Analysis of the Kolmogorov equation for filtered wall-turbulent flows

Transcription

1 J. Fluid Mech. (211), vol. 676, pp c Cambridge University Press 211 doi:1.117/s Analysis of the Kolmogorov equation for filtered wall-turbulent flows A. CIMARELLI AND E. DE ANGELIS DIEM, Università di Bologna, Via Fontanelle 4, Forlì, Italy (Received 9 May 21; revised 1 December 21; accepted 29 January 211; first published online 4 April 211) The analysis of the energy transfer mechanisms in a filtered wall-turbulent flow is traditionally accomplished via the turbulent kinetic energy balance, as in Härtel et al. (Phys. Fluids, vol. 6, 1994, p. 313) or via the analysis of the energy spectra, as in Domaradzki et al. (Phys. Fluids, vol. 6, 1994, p. 1583). However, a generalized Kolmogorov equation for channel flow has recently been proven successful in accounting for both spatial fluxes and energy transfer across the scales in a single framework by Marati, Casciola & Piva (J. Fluid Mech., vol. 521, 24, p. 191). In this context, the same machinery is applied for the first time to a filtered velocity field. The results will show what effects the subgrid scales have on the resolved motion in both physical and scale space, singling out the prominent role of the filter scale compared to the cross-over scale between production-dominated scales and inertial range, l c, and the reverse energy cascade region Ω B. Finally, we will briefly discuss how the filtered Kolmogorov equation can be used as a new tool for the assessment of large eddy simulation (LES) models. Classical purely dissipative eddy viscosity models will be analysed via an aprioriprocedure. Key words: turbulent flows, turbulence modelling 1. Introduction It is well established that in turbulent flows the energy-carrying structures are directly affected by the boundary conditions; hence, they are highly non-universal and generally anisotropic, while the small scales tend to be more homogeneous and isotropic than the large ones. Therefore, it is thought that relatively simple and universal models can be used to describe the last part of the energy spectrum, when, for example, a large eddy simulation (LES) approach is required for the computation of moderately large Reynolds number flows. The most important feature of such models should be their ability to accurately reproduce the energy transfer between resolved and unresolved scales, e.g. see Rogallo & Moin (1984). In this context, most of the commonly used LES models assume that the main role of the subgrid scales is to remove energy from the large resolved motion and dissipate it through the action of a diffusion mechanism analogous to the viscous forces, see Sagaut (21) and Kraichnan (1976), leading to the eddy viscosity concept. These assumptions are based on the idea of an inertial range in the spectrum of scales. Indeed, as asserted by the 4/5 law, Kolmogorov (1941), in the inertial range the energy flux is independent of the address for correspondence: e.deangelis@unibo.it

2 Analysis of the Kolmogorov equation for filtered wall-turbulent flows 377 scale under consideration, is from large to small scales and it is proportional to the viscous energy dissipation. This picture is claimed to be highly universal. No matter what the large-scale processes which feed turbulence are, the small scales of large Reynolds number flows are believed to behave according to this scenario and many LES models take inspiration from it. Indeed, in the early attempts, the Kolmogorov energy cascade concept was instrumental in proposing to reproduce the inertial range effects by defining suitable eddy viscosity (Smagorinsky 1963; Lilly 1967). On the other hand, the scale-invariance assumption of the inertial range was used both to postulate that the features of the subgrid velocity field are similar to those at large resolved scales (Bardina, Ferziger & Reynolds 198) and more recently to develop a dynamic procedure for the evaluation of an improved eddy viscosity coefficient (Germano et al. 1991). This kind of approach has given good results in homogeneous and in unbounded shear flows but less in wall turbulence. In fact, in the proximity of a solid wall the anisotropic turbulent production affects most of the turbulent eddies and the shear is sufficiently strong to hinder isotropy recovery even at small inertial scales, see e.g. Antonia, Djenidi & Spalart (1994), Casciola et al. (25) and Jacob et al. (28). Besides the above observations, we like to stress that the single most striking phenomenon observed in wall-bounded flows is the complete modification of the Richardson scenario up to a reverse energy cascade in the buffer layer, as quantified in Marati, Casciola & Piva (24). In these kinds of flows, indeed, a transfer through scales is coupled with spatial momentum flux increasing the complexity of the phenomena that have to be accounted for. In LES context, such a rich scenario has been classically addressed for the assessment of energy exchange between resolved and unresolved scales. As far as these ideas are concerned, we should mention the work of Domaradzki et al. (1994) where a spectral analysis of the energy transfer among wavenumbers belonging to the wall-parallel directions was performed for two datasets of confined turbulent flows, i.e. Rayleigh Bénard convection and a channel flow, at a moderate Reynolds number. This work considers the instantaneous interscale energy transfer between modes at a given distance from the wall. In order to assess the nonlinear triadic interactions, on the basis of the energy cascade, the spectrum is divided into welldefined regions and the transfer between these bands is studied. One of the main findings is that energy transfer processes are mostly between contiguous bands but due to non-local interactions, and this information should be valuably used when modelling subgrid stresses is addressed. In a complementary way, some issues related to the inhomogeneity of the channel flow are discussed by Härtel et al. (1994). In this contribution the authors analysed the behaviour of the subgrid stresses and their effects on the single-point turbulent kinetic energy balance, highlighting in particular the presence of a backward transfer in the buffer layer when a certain value of the cutoff filter is chosen. Similar aspects are also discussed in Piomelli, Yu & Adrian (1996), again for a channel flow, where an analysis of the transport equation for the subgrid stresses is performed. Forward and backward subgrid energy transfer is related to the dynamics of the quasi-streamwise vortices which live in the wall region, extrapolated via conditional averages of the velocity field. More in the context of computational issues for LES, Baggett, Jimenez & Kravchenko (1997) pointed out the importance of identifying the anisotropic dynamics in shear flows. Indeed, in wall-bounded flows, the lower bound limit for the size of anisotropic eddies decreases with wall distance, leading to a large number of degrees of freedom which should be computed explicitly when for example isotropic models are used.

