F Alizard

S Pirozzoli

M Bernardini

F Grasso

Optimal transient growth in compressible turbulent boundary layers

Keywords: compressible flows, compressible turbulence, turbulent boundary layers

The structure of zero-pressure-gradient compressible turbulent boundary layers is analysed using the tools of optimal transient growth theory. The approach relies on the extension to compressible flows of the theoretical framework originally developed by Reynolds & Hussain (J. Fluid Mech., vol. 52, 1972, pp. 263-288) for incompressible flows. The model is based on a density-weighted triple decomposition of the instantaneous field into the contributions of the mean flow, the organized (coherent) motions and the disorganized background turbulent fluctuations. The mean field and the eddy viscosity characterizing the incoherent fluctuations are here obtained from a direct numerical simulation database. Most temporally amplified modes (optimal modes) are found to be consistent with scaling laws of turbulent boundary layers for both inner and outer layers, as well as in the logarithmic region, where they exhibit a self-similar spreading. Four free-stream Mach numbers are considered: Ma ∞ = 0.2, 2, 3 and 4. Weak effects of compressibility on the characteristics length and the orientation angles are observed for both the inner-and the outer-layer modes. Furthermore, taking into account the effects of mean density variations, a universal behaviour is suggested for the optimal modes that populate the log layer, regardless of the Mach number. The relevance of the optimal modes in describing the near-wall layer dynamics and the eddies that populate the outer region is discussed.

Introduction

Since the pioneering study of [START_REF] Theodorsen | Mechanism of turbulence[END_REF] on horseshoe vortices, it has been known that coherent motions play a major role in the dynamics of wall turbulence. According to the mechanism proposed by [START_REF] Hamilton | Regeneration mechanisms of near-wall turbulence structures[END_REF] and later confirmed by the numerical experiment of [START_REF] Jiménez | The autonomous cycle of near-wall turbulence[END_REF], near-wall turbulence is characterized by a self-sustaining process, which relies on the formation and the subsequent breakdown of streamwise velocity streaks. The streaks, having a characteristic spanwise spacing of approximately 100 wall units [START_REF] Kline | The structure of turbulent boundary layers[END_REF], are induced by the advection of the mean velocity field by quasi-streamwise vortices, which are in turn generated by streak instability. While the dynamics and the typical structures of the near-wall layer are sufficiently well understood, much less is known about the structure of the outer layer, which has been the focus of substantial research in the past years. Recent experiments and numerical studies have revealed the existence of very long, meandering features consisting of regions of low-streamwise-momentum fluid flanked by higher-momentum fluid, commonly referred to as large-and very-large-scale motions (LSM, VLSM) in internal flows, and as superstructures in external flows [START_REF] Smits | High-Reynolds number wall turbulence[END_REF]. According to [START_REF] Kim | Very large-scale motion in the outer layer[END_REF], these motions are created by the organization of vortices into packets, formed by multiple hairpin vortices travelling at the same velocity.

The study of the spatial and temporal scales of coherent structures, and of their mutual interactions, is fundamental for developing low-order models of wall turbulence [START_REF] Panton | Overview of the self-sustaining mechanisms of wall turbulence[END_REF]. In this context, although turbulence requires the nonlinearity of the Navier-Stokes equations, there is strong evidence that the emergence of coherent structures and energy production mechanisms can be described through linear models [START_REF] Jiménez | How linear is wall-bounded turbulence?[END_REF]. For laminar shear flows, [START_REF] Schmid | Stability and Transition in Shear Flows[END_REF] have shown that linear stability theory based on transient growth provides some understanding of the early stages of amplification of the streaks. In the case of wall-bounded turbulent shear flows, [START_REF] Butler | Optimal perturbations and streak spacing in wall-bounded turbulent shear flow[END_REF] carried out an optimal transient growth analysis by linearizing the Navier-Stokes equations about the mean flow, and modelling the effect of the small-scale turbulence on the coherent part of motion by constraining the optimization time with the eddy turnover time. They showed that the most amplified modes have a characteristic spanwise wavelength of O(100) wall units, thus matching the spacing of streaks observed in experiments [START_REF] Kim | The production of turbulence near a smooth wall in a turbulent boundary layer[END_REF].

The importance of linear processes in the sustainment of near-wall turbulence in channel flow has been numerically investigated by [START_REF] Kim | A linear process in wall-bounded turbulent shear flows[END_REF]. By setting to zero the linear coupling term representing the transfer from wall-normal velocity to wall-normal vorticity, those authors showed suppression of linear transient growth, with subsequent strong reduction of turbulence, and disappearance of the streamwise vortices. They then concluded that, although near-wall turbulence features nonlinear processes (bursting events being an example), its maintenance relies on linear processes of transient growth. Other evidence for the importance of linear processes has been recently given by [START_REF] Alizard | Sensitivity analysis of optimal transient growth for turbulent boundary layers[END_REF], who developed a linear theory for the control of incompressible near-wall turbulence.

The temporal stability of the Orr-Sommerfeld and Squire equations in channels with turbulent mean velocity profiles and turbulent eddy viscosities was studied by del Alamo & Jiménez (2006), who, using a linear transient growth analysis, identified two maxima, one corresponding to the sublayer streaks and the other to large-scale global structures spanning the full channel. In the intermediate region, the authors also remark that the maximum gain reaches an almost constant value and the corresponding modes exhibit self-similarities, in agreement with the observed structures in the logarithmic layer. Similar results were obtained by [START_REF] Pujals | A note on optimal transient growth in turbulent channels flows[END_REF], who also found that the optimal growth rate in the outer region varies linearly with a turbulent Reynolds number based on the maximum velocity, the channel half-width and the maximum eddy viscosity. Optimal transient growth in incompressible turbulent boundary layers was studied by Cossu, Pujals & Depardon (2009). Using an analytical mean flow and the eddy viscosity model of [START_REF] Monkewitz | Self-consistent high-Reynolds-number asymptotics for zero-pressure-gradient turbulent boundary layers[END_REF], they showed that a double-peak structure of the optimal energy gain characterizes the organized turbulence dynamics.

In particular, in the near-wall region, streaky structures scale in wall units and are independent of the Reynolds number, while away from the wall they scale in outer units, and the optimal gain varies with the Reynolds number. They also found close similarity of the flow topology with data from direct numerical simulation (DNS) and experiments, thus confirming the effectiveness of optimal transient growth analysis in describing the initial stages of nonlinear processes of turbulence self-sustainment. However, as [START_REF] Cossu | Optimal transient growth and very large-scale structures in turbulent boundary layers[END_REF] pointed out, transient growth analysis may fail to yield quantitatively correct spanwise and streamwise wavelengths of the outer-layer structures.

The focus of the present study is on compressible wall-bounded turbulence. Early studies of compressible wall-bounded flows have led to the widely accepted notion that the main effect of flow compressibility resides in the incurred mean density variations, which can be compensated through suitable transformations to recover the incompressible statistics [START_REF] Morkovin | Effects of compressibility on turbulent flows[END_REF]. In general terms, the essential topology of the coherent structures is also believed to be very similar, and streaks and hairpins have been observed in experiments and DNS of supersonic boundary layers [START_REF] Spina | The physics of supersonic turbulent boundary layers[END_REF][START_REF] Smith | Visualization of the structure of supersonic turbulent boundary layers[END_REF][START_REF] Ringuette | Coherent structures in direct numerical simulation of turbulent boundary layers at Mach 3[END_REF][START_REF] Pirozzoli | Characterization of coherent vortical structures in a supersonic turbulent boundary layer[END_REF], 2010;[START_REF] Elsinga | Three-dimensional vortex organization in a high-Reynolds-number supersonic turbulent boundary layer[END_REF]. Sufficiently high Reynolds numbers to observe the emergence of superstructures and the formation of a genuine logarithmic layer have recently been attained in DNS (Pirozzoli & Bernardini 2011[START_REF] Jiménez | How linear is wall-bounded turbulence?[END_REF]. These studies have shown that the interaction mechanisms occurring between the large-scale motions of the outer layer and the near-wall streaks are similar to those of the incompressible regime and are characterized by phenomena of imprinting and amplitude modulation (Bernardini & Pirozzoli 2011a). The results of [START_REF] Lagha | A numerical study of compressible turbulent boundary layers[END_REF] further confirmed that supersonic and hypersonic turbulent boundary layers have close similarities with incompressible boundary layers, when accounting for the effect of mean density variations. In particular, consistent with the results of [START_REF] Kim | A linear process in wall-bounded turbulent shear flows[END_REF], they showed that if the linear coupling term is removed, near-wall turbulence cannot be sustained, thus indicating that linear transient growth plays an important role also in the compressible flow regime.

The experimental and theoretical evidence for the similarity between incompressible and compressible coherent structures, and their role in the wall-turbulence dynamics, have led us to develop a theoretical analysis of optimal transient growth, by extending to compressible flows the triple decomposition approach originally developed by [START_REF] Reynolds | The mechanics of an organized wave in turbulence shear flow. Part 3. Theoretical models and comparisons with experiments[END_REF] for low-speed flows, whereby the instantaneous flow field is separated into the contributions of mean flow, organized and background disorganized turbulent motions.

