REF] Horiuti | Nonequilibrium energy spectrum in the subgrid-scale oneequation model in large-eddy simulation[END_REF]. In Chap. 6 of the thesis, we will investigate the inhomogeneity-induced k -7/3 scaling correction.

In terms of turbulence modelling, [START_REF] Edeling | Data-free and data-driven RANS predictions with quantified uncertainty[END_REF] proposed a return-to-eddy-viscosity model, which explicitly considers the contributions from the departure of the small scales from the local equilibrium [START_REF] Duraisamy | Turbulence modeling in the age of data[END_REF]. [START_REF] Bos | Production and dissipation of kinetic energy in grid turbulence[END_REF] proposed a simple model in near-field grid turbulence. Vela-Martín (2022) discussed the nonequilibrium effect in subgrid-scale modelling.

Cascades in two-dimensional turbulence

When we consider turbulent flow confined in the two-dimensional plane x 2D = (x, y) ∈ Ω 2D , it exhibits qualitatively different properties from three-dimensional turbulence. We begin with the governing equations of two-dimensional velocity field u = (u x , u y ) and corresponding one-dimensional vorticity field ω perpendicular to x 2D . The Navier-Stokes equations are identical to the three-dimensional ones (1.1), and the kinetic energy

K ≡ 1 2 u • u Ω2D (1.50)
remains invariant in the inviscid and force-free limit. On the other hand, the vorticity equation reads

∂ω ∂t + (u • ∇)ω = ν∇ 2 ω + (∇ × f ) z , (1.51)
and there is no vortex stretching term as ω is perpendicular to u. Since the nonlinear term is conservative, the enstrophy

Ω ≡ 1 2 ω 2 Ω2D
(1.52)

1.3 Physical-space mechanism behind the energy cascade is also an inviscid invariant. This is the critical difference compared to the three-dimensional case (1.3), where there is vortex stretching (enstrophy production). In the following, we consider cascades of two invariants of two-dimensional turbulence: inverse energy cascade and forward enstrophy cascade as depicted in Fig. 1.8. For reference, see textbooks Lesieur (2008, § 8) and Davidson (2015, § 10) as well as review articles [START_REF] Kraichnan | Two-dimensional turbulence[END_REF][START_REF] Tabeling | Two-dimensional turbulence: a physicist approach[END_REF][START_REF] Kellay | Two-dimensional turbulence: a review of some recent experiments[END_REF][START_REF] Boffetta | Two-dimensional turbulence[END_REF][START_REF] Alexakis | Cascades and transitions in turbulent flows[END_REF]. First, we consider the enstrophy cascade. In Fourier space, the enstrophy spectrum can be expressed as k 2 E(k) and its equation reads

∂k 2 E(k) ∂t = k 2 F (k) + k 2 T (k) -2νk 4 E(k), (1.53)
which is obtained by multiplying the energy equation (1.21) by k 2 . Note that due to the inviscid conservation of enstrophy, the enstrophy transfer satisfies

ˆ∞ 0 k 2 T (k) dk = 0. (1.54)
By conducting a similar analysis for enstrophy as already done for energy in § 1.2.1, we can define the enstrophy dissipation rate

ω = 2ν ˆ∞ 0 k 4 E(k) dk , (1.55)
and corresponding length scale .56) By assuming that the statistics in the inertial range (k f k k ηω ) can be determined by the single parameter ω , a dimensional analysis yields

k ηω ≡ ω ν 3 1/6 . ( 1
E(k) = C K 2/3 ω k -3 for k f k k ηω . (1.57)
This is the k -3 scaling of the energy spectrum associated with the enstrophy cascade. Furthermore, due to the nonlocal nature of the enstrophy cascade, the k -3 scaling (1.57) is subject to a logarithmic correction (Kraichnan 1971b):

E(k) = C K 2/3 ω k -3 ln k/k f -1/3 for k f k k ηω . (1.58)
The ln k/k f correction acts to suppress the diverging integral of enstrophy: ´k2 E(k).

Next, we consider the inverse energy cascade. Thanks to Fjortoft's theorem [START_REF] Fjørtoft | On the changes in the spectral distribution of kinetic energy for twodimensional, nondivergent flow[END_REF] [Also see Lesieur (2008, § 8.3)], there is no energy transfer towards smaller scales (k > k f ) associated with the enstrophy cascade. Thus, the injected energy can only be transferred to larger scales (k < k f ). An analogy of the discussion for the three-dimensional turbulence in § 1.2.2 yields the same scaling for different wavenumber regime (Kraichnan 1967;Leith 1968):

E(k) = C K 2/3 k -5/3 for k k f . (1.59)
There are multiple candidates for the physical mechanism of the inverse energy cascade (Davidson 2015, § 10.1.4). Since the dissipation term does not act strongly on the largest wavenumbers, temporal evolution of E(k, t) is associated with energy accumulation at scales of the size of the flow domain.

Physical-space mechanism behind the energy cascade

The essence of Kolmogorov's 1941 theory (hereafter, K41 theory) is that one cannot distinguish the statistics of the small-scale structures of a given turbulence from different flow realisations. The smallscale universality of the K41 theory has set the paradigm of turbulence research for the last eighty years.

Chapter 1 : Introduction: the Energy Cascade Figure 1.9: Classical schematic of energy cascade with space-filling eddies for various scales, accounting for the K41 theory. Adapted from Frisch, Sulem, et al. (1978, Figure 1). A similar schematic with scale annotations can be found in Frisch (1995, Figure 7.2).

However, the scale-space description does not provide how the structures of various scales are generated and sustained in physical space. Thus, the following question:

How are the multi-scale structures in turbulent flow sustained? In particular, how does the energy cascade from large to small scales in physical space? has been one of the central questions in the turbulence research community, on which we still do not have a consensus. That means we are still far from the physical-space description of turbulence as a counterpart of the K41 theory in scale space. For example, Fig. 1.9 shows a schematic of the energy cascade by spacefilling vortices of various scales. This schematic may give the impression that the largest-scale vortices are "breaking up" as energy cascades down to smaller scales. This picture is common in the turbulence community, but to what extent is it true? This section reviews candidates for the energy cascade mechanism phenomenologically and quantitatively.

In particular, we focus on the two processes: vortex stretching ( § 1.3.1) and strain self-amplification ( § 1.3.2). In Appendix C, we review several other candidates for the energy cascade mechanism, such as instability, vortex reconnection, and spiral vortex scenario. In § 1.3.3, we briefly review the intermittency of turbulence in relation to the energy cascade.

Vortex stretching

Vortex stretching is the most commonly accepted mechanism of energy cascade. For a detailed review including quotations from the historical works, see Eyink (2007Eyink ( -2008, § IV (A)). We have two nonlinear terms in the vorticity equation (1.3): the vortex advection term (u • ∇)ω and the vortex stretching term (ω • ∇)u. It is the latter term we focus on in this subsection, as its name indicates the mechanism. By taking inner product of (1.3) and ω, we obtain the enstrophy equation .60) where ijk denotes the Levi-Civita symbol. The first term on the RHS, ω i S ij ω j , corresponds to the vortex stretching and is called the enstrophy production term. Figure 1.10 shows the schematic of vortex stretching between vortices of different scales. A similar schematic can be found in, for example, Tennekes and Lumley (1972, Figure 8.4) and Johnson (2021a, Figure 1). In the final paragraph of Taylor (1938), he wrote:

∂Ω ∂t + u j ∂Ω ∂x j = ω i S ij ω j + ν ∂ 2 Ω ∂x 2 k -ν ∂ω i ∂x j 2 + ω i ijk ∂f j ∂x k , ( 1
1.3 Physical-space mechanism behind the energy cascade Figure 1.10: Schematic of vortex stretching between large-scale vortices (red) and small-scale vortices (yellow). Aligned large-scale vortices with counter-rotating direction induce strain field between them; small-scale vortices are stretched in anti-parallel configuration to the large-scale vortices.

It seems that the stretching of vortex filaments must be regarded as the principal mechanical cause of the high rate of dissipation which is associated with turbulent motion.

In Onsager (1949, p. 282), a similar idea is presented as:

Now it is very reasonable to expect that a vortex line -of any line which is deformed by the motion of the liquid -will tend to increase in length as a result of more or less haphazard motion. This process tends to make the texture of the motion ever finer, and greatly accelerates the viscous dissipation.

Note that both Taylor (1938) and Onsager (1949) argue vortex stretching to be the main cause of energy dissipation by viscosity and not energy cascade. However, we remark that Onsager had the idea that the vortex stretching is related to the inertial range dynamics [START_REF] Eyink | Onsager and the theory of hydrodynamic turbulence[END_REF], § IV A). Although there are no published works, Onsager had been working on extending Taylor's idea to the "cascade" of energy, which he first used in the context of turbulence theory in a letter to Chia-Chiao Lin [START_REF] Eyink | Onsager and the theory of hydrodynamic turbulence[END_REF], Appendix B) and a one-page paragraph [START_REF] Onsager | The distribution of energy in turbulence[END_REF]). In the former, Onsager wrote

The distribution law (19) [the -5/3 spectrum] is compatible with the hypothesis that the mean rate of stretching of vortex lines is given by the average rate of deformation in the liquid.

In another place of the same letter, he remarked that The selection rule for the 'modulation' factor in each term of (8) [the nonlinear term of the Navier-Stokes equations in Fourier space] suggests a 'cascade' mechanism for the process of dissipation, and also furnishes a dynamical basis for an assumption which is usually made on dimensional grounds only.

In the textbook [START_REF] Tennekes | A First Course in Turbulence[END_REF], the vortex stretching is related to the energy cascade:

When vorticity finds itself in a strain-rate field, it is subject to stretching. On the basis of conservation of angular momentum, we expect that the vorticity in the direction of a positive strain rate is amplified, while the vorticity in the direction of a negative strain rate is attenuated. [...] Vortex stretching involves an exchange of energy, because the strain rate performs deformation work on the vortices that are being stretched. Tennekes and Lumley (1972, p. 257) is essentially same as Fig. 1.10. Although Tennekes and Lumley's remark was rather schematic, much study has been conducted since then. Numerical evidence of vortex stretching and generation of the fine-scale structures are reported (Melander and Hussain 1993;[START_REF] Jiménez | The structure of intense vorticity in isotropic turbulence[END_REF][START_REF] Nomura | The structure and dynamics of vorticity and rate of strain in incompressible homogeneous turbulence[END_REF]Pullin and Saffman 1998) and discussed in association with intermittency [START_REF] Siggia | Origin of intermittency in fully developed turbulence[END_REF]Buaria, Bodenschatz, et al. 2020). Also see Fig. 1.2. Recent analysis includes the large-deviation theory [START_REF] Johnson | Large-deviation statistics of vorticity stretching in isotropic turbulence[END_REF], Unstable Periodic Orbit (UPO) [START_REF] Lucas | Sustaining processes from recurrent flows in body-forced turbulence[END_REF][START_REF] Van Veen | Time-periodic inertial range dynamics[END_REF], and artificial suppression of vortex stretching [START_REF] Bos | Analysis of the normalized dissipation rate in inhomogeneous and unsteady turbulence Bibliography[END_REF]. [START_REF] Goto | A physical mechanism of the energy cascade in homogeneous isotropic turbulence[END_REF] proposed that the energy cascade occurs by stretching of smaller-scale vortices by the largescale strain induced by pairs of vortex tubes in anti-parallel configurations. This analysis was motivated by the picture previously proposed by Melander and Hussain (1993): The large-scale vortex tube generates smaller anti-parallel pairs of vortex around it and cascades the energy. Furthermore, the importance of vortices in the anti-parallel configurations was pointed out by Siggia (1985). [START_REF] Goto | A physical mechanism of the energy cascade in homogeneous isotropic turbulence[END_REF] defined the internal energy in each subdomain of the flow to be Galilei invariant and employed the Lagrangian framework to define its energy transfer rate. Goto (2012) and [START_REF] Goto | Hierarchy of antiparallel vortex tubes in spatially periodic turbulence at high Reynolds numbers[END_REF] investigated scaleby-scale vortex stretching between the anti-parallel vortex tubes of different scales in higher Reynolds numbers to find that the energy transfer due to vortex stretching is the most efficient at a scale ratio of about two. Similar finding was observed by other groups [START_REF] Eyink | Onsager and the theory of hydrodynamic turbulence[END_REF][START_REF] Leung | Geometry and interaction of structures in homogeneous isotropic turbulence[END_REF][START_REF] Doan | Scale locality of the energy cascade using real space quantities[END_REF] and also in wall-bounded flows [START_REF] Lozano-Durán | Multiscale analysis of the topological invariants in the logarithmic region of turbulent channels at a friction Reynolds number of 932[END_REF][START_REF] Motoori | Generation mechanism of a hierarchy of vortices in a turbulent boundary layer[END_REF]. From these findings, vortex stretching is a convincing candidate responsible for the physical mechanism of the scalelocal energy cascade. See Fig. 1.11 for the hierarchical vortex structures. Mathematical support for this finding is given in [START_REF] Gibbon | Dynamically stretched vortices as solutions of the 3D Navier-Stokes equations[END_REF], Yoneda (2020), [START_REF] Yoneda | Mathematical reformulation of the Kolmogorov-Richardson energy cascade in terms of vortex stretching[END_REF][START_REF] Tsuruhashi | Self-similar hierarchy of coherent tubular vortices in turbulence[END_REF]. See also Appendix D.

Strain self-amplification

Indeed, the vortex stretching mechanism has been the primary candidate for the energy cascade since 1940s. However, it has not been the consensus for the turbulence community (Tsinober 2009, § 6.2.1). In this subsection, we review another important candidate for the phenomena: self-amplification of strain. It was first proposed by Tsinober [START_REF] Tsinober | On depression of nonlinearity in turbulence[END_REF][START_REF] Galanti | Self-amplification of the field of velocity derivatives in quasiisotropic turbulence[END_REF][START_REF] Galanti | Self-amplification of the field of velocity derivatives in quasiisotropic turbulence[END_REF]. The governing equation of the rate-of-strain tensor reads (Nomura and Post 1998, (2.2)): .61) where the RHS terms consist of self-interaction, local interaction with vorticity, pressure Hessian, and viscous diffusion, respectively. Note that we omit the forcing term for simplicity. When multiplying (1.61)

∂S ij ∂t + u k ∂ ∂x k S ij = -S ik S kj - 1 4 ω i ω j -δ ij ω k ω k - ∂ 2 p ∂x i ∂x j + ν ∂ 2 S ij ∂x k ∂x k , ( 1
1 4 ∂|S| 2 ∂t + u j ∂ ∂x j |S| 2 = -S ij S jk S ki - 1 4 ω i S ij ω j -S ij ∂ 2 p ∂x i ∂x j + νS ij ∂ 2 S ij ∂x k ∂x k . (1.62)
The first term on the RHS, -S ij S jk S ki , is called the strain self-amplification term. Figure 1.12 shows a schematic of this energy cascade process where a strain-concentrated region (between two parallel planes) is subject to compression along with one axis (eigenvalue) and extension for the other two axes. This highlights the difference against the vortex stretching with one extension axis and two compression axes. For example, a similar schematic can be found in Johnson (2021a, Figure 2). The strain self-amplification process may be easily understood in the one-dimensional Burgers equation

∂u ∂t + u ∂u ∂x = 0.
(1.63) Figure 1.13 shows its temporal evolution, where an initially smooth velocity profile given by sin(2πx) leads to a finite-time singularity. Here, the same level of velocity gradient is confined in a smaller domain as time evolves. In other words, the solution amplifies the velocity gradient by itself. Thus, the energy is transferred to the smaller scales (higher Fourier modes) as time evolves, thanks to the self-amplification of the velocity gradient.

In Tsinober (2009, § 6.2.2), the importance of the strain self-amplification is emphasised for four reasons:

1. Nonlocal correlation between the strain and vorticity, 2. Energy dissipation is directly associated with strain S ij and not with vorticity ω i [see (1.9)],

3. Vortex stretching is essentially a process of strain-vorticity interactions, 4. Strain-dominated regions appear to be the most active in turbulence.

In [START_REF] Carbone | Is vortex stretching the main cause of the turbulent energy cascade?[END_REF], they answered to the title of the paper "Is vortex stretching the main cause of the turbulent energy cascade?" unfavourably by claiming that the strain self-amplification plays the dominant role in the average behaviour of the energy cascade. In contrast, vortex stretching plays a more important role in the fluctuations of the cascade about its average behaviour. Similarly, [START_REF] Johnson | Energy transfer from large to small scales in turbulence by multiscale nonlinear strain and vorticity interactions[END_REF] and Johnson (2021a) concluded that the contributions of the strain self-amplification to energy transfer are higher than that of vorticity stretching9 . We will review these recent articles in more detail in Appendix D. The investigations of the physical space mechanism point to a common problem in turbulence research; since the system is multi-scale, nonlinear and nonlocal, it is tremendously difficult to disentangle the effect of a particular structure or interaction from the rest of the dynamical features. For example, in Chap. 3, we illustrate that even at very low Reynolds numbers, the flow dynamics are determined by an intricate interplay between a large number of Fourier modes.

Intermittency and the energy cascade

In the previous section § 1.2, we discussed the scale-space description of turbulence. However, as we reviewed in this section, the physical-space description of energy cascade employs various flow structures, particularly vortices in various scales. The complex multi-scale interactions of such structures induce highly intermittent fine-scale structures. For example, see Fig. 1.2, where we observe dense clusters and voids of the fine-scale structures in the domain. The inhomogeneous nature of turbulence fragilizes the validity of the K41 theory, where the globally averaged energy dissipation rate determines the statistics. Kolmogorov himself corrected his 1941 theory in Kolmogorov (1962), where he proposed a refined similarity hypothesis with the energy dissipation rate locally averaged in space. This introduces corrections to the inertial range scaling, particularly for higher-order moments. The quest for an appropriate description of these higher-order moments (or heavy tails of probability distribution function), which are observed both in Eulerian and Lagrangian statistics [START_REF] Anselmet | High-order velocity structure functions in turbulent shear flows[END_REF][START_REF] Chevillard | Unified multifractal description of velocity increments statistics in turbulence: Intermittency and skewness[END_REF], has resulted in a wealth of models. The list of models for the intermittency includes the She-Leveque (She and Lévêque 1994) (or Log-Poisson [START_REF] Dubrulle | Intermittency in fully developed turbulence: Log-Poisson statistics and generalized scale covariance[END_REF])) model and multifractal descriptions [START_REF] Parisi | On the singularity structure of fully developed turbulence[END_REF], to name a few. For reference, Frisch (1995, § 8) is devoted to the intermittency of turbulence. We stress that in this thesis, where the energy cascade is its main topic, we mainly investigate secondorder quantities such as kinetic energy. For these lower-order statistics, the influence of intermittency is not dramatic [START_REF] Bos | Reynolds number effect on the velocity increment skewness in isotropic turbulence[END_REF]. Therefore, we will not focus on the influence of intermittency in the rest of this thesis.

Part I

Quasi-cyclic Temporal Fluctuations of Turbulence and its Minimal Model 2 Temporal Fluctuations in Turbulence

Turbulence is often referred to as an example of chaos. Here, the most important aspect of chaos is not the randomness but the sensitivity to the initial conditions, which means that two realisations of the same system with slightly different initial conditions can eventually evolve into completely different states. However, behind this sensitivity which makes it almost impossible to predict a turbulent flow in all of its details, sometimes there seems to be a more deterministic underlying mechanism when one considers the temporal fluctuations of turbulence. In this chapter, we review such organised structures in the temporal domain of turbulence. In § 2.1, we discuss the temporal Quasi-Cyclic Behaviour (QCB) of turbulence associated with the energy cascade. In § 2.2, we review reduced-order modelling of turbulence.

In particular, we focus on modelling developed turbulence and briefly address laminar-turbulent transition.

Temporal (quasi-)cyclic modulation associated with the energy cascade

In the Introduction ( § 1.3), we reviewed the concept and possible energy cascade mechanism in physical space. In this section, we discuss that this phenomenological picture also captures the temporal dynamics of turbulence. In particular, we review temporal Quasi-Cyclic Behaviour (QCB) observed in turbulence. In § 2.1.1, we discuss the consequence of the Local Equilibrium Hypothesis (LEH) breakdown. The imbalance between large-and small-scale energy associated with the energy cascade dynamics leads to an intrinsic unsteady nature, which can cause QCB in certain kinds of turbulent flows. In § 2.1.2, we first introduce the concept of the Unstable Periodic Orbit (UPO), which can be understood as a skeleton of turbulence embedded in complex fluctuations, followed by a concise review on its relation with the energy cascade. In § 2.1.3, we focus on a closed system where the top and bottom rid of cylindrical disks rotate in opposite directions to excite turbulence in the domain: von Kármán flow.

Intrinsic unsteadiness and quasi-cyclic behaviour of turbulence

The LEH ( § 1.2.1) sets the foundation of turbulence theory in equilibrium. However, as depicted in Fig. 1.7, the balance between the energy flux and dissipation does not hold. The imbalance between these large-and small-scale quantities reflects the intrinsic unsteadiness of turbulence. Figure 2.1(a) quantifies this time delay using the two-time correlation function. Similar results are reported in, for example, [START_REF] Pearson | Delayed correlation between turbulent energy injection and dissipation[END_REF], Figure 3), Cardesa, Vela-Martín, Dong, et al. (2015, Figure 2 (b)), and Wan et al. (2010). In particular, the earliest result is probably presented in Meneveau and Lund (1994, Figure 4) where a two-time energy correlation in fluid "blob" within the Lagrangian framework was computed. The correlation peak is always observed with a time delay, characterised by the typical time scale of the cascade, a function of the length scale or the size of the "blob" (Lumley 1992). These observations support the forward energy cascade and the causality between large and small scales of turbulence, leading to intrinsic unsteadiness [START_REF] Bos | Spectral imbalance and the normalized dissipation rate of turbulence[END_REF]). [START_REF] Cardesa | The temporal evolution of the energy flux across scales in homogeneous turbulence[END_REF] investigated the scale-by-scale energy flux and their time delay between different scales to find the scale-local (or step-wise) dynamics. [START_REF] Goto | Energy dissipation and flux laws for unsteady turbulence[END_REF] and Goto and Vassilicos (2016a) investigated the energy dissipation rate scaling in unsteady turbulence. We proceed to consider the recurrence of a kind of "cycle" in temporal fluctuations. A number of laminar and turbulent flows display QCB. An illustrative example is vortex shedding behind an obstacle. For low Reynolds number (Re), the so-called von Kármán vortex street behind a cylinder is perfectly periodic, which corresponds to a Stable Periodic Orbit (SPO) in phase space. Even when the flow becomes fully turbulent at higher Re, this periodicity is still present, though the stochastic nature of turbulent motion prevents the system from being perfectly periodic. Figure 2.1(b) shows the temporal evolution of turbulence in a periodic domain under a steady forcing (3.2) for three different Reynolds numbers. The counter-clockwise cycle between the energy input rate and energy dissipation rate indicates that turbulence undergoes two states: input surpasses dissipation and vice versa. This close-to-periodic motion, embedded in turbulent fluctuations, is what we will call QCB. The three similar curves suggest the robustness of the quasi-cyclic temporal evolution of turbulence. The QCB is also found in turbulent channel flow, where a Self-Sustaining Process (SSP) governs the dynamics [START_REF] Johansson | On the structure of turbulent channel flow[END_REF][START_REF] Luchik | Timescale and structure of ejections and bursts in turbulent channel flows[END_REF][START_REF] Hamilton | Regeneration mechanisms of near-wall turbulence structures[END_REF][START_REF] Hamilton | Regeneration mechanisms of near-wall turbulence structures[END_REF][START_REF] Panton | Overview of the self-sustaining mechanisms of wall turbulence[END_REF][START_REF] Hwang | Self-sustained process at large scales in turbulent channel flow[END_REF]. In particular, in small channel flow domains (the so-called minimal flow unit), close to periodic behaviour is observed (Jiménez and Moin 1991, Fig. 6). These observations suggest that such temporal dynamics might be a general feature of turbulence. We note here that the characteristic time scale of such QCB cannot be universal for different flow configurations, since it is determined by the largest-scale motion of the flow. This remark is related to the original "Landau's objection" to the K41 theory (Frisch 1995, § 6.4).

Periodic orbit as a skeleton of the energy cascade

Deterministic Navier-Stokes equations can be considered a dynamical system with a large number of degrees of freedom. Indeed, in its early days, dynamical system theory focused on low-dimensional models. However, fast-developing computational resources allow researchers to investigate various solutions of turbulence, such as steady, travelling-wave, and periodic solutions [START_REF] Kawahara | The significance of simple invariant solutions in turbulent flows[END_REF]). Figure 2.2 shows the first discovered Unstable Periodic Orbit (UPO) embedded in turbulent channel flow [START_REF] Kawahara | Periodic motion embedded in plane Couette turbulence: regeneration cycle and burst[END_REF]. These invariant solutions are low dimensional in phase space compared P vis denotes the energy injection via viscous force at the rotating disks, while bou and bulk denotes the energy dissipation at the boundary and bulk regions of the flow, respectively. Note that both panels visualise the system with smooth disks (without blades).

to turbulence but possess many key turbulence properties, such as universal statistics, coherent large-scale structures, and dynamical processes. However, the connection of the periodic solution with energy cascade dynamics has not been investigated until recently. In van Veen, Vela-Martín, et al. ( 2019), a UPO was identified in turbulence driven by a deterministic forcing1 . This UPO shows the energy cascade dynamics between vortices of different scales, similar to [START_REF] Goto | Hierarchy of antiparallel vortex tubes in spatially periodic turbulence at high Reynolds numbers[END_REF].

Quasi-cyclic behaviour in von Kármán flow: a closed system under constant forcing

In this subsection, we review the mechanism of such QCB in a realistic flow setup called von Kármán flow, a counter-rotating lid and bottom cover drive fluid in a cylinder. The content is based on the following: Araki, Ryo and Susumu Goto (2021). Quasiperiodic fluctuations of von Kármán turbulence driven by viscous stirring. Physical Review Fluids 6 (8), p. 084603.

Figure 2.3 shows the apparatus of the system with the distinctive large-scale structures in panel (a) and the schematic vortex structures in its meridional plane in panel (b). One of the intriguing properties of this system is the qualitatively different behaviour of the normalised energy dissipation rate C where the flow is driven by either smooth or rough disks, associated with viscous and inertial stirring, respectively [START_REF] Cadot | Energy injection in closed turbulent flows: Stirring through boundary layers versus inertial stirring[END_REF]. With inertial stirring, C is independent of the Reynolds number Re. On the other hand, the viscous stirring setup finds that C does depend on Re even in the high Re limit. However, when we consider only the bulk domain of the system, C remains approximately constant in this limit. The inertially-stirred von Kármán flow has been widely used in experimental investigations, as the system can realise high-Re turbulence. These studies consider fluid acceleration [START_REF] Mordant | Long time correlations in Lagrangian dynamics: a key to intermittency in turbulence[END_REF], possible singularities of the solution of the Navier-Stokes equation [START_REF] Saw | Experimental characterization of extreme events of inertial dissipation in a turbulent swirling flow[END_REF], intermittency with weak Kolmogorov-Kármán-Howarth-Monin formulation (Dubrulle 2019), and interscale energy transfer [START_REF] Knutsen | The inter-scale energy budget in a von Kármán mixing flow[END_REF]. Furthermore, the von Kármán flow with inertial stirring sometimes shows interesting behaviours such as bistable turbulent states [START_REF] Ravelet | Multistability and memory effect in a highly turbulent flow: Experimental evidence for a global bifurcation[END_REF]) and related slow reversal of the flow [START_REF] Torre | Slow dynamics in a turbulent von Kármán swirling flow[END_REF]. For the context of QCB, [START_REF] Labbé | Study of the von Kármán flow between coaxial corotating disks[END_REF] and [START_REF] Pinton | Power fluctuations in a closed turbulent shear flow[END_REF] 2.1 Temporal (quasi-)cyclic modulation associated with the energy cascade 1.30 1.32 To investigate the origin of such QCB in a wall-bounded flow with time-independent forcing, we conduct a Direct Numerical Simulation (DNS) of the von Kármán flow with smooth disks. Figure 2.4(a) shows the temporal evolution in terms of the viscous energy injection P vis (t) and dissipation (t) at Re = 1500, where the Reynolds number is defined by the radius of the rotating disk R and the period of its rotation T as

P vis ×10 -2 1.300 1.305 1.310 1.315 1.320 1.325 ×10 -2 (a) -2 -1 0 1 E tor -E tor ×10 -4 -2 -1 0 1 2 E pol -E pol
Re = 2πR 2 νT . (2.1)
Here, we notice a counter-intuitive clockwise cycle between P vis (t) and (t), which infers that the energy dissipation increases (decreases) after the energy input decreases (increases). We employ two decompositions to understand this paradox: boundary-bulk decomposition for the domain and toroidal-poloidal decomposition for the flow components.

For the boundary-bulk decomposition, we find that the energy cycle of the viscous-driven von Kármán flow consists of two different scenarios, as schematically shown in Fig. 2.4(b). In the vicinity of the rotating disks, there is an almost instantaneous balance between the viscous injection and dissipation of energy, resulting in close to constant kinetic energy. In the bulk of the flow, the energy injection is residual of the viscous energy injection and the dissipation at the boundary region. Here, we need the other decomposition of toroidal and poloidal components to understand the energy cycle in the bulk of the system. When we denote the toroidal and poloidal components of the kinetic energy by E tor and E pol , we find the following cycle in bulk: (i) When the toroidal energy E tor (t) is small, the toroidal velocity and toroidal-poloidal energy transfer are also small. Then, because of the large shear rates on the rotating disks, viscous stirring (and associated energy injection) develops the toroidal flow. (ii) It triggers the centrifugal instability to generate the energy transfer from toroidal to poloidal components with a finite time delay. (iii) Then, the poloidal energy E pol (t) increases with the phase difference against E tor (t).

(iv) The development of the vortical structures with the growing E pol (t) triggers the energy dissipation around them. (v) At the same time, E tor (t) decreases as E pol (t) develops due to the angular momentum transfer.

Following the above analysis, we conclude that the QCB of the viscous-driven von Kármán flow can be Chapter 2 : Temporal Fluctuations in Turbulence clearly observed in the exchange of energy between toroidal and poloidal components: E tor (t) and E pol (t).

