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1 This article was downloaded by: [University of Newcastle (Australia)] On: 16 June 2014, At: 16:17 Publisher: Taylor & Francis Informa Ltd Registered in England and Wales Registered Number: Registered office: Mortimer House, Mortimer Street, London W1T 3JH, UK Journal of Turbulence Publication details, including instructions for authors and subscription information: Breakdown of Kolmogorov's first similarity hypothesis in grid turbulence L. Djenidi a, S.F. Tardu b, R.A. Antonia a & L. Danaila c a Discipline of Mechanical Engineering, School of Engineering, University of Newcastle, Newcastle, Australia b Laboratoires des Ecoulements Geophysiques et Industriels, UMR 5519, University of Grenoble, Grenoble, France c CORIA UMR 6614, University of Rouen, Rouen, France Published online: 12 Jun To cite this article: L. Djenidi, S.F. Tardu, R.A. Antonia & L. Danaila (2014) Breakdown of Kolmogorov's first similarity hypothesis in grid turbulence, Journal of Turbulence, 15:9, , DOI: / To link to this article: PLEASE SCROLL DOWN FOR ARTICLE Taylor & Francis makes every effort to ensure the accuracy of all the information (the Content ) contained in the publications on our platform. However, Taylor & Francis, our agents, and our licensors make no representations or warranties whatsoever as to the accuracy, completeness, or suitability for any purpose of the Content. Any opinions and views expressed in this publication are the opinions and views of the authors, and are not the views of or endorsed by Taylor & Francis. The accuracy of the Content should not be relied upon and should be independently verified with primary sources of information. Taylor and Francis shall not be liable for any losses, actions, claims, proceedings, demands, costs, expenses, damages, and other liabilities whatsoever or howsoever caused arising directly or indirectly in connection with, in relation to or arising out of the use of the Content. This article may be used for research, teaching, and private study purposes. Any substantial or systematic reproduction, redistribution, reselling, loan, sub-licensing, systematic supply, or distribution in any form to anyone is expressly forbidden. Terms & Conditions of access and use can be found at

2 Journal of Turbulence, 2014 Vol. 15, No. 9, , Breakdown of Kolmogorov s first similarity hypothesis in grid turbulence L. Djenidi a,s.f.tardu b, R.A. Antonia a and L. Danaila c a Discipline of Mechanical Engineering, School of Engineering, University of Newcastle, Newcastle, Australia; b Laboratoires des Ecoulements Geophysiques et Industriels, UMR 5519, University of Grenoble, Grenoble, France; c CORIA UMR 6614, University of Rouen, Rouen, France (Received 7 January 2014; accepted 6 April 2014) Kolmogorov s first similarity hypothesis (or KSH1) stipulates that two-point statistics have a universal form which depends on two parameters, the kinematic viscosity ν and the mean energy dissipation rate ɛ. KSH1 is underpinned by two assumptions: the Reynolds number is very large and local isotropy holds. To disentangle the intricacies of these two requirements, we assess the validity of KSH1 in a flow where local isotropy is aprioritenable, i.e. decaying grid turbulence. The main question we address is how large should the Reynolds number be for KSH1 to be valid over a range of scales wider than, say, five Kolmogorov scales. To this end, direct numerical simulations based on the lattice Boltzmann method are carried out in low Reynolds number grid turbulence. The results show that when the Taylor microscale Reynolds number R λ drops below about 20, the Kolmogorov normalised spectra deviate from those at higher R λ ; the deviation increases with decreasing R λ. It is shown that at R λ 20, the contribution of the energy transfer in the scale-by-scale energy budget becomes smaller than the contributions from the viscous and (large-scale) non-homogeneous terms at all scales, but never vanishes, at least for the range of Reynolds investigated here. A phenomenological argument based on the ratio N between the energy-containing timescale and the dissipative range timescale leads to the condition R λ N 15 for KSH1 to hold. The numerical data indicate that N = 5, yielding R λ 20, thus confirming our numerical finding. The present results show that KSH1, unlike the second Kolmogorov similarity hypothesis (KSH2,) does not require the existence of an inertial range. While it may seem remarkable that KSH1 is validated at much lower Reynolds numbers than required for KSH2 in grid turbulence (R λ 1000,), KSH1 applies to small scales which include both dissipative scales and inertial range (if it exists). One can expect that, as the Reynolds number increases, the dissipative scales should satisfy KSH1 first; then, as the Reynolds number attains very high values, the inertial range is established in conformity with KSH2. Keywords: direct numerical simulation; homogeneous turbulence; isotropic turbulence 1. Introduction The first similarity hypothesis of Kolmogorov (or KSH1, [1]) stipulates that when the Reynolds number Re is large enough, and r L (r and L are the spatial separation between two points and the integral length scale, respectively), the statistics of δu(r) = u(x + r) u(x) (the longitudinal velocity structure function of u, the longitudinal velocity fluctuation) have a universal form which depends on ɛ, r and ν ( ɛ = ν 2 ( u i x j + u j x i ) 2 is Corresponding author. lyazid.djenidi@newcastle.edu.au C 2014 Taylor & Francis