The main goal of this work is to highlight the physical mechanisms associated with the formation of coherent structures in compressible boundary layers, thus addressing fundamental questions such as the following: (i) Do compressible turbulent boundary layers exhibit a double peak in the optimal energy growth? (ii) Does the spatial and temporal scale selection mechanism obey inner and outer scaling? (iii) What are the effects associated with flow compressibility on the energy budget of optimal modes and on the vorticity transfer mechanism in the inner and outer regions? (iv) How does the compressibility affect modes that populate the logarithmic region?

The paper is organized as follows. In § 2 we formulate the governing equations based on triple decomposition adapted to compressible turbulent flows, and discuss the closure for turbulent transport of momentum and energy associated with the coherent motions in boundary layers. In § 3 we develop the optimal energy growth analysis for the inner and the outer part of the boundary layer, including the energy budget and the vorticity transfer mechanism, and analyse the influence of the Mach number. Section 4 is devoted to drawing conclusions and perspectives.

Analysis of coherent motions

Governing equations

The Navier-Stokes equations for a compressible perfect gas are

∂ρ ∂t + ∂ ∂x j (ρu j ) = 0, (2.1) ∂ ∂t (ρu i ) + ∂ ∂x j (ρu i u j ) = - ∂ ∂x i p + ∂ ∂x j σ ij , (2.2) ∂ ∂t (ρe) + ∂ ∂x j (ρeu j ) = - ∂ ∂x j ψ j + σ ij ∂u i ∂x j -p ∂u i ∂x i , (2.3)
where ρ, p, u i and e are, respectively, density, pressure, the ith velocity component and internal energy. The viscous stress tensor (σ ij ) and the heat flux vector (ψ j ) components are defined as

σ ij = µ(T) ∂u i ∂x j + ∂u j ∂x i - 2 3 µ(T)δ ij ∂u k ∂x k , ψ j = - c p µ(T) Pr ∂T ∂x j ,          (2.4)
where δ ij , Pr and µ(T) are Kronecker's delta, the Prandtl number and the molecular viscosity (the latter is defined by assuming Sutherland's law). The temperature is defined as T = e/c v , and c p and c v are the specific heat at constant pressure and constant volume, respectively.

To study coherent motions in incompressible turbulent shear flows, [START_REF] Reynolds | The mechanics of an organized wave in turbulence shear flow. Part 3. Theoretical models and comparisons with experiments[END_REF] introduced a triple decomposition of the instantaneous fields into mean flow, organized motions and small-scale random motions. Hence, any variable φ is decomposed as

φ = φ + φ , + φ ,t , (2.5)
where (•), (•) , and (•) ,t represent, respectively, the long-time average, the contribution associated with organized motions and the random contribution due to the small scales. The coherent part is extracted from the instantaneous flow by phase averaging ( • ), and by assuming that the coherent and the random contributions are decorrelated with respect to time and phase, thus having

φ , = φ -φ, φ ,t = φ -φ . (2.6a,b)
Here we apply the incompressible triple decomposition to the density and the pressure fields, and introduce a mass-weighted triple decomposition for the velocity and the temperature fields

φ = φ + φ , + φ ,t , (2.7)
where φ stands for either velocity or temperature. Here φ, φ , and φ ,t represent, respectively, the mass-weighted time average, and the contributions of the coherent and incoherent motions evolving about the density-weighted mean flow. In the framework of mass averaging, in order to be decorrelated with respect to both time and phase, the organized and the random parts are defined as

φ , = ρφ ρ -φ, φ ,t = φ - ρφ ρ . (2.8a,b)
The equations governing the coherent dynamics are then obtained by applying the above triple decompositions to (2.1)-(2.3), after phase averaging and subtracting the time average:

∂ ∂t ρ , + ∂ ∂x k J k = 0, (2.9) ∂ ∂t ( ρ u , i ) + ∂ ∂x k ( ρ u k u , i ) + J k ∂ u i ∂x k = - ∂ ∂x i p , + ∂ ∂x k σ , ik - ∂ ∂x k ( r ik + r t ik ) -r ik + r t ik + H u i , (2.10) ∂ ∂t ( ρ e , ) + ∂ ∂x k ( ρ u k e , ) + J k ∂ e ∂x k = - ∂ψ , k ∂x k - ∂ ∂x k ( ψ T, k + ψ T,t k ) -ψ T, k + ψ T,t k + σ , ik -( r ik + r t ik ) -r ik + r t ik ∂u i ∂x k + σ ik -r ik + r t ik ∂u , i ∂x k -p ∂u , i ∂x i -p , ∂u i ∂x i + H e .
(2.11)

Here

J k = ρ , u k + ρ u , k , σ , ik = µ( T) ∂u , i ∂x k + ∂u , k ∂x i - 2 3 µ( T) ∂u , j ∂x j δ ik , ψ , k = c p Pr µ( T) ∂T , ∂x k , r t ik = ρu ,t i u ,t k , r ik = ρu , i u , k , ψ T,t k = ρe ,t u ,t k , ψ T, k = ρe , u , k ,                     
(2.12)

and H ui and H e account for higher-order forcing terms.

2.2. Closure Given our interest in the linear growth of coherent structures, the contribution of the nonlinear terms is neglected. In order to close (2.9)-(2.11) we use a Newtonian eddy model [START_REF] Reynolds | The mechanics of an organized wave in turbulence shear flow. Part 3. Theoretical models and comparisons with experiments[END_REF] to relate the Reynolds stress oscillation to the strain-rate oscillation via an isotropic eddy viscosity,

( r ik + r t ik ) -r ik + r t ik = -µ t ∂u , i ∂x k + ∂u , k ∂x i + 2 3 µ t ∂u , j ∂x j δ ik .
(2.13)

We also assume that, at the coherent level, the diffusion of internal energy is proportional to the coherent part of the Reynolds stress (i.e. we assume that the Reynolds analogy holds for the organized disturbances), thus obtaining

-( ψ T, k + ψ T,t k ) -ψ T, k + ψ
C f = 2τ w /(ρ ∞ U 2 ∞ ), Ma τ = u τ /(γ R T w ) 1/2 , Re τ = ρ w u τ δ/µ w = δ/δ v , δ is the 99 % boundary layer thickness, Re θ = ρ ∞ U ∞ θ /µ ∞ and θ is the momentum thickness.
The turbulent Prandtl number (Pr t ) has been assumed to be constant and equal to 0.9, also based on DNS results [START_REF] Duan | Direct numerical simulation of hypersonic turbulent boundary layers. Part 3. Effect of Mach number[END_REF]. Under the parallel flow assumption, and in the framework of the Newtonian eddy model, the turbulent viscosity is estimated as the ratio of the Reynolds stress (-ρu v = -ρu , v ,ρu ,t v ,t ) to the mean velocity gradient (µ t = -ρu v /(∂u/∂y)).

Table 1 reports the main properties of the turbulent boundary layers that we have investigated. Flow cases B1-B3 (mainly carried out for validation purposes) correspond to subsonic flow conditions, at various friction Reynolds numbers (Re τ ), whereas flow cases P1-P3 correspond to supersonic flow conditions at relatively low Re τ , at various friction Mach numbers (Ma τ ). The free-stream values are denoted with an ∞ subscript. The mean velocity profiles and the associated eddy viscosity for flow cases B1-B3 have been obtained using the model proposed by [START_REF] Monkewitz | Self-consistent high-Reynolds-number asymptotics for zero-pressure-gradient turbulent boundary layers[END_REF], whereas for flow cases P1-P3 we have used the DNS database of Pirozzoli & Bernardini (2011).

In figure 1 we show the mean streamwise velocity and temperature for the various flow cases, and in figure 2 we show the corresponding turbulent shear stress and turbulent viscosity profiles. It should be noted that the Van Driest transformation (Van Driest 1951) is applied to the mean velocity, u vd = u 0 (ρ/ρ w ) 1/2 du, and the turbulent shear stress is scaled with the density ratio, as customary in the compressible formalism. The data are given in wall units (denoted with a + superscript) based on the friction velocity u τ = (τ w /ρ w ) 1/2 , and the wall viscous length scale δ v = ν w /u τ .

Analysis of transient growth

Significant experimental effort has been devoted to determining the organized motion driven by a harmonic forcing. For instance, to extract the organized structure of the flow in the near wake of a circular cylinder, [START_REF] Cantwell | An experimental study of entrainment and transport in the turbulent near wake of a circular cylinder[END_REF] used an ensemble averaging process carried out on subpopulations, each one being associated with a particular phase interval (i.e. a phase average). In our approach, the organized motion is assumed to be driven by an initial condition. The coherent part of the motion can thus be investigated with an optimal transient growth analysis.

Let q = (ρ , , u , , v , , w , , T , ) be the unknown vector representing the coherent part of the motion. Next, (i) we assume that the mean flow is homogeneous in the streamwise and spanwise directions (ρ(y), u(y), 0, 0, T(y)), and (ii) we introduce a Fourier decomposition for q: q(x, y, z, t) = q (y, t)e i(αx+βz) + c.c.

(2.15) The asterisk denotes density-scaled quantities, reported in wall units. For the subsonic cases the fits given by [START_REF] Monkewitz | Self-consistent high-Reynolds-number asymptotics for zero-pressure-gradient turbulent boundary layers[END_REF] are shown. Refer to table 1 for nomenclature.

Here q is the mode amplitude (in the following, referred to as the Fourier mode or simply as the mode), and α and β are, respectively, the wavenumbers in the x and z directions (α = 2π/λ x , β = 2π/λ z ), and c.c. denotes the complex conjugate. Substituting (2.15) into the linearized equations (2.9)-(2.11), the resulting dynamical system for q is

∂q ∂t = L q , (2.16)
whose derivation is fully reported in appendix A.