We demonstrate the QCB in a range of Re (1000 ≤ Re ≤ 3000) in Fig. 2.4(b), claiming the robustness of our scenario. The mysterious relation between the viscous energy injection P vis (t) and dissipation (t), shown in Fig. 2.4(a), can be understood by the two different energy cycles in the boundary and bulk regions of the flow. In other words, while the phase of (t) is determined by the boundary energy cycle with instantaneous balance, that of P vis (t) is affected by the bulk energy cycle between E tor (t) and E pol (t) with a finite time delay. Here, we remark that this conclusion is related to the seminal experiments by [START_REF] Cadot | Energy injection in closed turbulent flows: Stirring through boundary layers versus inertial stirring[END_REF]; it is essential to distinguish the boundary and bulk regions to understand dynamics and statistics of the viscous-stirring von Kármán turbulence.

Note that the physical mechanism of this QCB is the energy exchange between the toroidal and poloidal components through centrifugal instability and the angular momentum transfer and is different from the energy exchange between the largest-scale vortices and smaller ones through the energy cascade as in box turbulence [START_REF] Yasuda | Quasi-cyclic evolution of turbulence driven by a steady force in a periodic cube[END_REF]Goto and Vassilicos 2016a;[START_REF] Goto | Hierarchy of antiparallel vortex tubes in spatially periodic turbulence at high Reynolds numbers[END_REF]). Nevertheless, since the non-negligible temporal fluctuations of the energy transfer rate lead to the temporal fluctuations of the inter-scale energy flux and dissipation rate, these cycles are equally important for describing the nonequilibrium statistics of turbulence. Thus, although turbulence in the examined range of Re is not fully developed, we speculate that the proposed picture may hold for higher Re.

Even though we have identified the key ingredients that seem to play a role in the observed QCB, we do not know how these ingredients exactly give rise to such dynamics. For this purpose, we would ideally develop a minimal model displaying the same qualitative dynamics.

Reduced-order modelling of turbulence

Turbulence is a complex problem with tens of millions of Degrees of Freedom (DoF) interacting in a complex manner. How can the essential DoFs (the nonlinearity determining the system) be extracted from this complexity? In this section, we review attempts to reproduce key turbulence characteristics with as small as possible number of DoF. In § 2.2.1, we introduce some important minimal models of physical phenomena. Historically, such a minimal model led to the discovery of chaos. In § 2.2.2, we focus on the minimal turbulence model. In particular, we discuss how to model highly complex behaviour in developed turbulence. In § 2.2.3, we focus on the minimal transition model from a laminar to a turbulent state.

Figure 2.5: Time series of the Lotka-Volterra system. Γ is a Homoclinic loop. W s and W u denote the stable and unstable manifolds, respectively, and their cross-section is Π 1 and Π 2 , respectively.

Minimal model of physical phenomena and chaos

Physicists love minimal descriptions of complex phenomena. One of the most famous examples is the Lotka-Volterra model [START_REF] Lotka | Analytical note on certain rhythmic relations in organic systems[END_REF], mimicking the evolution of the number of prey x and predator y:

       dx dt = ax -bxy, dy dt = cxy -dy.
(2.2)

(2.3)
Here, the four parameters denote the prey birth rate a, the predation rate b and c, and the predator death rate d. This minimalistic model for population dynamics represents the following physical processes: i)

The number of prey indivisuals x grows exponentially with a growth rate a, proportional to the current x. ii) Number of prey x decreases (and predator y increases) when they encounter. This is proportional to xy. iii) Number of predators y decreases when there is no prey, and this death rate is proportional to y. Figure 2.5 shows the behaviour of the Lotka-Volterra system. See Hirsch et al. (2012, § 11.2) and Guckenheimer and Holmes (2013, § 2.3) for more detailed analysis.

In 1963, Lorenz proposed a system consisting of three ODEs [START_REF] Lorenz | Deterministic nonperiodic flow[END_REF]:

               dx dt = σ(y -x), dy dt = rx -y -xz, dz dt = xy -bz.
(2.4)

(2.5) (2.6)
Here, the three variables (x, y, z) denote convection velocity and thermal gradient in two directions. The Lorenz model can be described as an extreme case of Galerkin truncation applied to the Rayleigh-Benard system. See § 2.2.2 for more details on Galerkin truncation. Because of this truncation, the nonlinear terms of the Lorenz model does not represent scale-by-scale interactions. The three parameters are the Prandtl number σ, Rayleigh number r, and the system size b. Figure 2.6(a) shows a trajectory of this system, indicating a set in which the solution is eventually trapped, called the Lorenz attractor. This Lorenz system unveils a deterministic phenomenon but is unpredictable because a small disturbance in the initial condition results in a completely different outcome. It is the first recognised example of deterministic Chapter 2 : Temporal Fluctuations in Turbulence chaos. See also Hirsch et al. (2012, § 14), [START_REF] Guckenheimer | Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields[END_REF]Holmes (2013, § 2.3), andStrogatz (2014, § 9). During the 1960s and 1970s, many low-dimensional models exhibiting chaos were invented, including the Rössler model, which is even simpler than the Lorenz model [START_REF] Rössler | An equation for continuous chaos[END_REF]Rössler 1979). Theoretical considerations by Silnikov allowed to predict that in the specific case of a dynamical system evolving on a homoclinic orbit with a saddle focus, chaos is expected for a well-defined set of parameters. After linearization, this dynamics can be described by [START_REF] Silnikov | A case of the existence of a denumerable set of periodic motions[END_REF]:

               dx dt = -ρx -ωy + F x (x, y, z), dy dt = ωx -ρy + F y (x, y, z), dz dt = γx + F z (x, y, z).
(2.7)

Here, x = (x, y, z) are the three variables and F i with i ∈ x are specific to the considered dynamical system. The interactions of the linearised system are governed by the real parameters ρ, ω, and γ. The Silnikov attractor is shown in Fig. 2.6(b). See also Hirsch et al. (2012, § 16.1) and Guckenheimer and Holmes (2013, § 6.5). A similar saddle-focus dynamics is also observed in the model we develop in Chap. 3. We conclude this subsection by noting that a data-scientific approach in both dynamical systems and chaos has recently been developed and has gained great attention [START_REF] Brunton | Discovering governing equations from data by sparse identification of nonlinear dynamical systems[END_REF]; Brunton and Kutz 2022).

Reduced-order models of developed turbulence

Next, we review minimal models of turbulence. Since turbulence has a large number of degrees of freedom, its low-dimensional description has been pursued by many researchers with great enthusiasm. There are many systematic approaches, such as proper orthogonal decomposition (POD) or Galerkin truncation. POD analysis enables one to extract modes which dominate the key dynamics of complex systems [START_REF] Berkooz | The proper orthogonal decomposition in the analysis of turbulent flows[END_REF][START_REF] Holmes | Turbulence, Coherent Structures, Dynamical Systems and Symmetry[END_REF]. This framework was pioneered by Lumley [START_REF] Bakewell | Viscous sublayer and adjacent wall region in turbulent pipe flow[END_REF][START_REF] Payne | Large eddy structure of the turbulent wake behind a circular cylinder[END_REF], and has been used to extract coherent (low-dimensional) structures in turbulent flows [START_REF] Arndt | The proper orthogonal decomposition of pressure fluctuations surrounding a turbulent jet[END_REF][START_REF] Galletti | Low-order modelling of laminar flow regimes past a confined square cylinder[END_REF][START_REF] Tinney | Low-dimensional characteristics of a transonic jet. Part 1. Proper orthogonal decomposition[END_REF][START_REF] Podvin | A proper-orthogonal-decomposition-based model for the wall layer of a turbulent channel flow[END_REF]) and to achieve optimal control [START_REF] Bergmann | Optimal rotary control of the cylinder wake using proper orthogonal decomposition reduced-order model[END_REF][START_REF] Van Doren | Reduced-order optimal control of water flooding using proper orthogonal decomposition[END_REF][START_REF] Liberge | Reduced order modelling method via proper orthogonal decomposition (POD) for flow around an oscillating cylinder[END_REF]. Reduced-Order Modelling (ROM) is often based on POD analysis [START_REF] Hall | Reduced-order modelling of unsteady smalldisturbance flows using a frequency-domain proper orthogonal decomposition technique[END_REF][START_REF] Ilak | Modeling of transitional channel flow using balanced proper orthogonal decomposition[END_REF][START_REF] Wang | Proper orthogonal decomposition closure models for turbulent flows: a numerical comparison[END_REF][START_REF] Xiao | Non-intrusive reduced order modelling of the Navier-Stokes equations[END_REF]. For a broad overview of recent advancements on data analysis, see [START_REF] Taira | Modal analysis of fluid flows: Applications and outlook[END_REF] and Taira, Hemati, et al. (2020).

Compared to this data-oriented approach, Galerkin truncation aims to approximate the discretised system by a finite number of basis functions Brunton and Kutz 2022, § 12. This approach aims to reproduce the large-scale dynamics of turbulence rather than its small-scale statistics. Kraichnan (1988) pioneered such an approach to obtain a reduced description of turbulence. [START_REF] Rempfer | On low-dimensional Galerkin models for fluid flow[END_REF] investigated the inconsistency between the bifurcations of the real system and truncated low-dimensional models. Both approaches are sometimes combined [START_REF] Couplet | Intermodal energy transfers in a proper orthogonal decomposition-Galerkin representation of a turbulent separated flow[END_REF][START_REF] Borggaard | Artificial viscosity proper orthogonal decomposition[END_REF]. Galerkin truncation-aided ROM includes transition in a parallel shear flow [START_REF] Eckhardt | Transition to turbulence in a shear flow[END_REF], flow past a step [START_REF] Couplet | Intermodal energy transfers in a proper orthogonal decomposition-Galerkin representation of a turbulent separated flow[END_REF], two-dimensional magnetohydrodynamics [START_REF] Carbone | Transition to turbulence in a five-mode Galerkin truncation of two-dimensional magnetohydrodynamics[END_REF]. POD and Galerkin truncation are widely used in turbulence research [START_REF] Berkooz | The proper orthogonal decomposition in the analysis of turbulent flows[END_REF][START_REF] Holmes | Turbulence, Coherent Structures, Dynamical Systems and Symmetry[END_REF].

For the SSP of wall-bounded flow discussed in § 2.1.1, a low-order model was first proposed in Waleffe (1997). Later, a Restricted Nonlinear (RNL) model was examined to investigate the maintenance mechanism of plane Couette flow [START_REF] Thomas | Self-sustaining turbulence in a restricted nonlinear model of plane Couette flow[END_REF][START_REF] Thomas | A minimal model of self-sustaining turbulence[END_REF]. Recently, Yalnız et al. ( 2021) constructed a ROM based on periodic orbits and probabilistic connection between them.

For more detail of ROM, [START_REF] Chevillard | Lagrangian dynamics and statistical geometric structure of turbulence[END_REF] and [START_REF] Chevillard | Local and nonlocal pressure Hessian effects in real and synthetic fluid turbulence[END_REF] scaling. Adapted from Ohkitani and Yamada (1989, Figure 1). [START_REF] Ohkitani | Temporal intermittency in the energy cascade process and local Lyapunov analysis in fully-developed model turbulence[END_REF] is called the GOY (Gledzer-Ohkitani-Yamaha) model

d dt + νk 2 n u n = i k n u * n+2 u * n+1 -bk n-1 u * n+1 u * n-1 + ck n-2 u * n-1 u * n-2 + f n , (2.8)
which describes the temporal evolution of a complex variable u n ∈ C of representing the velocity in Fourier space averaged over the nth shell. Note that u * n means the complex conjugate of u n . Here, k n denotes the wavenumber and f n is the forcing of the nth shell. Model parameters are the kinematic viscosity ν and the free parameters b and c. The nonlinear term for the shell n consists of the shells n + 2, n + 1, n -1 and n -2, representing the scale-local dominance of the nonlinear interactions of turbulence. This model reproduces space-averaged statistics of turbulence, such as kinetic energy spectrum (as shown in Fig. 2.7), velocity structure functions, intermittency, and so on. Note that earlier attempts include Obukhov (1971), where a step-by-step energy cascade model similar to a shell model was proposed, and [START_REF] Siggia | Origin of intermittency in fully developed turbulence[END_REF], where intermittency was discussed by the reduced equations. In real turbulence, an overwhelming number of nonlinear interactions govern chaotic behaviour. In contrast, the shell model retains nonlinear interactions only between neighbouring shells. This simplification reflects the scale-local nature of nonlinear interactions of real turbulence, as discussed in § 1.1.2. Thus, the similarity between the shell model and real turbulence supports the scale-local view of turbulence. For more details on shell models of turbulence, including Sabra model (L'vov et al. 1998), refer to a review article [START_REF] Biferale | Shell models of energy cascade in turbulence[END_REF]) and a textbook (Ditlevsen 2010).

Reduced-order model of laminar-turbulent transition

In the previous subsection, we reviewed low-order modelling of developed turbulence. Another important objective of fluid mechanics research is to understand the transition from a laminar state to turbulence. For the laminar-turbulent transition, refer to recent review articles [START_REF] Tuckerman | Patterns in wall-bounded shear flows[END_REF][START_REF] Graham | Exact coherent states and the nonlinear dynamics of wallbounded turbulent flows[END_REF][START_REF] Wu | New insights into turbulent spots[END_REF][START_REF] Avila | Transition to turbulence in pipe flow[END_REF]. [START_REF] Moehlis | A low-dimensional model for turbulent shear flows[END_REF] constructed a Fourier-mode-based model to find a subcritical and intermittent transition to turbulence with exponential lifetime distributions. [START_REF] Seshasayanan | Laminar-turbulent patterning in wall-bounded shear flows: a Galerkin model[END_REF] derived a Galerkin truncated model of plane Couette flow, confirming the original problem's boundary condition. Recently, a stochastic predator-prey model was proposed [START_REF] Shih | Ecological collapse and the emergence of travelling waves at the onset of shear turbulence[END_REF][START_REF] Goldenfeld | Turbulence as a problem in non-equilibrium statistical mechanics[END_REF][START_REF] Wang | Stochastic model for quasi-one-dimensional transitional turbulence with streamwise shear interactions[END_REF] to reproduce DP universal class hidden in laminar-turbulent transition [START_REF] Sano | A universal transition to turbulence in channel flow[END_REF][START_REF] Lemoult | Directed percolation phase transition to sustained turbulence in Couette flow[END_REF][START_REF] Klotz | Phase transition to turbulence in spatially extended shear flows[END_REF]. Figure 2.8 shows the space-time plot of the laminar-turbulent pattern generated [START_REF] Wang | Stochastic model for quasi-one-dimensional transitional turbulence with streamwise shear interactions[END_REF].

Minimal Model of Quasi-cyclic Behaviour

This chapter investigates the quasi-cyclic behaviour of turbulence observed in a turbulent flow and its minimal modelling. The contents are based on the following:

Araki, Ryo, Wouter J. T. Bos, and Susumu Goto (2023a). Minimal model of quasi-cyclic behaviour in turbulence driven by Taylor-Green forcing. Fluid Dynamics Research 55 (3), p. 035507. arXiv: 2112.03417 [flu-dyn].

In § 3.1, we investigate a quasi-cyclic temporal fluctuations observed in box turbulence. Then, in § 3.2, we construct a minimal model reproducing these features. § 3.3 investigates the detail of the model, including its temporal dynamics and bifurcation analysis. We conclude this chapter in § 3.4.

Observation of quasi-cyclic behaviour

To illustrate the features we want to reproduce and guide the formulation of a minimal model reproducing these features, we conduct numerical simulations of both turbulent and temporally periodic flows with the same type of forcing. More precisely, we conduct DNS of three-dimensional incompressible flow governed by the Navier-Stokes equations,

∂u ∂t + (u • ∇)u = -∇p + ν∇ 2 u + f , (3.1)
with a steady forcing of the two-dimensional Taylor-Green type [START_REF] Yasuda | Quasi-cyclic evolution of turbulence driven by a steady force in a periodic cube[END_REF][START_REF] Goto | Hierarchy of antiparallel vortex tubes in spatially periodic turbulence at high Reynolds numbers[END_REF][START_REF] Van Veen | Transitions in large eddy simulation of box turbulence[END_REF],

f = (-f 0 sin x cos y, f 0 cos x sin y, 0), (3.2)
and the continuity equation, ∇ • u = 0. Here, u, p, and f are the velocity, pressure, and forcing fields, respectively. The forcing amplitude f 0 is set to unity. The only control parameter is the kinematic viscosity ν. We employ a pseudo-spectral method in a (2π) 3 periodic box.

We use an in-house parallelised code [START_REF] Delache | Scale by scale anisotropy in freely decaying rotating turbulence[END_REF]) to conduct DNS. It employs a pseudo-spectral method with the 2/3 dealiasing rule for spatial discretisation and the Adams-Bashforth scheme in the time domain. The initial condition is generated in Fourier space by Rogallo's method [START_REF] Rogallo | Numerical Experiments in Homogeneous Turbulence[END_REF]).

We perform DNS in a (2π) 3 triply periodic box. We focus on two distinct flows: three-dimensional periodic and turbulent. The SPO is obtained by the DNS with 64 3 Fourier modes by adjusting the viscosity to ν = 0.102. This corresponds to the value of the Reynolds number (3.3) of Re = 5.83. We use 128 3 Fourier modes to simulate turbulent flow at ν = 0.02 (Re = 29.7). For the phase averaging procedure. Note that we discard the transient part from the analysis. This interval is approximately 1.08 × 10 4 T with T defined in (3.3). We define the Reynolds number and the characteristic timescale of large-scale flow as

Re ≡ √ f 0 k f 3/2 ν and T ≡ 1 k f f 0 = 0.840, (3.3)
Chapter 3 : Minimal Model of Quasi-cyclic Behaviour respectively. Here, k f = (±1, ±1, 0) is the wavevector of the forcing (3.2). Figure 3.1 shows the temporal evolution of the energy input rate P (t) ≡ f • u against the energy dissipation rate (t) given by ν |ω| 2 for various Reynolds numbers. Here, • denotes the spatial average and ω ≡ ∇ × u. This time delay between the large-and small-scale representatives (i.e. P (t) and (t)) reflects the causal nature of the energy cascade.

We apply a phase average to the complex time series of P (t) and (t) conditioned on the local maxima of P (t) in order to extract smooth, time-delayed oscillations shown in Fig. 3.1(b).

In the following, we explain the detailed procedure of the phase averaging shown in Fig. 3.1(b). We first describe the flow observed at Re = 29.7 (Re λ is about 90) in Fig. 3.2. Figure 3.2(a) shows isosurfaces of|ω| capturing small-scale structures, whereas the forcing-induced columnar vortices emerge by visualising the isosurfaces of ω < . Here, ω < ≡ ∇ × u < , which is obtained by applying a low-pass filter to the velocity field, defined as u < (x) ≡ ´dr G(r/r 0 )u(x + r) with G being the Gaussian function. We set r 0 = 2. To conduct the phase-averaging, first, we pick up the local maxima of P (t) [Fig. 3.3(a)] with the following two criteria: (i) It must be larger than P t + σ(P ) where • t and σ(•) denote the time average and the standard deviation, respectively. The horizontal pink line indicates this value in Fig. 3.3(a). (ii) The temporal gap between two consecutive local maxima must be larger than τ max /2 where τ max is the time for the second peak of the autocorrelation function of P (note that the first peak is at τ = 0). We denote the identified local maximum of P (t) and the corresponding time by P 0 and t 0 , respectively. Second, the segments of the time series of P (t) around the local maximum P 0 are overlapped, as shown in Fig. 3.3(b). We normalise the segments by P 0 to avoid overestimation due to huge intermittent peaks. Third, we compute the average over the overlapped and normalised time series to obtain the phase averaged time series P phase shown in Fig. 3.3(c).

We apply a similar procedure to (t). However, the time is shifted for t 0 , and (t) is normalised by P 0 so that we can evaluate the time delay and the relative amplitude difference between the two quantities. The pink vertical dashed line shows the time delay in Fig. 3.3(c), which is 2.80T . We denote the phase-averaged quantities by • phase . These results suggest that the QCB of turbulent flow driven by the steady body force (3.2) is robust. Such QCB is also shown in Fig. 12 of [START_REF] Goto | Hierarchy of antiparallel vortex tubes in spatially periodic turbulence at high Reynolds numbers[END_REF] for two different forcing types at even higher Re. The physical origin of QCB is rooted in the energy cascading process from larger to smaller scales. Since the coherent structures at these scales are composed of a large number of Fourier modes, to describe the QCB in terms of Fourier modes, we need to understand the underlying nonlinear interactions among them. However, identifying the direct cause of QCB from tens of thousands of excited Fourier modes seems illusory. Thus, we decrease Re to reduce the complexity of the flow. In Fig. 3.1(c), we show the phase-averaged plots of the parametric time series of P (t) and (t) for four different values of Re. The change in the shape of the parametric plots is gradual, suggesting that the quasi-cyclic orbit in the turbulent flow is continuously connected to an SPO at Re ≈ 5.83, which is also shown in Fig. 3.1(d) for comparison. As will be shown in Fig. 3.4 below, the SPO at Re = 5.83 is not the laminar solution of the system which corresponds to a purely two-dimensional structure resulting from a balance between viscous stress and the forcing (3.2). We emphasise that this SPO plays a key role in constructing our model. We find that the amplitude and the period of the periodic and quasi-cyclic flows monotonically increase when we decrease Re from 29.7 to 5.83. This does not prove that the dynamics are identical, but the turbulent QCB and periodic flow seem to share the same driving mechanism. Note that in high-Reynoldsnumber turbulence beyond Re = 30, the amplitude and period seem to saturate to values of the same order as in Fig. 3.1(b) (see Fig. 12 of [START_REF] Goto | Hierarchy of antiparallel vortex tubes in spatially periodic turbulence at high Reynolds numbers[END_REF]). In Fig. 3.4, we visualise the periodic flow (i.e. SPO) discussed in Fig. 3.1(d), similar to the threedimensional periodic solution reported in van Veen, Kawahara, et al. (2018, Fig. 5). We distinguish four large-scale columnar vortices associated with the Taylor-Green force (3.2) and counter-rotating pairs of smaller vortices perpendicular to them. Note that we do not perform low-pass filtering [see Fig. 3.2(a)].

3.1 Observation of quasi-cyclic behaviour

/P max (d) t = T 0 t = T 0 + 9.42T t = T 0 + 17.0T t = T 0 + 19.3T

Three-equation model of quasi-cyclic behaviour

3.2 Three-equation model of quasi-cyclic behaviour

Construction of the model

Our objective is to construct the simplest possible model capable of reproducing QCB while retaining a close connection with the structure of the Navier-Stokes equations (3.1). For this purpose, we recall that in a Fourier representation of (3.1), the individual modes q i for the ith wavevector k i are governed by [START_REF] Kraichnan | Irreversible statistical mechanics of incompressible hydromagnetic turbulence[END_REF]Kraichnan 1988),

∂ ∂t + ν|k i | 2 q i = j,m A ijm q j q m + f i , (3.4)
where f i is the forcing applied to the ith mode, and A ijm are the coupling constants resulting from the advection and pressure terms of (3.1). The nonlinear term associated with triad interactions rapidly yields an overwhelming complexity when the number of retained modes increases. Even in our SPO, a large number of modes are dynamically active. In order to develop an analytically tractable model, we use a coarse-graining approach where we group subsets of Fourier modes and represent each group by a single variable, leading to a sort of shell-model (Obukhov 1971;[START_REF] Ditlevsen | Turbulence and Shell Models[END_REF].

The shells or groups used in our model are not regrouping modes as a function of scale using a rigorous criterion but as a function of the type of nonlinear interactions and energetic content. Indeed, we investigate the Fourier decomposition of the SPO to find that only Fourier modes with wavevectors (k x , k y , k z ) of (±1, ±1, 0), (±1, 0, 0), (0, ±1, 0), (0, 0, ±2), (±1, ±1, ±2), (±2, 0, ±2), and (0, ±2, ±2)

are responsible for 98 % of its energy. We now show the detailed behaviour of the seven most energetic modes. Figure 3.5 shows |ω| distributions of these seven modes. Triangles indicate combinations of different modes where energy transfer via triad interactions is possible. Figure 3.5 also compares isosurfaces of |ω| of the sum of these seven primary energetic modes and that of all modes. We find similar principal structures: the large columnar vortices and the small counter-rotating pairs of vortices. Here, we denote the velocity field consisting of the forced mode by u X and the other six primary energetic modes by u Y . The corresponding vorticity fields are denoted by ω X and ω Y , respectively. We plot the time series of the energy of the forced and six primary modes of the SPO at Re = 5.83 in Fig. 3.6. Although the energy E 110 (t) of the forced mode dominates, which is approximately equal to the total energy E(t), we observe a distinctive difference between E(t) and E 110 (t) when the primary scale energies are excited. By summing up the contributions of these seven modes, we obtain E X+Y shown in Fig. 3.7(a). We also note that there are fast oscillations in E 100 (t) and E 010 (t). However, these two modes are compensated with each other, and such rapid dynamics are not visible in E X+Y (t) [Fig. 3.7(a)]. This observation explains why there are no fast oscillations in the time series shown in Fig. 3.10(b). Figure 3.7(a) illustrates that the time evolution of the kinetic energy is closely reproduced, retaining only these modes.

A close inspection of the seven modes shows that all the nonlinear interactions involve the forced mode and two of the six other modes (see Fig. 3.5). In the following, X ∈ R denotes the characteristic velocity of the forced modes k = (±1, ±1, 0) and Y ∈ R corresponds to that of the remaining modes in (3.5). At this point, we suppose that there are only these two classes of modes and that we represent each class by a single, real variable. Furthermore, we assume (3.4) to govern the interaction of these two variables, X and Y , yielding,

dX dt = -AY 2 -νK 2 X X + F, dY dt = +AXY -νK 2 Y Y, (3.6)
with a coefficient A > 0, typical wavenumbers K α > 0 with α ∈ {X, Y }, and a steady force F > 0.

The first term on the RHS of each equation represents the nonlinear coupling between X and Y . This interaction conserves the global energy, (X 2 + Y 2 )/2. Note that since we model the triadic nonlinear term of (3.4) by regrouping the modes into two families (see Fig. 3.5). For notation, we employ both X and Y as the principal variables of our model and as subscripts to denote quantities associated with these variables. An extensive parameter scan of the two-equation model shows that the model always converges to a steady solution, and we do not observe an SPO or QCB. In fact, linear stability analysis of the fixed points of (3.6) shows that there are only stable steady solutions.

Here, we show the results of the linear stability analysis of the fixed points of (3.6). There are two kinds of fixed points: namely,

X 1 = F νK 2 X , 0 and X 2 = νK 2 Y A , ± 1 A AF -ν 2 K 2 X K 2 Y , (3.7)
where X ≡ (X, Y ). Note that the fixed points X 2 exist only for ν < √ AF /K X K Y . The perturbation (x, y) in the vicinity of the fixed points (X, Y ) obeys

dx dt = -A Y 2 + 2Y y -νK 2 X X + x + F, (3.8
) where we have neglected second-order terms x2 , y 2 , and xy. The Jacobian matrix is then expressed as

dy dt = +A X Y + Xy + xY -νK 2 Y Y + y , ( 3
J =   -νK 2 X -2AY AY AX -νK 2 Y   , (3.10)
whose eigenvalues are

λ = - 1 2 -AX + ν K 2 X + K 2 Y ± 1 2 AX + ν K 2 X -K 2 Y 2 -8A 2 Y 2 .
(3.11)

The eigenvalues for X 1 are

λ (X1) 1 = -νK 2 X , λ (X1) 2 = AF νK 2 X -νK 2 Y , (3.12)
which are both negative for ν >

√ AF /K X K Y . Therefore, X 1 is stable for ν > √ AF /K X K Y ,

and a pitchfork bifurcation takes place at

ν = √ AF /K X K Y . Then, for ν < √ AF /K X K Y , X 2 exists, which is stable irrespective of ν because the eigenvalues are λ (X2) 1,2 = - νK 2 X 2 ± -8AF + ν 2 K 2 X 1 + 8K 2 Y 2 . (3.13)
Thus, retaining only this simple interaction between the forced and most energetic modes seems insufficient to reproduce QCB via supercritical bifurcations. Results of the parameter scan further suggest that the subcritical route to QCB is not present either.

The additional ingredient for QCB turns out to be a small-scale representative and its associated triad interaction terms. Schematic of three different scales: "forced", "primary", and "secondary". We visualise |ω| distributions of typical Fourier modes in each scale. The forced scale corresponds to k f = (±1, ±1, 0). In the primary scale, we visualise k = (0, 0, ±2) and (0, ±2, ±2) modes. For the secondary scale, we visualise k = (±3, ±1, 0) and (±2, ±2, ±2) modes for example. Note that the contributions of modes with all the possible sign combinations (±k x , ±k y , ±k z ) are gathered in the visualisations. Triangles denote triad interactions between different scales. For details of interactions between the forced and primary scales, see Fig. 3.5.

Three-equation model of quasi-cyclic behaviour

entirely contained in E X+Y , there is a visible difference between the full and partial energy dissipation rates. This reveals that the rest of the Fourier modes contribute significantly to the dynamics of the energy dissipation, representing the small scales. We denote the ensemble of these residual modes by Z.

The essential nonlinear interactions of Z form triads with one mode of the Y -ensemble and another mode from either the Z-ensemble or the forced mode X. These observations lead to a refined three-equation model,

dX dt = -A 1 Y 2 + A 3 Y Z -νK 2 X X + F, dY dt = +A 1 XY -A 2 Z 2 + A 4 XZ -νK 2 Y Y, (3.14
)

dZ dt = + A 2 Y Z -(A 3 + A 4 )XY -νK 2 Z Z,
which is represented by a schematic in Fig. 3.7(b). Here, A 1 , A 2 > 0 and A 3 , A 4 ∈ R are triad coefficients which retain the discrete Navier-Stokes structure (3.4). We choose the signs and the values of the triad coefficients such that the detailed balance holds in the energy transfer between the three scales. The signs of A 1 and A 2 are defined so that energy cascades towards small scales: from X to Y and Y to Z. This two-step energy cascade (for A 3 = A 4 = 0) is similar to the Obukhov two-stage cascade (Obukhov 1971). The triads with coefficients A 3 and A 4 represent the "non-local" interactions involving all three scales. Note that this system is different from, but is of the same level of complexity, as the well-known Lorenz [START_REF] Lorenz | Deterministic nonperiodic flow[END_REF] or Rössler models [START_REF] Rössler | An equation for continuous chaos[END_REF]Rössler 1979). An important difference is that each variable denotes a Fourier mode in the Lorenz model, while in our model, it represents a group of modes.