3 378 A. Cimarelli and E. De Angelis In the present paper we intend to extend these ideas by addressing the dynamics of the coupling between the subgrid and the resolved scales via the analysis of a filtered DNS dataset of a turbulent channel flow. The tool we propose for such a study is the same as used by Marati et al. (24) for the analysis of the spatial fluxes and energy cascade in the same flow, i.e. the evolution equation for the second-order structure function developed by Hill (22). The resulting generalized Kolmogorov equation describes the scale-dependent dynamics when spatial fluxes are present. Furthermore, it is possible to quantify how the framework of the classical Richardson energy cascade is modified by anisotropy and inhomogeneity. This is a crucial point in a context of subgrid energy transfer modelling. Here a preliminary analysis performed using an unfiltered Reynolds number dataset at Re τ = 298 enables the identification of specific ranges of scales at various distances from the wall where the relevant processes occur. A scale is evaluated as the characteristic one above which the dynamics of the fluctuations are governed by production, while in the buffer layer, the border of an area can be traced embracing the range of scales and distances from the wall where a reverse energy transfer occurs. An appropriate extension of the Kolmogorov equation to a filtered case, proposed here for the first time, adds those related to the subgrid scales to the processes accountable for in the turbulent channel flow. The resulting approach allows for the evaluation of the contributions due to the subgrid terms, at different values of the filter length, for different scales and wall distances. The analysis enables us to appreciate when subgrid stresses become relevant in the dynamics of the resolved motion and, if this is the case, the physical nature of their action that should be considered when modelling is proposed. In the context of LES modelling, the possibility of using the filtered Kolmogorov equation as a new tool for the assessment of proposed models will be outlined. Indeed, as pointed out in Meneveau (1994) for isotropic turbulence, the filtered Kolmogorov equation provides the correct framework to evaluate if the energy transfer rate is correctly reproduced. The paper is organized as follows. Section 2 is a description of the analysis of the Kolmogorov equation for the unfiltered turbulent channel flow data. The evolution equation for the second-order structure function of the filtered velocity field is discussed in 3. The two sub-sections included are devoted to the analysis of the scale budgets of the filtered field in the logarithmic and bulk regions and the buffer region, respectively. An outline of the assessment procedure for a few common LES models and the final remarks will conclude the paper. 2. Scale-by-scale analysis of the unfiltered dataset The considered dataset is the simulation of a turbulent channel flow at a friction Reynolds number Re τ = u τ h/ν = 298, where, as customary, u τ is the friction velocity, ν is the kinematic viscosity and h is half the channel height. The simulations have been carried out with a pseudo-spectral code for 5 large-eddy turnover times, defined as T = h/u c with U c the centreline velocity. All the statistics have been evaluated averaging over 1 configurations. Details of the numerical scheme can be found in Lundbladh, Henningson & Johansson (1992). The computational domain is 2πh 2h πh with grid points, respectively, corresponding to a resolution in wall units in the homogeneous directions of x + = z + =3.64, where the superscript + as customary implies non-dimensionalization in friction units. This very high resolution, quite unique for a turbulent channel flow, has been chosen, at the expense of the Reynolds number achievable, to capture the phenomena occurring in