3 Journal of Turbulence 597 the turbulent kinetic energy dissipation rate and ν the fluid kinematic viscosity (summation convention applies for double indices, angular brackets signify averaging). Two assumptions underpin KSH1: the Reynolds number is very large and the small scales are isotropic. One consequence of KSH1 is that second-order statistics can be expressed in a dimensionless form as (δu) 2 = u 2 Kf (r/η), (1) where v K and η are the Kolmogorov velocity (u K = (ν ɛ ) 1/4 ) and length (η = (ν 3 / ɛ ) 1/4 ) scales, respectively, and f a universal function; for convenience, (δu) 2 (r) is replaced by (δu) 2 in the remainder of the text. In spectral form, expression (1) becomes E u (k 1 ) = ( u 2 K η) F (k 1 η), (2) where E u (k 1 ) is the longitudinal velocity spectrum, k 1 the longitudinal wave number and F(k 1 η) a universal function. This universality has been verified by many experimental data and is well illustrated by the compilation of data obtained in several turbulent flows reported by [2]. Figure 1 (reproduced from [2]) shows one-dimensional (1D) longitudinal velocity spectra measured in turbulent boundary layers, wakes, jets, grid flows, ducts, pipes and oceanic flows, where the Reynolds number based on the Taylor microscale, R λ λu /ν, where λ ( ( u 2 / ( u/ x) 2 ) 1/2 ) and u is the root mean square value of the longitudinal velocity component, varies from 23 to At this stage, we must clearly indicate that the criterion we use to test whether KSH1 holds is the number (say, N η ) of Kolmogorov length scales over which either Equation (1) or (2) holds. We select here N η 5 for KSH1 to be valid. We should recall that most of the data originate from experiments, for which the accuracy of small-scale measurements is often problematical. The gradual departure from the collapse as k 1 η decreases below 0.1 is expected as it corresponds to the scales r L (L is the integral scale of the flow). There appears to be a relatively good collapse of the data for k 1 η 0.1, thus supporting KSH1 for the highest R λ represented here, but also at relatively small Reynolds numbers, provided effects due to inhomogeneity and/or anisotropy in the flow are negligible. The distributions comply well with KSH1 for k 1 η 0.1. It must be pointed out that no corrections at large k 1 η were applied and that isotropy was used to obtain ɛ (i.e. ɛ = ɛ iso = 15ν 0 (k 1η) 2 E u (k 1 )d(k 1 )) in all cases. One may recall that Figure 1 is presumably less sensitive to how accurately ɛ is determined, since the normalisation of k 1 and E u (k 1 η)involves ɛ 1/4. Nevertheless, leaving aside the uncertainty surrounding the determination of ɛ, Figure 1 provides strong support for KSH1 in the dissipative range when R λ is small. Recently, Djenidi and Antonia [3] exploited this similarity to develop a spectral method for estimating ɛ in various turbulent flows. Given that R λ for some of the data presented in Figure 1 is relatively small, and thus violates one of the conditions for KSH1, one may consider the collapse of the data in the dissipative range as rather remarkable. One may further wonder if such a collapse still holds at lower values of R λ than those reported in that figure. A closer look at the data in the high wave number region in Figure 1 reveals a scatter in the distributions. Interestingly, Mansour and Wray [4] showed that for a three-dimensional (3D) periodic box turbulence, the energy power spectrum, scaled on Kolmogorov variables, deviates from the universal Kolmogorov spectrum at high wave numbers at values of R λ smaller than about 20, suggesting not only