Determination of the parameters

The model (3.14) is a simplified representation of the SPO, where all Fourier modes are sorted into three scales; the forced mode X, the energetic modes Y directly draining energy from X through the A 1 interaction, and the small scale modes Z which couple through the local direct cascade interaction A 2 with Y . There are also scale non-local interactions represented by A 3 and A 4 . Even though such a representation of the flow discards details of the actual flow obtained by the DNS, we will fit the model parameters to the DNS data to assess how the model can reproduce actual flow properties. We can fit six out of eight model constants in (3.14) by comparing them to the DNS of the periodic flow:

A i with i = 1, 2, 3, 4, K 2 α
, where α ∈ {X, Y, Z}, and F . To do so, we use the energy equations associated with (3.14),

dE X dt = T X -X + P, dE Y dt = T Y -Y , (3.15) dE Z dt = T Z -Z .
Here, E α ≡ α 2 /2 is the energy,

T X ≡ -A 1 XY 2 + A 3 XY Z, T Y ≡ +A 1 XY 2 -A 2 Y Z 2 + A 4 XY Z, (3.16
)

T Z ≡ + A 2 Y Z 2 -(A 3 + A 4 )XY Z
are the energy transfer terms, α ≡ 2νK 2 α E α is the energy dissipation rate, and P ≡ F X is the energy input rate. The model parameters are determined by their corresponding quantities of the SPO obtained by DNS. The resulting values are In the following, we discuss the detailed procedure of the parameter fitting (3.17) of the three-equation model (3.14). Figure 3.8(a) shows the time evolution of the forcing coefficient defined by

A 1 = 0.4, A 2 = 4, F = 0.7, K 2 X = 2, K 2 Y = 5,
F (t) ≡ P √ 2E X . (3.18)
We estimate the model parameter F = 0.7, since the time average F (t) t = 0.696. The periodic drops of F (t) are associated with a phase-desynchronisation between the forcing and the forcing-induced velocity field, u X .

We also compute the scale factors

K 2 α (t) ≡ α 2νE α (α ∈ {X, Y, Z}). (3.19)
T X (t) ≈ -A 1 XY 2 , T Y (t) ≈ +A 1 XY 2 -A 2 Y Z 2 , (3.20) T Z (t) ≈ + A 2 Y Z 2 .
Figure 3.9(a) shows their time series by the DNS of the SPO. T X (t) < 0 supports the energy cascade picture; the forced scale X is transferring energy to smaller scales (Y, Z) on average. Similarly, T Y (t), T Z (t) > 0 means that these smaller scales receive energy from the larger scales. We then evaluate the time-dependent coefficients,

A 1 (t) ≈ - T X XY 2 = - 1 2 √ 2 T X √ E X E Y , (3.21) A 2 (t) ≈ T Z Y Z 2 = 1 2 √ 2 T Z √ E Y E Z . (3.22)
Again, we neglect the scale non-local interactions (A 3 = A 4 = 0) in these expressions. The result is shown in Fig. 3.9(b), and we estimate A 1 = 0.4 and A 2 = 4 as the model parameters from the time-averaged values A 1 (t) t = 0.440 and A 1 (t) t = 4.04, respectively.

The above argument allows us to determine the model parameters in (3.17). The non-local interaction coefficients A 3 and A 4 are left to be determined. In § 3.2.2, we vary these two parameters to investigate the model properties.

The energy flux coefficients A 1 and A 2 are determined by the energy transfer terms T α in (3.16) while ignoring the nonlocal coefficients A 3 and A 4 (3.21-3.22). The forcing coefficient F is evaluated by P and the X-scale energy E X (3.18). The squared characteristic wavenumber K 2 α is set by E α and α in each scale (3.19). Note that K X = √ 2 of the model parameter can be related to k f = √ 2 of the forcing (3.2) of the DNS. We remark here that our parameter choice (3.17) supports the energy cascade picture with T X (t) < 0: the forced scale X transfers its energy to smaller scales (Y, Z) on average. And T Y (t), T Z (t) > 0 means that the smaller scales receive energy from the larger scales. The undetermined parameters of the model are the scale non-local interaction coefficients A 3 and A 4 , which can be freely chosen. The only control parameter is Re ≡ 1/ν. We numerically integrate the model with a fourth-order Runge-Kutta scheme and ∆t = 0.01 starting from random initial conditions. See [START_REF] Rackauckas | DifferentialEquations.jl -A performant and feature-rich ecosystem for solving differential equations in Julia[END_REF] for the solver information. Our numerical simulations seem to indicate that no periodic solutions exist without the complete non-local interactions: A 3 = 0, A 4 = 0, or A 3 + A 4 = 0. Conversely, periodic behaviour is observed for a wide range of values when A 3 = 0, A 4 = 0, and A 3 + A 4 = 0. This observation emphasises the importance of non-local triad interactions for periodic behaviour. 3.17) and (A 3 , A 4 ) = (0.5, -0.95). Since the definitions of Re are different in the model and DNS, we have chosen a Reynolds number in the model, which allows qualitatively reproducing the DNS results. We compute two quantities. One is E X-X0 ≡ (X -X 0 )

Comparison between the model and the direct numerical simulation result

2 /2, which is the fluctuating energy of the forced mode around the laminar base flow X 0 ≡ F Re /K 2 X . The other quantity E Y +Z ≡ Y 2 /2 + Z 2 /2 is the energy of the rest of the modes. We compare them to the corresponding quantities in the DNS of the SPO [Fig. 3.10(b)], where the base flow is

u 0 ≡ f /2ν k f 2 , the forced-mode fluctuating energy is E X-X0 ≡ |u X -u 0 | 2 /2
, and E Y +Z is defined by the energy possessed by the non-forced modes. We can observe similar periodic behaviour of E X-X0 and E Y +Z in the model (3.14) and in the SPO driven by the steady forcing (3.2). In particular, there are predator-prey-like exponential growth and decay in both systems. Although fast oscillations are observed in the model but not in the DNS, a close analysis of the DNS of the SPO reveals the presence of rapid oscillations in specific Fourier modes. These oscillations are compensated by modes that display the same energy oscillations with an opposite phase and do not appear in Fig. 3.10(b). We stress that this SPO is independent of the exact We observe a chaotic state of the model by varying Re from 14.05 to 14.1 while keeping the model parameters as in Fig. 3.10(a). Figure 3.11(a) shows the orbits in phase space for both the periodic (at Re = 14.05) and chaotic (at Re = 14.1) cases. The chaotic solution remains close to the SPO as it shows chaotic QCB and is permanent as in the turbulence investigated in § 3.1. Thus, the same model reproduces SPO and chaotic QCB. Incidentally, the SPO resembles a Silnikov homoclinic orbit [START_REF] Silnikov | A case of the existence of a denumerable set of periodic motions[END_REF]. We also plot a simpler periodic orbit at Re = 12.5 in this figure. It is almost two-dimensional as opposed to the complex three-dimensional periodic and chaotic orbits, suggesting a possible connection with a two-dimensional periodic orbit in the same forcing configuration (van Veen, Kawahara, et al. 2018, Fig. 4). However, we do not focus on this orbit as it is not directly connected to a chaotic one.

To further assess the behaviour of the system, we draw the bifurcation diagram in Fig. 3.11(b) with the same parameter set as in Fig. 3.10(a) and Fig. 3.11(a). We observe a supercritical transition from periodic to chaotic solutions at a critical Reynolds number Re cr ∈ [14.060, 14.061] [inset of Fig. 3.11(b)], and, as observed in Fig. 3.11(a), the chaotic orbit remains close to the SPO. We note that the solution becomes periodic again when we further increase Re beyond the range of Fig. 3.11(b), probably because the model contains only a small number of degrees of freedom. The inset of Fig. 3.11(b) shows that there is a hysteresis in the range Re ∈ [13.82, 14.03], below Re cr , which corresponds to a subcritical bifurcation from a periodic solution to another periodic solution shown in Fig. 3.10(a). The appearance of the multiplicity of local extrema corresponds to the spiraling behaviour of the orbit in phase space. Thus, although the bifurcations from the trivial steady solution to the SPO are rather complicated, that from the SPO to chaos with QCB is simple. Although there is no clear scenario for the route to turbulence with QCB, the present model results may give us a hint to describe the route in real turbulence.

Subcritical bifurcation to chaos

Since it is well known that, in some cases, turbulence appears via a subcritical transition, here we demonstrate that our model also expresses such a route to chaos. We stress that we cannot use the strategy above to determine the model parameters since there is no SPO in such a system. Instead, by varying the undetermined parameters of the model, we observe transient chaos at (A 3 , A 4 ) = (0.4, -0.5) as shown in Fig. 3.12(a). The corresponding bifurcation diagram in Fig. 3.12(b) shows a subcritical bifurcation between steady and chaotic solutions around Re ≈ 32.3. There are bi-stable states for 33 Re( 35) of steady and chaotic solutions. The inset of Fig. 3.12(b) shows that there are multiple windows of periodic solutions in the chaotic regime, probably due to the limited number of degrees of freedom of the model (3.14). The transient behaviour in Fig. 3.12(a) reminds us of the sudden relaminarisation observed in a linearly forced turbulence [START_REF] Linkmann | Sudden relaminarization and lifetimes in forced isotropic turbulence[END_REF], turbulent Kolmogorov flow (van Veen and Goto 2016), pipe flow [START_REF] Hof | Finite lifetime of turbulence in shear flows[END_REF], and even in the Lorenz system [START_REF] Yorke | Metastable chaos: The transition to sustained chaotic behavior in the Lorenz model[END_REF][START_REF] Maslennikov | Dynamic boundary crisis in the Lorenz-type map[END_REF]. We evaluate the survival probability P Re (t), representing how likely the solution remains in a chaotic regime at a given time t, to investigate this phenomenon. To evaluate P Re (t), we identify the relaminarisation time t r by the first time when the local maxima of oscillating energy

E Y -Y0 ≡ (Y -Y 0 )
2 /2 becomes smaller than a threshold δ = 1 × 10 -3 . Here, Y 0 is the stable and steady solution. Then, the probability P Re (t) for given t can be evaluated by the ratio of a number of samples with t r < t against the number of the whole sample. We plot P Re (t) in Fig. 3.13(a) to find that an exponential scaling, studies [START_REF] Linkmann | Sudden relaminarization and lifetimes in forced isotropic turbulence[END_REF], the exponential scaling (3.24) of τ (Re) differs from a superexponential behaviour observed in [START_REF] Linkmann | Sudden relaminarization and lifetimes in forced isotropic turbulence[END_REF]. This qualitative difference may also be caused by the minimal number of degrees of freedom in the model. Note that the Taylor-Green forcing (3.2) in the DNS does not permit such a transition since the laminar base flow u 0 ≡ f /2ν k f 2 is linearly unstable. However, the steady Kolmogorov forcing with a linearly stable laminar base flow exhibits sudden relaminarisations (van Veen and Goto 2016). Thus, we can speculate that the model can reflect different forcing setups applied to the Navier-Stokes equations by varying the parameters (A 3 , A 4 ).

P Re (t) ∝ exp - t τ (Re) , ( 3

Concluding remarks

The present investigation attempts to construct a minimal model of turbulence with quasi-cyclic behaviour (QCB) in a steady-force driven flow while keeping the structure of the Navier-Stokes equations. First, through the DNS of Navier-Stokes turbulence, we show that QCB in high-Re turbulence is continuously connected to an SPO at small Re by extracting the intrinsic periodicity of QCB via a phase averaging technique ( § 3.1). Next, we conduct a mode-by-mode analysis of the SPO to identify the flow's forced, primary energetic, and secondary scales. We propose the three-equation model (3.14) describing the evolution of such three distinct scales ( § 3.2.1). By adjusting the model parameters, we observe that the model reproduces an SPO similar to that of the DNS ( § 3.2.2). We emphasise that scale non-local nonlinear interactions (interactions involving three separate scales) are mandatory for reproducing these dynamics. Then, we conduct a bifurcation analysis to show that the model also exhibits chaotic QCB via a supercritical bifurcation, which is continuously connected to the SPO ( § 3.3.1). Thus, we conclude that the proposed model reproduces turbulent QCB and its relation to an SPO using a minimum number of degrees of freedom.

Further analysis of the model by varying the undetermined parameters yields transient chaos with sudden relaminarisation, which is also observed in turbulent flow with different forcing setups ( § 3.3.2). Thus, we speculate that the present model can be a minimal model for certain features of turbulence.

An outstanding open question is how QCB survives in spatially extended flows. How will the global dynamics change when the forcing is applied to scales smaller than the domain size? In other words, how will the modes larger than the forced scale alter QCB turbulence, and how can we model it? Investigating the relation between space and scale locality and temporal dynamics of turbulence is left for further research.

Part II

Space-local Navier-Stokes Turbulence 4 Physical-space Locality of Turbulence

Locality is one of the most fundamental, often implicitly assumed, properties in a number of fields of physics. For example, we do not consider the universal gravitation due to the moon when we compute the velocity of an apple falling from a tree in Cambridge. Generally, we implicitly assume that two nearby objects are more strongly correlated (or affected by each other) than distant ones. Here, the terms nearby and distant refer to both physical space and scale-space sense; one needs not to consider the effect of the moon on the falling apple, and similarly, one can model the falling apple without being bothered by its O 10 23 molecules constituting it. We already discussed the scale-local nature of the energy cascade in § 1.1.2. In this section, we focus on the other locality of turbulence, in the physical space. Although much less attention has been paid to the physical-space locality, this idea may have important implications for the understanding and modelling of turbulence, particularly its physical-space mechanism. In § 4.1, we review previous studies on this subject. In § 4.2, we define the space-local velocity field u L , derived from the Biot-Savart law between the velocity and the vorticity fields. It can be understood as the velocity field induced by the neighbouring vorticity field and is the central quantity of interest in this and the next chapter. In § 4.3, we conduct a post-process analysis of a turbulent flow dataset with u L . We investigate how the scale-space property of turbulence is altered by u L . We conclude this chapter in § 4.4.

Previous studies on physical-space locality

The scale-space locality states that the nonlinear interactions between adjacent scales are responsible for the small-scale universality of turbulence. However, such a discussion on scale space does not explain how local the nonlinear interactions are in physical space. In this section, we review the locality of turbulence in physical space. The physical-space locality of the energy flux was first considered by Kraichnan (1974). Since then, several formulations of the space-local energy flux have been proposed; Meneveau employed wavelets [START_REF] Meneveau | Analysis of turbulence in the orthonormal wavelet representation[END_REF] and Lagrangian correlation framework to locally track the energy flux in space [START_REF] Meneveau | On the Lagrangian nature of the turbulence energy cascade[END_REF]. Eyink used a refined similarity hypothesis involving a spatial length scale [START_REF] Eyink | Local energy flux and the refined similarity hypothesis[END_REF], as well as developing a multi-scale gradient expansion, which decomposes the turbulent stress tensor into multi-scale and multi-order spatial derivatives (Eyink 2005;[START_REF] Eyink | Onsager and the theory of hydrodynamic turbulence[END_REF][START_REF] Eyink | Localness of energy cascade in hydrodynamic turbulence. I. Smooth coarse graining[END_REF]. Tsinober discussed spatially concentrated vorticity and its nonlocal interactions with background turbulence [START_REF] Tsinober | Is concentrated vorticity that important?[END_REF]. "Five-dimensional" (three-dimensional space, scale, and time) analysis of energy cascade revealed an emergence and disappearance of fluid structures within the larger-and smaller-scale structures, respectively [START_REF] Cardesa | The turbulent cascade in five dimensions[END_REF]. [START_REF] Doan | Scale locality of the energy cascade using real space quantities[END_REF] investigated the scale-local energy cascade in terms of vortex stretching in real space. More recent work found a powerlaw correlation between the filtered strain rate and the space-local energy flux [START_REF] Alexakis | Local energy flux of turbulent flows[END_REF]. In Vela-Martín and Jiménez (2021), the irreversibility of turbulence and preference for direct energy cascade is discussed with its space-local property. We note that these questions are not only of fundamental importance but are also relevant to turbulence modelling, particularly in refining sub-grid scale models (Borue and Orszag 1998).

Recently, the combined analysis of both position in space and scale has been conducted using the Kármán-Howarth-Monin-Hill equation (Valente and Vassilicos 2015;[START_REF] Yasuda | Spatio-temporal intermittency of the turbulent energy cascade[END_REF]. It is a rigorous equation derived from the Navier-Stokes equation and describes the evolution of "energy" of velocity at two points, xr/2 and x + r/2, separated by distance r = |r| centered at x. Thus, this equation can handle the scale r and space x simultaneously. Pressure and its Hessian are also important to describe the physical-space nonlocality of turbulence [START_REF] She | Structure and dynamics of homogeneous turbulence: models and simulations[END_REF]. Several investigations report its role in decaying turbulence [START_REF] Kishiba | Physical-space nonlocality in decaying isotropic turbulence[END_REF], relation with finite time blow-up [START_REF] Ohkitani | Nonlocal nature of vortex stretching in an inviscid fluid[END_REF], rotation [START_REF] Nomura | The structure and dynamics of vorticity and rate of strain in incompressible homogeneous turbulence[END_REF], and the role of local and nonlocal contributions to the influence of the pressure Hessian [START_REF] Chevillard | Local and nonlocal pressure Hessian effects in real and synthetic fluid turbulence[END_REF].

In decaying isotropic turbulence, analysis of the spatial local/nonlocal contributions of the strain-rate tensor on the vorticity revealed that considerable (about 40 % for energy dissipation and 50 % for enstrophy production) contributions comes from nonlocal regions in which the separation from the source vorticity is larger than 2λ, where λ is the Taylor microscopic length scale [START_REF] Kishiba | Physical-space nonlocality in decaying isotropic turbulence[END_REF]). Geometric analysis of the nonlinear interactions of turbulence was performed in real space to show how purely geometric properties contribute to the forward energy cascade (Ballouz and Ouellette 2020). In the framework of the toy model, Tanogami and Sasa considered an XY spin model with space-local interactions to consider its energy cascade and universal scaling [START_REF] Tanogami | XY model for cascade transfer[END_REF].

The most important work regarding our methodology in this study is developed by Hamlington and his colleagues [START_REF] Hamlington | Physics-Based Turbulence Anisotropy Closure Including Nonlocal and Nonequilibrium Effects in Turbulent Flows[END_REF]. They proposed the local/nonlocal decomposition of the strain-rate tensor to investigate the alignment between the vorticity and eigenvectors of the strain-rate tensor (Hamlington, Schumacher, et al. 2008a;Hamlington, Schumacher, et al. 2008b). They revealed that the most extensional eigenvector of the nonlocal strain-rate tensor aligns with vorticity. In industry-oriented Computational Fluid Dynamics (CFD), assessing the physical space locality of turbulence is important. Industrial CFD often solves flow in/around complex structures such as mixers with rotating blades or aeroplanes. In such a case, the numerical algorithm should be local in space or sparsely connected to a distant fluid element for high scalability. For example, the recent growth of interest in the Lattice Boltzmann Method (LBM) relates to its space-locality (Krüger et al. 2016, §2.4). The LBM considers the temporal evolution of distributions of "particles" in nodes spanning the domain, based on the Boltzmann equation. The nonlinear terms are then evaluated locally (node-wise), and this property contributes to the high scalability of the scheme. shows the distributions on the plane denoted by the grey plane in the other panels. Adapted from Buaria, Pumir, and Bodenschatz (2020, Figure 1).

Previous studies on physical-space locality

Space-local filtering on the velocity field

Space-local filtering on the velocity field

In this section, we formulate the space-local description we employ in the subsequent investigations. In § 4.2.1, we review the nonlocal relation between vorticity and velocity fields connected by the Biot-Savart law to define the space-local velocity field by truncating the Biot-Savart law in § 4.2.2. In § 4.2.3, we discuss the physical-space locality of the pressure gradient term in the Navier-Stokes equations and how it relates to the space-local velocity field.

Nonlocal relation between vorticity and velocity fields

In this subsection, we consider the physical-space locality in the macroscopic description of fluid. We consider the incompressible three-dimensional Navier-Stokes equations,

       ∂u ∂t = -ω × u -∇ p + u 2 2 + ν∇ 2 u + f , ∇ • u = 0. (4.1) (4.2)
Here, u and ω ≡ ∇ × u are velocity and vorticity field, respectively. The forcing field driving the flow is denoted by f . The pressure term divided by the constant density is p, and ν is the kinematic viscosity determining the flow property. Except for the pressure gradient term, all the terms on the RHS of (4.1) can be evaluated locally in physical space. By solving the Poisson equation,

∇ 2 p = - ∂u i ∂x j ∂u j ∂x i , (4.3)
one obtains the pressure field, p(x, t) = p 0 (x, t) + 1 4π

ˆΩ ∂u i ∂x j ∂u j ∂x i (x , t) dx |x -x | , (4.4)
where Ω denotes the domain. The harmonic part p 0 satisfies ∇ 2 p 0 = 0 and depends on the boundary conditions. The pressure field, including the boundary condition and the spatial integral, is thus determined nonlocally in terms of physical space. See § 4.2.3 for a more detailed discussion.

The curl of (4.1) gives the vorticity equation,

∂ω ∂t = -u • ∇ω + ω • ∇u + ν∇ 2 ω + ∇ × f , (4.5)
which gives an alternative point of view on the physical space locality. Since the pressure term vanishes by the identity ∇ × ∇p = 0, the curl operator seems to eliminate the spatial nonlocality. However, there is a spatial nonlocality associated with the curl operator itself. Although one can compute the curl of a given velocity field ∇ × u locally in space, the opposite procedure requires spatially nonlocal information. Evaluation of u from known ω can be done by the Biot-Savart law (Majda and Bertozzi 2002, § 2.4),

u(x) = 1 4π ˆΩ d 3 x ω(x ) × x -x |x -x | 3 , (4.6)
which includes the spatial integral as in the pressure Poisson equation (4.4). It leads us to a new perception of (4.5); temporal evolution of vorticity is described by the advection and stretching due to the velocity induced by the whole vorticity field, along with the viscous damping and the forcing. In this sense, we understand that the nonlinear term of the vorticity equation is spatially nonlocal. In § 4.2.3, we discuss the relationship between the two physical-space nonlocalities in pressure (4.4) and velocity (4.6).

Space-local velocity field

Here, we define the space-local velocity field with the superscript

• L , u L (x) ≡ 1 4π ˆr≤R d 3 x ω(x ) × x -x |x -x | 3 , (4.7)
where r ≡ xx . Equation (4.7) involves truncation of the spatial integral of the Biot-Savart law (4.6) at a sphere of radius R centered at x. Expression (4.7) was first proposed to investigate the alignment between the vorticity vector and the eigenvectors of the strain-rate tensor (Hamlington, Schumacher, et al. 2008a;Hamlington, Schumacher, et al. 2008b;[START_REF] Hamlington | Physics-Based Turbulence Anisotropy Closure Including Nonlocal and Nonequilibrium Effects in Turbulent Flows[END_REF]. Figure 4.4 shows the schematic of the space-local velocity field u L .

Next, we consider expressing (4.7) in Fourier space. The procedure below is simpler than the one provided in the Supplemental Note of Buaria, Pumir, and Bodenschatz (2020). To this end, we introduce the threedimensional top-hat (box) filter with radius R,

B(r; R) =      3 4πR 3 |r| ≤ R 0 |r| > R , (4.8)
by using the fact that the volume of the sphere of radius R is 4πR 3 /3 and ˆB(r; R) dr = 1. (4.9)

Here, r ≡ xx denotes the vector between two points. We introduce spherical coordinates

x =     x y z     =     r sin θ cos φ r sin θ sin φ r cos θ     , (4.10)
where r ≥ 0 denotes the radius, φ ∈ [0, 2π) the angle between the projection of r vector onto the (x, y) plane and the x axis, and θ ∈ [0, π] the angle between r and the z axis. The Fourier representation of (4.8) From (4.11) to (4.12), we can choose arbitrary expression for k so that we set k = (0, 0, k). From (4.12) to (4.13), we exchange the variable by ζ =cos θ, then dζ = sin θ dθ and θ ∈

[0, π] → ζ ∈ [-1, 1].
From (4.13) to (4.14), we rewrite using sin θ = i e -iθe iθ /2. We change the variable from r to kr in (4.15). This integral is solved as ´X 0 dx x sin x = sin X -X cos X.

Thus, we obtain the following Fourier-space expression of the space-local velocity field

       u L (k) ≡ 1 -B(kR) u(k), B (
f L i (x; R) = 1 2π ˆ 1 -B(kR) f i (k)e ik•x dk , (4.20)
with the filter function (4.19). Figure 4.5(a) shows the profile of f L x (x; R) in x coordinate at the edge of the computational domain, y = z = -π. The colour gradient from dark to light corresponds to 0 ≤ R ≤ 2L f . At R = 0, the profile is zero everywhere, f L x (x; 0) = 0. By increasing R, it converges to the original profile lim R→∞ f L x (x; R) = f x (x). However, we note that the convergence is not monotonous as we observe overshoot in the intermediate R values. This property is also depicted in Fig. 4.5(b) with the "energy" of the space-local forcing field f L 2 /2 against R. We observe overshoot and eventual convergence to the energy of the original forcing field, f 2 /2.

Physical-space nonlocality of the pressure field

Here, we consider the physical-space nonlocality associated with the pressure field. Indeed, the solution of the Poisson equation ( 4.4) contains a spatial integral. When we restrict the integral range, as we have done for the Biot-Savart law between vorticity and velocity fields (4.7), how different are these two fields: space-local pressure field and space-local velocity field? Regarding this problem, it is important to note that the role of the pressure gradient term is to sustain the incompressibility of the fluid. Indeed, the Poisson equation represents the velocity redistribution on the same scale |k| in Fourier space. Thus, it does not directly contribute to the nonlinear interactions involving multiple scales. However, it is possible that the space-local pressure field indirectly affects the nonlinearity or the energy cascade. Theoretical and numerical quantification of its role, including the comparison with a similar analysis through the pressure Hessian [START_REF] Chevillard | Local and nonlocal pressure Hessian effects in real and synthetic fluid turbulence[END_REF], would constitute an important subject of future research interest.

Space locality in turbulence spectra

In this section, we investigate the physical-space locality of Navier-Stokes turbulence using Fourier spectra. In § 4.3.1, we address the DNS configurations. In § 4.3.2, we apply the space-local filter, introduced in § 4.2, to re-define space-local contributions of various spectra in Fourier space.

L f ≡ 2π/ k f = 2π/ √ 3 (4.22)
and the characteristic time scale

T f = 1/ k f f 0 = 1/ 4 √ 3 (4.23)

Spectra with the space-local restrictions

In Fig. 4.8(a), we plot the space-local nonlinear energy transfer

T L (k) ≡ ˆN L i (k)u * i (k) dΩ k , ( 4

Concluding remarks

with the space-local nonlinear term,

N L (x) ≡ -ω(x) × u L (x). (4.25)
Note that we add the superscript • L to distinguish the quantity responsible for the space-local contributions of the original governing equation. One immediate finding is a collapse in the small scale, indicating that the small-scale transfer is not affected by the space-local cutoff. On the contrary, there is a strong R dependency on the larger scale. In particular, the large negative regime is different from the original turbulence data, which exhibits a plateau at zero (corresponding to the inertial range) in the same regime. By increasing R, T L (k) collapses on the original energy transfer spectrum T (k). These observations are consistent with the rough correspondence between large (small) wavenumber and small (large) physicalspace scale. By integrating T L (k), one can evaluate the space-local energy flux Π L (k) ≡ -´k 0 T L (p) dp.

In the original turbulence, the energy transfer is summed up to zero, thus Π(∞) ≡ ´∞ 0 T (k) dk = 0. However, the space-local transfer is not a conserved quantity: Π L (∞) ≡ ´∞ 0 T L (k) dk = 0. This is evident in inset of Fig. 4.8(a), which plots Π L (∞)/ . This quantity approaches unity for small values of R, indicating that almost no energy is transferred by T L (k). The convergence property Π L (∞)/ → 0 as R is consistent with the property of the original Navier-Stokes turbulence. Another finding from Fig. 4.8(a) is a trend on the zero-crossing wavenumber k 0 (R), defined by T L (k 0 ) = 0. Since k 0 (R) corresponds to the peak of Π L (k), a decreasing trend of k 0 (R) indicates that the space-local constraint inhibits the large-scale transfer and flux. Note that the overshoot in T L (k) against T (k) at large values of R is due to the convergence property of the space-local filter. Overall, the space-local contributions of the energy transfer T L (k) exhibit a qualitatively different profile against the one in the space-local turbulence T (k), in particular at the large scales k f < k < 2π/R. This observation indicates that the space-local system (see next chapter) and the space-local contributions of the original system are not identical. In Fig. 4.8(b), we plot the space-local energy dissipation rate spectrum

L (k) ≡ 2ν ˆk 0 p 2 E L (p) dp . (4.26)
Note that its integral defines the space-local contributions of the energy dissipation rate L ≡ 2ν ´∞ 0 k 2 E L (k) dk.

We observe similar properties as in Fig. 4.8(a); while the small-scale profile is preserved, the large-scale profile is affected by R. However, since the small scale primarily defines the energy dissipation rate thanks to the k 2 coefficient, the R dependency on L (R) is small compared to Π L (R), as shown in the inset of Fig. 4.8(b). Thus, we can conclude that energy dissipation is a spatially local phenomenon.

Concluding remarks

The research question posed in this Part II is the physical-space locality of turbulence, which has not been focused on compared to the locality in scale. In this chapter, we reviewed the physical-space locality of turbulence in the literature ( § 4.1), and we defined the space-local velocity field ( § 4.2) to analyse the turbulence dataset ( § 4.3). Here, the space-local velocity (4.7) is defined by the truncated Biot-Savart law and interpreted as "locally induced velocity by the neighbouring vorticity field". The main results, summarised in Fig. 4.8, are consistent with our intuition that the space-local structures correspond to small-scale structures. We find that the space-local filter strongly alters the energy transfer rate at large scales. Contrarily, transfer at the small scales and the energy dissipation, dominated by the smallest scales of the flow, remains almost unaltered under the filtering.

Here, we note that the analysis conducted in this chapter examines the space-local contributions in the turbulence dataset. In the next chapter, we will work on the same research question from an alternative point of view. We will modify the governing equations in a space-local sense to see how the space-local turbulence driven by such a system behave differently against the original turbulence we investigated in this chapter.