4 Analysis of the Kolmogorov equation for filtered wall-turbulent flows 379 the dynamics of the velocity field up to the dissipative scale and to better appreciate the alterations taking place in the statistics when the small-scale effects are removed by an explicit filtering. A detailed analysis of the single-point statistics of the present dataset can be found in Saikrishnan et al. (211). Because of the inhomogeneity of the channel flow, the balance equation for the turbulent kinetic energy, q = u i u i /2, has to account also for the spatial energy redistribution. Due to the symmetries of the domain the corresponding spatial flux reduces to a single scalar component φ =( qv + pv /ρ νd q /dy), which presents three different contributions, one inertial, one due to pressure and a viscous one. Hence, for this particular case the equation for q reads dφ dy = uv du dy ɛ, (2.1) where the divergence of the flux balances with production uv du/dy and pseudodissipation ɛ = ν( u i / x j )( u i / x j ). The y-distribution of the terms in (2.1) allows for the identification of different regions in the flow. The first one close to the wall, dominated by dissipation, is normally referred to as the viscous sublayer. In turn, the intermediate region where production is predominant is generally called the buffer layer. Above this region there is a zone, traditionally called the logarithmic layer, where for very high Reynolds number flows an overlap between an inner and an outer scaling appears and production and dissipation balance (Pope 2). This asymptotic state is not reached at the present Reynolds number; however, in the rest of the paper we will conventionally refer to the region 4 <y + < 18 as logarithmic, its relevant features being that the production approximatively balances with dissipation, see figure 3(a), and the rate of spatial energy flux is nearly zero. This results in a constant spatial flux of energy which crosses this equilibrium layer feeding the fluctuations in the bulk central region of the channel where production vanishes. A somewhat complementary class of processes is instead identified by the classical description of homogeneous turbulence where, according to Kolmogorov theory, the most relevant physical aspect is the energy transfer across the scales. In addition, for inhomogeneous flows, the relevant processes such as production, dissipation and spatial flux take place in various ranges of scales that may vary appreciably in the different flow regions. As a consequence, a full understanding of these interacting phenomena requires a detailed description of the processes occurring simultaneously in physical and scale space. To analyse the energy content of a given scale as a function of the spatial position, we study the second-order structure function δu 2, where δu 2 = δu i δu i and δu i = u i (x s + r s ) u i (x s ) is the fluctuating velocity increment, that can be thought of as a scale energy at r = r s r s. For the case of a channel flow, δu 2 is a function of the separation vector r i and of the mid-point X ci =1/2(x i + x i ), allowing us to describe the scale-dependent energy processes in the presence of inhomogeneity. Its governing equation in wall flows is the generalized Kolmogorov equation developed by Hill (22), which for a flow with longitudinal mean velocity U(y) reads δu 2 δu i r i + δu2 δu r x +2 δuδv ( ) du + v δu 2 dy Y c = 4 ɛ +2ν 2 δu 2 r i r i 2 ρ δpδv Y c + ν 2 2 δu 2 Y c 2, (2.2)

5 38 A. Cimarelli and E. De Angelis where the asterisk denotes a mid-point average, i.e. u i =(u i (x s)+u i (x s ))/2 and the Y c dependence is associated with inhomogeneity. The numerical analysis of (2.2) in a turbulent channel flow was already performed in Marati et al. (24) on a lower Reynolds number dataset. The complete scale analysis on the current dataset can be found in Saikrishnan et al. (211); however, in the present section some of those results are reported in an appropriately revisited form that will be instrumental for the comprehension of the filtered data analysis. The generalized equation (2.2) can also be rewritten as r Φ r (r,y c )+ d Φ c (r,y c )=s(r,y c ), (2.3) dy c where Φ r = δu 2 δu + δu 2 δu 2ν r δu 2, Φ c = v δu 2 +2 δpδv /ρ νd δu 2 /2dY c and s = 2 δuδv (du/dy) 4 ɛ. Two kinds of scale-energy fluxes appear, namely Φ r which identifies the transfer through the scales and Φ c which is the flux of scale energy in physical space. These fluxes balance with a source term s which accounts for energy production and dissipation. When homogeneous and isotropic conditions are recovered, (2.3) reduces to δu 2 δu i / r i = 4 ɛ which is a relation analogous to the 4/5 law. This states that the energy flux through the scales equals the rate of energy dissipation and therefore is constant and from large to small scales. There is no direct energy injection and extraction, see Frisch (1995). Such a law is associated with the phenomenology of the Richardson energy cascade, and the range of scales where it holds will be hereafter referred to as the inertial range. To highlight the scale processes, it is useful to consider the r-averaged form of (2.2). In a channel flow however the classical integration over a ball of radius r, seenie& Tanveer (1999), is unfeasible due to the strong y-dependence of the flow s quantities. Hence, after setting r y =, the r-average is performed on a square domain of side r belonging to wall parallel planes, Σ, seemaratiet al. (24) for the details. This procedure allows us to analyse the terms in (2.2) as a function of the single scale parameter r and of the wall distance Y c. Considering that the second term on the left-hand side of (2.2) vanishes since δu = for r y =, the r-averaged form of (2.2) follows as T r + Π + T c = E + D r + P + D c, (2.4) where T r = 1 δu 2 δu i dr r 2 x dr z, D r = 1 2ν 2 δu 2 dr Σ r i r 2 x dr z (2.5) Σ r i r i are the contributions to the scale-space energy transfer due to the inertial fluctuations and viscous diffusion; T c = 1 r 2 P = 1 r 2 Σ Σ v δu 2 dr x dr z, D c = 1 ν 2 δu 2 dr Y c r 2 2 x dr z, Σ 2 Y c 2 δpδv ρ Y c dr x dr z (2.6) are the inhomogeneous contributions to the spatial flux related to the inertial fluctuations, viscous diffusion and pressure velocity correlation; Π = 1 ( ) du 2 δuδv dr r 2 x dr z, E = 1 4 ɛ dr Σ dy r 2 x dr z (2.7) Σ are the energy production by mean shear and the rate of energy dissipation. It is useful to group together some terms of (2.4) in a sort of effective production,