4 598 L. Djenidi et al. Figure 1. One-dimensional longitudinal velocity spectra measured in various turbulent flows where R λ varies from 23 to E 11 (k 1 ) corresponds to E u (k 1 ) in the present manuscript. Adapted from Saddoughi and Veeravalli (1994) [2]. This figure has been reproduced by kind permission of Cambridge University Press. that a breakdown of the Kolmogorov scaling has occurred, but also that a possible minimum (critical) value of R λ exists, below which the Kolmogorov normalisation ceases to hold. The present work aims at extending the work of Mansour and Wray [4] to decaying grid turbulence. In particular, it attempts at confirming whether the breakdown of the

5 Journal of Turbulence 599 Kolmogorov normalisation occurs at low Reynolds numbers and, if so, to estimate the critical value of R λ, called (R λ ) c, below which it occurs. Section 2 is dedicated to the description of the numerical method. In Section 3, the data are also analysed with the view to determining whether local isotropy is valid, and how the velocity second-order statistics behave at very low R λ. We will show that, at least for the investigated flow, (R λ ) c 20. Section 4 provides a physical insight into the KSH1 breakdown, through the competition between the viscous, large-scale and energy transfer terms. Section 5 is devoted to predictions for the energy spectrum in the far dissipative range, when the Reynolds number is very low. We will conclude in Section Numerical procedure 2.1. Lattice Boltzmann method Direct numerical simulations are carried out using the lattice Boltzmann method (LBM). Rather than solving the governing fluid equations (Navier Stokes equations), LBM solves the Boltzmann equation on a lattice [5]. The method was successfully used to simulate turbulent flows [6,7]. Note that unless otherwise specified, all quantities are expressed in lattice units or made non-dimensional. Details on the LBM can be found in [8] or[9]; the implementation of the LBM for this flow is described in [6] and [10] Computational domain and boundary conditions The computational uniform Cartesian mesh consists of mesh points with x = y = z = 1(xis the longitudinal direction and y and z are the lateral directions). The turbulence-generating grid is made up of 6 6 floating flat square elements in an aligned arrangement (see [10]). Each element is represented by mesh points and the mesh spacing (M) between the centre of two elements is 40 mesh points (i.e. 2D), yielding a grid solidity of The downstream distance extends to x/d = 70 (equivalently x/m = 35), where the origin of x is the grid plane and D = 20 mesh points is the grid element s side length. Note that the same distance would require far too large a number of mesh points if a grid made of vertical and horizontal bars and with the same solidity were to be used. This is the main reason why the square elements were chosen. Periodic conditions are applied in the y- and z-directions. At the inlet, a uniform velocity (U 0 = 0.05, and V 0 = W 0 = 0) is imposed, and a convective boundary condition is applied at the outlet. It was observed that the convective condition affected marginally the simulation results within a distance of less than 2D upstream of the outlet. A no-slip condition at the grid elements is implemented with a bounce-back scheme ([9]). Three values of R M UM/ν are used: 3200, 1600 and 400. Although the steady-state solution is reached after 10,000 iterations, the first velocity field is saved at the 50,000th iteration. Subsequently, 40 velocity fields are recorded, each separated by about 15,000 iterations (about five times λ/u ) to ensure that two consecutive fields are uncorrelated. In order to avoid the occurrence of instabilities where the magnitude of the local strain rate could be large, mainly around the grid, a large eddy simulation (LES) scheme with a filter size equal to the mesh resolution was introduced. The LES scheme is based on the Smagorinsky model and developed for LBM by Hou et al. [11]); details of its implementation can be found in [6] and [10]. Note that for R M = 1800 and 400, no LES scheme is used. Also, for the case R M = 400, the number of mesh points in the x-direction is 2240, thus extending the downstream length by