This chapter analyses the modified Navier-Stokes equations, in which the nonlinear term is restricted in the space-local sense. The content is based on the following: Araki, Ryo, Wouter J. T. Bos, and Susumu Goto (2023b). Space-local Navier-Stokes turbulence. arXiv preprint. arXiv: 2308.07255 [flu-dyn].

In § 5.1, we define a new system called space-local Navier-Stokes equations, in which the nonlinear interactions are restricted locally in physical space. We also discuss some basic properties of the new system, which may differ from the original Navier-Stokes equations. In § 5.2, we review details of direct numerical simulation of the space-local system. In particular, we focus on its temporal dynamics. In § 5.3 and § 5.4, we investigate statistics of the space-local system. We focus on two different scaling regimes, observed in scales larger ( § 5.3) and smaller ( § 5.4) than the space-local filter size, respectively. We conclude this chapter in § 5.5.

Space-local Navier-Stokes equations

Using the space-local velocity field (4.7-4.19), we here define the modified Navier-Stokes equations with restricted nonlinearity in the space-local sense. The space-local vorticity equation becomes

∂ω ∂t + u L • ∇ω = ω • ∇u L + ν∇ 2 ω + ∇ × f , (5.1)
where the velocity u in the nonlinear terms is substituted by the space-local one u L . This equation is space-locally closed, as the evolution of ω at point x is described by u L , locally determined in the sphere of radius R centered at x. We here examine the basic properties of (5.1). First, we remark that the space-local system remains incompressible; namely, ∇ • u = 0. Here, u is governed by the space-local Navier-Stokes equations

∂u ∂t = -ω × u L -∇ p + u 2 2 + ν∇ 2 u + f , (5.2)
corresponding to (5.1). The incompressibility of u immediately follows from that of the space-local velocity field,

∇ • u L = 0 ↔ ik j u L j = i 1 -B(kR) k j u j = 0. (5.3)
Second, the space-local system violates the Galilean invariance of the original Navier-Stokes equations.

It follows from the property of the space-local filter function (4.19): (5.4) stating that the k = 0 mode flow is purely nonlocal and is eliminated by filtering. Third, by taking the inner product of (5.2) and u and integrating over space, we obtain the energy (5.7)

lim k 0 1 -B(kR) = 0,
The additional term NL is associated with the spatially nonlocal contributions of the nonlinear term. It disappears in the R ∞ limit, since

lim R ∞ NL = ˆdx [ω × u] • u = 0.
(5.8) See § 4.2.3 for physical-space locality of pressure gradient term in the Navier-Stokes equations and how it relates to the space-local velocity field.

Direct numerical simulation of the space-local Navier-Stokes equations

In this section, we address details of the DNS of the space-local Navier-Stokes equations. In § 5.2.1, we provide the implementation of the space-local nonlinear term. In § 5.2.2, we report the temporal evolution of the space-local system.

Numerical evaluation of the space-local nonlinear term

Here, we address the numerical implementation of the space-local nonlinear term. The space-local vorticity equation reads, without forcing and damping, 5.9) and its Fourier transform is

∂ω(x) ∂t = -∇ × ω × u L (x), (
∂ω(k) ∂t = -ik × ω × u L (k).
(5.10)

We can retrieve the Euler equations by uncurling (5.10) as,

∂u ∂t = - ik k 2 × ik × ω × u L , = -P ij ω × u L j ,
(5.11) (5.12) where

P ij = δ ij -k i k j /k 2 .
Using this formulation, the space-local nonlinear term in the Fourier domain can be computed using standard pseudo-spectral procedures.

Temporal evolution of space-local Navier-Stokes turbulence

We This transient regime appears as a peak in the time series. We observe a higher peak for flows with smaller values of R because the nonlinearity is more suppressed for small R.

After the transient stage, the flow reaches a state where P (t)/ (t) seems to fluctuate around a constant value which depends on R. It therefore differs from a statistically steady state of the original Navier-Stokes turbulence, as there is no statistical balance between the injection and the dissipation of energy:

P (t) t = (t) t .
Here, • t denotes the time average. In Fig. 5.1(b), we plot P (t) -NL (t) -(t), i.e. the RHS of the energy equation (5.5), to assess the temporal evolution of the space-local Navier-Stokes turbulence. For the original Navier-Stokes turbulence, we observe P (t) -NL (t) -(t) t = 0 with NL = 0 (5.8). When the value of R is finite, we still observe fluctuations around zero after the initial transient regime depicted in Fig. 5. 1(a). This observation indicates that the space-local Navier-Stokes turbulence eventually establishes the statistically steady state, as the three terms on the RHS of the energy equation ( 5.5) attain a statistical balance and thus dE(t) dt t = 0.

We note that for the smallest value of R/L f = 0.1, we observe a large fluctuation amplitude indicating a significant instantaneous imbalance between the three terms. However, since there are both positive and negative values, we speculate that there is a statistical balance in the long enough time series. Unfortunately, due to the energy accumulation at large scales, long enough computation to evaluate the statistical convergence is beyond our computational capacity. Figure 5.1(c) plots the time series of NL (t), illustrating that NL → 0 for R ∞ as discussed in (5.8) of § 5.1. Note that NL (t) does not always act as an alternative energy dissipation as it can be negative. Figure 5.2 shows the vortical structures in space-local Navier-Stokes turbulence at R = 0.2L f and R = 0.6L f . By comparing them with the original Navier-Stokes turbulence in Fig. 4.6(b), we can see that the space-local turbulence has much less fine-scale structures (in blue), even though smaller thresholds are employed for the visualization. It supports our finding that the nonlinear interactions and energy cascade are weakened due to the space-local restrictions in the nonlinear term. Next, we compare the two panels of Fig. 5.2 with the same isosurface thresholds. For R = 0.2L f in panel (a), there are distinctive large-scale structures (in red and yellow), while the small-scale structures (in blue) are barely visible. This indicates that more energy remains at these large-scale structures because less energy cascades towards scales smaller than R. For R = 0.6L f in panel (b), we do not observe the strong large-scale structures (in red), whereas the small-scale structures are more active. This is also consistent with the picture that less energy is retained at large scales as the energy cascade becomes more efficient for larger R because the space-local restrictions become less significant. More quantitative arguments are developed in terms of the energy spectrum shown in Fig. 5.4. First, panel (a) shows the flow at R = 0, which means that the nonlinear term is zero. Note that this result is obtained in 128 3 box. Thus, the flow exhibits no nonlinear interactions. The flow rapidly loses its energy in k = k f while the energy is accumulated at k = k f . The energy spectrum eventually becomes like the delta function.

Second, the panel (b) shows the result at R = 0.2L f . At first, energy accumulates at k = k f , and the small scales (k k f ) become less energetic. These changes are due to the sudden reduction of the energy cascade. Another interesting observation is an eventual accumulation of energy in scales larger than the forcing (k < k f ), suggesting a possible inverse energy cascade. However, since a further analysis of this property requires a sufficient scale separation between the system size and the forcing scale, we focus on the smaller-scale (k > k f ) scaling regime, which seems to consist of three ranges: a range with the energy spectrum with a power law steeper than k -5/3 , an E(k) ∝ k -5/3 scaling range associated with the Kolmogorov similarity, and a dissipation range. Third, the panel (c) shows the R = L f result. The flow exhibits almost equivalent statistics against the original turbulence, although we observe a slight energy accumulation in k < k f . We discuss the full statistical recovery and assimilation of the phase-space orbit as future perspectives in Chap. 5.5.

Space-local nonlinear interactions and robust Kolmogorov's 1941 scaling

In the remainder of this chapter, we focus on the small-scale (k > k f ) scaling, which seems to consist of three regions: a range with scaling steeper than k -5/3 , a k -5/3 scaling range associated with the Kolmogorov theory, and a dissipation range. See § 5.2.2 for more DNS details, including the time series and visualization. For the remainder of this chapter, we focus on the first two scaling regimes for k > k f . To investigate the R-dependence of these regimes, we plot the instantaneous energy spectrum at different values of R in Fig. 5.4. The flows are evaluated after the transient when the energy cascade adapts to the truncation of the nonlinear interactions. We use the same snapshots in the remainder of this chapter. In Fig. 5.4, the spectra are normalized by the high-pass filtered energy dissipation rate

> ≡ 2ν ˆ∞ 2π/R k 2 E(k) dk , (5.13)
for which we remove the contributions from the direct energy dissipation in wavenumber range k < 2π/R without energy cascade. Accordingly, the modified Kolmogorov length

η > ≡ ν 3 / > 1/4 (5.14)
is used to normalize the wavenumber.

In the small scales 2π/R k, the normalized energy spectra collapse onto the Kolmogorov spectrum. We also observe that E(k) ∝ k -5/3 law in 2π/R k k η > (= 2π/η > ) is robust. Here, 2π/R is the wavenumber corresponding to the space-local domain of radius R. This result indicates that the system with only spatially local nonlinear interactions (parametrized by R) can sustain energy cascade in scales smaller than R. We stress the nontriviality of this result; the space-local structures in physical space within a sphere of radius R are not equivalent to the small-scale structures in Fourier space for k ≥ 2π/R. Similarly, the space-local domain of radius R contains (partial) information of all the Fourier modes, not only k ≥ 2π/R. Thus, there is no one-to-one correspondence between the space-local and the small-scale structures. Overall, Fig. 5.4 confirms that the observations are consistent with the original space-local assumption in Kolmogorov (1941a).

Space-local nonlinear interactions and enstrophy-conserving scaling

In this section, we investigate the enstrophy scaling to understand the alternative scaling observed in the large scales k f k 2π/R in Fig. 5.4. In § 5.4.1, we first discuss the enstrophy scaling in the Navier-Stokes turbulence. Then, in § 5.4.2, we investigate the enstrophy scaling in the space-local Navier-Stokes turbulence.

Enstrophy balance and its scaling

Here, we investigate the enstrophy balance scaling of the original Navier-Stokes turbulence in Fourier space. We begin with the enstrophy balance equation 5.15) where k 2 E(k) denotes the enstrophy spectrum. There are two nonlinear terms, namely, the enstrophy production 5.16) and the enstrophy transfer (5.17) associated with the stretching and advection term in the vorticity equation (5.1), respectively. Here, • * and ´dΩ k denote the complex conjugate and the integral over spherical shells of radius k, respectively. The third term on the RHS denotes the enstrophy dissipation, and the fourth term is the enstrophy

∂ ∂t k 2 E(k) = T ω (k) + S ω (k) -2νk 4 E(k) + F ω (k), (
S ω (k) = ˆ ω j ∂ j u i (k)ω * i (k) dΩ k (
T ω (k) = ˆ- u j ∂ j ω i (k)ω * i (k) dΩ k ,
10 -2 10 -1 1 kη 10 -4 10 -3 10 -2 10 -1 1
Enstrophy budget

F ω (k) = ˆ ijk ∂ j f k (k)ω * i (k) dΩ k .
(5.18)

The large-scale integral of (5.15) defines the enstrophy budget equation (5.19) which is equivalent to (5.27). Figure 5.5 shows the three terms of (5.19): the cumulative enstrophy production (5.20) the enstrophy flux (5.21) and the cumulative enstrophy dissipation (5.22) In the inertial range k f k k η (= 2π/η), we observe a balanced scaling of (5.23) associated with the plateau in the energy flux Π(k) ∝ k 0 [START_REF] Davidson | On the generation and flux of enstrophy in isotropic turbulence[END_REF][START_REF] Sadhukhan | Enstrophy transfers in helical turbulence[END_REF].

∂ ∂t ˆk 0 p 2 E(p) dp + Π ω (k) = V < ω (k) -< ω (k) + ˆk 0 F ω (p) dp ,
V < ω (k) = ˆk 0 S ω (p) dp ,
Π ω (k) = - ˆk 0 T ω (p) dp ,
< ω (k) = 2ν ˆk 0 p 4 E(p) dp .
Π ω (k) = V < ω (k) ∝ k 2 ,
Given that T (k) ∝ -∂Π(k) ∂k , where T (k) is the energy transfer, V < ω (k) and Π ω (k) can be expressed as

-k 2 ∂Π ∂k = - ∂ ∂k k 2 Π(k) + 2kΠ(k) = T ω (k) + V < ω (k).
(5.24)

5.4 Space-local nonlinear interactions and enstrophy-conserving scaling The inertial range of the enstrophy budget equation (5.19), shown in Fig. 5.5, can be understood as the scale-by-scale balance between the cumulative enstrophy production and enstrophy flux (5.23). Although no conservative enstrophy cascade exists, enstrophy is transferred from larger to smaller scales. More precisely, the enstrophy transferred to scale k from a larger scale is further transferred towards a smaller scale, along with the enstrophy generated at that scale. In Fig. 5.6, we show the balance of V < ω (k)/Π ω (k) for both the original and space-local Navier-Stokes turbulence. For the latter, the enstrophy production and transfer are defined by

10 -2 10 -1 1 kη > 10 -2 10 -1 1 10 |V < ω (k)/Π ω (k)| R ↓
       S ω (k) = ˆ ω j ∂ j u L i (k)ω * i (k) dΩ k , T ω (k) = ˆ- u L j ∂ j ω i (k)ω * i (k) dΩ k ,
(5.25) (5.26) where u in (5.16) and ( 5.17) is substituted by u L .

Enstrophy-conserving scaling in scales larger than the space-local filter size

To understand the steeper scaling of the spectrum in k f k 2π/R, we investigate the enstrophy balance in space-local Navier-Stokes turbulence. We consider the large-scale enstrophy budget equation 5.27) of the Navier-Stokes equations. In three-dimensional isotropic turbulence, there is a balance between the cumulative enstrophy production V < ω (k) and the enstrophy flux Π ω (k) in the inertial range [START_REF] Davidson | On the generation and flux of enstrophy in isotropic turbulence[END_REF][START_REF] Sadhukhan | Enstrophy transfers in helical turbulence[END_REF]. These terms correspond to the vortex stretching and advection terms of the vorticity equation (5.1). For the definition, derivation, and scaling of (5.27), see § 5.4.1. Figure 5.6 shows the ratio V < ω (k)/Π ω (k) between the cumulative enstrophy production rate V < ω (k) and the enstrophy flux Π ω (k). These two terms are balanced in the inertial range of the unmodified Navier-Stokes turbulence (corresponding to the R ∞ limit), and V < ω (k)/Π ω (k) should thus be unity. See also (5.32). For both panels, the wavenumber is normalized by R so that different 2π/R collapses onto kR = 2π. Note that the spectrum of the original turbulence is not shown, since it corresponds to the R ∞ limit. The horizontal dashed line denotes a plateau. The same colourmap is employed as in Fig. 5.4. Fig. 5.5. In space-local Navier-Stokes turbulence, the ratio becomes considerably smaller than unity for k f k 2π/R as R 0, which is, as we will argue now, associated with the suppression of enstrophy production. We consider the global enstrophy production rate, 5.28) of space-local Navier-Stokes turbulence. Here, F -1 [•] and • denote the inverse Fourier transform and the spatial average, respectively. In the limit of kR 1, a Taylor expansion of the space-local filter function (4.19) yields

∂ ∂t ˆk 0 p 2 E(p) dp + Π ω (k) = V < ω (k) -< ω (k) + ˆk 0 F ω (p) dp (
V ω = ω i ∂u L i (x) ∂x j ω j = ω i ω j ∂ ∂x j F -1 u L i (k) , (
1 -B(kR) ≈ (kR) 2 10 + O (kR) 3 .
(5.29) Thus, in this limit, the total enstrophy production rate (5.28) scales as

V ω ≈ R 2 10 ω i ω j ∂ ∂x j F -1 k 2 u i (k)
for kR 1.

( 5.30) This relation states that the enstrophy production at a given scale k is weakened by decreasing the radius of the space-local domain R, and is consistent with Fig. 5.6. From these numerical and theoretical observations, we conjecture that space-local Navier-Stokes turbulence is asymptotically equivalent to turbulence without vortex stretching [START_REF] Bos | Analysis of the normalized dissipation rate in inhomogeneous and unsteady turbulence Bibliography[END_REF][START_REF] Wu | Cascades of enstrophy and helicity in turbulence without vortex stretching[END_REF] in the kR 1 limit. This system has intermediate properties between two-and three-dimensional Navier-Stokes turbulence, as enstrophy and helicity are conserved in the inviscid limit. Note that the nonlinearity of the two-dimensional Navier-Stokes equations conserves energy and enstrophy, while the three-dimensional system conserves energy and helicity. We numerically verify this speculation in Fig. 5.7. We first show the enstrophy flux Π ω (k) normalized by the enstrophy injection rate 5.31) in Fig. 5.7(a). The wavenumber is normalized by R so that the flux is horizontally shifted. Under the space-local constraint, we observe shallower scaling in the large scales k 2π/R, indicating a constant

P ω ≡ (∇ × f ) • ω (

Concluding remarks

enstrophy flux Π ω (k) ∝ k 0 in the asymptotic limit of R 0. Since there is no enstrophy production in this limit, the magnitude of Π ω (k) in this regime is of the order of the total enstrophy injection P ω by the forcing (5.31). In an enstrophy-conserving system, the energy spectrum exhibits an asymptotic scaling of E(k) ∝ k -3 , for example, see Bos (2021, Fig. 1). Figure 5.7(b) shows the compensated energy spectrum according to

E(k) ∼ P 2/3 ω k -3 , (5.32)
associated with the conservative enstrophy cascade picture. Although we do not observe a clear plateau even for the smallest value of the R = 0.1L f snapshot, the large-scale behaviour in R 0 limit is not contradicting the asymptotic enstrophy conservation discussed in Fig. 5.7(a). Plausibly, we may observe clearer k -3 scaling with large enough separation between k f and 2π/R, which would require much larger computational capacity. Furthermore, logarithmic corrections can also affect this scaling (Kraichnan 1971b; Wu and Bos 2022).

Concluding remarks

The scale locality of nonlinear interactions in three-dimensional turbulence has received considerable attention in the turbulence community, unlike locality in physical space. The present study aims to understand how spatially local and nonlocal nonlinear interactions contribute to the small-scale universality of turbulence. To this end, we considered the space-local velocity field (4.7) induced by the space-local vorticity field. Here, space-local is defined by the contributions contained in a spherical region of radius R around the considered point in space. We use this velocity to define a variant of the vorticity equation (5.1) in which the nonlinear term is determined in the space-local sense. We conducted DNS of this space-local flow in a 2π-periodic box driven by the steady forcing (4.21). The wavenumber range of the energy spectrum E(k) of space-local Navier-Stokes turbulence can be decomposed into the following regions: (i) a possible inversely energy cascading range in k k f , (ii) E(k) ∝ k -3 scaling range associated with a conservative enstrophy cascade in the asymptotic limit of R 0 in k f k 2π/R, (iii) E(k) ∝ k -5/3 scaling range in 2π k k η > , and (iv) a dissipation range. Here, k f , 2π/R, and k η > denote the characteristic wavenumbers of the forcing, physical-space locality, and dissipation, respectively. These regimes are schematically summarized in Fig. 5.8. When we focus on the inertial range k f k k η > [regions (ii) and (iii)], the space-local Navier-Stokes equations consolidate the robustness of Kolmogorov similarity with energy cascade in its small-scale part [regions (iii) and (iv)]. This finding suggests that the nonlinear interactions of three-dimensional turbulence are local in physical space as well as in scale space. The physical-space locality of the nonlinear interactions is consistent with Kolmogorov (1941a)'s hypothesis, where the spatially local domain was considered. The large-scale part [region (ii)] behaves asymptotically as turbulence without vortex stretching, which is explained by the suppressed enstrophy production and corresponds to a constant enstrophy flux. We note that this spectral shape with two (asymptotic) scaling ranges is similar to the Nastrom-Gage spectrum of atmospheric turbulence [START_REF] Nastrom | Kinetic energy spectrum of large-and mesoscale atmospheric processes[END_REF]. In that case, the enstrophy-conserving range with the k -3 scaling corresponds to close to two-dimensional turbulence while the k -5/3 scaling is recovered in the small scales. Extensive investigations with higher resolution and wider scaling range between k f and 2π/R are needed to confirm the E(k) ∝ k -3 scaling and its intersection with the E(k) ∝ k -5/3 scaling. Furthermore, the nature of a possible inverse cascade of the injected energy with small finite values of R, as in Fig. 5.4, is not investigated in the current study. Indeed, three-dimensional turbulence can exhibit inverse cascades if the nonlinear term is modified [START_REF] Biferale | Inverse energy cascade in three-dimensional isotropic turbulence[END_REF][START_REF] Frisch | Turbulence in noninteger dimensions by fractal Fourier decimation[END_REF][START_REF] Wu | Cascades of enstrophy and helicity in turbulence without vortex stretching[END_REF]). An alternative configuration with a much larger scale separation between the system size and the forcing would make it possible to investigate the behaviour of the system in the k k f regime. Two-dimensional space-local turbulence may exhibit qualitatively different properties compared to the three-dimensional case since the two-dimensional Navier-Stokes equations are governed by long-range (spatially nonlocal) interactions. Recent investigations show that space locality is important in the dynamics of the large-scale condensation in two-dimensional turbulence [START_REF] Svirsky | Two-dimensional turbulence with local interactions: statistics of the condensate[END_REF].

log k log E(k) ∝ k -3 ln(k/k f ) -1/3 ∝ k -5/3 k f k * ∼ R -1 k η > (i) (ii) (iii) (iv)
Inverse energy cascade or equipartition

Enstrophy cascade

Energy cascade We begin this chapter by reviewing the effect of large-scale modulation on the energy spectrum in § 6.1. In § 6.2, we discuss the Derivation of the nonequilibrium inhomogeneous scaling correction in a self-contained manner. In § 6.3, we examine the derived scaling correction from the numerical data. We conclude this chapter in § 6.4.

Nonequilibrium scaling in turbulence with large-scale modulation

As we reviewed in § 1.2, Kolmogorov postulated in 1941 that the small scales of turbulent flows away from boundaries can be considered universal if the Reynolds number is sufficiently large (Kolmogorov 1941a). The small scales are then supposed to be in equilibrium, and the energy spectrum satisfies,

E(k, x, t) ∼ (x, t) 2/3 k -5/3 , (6.1)
where is the average energy dissipation rate. For this expression to hold, the wavenumber k should be sufficiently large compared to L(x, t) -1 , the inverse of the length scale characterising the largest scales of the flow, and sufficiently small compared to the inverse of the Kolmogorov-scale η(x, t) -1 (with η = ν 3/4 -1/4 ), associated with the smallest scale of the flow.

In expression (6.1), the time and space dependence of E and need some particular attention. Theoretically, the most convenient flow type for investigating inertial range scaling is an infinitely large and statistically stationary flow without boundaries. Since all practical flows are limited in size and lifetime, the dissipation rate will be dependent, even on average, on either position x or time t, or both. Expression (6.1) will therefore hold only locally in subdomains of space and time-intervals large enough compared to the considered length and time scales. Indeed, the assumptions allowing the simple prediction (6.1) are that flow at the scales k can be considered locally isotropic, stationary, and homogeneous. This local equilibrium hypothesis is introduced in § 1.2.1.

The criterion k L -1 represents the implicit assumption that the influence of anisotropy, instationarity and inhomogeneity decreases as a function of scale. As an illustration, let us discuss the influence of statistical instationarity on the behaviour of the small scales. This subject was addressed by Yoshizawa (1994), who proposed that the influence of instationarity at large wavenumbers can be described as a perturbation on the energy spectrum as

E(k, x, t) = E 0 (k, x, t) + E 1 (k, x, t), (6.2) Figure 6.1: (a)
The zeroth E 0 (k), first E + 1 (k) , and second E + 2 (k) order contributions of the energy spectrum. The dotted lines denote the corresponding k -5/3 , k -7/3 , and k -9/3 scalings, respectively. The superscript + denotes the data associated with the positive sign of the time derivatives of the energy dissipation rate of each order. (b) The first order contributions of the energy spectrum E 1 (k). Both the positive (denoted by the superscript +) and negative (-) contributions are shown. The dotted lines denote the k -7/3 scaling. Adapted from Horiuti and Tamaki (2013, Figure 2).

where the equilibrium part E 0 is given by (6.1) and the perturbation scales as

E T 1 (k, x, t) = C Y d (x, t) dt (x, t) -2/3 k -7/3 , (6.3)
where the superscript T denotes that we consider perturbations due to instationarity. Numerical evidence of this scaling was first obtained by [START_REF] Horiuti | Multimode stretched spiral vortex and nonequilibrium energy spectrum in homogeneous shear flow turbulence[END_REF] for the case of homogeneous shear flow and by [START_REF] Horiuti | Nonequilibrium energy spectrum in the subgrid-scale oneequation model in large-eddy simulation[END_REF] for statistically isotropic turbulence in a periodic box. Figure 6.1 shows the equilibrium and nonequilibrium energy spectrum in the temporal fluctuations sense [START_REF] Horiuti | Nonequilibrium energy spectrum in the subgrid-scale oneequation model in large-eddy simulation[END_REF]. Further theoretical discussion and a more straightforward derivation of (6.3) can be found in [START_REF] Rubinstein | Self-similar turbulence evolution and the dissipation rate transport equation[END_REF], [START_REF] Woodruff | Multiple-scale perturbation analysis of slowly evolving turbulence[END_REF], and [START_REF] Bos | Dissipation in unsteady turbulence[END_REF].

The effect of large-scale temporal fluctuations on the kinetic energy spectrum is thus proportional to k -7/3 , which decays more rapidly than the equilibrium spectrum (6.1) with the k -5/3 scaling. For anisotropic turbulence generated by mean-shear S = ∂u x /∂z an expression for the shear-stress spectrum similar to (6.3) was predicted by [START_REF] Lumley | Similarity and the turbulent energy spectrum[END_REF],

E A 1 (k, x, t) = C L S (x, t) 1/3 k -7/3 . (6.4)
where the superscript A denotes that we consider perturbations due to anisotropy. This scaling was first confirmed in atmospheric measurements [START_REF] Wyngaard | Cospectral similarity in the atmospheric surface layer[END_REF], then in closure theory [START_REF] Leslie | Developments in the theory of turbulence[END_REF][START_REF] Bertoglio | A stochastic subgrid model for sheared turbulence[END_REF]. Later in wind-tunnel experiments [START_REF] Saddoughi | Local isotropy in turbulent boundary layers at high Reynolds number[END_REF] and more recently in Direct Numerical simulations and by a perturbative analysis [START_REF] Ishihara | Anisotropic velocity correlation spectrum at small scales in a homogeneous turbulent shear flow[END_REF]. This last work also considered other types of anisotropy in shear flow, which were also shown to decay proportionally to k -7/3 , as shown in Fig. 6.2. These measures concern thus anisotropy induced by velocity gradients on the level of second-order statistics.

In comparison to unsteady or anisotropic effects, the influence of inhomogeneity has received little attention. Historically, sophisticated models are proposed for inhomogeneous spectral dynamics, based on the Test Field Model (Kraichnan 1971a) or the Eddy-Damped Quasi-Normal Approximation [START_REF] Laporta | A model for inhomogeneous turbulence based on twopoint correlations[END_REF][START_REF] Parpais | A spectral closure for inhomogeneous turbulence applied to turbulent confined flow[END_REF]. Using Karhunen-Loeve eigenfunctions, it was illustrated that Kolmogorov's equilibrium spectrum can be observed in statistically inhomogeneous flows [START_REF] Knight | Kolmogorov inertial range for inhomogeneous turbulent flows[END_REF][START_REF] Moser | Kolmogorov inertial range spectra for inhomogeneous turbulence[END_REF][START_REF] Liao | Kolmogorov's hypotheses and global energy spectrum of turbulence[END_REF]. By using the SO(3) symmetry group decomposition, [START_REF] Kurien | Scaling structure of the velocity statistics in atmospheric boundary layers[END_REF] showed that structure functions contain a subdominant scaling component associated ii (k)/2; dotted line) energy spectra. The dashed anddash-dotted lines show -5/3 and -7/3 scalings, respectively. with inhomogeneity. In the remainder of this chapter, we will focus on this inhomogeneity-induced scaling correction.

Derivation of the nonequilibrium inhomogeneous scaling correction

The main difficulty in the present investigation comes from the fact that we investigate a multi-scale description (the energy spectrum) in an inhomogeneous setting. To simplify as far as possible, we restrict ourselves to a fairly simple setting, where the (statistical) inhomogeneity is periodic in space, and the flow is stationary and far away from boundaries. Before addressing the inhomogeneous multi-scale description, we will first consider the pointwise energy balance of the flow.

Kinetic-energy budget in inhomogeneous turbulence

We consider a statistically inhomogeneous flow kept in a statistically stationary state by a steady forcing f (z). The forcing in the present manuscript consists of a unidirectional steady body force in the xdirection with a sinusoidal dependence in the z-direction. The Navier-Stokes equations for this specific system write DU (x, t)

Dt = -∇P(x, t) + ν∆U (x, t) + f (z)e x , (6.5)
where D/Dt is the material derivative, P is the pressure (divided by density) ensuring incompressibility ∇ • U = 0, and e x denotes the unit vector in the x-direction.

The equations for the mean flow and the kinetic energy of the fluctuations can be derived by introducing the Reynolds decomposition U = U + u, where U ens is the ensemble-averaged velocity and u = (u, v, w) the fluctuation. The specific forcing considered in the present investigation leads to a mean flow U (x, t) = U (z)e x . Then, the kinetic energy corresponding to the mean flow can be written as

K U (z) = U (z) 2 /2, (6.6)
and the kinetic energy of the fluctuations is

K(z) = 1 2 u 2 (z) + w 2 (z) + w 2 (z) . (6.7)
Chapter 6 : Inhomogeneous Scaling Correction of Energy Spectrum

The equation for the mean-velocity U (z) reduces to,

DU (z) Dt = - ∂ ∂z uw (z) + f (z) + ν ∂ 2 U (z) ∂z 2 = 0. (6.8)
The details are, for instance, provided in [START_REF] Bos | Production and dissipation of kinetic energy in grid turbulence[END_REF]. The equation for the turbulent kinetic energy writes, in a steady state,

DK(z) Dt = p(z) -(z) + d(z) = 0, (6.9)
where the production p(z), dissipation (z), and diffusion d(z) terms are given by

p(z) = -uw (z) ∂U (z) ∂z , (z) = ν ∂u i ∂x j ∂u i ∂x j (z), d(z) = - ∂ ∂z Pw (z) + u i u i w (z) -ν ∂K(z) ∂z , (6.10)
(6.11) (6.12)

respectively. The first term p(z) represents the production of turbulent kinetic energy through the interaction of the turbulent fluctuations with the mean-velocity gradient ∂U (z) ∂z . The viscous dissipation term (z) involves the gradients of the fluctuating velocity.