Introducing the ratio of the total energy associated with q (including the kinetic energy as well as the internal energy) to the total energy at the initial time, the optimal temporal energy growth over all possible q (t=0) is defined as

G(α, β, t) = sup q (t=0) q q (t=0)
.

(2.17)

Here the norm (•) is based on the inner product defined by [START_REF] Chu | On the energy transfer to small disturbances in fluid flow (Part 1)[END_REF] and [START_REF] Hanifi | Transient growth in compressible boundary layer[END_REF], (2.18) where (•) h denotes the transconjugate, L y is the upper boundary location and D is the diagonal matrix

(q k , q l ) = L y 0 q h k D q l dy,
(T/[ργ M 2 ∞ ], ρ, ρ, ρ, ρ/[γ (γ -1)TM 2 ∞ ]
). In the analysis that follows, we consider the supremum of G over all possible t for a given wavenumber pair (α, β),

G max (α, β) = sup t G(α, β, t), (2.19)
and the supremum of G max over all possible (α, β),

G opt = sup α,β G max (α, β).
(2.20)

Modes attaining maximum transient amplification G max for a given (α, β) pair will hereafter be referred to as optimal modes, and modes attaining G opt will be referred to as absolute optimal modes. We also define t max and t opt as the times at which a given mode attains G max and G opt , respectively.

In the laminar regime, these quantities are usually computed through singular-value decomposition (SVD) of the operator e L t [START_REF] Schmid | Stability and Transition in Shear Flows[END_REF]. This approach is here extended to the turbulent case to solve the dynamical system (2.16). The initial optimal coherent mode is taken as the right singular vector associated with the largest singular value of e L t . The dynamical system (2.16) is discretized by using a spectral collocation method along the normal direction. In order not to impose boundary conditions on the density at the wall and at the far field, we use a staggered mesh (as usually done in the incompressible case), whereby a Gauss-Lobatto grid is employed for the velocity and the temperature components of the modes, whereas a Gauss grid is used for the density component [START_REF] Malik | Numerical methods for hypersonic boundary layer stability[END_REF]). The velocity and the temperature are set to zero at the wall, and assumed to vanish at the upper boundary. Similar boundary conditions are used for laminar adiabatic supersonic boundary layers by [START_REF] Hanifi | Transient growth in compressible boundary layer[END_REF] and [START_REF] Malik | Numerical methods for hypersonic boundary layer stability[END_REF] for both a transient and an asymptotic stability analysis. The SVD of the discrete exponential operator is based on a discrete adjoint operator obtained with a weighted matrix derived from spectral integration [START_REF] Hanifi | Transient growth in compressible boundary layer[END_REF]. Finally, the matrix exponentials are computed using the EXPOKIT library [START_REF] Sidje | EXPOKIT: a software package for computing matrix exponentials[END_REF]. A grid sensitivity analysis has been carried out, but for the sake of conciseness we only present the results obtained with the finest grid having 500 collocation points in the wall-normal direction.

The mean flow and the turbulent viscosity are obtained by interpolation of DNS statistics (Bernardini & Pirozzoli 2011b;Pirozzoli & Bernardini 2011), using a cubic spline. The optimal energy growth associated with the dynamical system (2.16) has been computed for a wide range of streamwise and spanwise wavenumbers. The code has been validated for the linear transient growth of a laminar supersonic adiabatic boundary layer, obtaining favourable comparison with the results of [START_REF] Hanifi | Transient growth in compressible boundary layer[END_REF] (see appendix C). The code has also been validated for a turbulent subsonic boundary layer against the incompressible results of [START_REF] Cossu | Optimal transient growth and very large-scale structures in turbulent boundary layers[END_REF] (the results will be shown later on). Refer to table 1 for nomenclature.

3. Optimal transient growth

Optimal energy gain

The main goal here is to understand the influence of compressibility and mode orientation on the optimal energy growth, by considering several wavevectors (α, β).

In figure 3 we show the maximum total energy amplification (attained at the optimal time) of infinitely long structures (α = 0) as a function of the spanwise wavelength, in inner (figure 3a) and outer (figure 3b) units. We find that G max exhibits two peaks, associated with maximum amplification in the inner and the outer layer, as also observed in turbulent incompressible boundary layers [START_REF] Cossu | Optimal transient growth and very large-scale structures in turbulent boundary layers[END_REF]) and in channels (del Alamo & Jiménez 2006;[START_REF] Pujals | A note on optimal transient growth in turbulent channels flows[END_REF]. The inner peak of G max scales in inner units, and corresponds to a spanwise wavelength λ + z ≈ 81 (λ + z = λ z /δ v ). The outer peak scales in outer units, and corresponds to a spanwise wavelength λ z ≈ 8.5δ, where δ is the 99 % boundary layer thickness. These values are consistent with those obtained by [START_REF] Cossu | Optimal transient growth and very large-scale structures in turbulent boundary layers[END_REF] for incompressible boundary layers. The distance from the wall (y max ) at which the maximum gain in total energy occurs also confirms the occurrence of the double scaling, as shown in figure 4, where it is reported as a function of the spanwise wavelength. In particular, the inner peak occurs at y + max ≈ 10, whereas the outer peak is found at y max /δ ≈ 0.8δ. Overall, it appears that scale separation is driven by variation of Re τ , rather than by variation of the Mach number. In the intermediate range of spanwise wavenumbers, we also observe a region where the maximum growth is nearly constant. This property has also been observed in optimal transient growth calculations of channel flows and boundary layers (del Alamo & Jiménez 2006;[START_REF] Cossu | Optimal transient growth and very large-scale structures in turbulent boundary layers[END_REF][START_REF] Pujals | A note on optimal transient growth in turbulent channels flows[END_REF]. Recently, Hwang & Cossu (2010a) have shown that this region is associated with the amplification of log-layer structures.

Inner-layer optimal modes

We now focus on the near-wall peak of G max , which is shown in figure 5 as a function of the spanwise wavelength for several values of the streamwise wavelength. As noted above, the absolute optimal growth is found to be associated with infinitely elongated coherent structures having a spanwise size of approximately 80 wall units. Figure 5 further suggests that the most amplified inner-layer modes have proportional values of λ x and λ z , and hence the longer optimal modes are also wider. In figure 6 we show the velocity and temperature maps corresponding to the absolute optimal mode in the cross-stream plane. The arrows in the figure represent the velocity vectors at the initial time. As in the incompressible regime, a peculiar lift-up effect [START_REF] Landahl | A note on an algebraic instability of inviscid parallel shear flows[END_REF]) is observed, which is seen to be responsible for the optimal growth. Figure 6 also shows that the temperature pattern is strictly related to the velocity pattern, and positive (respectively, negative) temperature fluctuations strongly correlate with low-speed (respectively, high-speed) streaks. This is in agreement with the well-known tendency of velocity and temperature fluctuations to be negatively correlated in turbulent supersonic boundary layers Pirozzoli & Bernardini (2011). The temperature here seems to have a relatively passive role, while the streamwise vortices at the initial time act to redistribute momentum and temperature through sweeps and ejections.

The maps of G max and t max for two-dimensional optimal modes are shown in figure 7 for all the flow cases. The maximum transient amplification (figure 7a) is found to be more affected by Mach number variation rather than by Reynolds number variation, and G opt is found to vary from 2.8 at Ma ∞ = 0.2 to 3.2 at Ma ∞ = 4. However, the spanwise length scale of the optimal mode is weakly affected by compressibility, and λ + z ≈ 81.5 in all cases, as also found in the incompressible case [START_REF] Cossu | Optimal transient growth and very large-scale structures in turbulent boundary layers[END_REF]). It is worth pointing out that the length scale recovered with the present linear model is also consistent with the size of the streaks found in DNS of supersonic turbulent boundary layers [START_REF] Pirozzoli | Probing high-Reynolds-number effectis in numerical boundary layers[END_REF]. Figure 7(b) shows that the typical time scale of the structures attaining the maximum amplification is nearly independent of the Mach number, and it varies linearly with the spanwise wavelength.

Figure 8(a) shows iso-lines of the wall distance at which G max is attained in the P1 flow case, and figure 8(b) shows the corresponding optimal times t max as a function of the streamwise and spanwise wavenumbers. It is found that both y max and t max increase as α and β decrease. We recall that in wall-bounded turbulent shear flows coherent structures are disrupted on a time scale of the order of the eddy turnover time τ = k/ε, where k = ρ u k u k /2 is the turbulent kinetic energy and ε = µ(∂u i /∂x j )(∂u i /∂x j ) is its dissipation rate [START_REF] Lee | Structure of turbulence at high shear rate[END_REF][START_REF] Butler | Optimal perturbations and streak spacing in wall-bounded turbulent shear flow[END_REF]. Hence, only . Inner-layer modes: eddy turnover time as a function of wall distance for flow case P1. The dashed line denotes the optimal time of infinitely elongated modes (t max (0, β)) at the corresponding distances (y max (0, β)), for λ + z from 40 to 200. The asterisk denotes the absolute optimal mode. coherent modes having temporal scale less than the eddy turnover time are physically acceptable solutions. In figure 9 we then report the distribution of τ as a function of the wall distance, together with the optimal time of infinitely elongated modes (t max (0, β)), which, based on the previous considerations, have the longer lifetimes, at the corresponding distances (y max (0, β)). The eddy turnover time is estimated from the DNS database of Pirozzoli & Bernardini (2011). Figure 9 shows that t max does not exceed the eddy turnover time for the most amplified optimal modes that we have analysed. This leads to the conclusion that the coherent modes retrieved through the transient growth analysis are physically relevant.