6 600 L. Djenidi et al. Table 1. Values of the ratio x/d representing the length of the computational domain, as well as the number of Kolmogorov scales per mesh unit, x/η. R M x/d x/η Figure 2. Streamwise evolution of R λ for different values of R M. The inset represents values of R λ for experimental data. about 40% when compared to that for the other two cases. The mesh resolution shown in Table 1 is fine enough for the LBM to resolve the small-scale motion accurately. Figure 2 shows the variations of R λ with the downstream distance x/m for the three R M values. For comparison, single hot-wire data taken downstream of a grid consisting of vertical and horizontal square bars (d = 4.76 mm, M = 25 mm) and downstream of a interwoven wires (d = 1 mm and M = 5 mm) are shown in the inset of Figure 2. Aftera sharp increase, R λ decreases monotonically, first rapidly over a short distance and then at a reduced rate. The maximum of R λ occurs at x/m = 1 for the three cases. The ratio of the streamwise and transverse velocity variance (u 2 /v 2 ) is about 1.15 and is constant for x/m > 5 (e.g. [12]). 3. Assessment of KSH1 in LBM simulations of decaying grid turbulence In this section, we first show that local isotropy is tenable for the (low) Reynolds numbers investigated here. Figure 3 reports both Eu (k 1 ) (calculated from the time series signals of u using Taylor s hypothesis) and Ev (k 2 ) (calculated from the spatial signal of v in the lateral direction). The asterisk denotes normalisation by Kolmogorov scales. Noticeable in this figure is the fact

7 Journal of Turbulence 601 Figure 3. Distributions of E u (k 1 )ande v (k 2 ). : Comte-Bellot and Corrsin (1971), R λ = 41. Symbols: E u (k 1 ), (R M = 3200), (R M = 1600), (R M = 400). Lines: E v (k 2 ), solid: R M = 3200, dashed: R M = 1600, dash-dotted: R M = 400. that Eu (k 1 ) = E v (k 2 ), i.e. the local isotropy is valid despite the small values of R λ, and that Eu (k 1 ) and E v (k 2 ) can be used interchangeably. We now proceed to investigate the validity of KSH1 in the spectral space. As already observed in Figure 1, when normalised with the Kolmogorov length and velocity scales, the velocity spectra collapse on a universal curve for high wave numbers in accord with KSH1, although one of the main requirements for KSH1 (R λ is large) is violated. However, Figure 3 shows that when R λ becomes small, about 20 or smaller, the Kolmogorov normalised spectra do not follow the spectrum of Comte-Bellot and Corrsin (1971, hereafter denoted as CBC) in the dissipative range (k1 0.4); the CBC spectrum is used as a reference for which KSH1 is tenable. There is a systematic shift towards high wave numbers as R λ decreases. This low Reynolds number dependence of the spectra is similar to that observed in the 3D velocity spectra of Mansour and Wray [4] in a 3D decaying periodic box turbulence at R λ = 15 and 20. Similar results are obtained for Kolmogorov-normalised second-order structure functions (not shown here). For R λ 20, the collapse at small scales r 5 is noticeable, whereas for lower Reynolds numbers, there is no collapse in the dissipative range, thus confirming the KSH1 breakdown observed for spectra. Therefore, the critical Reynolds number below which KSH1 is not tenable in this flow is (R λ ) c 20. In the following, we will study the phenomenology of the flow, when the Reynolds number is either slightly larger or smaller than (R λ ) c 20. Since the validity of KSH1 is investigated for second-order statistics, it is natural to explore the dynamical transport equation for these quantities, with the aim to quantify the balance between energy transferred through different effects (molecular, large-scale). 4. Dynamical and phenomenological viewpoint on the KSH1 breakdown In this section, the critical value of (R λ ) c 20 below which KSH1 breaks down is corroborated by the analysis of the relative physical effects at play in energy transfers. The spectra of Figure 3 show that there is no indiscernible scaling range between the large scales and the dissipative scales, implying that the energy transfer, represented by T(k) in the spectral