In statistically homogeneous flows, production and dissipation are the only terms appearing in the turbulent kinetic energy balance. In statistically inhomogeneous flows, we also have spatial diffusion of turbulent kinetic energy d(z). The diffusion contains contributions associated with the turbulent fluctuations of the velocity and pressure (first two terms) and a contribution through viscous diffusion (the last term). This viscous part of the diffusion is generally negligible compared to the contribution of the other two terms and will be dropped in the following.

The main question in the present investigation is how such inhomogeneous redistribution processes d(z) affect the scaling of the kinetic energy spectrum E(k, x) in the inertial range of high Reynolds number turbulence.

Fourier-analysis of inhomogeneous turbulence

The use of energy spectra in general turbulent flows needs some justification. In principle, Fourier modes are associated with infinite or periodic domains. This property would exclude using spatial Fourier analysis of any realistic, non-periodic flow. However, a closer look at the lengthscales involved in turbulent flows permits invoking an assumption of scale separation, allowing us to get around this problem. Indeed, the theoretical basis for practical Fourier modelling of non-periodic turbulent flows can be found in various works (see [START_REF] Jeandel | Modeling methods in physical and spectral space[END_REF][START_REF] Yoshizawa | Statistical analysis of the deviation of the Reynolds stress from its eddy-viscosity representation[END_REF][START_REF] Bertoglio | A simplified spectral closure for inhomogeneous turbulence: application to the boundary layer[END_REF][START_REF] Laporta | A model for inhomogeneous turbulence based on twopoint correlations[END_REF][START_REF] Besnard | Spectral transport model for turbulence[END_REF]. In practice, to develop a spectral description of inhomogeneous flows, one needs to introduce a lengthscale L characterising the inhomogeneity of the flow geometry. Then, one can consider Fourier spectra associated with scales r ∼ k -1 small compared to L.

In the present investigation, we consider a spatially periodic flow without solid boundaries or obstacles to avoid most of these complications. Furthermore, we consider statistically stationary turbulence with a single inhomogeneous direction z to derive corrections due to statistical inhomogeneity. An advantage of the present configuration, where only one inhomogeneous direction is present, is that we can compute energy spectra in planes perpendicular to the z-axis. We thus define

E(k ⊥ , z) ≡ 1 2 ˆui (k ⊥ , z)u * i (k ⊥ , z) dA(k ⊥ ) , (6.13)
where A(k ⊥ ) denotes a wavenumber-shell of radius k ⊥ in the k x , k y plane. The velocity field in (6.13) is defined by the two-dimensional Fourier transform,

u i (k ⊥ , z) ≡ ˆe-i(kxx+kyy) u i (x, y, z) dx dy . (6.14)

Derivation of the nonequilibrium inhomogeneous scaling correction

The resulting energy spectrum E(k ⊥ , z) is a function of a perpendicular wavenumber k ⊥ = k 2 x + k 2 y and a vertical coordinate z. We note that if isotropy is restored in small scales, E(k ⊥ , z) is expected to scale like the three-dimensional spectrum E(k, z) (see (6.17) for the definition). In the following subsections in § 6.2, we will keep the notation E(k, z) for the sake of generality, but it should be kept in mind that the scaling of E(k, z) and E(k ⊥ , z) should be equivalent in statistically isotropic flow at large k.

Governing equation and modelling

The derivation in this subsection closely follows the rationale used to derive the instationary correction presented in [START_REF] Bos | Dissipation in unsteady turbulence[END_REF]. This same methodology is here applied to the evolution-equation of the energy spectrum in inhomogeneous turbulence.

The kinetic energy spectrum is associated with the turbulent kinetic energy by the relation

ˆE(k, z) dk = K(z). (6.15)
The evolution equation for E(k, z) is the multi-scale extension of equation (6.9). This equation reads, for the case of a unidirectional mean flow U (z)e x as in (6.5),

DE(k, z) Dt = P (k, z) production -2νk 2 E(k, z) dissipation + T (k, z) transfer + D(k, z) diffusion . (6.16)
Next, we define the spectrum and the governing equations for the energy spectrum tensor in inhomogeneous flow (see (6.16)). The generalised spectrum

E(k, x) = E ii (k, x) is defined by E(k, x) ≡ 1 2 ¨e-ik•r u i x + r 2 u i x - r 2 dr dΩ k , (6.17)
where ´dΩ k denotes the integral over spherical shells of radius k. The brackets in this subsection denote the ensemble average. For statistically homogeneous turbulence, this definition is equivalent to the expression

E(k) ≡ ˆ1 2 u i (k)u * i (k) dΩ k . (6.18)
The evolution equation for E ij (k, x, t) formally reads

∂E ij (k, x, t) ∂t = 1 2 ˆ ˆe-ik•r Ψ ij (x + r/2, x -r/2, t) dr dΩ k . (6.19)
For the tensor on the RHS, we have

Ψ ij (x 1 , x 2 ) =ν ∇ 2 1 + ∇ 2 2 R ij (x 1 , x 2 ) - ∂ ∂x 1n U n (x 1 )R ij (x 1 , x 2 ) + ∂ ∂x 2n U n (x 2 )R ij (x 1 , x 2 ) + ∂ ∂x 1n U i (x 1 )R nj (x 1 , x 2 ) + ∂ ∂x 2n U j (x 2 )R ni (x 1 , x 2 ) + ∂ ∂x 1i P(x 1 )u j (x 2 ) + ∂ ∂x 2j P(x 2 )u i (x 1 ) + ∂ ∂x 1n u i (x 1 )u n (x 1 )u j (x 2 ) + ∂ ∂x 2n u j (x 2 )u n (x 2 )u i (x 1 ) . (6.20)
In this expression and the following, the argument t for time is omitted for visibility. The two-point velocity tensor is defined by

R ij (x 1 , x 2 ) ≡ u i (x 1 )u j (x 2 ) , (6.21)
Chapter 6 : Inhomogeneous Scaling Correction of Energy Spectrum and the equation (6.20) is completed by incompressibility conditions for the mean field and the fluctuations. Both inhomogeneous turbulence diffusion and spectral transfer are associated with the last two lines of expression (6.20). An assumption of weak inhomogeneity must be invoked to dissociate them to obtain a closed expression. Even retaining only the leading order terms in an expansion about inhomogeneity, the resulting equations become quite cumbersome (see [START_REF] Laporta | Etude spectrale et modélisation de la turbulence inhomogène[END_REF] and [START_REF] Besnard | Spectral transport model for turbulence[END_REF]). Subsequently, the different terms in (6.20) need to be modelled to close the triple correlations. We will not proceed in this direction and will directly model them by their physical effects. See (6.22)-(6.25). Except for the viscous dissipation, all the terms in (6.16) are unclosed. In the following, we discuss the different physics and contributions to propose simple models for them. Since the flow is statistically stationary and the mean flow is unidirectional, the material derivative on the left-hand side of (6.16) is zero. The first term on the RHS, P (k, z), represents the terms directly proportional to the mean-velocity gradient. It contains two contributions: the production of turbulent kinetic energy and a linear transfer term [START_REF] Cambon | Spectral modelling of homogeneous nonisotropic turbulence[END_REF][START_REF] Briard | Advanced spectral anisotropic modelling for shear flows[END_REF]). These terms are mainly important at large scales and become zero at points in space where the velocity gradient vanishes. The order of magnitude of the production term can be estimated by [START_REF] Tennekes | A First Course in Turbulence[END_REF],

Π(k, z) Φ(k, z) log(k) z log[E(k, z)]
P (k, z) ∼ ∂U (z) ∂z 2 τ (k, z)E(k, z), (6.22)
with the time scale τ (k, z) ∼ (z) -1/3 k -2/3 in the inertial range. The integral of P (k, z) over wavenumbers yields p(z) in (6.9). Here, (z) denotes the profile of the dissipation of kinetic energy through viscous stresses (see (6.11)) and is obtained by the integral of the second term on the RHS of (6.16). At large Reynolds numbers, this term is significant only at large wavenumbers. It is thus this term which is responsible for energy transfer between the mean velocity field U (z) and the turbulent kinetic energy. The nonlinear transfer T (k, z) represents the energy flux and is a redistributive term in scale space; thus, its integral over all wavenumbers yields zero. The last term D(k, z) represents the diffusion, or transport, through turbulent fluctuations and viscous diffusion. Note that this term is zero in statistically homogeneous turbulence. The term D(k, z) is also a redistribution term like T (k, z), but in physical space. Its integral over wavenumbers corresponds to d(z) in (6.9). Both T (k, z) and D(k, z) are a function of triple correlations between Fourier modes at different wavelengths. There is no exact expression of these quantities as a closed function of the kinetic energy spectrum E(k, z). At this moment, we will therefore introduce modelling assumptions. Sophisticated models exist for inhomogeneous spectral dynamics, based on the Test Field Model (Kraichnan 1971a) or the Eddy-Damped Quasi-Normal Approximation [START_REF] Laporta | A model for inhomogeneous turbulence based on twopoint correlations[END_REF][START_REF] Parpais | A spectral closure for inhomogeneous turbulence applied to turbulent confined flow[END_REF]. However, the resulting closures are quite complicated and do not allow a straightforward analytical perturbation treatment. Therefore, our approach uses simple models that reproduce their main physical features: energy redistribution in scale space for T (k, z) and in physical space for D(k, z), respectively.

Derivation of the nonequilibrium inhomogeneous scaling correction

We use diffusion approximations for both terms,

T (k, z) = - ∂ ∂k Π(k, z), D(k, z) = - ∂ ∂z Φ(k, z), (6.23) (6.24)
where Π(k, z) and Φ(k, z) are turbulent fluxes in wavenumber and physical space, respectively. Figure 6.3 schematically depicts these two fluxes in (k, z) space. In the absence of inhomogeneity, the flux Φ(k, z) is zero. In the inhomogeneous case, the presence of this flux will affect the kinetic energy spectrum E(k, z).

We model both fluxes using a gradient-diffusion approximation,

Π(k, z) = -ρ(k, z) ∂ k -2 E(k, z) ∂k (6.25) with ρ(k, z) ∼ k 11/2 E(k, z) 1/2
being a turbulent energy diffusion in Fourier space, and

Φ(k, z) = -µ(z) ∂E(k, z) ∂z , (6.26)
where µ(z) is a turbulent diffusivity in real space (see (6.38)). We have effectively decoupled (and simplified) the transfer terms in scale and physical space. Indeed, both Φ(k, z) and Π(k, z) are determined by the same triple velocity and velocity-pressure correlations (see (6.20)). Decomposing the physical space-scale space flux is a major assumption which seems necessary to obtain an analytically tractable model of energy transfer in inhomogeneous turbulence. The model for Π(k, z) (6.25) is known as the Leith model [START_REF] Leith | Diffusion approximation to inertial energy transfer in isotropic turbulence[END_REF][START_REF] Rubinstein | Reassessment of the Classical Turbulence Closures[END_REF]. This model tends to homogenise the kinetic energy in spectral space towards equipartition among wave vectors, corresponding to an energy spectrum proportional to k 2 . The gradient-diffusion model for the diffusion (6.26) tends to homogenise the energy distribution in physical space and is used in [START_REF] Besnard | Spectral transport model for turbulence[END_REF], [START_REF] Touil | The decay of turbulence in a bounded domain[END_REF][START_REF] Cadiou | A two-scale second-moment turbulence closure based on weighted spectrum integration[END_REF], for instance. Eddy viscosity models are obviously simplified representations of the real transfer terms. For instance, see Pope (2000, § 10) for extensive discussions. However, we think that this kind of modelling is a useful first step before turning to more sophisticated modelling approaches.

Linear perturbation analysis and scaling predictions

Our goal is to derive a prediction for inertial range scaling at large Reynolds numbers in the limit of weak inhomogeneity, where the influence of inhomogeneity can be treated as a perturbation. In the following, the leading order contributions and perturbations are indicated by a subscript 0 and 1, respectively. We define an inertial range L -1 k η -1 with the length L representing the typical length of the largest and energy-containing scales of the flow. Furthermore, in our description, it is associated with the longest wavelength in our flow domain and is chosen constant. We will define this length scale more precisely later, in § 6.2.5. We now define the equilibrium about which we expand the equations. To do so, we consider the decomposition

E(k, z) = E 0 (k, z) + E 1 (k, z) (6.27) with |E 1 | |E 0 |.
The other quantities, such as Π(k, z) and Φ(k, z), are decomposed in the same manner.

We recall here that in addition to these two contributions to the energy spectrum, the flow also contains the time-averaged velocity profile, which consists of a single wave vector in the z-direction in the present case (6.6). This mean flow is not present in the inertial range, on which we will focus in the following. Therefore, in the remainder of this section, we can focus on the contributions E 0 (k, z) and E 1 (k, z).

For very high Reynolds numbers in the limit of vanishing inhomogeneity, we assume that the equilibrium contributions to the kinetic energy balance (6.16) do not depend on the inhomogeneous turbulent diffusion D(k, z). By integrating the balance between the transfer and dissipation terms in (6.16) from k to ∞, we find

ˆ∞ k T (p, z) dp = ˆ∞ k 2νp 2 E(p, z) dp (6.28)
Chapter 6 : Inhomogeneous Scaling Correction of Energy Spectrum or, using expression (6.23) and the equilibrium/nonequilibrium decomposition, Π 0 (k, z) = (z). (6.29) Indeed, this corresponds to the equilibrium between the energy flux and the energy dissipation rate, essential to the inertial range description of Kolmogorov (1941a). The constant flux solution of the Leith model is consistent with this framework and is given by (6.30) This expression defines our equilibrium solution. We now assess the influence of the inhomogeneity of (z) on this scaling as a perturbation.

E 0 (k, z) ∼ (z) 2/3 k -5/3 .
In the following, we consider the terms in the balance equation (6.16) for the nonequilibrium contributions: E 1 (k, z) and Π 1 (k, z). The order of magnitude of the production term (6.22) and the diffusion-gradient modelling with the flux (6.26) leads us to deduce that D(k, z) P (k, z) at k L -1 . Therefore, the firstorder perturbation to the equilibrium scaling in the inertial range is due to the inhomogeneous diffusion D(k, z). Then, in the inertial range, we have

T (k, z) = -D(k, z), (6.31)
and

- ∂ ∂k Π 1 (k, z) = ∂ ∂z Φ 0 (k, z), (6.32)
since ∂Π 0 (k, z) ∂k = 0. Thus, the first-order correction of the nonlinear transfer balances the zerothorder contribution of the inhomogeneous diffusion. The first-order perturbation to the nonlinear flux Π 1 (k, z) is evaluated as [START_REF] Rubinstein | Self-similar turbulence evolution and the dissipation rate transport equation[END_REF]) (6.33) where δΠ/δE E0 is the Fréchet derivative of the total flux Π evaluated at E(k, z) = E 0 (k, z). In the inertial range, assuming E 1 to scale as a power law, this yields the scaling,

Π 1 (k, z) = E 1 (k, z) δΠ δE E0 ,
Π 1 (k, z) ∼ (z) E 1 (k, z) E 0 (k, z) . (6.34)
Note that we obtain (6.34) not only for the Leith model, but also for most of the other classical closures such as the Kovaznay and Heisenberg model [START_REF] Rubinstein | Reassessment of the Classical Turbulence Closures[END_REF]. Integrating (6.32) from k to ∞, we have

Π 1 (k, z) = ∂ ∂z ˆ∞ k Φ 0 (k, z) dk . (6.35)
By combining this with (6.26) and (6.34), we obtain

E 1 (k, z) ∼ - E 0 (k, z) (z) ∂ ∂z µ(z) ∂ ´∞ k E 0 (p, z) dp ∂z .
(6.36) Substituting (6.30), the above expression gives 6.37) where the subscripts denote derivatives with respect to z, for example, zz = ∂ 2 (z)/∂z 2 . We will model

E 1 (k, z) ∼ -µ(z) (z) 1/3 k -7/3 2 3 zz (z) (z) + 2 3 µ z (z) µ(z) z (z) (z) - 2 9 z (z) (z) , (

Assessment of the inhomogeneous scaling correction

the unknown eddy diffusivity in its simplest way, µ(z) ∼ L 4/3 (z) 1/3 . (6.38) Doing so, we obtain (6.39) Note that although all the terms involving z and µ z vanish exactly for the current definition of µ(z) in (6.38), this might not be the case for arbitrary choices of µ(z).

E 1 (k, z) ∼ - zz (z)L 4/3 (z) 1/3 k -7/3 .

Case of a sinusoidal dissipation profile

The comparison of expressions (6.30) and (6.39) indicates that the inhomogeneous contribution (∝ k -7/3 ) is subdominant compared to the equilibrium energy spectrum (∝ k -5/3 ) at large wavenumbers. Furthermore, the expression is proportional to the second spatial derivative of the dissipation rate zz (z) and can thus be both positive and negative. Let us illustrate the implication of this expression by considering a large-scale inhomogeneity characterised by a cosine function with a characteristic wavelength of order L, (z) = + ˜ cos z/L , (6.40) with ˜ . We consider L, first introduced in § 6.2.4, to be of the order of and proportional to the characteristic large-scale length of the flow. Substituting this expression for (z) in (6.39), we find (6.42) where the superscript X indicates the perturbations due to inhomogeneity.

E 1 (k, z) = E X 1 (k) cos z/L (6.41) with E X 1 (k) = C A ˜ -1/3 L -2/3 k -7/3 ,
Let us now assume that both the equilibrium spectrum E 0 (k, z) and E 1 (k, z) extend from k = L -1 to ∞.

Integrating the spectra in this range, we find that K 0 (z) ∼ L 2/3 (z) 2/3 (6.43) and

K 1 (z) ∼ - zz (z)L 2 K 0 (z). (6.44)
Comparing these last two expressions illustrates that the formal expansion parameter in our system is

γ = zz (z)L 2 . (6.45)
The main analytical results of the present investigation [(6.42)-(6.44)] are obtained by phenomenological modelling based on gradient-diffusion assumptions of nonlinear transfer in both physical and scale space. The models and their consequences are, at best, crude approximations of the intricate nonlinear interactions in the actual flow. Therefore, The resulting expressions need verification by experiments or direct numerical simulations.

6.3 Assessment of the inhomogeneous scaling correction

Numerical setup

In order to verify the theoretical predictions, in particular expression (6.42), we carry out DNS of threedimensional Kolmogorov flow in a triple-periodic box. Such flow has the convenient properties of being statistically inhomogeneous in one direction and free of solid boundaries. Furthermore, its properties have been widely investigated numerically [START_REF] Borue | Numerical study of three-dimensional Kolmogorov flow at high Reynolds numbers[END_REF][START_REF] Musacchio | Turbulent channel without boundaries: The periodic Kolmogorov flow[END_REF][START_REF] Wu | A quadratic Reynolds stress development for the turbulent Kolmogorov flow[END_REF].

The dynamics of the Kolmogorov flow in the present investigation are governed by (6.5) with

f (z) = sin k f z . (6.46)
The numerical domain is a cube of size 2π. These choices imply that the forcing wavelength is equal to the width of the cubic domain, and we set k f = L -1 = 1. Simulations are carried out using a standard pseudo-spectral solver [START_REF] Delache | Scale by scale anisotropy in freely decaying rotating turbulence[END_REF]) with a third-order Adams-Bashfort time-integration scheme.

The details of the simulations are reported in table 6.1. Since we focus on the effect of inhomogeneity, we attempt to obtain statistics in a steady state over a long-enough time interval to allow the effects of the temporal variations to become as small as possible (see the last column in table 6.1).

Visualisation and dissipation profile

In the following, we will discuss the simulation at the highest considered Reynolds number Re λ = 184. A flow visualisation is shown in figure 6. 4(a) with the x-component of the velocity field U x (x, t). The influence of the large-scale mean flow, proportional to the sinusoidal forcing along the z axis, is distinguishable. Profile of kinetic energy

K(z) K 0 (z) K 1 (z) (b)
Figure 6.5: (a) Instantaneous profile of (z, t) = (x, t) ⊥ . Time-averaged profile (z) = (x, t) ⊥,t is also shown. The red dashed line denotes (z), a sinusoidal fitting of (z) by (6.40). (b) Time-averaged profile of kinetic energy with fluctuating velocity K(z) and its equilibrium K 0 (z) and nonequilibrium K 1 (z) contributions. See the main text for the definition.

the horizontal average of a snapshot, as shown in figure 6. 4(a). Its time average, U (z) = U x (x, t) ⊥,t , is also shown in figure 6.4(b) with a smooth sinusoidal profile. In figure 6.5(a), the instantaneous profile of the energy dissipation rate (z, t) = (x, t) ⊥ is shown along with its time average (z) = (x, t) ⊥,t . The instantaneous profile shows large fluctuations in comparison to the velocity profile (figure 6.4(b)). Its time average, in contrast, shows a smooth sinusoidal profile. This property allows us to use the approximations in § 6.2.5. As expected, the dissipation peaks at values where the mean velocity gradient is strongest (at z = 0 and ±π). For numerical convenience, we perform a sinusoidal fitting (z) introduced in (6.40). This profile is also shown in figure 6.5(a). Figure 6. 5 (b) shows the kinetic energy profile of the fluctuating velocity field. The fluctuating energy profile is defined by K(z, t) = K(z, t) -K U (z), where the total energy is K(z, t) = U i (z, t)U i (z, t)/2 and the mean flow energy is K U (z) = U (z) 2 /2. We consider the decomposition, see (6.43)-(6.44), K(z, t) = K 0 (z, t) + K 1 (z, t). (6.47)

In figure 6.5 (b), we observe that the equilibrium K 0 (z) = K 0 (z, t) t and the nonequilibrium K 1 (z) = K 1 (z, t) t profiles share the same phase, consistent with the prediction that the spectrum

E 1 (k, z) is proportional to -∂ 2 (z) ∂z 2 .

Equilibrium and nonequilibrium spectra

Here, we investigate the different normalisation procedures. We state that the energy spectrum is decomposed into equilibrium (labelled by the subscript 0) and nonequilibrium (labelled by the subscript 1) contributions as

E(k, z) = E 0 (k, z) + E 1 (k, z). (6.48)
We further assume that the nonequilibrium contributions are zero-mean,

E 1 (k, z) z = 0. (6.49)
We have therefore In order to compute the nonequilibrium contributions E 1 (k, z) = E(k, z) -E 0 (k, z), we need to know the z-dependence of E 0 (k, z). For this purpose, we use self-similarity assumptions and Kolmogorov's equilibrium hypothesis. Scaling ranges in turbulence spectra appear when scale separation is attained, i.e., at sufficiently high Reynolds numbers. In general, one can write the energy spectrum to scale as,

E(k, z) z = E 0 (k, z) z . ( 6
E 0 (k, z) ∼ (z) 2/3 k -5/3 f L [kL]f η [kη(z)],
(6.51) at high-Reynolds numbers. We have two nondimensional functions in (6.51); the f L determines the shape of the spectrum for small k (large-scale) and f η for large k (small-scale), respectively. These functions satisfy the framework of Kolmogorov (1941a),

lim x→0 f η [x] = lim x→∞ f L [x] = 1. (6.52)
Therefore, we retrieve (6.1) for scales L -1 k η -1 in the limit of L/η → ∞. Multiplying both sides of (6.51) by η -5/3 and dividing by 2/3 , we obtain

E 0 (k, z) (z) 1/4 ν 5/4 = F η [kη(z)]f L [kL], (6.53) with F η [kη(z)] = (kη) -5/3 f η [kη(z)]. (6.54)
Since F (kL) tends to unity for k 1/L, the equilibrium spectra E 0 (k, z) should collapse when normalised by (6.53) for any z, for large kL. Similarly, if the large scales are characterised by a length scale L, we can propose an alternative normalisation for (6.53), It should scale purely as a function of kL for kη 1. There are, therefore, two normalisation possibilities. One focuses on the high wavenumber limit of the inertial range close to the dissipation range (6.53)-(6.54), the other one on the low k range close to the energy-range (6.55)-(6.56). In the limit of infinite Reynolds number, we should find them to be equivalent (6.57) Figure 6.6 plots the absolute value of the nonequilibrium energy spectrum E 1 (k ⊥ , z) for these two normalisations. It is observed that using F [kL] we reveal a larger inertial range. We will therefore use this normalisation in the present investigation.

E 0 (k, z) (z) 2/3 L 5/3 ∼ F L [kL]f [kη], (
Since the kinetic energy is dominantly determined by large scales, (6.55) allows us to determine the equilibrium kinetic energy profile,

K 0 (z) = ˆE0 (k, z) dk = C L (z) 2/3 L 2/3 (6.58) with C L = ´FL [x] dx.
Then, we define the decomposition (6.59) where K 0 (z) z = K(z) z follows from the assumption that K 1 (z) z = 0. By employing the decomposition for the energy dissipation rate profile (6.60) it follows from (6.58) that

K 0 (z) = K(z) z + K 0 (z),
(z) 2/3 = (z) 2/3 z + (z) 2/3 ,
K 0 (z) K 0 (z) z = (z) 2/3 (z) 2/3 z , (6.61)
and by (6.59),

K 0 (z) =   1 + (z) 2/3 (z) 2/3 z   K(z) z . (6.62)
Since all the terms on the RHS are known, one can evaluate the nonequilibrium kinetic energy profile K 1 (z) = K(z) -K 0 (z) (see (6.47)).

Figure 6.7 shows the isotropic energy spectrum E(k, t) (see (6.18) for the definition) at three different Taylor-length Reynolds numbers. For simplicity, we denote its time-average by E(k) = E(k, t) t . Normalisation using ν and = (x, t) Ω,t allows an excellent collapse for large values of k.

Next, we assess energy spectra in statistically homogeneous planes perpendicular to the z axis, as defined in (6.13). In the following, we analyse the time-averaged inhomogeneous energy spectrum E(k ⊥ , z) = E(k ⊥ , z, t) t in a statistically steady state (see table 6.1). Figure 6.8(a) shows E(k ⊥ , z) nondimensionalised by (z) 2/3 L 2/3 . The fluctuations at small scales are small, and variations are barely visible.

In this study, we employ large-scale normalisation in (6.55) and evaluate

f L [k ⊥ L] ≡ E(k ⊥ , z) (z) -2/3 k 5/3 ⊥ z , (6.63)
as shown in figure 6.8(a). Note that this expression is valid for k ⊥ η 1 where f η (k ⊥ η) tends to a constant value. Then, the equilibrium spectrum can be defined as

E 0 (k ⊥ , z) ≡ (z) 2/3 f L [k ⊥ L]k -5/3 ⊥ .
(6.64)

Now, we can evaluate the nonequilibrium spectrum by E 1 (k ⊥ , z) ≡ E(k ⊥ , z) -E 0 (k ⊥ , z). Note that i) this quantity is defined by the time-averaged spectra and ii) since this quantity can be regarded as a perturbation of E(k ⊥ , z) around E 0 (k ⊥ , z), it can be both positive and negative.

E + 1 (k ⊥ ), E - 1 (k ⊥ ) (b)
Figure 6.8: (a) Nondimensionalised two-dimensional energy spectrum.

Note that E(k ⊥ , z) = E(k ⊥ , z, t) t . Dark (light) colour represents the small (large) value of z coordinate. The thick black line denotes (6.63), the average over z coordinate. (b) Time-averaged nonequilibrium energy spectrum with specific signs:

E + 1 (k ⊥ ) = E 1 (k ⊥ , z) > 0 z and E - 1 (k ⊥ ) = E 1 (k ⊥ , t) < 0 z . Red dashed lines denote the k -7/3 ⊥ slope.

Assessment of the inhomogeneous scaling correction

-π 0 π z -3 -2 -1 0 1 2 3 E 1 (k ⊥ , z)/σ(E 1 )
k ⊥ 0 50 100 Figure 6.9: z profile of E 1 (k ⊥ , z) at different k ⊥ normalised by their standard deviation σ(E 1 ). The darker (lighter) colour denotes smaller (larger k ⊥ ).

and figure 2 of [START_REF] Horiuti | Nonequilibrium energy spectrum in the subgrid-scale oneequation model in large-eddy simulation[END_REF]. The scaling is consistent with the one derived in § 6.2.4,

E 1 (k ⊥ , z) z ∝ k -7/3 ⊥ .
(6.65) Figure 6.9 shows the z profile of E 1 (k ⊥ , z) for 0 ≤ k ⊥ ≤ 100. This profile is in phase with (z) shown in Fig. 6.5(a) and thus consistent with the theoretical prediction (6.39). . At larger Re λ , the slope approaches the k -7/3 ⊥ scaling. At the same time, the spectrum in the higher k ⊥ range exhibits a bump associated with shallower scaling than k -7/3 ⊥ . We plot the compensated spectra in figure 6.10(b). Although the scaling range extends for less than a decade, the emergence of the k -7/3 ⊥ scaling range is well captured using this normalisation. In figure 6.6, we confirm that the bump in the compensated spectra is due to our choice of the nondimensional function (6.51).

Concluding remarks

The numerical simulations in the previous section support our prediction, (6.66) of the energy spectrum for turbulence with inhomogeneity in the z direction. This scaling quantifies the influence of spatial inhomogeneity in wavenumber space. In particular, the special case where the dissipation fluctuates as a sinusoidal function around a mean value, discussed in §6.2.5, gives us a useful estimate of the influence of inhomogeneity (6.41)-(6.42). Indeed, introducing an average dissipation and smooth spatial fluctuations ˜ around , so that zz /˜ ∼ L -2 , we obtain that (6.67) This expression shows that the influence of large-scale inhomogeneity is negligible for First, we define the normalised energy dissipation rate in § 7.1 and discuss its relation with the equilibrium energy spectrum. The derivation of the inhomogeneous or unsteady scaling is provided in § 7.2, followed by their numerical assessment in § 7.3. We further discuss the choice of the length scale and its influence in § 7.4. We then conduct the linearisation of the scaling in § 7.5 to evaluate the scaling exponent. In § 7.6, we remark the influence of the large-scale energy distributions on the dissipation rate scaling, before concluding the chapter in § 7.7.