The contributions to the growth of the absolute optimal mode are separately shown in figure 10 reported, namely

u * = ρ ρ w u , v * = ρ ρ w v . (3.1a,b)
In figure 10(d) the unscaled velocity is shown for comparison, limited to flow case P3. In agreement with Morkovin's hypothesis, figure 10(d) shows the similarity of the kinematic components of the optimal mode, and better collapse is achieved through density scaling. On the other hand, temperature exhibits a peak that increases with the Mach number. In this respect we recall that, based on the chosen energy norm given in (2.18), G incorporates the contribution of both the kinetic energy and the internal energy. Hence, temperature growth is the main cause for the overall increase of the optimal gain with the Mach number observed in figure 7.

To quantify the kinetic energy contribution to the energy gain, we introduce the following metric:

G k max (α, β) = L y 0 ρ u h u dy (t=t max ) L y 0 ρ u h u dy (t=0) , (3.2)
where u = (u , v , w ). The distribution of sup β G max (α, β) and its kinetic counterpart is reported in figure 11 as a function of the streamwise wavenumber. As one would expect, we note that kinetic energy contributes most to G max , and its increase with respect to the incompressible case is mainly associated with the amplification of the internal energy. On the other hand, the kinetic energy contribution is weakly affected by the Mach number, and the curves nearly collapse for Mach number up to 3.

To further elucidate the mechanisms of energy transfer between the kinetic and the internal energy, we have analysed the corresponding energy budgets for the absolute optimal mode. Let E be the integrated total energy in the wall-normal direction

E = E k + E in , (3.3)
where E k and E in are the kinetic energy and the internal energy, respectively. The evolution of E is described by [START_REF] Malik | Nonmodal energy growth and optimal perturbations in compressible plane Couette flow[END_REF])

dE dt = L y 0 q h DL q dy + c.c., (3.4)
where the energy is normalized by the total energy associated with the initial optimal coherent mode. The budget of the kinetic energy is written in the form

dE k dt = P k -D k v -D k t , (3.5)
where P k , D k v and D k t represent, respectively, the production due to mean shear, the dissipation associated with molecular viscosity and the turbulent dissipation at the small scales (see appendix B for their definition). The contributions to (3.5) are shown in figure 12. The evolution of the total energy is found to be mainly driven by its kinetic part, and in particular by production due to mean shear, P k . Furthermore, the time rate of change of the energy of the optimal inner mode attains its maximum on a time scale corresponding to the peak production. One may note that the optimal gain (for which dE /dt = 0) corresponds to an equilibrium state where dissipation is balanced by production, and as previously noted this mechanism is weakly affected by the Mach number. We also note that the dissipation caused by molecular diffusion and caused by turbulent coherent motions are of the same order of magnitude. It is then evident that the latter contribution cannot be neglected when carrying out the linear stability analysis of a turbulent mean flow, thus corroborating the statements of del Alamo & Jiménez ( 2006 case. Figure 13 shows the wall-normal profile of the local kinetic energy production (p k (y) = -ρ u h v du/dy), at the time at which P k attains its maximum. It is found that p k attains a peak at y + ≈ 10, which well matches the innermost layer distribution found in DNS (Pirozzoli & Bernardini 2011), upon suitable rescaling, thus supporting the important role played by coherent motions in the establishment of the turbulence kinetic energy peak. Iso-contour lines of the streamwise (ω x ), wall-normal (ω y ) and spanwise (ω z ) vorticity components of the absolute optimal mode and an oblique optimal mode sup β G max (0.08/δ v , β) are shown in figure 14, in a plane orthogonal to the associated wavevector. Vortical structures associated with the absolute optimal mode (figure 14a-c) are two-dimensional, and are stretched in the streamwise direction, undergoing a lift-up mechanism whereby vorticity is transferred from the streamwise to the wall-normal direction. Vortical structures associated with a three-dimensional optimal mode are shown in figure 14(d-f ) for λ + x = 78, λ + z = 55, which corresponds (see figure 5) to sup β G max (0.08/δ v , β). This particular three-dimensional mode (similarly to many others, not shown) is dominated by tilting in the shear direction, according to the Orr mechanism. The linear model thus incorporates the two main mechanisms associated with the intensification of vortical structures in turbulent shear flows. Similar observations also hold at higher Mach number (the figures are not shown for brevity). FIGURE 14. Inner-layer modes, flow case P1: vorticity contours in planes normal to the wavevector for (a-c) the absolute optimal (two-dimensional) mode and (d-f ) the optimal mode having λ + x = 78, λ + z = 55. Axis label s denotes the wall-parallel coordinate normal to the wavevector. The streamwise, wall-normal and spanwise vorticity components are shown in panels (a,d), (b,e) and (c,f ), respectively. Solid lines correspond to the initial time, and dashed lines to the optimal time.

To characterize the influence of compressibility on the lift-up mechanism, the time evolution of the integrated enstrophy ratios [START_REF] Guegan | Optimal disturbances in swept Hiemenz flow[END_REF][START_REF] Alizard | Transient growth in a right-angled streamwise corner[END_REF]), r = Ω / i Ω i , where Ω i = D ω 2 i (t) dD, are analysed for the absolute optimal mode. Figure 15 confirms that the initial optimal vorticity field is mainly oriented in the streamwise direction, whereas the wall-normal and the spanwise vorticity components are amplified under the action of the mean shear. Figure 15 also shows the collapse of all the curves, thus confirming that the vorticity transfer mechanism is weakly affected by the Mach number. FIGURE 15. Inner-layer modes: time evolution of the enstrophy ratios for the absolute optimal mode. Refer to table 1 for nomenclature.

Outer-layer optimal modes

To characterize the properties of the most amplified outer-layer modes, we have also computed the singular vectors associated with the maximum kinetic energy amplification G k max (α, β), for a wide range of streamwise and spanwise wavelengths. Note that the kinetic energy contribution is only retained to compare the supersonic cases with the subsonic ones. In figure 16 we show the kinetic energy contribution to the maximum energy gain as a function of the spanwise wavelength for different streamwise wavelengths, and mark with asterisks the values of sup β G k max (α, β). Figure 16 indicates that the most amplified outer-layer structures correspond to a particular combination of (λ x , λ z ) that weakly depends on the Mach number. As for the inner-layer modes, it is found that the absolute optimal modes correspond to infinitely elongated streamwise structures, whose typical spanwise size is approximately 8δ.

In figure 17 we show the velocity and temperature maps corresponding to the outer optimal mode in the cross-stream plane, for flow case P1. The arrows in the figure represent the velocity vectors at the initial time. As in the incompressible regime, the optimal initial coherent structures are found to be associated with streamwise vortices that lead to streamwise streaks. Strong negative correlation of u , and T , is found, as in the inner layer. Similar results, not shown, are also obtained for the P2 and P3 flow cases. The contributions to the growth of the absolute optimal mode are separately shown in figure 18, at the initial time (figure 18b) and at the optimal time (figure 18a,c), where density-scaled velocity components are reported for the supersonic cases. As in the inner layer, and in agreement with Morkovin's hypothesis, similarity still holds for the kinematic components of the optimal outer mode, and good collapse is achieved through density scaling.

Regarding the energy budgets (reported in figure 19), we find that, on a short time scale (t/t opt ≈ 0.2-0.4), the coherent dynamics is strongly affected by the internal energy. When the energy attains its optimal gain (t/t opt = 1), an equilibrium state is reached, and the production of kinetic energy balances its dissipation, as also observed for the inner peak. When this state is reached, the role played by the internal energy becomes marginal. Turbulent dissipation is found to be much larger in this case than the molecular one, as expected in the outer layer.

To verify the physical relevance of the optimal outer coherent modes on the innerlayer dynamics, in figure 20 we show their amplitude as a function of the inner-scaled wall distance. As also noted by [START_REF] Pujals | A note on optimal transient growth in turbulent channels flows[END_REF] for incompressible channel flows, we find collapse of the curves up to y + ≈ 100 for all the flow cases. Furthermore, the velocity and the temperature amplitudes vary linearly up to y + ≈ 10, and the amplitude of the mode is found to be proportional of the mean velocity. A similar signature was observed by [START_REF] Cossu | Optimal transient growth and very large-scale structures in turbulent boundary layers[END_REF].

Regarding the structural properties of outer-layer eddies, DNS and experiments [START_REF] Ganapathisubramani | Large-scale motions in a supersonic turbulent boundary layer[END_REF]Pirozzoli & Bernardini 2011) show that the organized structures have a rather universal behaviour, in that their characteristic scales and orientations are weakly affected by compressibility. In particular, in agreement with low-speed turbulent boundary layers ( at Ma ∞ = 2 show that the typical size of outer-layer eddies is approximately 1δ in the streamwise direction, and approximately 0.5δ in the spanwise direction, and their typical streamwise inclination angle is ≈ 12 • -14 • .

To quantitatively characterize the orientation of outer-layer modes educed from transient growth analysis, we consider their projected angle in the x-y plane, estimated from the wall-normal position ( y max ) where the streamwise velocity is maximum at the optimal time, and half the streamwise wavelength, θ xy = tan -1 (2y max /λ x ).

