text:
M Obligado

J C Vassilicos

The non-equilibrium part of the inertial range in decaying homogeneous turbulence

Keywords: 47, 27, Ak -Turbulent flows, fundamentals 47, 27, Gs -Isotropic turbulence; homogeneous turbulence 47, 27, Jv -High-Reynolds-number turbulence

The non-equilibrium part of the inertial range in decaying homogeneous turbulence

M. Obligado, J.

Introduction. -The Kolmogorov phenomenology relies on the presence of a statistically stationary, i.e. equilibrium, cascade of energy where large-scale energy input rate balances turbulence dissipation rate effectively instantaneously. Assuming homogeneous or locally homogeneous turbulence, and assuming external power input to be either absent or limited to wavenumbers much smaller than k, the interscale energy balance can be expressed as (see [START_REF] Goto | Local equilibrium hypothesis and taylor's dissipation law[END_REF][START_REF] Goto | Unsteady turbulence cascades[END_REF][START_REF] Sagaut | Homogeneous turbulence dynamics[END_REF]),

∂ ∂t K > (k, t) = Π(k, t) -ε > (k, t) (1)
where

K > (k, t) = ∞ k E(k , t
)dk is the high-pass filtered kinetic energy (with E(k, t) the energy spectrum), Π(k, t) is the interscale energy flux and ε > (k, t) is the high-pass filtered turbulence dissipation rate.

The Kolmogorov equilibrium hypothesis is [START_REF] Goto | Local equilibrium hypothesis and taylor's dissipation law[END_REF][START_REF] Goto | Unsteady turbulence cascades[END_REF][START_REF] Tennekes | A first course in turbulence[END_REF]). The size of the largest turbulent eddies is typically accessed via the integral length-scale L, and if this equilibrium hypothesis can be extended to these largest eddies, i.e. to wavenumbers comparable to 2π/L, then

| ∂ ∂t K > (k, t)| ε > (k, t) (see
ε ≈ Π(2π/L, t) ∼ K 3/2 /L (2)
where the scaling K 3/2 /L of Π(2π/L, t) comes from the expectation that there are no direct viscosity effects at the largest scales and that the memory of initial/inlet conditions has had time to fade away (K is the turbulent kinetic energy). The resulting scaling ε = C ε K 3/2 /L where C ε = Const (sometimes refered to as zeroth law of turbulence or Taylor-Kolmogorov turbulence dissipation scaling) is perhaps the most important consequence of the equilibrium cascade phenomenology as it has a very wide range of implications in turbulence theory and modeling (see [START_REF] Tennekes | A first course in turbulence[END_REF][START_REF] Vassilicos | Dissipation in turbulent flows[END_REF]). However, a different dissipation scaling has been found over the past ten years in various turbulent flows, including forced periodic turbulence and decaying periodic turbulence [START_REF] Goto | Energy dissipation and flux laws for unsteady turbulence[END_REF][START_REF] Goto | Local equilibrium hypothesis and taylor's dissipation law[END_REF][START_REF] Goto | Unsteady turbulence cascades[END_REF], various types of grid-generated turbulence [START_REF] Vassilicos | Dissipation in turbulent flows[END_REF] and various turbulent shear flows [START_REF] Cafiero | Non-equilibrium turbulence scalings and self-similarity in turbulent planar jets[END_REF][START_REF] Nedić | Dissipation scaling in constant-pressure turbulent boundary layers[END_REF][START_REF] Obligado | Nonequilibrium scalings of turbulent wakes[END_REF][START_REF] Vassilicos | Dissipation in turbulent flows[END_REF]:

ε ∼ U 0 L 0 K/L 2 , (3)
or equivalently

C ε ∼ Re I /Re L ∼ √ Re I /Re λ where R L = √ KL/ν, Re λ = √
Kλ/ν and Re I = U 0 L 0 /ν, U 0 and L 0 being initial/inlet velocity and length scales respectively (ν is the fluid's kinematic viscosity and λ the Taylor length-scale). One can expect this scaling to be a consequence of a cascade which is out of equilibrium, at least p-1 arXiv:1910.04444v1 [physics.flu-dyn] 10 Oct 2019 in the upper part of the inertial range (closer to L), and this was indeed confirmed by Goto & Vassilicos (2016) [START_REF] Goto | Unsteady turbulence cascades[END_REF] in direct numerical simulations of freely decaying periodic turbulence. These authors also found, in agreement with grid-generated turbulence results (see [START_REF] Vassilicos | Dissipation in turbulent flows[END_REF]), that the classical dissipation scaling ε ∼ K 3/2 /L (C ε = Const) appears rather suddenly after a number of turnover times and as the local Reynolds numbers R L and Re λ decrease. They also reported that this classical dissipation scaling is not a reflection of a Kolmogorov equilibrium cascade but in fact coexists with non-equilibrium over a wide range of scales, including those inertial scales closer to L, as in the model of Bos et al [START_REF] Wjt Bos | Spectral imbalance and normalized dissipation rate of turbulence[END_REF]. Bos et al [START_REF] Wjt Bos | Spectral imbalance and normalized dissipation rate of turbulence[END_REF] explained this non-equilibrium version of C ε = Const by relating it to its equilibrium version and Goto & Vassilicos (2016) [START_REF] Goto | Unsteady turbulence cascades[END_REF] explained it by noting that the three terms in equation (1) remain numerically comparable as they decay in time, and proportional to each other for kL fixed close to 2π.