E(k, z) = C K (z) 2/3 k -5/3 -C A zz (z)L 4/3 (z) 1/3 k -7/3 ,
E X 1 (k, z) E 0 (k, z) ∼ ˜ (kL) -2/3 .
k L -1 ˜ 3/2 . ( 6

Fluctuations of the normalised energy dissipation rate

We begin our discussion, again, with the energy spectrum

E 0 (k, x, t) = C K (x, t) 2/3 k -5/3 (7.1)
in the equilibrium inertial range of high Reynolds number turbulence. See § 1.2 for more details on the equilibrium description of turbulence. Its integral from L -1 to ∞ defines the equilibrium kinetic energy

K 0 (x, t) = 3 2 C K (x, t) 2/3 L(x, t) 2/3 , (7.2)
which is thus a function of and the characteristic length scale of the largest scales L(x, t). Note that L(x, t) can be a function of position and time, whereas k is not. For the influence of the large scales k < L -1 , refer to § 7.6. By introducing the typical velocity fluctuations U = 2K/3, relation (7.2) leads to

(x, t) = C -3/2 K U (x, t) 3 L(x, t) , (7.3)
which is Taylor's dissipation law [START_REF] Taylor | Statistical theory of turbulence[END_REF] 3). (b) The theoretical prediction along with numerical and experimental data. The numerical data is taken from panel (a). Adapted from Bos and Rubinstein (2017, Figure 1).

By comparing the expressions (7.3) and (7.4), the normalised dissipation rate should be constant and

C (x, t) = C -3/2 K , (7.5)
irrespective of the Reynolds number. Note that this relation should hold in the average sense since C is a function of space and time by definition (7.4). See § 1.2.3 and Fig. 1.6 for more details.

In the following, we review the scaling of fluctuations of C . In Fig. 7. Goto and Vassilicos 2015)

Re λ (x, t) ≡ 20 3 K(x, t) ν (x, t) . (7.9)
In particular, we consider the effect of statistical inhomogeneity and nonstationarity.

Derivation of the normalised energy dissipation rate scaling

In this section, we theoretically analyse the normalised energy dissipation rate scaling. We first focus on the corrections to the kinetic energy spectrum in § 7.2.1, then conduct linear perturbation analysis in § 7.2.2. See [START_REF] Fang | An EDQNM study of the dissipation rate in isotropic nonequilibrium turbulence[END_REF] for the same analysis in an EDQNM approach.

Corrections to the kinetic energy spectrum

Here, we introduce several quantities discussed in Chap. 6. We considered the equilibrium/nonequilibrium decomposition of the kinetic energy spectrum

E(k, x, t) = E 0 (k, x, t) + E 1 (k, x, t) = C K (x, t) 2/3 k -5/3 + A(x, t)k -7/3 , (7.10)
where both spatial and temporal corrections are associated with a steeper scaling ∼ k -7/3 , as discussed in § 6. Here, the coefficient of the perturbation spectrum reads

A T (x, t) = C Y ˙ (x, t) (x, t) 2/3 (7.11)
for temporal correction (6.3) and

A X (x, t) = -C A zz (z, t)L(z, t) 4/3 (z, t)
(x, t) ≡ ˆη-1 L -1 E(k, x, t) dk = K 0 (x, t) + K 1 (x, t). (7.13)
Here, in the limit of high-Reynolds number or L/η → ∞, the integral in (7.13) is determined by its lower bound k ∼ L -1 . Thus, the integral to determine the kinetic energy does not depend on the upper bound of the spectrum k ∼ η -1 associated with viscosity. For the influence of the large scales k < L -1 , refer to § 7.6. On the contrary, the integral to determine the kinetic energy dissipation rate depends on the viscosity. For example, when we compute the ratio of the nonequilibrium against the equilibrium energy dissipation rate, it yields

1 0 = ´η-1 L -1 2νk 2 E 1 (k, x, t) dk ´η-1 L -1 2νk 2 E 0 (k, x, t) dk = 2A η -2/3 -L -2/3 C K 2/3 η -4/3 -L -4/3 ∼ η 2/3
1.

(7.14) (7.15)

Note that we consider L/η → ∞ limit in the second line. Since η ∼ L Re -3/2 λ → 0 for a large Reynolds number limit, the perturbation component 1 does not significantly contribute to the total kinetic energy dissipation. In other words, we have ≈ 0 (7.16) for a large Reynolds number limit.

Perturbation analysis of the energy dissipation rate

Since the nonequilibrium energy dissipation rate does not significantly affect the total dissipation, we will consider the following decomposition (x, t) = (x, t) + ˜ (x, t) (7.17)

with |˜ | / 1. Here, (x, t) denotes the equilibrium component, and ˜ (x, t) is a small periodic perturbation in both temporal and spatial domains. To that, we assume a spatially homogeneous flow for temporal perturbations and a temporally steady flow for spatial perturbations, respectively. Thus, the bracket • in (7.17) denotes plane-temporal average for steady flow

⊥,t (z) = 1 + α sin(qz) (7.18)
and spatial average for homogeneous flow

Ω (t) = 1 + α sin(ωt) , (7.19)
respectively. Here, α denotes the amplitude of the small perturbations, and we employ a sinusoidal function to express the periodic perturbations. In the following, we employ the general notation (x, t) for both Ω (t) and ⊥,t (z). We will provide specific expressions for both cases and omit the bracket symbol where there is no ambiguity. Then, the equilibrium/nonequilibrium kinetic energy profile is expressed by

K(x, t) ≡ K 0 (x, t) + K 1 (x, t) = 3 2 C K (x, t) 2/3 L 2/3 + 3 4 A(x, t)L 4/3 = 3 2 C K (x, t) 2/3 L 2/3
C k f (x, t) ≡ (x, t)L 2K(x, t)/3 3/2 = C -3/2 K 1 + ∆(x, t) -3/2 .
(7.24) (7.25)

Here, we fix the characteristic length scale L to be the inverse of the forced wavenumber: L = k -1 f . To explicitly distinguish this choice, we employ the superscript • k f . See § 7.4 for an alternative choice.

Next, we consider the Reynolds number. To obtain a perturbative expression, we introduce a global Reynolds number

Re 0 ≡ 20 3 K ν . (7.26)
We can express the standard Taylor-scale Reynolds number (7.9) by Re 0 as

Re λ (x, t) = K(x, t) K (x, t) 1/2 Re 0 = 1 + ∆(x, t) (x, t) 1/6
Re 0 .

(7.27) (7.28) Now, we have expressions for both C k f (7.25) and Re λ (7.28).

For the spatial inhomogeneity scaling, assuming statistical stationarity, we obtain the following expressions:

C k f (z) = C -3/2 K 1 + ∆(z) -3/2 and Re λ (z) = 1 + ∆(z) (z) 1/6
Re 0 (7.29) with

∆(z) = (qX) 2 α sin(qz) 1 + α sin(qz) where X = L C A 2C K 1/2 . (7.30)
Similarly, for the temporal unsteady scaling, assuming statistical homogeneity, we obtain the following expressions:

C k f (t) = C -3/2 K 1 + ∆(t) -3/2 and Re λ (t) = 1 + ∆(z) (t) 1/6
Re 0 (7.31) with

∆(t) = ωT α cos(ωt) 1 + α sin(ωt) 4/3 where T = C Y 2C K L 2/3 1/3 . (7.32)
Note that the ratios (z)/ and (t)/ are evaluated from (7.18) and (7.19), respectively. Furthermore, to complete these expressions, we need to fix the free parameters α , qX, and ωT . Note that the angular velocity ω can be related to the characteristic time scale of temporal fluctuations, as discussed in § 3.1 with Fig. 3.1. However, due to the difference in the external forcing configurations 1 , it is not possible to perform direct comparison between the two flows.

Numerical assessment of the dissipation rate scaling

To assess the theoretical discussion in the previous section, we employ the same Kolmogorov flow dataset as in Chap. 6. The advantage of this setup is that due to the unidirectional forcing, f (z) = sin k f z with k f = 1, the dataset is statistically inhomogeneous in this direction. Furthermore, the flow exhibits large-scale quasi-cyclic temporal fluctuations where the time scale is comparable to the integral one, as we have investigated in detail for a different forcing setup in Chap. 3. Thus, we use the Kolmogorov flow dataset to analyse the spatial and temporal scalings simultaneously. For details of the numerical scheme and dataset, refer to § 6. In this section, we investigate the spatial and temporal fluctuations in § 7. 3.1 and § 7.3.2, respectively. Note that in this section, we choose the forcing length k f = 1 as the inverse of the characteristic length scale of the largest scales: k f = L -1 . We will discuss an alternative choice of the characteristic length scale in § 7.4.

Spatial fluctuations and their scaling

We have already provided the instantaneous, time-averaged, and trigonometrically fitted energy dissipation rate profile (6.40) in Fig. 6.5. Note that (z) is a shortened notation of (x, t) ⊥,t , where the subscript denotes the average over (x, y) plane and time. Re λ (z) for the three datasets by normalising both quantities by their average. The dataset information is summarised in Table 6.1. By overlapping the theoretical relation (7.29), we observe a nice collapse between the theory and the numerical results. Here, we set α = 0.13 from the relative amplitude of the fitted energy dissipation rate profile (z) in Fig. 6.5(a). The other free parameter qX is treated as a fitting parameter and set to be unity. Both quantities are divided by their time-averaged values.

Temporal fluctuations and their scaling

Choice of the length scale

In the previous sections, we used the characteristic length scale L = k -1 f associated with the forced wavenumber and the peak of the energy spectrum. In this section, we consider an alternative definition to see how the choice of different length scales affects the scaling. Here, we define the integral length scale (7.34) and K(x, t) defined by (7.13). It is important to note that the alternative choice of the characteristic length scale does not affect the integral range from L -1 to η. For the equilibrium/nonequilibrium decomposition, we find that

L(x, t) = 3π 4 I(x, t) K(x, t) (7.33) with I(x, t) = ˆη-1 L -1 k -1 E(k, x, t) dk ,
K 0 (x, t) = 3 2 C K (x, t) 2/3 L 2/3 , K 1 (x, t) = 3 4 A(x, t)L 4/3 , (7.35) and I 0 (x, t) = 3 5 C K (x, t) 2/3 L 5/3 , I 1 (x, t) = 3 7 A(x, t)L 7/3 , (7.36) so that I(x, t) = 2 5 L K 0 (x, t) + 10 7 K 1 (x, t) .
(7.37) Thus, we have with ∆(x, t) defined in (7.23). This relation illustrates that the equilibrium value of L(x, t), i.e. setting K 1 = 0, is proportional to L = k -1 f , but the nonequilibrium contributions vary as a function of x and t. By replacing L in (7.4), the normalised energy dissipation rate reads

L(x, t) = L 3π 4 1 + 10 7 K 1 (x, t)/K 0 (x, t) 1 + K 1 (x, t)/K 0 (x, t) = L 3π 4 1 + 10 7 ∆(x, t) 1 + ∆(x,
C L (x, t) ≡ (x, t)L(x, t) 2K(x, t)/3 3/2 = 3π 4 C -3/2 K 1 + 10 7 ∆(x, t) 1 + ∆(x, t) -5/2 . (7.40) (7.41)
To explicitly distinguish this choice, we employ the superscript • L in the following. Again, our alternative choice of the characteristic length scale does not affect the expression of K (7.22), which is defined by the integral from L -1 to η -1 . Thus, the normalised energy dissipation (7.25) becomes (7.42) where

C L (x, t) C L = 1 + 10 7 ∆(x, t) 1 + ∆(x, t) -5/2 ,
C L = 3πC -3/2 K
/4 is also modified from (7.5). We can consider C L (z)/ C L and C L (t)/ C L according to (7.29) and (7.31), respectively. Figure 7.6 shows the C L -Re λ scaling based on the integral length scale L. A good agreement between the theoretical relation and numerical data is observed for both panels.

Linearisation of the parametric plots between C and Re λ

In this section, we conduct linearisation of the dissipation scaling to discuss the scaling coefficient. In § 7.5.1, we consider the scaling derived in § 7.2.2 with the forcing length scale L = k -1 f . In § 7.5.2, we consider the scaling of § 7.4, associated with the integral length scale L. k f (z) (7.25) and Re λ (z) (7.28), we obtain For the inhomogeneous modulation case, we employ A X (x, t) (7.12) and (z) (7.18) to obtain

C k f (x, t) = C -3/2 K 1 + ∆(x, t) -3/2 = C -3/2 K 1 - 3 2 ∆(x, t) and Re λ (x, t) = 1 + ∆(x, t) (x, t) 1/6 Re 0 = 1 + ∆(x, t) 1 + ˜ (x, t) 1/6 Re 0 = 1 + ∆(x, t) 1 + 1 6 ˜ (x, t) Re 0 = 1 + ∆(x, t) + 1 6 ˜ (x,
C k f (z) C k f = 1 - 3 2 α (qX) 2 sin(qz) and Re λ (z) Re λ = 1 + α 1 + 1 6(qX) 2 (qX)
2 sin(qz).

(7.49) (7.50) See (7.30) for the definition of X. Thus, the parametric plot of C k f (z)-Re λ (z) exhibit a line with its slope close to -3/2 in Fig. 7. 3(b). However, the slope should be somewhat steeper than -3/2, since the variations around the equilibrium value exhibit

δC k f (z) 0 C k f ≡ δz C k f dC k f (z) dz z=0 = - 3 2 α q 3 X 2 , δ Re λ (z) 0 Re λ ≡ δz Re λ d Re λ (z) dz z=0 = α 1 + 1 6(qX) 2 q 3 X 2 , (7.51) (7.52) and thus δC k f (z) δ Re λ (z) 0 = - 3 2 Re λ C k f 1 1 + 1 6(qX) 2 ≈ - 3 2 1 - 1 6 (qX) 2 .
(7.53) (7.54)

For the temporal modulation case, we employ A T (x, t) (7.11) and (t) (7.19) to obtain By assuming a small amplitude of K 1 against K 0 or K 1 /K 0 1 in (7.39), we can linearise the integral length scale to find

C k f (t) C k f = 1 - 3
L(x, t) = L 3π 4 1 + 10 7 K 1 (x, t)/K 0 (x, t) 1 + K 1 (x, t)/K 0 (x, t) ∼ L 3π 4 1 + 3 7 ∆(x, t) .
(7.57) (7.58)

By employing the linearised L(x, t) and K(x, t) in the definition of the normalised energy dissipation rate (7.40), we obtain the alternative expressions:

C L (x, t) ≡ (x, t)L(x, t) 2K(x, t)/3 3/2 ∼ C L 1 - 3 2 ∆(x, t) 1 + 3 7 ∆(x, t) , C L (x, t) C L ∼ 1 - 15

Influence of the large-scale energy distribution

In this section, we discuss the influence of the large-scale energy distribution on the dissipation rate scaling. To this end, we consider the following decomposition of the energy spectrum

E(k, x, t) = C K (x, t) 2/3 k -5/3 F (kL) E0(k,x,t) + A(x, t)k -7/3 G(kL) E1(k,x,t) , (7.64)
which is a refined expression of (7.10). Here, two nondimensional functions

F (x) = x s+5/3 for x < 1 1 for x ≥ 1 (7.65) (7.66) and G(x) = x s +7/3 for x < 1 1 for x ≥ 1
(7.67) (7.68) determine the behaviour of the energy spectrum on large scales as

E 0 ∝ k s and E 1 ∝ k s for k < L -1 . (7.69)
Chapter 7 : Dissipation Rate Scaling in Inhomogeneous or Unsteady Turbulence

log k log E(k) ∝ k s ∝ k -5/3 ∝ k s ∝ k -7/3 1/L 1/η E 0 (k) E 1 (k) Figure 7.7: Schematic of equilibrium E 0 (k) and nonequilibrium E 1 (k) energy spectra with their large-scale distribution for k < L -1 . They exhibit E 0 (k) ∝ k -5/3 and E 1 (k) ∝ k -7/3 scalings for the inertial range L -1 k k η , respectively. Their large-scale destribution is denoted by E 0 (k) ∝ k s and E 1 (k) ∝ k s , respectively. Their integral defines K(x, t) = 3 2 C K (x, t) 2/3 L 2/3 ˆF (x)x -5/3 dx K0(x,t) + 3 4 A(x, t)L 4/3 ˆG(x)x -7/3 dx K1(x,t) = 3 2 C K (x, t) 2/3 L 2/3 ˆF (x)x -5/3 dx 1 + ∆(x, t) ´G(x)x -7/3 dx ´F (x)x -5/3 dx , (7.70) (7.71) and I(x, t) = 3 5 C K (x, t) 2/3 L 5/3 ˆF (x)x -8/3 dx I0(x,t) + 3 7 A(x, t)L 7/3 ˆG(x)x -10/3 dx I1(x,t) = 3 5 C K (x, t) 2/3 L 5/3 ˆF (x)x -8/3 dx 1 + 10 7 ∆(x, t) ´G(x)x -10/3 dx ´F (x)x -8/3 dx , (7.72) (7.73)
respectively. Note that we consider the case with the integral length scale ( § 7.5.2). By substituting these additional integrals into the normalised energy dissipation rate (7.41), one obtains

C L (x, t) ≡ (x, t)L(x, t) 2K(x, t)/3 3/2 = 3π 4 2 3 -3/2 (x, t)I(x, t)K(x, t) -5/2 ∝ 1 + ∆(x, t) α .
(7.74) (7.75) (7.76)

Concluding remarks

The scaling exponent reads When we assume a sharp cutoff of the energy spectrum for k < L -1 , it corresponds to s = s = ∞, and we obtain α(∞, ∞) = -15/14. Note that this exponent value is consistent with the linear perturbation analysis in § 7.5.2 and the current numerical configurations where the forcing scale and the box size are comparable. In fact, the above expression reduces to α(s, s ) = -15/14 when s = s is assumed. For example, s = s = 2 corresponding to the equipartition of the energy would be a reasonable candidate. However, we can generally consider s = s . For this case, the exponent α(s, s ) can vary from -15/14. Further analysis on this aspect remains in future research.

α(s, s ) = - 5 2 + ´G(x)x -10/3 dx ´F (x)x -5/3 dx ´G(x)x -7/3 dx ´F (x)x -8/3 dx = - 5 2 + ´1 0 x s -1 dx + ´∞ 1 x -10/3 dx ´1 0 x s dx + ´∞ 1 x -5/3 dx ´1 0 x s dx + ´∞ 1 x -7/3 dx ´1 0 x s-1 dx + ´∞ 1 x -8/3 dx = - 5

Concluding remarks

The scaling of the normalised energy dissipation rate C , and its deviations from a constant value, have received a large amount of attention. Our objective in this chapter was to propose a unified view on spatial and temporal scaling, where different behaviour has been reported. We begin from the equilibrium energy spectrum, observed in a wide range of turbulent flows, even when the large scales are in nonequilibrium.

We consider perturbations to the equilibrium spectrum and derive the nonequilibrium scaling of C for both inhomogeneous and unsteady flow configurations. These theoretical predictions are tested against a high Reynolds number Kolmogorov flow, where the large scales have both statistical inhomogeneity and unsteadiness. The comparison shows that the theoretical results are consistent with the numerical data. We test two different length scales characterising the macroscopic properties of the flow and discuss the difference in the resulting scaling exponent, as well as the influence of the large-scale energy distribution.

Conclusions and Perspectives

This chapter summarises the thesis and discusses perspectives for future research.

Intrinsic quasi-cyclic temporal nature of turbulence

The main topic of Part I of the thesis is quasi-cyclic temporal fluctuations (QCB: Quasi-Cyclic Behaviour) observed in turbulence. These dynamics are driven by the energy cascade and resulting imbalance between large-and small-scales. This phenomenon seems to be robust in a number of flow configurations. In particular, we focus on the energy cycle in the von Kármán flow ( § 2.1) and the minimal model of QCB in box turbulence maintained by the steady forcing (Chap. 3). In fact, the von Kármán flow can be considered to be a physical realisation of a steady force-driven box turbulence. Thus, it is tempting to relate the bulk of the von Kármán flow to the dynamics of box turbulence. We also investigate the bifurcation property of the model to discuss its similarity with the real laminarturbulent transition. Here, we observe a model trajectory as in Fig. 3.11(a), which looks like the Silnikov bifurcation shown in Fig. 2.6(b). Although this type of bifurcation can be observed in various fields, it has not been observed in laminar-turbulent transition. It would be interesting to assess the flow further to find such a bifurcation.

Localities of turbulence

The main topic of Part II is the physical-space locality of turbulence. Although it is common to discuss the locality in scale space for turbulence research, the physical space locality should not be overlooked; Kolmogorov developed his 1941 theory in "sufficiently small" four-dimensional domain in space and time (Kolmogorov 1941a). There are multiple approaches to investigating the physical-space locality of turbulence, and we employ the space-local velocity field ( § 4.2) in this thesis. It is defined by truncating the Biot-Savart law between the velocity and vorticity fields (Hamlington, Schumacher, et al. 2008a). We conduct both post-process (Chap. 4) and in-situ (Chap. 5) analysis, and the novelty is in the latter where we define the space-local Navier-Stokes equations.

The main findings of the spatially restricted system are two fold. First, we confirm that the nonlinear interactions of the Navier-Stokes turbulence are spatially local. In other words, when the nonlinearity is restricted by a parameter R (radius of the space-local domain), the energy spectrum in k 2π/R obeys E(k) ∝ k -5/3 of Kolmogorov's 1941 theory. This result validates his idea of the "sufficiently small" domain in physical space. Second, due to the suppressed enstrophy production in the large scales, the space-local Chapter 8 : Conclusions and Perspectives Navier-Stokes turbulence can be asymptotically equivalent to turbulence without vortex stretching [START_REF] Bos | Analysis of the normalized dissipation rate in inhomogeneous and unsteady turbulence Bibliography[END_REF][START_REF] Wu | Cascades of enstrophy and helicity in turbulence without vortex stretching[END_REF]. Thus, in k 2π/R, we asymptotically observe conserved enstrophy flux and the corresponding scaling. Since the space-local Navier-Stokes equations is an original approach, many questions must be explored. First, the comparison with different approaches on the physical-space locality should be further investigated. See § 4.2.3 for more details on the relation with the nonlocality of the pressure field. Second, the multi-scale nonlinear interactions under the space-local restriction can differ greatly from original Navier-Stokes turbulence. In particular, the energy cascade mechanisms reviewed in § 1.3 and their spatial local/nonlocal properties would be important to improve our understanding of the energy cascade. Third, it is interesting to look at the problem of chaos synchronisation from the space-local point of view. In past works [START_REF] Yoshida | Regeneration of small eddies by data assimilation in turbulence[END_REF][START_REF] Lalescu | Synchronization of chaos in fully developed turbulence[END_REF], synchronisation of two turbulent trajectories was investigated by varying the characteristic wavenumber below which the velocity field is assimilated. How do we observe synchronisation when translating this setup and assimilating the space-local velocity field? Last, the same space-local formulation of the two-dimensional Navier-Stokes equations, where the spatially nonlocal interactions dominate the flow, would be of interest to quantify the importance of its spacelocal nonlinear interactions. Furthermore, the large-scale condensation in two-and three-dimensional space-local Navier-Stokes turbulence should be compared and quantified.

Nonequilibrium scalings in turbulence

The main topic of Part III is nonequilibrium scaling of the energy spectrum and energy dissipation rate. Here, equilibrium and nonequilibrium of turbulence are defined by the instantaneous balance between the energy flux and energy dissipation rate. Indeed, the energy arrived at the smallest scales by the forward energy cascade is instantly dissipated. On the contrary, this balance is broken at large scales, and thus, turbulence is in a strong nonequilibrium state. In this thesis, we treat such large-scale nonequilibrium contributions by a perturbation to the equilibrium component of the flow. We analyse the effect of large-scale statistical inhomogeneity on the inertial range scaling to reveal the presence of corrections to the equilibrium spectrum, proportional to k -7/3 (Chap. 6). Since the correction is steeper than the equilibrium k -5/3 scaling, the nonequilibrium scaling decays faster than the equilibrium scaling towards the smaller scales. This result indeed confirms the equilibrium nature (local homogeneity) of the small scales of turbulence. For the perspectives, the universality of the nonequilibrium k -7/3 scaling should be further investigated. Here, the term universality refers to the origin of the large-scale nonequilibrium. In fact, it has been reported that the large-scale unsteadiness (Yoshizawa 1994) and anisotropy [START_REF] Ishihara | Anisotropic velocity correlation spectrum at small scales in a homogeneous turbulent shear flow[END_REF] results in the same k -7/3 scaling in their nonequilibrium component. We apply the same methodology to the normalised energy dissipation rate C and the Reynolds number Re λ (Chap. 7). Here, we extend our target to unsteady flow configurations along with the inhomogeneous flow, and our analysis reveals the same scaling for both cases. The crucial difference is that the inhomogeneous C (z)-Re λ (z) scaling consists a line-like profile, while the unsteady C (t)-Re λ (t) scaling forms an ellipse-like shape ( § 7.2). These theoretical results are tested against the high Reynolds number turbulence dataset ( § 7.3) as well as alternative formulation using the different macroscopic length scale ( § 7.4). These results can be extended to more complex flow geometries, for example, in the presence of the wall. The simple parametric expressions for the unsteady and inhomogeneous corrections to the normalised energy dissipation rate allow the improvement of turbulence models, where C is generally assumed to be constant. Here, the first term can be combined into the nonlinear term in the divergence form. The second term becomes (A.13) due to the identity between the symmetric and antisymmetric tensor

Appendix

-2ν ∂u i ∂x j S ij = -2ν S ij + Ω ij S ij = -2νS ij S ij ,
S ij Ω ij = 0. (A.14)
The rightmost-hand side of (A.13) is equivalent to the proper definition of the energy dissipation rate (A.2).

B Betchov's relation

In this section, we derive the Betchov's relation [START_REF] Betchov | An inequality concerning the production of vorticity in isotropic turbulence[END_REF]; Davidson (2015, § 5.3.6)):

ω i S ij ω j = - 4 3 S ij S jk S ki . (B.1)
We start with the velocity gradient tensor and its symmetric/anti-symmetric decomposition,

A ij = ∂u i ∂x j = S ij + Ω ij =      a 0 0 0 b 0 0 0 c      + 1 2     0 -ω c ω b ω c 0 -ω a -ω b ω a 0     , (B.2)
which is obtained by aligning the coordinate to the principal axes of the strain-rate tensor S ij and by where P, Q, and R are three invariants of A with Betchov derived the divergence expressions

P = λ 1 + λ 2 + λ 3 = A ii = 0, Q = λ 1 λ 2 + λ 2 λ 3 + λ 3 λ 1 = 1 2 P 2 - 1 2 A ji A ij = - 1 2 A ji A ij , R = λ 1 λ 2 λ 3 = det A ij = 1 3 A ij A jk A ki .
Q = - 1 2 ∂ ∂x j u i ∂u j ∂x i , R = 1 3 ∂ ∂x i ∂u i ∂x j ∂u j ∂x k u k - 1 2 u i ∂u k ∂x j ∂u j ∂x k . (B.13) (B.14)
One can assume the divergence free in homogeneous turbulence: Q = 0 and R = 0. Thus, (B.9, B.12) becomes, in the average sense, (1985), and [START_REF] Kawahara | Wrap, tilt and stretch of vorticity lines around a strong thin straight vortex tube in a simple shear flow[END_REF]. Adapted from Horiuti and Fujisawa (2008, Figure 1). See Hino (2020, § 4.11.1) for a review (in Japanese). the scale-local interactions between them. The novelty of this study is that they reformulate the universal scaling without directly using the similarity hypotheses.

ω 2 = 2 S ij S ij , ω i S ij ω j = - 4
K ′ /K K ′ /K K ′ /K K ′ /K T (K, K ′ , t) T (K, K ′ , t) T (K, K ′ , t) T (K, K ′ , t)
To quantify the scale-space property of energy transfer associated with vortex stretching, they defined

T (K, K ) ≡ 1 K 2 ˆωi (K) ∂u i (K ) ∂x j ω j (K) dΩ . (D.1)
Here, the argument K and K denote the bandpass-filter defined by

u i (K) ≡ F -1 χ [K/ √ 2, √ 2K) F[u i ] ,
and

ω i (K) ≡ F -1 χ [K/ √ 2, √ 2K) F[ω i ] , (D.2)
where F and F -1 denote forward and inverse Fourier transformation, respectively. The bandpass-filter

in k ∈ K/ √ 2, √ 2K is denoted by χ [K/ √ 2, √ 2K
) . Since ´ωi (K) ∂u i (K ) ∂x j ω j (K) dΩ represents the enstrophy production rate at scale K due to velocity gradient at K , (D.1) with the coefficient K -2 denotes the energy transfer due to vortex stretching from scale K to K. The collapsed curves in this panel indicate that the energy transfer profile is self-similar irrespective of K (receives energy). This observation is consistent with the inertial range statistics of the K41 theory. Furthermore, the positive peak of the spectrum at K /K ≈ 0.5 (where the red dashed line denotes K /K = 0.58) states that the most significant energy transfer is from as Appendix twice as large scales. At the same time, the energy transfer to the smaller scale is most significant at the negative peak around K /K ≈ 2 (where the blue dashed line denotes K /K = 1.7), corresponds to scales as half as small as the original scale K. These properties also support the scale-local nature of the energy cascade. [START_REF] Carbone | Is vortex stretching the main cause of the turbulent energy cascade?[END_REF] They investigated the Kármán-Howarth equation

∂ t K = -∂ r • T + 2ν∂ 2 r K -2 + W, (D.3)
which governs the ensemble-averaged turbulent kinetic energy K(x, r, t) ≡ ∆u(x, r, t) 2 /2 (D.4)

of the velocity increment ∆u(x, r, t) ≡ u(x + r/2, t)u(xr/2, t). Here, the nonlinear energy flux term is defined by

∂ r • T ≡ 1 2 ∂ r • ∆u 2 ∆u . (D.5)
The other terms of (D. The terms with subscript "1" involve quantities at a single scale: the filter length scale . The above Π s1 (D.16) and Π ω1 (D.17) are strain self-amplification and vortex stretching due to strain at . Note that these terms also appear in the restricted Euler equation for the magnitude of the filtered velocity gradient tensor D Dt Thus, the author discussed that:

The same processes responsible for increasing the filtered velocity gradient magnitude also redistribute energy to sub-filter scales, thus connecting the restricted Euler singularity with the energy cascade.