Figure 21 shows θ xy as a function of the spanwise wavelength, for various streamwise wavelengths. In the incompressible case the inclination angle is not affected by the Reynolds number, and collapse of flow cases B1-B3 is observed in a wide range of wavenumbers. The results of the P1 flow case are also in excellent agreement with the subsonic one, whereas deviations are observed for flow cases P2 and P3.

Figure 21(a,b) further shows that the inclination angle is a decreasing functions of λ x and λ z . Hence, larger structures have a tendency to lean towards the wall, whereas smaller structures have a tendency to be tilted with respect to it. As the Mach number increases, the inclination angle increases, and the location where the velocity is maximum shifts away from the wall, meaning that organized structures have greater tendency to be tilted and lifted up from the wall. These results are consistent with the DNS data of [START_REF] Duan | Direct numerical simulation of hypersonic turbulent boundary layers. Part 3. Effect of Mach number[END_REF], who explored the effect of the Mach number on turbulent boundary layers up to Ma ∞ = 12. For flow case P1, we find that the coherent structures associated with G k max (2π/λ x , 2π/λ z ), λ x = 2.3δ and λ z = 1.5δ, have an inclination angle of θ xy ≈ 15 • . This particular optimal mode is then compatible with the DNS findings (Pirozzoli & Bernardini 2011). The streamwise velocity maps for this mode are shown in figure 22 at various stages of its evolution, FIGURE 20. Outer-layer modes: normalized amplitude of (a) streamwise velocity and (b) temperature for the absolute optimal mode based on the reference subsonic case B1. The solid line represents the mean streamwise velocity. Refer to table 1 for nomenclature.

from t = 0 to t max . Notably, the distance at which the velocity is maximum is less than for infinitely elongated streaks (see figure 18a). This is consistent with the experiments of [START_REF] Ganapathisubramani | Large-scale motions in a supersonic turbulent boundary layer[END_REF], who observed an increasing trend of the streamwise length scale with the wall distance. All the subsonic cases are collapsing. Refer to table 1 for In figure 23 we show the time evolution of the enstrophy ratios for the absolute optimal outer-layer mode and for the three-dimensional 'DNS-compatible' mode, having λ x = 2.3δ and λ z = 1.5δ. Figure 23(a) shows that a similar vorticity transfer mechanism holds as in the inner layer. For the three-dimensional modes (see figure 23(b) for the P1 flow case), in a first stage (shaded in grey), the mode is reoriented along the shear direction under the action of the Orr mechanism. Then, the mode undergoes a lift-up mechanism, whereby streamwise vorticity is primarily reoriented to the wall-normal direction. Hence, the most relevant structures compatible with experiments and DNS appear to combine the effects of the Orr and of the lift-up mechanisms [START_REF] Farrell | Optimal excitation of three-dimensional perturbations in viscous constant shear flow[END_REF]. In figure 24 we show the streamwise velocity contours at various times in panels (a-c) and the time evolution of the enstrophy ratios in panel (d) for the optimal three-dimensional mode reaching sup β G k max (2π/λ x , 2π/λ z ) for λ x = 2.3δ. Figures 22 and24 show that the optimal structure and the most compatible one exhibit a similar pattern. However, the optimal structure has an inclination angle that is nearly twice the value observed in experiments and in DNS, and experiences Orr/lift-up mechanisms on a longer time scale (figure 24d). A similar behaviour is recovered at all Mach numbers.

0 1 2 0 1 2 0 1 2 1 2 (a) ( b) ( c)

Log-layer optimal modes

The amplification curves in figure 3 showed the existence of a log-layer region between the inner-and outer-layer peaks, where the total energy gain attains a near plateau, and which is the subject of the present section. In the past a lot of interest has devoted to the logarithmic part of the wall layer. According to the classical scenario [START_REF] Townsend | The Structure of Turbulent Shear Flow[END_REF]), this region is populated by a host of self-similar attached eddies, whose size is proportional to the wall distance. One of the consequences of this assumption is that the variances of the wall-parallel velocity fluctuations should scale logarithmically with the wall distance. Townsend's attached-eddy hypothesis has received support from the experiments of Tomkins [START_REF] Tomkins | Spanwise structure and scale growth in turbulent boundary layers[END_REF], who noted that the spanwise length scale of eddies in the log layer varies linearly with the wall distance, and logarithmic decay of the velocity variances has been recovered in a number of studies (see [START_REF] Marusic | On the logarithmic region in wall turbulence[END_REF] for an overview). The turbulence self-sustainment processes in the log layer were studied through filtered DNS in minimal boxes by [START_REF] Hwang | Self-sustained processes in the logarithmic layer of turbulent channel flows[END_REF]. They found that the characteristic spanwise scales associated with the surviving motions follow a linear scaling with the wall distance. More recently, [START_REF] Moarref | Model-based scaling of the streamwise energy density in high-Reynolds-number turbulent channels[END_REF] confirmed the important role of linear processes in establishing a self-similar hierarchy of coherent eddies in the log layer. Their results further supported the hypothesis that linear amplification of optimal modes in the logarithmic layer leads to streaky motions, which may regenerate after bursting, and which exhibit a universal behaviour, when scaled by their spanwise extent. Modes belonging to the intermediate region are hereafter referred to as log-layer optimal modes.

In order to verify if supersonic boundary layers also exhibit geometrical similarity, we study the dependence of the log-layer optimal modes on the spanwise wavelength, limiting our analysis to infinitely elongated modes (α = 0) that are the most amplified.

The kinetic energy amplification curves are shown in figure 25 as a function of the spanwise wavenumber, in a linear scale. As previously noticed in § 3.2, the G k max data fall on a nearly universal curve close to the inner peak, and tend to form a plateau as the Reynolds number increases. Figure 25 also shows a slight shift of the minimum value for G k max when increasing the Mach number from 3 to 4. To characterize the vertical size of the optimal log-layer modes, we define a typical length scale (Λ y ) as the wall distance at which the cumulative modal kinetic energy is 80 % of the total, i.e. similarity of the log-layer modes, in agreement with Townsend's attached-eddy model (Townsend 1976). Finally, figure 26(c) indicates that the optimal time for these modes also varies linearly with λ z . This finding is consistent with the claim of del Alamo et al. (2006) that the lifetime of the log-layer eddies is proportional to their size. Finally, figure 26(a,c) shows that, for subsonic Mach numbers, modes larger than λ z ≈ 0.43δ (corresponding to λ + z = 950, 1700 and 2850 for B1, B2 and B3, respectively) deviate from the universal behaviour and they are mostly affected by the outer motion.

Similar data are shown in figure 27 for the supersonic flow cases. In fact, figure 27(a) confirms that the vertical length scale of the optimal modes scales linearly with the spanwise wavelength also in supersonic boundary layers. The modal velocity and temperature distributions, shown in figure 27(b), do exhibit geometrical similarity, thus suggesting that Townsend's attached-eddy model also applies to supersonic boundary layers. The rescaled optimal modes also bear strong similarities with the buffer-layer modes, indicating that kinetic energy grows through a similar lift-up mechanism. The optimal time scale, shown in figure 27(c), also exhibits a behaviour similar to the subsonic case, regardless of the Mach number. As found in subsonic cases, Λ y and t max deviate from the linear behaviour for a spanwise size λ z ≈ 0.45δ (λ + z ≈ 500) for P1 and λ z ≈ 0.69δ (λ + z ≈ 350) for P2 and P3. The observed increase in the vertical length scale seen in figure 27 can help to explain the increased growth in figure 25. Indeed, we can define an effective turbulent Reynolds number as

Re T = Λ 2 y ν T t max = Λ y u τ t max = Λ + y t + max , (3.7)
having assumed ν T ∼ u τ Λ y . From the data shown in figure 27(a,c) it follows that Re T is approximately constant (as expected) in the log layer. In particular, figure 27(d) shows that the effective turbulent Reynolds number varies from approximately 1 for subsonic flow cases to ≈1.6 for P3.

From the above discussion, the change in the characteristic spanwise scale associated with the deviation from a linear behaviour may be attributed mainly to a Reynolds-number effect, owing to lack of scale separation in P2 and P3. However, the slight increase of Λ y in P3 as compared to P2 (see figure 27a) is probably associated with a Mach-number effect, since the friction Reynolds number is approximately the same for the two flow cases. Trying to further unravel Reynolds-and Mach-number effects, in figure 28 we show optimal log-layer modes for the P1-P3 flow cases, with energy mainly concentrated in the overlap layer. Figure 28(a) shows a similar shape of the mode profiles in the inner and the outer layers, with differences concentrated within the alleged overlap layer. Differences between P2 and P3 are found to be very nearly compensated through the Howarth-Dorodnitsyn transformation: ỹ = y 0 (ρ/ρ w ) dy [START_REF] He | An asymptotic two-layer model for supersonic turbulent boundary layers[END_REF]. As shown in figure 28(b), the mode shapes in the transformed vertical coordinate are nearly the same for flow cases P2 and P3. Hence, for given friction Reynolds number, mean density gradients seem to incorporate well the (small) effects of Mach-number variations. However, differences with respect to the P1 flow case are still present, which may be a consequence of lack of scale separation between the inner and outer layers in P2 and P3.