The question arises whether this non-equilibrium, in particular at the inertial scales closer to L, persists as the Reynolds number increases towards infinity. This question is addressed in this paper in the context of freely decaying homogeneous isotropic turbulence (HIT) by using published wind tunnel data and Lundgren's [START_REF] Thomas S Lundgren | Kolmogorov two-thirds law by matched asymptotic expansion[END_REF][START_REF] Thomas S Lundgren | Kolmogorov turbulence by matched asymptotic expansions[END_REF] matched asymptotic expansions for the second and third order structure functions.

Finite Reynolds number effects. -We work with the Kármán-Howarth equation for decaying HIT in a rearranged version of the form found in [START_REF] Danaila | A generalization of yaglom's equation which accounts for the large-scale forcing in heated decaying turbulence[END_REF]13]:

-εF = - (δu) 3 r - 4 5 ε + 6ν r ∂ ∂r (δu) 2 (4)
where F ≡ -3 εr 5 r 0 dr r 4 ∂ ∂t (δu) 2 is the nonstationarity/non-equilibrium term. (δu) 2 and (δu) 3 are, respectively, the second and third order longitudinal structure functions defined in terms of δu = u(x + r, t)u(x, t) where u is the fluctuating velocity component in the direction of the x axis, and r is the distance between points x and x + r on that axis. This equation is the physical space equivalent of (1) where -εF and -

(δu) 3 r correspond to ∂ ∂t K > (k, t
) and Π(k, t) respectively. Finite Reynolds number (FRN) effects come from the non-stationarity/non-equilibrium term -εF and from the viscous term 6ν r ∂ ∂r (δu) 2 [START_REF] Danaila | A generalization of yaglom's equation which accounts for the large-scale forcing in heated decaying turbulence[END_REF]. It has been proved rigorously [START_REF] Laizet | Interscale energy transfer in decaying turbulence and vorticitystrain-rate dynamics in grid-generated turbulence[END_REF][START_REF] Valente | The energy cascade in gridgenerated non-equilibrium decaying turbulence[END_REF] using simple kinematic constraints on the basis that K is finite that 6ν It is easily seen that f (r) → 4/3 and F (r) → 4/5 in the limit r L because (δu) 2 → 4 3 K in that limit. From a first order Taylor expansion and ε = 15ν ( ∂u ∂x ) 2 , (δu) 2 ≈ r 2 ε 15ν for r λ [START_REF] Ingram | Statistical theory of turbulence. iv. diffusion in a turbulent air stream[END_REF]. Differentiating both sides with respect to time, we obtain

f ≈ r λ 2 λ 2 15ν - d dt ε ε which implies f ≈ Const( r λ ) 2 for r
λ if K decays as a power of time; F ≈ 3 7 Const( r λ ) 2 follows under these same conditions.

We can expect the non-stationarity functions f and F to increase from 0 at r = 0 to 4/3 and 4/5, respectively, at r L. Incidentally, if f is a monotonically increasing function of r, so is F . (Proof:

∂ ∂r F > 0 if r 5 f > 5 r 0 dr r 4 f (r ) which follows from 5 r 0 dr r 4 f (r ) < 5 r 0 dr r 4 f (r) = r 5 f given that f (r) > f (r ) for r > r .)
We might expect a tendency towards equilibrium at the smaller inertial scales, but what are the values of f and F for r smaller than but comparable to L? The answer to this question can inform us about the degree of nonequilibrium at the larger inertial scales.