The subscript "2" in the third and fourth term on the RHS of (D.15) represents the nonlinear interactions between the strain rate filtered at and velocity gradient at scales smaller than . Similar to those with subscript "1", Π s2 (D.18) denotes the strain self-amplification, but this time the amplification of the small-scale strain by the larger-scale strain. The other term, Π ω2 (D.19), represents the stretching of the small-scale vorticity by the large-scale strain. The final term Π c denotes the interactions between the larger-scale strain and smaller-scale strain-vorticity covariance. Figure D.2(b) shows the distribution of (D.16-D.20) against the filter length scale size. In the inertial range, we observe each contribution takes a plateau, supporting the statistically self-similar nature of turbulence. The most important indication of the figure is that Π s1 takes the largest value among the five contributions, roughly one-third of the total energy flux. This claims that the single-scale strain self-amplification is the most significant in the energy cascade. Then, almost indistinguishable multiscale strain self-amplification and vortex stretching follow. The single-scale vortex stretching has weaker contributions than these three terms. The covariance term does not play a role in the inertial range. [START_REF] Mckeown | Energy transfer and vortex structures: Visualizing the incompressible turbulent energy cascade[END_REF] They defined the Fourier band

I P = k x , k y , k z , such that 2 P -1 k f / √ 2 ≤ |k| ≤ 2 P -1 k f × √ 2 , (D.22)
where k f is the forcing wavenumber, to evaluate the physical-space velocity corresponds to the band P : Here, a larger value of P corresponds to a larger Fourier shell or small scale. Now, one can define the energy transfer from shell (Fourier band) K to shell Q due to the interaction with the shell P : Note that this quantity also depends on x, allowing one to investigate the distribution of the energy transfer rate in physical space. By integrating T 3 in space, scale-to-scale energy transfer (with a specific mediator scale) can be evaluated in the global sense: T 3 (K, P, Q) = ˆT3 (K, P, Q, x) dx .

T 3 (K, P, Q, x) = -u K • (u P • ∇)u Q . (D.24)
(D.25)

From the space-dependent T 3 (K, P, Q, x) (D.24) The "mediator" shell P can be averaged out to define (D.26) corresponds to the energy transfer from shell K to Q with the aid of the whole velocity field. By further integrating over the "receiver" shell Q, one obtains the energy transferred from shell K:

T 2 (K, Q, x) = P T 3 (K, P, Q, x) = -u K • (u • ∇)u Q ,
T 1 (K, x) = Q T 2 (K, Q, x) = -u K • (u • ∇)u. (D.27)
Note that the space-averaged T 2 (K, Q) and T 1 (K) can be defined accordingly. The energy flux through the shell k can then be obtained by

Π(k) = - k K=0
T 1 (K).

(D.28) .3(a) shows the two-dimensional energy transfer spectrum T 2 (K, Q) and its scaling in terms of the receiving wavenumber scale Q. First, we notice that T 2 (K, Q) > 0 for K < Q and T 2 (K, Q) < 0 for K > Q. Since the shell K is the "giver" and Q is the "receiver", they correspond to forward energy transfer, positive energy transfer from larger to smaller scales and negative energy transfer from smaller to larger scales2 . In the following, we focus on the upper triangle of Fig. D.3(a) or Q > K. For the magnitude of transfer, the local interactions T 2 (K, K + 1) tend to take large values and the decay of T 2 (K, Q) for fixed K and increased Q is fast. These properties are consistent with the scale-local nature of the energy

Acronyms

  DNS Direct Numerical Simulation (defined on page 7) DoF Degrees of Freedom (defined on page 32) HIT Homogeneous and Isotropic Turbulence (defined on page 13) K41 Kolmogorov 1941 (theory) (defined on page 19) LBM Lattice Boltzmann Method (defined on page 58) LEH Local Equilibrium Hypothesis (defined on page 12) PDF Probability Distribution Function (defined on page 65) POD Proper Orthogonal Decomposition (defined on page 34) QCB Quasi-Cyclic Behaviour (defined on page 28) RHS Right Hand Side (defined on page 7) SPO Stable Periodic Orbit (defined on page 28) SSA Strain Self-Amplification (defined on page 124) UPO Unstable Periodic Orbit (defined on page 22) VS Vortex Stretching (defined on page 124) Greek characters δ ij Kronecker's delta function (defined on page 10) (k) Cumulative energy dissipation spectrum at wavenumber k (defined on page 13) Energy dissipation rate (defined on page 7) ijk Levi-Civita symbol (permutation symbol) (defined on page 20) η Kolmogorov length scale (defined on page 14) λ Taylor micro length scale (defined on page 65) ν Kinematic viscosity (defined on page 6) Ω Domain of the flow (defined on page 6) Ω k Spherical shell of radius k (defined on page 10) Π(k) Energy flux spectrum at wavenumber k (defined on page 12) ω Vorticity field = (ω x , ω y , ω z ) (defined on page 7) Ω Rate-of-rotation tensor (defined on page 8

pX

  C K Kolmogorov constant (defined on page 14) E(k) Energy spectrum at wavenumber k (defined on page 10) F (k) Energy injection spectrum at wavenumber k (defined on page 10) Pressure field (defined on page 6) P ij (k) Orthogonal projection tensor for vector k (defined on page 10) R Radius of the space-local domain (defined on page 62) T (k) Energy transfer spectrum at wavenumber k (defined on page 10) Characteristic velocity of the forced modes (defined on page 43) Y Characteristic velocity of the energetic modes (defined on page 43) Z Characteristic velocity of the residual modes (defined on page 47)

Figure 1 :

 1 Figure 1: Overview of the construction of the thesis. For the annotated references, see "List of Publications" on p. v.

Figure 1 . 1 :

 11 Figure 1.1: (a) A sketch of pouring water into a pool with resulting turbulent eddies by Leonardo da Vinci. Adapted from Royal Collection at Windsor, RCIN 912660v. (b) The Starry Night by Vincent van Gogh. Adapted from The Museum of Modern Art, 472.1941.

Figure 1 . 2 :

 12 Figure 1.2: Visualisation of enstrophy (cyan) and energy dissipation (red) in turbulence obtained by the DNS of 8192 3 resolutions with the Taylor-scale Reynolds number of R λ = 650. The centre of the domain visualised in panel (a) is successively zoomed in for panels (b) and (c). The sub-labels denote the approximate domain sizes in the Kolmogorov length scale η. In each panel, the contour threshold is denoted by C at the left-bottom corner. Adapted from Buaria, Pumir, Bodenschatz, and Yeung (2019, Figure 1).

Figure 1 . 3 :

 13 Figure 1.3: (a) Schematic of the mode-to-mode energy transfer from giver p to receiver k with the aid of mediator q. The figure is drawn after Verma (2019, Figure 4.3). (b) The surface of function P (k|p) interpreted as the energy transfer between wavenumber bands k and p. Adapted from Domaradzki and Rogallo (1990, Figure 6).

Figure 1 . 4 :

 14 Figure 1.4: Schematic of the local equilibrium hypothesis. The horizontal and vertical axis denotes the wavenumber k in log form and the energy budget, respectively. Energy flux spectrum Π(k) and cumulative energy dissipation spectrum (k) are shown along with a schematic picture of vortices of different scales. The vertical lines with a horizontal arrow represent Taylor's dissipation law and the local equilibrium hypothesis, respectively.

1. 2 Figure 1 . 5 :

 215 Figure 1.5: Energy spectrum in various types of turbulence normalised by the small-scale representatives: averaged energy dissipation rate and kinematic viscosity of the fluid ν. Values in the legend denote the Taylor-scale based Reynolds number R λ = u λ/ν, where u is the standard deviation of a single velocity component and λ is the Taylor length scale. Adapted from Goto (2018, Figure 1).

Figure 1 . 6 :

 16 Figure 1.6: Normalised energy dissipation rate C as a function of the Taylor-scale Reynolds number Re λ . Adapted from Sreenivasan (1984, Figure 1).

Figure 1 . 7 :

 17 Figure 1.7: Time series of the energy dissipation rate > (k, t) (thick grey line) high-pass filtered at given k and the energy loss rate Π(k, t) (thin black line) at scale k for five different values of k, obtained in turbulence with averaged Taylor-scale Reynolds number of R λ = 490. The value of k increases from panel (a) to (e). Adapted from Goto and Vassilicos (2016a, Figure 5).

Figure 1 .

 1 Figure 1.8: A schematic of the energy spectrum of two-dimensional Navier-Stokes turbulence. Dotted lines in the largest scales represent energy accumulation. Dashed arrows denote k -5/3 scaling of the inverse energy cascade and k -3 (plus logarithmic correction) scaling of the forward enstrophy cascade, respectively. Vertical dash-dotted lines denote the forcing wavenumber k f and the Kolmogorov wavenumber associated with enstrophy dissipation k ηω , respectively.

Chapter 1 :

 1 Figure 1.11: (a) Visualisation of coarse-grained vortex structures with the larger (red) and the smaller (yellow) scales. (b) Same visualisation as panel (a) with a different angle. (c) Cropped domain in panel (b) (blue frame). Arrows denote the large-scale velocity field for panels (b) and (c). Adapted from Goto, Saito, et al. (2017, Figure 3).

Figure 1 .

 1 Figure 1.13: Temporal evolution of the one-dimensional inviscid Burgers equation. Simulation is conducted by an open-source project CFD Julia (Pawar and San 2019). Time evolves from dark to light colours.

Figure 2 . 1 :

 21 Figure 2.1: (a) Two-time correlation function of (a-a) self correlation of the turbulent velocity u (t) = 2/3K(t) defined by the kinetic energy. (a-b) the turbulent velocity u (t) and the energy dissipation rate (t). (a-c) The normalised energy dissipation rate C (t) and the Taylor-scale Reynolds number R λ = u λ/ν. Adapted from Goto and Vassilicos (2016a, Figure 3 (b)). (b) Time series of the energy input rate Π(t) against the energy dissipation rate (t) in turbulence driven by a steady body force (3.2). Different Reynolds numbers are overlapped; the value increases from the thin red curve, then the black curve and the dotted blue line corresponds to the highest Re. Time evolves counter-clockwise. Adapted from Goto, Saito, et al. (2017, Figure 15 (b)).

2. 1

 1 Temporal (quasi-)cyclic modulation associated with the energy cascade

Figure 2 . 2 :Figure 2 . 3 :

 2223 Figure 2.2: The projection of an Unstable Periodic Orbit (UPO) onto the 2D portrait of the normalised energy input rate I and the energy dissipation rate (t). A periodic orbit is denoted by the red closed line.The turbulence time series is also shown in the grey line, where the green dots denote the constant time interval. The yellow line is a cut of the turbulent trajectory and shows a typical behaviour of turbulence approaching the periodic orbit. Adapted from Kawahara and Kida (2001, Figure1).

Figure 2 . 4 :

 24 Figure 2.4: Quasi-cyclic temporal evolution in von Kármán flow driven by the viscous forcing at Re = 1500. (a) The energy input rate P vis (t) and energy dissipation rate (t). Adapted from Araki and Goto (2021, Figure 2 (b)). (b) The toroidal energy E tor (t) and the The poloidal energy E pol (t) for different Reynolds numbers. Adapted from Araki and Goto (2021, Figure7). For both panels, the colour gradient from dark to light represents the temporal evolution, and two consecutive dots denote the time interval of T .

Figure 2

 2 Figure 2.6: Attractors of minimal models of chaos: (a) Lorenz model and (b) Silnikov orbit adapted from Silnikov and Silnikov (2007). (a) Two orbits of slightly different initial conditions are shown. (b)Γ is a Homoclinic loop. W s and W u denote the stable and unstable manifolds, respectively, and their cross-section is Π 1 and Π 2 , respectively.

Figure 2 . 7 :

 27 Figure 2.7: The time-averaged energy spectrum of the GOY shell model. The solid line denotes the k -5/3scaling. Adapted from Ohkitani and Yamada (1989, Figure1).

Figure 2

 2 Figure 2.8: Space-time plot of the laminar-turbulent transition observed in the predator-prey model. The blue and yellow regions denote the turbulent and laminar regions, respectively. The variable U is equivalent to the Reynolds number, and U c denotes the critical value. Adapted from Wang, Shih, et al. (2022, Figure 4).

Figure 3 .

 3 1(a) shows the turbulent time series, where the time-averaged Taylor scale-based Reynolds number Re λ t is about 90. Note that • t denotes the time average. We also show the snapshot of this flow in Fig.3.2(a). The time series exhibits QCB in a counter-clockwise direction behind the chaotic fluctuations.

Figure 3 .

 3 2(b) shows the temporal evolution of the energy input rate P (t) and the energy dissipation rate (t). Both time signals exhibit significant fluctuations with a clear time-delayed correlation. See Fig.3.1(a) for the two-dimensional projection of the same time series.

  Figure 3.1(b) is a parametric plot of Fig. 3.3(c).

Figure 3 . 1 :

 31 Figure 3.1: (a) Parametric plots of the instantaneous values of the energy dissipation rate (t) and the energy input rate P (t) for the turbulent flow at Re = 29.7 for 50T [see Fig. 3.2(b)]. (b) Phase-averaged values P phase and phase at Re = 29.7 [same as in (a)] for 20T . (c) Parametric plots of the phase averaged values P phase and phase at four different values of Re (29.7, 11.9, 8.49, and 6.61). Note that the orbit in panel (b) at Re = 29.7 is re-plotted in panel (c). (d) The SPO at Re = 5.83 is shown with a coloured line. Note that this orbit is also shown in panel (c). The gap between two consecutive dots corresponds to 5T for all panels. In panels (a), (b), and (d), the time evolves from dark to light colours. Four black cross symbols in panel (d) denote the instances shown in Fig. 3.4.

Chapter 3 :Figure 3

 33 Figure 3.2: (a) A snapshot of the turbulent flow at Re = 29.7 (Re λ ≈ 90). Isosurfaces of |ω| = 20 (blue) and low-pass filtered ω < = 4 (red) are visualised. (b) Time series of energy input rate P (t) and energy dissipation rate (t) of the same flow. The red rectangle denotes the time interval examined in Fig. 3.1(a).

Figure 3 . 4 :

 34 Figure 3.4: Visualisation of vortical structures of the SPO (at Re = 5.83) at four instances with isosurface of |ω| = 5. See Fig. 3.1(d) for the corresponding instances.

Chapter 3 :Figure 3 . 5 :

 335 Figure 3.5: Schematic of forced (centre) plus six primary energetic (surrounding) Fourier modes in the SPO at Re = 5.83. Visualisations show distributions of |ω| at the same instance. The three-digit numbers on the visualisations indicate three components of wavevector k x k y k z . Note that all the possible sign combinations (±k x , ±k y , ±k z ) are gathered. Triangles denote the possible triad interactions. On top, we compare the isosurfaces of |ω X + ω Y | with |ω| of the full flow. Here, ω X and ω Y denote the vorticity of the forced and the primary energetic modes, respectively.

Figure 3

 3 Figure 3.7: (a) Time series of energy E(t) and energy dissipation rate (t) computed from all modes (solid lines) and those of the forced plus the primary energetic modes, denoted by (•) X+Y (dashed lines). (b) Schematic of three different scales: "forced", "primary", and "secondary". We visualise |ω| distributions of typical Fourier modes in each scale. The forced scale corresponds to k f = (±1, ±1, 0). In the primary scale, we visualise k = (0, 0, ±2) and (0, ±2, ±2) modes. For the secondary scale, we visualise k = (±3, ±1, 0) and (±2, ±2, ±2) modes for example. Note that the contributions of modes with all the possible sign combinations (±k x , ±k y , ±k z ) are gathered in the visualisations. Triangles denote triad interactions between different scales. For details of interactions between the forced and primary scales, see Fig. 3.5.

Figure 3 . 8 :Figure 3 . 9 :

 3839 Figure 3.8: Time series of (a) forcing coefficient F (t) and (b) scale coefficients K 2 α (t) in the DNS of the SPO.

Figure 3 .

 3 Figure 3.8(b) shows their temporal evolutions. The forced scale factor K 2 X (t) = 2 is constant, since it corresponds to k f = (±1, ±1, 0) mode. On the other hand, K 2 Y (t) and K 2 Z (t) fluctuate, reflecting the competition of different Fourier modes in these scales. We estimate the model parameters by K 2 Y = 5 and K 2 Z = 15, since K 2 Y (t) t = 4.97 and K 2 Z (t) t = 15.4, respectively.To obtain rough estimates of the scale local coefficients A 1 and A 2 , we compute the average energy transfer rate from X to Y and Y to Z while ignoring the scale non-local interactions by setting A 3 = A 4 = 0. In

Figure 3 .

 3 Figure 3.10 compares the SPO obtained by the model and the DNS.Figure3.10(a) shows the time series of the model with the parameters (3.17) and (A 3 , A 4 ) = (0.5, -0.95). Since the definitions of Re are different in the model and DNS, we have chosen a Reynolds number in the model, which allows qualitatively reproducing the DNS results. We compute two quantities. One is E X-X0 ≡ (X -X 0 )2 /2, which is the fluctuating energy of the forced mode around the laminar base flow X 0 ≡ F Re /K 2 X . The other quantity E Y +Z ≡ Y 2 /2 + Z 2 /2 is the energy of the rest of the modes. We compare them to the corresponding quantities in the DNS of theSPO [Fig. 3.10(b)], where the base flow is u 0 ≡ f /2ν k f 2 , the forced-mode

Figure 3 .

 3 Figure 3.10 compares the SPO obtained by the model and the DNS.Figure3.10(a) shows the time series of the model with the parameters (3.17) and (A 3 , A 4 ) = (0.5, -0.95). Since the definitions of Re are different in the model and DNS, we have chosen a Reynolds number in the model, which allows qualitatively reproducing the DNS results. We compute two quantities. One is E X-X0 ≡ (X -X 0 )2 /2, which is the fluctuating energy of the forced mode around the laminar base flow X 0 ≡ F Re /K 2 X . The other quantity E Y +Z ≡ Y 2 /2 + Z 2 /2 is the energy of the rest of the modes. We compare them to the corresponding quantities in the DNS of theSPO [Fig. 3.10(b)], where the base flow is u 0 ≡ f /2ν k f 2 , the forced-mode

Figure 3 .

 3 Figure 3.10: Time series of fluctuating energy E X-X0 (t) of the forced scale and residual energy E Y +Z (t) of periodic solutions of (a) model (3.14) at Re = 14.05 and (b) the Navier-Stokes equations (3.1) at Re = 5.83. Parameters of the model are (3.17) and (A 3 , A 4 ) = (0.5, -0.95). Note that time in panel (b) is normalised by T .

Figure 3 .

 3 Figure 3.11: (a) Simple periodic (Re = 12.50), complex periodic (Re = 14.05) [Fig. 3.10(a)], and chaotic (Re = 14.1) orbits of the model (3.14). The parameters are the same as in Fig. 3.10(a). The chaotic orbit is tracked over 100 periods. The arrow indicates the direction of the orbit. (b) Bifurcation diagram of the model with changing Re for the same parameters as in Fig. 3.10(a). We plot the local extrema of Y . We have determined the periodicity by Poincaré analysis. The black vertical dotted line corresponds to Re = 12.50 used in panel (a). Inset: close-up in the range shown by the red rectangle in the main plot. The red vertical dashed line corresponds to Re = 14.05 used for panel (a) and Fig.3.10(a). For both panels, black, purple, orange, and yellow data denote steady, simple periodic, complex periodic, and chaotic solutions, respectively.

Figure 3 .

 3 Figure 3.12: (a) Time series of (X, Y, Z) of the model (3.14) with parameters (3.17) and (A 3 , A 4 ) = (0.4, -0.5) at Re = 32. A random initial condition is used. (b) The bifurcation diagram for the same parameter set. The red vertical dashed line denotes Re = 32, which is used for Fig. 3.12(a). Inset: closeup of the diagram in the range shown by the red rectangle in the main plot.

Figure 3 .

 3 Figure 3.13: (a) Survival probability P Re (t) of the transient chaos of the model (3.14) evaluated from 10,000 samples for each Re. The parameter set is the same as in Fig.3.12. Dashed line denotes exponential fitting by (3.23) using 0.01 ≤ P Re (t) ≤ 0.9. (b) The escape rate 1/τ as a function of Re. The dashed line denotes the exponential fitting by(3.24).

Figure 4 . 1 :

 41 Figure 4.1: Distribution of the vortex stretching contributions from the background strain S B ij ω j against the local strain S R ij ω j . Adapted from Hamlington, Schumacher, et al. (2008b, Figure 4).

Figure 4 .

 4 1 shows the distributions of background to local vortex stretching ratio. Buaria and his colleagues used this local/nonlocal decomposition in Direct Numerical Simulation (DNS) to find the self-attenuation property of the intense vorticity due to the locally induced strain rate (Buaria, Pumir, and Bodenschatz 2020). See Figure4.2 for the visualisations. They extend the investigation in the eigenvector alignment framework[START_REF] Buaria | Nonlocal amplification of intense vorticity in turbulent flows[END_REF]. In this part of the thesis, we employ the same space-local filtering in the Navier-Stokes equations to investigate the spatial locality of turbulence. Figure7.7 shows the timeline of research on the spatial locality of turbulence.

Figure 4 . 2 :

 42 Figure 4.2: Space-local vortex stretching term and the self-attenuation of intense vortex. (a-c) shows the enstrophy. (d-f) shows the positive and negative enstrophy production due to the total strain. (g-i) shows the positive and negative enstrophy production due to the space-local strain. (a, d, g) shows moderate structures with a smaller threshold. (b, e, g) shows the intense structures with a larger threshold. (c, f, i)shows the distributions on the plane denoted by the grey plane in the other panels. Adapted from Buaria, Pumir, and Bodenschatz (2020, Figure1).

Chapter 4 :Figure 4 . 4 :

 444 Figure 4.4: Schematic of the space-local velocity field u L , defined by the local vorticity field inside the radius R. The shadowed region denotes the nonlocal domain outside the radius R.

Figure 4 . 5 :

 45 Figure 4.5: (a) Profile of the space-local component of the forcing field f L x against the x coordinate at the edge of the computational domain, y = z = -π. Dark (light) colour denotes a small (large) value of R in 0 ≤ R ≤ 2L f . The black dashed line represents the original f x profile. (b) "Energy" of the space-local forcing field f L 2 /2 as a function of R/L f . The black dashed line represents the one of the original forcing field f 2 /2.

Figure 4 . 5

 45 Figure 4.5 shows the validation of the space-local function using the space-local forcing field,

4. 3 . 1

 31 Direct numerical simulation with the three-dimensional Taylor-Green forcing We conduct DNS of the space-local Navier-Stokes equations (5.2) in a triply periodic cube of size 2π. Throughout this part of the thesis, we report the result of the three-dimensional Taylor-Green forcing coefficient f 0 = 1.

Figure 4 .

 4 6(a) visualizes positive and negative isosurfaces of the xcomponent of the forcing field(4.21). It consists of large-scale spherical objects, which correspond to the vortical structures in the steady flow at low Reynolds numbers as well as the largest vortices in turbulence at high Reynolds numbers. We define the characteristic length scale

Figure 4

 4 Figure 4.6: (a) Positive (red) and negative (blue) isosurfaces of x-component of the forcing field, f x = ±0.5. See (4.21) for the definition. (b) An instantaneous snapshot of vortical structures. Isosurfaces of vorticity magnitude in low-pass filtered ω < = 4 for k ≤ 3 (red), band-pass filtered ω ≶ = 6 for 3 < k ≤ 6 (yellow), and original |ω| = 100 (blue) are shown. Low-pass and band-pass filtering are applied in the Fourier-space velocity field.

Figure 4 .

 4 7 shows the time-averaged statistics of turbulent flow. Panel (a) shows the normalized energy spectrum E(k) with the universal scaling E(k) ∝ k -5/3 for about an order of scale. Panel (b) shows the normalized Probability Distribution Function (PDF) of vorticity components. The figure indicates the small-scale isotropy of the flow, in agreement with the K41 theory discussed in § 1.2.2. Its wide tail represents the intermittency of the small scale of turbulence.

Figure 4 Figure 4 .

 44 Figure 4.7: Time-averaged statistics of the turbulent flow. (a) Normalized energy spectrum E(k) with the -5/3 scaling denoted by the red dashed line. (b) Normalized Probability Distribution Function (PDF) of vorticity components (ω x , ω y , ω z ).

  ω × u L • u.

  conduct DNS of the space-local nonlinear term by taking advantage of the Fourier expression of the space-local Navier-Stokes equations. See § 4.3.1 for details of the DNS configurations. We first conduct a DNS of developed turbulent flow governed by the original Navier-Stokes equations. Then, we launch the DNS of flow governed by the space-local Navier-Stokes equations from a snapshot of this developed turbulent flow.

5. 2 Figure 5 . 1 :Figure 5 . 2 :

 25152 Figure 5.1: Time series of (a) energy input rate P (t), (b) RHS of the energy equation (5.5), and (c) spatially nonlocal nonlinear contributions NL (t) of space-local Navier-Stokes turbulence at different values of R. For all panels, time is normalized by T f : the characteristic timescale of the forcing (4.23). For panels (a) and (c), both quantities are normalised by the energy dissipation rate (t). The horizontal grey dashed line in panel (b) denotes y = 0. See the caption of Fig. 5.4 for the parameter list for R corresponding to each curve with a different colour.

Figure 5 .

 5 Figure 5.1(a) shows the time series of P (t)/ (t), the ratio between input and dissipation rates of energy. By switching the governing equation from original to space-local Navier-Stokes equations, the energy input rate surpasses the energy dissipation rate due to the weakened nonlinearity and energy cascade. This transient regime appears as a peak in the time series. We observe a higher peak for flows with smaller values of R because the nonlinearity is more suppressed for small R. After the transient stage, the flow reaches a state where P (t)/ (t) seems to fluctuate around a constant value which depends on R. It therefore differs from a statistically steady state of the original Navier-Stokes turbulence, as there is no statistical balance between the injection and the dissipation of energy: P (t) t = (t) t . Here, • t denotes the time average. In Fig.5.1(b), we plot P (t) -NL (t) -(t), i.e. the RHS of the energy equation(5.5), to assess the temporal evolution of the space-local Navier-Stokes turbulence. For the original Navier-Stokes turbulence, we observe P (t) -NL (t) -(t) t = 0 with NL = 0 (5.8). When the value of R is finite, we still observe fluctuations around zero after the initial transient regime depicted in Fig.5.1(a). This observation indicates that the space-local Navier-Stokes turbulence eventually establishes the statistically steady state, as the three terms on the RHS of the energy equation (5.5) attain a statistical balance and thus dE(t) dt

5. 2 Figure 5 . 3 :

 253 Figure 5.3: Time evolution of energy spectrum E(k, t; R) in the space-local system at different values of R: (a) R = 0, (b) R = 0.2L f , and (c) R = L f . Time evolves from dark to light colours with an interval of 5T . The thick grey line represents the time-averaged energy spectrum of the original Navier-Stokes turbulence. The red dashed and black dotted lines denote the k -5/3 and k -3 scalings, respectively. Panel (a) employs the data from 128 3 box simulation while the other panels (b-c) show 512 3 box data.

Figure 5 .

 5 [START_REF] Araki | Space-local Navier-Stokes turbulence[END_REF] shows the temporal evolution of the energy spectrum at three different values of R.

5. 4 Figure 5 . 4 :

 454 Figure 5.4: Normalized instantaneous energy spectrum E(k) at different values of R. Normalization is performed by the energy dissipation rate > and the Kolmogorov length scale η > (5.13) with high-pass filter. Vertical dashed lines denote 2π/R and normalized by η ≡ ν 3 / 1/4 of the original turbulence. The dark (light) colour represents small (large) values of R: R/L f = 0.1, 0.2, 0.4, 0.6, 0.8, 1, and the lightest colour represents the original turbulence. The red dashed and black dotted lines denote the k -5/3 and k -3 scalings, respectively.

Figure 5 . 5 :

 55 Figure 5.5: Three terms from the enstrophy budget equation (5.19): enstrophy flux Π ω (k), cumulative enstrophy production V < ω (k), and cumulative enstrophy dissipation rate ω (k). Each spectrum and the wavenumber are normalized by the enstrophy dissipation rate < ω = 2ν ´∞ 0 k 4 E(k) dk and the Kolmogorov length scale η, respectively. Vertical solid and dashed lines denote k f and k η , respectively.

Figure 5 . 6 :

 56 Figure 5.6: Ratio of cumulative enstrophy production V < ω (k) and enstrophy flux Π ω (k). The horizontal dashed line denotes V < ω (k) = Π ω (k). The same colourmap and the vertical dashed lines are employed as in Fig. 5.4.

Figure 5

 5 Figure 5.7: (a) Enstrophy flux Π ω (k) normalized by the enstrophy input rate P ω . The red dashed line denotes k 2 scaling. (b) Compensated energy spectrum E(k) based on the scaling(5.32). For both panels, the wavenumber is normalized by R so that different 2π/R collapses onto kR = 2π. Note that the spectrum of the original turbulence is not shown, since it corresponds to the R ∞ limit. The horizontal dashed line denotes a plateau. The same colourmap is employed as in Fig.5.4.

Figure 5 .

 5 Figure 5.8: A schematic of the energy spectrum associated with space-local Navier-Stokes turbulence.Dashed arrows denote k -3 (plus logarithmic correction) and k -5/3 scaling, respectively. Vertical dashdotted lines denote: the forcing wavenumber k f , the intersecting wavenumber k * of the two scalings, and the Kolmogorov wavenumber k η > , respectively. Annotations (i)-(iv) correspond to different scaling regions divided by the characteristic wavenumbers.

6. 2

 2 Figure 6.2: Isotropic (E ii (k)/2, solid line) and anisotropic (E 12 (k); dashed line and -E 12ii (k)/2; dotted line) energy spectra. The dashed anddash-dotted lines show -5/3 and -7/3 scalings, respectively.

Figure 6 . 3 :

 63 Figure 6.3: A schematic of the energy spectrum in (k, z) coordinates. Two arrows denote the direction of energy fluxes in wavenumber and physical space, respectively.

  Figure 6.4: (a) Instantaneous distributions of U x (x, t) at Re λ = 184. Blue (red) corresponds to the negative (positive) value of U x . (b) Instantaneous profiles of U x (z, t) = U x (x, t) ⊥ in grey. Timeaveraged profile U (z) = U x (x, t) ⊥,t is indicated by a thick line.

  Figure 6.4(b) shows the instantaneous profile of U x (z, t) = U x (x, t) ⊥ . The single curve corresponds to 6.3 Assessment of the inhomogeneous scaling correction

  Figure 6.6: Absolute value of the time-averaged nonequilibrium energy spectrum E 1 (k ⊥ , z) = E 1 (k ⊥ , z, t) t for the highest Reynolds number dataset. Different nondimensionalised functions are employed to compute the nonequilibrium spectrum; panel (a) with (6.54) and (b) with (6.56), respectively. The black solid and red dashed lines denote the z-average and k -7/3 ⊥ scaling, respectively.