It is clear that, based on the present data, making definite statements regarding Reynolds-and Mach-number effects on log-layer modes is quite difficult, mainly given the limited range of available Reynolds numbers, and the difficulty of changing the two parameters separately. It is also far from clear which Reynolds number should be kept constant to isolate 'compressibility' from 'Reynolds-number' effects. Hence, a much more complete dataset would be needed to further strengthen or refute our findings.

Conclusions

The evolution of coherent structures in compressible turbulent boundary layers with free-stream Mach number from 0.2 to 4 has been analysed. The approach relies on optimal transient growth analysis whereby the linearized evolutionary equations of the organized structures are obtained by extending to compressible flows the triple decomposition procedure developed by [START_REF] Reynolds | The mechanics of an organized wave in turbulence shear flow. Part 3. Theoretical models and comparisons with experiments[END_REF] for strictly incompressible flow. A linear dynamical system results for the organized (coherent) motions, which is closed by prescribing the mean field, and introducing an isotropic eddy viscosity model to describe the turbulent transport of momentum and internal energy by the large scales. For closure purposes the DNS database of Pirozzoli & Bernardini (2011) has been used for the supersonic flow cases, whereas the explicit closures of [START_REF] Monkewitz | Self-consistent high-Reynolds-number asymptotics for zero-pressure-gradient turbulent boundary layers[END_REF] have been used for the subsonic cases, here mainly considered for validation purposes. Although closure models based on the assumption of isotropy may suffer from some shortcomings, they nevertheless can also serve as a basis to compare our results with previous incompressible analyses [START_REF] Cossu | Optimal transient growth and very large-scale structures in turbulent boundary layers[END_REF].

An optimal transient growth analysis has been developed based on the definition of an energy norm containing the contributions of both kinetic energy and internal energy. Optimal transient modes have been calculated for a wide range of streamwise and spanwise wavenumbers, with the main finding that absolute optimal modes are always two-dimensional. Optimal modes naturally split themselves into two families, scaling in inner and outer units, respectively, as previously also found in the incompressible regime [START_REF] Cossu | Optimal transient growth and very large-scale structures in turbulent boundary layers[END_REF], and consistent with the two-layer nature of turbulent wall layers. The scale separation effect is here found to be mainly dependent on the friction Reynolds number, rather than on the Mach number. In the intermediate range of spanwise wavenumbers, the maximum growth attains a plateau at all Mach numbers, as also found in incompressible channel flow (del Alamo & Jiménez 2006). In this region, the optimal structures are mainly localized in the log layer.

Regarding the inner-layer modes, near collapse of the typical spanwise size of the most amplified modes is observed across the Mach-number range, with an absolute peak occurring at λ + z = 81.5, which matches well the value expected from the near-wall streaks, thus supporting their universality. A similar pattern is observed for the temperature component of the transient growing modes, which is found to be negatively correlated with the streamwise velocity component. The lift-up mechanism underlying the formation of velocity/temperature streaks is explained in terms of vorticity transfer from the streamwise to the spanwise component. As predicted by the classical Morkovin hypothesis, we find that amplitude changes with the Mach number can be effectively compensated through the use of density scaling. Furthermore, the variation of the optimal growth rate with the Mach number is found to be mainly an additive effect linked to increased aerodynamic heating. Hence, no major dynamical effect of compressibility on the mechanisms of transient growth is observed, at least at the Mach numbers under scrutiny. This behaviour is consistent with observations based on DNS (Bernardini & Pirozzoli 2011b;Pirozzoli & Bernardini 2011).

Amplified modes that populate the outer part of the wall layer have also been identified. Absolute optimal growth is attained by coherent structures that are infinitely elongated in the streamwise direction, and having spanwise size λ z ≈ 8.5δ. This mode is found to be a robust feature, being very weakly dependent on the Reynolds number and on the Mach number. As for the inner peak, a lift-up mechanism is responsible for the growth of the absolute optimal mode, with a similar transfer of vorticity from the streamwise to the spanwise components. All these features agree well with those observed in the incompressible limit [START_REF] Cossu | Optimal transient growth and very large-scale structures in turbulent boundary layers[END_REF]. As in the inner layer, a passive role of temperature is highlighted, and increasing internal energy contribution to the total energy budget is observed. Besides the absolute (two-dimensional) optimal modes, we have also characterized the structure of the three-dimensional modes corresponding to given streamwise and spanwise spatial scales. The general conclusion is that smaller coherent structures have higher inclination angles with respect to the wall, as compared to larger eddies.

Regarding the most amplified log-layer optimal modes, we show that they are streamwise uniform, and exhibit a self-similar behaviour. In particular, both vertical size and characteristic time scale vary linearly with the spanwise wavelength. This observation leads to the conclusion that the idea of geometrically similar attached eddies proposed by [START_REF] Townsend | The Structure of Turbulent Shear Flow[END_REF] also applies to supersonic boundary layers. Furthermore, a universal behaviour of optimal log-layer modes is suggested when accounting for the effects of mean density variations. However, further investigations at higher Mach number and higher Reynolds number are needed in order to draw firm conclusions. The fundamental issue that may be raised regarding the present study is whether the optimal growing modes educed through the linear analysis have physical significance. Regarding the inner-layer modes, it is far from questionable that the absolute optimal modes have strong similarity with coherent structures observed in experiments and DNS, which are in fact mainly elongated in the streamwise direction, and have a spanwise size of the order of 100 wall units. Much less clear is the relevance of the observed outer-layer transient growing modes. In fact, large discrepancies exist between the absolute optimal outer modes and the superstructures that populate the outer part of turbulent boundary layers. For instance, the amplitude of the outer optimal modes is reached at the edge of the boundary layer, whereas very-large-scale outer eddies are mainly energetic at a distance of approximately 0.3δ from the wall [START_REF] Hutchins | Evidence of very long meandering features in the logarithmic region of turbulent boundary layers[END_REF]. Furthermore, both the spanwise and the streamwise size of the absolute optimal outer modes is much larger than for eddies educed in DNS and experiments. However, a host of transient growing modes exist other than the absolute optimal ones, which has not been analysed in detail before. In particular, for boundary layers at Ma ∞ = 2 we identify optimal modes having λ x ≈ 2.3δ and λ z ≈ 1.5δ (close to the typically quoted eddy integral length scales), which have a typical inclination angle of approximatively θ xy ≈ 15 • with respect to the streamwise direction, and which also exhibit anticorrelation between the streamwise and the temperature component.

Our findings seem to support the view that very-large-scale structures in supersonic turbulent boundary layers are the result of an autonomous self-sustained process associated with the amplification and regeneration of optimal modes, rather than the agglomeration of smaller structures (Hwang & Cossu 2010b). However, our analysis admittedly fails to describe the underlying selection mechanism of the most representative mode. In fact, even though modes resembling the 'real' ones are recovered among the many suboptimal three-dimensional modes, the most amplified outer mode is still much different from the typical structures observed in turbulent wall layers. In our understanding, this is an inherent limitation of linear analysis, as structures that are selected in physical reality are the eventual result of nonlinear effects and/or secondary instability processes, which are necessarily disregarded by modal analysis.

Regarding the effects of the free-stream Mach number, the general conclusion that can be drawn from the present data is that they are typically small and confined to increasing the amplitude of the temperature fluctuations, while the kinematic field is virtually unaffected. In particular, compressibility effects are seen to be mainly a consequence of changes in the mean density. In this sense, the present results provide further confirmation of the validity of Morkovin's hypothesis, at least up to Ma ∞ = 4, which is the upper limit investigated here. Notably, this result is at odds with the classical results of linear stability analysis of laminar boundary layers, which shows distinctly different nature of supersonic boundary layers with respect to their lowspeed counterpart, also as far as the transition process is concerned [START_REF] Reshotko | Boundary-layer stability and transition[END_REF]). Studies of transient growth at yet higher Mach number are of course desirable to establish the possible onset of genuine compressibility effects.

We believe that further verification of the robustness of optimal modes computed here can be obtained by considering more accurate modelling of the terms requiring closure, and in this sense full usage of the DNS data may be considered in future work. Specifically, some effort is currently in progress regarding the analysis of different terms involved in the evolution equations for the oscillating background Reynolds stress. We believe that, in perspective, the application of the present structural approach to more complex flows, such as supersonic inlets and shock wave-turbulent boundary layer interactions, could be considered to develop low-order models that accurately describe the organized part of the motion, thus opening interesting avenues to flow control.

  FIGURE 1. (a)The Van Driest-transformed mean velocity and (b) the Favre-averaged mean temperature. For the subsonic cases the mean velocity profiles proposed by[START_REF] Monkewitz | Self-consistent high-Reynolds-number asymptotics for zero-pressure-gradient turbulent boundary layers[END_REF] are shown. Refer to table 1 for nomenclature.

FIGURE 3 .

 3 FIGURE 3. Maximum total energy amplification (G max ) as a function of the spanwise wavelength in (a) inner and (b) outer units, for infinitely elongated structures (α = 0). Refer to table 1 for nomenclature.

FIGURE 4 .

 4 FIGURE 4. Wall-normal position (y max ) of the peak of G max as a function of the spanwise wavelength in (a)inner and (b) outer units, for infinitely elongated structures (α = 0). Refer to table 1 for nomenclature.

  FIGURE 6. Inner-layer modes, P1 flow case: contours of (a) streamwise velocity and (b) temperature in the cross-stream plane, with superimposed cross-stream velocity vectors, for the absolute optimal mode (α = 0, λ + z = 81).