Experimental data. -We use published wind tunnel measurements of (δu) 2 (r) and (δu) 3 (r) for approximate HIT to calculate F (r) from ( 4) and then f (r) from f = 1 3r 4 d dr r 5 F . The only published data sets that we could find for wind tunnel measurements of both (δu) 2 and (δu) 3 in approximate HIT conditions are from the grid turbulence experiments of Zhou & Antonia [START_REF] Zhou | Reynolds number dependence of the small-scale structure of grid turbulence[END_REF], Malecot [START_REF] Malécot | Intermittence en turbulence 3D: statistique de la vitesse et de la vorticité[END_REF] and Bourgoin et al [START_REF] Bourgoin | Investigation of the smallscale statistics of turbulence in the modane s1ma wind tunnel[END_REF] and from the Modane wind tunnel measurements of [START_REF] Gagne | Reynolds dependence of third-order velocity structure functions[END_REF]. Note that the data of Bourgoin et al [START_REF] Bourgoin | Investigation of the smallscale statistics of turbulence in the modane s1ma wind tunnel[END_REF] have been obtained in the same Modane wind tunnel as [START_REF] Gagne | Reynolds dependence of third-order velocity structure functions[END_REF]. The particular datasets that we selected from these publications span the widest possible range of Reynolds numbers based on Re λ . Table 1 summarises the turbulence parameters for each dataset. All these data have been obtained with hot-wire anemometry (the specific details of the experimental setups for each value of Re λ are given in the corresponding references), although for the two larger values of Re λ the Kolmogorov length scale (η = (ν 3 /ε) 1/4 ) was not fully resolved. In all cases, the conversion from time to space was done via the Taylor hypothesis. The Taylor lengthscale is defined from λ 2 =< u 2 > / < (∂u/∂x) 2 >, and Re λ is calculated as Re λ = u λ/ν where u ≡< u 2 > 1/2 . The turbulence dissipation rate, assuming HIT, can be obtained from ε = 15ν < u 2 > /λ 2 . For the two larger values of Re λ , where η was not resolved, ε and λ have been obtained via the compensated second order structure function [START_REF] Zhou | Reynolds number dependence of the small-scale structure of grid turbulence[END_REF], M a for Malecot [START_REF] Malécot | Intermittence en turbulence 3D: statistique de la vitesse et de la vorticité[END_REF], M I for the Modane inflatable grid-turbulence experiment [START_REF] Bourgoin | Investigation of the smallscale statistics of turbulence in the modane s1ma wind tunnel[END_REF] and M o for the Modane measurements of Gagne et al [START_REF] Gagne | Reynolds dependence of third-order velocity structure functions[END_REF] the integral scale too but without FRN effects coming from C ε . F is found to be larger than 0.2 and f larger than 0.4 for all r larger or equal to L/5 in all data sets. In the case of our highest Reynolds number, Re λ = 2260, r = 40λ corresponds to r ≈ L/4, which shows that L/5 is quite a small scale for all our data and well within the smallest scales of the inertial range for most if not all of them. We must conclude that most if not all the inertial range is not in Kolmogorov equilibrium in all our data sets, and definitely not in the part of the inertial range closer to its upper bound L. Justifying C ε = Const on the basis of such equilibrium at the Reynolds numbers here is definitely questionable. Note the much better collapse with λRe λ than with L for both F and f in figure 1. This might suggest that the lack of collapse with L is caused by the residual Reynolds number dependence of C ε at the smaller Re λ values. Extrapolating to Re λ larger than 2260 might suggest that F and f remain larger than 0.2 and 0.4, respectively, for r ≥ L/5 as Re λ grows beyond 2260. A non-stationarity function f larger than 0.4 means that -∂ ∂t (δu) 2 > 0.4ε. Hence, the current state of experimental evidence for HIT does not rule out, and in fact might even suggest, that the upper part of the inertial range between L and an order of magnitude smaller than L is out of equilibrium at all Reynolds numbers. This conclusion is supported by Lundgren's [START_REF] Thomas S Lundgren | Kolmogorov two-thirds law by matched asymptotic expansion[END_REF][START_REF] Thomas S Lundgren | Kolmogorov turbulence by matched asymptotic expansions[END_REF] matched asymptotic expansions for the second and third order structure functions which we examine next.