8 602 L. Djenidi et al. space, is likely to be weak, if not negligible. One can then hypothesise that the breakdown of the Kolmogorov normalisation is related to the level of the energy transfer between the large and small scales. To assess this supposition, we investigate the scale-by-scale (sbs) energy budget. General similarity solutions of the sbs energy budget equation are first investigated (Section 4.1). The results will be presented in Section General self-similar solutions to scale-by-scale energy budget The transport equation of the second-order moment, (δq) 2 = (δu i )(δu i ), is as follows [13]: 2ν d (δq)2 ) dr (δu)(δq) 2 +I q = 4 ɛ r, (3) 3 where I q represents the contribution of the non-homogeneous motion (here, the large-scale decay), and, for grid turbulence, is given by I q = U r 2 r 0 s 2 (δq)2 ds, (4) x wheres is a dummy variable and U is the longitudinal mean velocity. Equation (3) requires only local isotropy and essentially relates (δq) 2 to δu(δq) 2. At large r, Equation (3) correctly represents the one-point energy budget (i.e. ɛ = 1 d q 2 ) for spatially decaying 2 dx homogeneous and isotropic turbulence. In the following, we investigate how Equation (3) is compatible with KSH1. For r 0, only two terms are balanced in Equation (3), which reduces to 2ν d (δq)2 ) dr 4 ɛ r. (5) 3 As the separation r increases, the next terms which become non-negligible are δu(δq) 2 and I q. Finally, for very large r, the only remaining term is I q. This supposition is in agreement with the experimental data shown in Figure 4 (see also Figure 12 of [14]). (However, the latter concerns the spectral space and particular care should be paid to the correspondence between real and spectral spaces.) As already shown in [15], one possible similarity solution of Equation (5) requires that the mean energy q 2 u i u i decays according to a power-law behaviour, namely, q 2 x n. This requirement is satisfied approximately by the present simulations, with n We first briefly recall the conditions under which Equation (5) can satisfy similarity. Following [16,17], (δq) 2 =Qf ( r L), (6) where L is a characteristic length scale and Q (with dimensions of velocity squared) is a scale that characterises (δq) 2 ; L and Q depend only on x. Scales L and Q are not yet known. The dimensionless function f depends not only on r/l, but also on initial conditions (for simplicity of expression, this latter dependence is not indicated). After substituting

9 Journal of Turbulence 603 Figure 4. Terms in Equation (3) divided by ( 4 ɛ r), for R 3 M = 1600 at x/m = 55, R λ = 11, normalised with respect to λ scale. Black symbols: calculated from the single-point time series; open symbols: calculated from the actual spatial series; circles: A = 2ν d (δq)2 ) /( 4 ɛ r); stars: B = dr 3 (δu)(δq) 2 /( 4 ɛ r); squares: C = I 3 q/( 4 ɛ r)); diamonds: A + B + C. Dashed lines and the same 3 symbols as for LBM data: measurements (R M and R λ = 43, [18]). Equation (6) into Equation (5), we obtain 2 νq L f = 4 3 ɛ L r L, (7) where a prime denotes differentiation with respect to r/l. In Equation (7), the dependence of f on r r/l is implicitly assumed. After multiplicationby(l/νq), Equation (7) could be written as [2]f = 4 3 [ ] ɛ L 2 r. (8) νq Each term in the latter expression has been written as a product between a term which depends on x only (inside square brackets) and a term which depends on r. Kolmogorov similarity (first hypothesis) requires that the two terms in Equation (8) that depend on x must evolve in the streamwise direction in the same way, i.e. ɛ L 2 Q(x) = const. (9) Since here we are considering Kolmogorov similarity, which focuses on small scales, then the velocity similarity scale should be the Kolmogorov velocity u K (Equation (1) in this paper). Equation (9) then implies that the similarity length scale is L η, the Kolmogorov scale.