  6.55) with F L [kL] = (kL) -5/3 f L [kL]. (6.56)

6. 3

 3 Assessment of the inhomogeneous scaling correction in the inertial range, since lim x→∞ F L [x] = lim x→0 F η [x] = x -5/3 .

Figure 6

 6 Figure 6.7: Time-averaged three-dimensional isotropic energy spectrum E(k) = E(k, t) t , normalised by Kolmogorov variables. Results are shown at Re λ = 69.6, 113 and 184 (see table 6.1). The red dashed line denotes the k -5/3 scaling for reference.

Figure 6 .

 6 Figure 6.10(a) compares E 1 (k ⊥ , z) z for three different Reynolds numbers as in figure 6.7. For smaller values of Re λ , the spectrum exhibits steeper scaling than k -7/3 ⊥

  Figure 6.10: (a) Absolute value of the time-averaged nonequilibrium energy spectrum E 1 (k ⊥ , z) = E 1 (k ⊥ , z, t) t for three values of the Taylor-length Reynolds numbers. The red dashed line represents the k -7/3 ⊥ scaling. (b) Compensated spectrum of the panel (a). The red dashed line denotes the compensated k -7/3 ⊥ scaling.

7. 2

 2 Figure 7.2: Scatter plot of C and Re λ for two different inlet Reynolds numbers: Re = 1.2 × 10 4 and Re = 1.5 × 10 4 . "SFV" denotes "Small Field of View" and 20 denotes the measurement position. The parameters G and H denote the distance between the prisms and the prism width, respectively. Adapted from Chen et al. (2021, Figure 20(f)).

  Figure 7.3: (a) z-profile of the normalised energy dissipation rate C k f (z) and the Reynolds number Re λ (z). (b) Parametric plot of C k f (z) as a function of Re λ (z). Both quantities are divided by their average values: C k f and Re λ , respectively. Results for three Re λ are shown. The red dashed line indicates a theoretical relation (7.29) with α = 0.13 and qX = 1.

  Figure 7.3(a) shows the z-profile of C k f (z) and Re λ (z) displaying a clear anti-phase relationship. In Fig. 7.3(b), we overlap parametric plots of C k f (z)-

Figure 7 Figure 7 . 4 :

 774 Figure 7.4(a) shows temporal evolution of Re λ (t), and C k f (t). We observe a similar anti-phase trend, as in the inhomogeneous case in Fig. 7.3(a), in complex fluctuations. The normalised parametric plot in Fig. 7.4(b) shows wider distributions than in Fig. 7.3(b). The theoretical relation (7.31) represents an ellipse-like trajectory, which indeed captures the characteristics of the numerical data. Here, we set

3 . 3 )

 33 to overlap the phase-averaged time series over the original turbulent time series. From the relative amplitude of the phase-averaged time series C k f phase , we set α = 0.19. This value is used to draw the theoretical relation in Fig. 7.4(b).

  Figure 7.6: C L -Re λ scaling based on the integral length scale L. Panels (a) and (b) corresponds to Fig. 7.3(b) and Fig. 7.4(b), respectively. The fitting parameters for the theoretical relation are: (a) α = 0.13 and qX = 1 and (b): α = 0.13 and ωT = 0.5.

Chapter 7 :

 7 Dissipation Rate Scaling in Inhomogeneous or Unsteady Turbulence 7.5.1 The -3/2 scaling with the forcing length scale By linearising C

  The minimal model involves three variables representing the forcing, energy-containing, and small scales representing the energy dissipation rate of the actual flow. This model reproduces several properties of the actual flow, including similar periodic (laminar) and quasi-cyclic chaotic (turbulent) solutions. Here, a new question arises for the robustness of the QCB in different flow configurations, particularly for spatially extended flows. Since the QCB considered in this thesis are excited in minimal flow domains: between a pair of rotating disks or (k x , k y ) = (±1, ±1) forcing Fourier mode in the (2π) 3 box. What would happen in a much larger flow domain with many large-scale structures? How do the nearby and far away large-scale structures interact with each other regarding QCB? In fact, this question is related to Part II of the thesis.

B

  Betchov's relation using ω = (ω a , ω b , ω c ). We set a ≥ b ≥ c. Note that the incompressibility readsA ii = S ii = a + b + c = 0, (B.3)and we denote the three eigenvalues of A by λ 1 , λ 2 , and λ 3 . The characteristic equation of A reads A -P A 2 + QA -RI = 0, (B.4)

  that the RHS of R obeys from the trace of (B.4). By substituting (B.2) into (B.6, B.7), we obtain use the identity (B.3) from the first to the second line.

3 S

 3 ij S jk S ji . importance of these two equations cannot be overestimated; Equation (B.15) leads to the estimation of the kinetic energy dissipation rate≡ 2ν S ij S ij = ν ω 2 , (B.17)and (B.16) connects the enstrophy production (LHS) and the strain self-amplification (RHS).

Figure C. 3 :

 3 Figure C.3: Visualisation of vorticity magnitude isosurface during the vortex reconnection event at Re = 2000. Colours denote the axial vorticity magnitude. Sublabels denote the nondimensional time t * . Adapted from Hussain and Duraisamy (2011, Figure 2).

Figure C. 4 :

 4 Figure C.4: Schematic of (generalised) Lundgren's spiral vortex model. A pair of vortex sheets wrap around a vortex tube. The arrows denote the vorticity vectors: (a) Mode 1 investigated by Lundgren (1982), (b) Mode 2 by Horiuti and Takagi (2005), (c) Mode 3 by[START_REF] Pearson | Evolution of the flow field associated with a streamwise diffusing vortex[END_REF],[START_REF] Moore | The interaction of a diffusing line vortex and an aligned shear flow[END_REF], and[START_REF] Kawahara | Wrap, tilt and stretch of vorticity lines around a strong thin straight vortex tube in a simple shear flow[END_REF]. Adapted fromHoriuti and Fujisawa (2008, Figure 1). See Hino (2020, § 4.11.1) for a review (in Japanese).

Figure D. 1 :

 1 Figure D.1: Scale-by-scale energy transfer T (K, K , t) from wavenumber K to K due to the vortex stretching term. In each panel, darker and thinner curves correspond to larger K, and lighter and thicker curves denote smaller values of K. From panel (a) to (d), the different ranges of K are shown; Panels (a) show the smallest values of K and represent the energy transfer in the forcing region. The range of K successively increases: (b) represents the inertial range, (c) the dissipation range, and (d) the fardissipation range. The red and blue dotted lines denote the characteristic wavenumber for the dominant energy loss and gain, respectively. Adapted from Yoneda et al. (2022, Figure 1).

  Figure D.1 shows this energy transfer spectrum in high-Reynolds-number turbulence (Taylor-scale Reynolds number Re λ ≈ 280) snapshot as a function of K /K. Four panels of Fig. D.1 correspond to different spectral regimes in turbulence, and we focus on panel (b): inertial range.

u

  3) are the average kinetic energy dissipation rate and the power input W , respectively. To investigate the energy flux ∂ r • T for different scales, they introduced the coarse-graining = u + u , u(x, t) ≡ ˆR G y u(xy, t) dy , an isotropic filter kernel with length scale r. With this decomposition and assuming isotropy, one can define the energy flux at scale r:∂ r • T = L S • S : S -≡ ∇ u + ∇ u /2and ω ≡ ∇ × u are filtered strain-rate tensor and vorticity, respectively, and Eq. (D.9) is an operator in isotropic configuration and F denotes the residual contributions. The S • S : S and ω ω : S are the strain self-amplification (SSA) and the vortex stretching (VS) terms, respectively.

  Figure D.2: (a) Nonlinear energy flux with its different contributions: DNS result (black line with dots), ∂ r •T defined in (D.8) (red line), SSA (green line) and VS (blue line) contributions. The difference between the DNS result (black line with dots) and ∂ r • T (red line) denotes the residual F in (D.8). Adapted from Carbone and Bragg (2020, Figure 1(a)). (b) The mean contributions to the energy cascade rate Π x for different contributions against the filter size. See the main text for the definition and interpretation of each term. The total energy cascade rate is denoted by Π x and equals one after the normalisation. The grey lines denote the approximate range of the inertial range. The horizontal dashed lines denote the magnitude of each contribution in the inertial range. Adapted from Johnson (2021a, Figure 8(b)).

u

  P (x, t) = (kx,ky,kz)∈IP û k x , k y , k z exp i k x x + k y y + k z z .(D.23)

  Figure D.3: (a) Two-dimensional instantaneous energy transfer spectrum normalised by the average energy dissipation rate T 2 (K, Q)/ in homogeneous isotropic turbulence at Re λ = 210. Adapted from McKeown, Pumir, et al. (2022, Figure 2(a)). (b) The integrated energy transfer scaling T 1,Q = T 2 (1, Q) (blue plus symbols) and T 2,Q = T 2 (2, Q) (red cross symbols). Note that the shortened notation T K,Q = T 2 (K, Q) is employed in the paper. The straight dashed line shows the theoretical prediction T K,Q ∝ 2 -4Q/3 . Adapted from McKeown, Pumir, et al. (2022, Figure 2(b)).

Table of Contents

 of Energy cascade in three-dimensional turbulence . . . . . . . . . . . . . . . . . . . . . . . . 5 1.1.1 Inviscid invariants and their cascades . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.1.2 Scale-local nature of the energy cascade . . . . . . . . . . . . . . . . . . . . . . . . 9 1.2 Equilibrium and nonequilibrium aspects of turbulence . . . . . . . . . . . . . . . . . . . . Physical-space mechanism behind the energy cascade . . . . . . . . . . . . . . . . . . . . . 1.3.1 Vortex stretching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.2 Strain self-amplification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.3 Intermittency and the energy cascade . . . . . . . . . . . . . . . . . . . . . . . . . Temporal (quasi-)cyclic modulation associated with the energy cascade . . . . . . . . . . .

	Abstract	i
	List of Publications	v
	List of Figures	vii
	List of Tables	ix
	List of Frequently Used Symbols	xi
	Table of Contents	xv
	Overview of the Thesis	1
	1 Introduction: the Energy Cascade	5
	1.1 I Quasi-cyclic Temporal Fluctuations of Turbulence and its Minimal Model
	25
	2 Temporal Fluctuations in Turbulence	27
	2.1

1.2.1 Local equilibrium hypothesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.2 Kolmogorov-Onsager theory of the small-scale universality . . . . . . . . . . . . . 1.2.3 Taylor's dissipation law and its violation . . . . . . . . . . . . . . . . . . . . . . . . 1.2.4 Cascades in two-dimensional turbulence . . . . . . . . . . . . . . . . . . . . . . . . 1.3 2.1.1 Intrinsic unsteadiness and quasi-cyclic behaviour of turbulence . . . . . . . . . . . 2.1.2 Periodic orbit as a skeleton of the energy cascade . . . . . . . . . . . . . . . . . . . 2.1.3 Quasi-cyclic behaviour in von Kármán flow: a closed system under constant forcing 2.2 Reduced-order modelling of turbulence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Minimal model of physical phenomena and chaos . . . . . . . . . . . . . . . . . . . 2.2.2 Reduced-order models of developed turbulence . . . . . . . . . . . . . . . . . . . . 2.2.3 Reduced-order model of laminar-turbulent transition . . . . . . . . . . . . . . . . .

Table 4 .

 4

1: DNS setting and statistical quantities of the DNS of the original Navier-Stokes equations. The parameters are the resolution of computational domain N and kinematic viscosity ν. The statistical quantities are evaluated by time average of: the fluctuating isotropic RMS velocity u (t) ≡ 2K (t)/3, which is defined by the fluctuating energy K (t) ≡ u i u i /2 where • denotes the spatial average and u i (x, t) denotes the temporal fluctuating velocity field; the Taylor microscale λ(t) ≡ u (t) 15ν/ (t) where the energy dissipation rate is evaluated by (t) = ν ω i ω i ; the Taylor-length Reynolds number Re λ (t) ≡ u (t)λ(t)/ν; the simulation time in the statistically steady state T total as a function of T f . of the forcing, respectively, where the forced wave vector is k f ≡ (±1, ±1, ±1) .

Table 4 .

 4 1 summarizes the DNS setting and the statistical quantities, and Fig.4.6(b) shows the vortical structures in a snapshot of developed turbulence generated by the original Navier-Stokes equations. The observations are consistent with higher Reynolds number turbulence visualizations presented, for example, in[START_REF] Goto | Hierarchy of antiparallel vortex tubes in spatially periodic turbulence at high Reynolds numbers[END_REF].

Table 6 .

 6 1: DNS parameters and statistical quantities.

	Chapter 6 : Inhomogeneous Scaling Correction of Energy Spectrum
	N	ν	u	λ	Re λ T total /T
	128 0.07 1.31 0.371 69.6	959
	256 0.028 1.35 0.233 113	645
	512 0.01 1.33 0.138 184	170

  This chapter investigates the scaling of the normalised energy dissipation rate of spatially inhomogeneous and temporally unsteady turbulence. The content is based on the following: Bos, Wouter J. T. and Ryo Araki (in preparation). Analysis of the normalized dissipation rate in inhomogeneous and unsteady turbulence.

	7 Dissipation Rate Scaling in Inhomogeneous
	or Unsteady Turbulence
	.68)
	Therefore, if this requirement is fulfilled in a statistically stationary flow, far enough away from walls,
	Kolmogorov's equilibrium spectrum is expected to be dominant compared to the contributions associated
	with spatial inhomogeneity.

  Taylor-scale Reynolds number Re λ (t) dependency of the nondimenional energy dissipation rate C (t). (a) In forced turbulence for different Reynolds numbers. Adapted from Goto and Vassilicos (2015, Figure

	Chapter 7 : Dissipation Rate Scaling in Inhomogeneous or Unsteady Turbulence
	(a)	(b)
	Figure 7.1:

. See § 1.2.3 for an alternative derivation, validation in the statistical sense and violation due to the nonequilibrium nature of turbulence. Now, we define the normalised energy dissipation rate C (x, t) ≡ (x, t)L(x, t) U (x, t) 3 . (7.4)

  Chapter 7 : Dissipation Rate Scaling in Inhomogeneous or Unsteady Turbulence the energy spectrum yields the decomposition of the kinetic energy K

1/3 (7.12)

for spatial correction (6.39), respectively. Note that we employ shorthand notations ( •) = d(•) dt for the temporal derivative and (•) zz = ∂ 2 (•) ∂z 2 for the second spatial derivative, respectively. The integral of

  Linearisation of the parametric plots between C and Re λ

	0.90 0.95 1.15 (a) 1.00 1.05 1.10 C L (z)/ C L	1.0 Re λ (z)/ Re λ 7.5 0.9 1.1 Re λ 69.6 113 184	C L (t)/ C L	0.8 1.4 1.0 1.2	0.8	1.0 Re λ (t)/ Re λ	1.2 Re λ 69.6 113 184
								(7.38)
			t)	,			(7.39)

  t) Re 0 .

	(7.43)
	(7.44)
	(7.45)
	(7.46)
	(7.47)
	(7.48)

  Influence of the large-scale energy distribution 7.5.2 The -15/14 scaling with the integral length scale

		2	α ωT cos(ωt) and	(7.55)
	Re λ (t) Re λ	= 1 + α ωT cos(ωt) +	1 6	sin(ωt) .	(7.56)

See (7.32)

for the definition of T . Thus, the parametric plot of C k f (t)-Re λ (t) exhibit a ellipse with its major axis have -3/2 slope in Figs. 7.4(b) and 7.5.

7.6

  sum yields the viscous term of the equation of u i u k :where on the RHS, we take k → i and divide by 2 to make it the viscous term of the energy equation,K = u i u i /2. Another relation ∂ 2 u i u i ∂x j ∂x j

							ν u i	∂ 2 u k ∂x j ∂x j	+ u k	∂ 2 u i ∂x j ∂x j	.	(A.7)
	Then, we take advantage of the following two identities
			u i	∂ 2 u k ∂x j ∂x j	=	∂ ∂x j	∂u i u k ∂x j	-u k	∂u i ∂x j	-	∂u i ∂x j	∂u k ∂x j	,	(A.8)
			u k	∂ 2 u i ∂x j ∂x j	=	∂ ∂x j	u k	∂u i ∂x j	-	∂u i ∂x j	∂u k ∂x j	.	(A.9)
	Then, (A.7) becomes
																
	ν	 ∂ ∂x j	∂u i u k ∂x j	-2	∂u i ∂x j	∂u k ∂x j	 → ν	 ∂ ∂x j	1 2	∂u i u i ∂x j	-	∂u i ∂x j	∂u i ∂x j	 ,	(A.10)
									= 4	∂ ∂x j	S ij u i -2	∂u i ∂x j	∂u j ∂x i	(A.11)
	turns (A.10) into															
				ν	 2	∂ ∂x j		S ij u i -	∂u i ∂x j	∂u j ∂x i	+	∂u i ∂x j	 .	(A.12)

During its propagation, the poem is often converted into the four-line format. In[START_REF] Hunt | Thin shear layers-the key to turbulence structure[END_REF], the poem is introduced as:

The term "cascade" was first used by Onsager in 1945[START_REF] Onsager | The distribution of energy in turbulence[END_REF][START_REF] Eyink | Onsager and the theory of hydrodynamic turbulence[END_REF]. See § 1.3.1 for more details.

Note that it is not possible to describe P (k|p) in Fig.1.3(b) as the energy transfer from wavenumber p to k, as depicted

Here L and U dare the typical length and velocity scales for the flow as a whole.

It is sometimes called the Taylor-Kolmogorov relation (or dissipation law) (Rubinstein and Clark 2017), asKolmogorov (1941b) first derived this dissipation law. See alsoGoto and Vassilicos (2016a) for historical remark.

Also, see the same author's Physics Today article(Johnson 2021b).

We employ the same two-dimensional Taylor-Green forcing in Chap. 3; see(3.2).

contained by the forced and primary modes. While the energy is almost

We employ the two-dimensional Taylor-Green forcing (3.2) in § 3 and the Kolmogorov forcing (6.46) in the following sections of this chapter, respectively.

This study is inspired by the one investigating the global singularity of the three-dimensional Navier-Stokes equations by a kind of shell model and its energy-conserving iterative cascades(Tao

2016).

Note that there is a relationT 2 (K, Q) + T 2 (Q, K) = 0 for spatially integrated spectrum. For space-dependent spectrum, T 2 (K, Q, x) + T 2 (Q, K, x) = 0.

Acknowledgements

Acknowledgements iii

List of Publications

List of Tables Chapter 4 : Physical-space Locality of Turbulence K41 theory Kolmogorov (1941a), Kolmogorov (1941b), and Kolmogorov (1941c) Pressure (Hessian) [START_REF] She | Structure and dynamics of homogeneous turbulence: models and simulations[END_REF] Ohkitani and [START_REF] Ohkitani | Nonlocal nature of vortex stretching in an inviscid fluid[END_REF] Nomura and Post (1998) [START_REF] Chevillard | Local and nonlocal pressure Hessian effects in real and synthetic fluid turbulence[END_REF] Kármán-Howarth-Monin-Hill [START_REF] Hill | Exact second-order structure-function relationships[END_REF] Valente and [START_REF] Vassilicos | Dissipation in turbulent flows[END_REF] Yasuda and Vassilicos ( 2018) Dubrulle (2019) Energy flux Kraichnan (1974) Eyink (1995) Eyink (2005) Eyink ( 2006) [START_REF] Alexakis | Local energy flux of turbulent flows[END_REF] Space local/nonlocal strain Hamlington, Schumacher, et al. (2008a) Hamlington, Schumacher, et al. (2008b) Buaria, Pumir, and Bodenschatz (2020) Buaria and Pumir (2021)

Appendix

A Pseudo-energy dissipation rate

In this section, we consider the so-called pseudo-energy dissipation rate

and how it is different from the proper energy dissipation rate

Note that we omit the subscript x from the bracket symbol for readability.

First, we rewrite the Navier-Stokes equations (1.1) in the index notation

Note that we do not nondimensionalise the system (ν → 1/ Re). By taking the inner product with u i and integrate over space, one obtains

(A.4) The integral of the nonlinear term and the first part of the viscous term become zero, thanks to the divergence theorem. In other words, they are conservative terms. Thus, the integrated energy equation can be written as ∂ ∂t

yielding the pseudo energy dissipation rate (A.1). However, this is not the correct expression of the energy dissipation rate since (A.1) contains the conservative contributions. The objective of this section is to derive the correct expression (A.2) from (A.3) and highlight the difference against the pseudo expression (A.1). The contents of this section heavily rely on Mortensen (2020, Chapter 6), while notations and configurations are simplified.

In the following, we consider the viscous term in (A.

C.1 Instability

Although vortex stretching ( § 1.3.1) and the strain self-amplification ( § 1.3.2) are the main contenders for the energy cascade mechanism, there are several other candidates. In this subsection, we focus on the "instability-driven cascade" picture. To begin our discussion, we quote the full paragraph in which Richardson wrote down his poem (Richardson 1922, p. 66):

On the other hand we find that convectional motions are hindered by the formation of small eddies resembling those due to dynamical instability. Thus C. K. M. Douglas writing of observations from aeroplanes remarks: "The upward currents of large cumuli give rise to much turbulence within, below, and around the clouds, and the structure of the clouds is often very complex." One gets a similar impression when making a drawing of a rising cumulus from a fixed point; the details change before the sketch can be completed. We realize thus that: big whirls have little whirls that feed on their velocity, and little whirls have lesser whirls and so on to viscosity -in the molecular sense.

Here, he referred to "dynamical instability" as the generation mechanism of the small eddies. In [START_REF] Vincent | The dynamics of vorticity tubes in homogeneous turbulence[END_REF], instability was claimed as the key energy cascade mechanism. In this study, the authors investigated the early-days DNS result of both statistically steady homogeneous and decaying turbulence from visualisations. They claimed that the roll-up of the vortex sheet forming the vortex tube is the essential mechanism of the energy cascade. Although this process contradicts Richardson's step-by-step cascade picture, the authors describe the vortex sheet roll-up as "a one-step process, with a strong correlation between small and large scales" [START_REF] Vincent | The dynamics of vorticity tubes in homogeneous turbulence[END_REF]. However, note that this study is based on the visualisation of DNS, and no quantitative evidence was presented. Furthermore, it seems that this paper is referred to as one of the first numerical pieces of evidence of the strong fine-scale vortex filaments in turbulence and not in the context of the physical mechanism of the energy cascade. In [START_REF] Brenner | Potential singularity mechanism for the Euler equations[END_REF], the iterative vortex mechanism between a pair of vortex tubes and sheets are discussed. As schematically shown in Fig. C.2(a), this process is a candidate of the singular solution of the Euler equation 1 . This instability process in the inviscid situation can be related to vortex reconnection in a viscous fluid (see § C.2). Recently, another mechanism driven by the instability has been reported as a candidate for the energy cascade [START_REF] Mckeown | Cascade leading to the emergence of small structures in vortex ring collisions[END_REF]McKeown, Ostilla-Mónico, et al. 2020); It is the elliptical instability observed in the secondary (and beyond) vortex structures of the face-to-face vortex ring collision. Figure C.2(b) visualises the development of the finer vortex structures in a pair of interacting vortex tubes. The collided vortex rings instantaneously break down into secondary filaments; their interactions lead to tertiary fine-scale structures. This cascade of energy from the large vortex rings to the fine-scale vortex filaments. The authors reminded the similarity of this process with the scale-by-scale vortex stretching (see § 1.3.1). However, they simultaneously remark that this process is "reminiscent of, but perhaps different from, Richardson's initial proposal [START_REF] Richardson | Weather Prediction by Numerical Process[END_REF])" [START_REF] Mckeown | Cascade leading to the emergence of small structures in vortex ring collisions[END_REF].

C.2 Vortex reconnection

In this subsection, we focus on the vortex reconnection [START_REF] Kida | Vortex reconnection[END_REF]Yao and Hussain 2022) and its contribution to the energy cascade. The vortex reconnection event, shown in An alternative picture proposed in Melander and Hussain (1993) is that the large-scale vortex tube generates the secondary small-scale anti-parallel pairs around it. This process can also be iterative at a large enough Reynolds number and further investigated in [START_REF] Goto | A physical mechanism of the energy cascade in homogeneous isotropic turbulence[END_REF]. The latest development of this mechanism can be found in a review article (Yao and Hussain 2022).

C.3 Lundgren's spiral vortex scenario

In 1982, Lundgren proposed an innovative energy cascade model. This is Lundgren's spiral vortex scenario (model), which we review in this subsection. This model is schematically described as (Davidson 2015, § 6.4.3 (p. 357)):

He considered an unsteady, stretched, Burgers-like vortex filament whose internal structure is not axisymmetric but spiralled, like a sheet of paper rolled up into a scroll [...].

The schematic is shown in Fig. C.4. A similar schematic can be found in, for example, Horiuti and Ozawa (2011, Figure 2) and Davidson (2015, Figure 6.32). This model is first proposed as a purely mathematical model. In the same year, when Lundgren proposed his refined model (Lundgren 1993), Pullin and Saffman investigated the vorticity and the moments of velocity gradient by the Lundgren's model [START_REF] Pullin | On the Lundgren-Townsend model of turbulent fine scales[END_REF]. Also, in the same year, Moffatt discussed that the spiral vortex could explain the K41 scaling without the hierarchical structures of different scales [START_REF] Moffatt | Spiral structures in turbulent flow[END_REF]. The review article (Pullin and Saffman 1998, § 6) summarises the importance of Lundgren's model as:

Appendix

It appears to be the only known repeatable result making a definite analytical connection between a k -5/3 -type spectrum and an approximate solution of the Navier-Stokes equations [...].

The first experimental evidence of spiral vortex and corresponding E(k) ∝ k -5/3 scaling is reported two decades after the first paper [START_REF] Cuypers | Vortex burst as a source of turbulence[END_REF]. Japanese researchers investigated the spiral vortex in DNS with HIT and homogeneous shear flows [START_REF] Horiuti | The multi-mode stretched spiral vortex in homogeneous isotropic turbulence[END_REF][START_REF] Horiuti | Multimode stretched spiral vortex and nonequilibrium energy spectrum in homogeneous shear flow turbulence[END_REF]. Kerr (2013) discussed that a chain of spiral vortex rings generated from anti-parallel vortex tubes exhibits the 3D turbulence statistics.

D Quantitative comparison of different cascade mechanisms

We have reviewed many (but not all) candidates for forward energy cascade mechanisms in threedimensional turbulence in § 1.3 and Appendix C. However, our discussion was limited on a phenomenological level. Now, we would ask a question: What is the physical mechanism of energy cascade driving turbulence? The difficulties regarding this problem are twofold, as summarised by Vela-Martín (2019, § 1.1.3). First, "eddy" cannot be locally determined regarding space and scale. For example, applying a bandpass filter isolates vortex structures of a given scale (see Fig. 1.11) distributed in the entire domain. The opposite procedure of extracting the subspace of the flow domain can isolate the spatially local vortex structures, but their size spans a wide range of scales. Second, there is no definitive way of determining energy flux. One can define one's energy flux by focusing on different quantities, terms in the governing equations, and types of nonlinear interactions. Indeed, as Tsinober stated (Tsinober 2009, § 5.4.2):

[...] the energy exchange/transfer is decomposition dependent.

In this subsection, we review several recent articles: [START_REF] Goto | Hierarchy of antiparallel vortex tubes in spatially periodic turbulence at high Reynolds numbers[END_REF] In fact, we would encounter several different definitions of energy flux in this subsection. Thus, our objective in this subsection is not to claim the answer to the question but to pause a current picture of our understanding. Note that we maintain the notations in the original articles. [START_REF] Goto | Hierarchy of antiparallel vortex tubes in spatially periodic turbulence at high Reynolds numbers[END_REF] and [START_REF] Yoneda | Mathematical reformulation of the Kolmogorov-Richardson energy cascade in terms of vortex stretching[END_REF] The hierarchical vortex structures in anti-parallel configurations (see Fig. 1.11) reported in [START_REF] Goto | Hierarchy of antiparallel vortex tubes in spatially periodic turbulence at high Reynolds numbers[END_REF] suggests that the vortex structures at scale are generated by the strain-rate field induced by the large-scale (for example, 2 ) vortex structures. In contrast, they induce the strain-rate field by themselves, which leads to finer-scale vortices (at scale /2). The scale-local nature of vortex stretching is consistent with the scale-local energy cascade picture. In [START_REF] Yoneda | Mathematical reformulation of the Kolmogorov-Richardson energy cascade in terms of vortex stretching[END_REF], a new mathematical derivation of the Kolmogorov's -5/3 law was presented based on the hierarchical vortex structures and Appendix cascade. derived in Kraichnan (1971b, Eq. (2.19)). Also see Kraichnan (1966), [START_REF] Zhou | Interacting scales and energy transfer in isotropic turbulence[END_REF], and [START_REF] Aluie | Localness of energy cascade in hydrodynamic turbulence. II. Sharp spectral filter[END_REF].

The authors investigated the possible energy cascade mechanism based on the visual observation and correlation of the space-dependent energy transfer spectrum T 2 (K, Q) with the magnitude of vorticity and strain. They compared i) Homogeneous Isotropic Turbulence (HIT) driven by a stochastic forcing, ii) anti-parallel and perpendicular vortex tubes, and iii) turbulence driven by a deterministic forcing, also used in [START_REF] Goto | Hierarchy of antiparallel vortex tubes in spatially periodic turbulence at high Reynolds numbers[END_REF]. From the HIT results, they concluded that:

[...] the enticing energy cascade proposed by [START_REF] Tennekes | A First Course in Turbulence[END_REF], resting on a description of the action of strain and vorticity at neighbouring scales, is oversimplified; [...].

Furthermore, they claimed that in the simplified setup, such as anti-parallel vortex tubes or even turbulence driven by a columnar forcing (two-dimensional Taylor-Green forcing), one cannot correctly capture the energy transfer in HIT: [...] the remarkably simple picture, observed in the interaction of antiparallel vortex tubes and in the configuration studied by [START_REF] Goto | Hierarchy of antiparallel vortex tubes in spatially periodic turbulence at high Reynolds numbers[END_REF], appears to be the result of a strong influence of the large-scale forcing (or initial conditions), [...]. This makes both flows ideal for the study of the energy transfer associated to vortex stretching, but less applicable to the general case of the turbulent cascade.