Grid Re λ U ∞ (m/s) L (cm) η (mm) ε (m 2 s -3 ) λ (cm) u (m/
Extrapolation to even higher Re λ . Lundgren [START_REF] Thomas S Lundgren | Kolmogorov two-thirds law by matched asymptotic expansion[END_REF][START_REF] Thomas S Lundgren | Kolmogorov turbulence by matched asymptotic expansions[END_REF] obtained the following forms for (δu) 2 and (δu) 3 from the Navier-Stokes equation, and in particular the Kármán-Howarth equation, by using the method of matched asymptotic expansions in the range η r L:

(δu) 2 (εr) 2/3 ≈ C[1 -A 2 (r/L) 2/3 -B 2 (r/η) -4/3 ] (5) - (δu) 3 (εr) ≈ 4/5 -A 3 (r/L) 2/3 -B 3 (r/η) -4/3 (6)
where A 2 , B 2 , A 3 and B 3 are dimensionless constants of order 1, and C = 2 from widely accepted experimental The colour code for Re λ indicated in (a) is the same in all the plots in this paper and the data sets used to obtain F and f are described in Table 1.

evidence to date (e.g. see Pope 2000 [START_REF] Stephen B Pope | Turbulent flows. Turbulent Flows[END_REF]). It is worth mentioning that these corrections to the Kolmogorov equilibrium scalings of (δu) 2 are different from the nonequilibrium correction obtained by Yoshizawa [START_REF] Yoshizawa | Nonequilibrium effect of the turbulentenergy-production process on the inertial-range energy spectrum[END_REF] from his spectral closure theory and by Bos & Rubinstein [START_REF] Bos | Dissipation in unsteady turbulence[END_REF] who obtained the same correction as Yoshizawa by using Kovaznays spectral closure model. Whilst the correction of these authors scales as r 2/3 similarly to the correction A 2 (r/L) 2/3 in (5), it also scales with dε/dt which is not generally the case in (5) (unless we assume, for example, that the K -ε equation for dε/dt and C ε = Const hold true). More importantly, however, the non-equilibrium correction of Yoshizawa and Bos & Rubinstein does not capture the B 2 (r/η) -4/3 correction in (5) which is essen- Fig. 2: Functions (a) F and (b) f , both plotted vs r/λ. The colour code for different Re λ is the same as in figure 1 and the data sets used to obtain F and f are described in Table 1.

tial for our conclusions as the special role of the Taylor microscale described below cannot be captured without it.

Using (1), formulae ( 4) and ( 5) lead to

F ≈[A 3 (r/L) 2/3 -4CRe -2 λ (r/L) -4/3 + 8CA 2 Re -2 λ (r/L) -2/3 ]- B 3 Re -2/3 λ (r/λ) -4/3 [1 + 4C B 2 B 3 Re -2 λ (r/L) -4/3 ]. (7)
This expression for F implies that F ≈ A 3 (r/L) 2/3 in the limit Re λ → ∞ as r/L is kept constant or as r and t are both kept constant relative to laboratory length and time scales. In other words, the higher part of the inertial range does not tend towards equilibrium as Re λ → ∞, it keeps its level of non-equilibrium unchanged. This observation does not contradict the fact that, as Re λ → ∞ and L/λ → ∞, an approximate equilibrium range does develop in the limit where r/L → 0. Indeed, for fixed r/λ, F does tend to 0 as Re λ → ∞, and our data show evidence in agreement with this in figure 2: note how both F and f decrease, presumably towards zero, as Re λ increases while keeping r/λ constant. However, this asymptotic equilibrium part of the inertial range is so far from L, and increasingly so with increasing Re λ , that it cannot be used as a basis for C ε = Const.

To clarify this distinction between the asymptotic equilibrium scales at the lower end of the inertial range and the non-equilibrium scales at upper end of the inertial range we calculate the value r max of r where Lundgren's formulae for <(δu) 2 > (εr) 2/3 and -

(δu) 3 (εr)
have a maximum. As already noted in [START_REF] Thomas S Lundgren | Kolmogorov turbulence by matched asymptotic expansions[END_REF], the maximum is at r max ∼ λ for both (εr) , as shown in figures 3a&b. To our knowledge, this is the first time that this prediction is confirmed for the second order structure function and the first time that it is confirmed over such a wide range of Reynolds num- (data from only two Reynolds numbers from Yves Gagne's Modane measurements were used in [START_REF] Thomas S Lundgren | Kolmogorov turbulence by matched asymptotic expansions[END_REF]).

It may be worth pointing out that the empirical formula for (δu) 2 in [START_REF] Ra | Approach to the 4/5 law in homogeneous isotropic turbulence[END_REF][START_REF] Kurien | Anisotropic scaling contributions to high-order structure functions in high-reynolds-number turbulence[END_REF], namely

(δu) 2 (εη) 2/3 = (r/η) 2 (1 + r/L) -2/3 15[1 + D(r/η) 2 ] 2/3 (8)
where 30D 2/3 = 1, leads to a maximum of (δu) 2

(εr) 2/3 at r max ≈ 2 5/6 C ε λ for L/η 1 (using ε = C ε u 3 /L = 15νu 2 /λ 2 to relate λ to L via 15L/λ = C ε Re λ ). This empirical formula is designed by construction to give < (δu) 2 >≈ 2(εr) 2/3 in the range η r L, and the right kinematic behaviours at r/L 1 and r/η 1. It is interesting that it combines dependencies on L and η in exactly the right way for ∂ ∂r (δu) 2 (εr) 2/3 = 0 to yield r max ∼ λ, in agreement with Lundgren's prediction.