10 604 L. Djenidi et al. Note that when the focus is over the whole range of scales, or when either of the terms δu(δq) 2 or I q cannot be neglected, then the complete Equation (3) should be considered and the equilibrium similarity is a possible solution with L = λ and Q q 2 (the reader is referred to [15] for further details on this development). Although equilibrium similarity and KSH1 may be compatible, they apply to different ranges of scales (overlapping at the level of the smallest scales). Our previous analysis points to a necessary condition for KSH1 to be valid, i.e. both terms (δu)(δq) 2 and I q should be negligible with respect to the other two terms (involving the kinematic viscosity and the mean energy dissipation rate ɛ ). When the Reynolds number becomes sufficiently small, so that there is no scale separation between viscous and large-scale terms, then all terms in Equation (3) are non-negligible, and if some similarity was to apply, then the equilibrium similarity has ultimately the best prospect to hold Results An example of the sbs budget is shown in Figure 4, for two types of data: - Experimental data, for R λ = 43 (dashed lines and symbols). - LBM data at R M = 1600 (R λ = 11, at x/m = 55). Because of the very low Reynolds numbers, and with the arguments developed in the previous subsection, the involved terms are λ-normalised. The LBM budget is calculated from two sets of data: a single-point time series using Taylor s hypothesis, which replicates hot-wire measurements (U o dt corresponds to the spatial separation r, dt is the time lag), and from the actual spatial signals which allow spatial derivatives to be calculated directly. The first set is a single-point time series using Taylor s hypothesis, (e.g. (q(u o t) q(u o t + U o dt))2 ), which replicates hot-wire measurements. The second set of data is from the actual spatial signal (e.g. (q(x) q(x + r)) 2 ) which allows the derivatives to be calculated directly. For the first set of data, q 2 (U o t + U o dt) remains constant as dt increases, while for the second set of data, q 2 (x + r) decreases as r increases. Thus, along the separation r, it is expected that (q(u o t) q(u o t + U o dt)) 2 ) differs from (q(x) q(x + r)) 2 when the separation r is relatively large for the inhomogeneity to affect (δq) 2 and I q. This occurs for r 3λ. Altogether, the budgets calculated from the two signals are in reasonably good agreement. As expected, the budget is dominated by the viscous term at small r and by the largescale term at large r. Also, the transport equation of (δq) 2 is relatively well balanced at all scales. While the measured sbs budget exhibits similar features to the LBM sbs budget (notice, for instance, that the maximum of (δu)(δq) 2 max occurs at the same separation r in terms of λ), there are also some significant differences. For the measurements, (δu)(δq) 2 I q for r r c, where r c is the separation at which (δu)(δq) 2 =I q. For the LBM data, (δu)(δq) 2 I q for all r, which implies that as the Reynolds number decreases beyond a critical value, the contribution of (δu)(δq) 2 to the energy budget is the smallest at all scales. This result is fully supported by all the LBM simulations for which R λ 20. These results indicate that for R λ 20, below which the Kolmogorov normalisation breaks down, the contribution of (δu)(δq) 2 is smaller than the other two terms at all scales. We have also emphasised that the maximum of (δu)(δq) 2 occurs at r λ. Since

11 Journal of Turbulence 605 λ/η R 1/2 λ, the requirement for KSH1 to be valid translates to λ/η N η 5, therefore, R λ 25, in agreement with our findings. Moreover, using a phenomenological argument, Antonia et al. [17] showed that can also be estimated using a phenomenological argument. The ratio u 2 / ɛ may be interpreted as a characteristic time of the energy-containing eddies, namely, τ E u2 ɛ, (10) whereas the dissipative range time scale is given by the Kolmogorov time τ D For Kolmogorov scaling to apply, one expects that ( ) ν 1/2. (11) ɛ τ D τ E. (12) To quantify this inequality, we introduce the factor N between the two characteristic times, namely, τ D τ E N, (13) with N 10, corresponding to one order of magnitude. Therefore, ( ) ν 1/2 1 u 2 ɛ N ɛ, (14) which finally leads to the value of the threshold Reynolds number for which the ratio of the characteristic times is larger or equal to N, (R λ ) t, namely, (R λ ) t N 15 39, (15) is the critical value below which the Kolmogorov scaling is no longer tenable. However, the present simulations point to a critical value of about 20 for R λ. This implies that N is 5, which is confirmed by Figure 5, where the streamwise evolution of the ratio τ E /τ D is displayed for R M = 3200, 1600 and 400. As R λ decreases, N decreases in agreement with (15). For R M = 3200 and x/d 40 (R λ 19), the ratio is indeed about 5, whereas for R M = 400 (R λ 3.5), the ratio is almost equal to Predictions of the spectrum in the far dissipative range for low Reynolds numbers The same analysis holds in spectral space (actually, the equilibrium similarity had first been developed in spectral space [16]). Obviously, the same conclusions are valid. For low Reynolds numbers, at which the non-linear spectral energy transfer cannot be neglected, equilibrium similarity has the best prospect to describe the involved statistics. The similarity scale is L [ν(t t o )] 1/2,orL λ, and the velocity similarity scale is u 2. Note that