FIGURE 7 .

 7 FIGURE 7. Inner-layer modes: (a) maximum energy gain and (b) corresponding time t max as functions of the spanwise wavelength for α = 0. Refer to table 1 for nomenclature.

FIGURE 8 .

 8 FIGURE 8. Inner-layer modes, flow case P1: maps of (a) wall distance and (b) time t max at which amplification is maximum.

FIGURE 10 .

 10 FIGURE 10. Inner-layer modes, amplitude of the components of the absolute optimal modes, scaled with respect to their initial value: (a) u * at t opt ; (b) v * at the initial time; (c) T at t opt ; and (d) u * at t opt , limited to flow case P3 (open symbols indicate that density scaling is not used).

FIGURE 12 .

 12 FIGURE 12. Inner-layer modes: energy budget of the optimal mode for (a) flow case B1, (b) flow case P1, (c) flow case P2 and (d) flow case P3. Nomenclature: solid, dE k /dt; long dashed, P k ; short dashed, D k v ; dot-dashed,D k t ; circles, dE /dt; triangles, P k + D k v + D k t

FIGURE 13 .

 13 FIGURE 13. Inner-layer modes, distribution of the local kinetic energy production, p k (y), normalized by its initial value, at the time at which P k attains its maximum. The DNS data ofPirozzoli & Bernardini (2011) are shown with a solid line, in arbitrary units. Refer to table 1 for nomenclature.

FIGURE 16 .

 16 FIGURE 16. Outer-layer modes: optimal gain G k max as a function of the spanwise wavelength for different streamwise wavelengths for (a) flow case B1 and (b) flow case P1. The asterisks denote the positions of sup β G k max (α, β).

  FIGURE 17. Outer-layer modes, P1 flow case: contours of (a) streamwise velocity and (b) temperature in the cross-stream plane, with superimposed cross-stream velocity vectors, for the absolute optimal mode (α = 0, λ z = 8.5δ).

FIGURE 18 .

 18 FIGURE 18. Outer-layer modes, amplitude of the components of the absolute optimal mode, scaled with respect to their initial value: (a) u * at t opt ; (b) v * at the initial time; and (c) T at t opt .

FIGURE 19 .

 19 FIGURE 19. Outer-layer modes: energy budget for the optimal mode for (a) flow case B1, (b) flow case P1, (c) flow case P2 and (d) flow case P3. Nomenclature: solid, dE k /dt; long dashed, P k ; short dashed, D k v ; dot-dashed, D k t ; circles, dE /dt; triangles, P k + D k v + D k t

FIGURE 21 .

 21 FIGURE 21. Outer-layer modes: (a) inclination angle of modes attaining G k max , shown as a function of the spanwise wavelength for various streamwise wavelengths; (b) inclination angles as a function of λ z for modes attaining sup β G k max (α, β).All the subsonic cases are collapsing. Refer to table1 for

FIGURE 22 .FIGURE 23 .

 2223 FIGURE 22. Outer-layer modes, flow case P1: streamwise velocity contours for the mode associated with λ x = 2.3δ and λ z = 1.5δ (having θ xy = 15 • ), at (a) t/t max = 0, (b) t/t max = 0.27 and (c) t/t max = 1. Solid lines represent positive values and dashed lines negative values.

  FIGURE 24. Outer-layer modes, flow case P1: (a-c) streamwise velocity contours for the mode associated with λ x = 2.3δ and λ z = 2δ (having θ xy = 27 • ), at (a) t/t max = 0, (b) t/t max = 0.67 and (c) t/t max = 1 (solid lines represent positive values and dashed lines negative values); and (d) time evolution of the enstrophy ratios. The grey area in panel (d) represents the time necessary for the structure to reach the pattern shown in panel (b).

FIGURE 25 .

 25 FIGURE 25. Maximum kinetic energy gain for infinitely elongated optimal modes (α = 0) as a function of the spanwise wavelength. Refer to table 1 for nomenclature of lines.

  all quantities are evaluated at the time at which kinetic energy is maximum. The relevance of this length scale for describing self-similarity of the log layer is first established by analysing the subsonic flow cases B1-B3. The results are summarized in figure26, which in fact shows linear variation of Λ y on λ z , regardless of the Reynolds number. The corresponding optimal modes, shown in figure26(b) as a function of the wall distance scaled by the spanwise wavelength, support geometrical log-layer optimal modes (for α = 0). (a) Vertical length scale as a function of spanwise wavelength for flow case B1-B3. (b) Mode amplitudes as a function of wall distance scaled by spanwise wavelength: B1 (λ + z = 770), B2 (λ + z = 863) and B3 (λ + z = 893, 1250, 2084), where u is taken at t = t max , and v at t = 0. (c) Optimal time as a function of the spanwise wavelength. The solid lines indicate the linear fit of the data. Refer to table 1 for nomenclature.

FIGURE 27 .

 27 FIGURE 27. Supersonic log-layer optimal modes (for α = 0). (a) Vertical length scale as a function of spanwise wavelength for flow cases P1-P3. (b) Mode amplitudes as a function of wall distance scaled by spanwise wavelength (for λ + z = 331, 397, 441), for flow case P1, where u and T are taken at t = t max , and v at t = 0. (c) Distribution of t max as a function of the spanwise size in inner units for flow cases P1-P3. (d) Distribution of Λ + y as a function of t + max for B1-B3 and P3. The solid lines denote the linear fit of the supersonic data. Refer to table 1 for nomenclature.

FIGURE 28 .

 28 FIGURE 28. Optimal log-layer modes for flow cases P1 (corresponding to λ + z = 441), P2 and P3 (corresponding to λ + z = 340), taken at t = t max . (a) Mode amplitudes as a function of wall distance scaled by spanwise wavelength. (b) Mode amplitudes as a function of density-weighted wall distance scaled by spanwise wavelength. In panel (a) the vertical lines denote the density-averaged wall-normal positions associated with ỹ+ = 20 and ỹ+ = 100 for P1-P3. Refer to table 1 for line nomenclature.

  FIGURE 11. Inner-layer modes: distributions of (a) sup β G max (α, β) and (b) kinetic energy contribution as a function of the streamwise wavenumber. Refer to table 1 for nomenclature.

	(a)	3.2		( b)	3.2
		3.0			3.0
		2.8			2.8
		2.6			2.6
		2.4			2.4
		2.2			2.2
		2.0			2.0
		1.8			1.8
		1.6	0	0.02 0.04 0.06 0.08 0.10	1.6	0	0.02 0.04 0.06 0.08 0.10

† Email address for correspondence: frederic.alizard@cnam.fr

Acknowledgements

We acknowledge that some of the results reported in this paper have been achieved using the PRACE Research Infrastructure resource FERMI based at CINECA, Casalecchio di Reno, Italy, and the HPC resources of IDRIS and CCRT under the allocation x20142a7151 by GENCI (Grand Equipement National de Calcul Intensif).

Appendix A. Linearized equations for parallel flows

The governing linear equations for the mode amplitude vector q are ∂ρ ∂t + iα(ρ u + ρu ) + ∂ρv ∂y + iβρw = 0, (A 1) with T = e /C v and µ tot = µ + µ t .

The production and dissipation terms associated with the kinetic energy budget are

A similar expression for D k t is obtained, by replacing µ with the eddy viscosity.

Appendix C. Validation of the optimal transient growth code

In order to prove the capability of the present approach to reproduce classical results of stability theory, we have analysed a laminar boundary layer under the same flow conditions as [START_REF] Hanifi | Transient growth in compressible boundary layer[END_REF], namely Ma ∞ = 2.5, Re δ = 2550, with adiabatic wall. In figure 29(a), we show the growth rate as a function of time for α = 0 and β = 0.1, and in figure 29(b) the maximum growth rate as a function of the spanwise wavelength, assuming either laminar flow or fully turbulent flow, whereby the model terms are set to zero. The results confirm the validity of our approach and show that the neglected terms have very limited influence in a laminar regime. [START_REF] Hanifi | Transient growth in compressible boundary layer[END_REF]; full line, a laminar flow is assumed; dotted line, a turbulent flow is assumed with µ t = 0.


Anthony Leonard

T Leweke

C H K Williamson

G R S Assi

R N Govardhan

K Hourigan

J R Meneghini

C Norberg

J N Sørensen

M C Thompson

M S Triantafyllou

G Bouchet

C Eloy

P Meunier

Anatol Roshko of the California Institute of Technology. Over the years, Anatol made a remarkable number of seminal contributions to fluid mechanics, beginning with his 1952 Ph.D. thesis entitled "On the Development of Turbulent Wakes from Vortex Streets", in which he described, for the first time and with deep understanding, the various regimes for flow past a cylinder at Reynolds numbers from 40 to 10000. He then followed with a steady stream of insight works on wakes, bluff body flows, and free shear layers.

From the mid 1980's until more than twenty years after his official retirement in the mid 1990's, he turned his attention to forced and freely vibrating cylinder flows, and we remember his spirited participation in our June 2000 IUTAM meeting in Carry-le-Rouet.

This volume is dedicated to the memory of Anatol, a gentle, soft-spoken, legendary figure in the field of fluid mechanics.