Conclusions. -Given that the inertial range is, by definition, a range where the viscosity is effectively absent, it must be defined as λ r L. Our data and Lundgren's [START_REF] Thomas S Lundgren | Kolmogorov two-thirds law by matched asymptotic expansion[END_REF][START_REF] Thomas S Lundgren | Kolmogorov turbulence by matched asymptotic expansions[END_REF] formulae [START_REF] Cafiero | Non-equilibrium turbulence scalings and self-similarity in turbulent planar jets[END_REF] and [START_REF] Danaila | A generalization of yaglom's equation which accounts for the large-scale forcing in heated decaying turbulence[END_REF] show that (δu) 2 and (δu) 3 are closest to their Kolmogorov equilibrium predictions at the lower end of the inertial range, i.e. r = r max ∼ λ. Both (δu) 2 and (δu) 3 increasingly deviate away from their Kolmogorov equilibrium prediction as r increases in the inertial range. The matched asymptotic expansions (5) and [START_REF] Danaila | A generalization of yaglom's equation which accounts for the large-scale forcing in heated decaying turbulence[END_REF] imply that (δu) 2 and (δu) 3 tend towards C(εr) 2/3 and -4 5 εr, respectively, as Re λ → ∞ for fixed r/λ but not for fixed r/L. For fixed r/L smaller that 1, they tend to

The upper end of the range is therefore always significantly out of equilibrium whatever the Reynolds number.

The wind tunnel data that we have compiled here for HIT support this conclusion and suggest that the upper end of the inertial range where r is up to one order of magnitude smaller than L is such that -∂ ∂t (δu) 2 is comparable to ε. Any justification of C ε = Const in terms of Kolmogorov equilibrium is therefore, at the very least, seriously questionable in decaying HIT, at any Reynolds number. * * *

3 r 5 r0 4 d dr r 5 F

 3545 viscous term contributes no significant FRN effects at scales r λ. We define the non-stationarity function f by ∂ ∂t (δu) 2 ≡ -εf . This function provides a direct comparison between ∂ ∂t (δu) 2 and ε across scales. It is related to the non-stationarity function F by F = dr r 4 f and f = 1 3r . Note that both f and F are dimensionless and non-negative.

(δu) 2 (

 2 εr) 2/3 (Re λ = 380) or by matching the shape of (δu)3 (εr) at r close to η to other data (Re λ = 2260). Finally, the lengthscale L is calculated from the autocorrelationR uu as L = ∞ 0 R uu dr.Results. -In figure1we plot the non-stationarity functions F and f as functions of r/L and r/(λRe λ ). Note the relation 15L = C ε λRe λ in HIT, hence λRe λ represents

Fig. 1 :

 1 Fig. 1: Top plots: non-stationarity function F vs (a) r/L and (b) r/(λRe λ ). Bottom plots: non-stationarity function f vs (c) r/L and (d) r/(λRe λ ). The colour code for Re λ indicated in (a) is the same in all the plots in this paper and the data sets used to obtain F and f are described in Table1.

(δu) 2

 2

3 (

 3 εr) . Our data confirm this prediction and give r max ≈ 2λ for

(δu) 2 (

 2 εr) 2/3 and r max ≈ 1.5λ for -(δu)3

Fig. 3 :

 3 Fig. 3: Compensated (a) second-order structure function (δu) 2 (εr) 2/3 and (b) third-order structure function (δu) 3 εr, both plotted vs r/λ.

(δu) 2 =

 2 C(εr) 2/3 [1 -A 2 (r/L) 2/3 ] (9)and(δu)3 = -(εr)[4/5 -A 3 (r/L) 2/3 ].

Table 1 :

 1

	s)

Experimental parameters : Reynolds number based on Taylor micro-scale (Re λ ), incoming wind velocity (U∞), integral length scale (L), Kolmogorov scale (η), energy dissipation rate (ε), Taylor micro-scale (λ) and standard deviation of fluctuating streamwise velocity (u ). ZA stands for Zhou & Antonia

This work was motivated by a discussion with Claude Cambon during a dinner held in his honour in Lyon to mark his 65th year.