12 606 L. Djenidi et al. Figure 5. Timescale ratio τ E /τ D. Solid line: R M = 3200; dashed line: R M = 1600; dash dotted line: R M = 400. George [16] pointed out that the λ-scaling may be non-universal, in the sense that it depends on initial conditions (e.g. R M ). Interestingly, the collapse of all spectra represented in Figure 6 is reasonable. At this low Reynolds number, this equilibrium similarity is not too surprising. Very few studies are dedicated to the final period of the decay, and most of these suppose that the energy transfer (in either real or spectral space) is negligible. Ling and Huang (1970) [14] focused on weak turbulence (with R λ 3 30) and showed experimentally that the spectral transfer was weak, but not negligible. Under these conditions, their normalised Figure 6. Velocity spectra E u normalised by u 2 and [ν(t t 0 )] 1/2 at x/m = 55. : R M = 400, R λ = 3.5; : R M = 1600, R λ = 11; : R M = 3200, R λ = 19. The solid line represents Equation (17).

13 Journal of Turbulence 607 spectrum with respect to the diffusion length λ and u 2 can be written in the form E u + (k+ 1 ) = E u (k 1 ) u 2, (16) [ν(t t 0 )] 1/2 where k + 1 = k 1[ν(t t 0 )] 1/2 (Ling and Huang defined [ν(t t 0 )] 1/2 as a diffusion length) can take the analytical form E + u (k+ 1 ) = αe αk+ 1, (17) with α Expression (17) differs from the usual spectral form f (k + 1 ) = βe αk+2 1, (18) when the spectral transfer is assumed to be negligible [19]. Although the Reynolds number of our simulations is larger than 3, it falls within the range of Reynolds numbers investigated by Ling and Huang [14]. The present data follow Equation (17) rather well, as seen in Figure 6, which shows the LBM velocity spectra E u + (k+ 1 ) with the theoretical distribution calculated from Equation (17) with α = 3.16 ([14]), t = xu, the origin t o was arbitrarily set to zero. The numerical data all lie close to the theoretical line (17), although systematic deviations from the line occur at the largest k + 1. The data for the lowest R λ deviate most from Equation (17), which may be attributed to a smaller spectral transfer. Further insight into the validity of Equation (17) can be gained by considering the following model for the spectrum at large values of k 1 : E u + (k+ 1 ) e αk+p 1. (19) We aim at determining the value of the exponent p. Since isotropy is valid, the behaviour of the 1D spectrum is similar to that of the 3D spectrum, for which the transport equation reads [20] E(k) = T (k) 2νk 2 E(k), (20) t where T(k) is the spectral energy transfer function (E(k) and T(k) are functions of the time t, but for convenience, the dependence on t has been dropped). At large wave numbers, E(k) t 0(seeFigure12of[14]), therefore, T(k) = 2νk 2 E(k). The transfer term at a wave number k is modelled as thek-derivative of the energy flux. The latter is the energy ke(k)) transferred during a characteristic time at that wave number τ(k) [21]. Under these conditions, the analytical solution for large values of k is ( E(k) small-scale exp α k 0 ) sτ(s)ds, (21) where τ(k)is[22], [21], namely, [ k 1/2 τ(k) s E(s)ds] 2. (22) 0

14 608 L. Djenidi et al. By modelling the spectrum E(k) as(e.g.[23]) - E(k) k α s1 for k k L (with α s1 > 0); and - E(k) ɛ α ɛ k α s2 for k L k k D (with α s2 < 0 and α ɛ 2/3); - exponential decay for E(k) fork D k. Here, k L represents the wave number corresponding to the integral scale, and k D is the upper bound of the restricted scaling range. By following the calculations developed in CBC, but with explicit consideration of the exponents α s1 and α s2 (α s2 is allowed to be different from the asymptotic value of 5/3), it is straightforward to show that the decay exponent of the kinetic energy n (i.e. q 2 x n ) is given by n = α ɛ (α s1 + 1) α s1 α s2 α ɛ (α s1 + 1). (23) In developing Equation (23), the exponential decay region was assumed not to contribute to the kinetic energy (which is the integral of the spectrum). In our simulations, n 1.56, α s Equation (23) then yields α s2 1, also in agreement with Figure 10 in Mydlarski and Warhaft [24]. With all these values, Equation (21) provides p = (3 + α s2) = 1 + α s2, (24) 2 with α s2 [0.8, 1], p [0.9 1], which is in reasonable agreement with Equation (17). We recall here that Equation (17) was obtained in [14] exclusively on the basis of experimental data. For increasing values of α s2 towards the asymptotic value of 5/3, p gradually increases and the upper limit is 4/3 (Pao s spectrum). Note that this superior limit takes into account the existence of the energy transfer term. In summary, the behaviour of the spectrum in the far dissipation range can be modelled adequately via the local power law in the restricted scaling range, the crux of this approach being the retention of the (non-negligible) transfer term, as well as its modelling via the characteristic time scale. 6. Conclusions Direct numerical simulations based on the LBM were carried out for low R λ turbulence downstream of a grid made up of flat square elements to investigate the effect of low Reynolds number on Kolmogorov scaling. It is found that when the Reynolds number R λ falls below about 20, the Kolmogorov normalised spectra deviate from those at higher R λ ; the deviation increases with decreasing R λ. Interestingly, local isotropy is still reasonably well satisfied during the Kolmogorov breakdown, implying that the main reason for this breakdown lies elsewhere. The present results show that KSH1, which this paper focuses on, does not require the existence of an inertial range. While it may seem remarkable that KSH1 is validated at a much lower Reynolds numbers than required for KSH2 [25] in grid turbulence (R λ > 1000, [18]), this may not be too surprising. Indeed, KSH1 applies to small scales which include both inertial range (if it exists) and dissipative scales; however, the latter are less impacted on by the large scales than the former. One can then expect that, as the Reynolds