PREFACE

This Special Issue consists of selected papers from the Seventh Conference on Bluff Body Wakes and Vortex-Induced Vibrations, held in Carry-le-Rouet near Marseille, France, during the period 3-6 July 2018. The present conference (BBVIV-7) was the seventh in the series, following on from the Forum on Advances in the Understanding of Bluff Body Wakes and Vortex-Induced Vibrations (BBVIV-1), which was part of the 1998 ASME-FED Summer Meeting in Washington, DC, USA; the IUTAM Symposium on Bluff Body Wakes and Vortex-Induced Vibrations (BBVIV-2) held in Marseille in June 2000; the Conferences on Bluff Body Wakes and Vortex-Induced Vibrations held in Port Douglas, Queensland, Australia, in December 2002 (BBVIV-3), on Santorini Island, Greece, in June 2005 (BBVIV-4), and in Costa do Sauípe, Bahia, Brazil, in December 2007 (BBVIV-5); and the IUTAM Symposium on Bluff Body Wakes and Vortex-Induced Vibrations (BBVIV-6) held on Capri Island, Italy, in June 2010. This series of conferences is designed to provide a stimulating and constructive forum for researchers specializing in the areas of flows around bluff bodies that are either fixed or undergoing vortex-induced vibrations (VIV).

The papers in this volume were selected by the Scientific Committee from amongst the oral presentations at the conference and have all been peer-reviewed. The meeting attracted 99 participants from 21 countries, including many of the most active researchers in the field. The event took place over four days and consisted of single plenary sessions with 6 keynote lectures, 44 oral presentations, 33 poster presentations, as well as one evening lecture. A total of 102 abstracts were received and reviewed by the Scientific Committee and the Chairmen. It is believed that the high standard of the presentations and the science set at the previous six conferences was maintained.

A full set of papers, of both oral and poster presentations, was provided to each participant at the conference site -a detailed list is given below. The presentations were divided into the following sessions. As with all the BBVIV conferences, one might imagine topics to be limited to classical cylinder bluff body wakes, and vortex-induced vibrations of cylinders. However, behind these titles are a great number of flows which are distinctly different to the classics, as seen in the list above, and by browsing through this volume of papers from our latest conference in Marseille. Ever more research is now being directed towards more exotic flows and fundamentally three-dimensional (3D) shapes, such as: spheres; cylinders and balls running down an incline; wakes and vortex dynamics of rotating bodies; sports ball dynamics; airfoils vibrating at high angles of attack; swimming, flying and propulsion; control and flexibility of bodies. These problems extend much further than the classical cylinder-bluff body flows.

Part of the large increase of studies of 3D body wakes comes from the immense computational power that exists in comparison with our earlier conference in the same location, Carry-le-Rouet, 18 years ago. Stability of wakes in general has been a subject of interest in a number of papers, for a great many fluid flow configurations. Following from recent conferences, there has been more work to reduce VIV, at the same time reducing drag, and this remains relevant. The dynamics of riser tubes continue to be an important topic. Also, the ability to study more complex flows has nowhere been made clearer than by the prolific group of researchers at Monash University (under Hourigan, Thompson, Sheridan, Blackburn), and they are contributing an impressive number of careful studies at high resolution. Experimental, numerical and analytical approaches to all the problems listed above were presented at the conference, and it is quite evident that a great deal remains to be done to understand these varied flows and phenomena.

Our Invited Speakers delivered particularly impressive and interesting presentations. Our "Opening Speaker" on Tuesday afternoon was Tony Leonard, who gave a lovely talk on "Vortex-Induced Vibrations with Anatol Roshko", which reviewed some of the classical important work of Anatol, including some of his own more recent results on wakes and VIV, in collaboration with Anatol. Tony dedicated this special Proceedings issue to Anatol Roshko, which was met with full support of the participants. Gustavo Assi gave a seminar on his control of fixed or spinning cylinders. Sébastien Michelin looked at the highly topical problem of energy harvesting in the case of flexible cables. Lex Smits talked about his group's research into propulsion inspired by biology, which was extensive and fascinating. Jens Sørensen and a strong group of collaborators looked at the frequencies of wakes behind disks and wind turbine rotors. We are grateful to David Sumner for presenting the final invited seminar at the conference, based on his excellent comprehensive study of wake interference between two finite cylinders.

Our choice to select Carry-le-Rouet, along the Mediterranean coast of France to the west of Marseille, as the venue for BBVIV-7, led to a record number of applicants to participate in the event. We were not disappointed. The conference was held at a family resort, "Club Vacanciel", in a magnificent location overlooking the blue sea. The hotel possessed a sufficient level of isolation to ensure the scientific sessions were always very well attended, and ample time was available outside the formal sessions for participants to meet in a convivial atmosphere. The hospitality, meals and generous availability of the wine and rosé bottles along the lunch and dinner tables led the participants to indulge in profound discussions, especially after the mealtimes, in the restaurant and cafe further up the small hill.

We took the opportunity to ask a few of the international students what were their most memorable impressions coming from their attendance of BBVIV-7. Here are some of the comments:

• "In my opinion, the most memorable occasions at BBVIV-7 were the scheduled events (after the talks, of course!). The first night was the Cuban jazz concert. That was a fantastic atmosphere, with everyone sitting around the patio area at the bar listening to the band play good music. It was a very relaxed ambiance, and everyone was enjoying some wine/beer/mojitos while catching up with old friends (for the BBVIV veterans) or meeting new ones (for BBVIV newbies like my group)." • "The food was amazing at the really nice restaurant in the museum, and it was really cool that the world championship of pétanque was going on about 100 meters away. I recall picking a seat next to the windows so I could watch the game while eating my dinner." • "On the final day, most of the participants joined together with the resort staff and watched the FIFA World Cup game that afternoon. It was particularly exciting because France was playing so the staff were dressed up with painted faces." • "We played an impromptu game of pétanque on the last evening. Thomas was about 10 times better than any of the rest of us."

Since our BBVIV-2 held 18 years ago at the same resort, we have always made arrangements so that participants and their friends and family could use a free half day to decide for themselves what to explore. Some of the group saw "Les Calanques" (the beautiful steep inlets along the coast), or relaxed in places outside Marseille, famous for their painters (for example Aixen-Provence), or simply remained at our private beach ... Ultimately the groups met under their own steam for a reception at MuCEM museum near the Old Port in Marseille on the Thursday evening, followed by a wonderful banquet at La Table . This constituted a beautiful and fitting end to our conference.

As is customary at these BBVIV conferences: All participants came away from the meeting in Carry-le-Rouet with both the intellect and the senses stimulated, and with new insights into wakes and VIV.

Thomas Leweke

•

  Fundamentals (5 talks) • Wakes of Three-Dimensional Bodies (8 talks, 5 posters) • Wake Instability (2 talks) • Wake Interference (3 talks, 3 posters) • Wakes and Acoustics (2 talks) • Vortex-Induced Vibrations (3 talks, 8 posters) • Vortex-Induced Vibrations of Spheres (3 talks) • Vortex-Induced Vibrations of Flexible Structures (5 talks, 4 posters) • Swimming, Flying and Propulsion (3 talks, 2 posters) • Airfoil Aerodynamics (3 talks, 2 posters) • Control (7 talks, 9 posters)

BBVIV-7 group photo -4July

ACKNOWLEDGEMENTS

We are grateful for the support of this conference from the following organisations: Institut de Recherche sur les Phénomènes Hors Equilibre (IRPHE), Marseille

List of all conference presentations

Airfoil Aerodynamics

Barman, D., Prahallada, J., Govardhan, R. N.

Heave mode stall flutter in a linear cascade at low Reynolds numbers Branger, H., Paillard, B., Roy, S., Luneau, C., Perrot, E., Bourras, D. Experimental and numerical study on a model of offshore vertical axis wind turbine with pitching blades Durán Venegas, E., Le Dizès, S., Eloy, C.

A coupled model for flexible rotors (Durán Venegas et al, 2019, this Feedback control of bluff-body wakes using flaps Castro Hebrero, F., D'Adamo, J., Sosa, R., Artana, G.

Control of cylinder oscillations by plasma actuators Cicolin, M. M., Serson, D., Meneghini, J. R., Assi, G. R. S.

Experimental and numeric study of the control of flow over a circular cylinder using control rods Clapperton, B. L., Bearman, P. W.

Control of circular cylinder flow using distributed jets Desai, A., Chopra, G., Mittal, S.

Span-wise distribution of surface pressure on a circular cylinder in the regime of drag crisis Gao, D., Chen, W., Chen, G., Li, H.

Lift suppression, drag reduction and vortex shedding alleviation of a circular cylinder by active jets García-Ortiz, J. H., Parras, L., Del Pino, C., Blanco-Rodríguez, F. J.

On the effect of low blowing ratio continuous jets on wingtip vortex characteristics Huang, Y.-W., Chen, W.-L.

Experimental research on passive jet control of flow around a rectangular cross section column Jaiswal, Y., Sahu, T. R., Furquan, M., Mittal, S.

Flow-Induced Vibration of a cylinder with rigid splitter plate (Sahu et al., 2019, this issue) McQueen, T., Zhao, J., Sheridan, J., Thompson, M. C.

Control of Vortex-Induced Vibrations of a sphere Patino, G., Gioria, R., Assi, G. R. S, Meneghini, J. R.

Sensitivity analysis applied to the flow around the water line of a ship at low and high Reynolds numbers Sicot, C., Paillé, F., Barros, D., Borée, J.

Unsteady load measurements: analysis of 2D and 3D cylinder wakes Van Hinsberg, N.

On the aerodynamic loading on rounded square-section cylinders at very high Reynolds numbers