15 Journal of Turbulence 609 number increases, the dissipative scales should satisfy KSH1 first; then, as the Reynolds number attains very high values, the inertial range is established in conformity with KSH2. A further investigation of the similarity solutions for the sbs energy budget highlighted the following: KSH1 may hold for small scales, if and only if the energy transferred by both turbulence and large scales is negligible with respect to the molecular effects. Another similarity solution (called equilibrium similarity, for which the characteristic velocity is the variance of the velocity field; the characteristic scale is the Taylor microscale) is valid when all terms in the energy budget are of the same order of magnitude. This latter scenario is most likely to be valid at low Reynolds numbers. An analysis of the sbs energy budget revealed that the breakdown occurs when the contribution of the energy transfer becomes smaller, irrespectively of the scale, than the contributions from the viscous and non-homogeneous terms, implying a very weak scale separation. Using a phenomenological argument, Antonia et al. [17] showed that the requirement on R λ for the Kolmogorov scaling to hold is R λ N 15, where N is the ratio between the ( energy-containing timescale (τ E u2 ɛ ) and the dissipative range timescale (τ ν 1/2). D ɛ ) Assuming N = 10, these authors show that R λ = 39 is the critical value below which the Kolmogorov scaling is no longer tenable. However, the present simulations point to a critical value of about 20 for R λ. This supports a phenomenological argument which establishes the condition R λ N 15 for Kolmogorov scaling to hold, where N is the ratio between the energy-containing timescale and the dissipative range timescale. The numerical results support a value of 5 for N, yielding R λ 20. For R λ (R λ ) c, we provided an analytical expression of the spectra in the far dissipative range. The crux of this development is modelling the energy transfer term as a function of the characteristic time, in which the strain imposed by larger scales is considered. Acknowledgements The financial support of the Australian Research Council is gratefully acknowledged. References [1] A. Kolmogorov, On the degeneration (decay) of isotropic turbulence in an incompressible viscous fluid, Dokl. Akad. Nauk SSSR 31 (1941), pp [2] S.G. Saddoughi and S.V. Veeravalli, Local isotropy in turbulent boundary layers at high Reynolds number, J. Fluid Mech. 268 (1994), pp [3] L. Djenidi and R.A. Antonia, A spectral chart method for estimating the mean kinetic energy dissipation rate, Exp. Fluids 53 (2012) pp [4] N.N. Mansour and A.A. Wray, Decay of isotropic turbulence at low Reynolds number, Phys. Fluids 6 (1994), pp [5] U. Frisch, B. Hasslacher, and Y. Pomeau, Lattice gas automata for the Navier Stokes equations, Phys. Rev. Lett. 56 (1986), pp [6] L. Djenidi, Lattice Boltzmann simulation of grid-generated turbulence, J. Fluid Mech. 552 (2006), pp [7] L. Djenidi, Study of the structure of a turbulent crossbar near-wake by means of Lattice Boltzmann, Phys. Rev. E 77 (2008),

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