6 (a) Analysis of the Kolmogorov equation for filtered wall-turbulent flows 381 (b) r + r + Figure 1. Scale-energy balance 2.8 in the log layer Y c + = 16 (a) and in the buffer layer Y c + =2 (b). The values of Π e (solid line), E e (dashed line) and T r (dash-dotted line) are expressed in viscous units; the same convention is used in all the plots of the paper Π e = Π + T c P, and modified dissipation rate, E e = E + D r + D c. Therefore, the r-averaged balance can be rewritten as Π e (r, Y c )+T r (r, Y c )=E e (r, Y c ), (2.8) whose analysis permits us to characterize the different regions of the channel in terms of scale-by-scale dynamics. In the logarithmic layer, the large-scale production range is followed by a range dominated by the inertial energy cascade which is closed by viscous diffusion, see figure 1(a). A cross-over scale l c which splits the space of scales into an inertial range at small r from a production-dominated range at large r is identified as Π(l c,y c )=T r (l c,y c ). l c is dimensionally related to the shear scale L s = ɛ/ S 3, with S =du/dy, which is found to be crucial for the small-scale isotropy recovery, see Casciola et al. (23, 25, 27), and for the subgrid scale stress modellization, see Gualtieri et al. (27). For large Reynolds number flows, in the logarithmic region of the boundary layer, where the production and dissipation are in equilibrium, the shear scale is expected to behave linearly with the distance from the wall, namely L s = κy. Instead, in the buffer layer, shown in figure 1(b), a direct energy cascade at small scales and an inverse energy cascade at large scales exist, see also figure 2(b). More details about the reverse energy cascade can be found in Marati et al. (24) and Domaradzki et al. (1994), where it is argued that such a process can be related to the dynamics of the coherent structures which live in this region. These are responsible to build up Reynolds stresses with a regeneration cycle of large structures and breakdown to small scales, see Robinson (1991) and Hamilton, Kim & Waleffe (1995). In a context of LES, the so-called subgrid dissipation is interpreted as the rate of energy flux between the scales larger and smaller than the filter scale. Assuming isotropy and homogeneity recovery at small scales of any flow for a sufficiently high Reynolds number, most of the LES models attempt to reproduce the energy cascade as described by the 4/5 law. But in wall turbulence the scale dynamics are dominated by the spatial energy flux and production by mean shear, and these processes strongly modify the energy cascade up to a reverse energy cascade in the buffer layer. Therefore, the assumption of an inertial range without production and spatial fluxes drops out when dealing with wall flows and should be taken into account in LES models. In order to rationalize the present scenario, it is useful to identify the various regions of the (r +,Y c + )-plane where the relevant energy processes take place. Firstly the curve

7 382 A. Cimarelli and E. De Angelis (a) 21 (b) r + 1 r Y + c Y + c Figure 2. (a) Characteristic length scales in the channel: the cross-over scale l c + (circle), the reverse energy cascade region Ω B (grey region), whose edge is the scale l B +, and the Kolmogorov scale η + (inset plot). (b) The energy cascade, T r (r +,Y c + ), as a function of the distance from the wall and of the scale. The uniformly spaced contour levels are shown by light and dark grey colours for positive and negative values, respectively. l c + (Y c + ) divides the plane into a production-dominated region at large scales and a transfer-dominated one at smaller scales. In addition, in the production-dominated area it is important to highlight the region Ω B where the energy cascade term T r changes sign leading to a reverse energy cascade from small to large scales. The edge of this region is the scale l + B (Y c + ) which splits the space of scales into a forward energy cascade at smaller r + from a reverse energy cascade at larger r +. This reverse energy cascade will be shown to be responsible for the backward energy transfer observed in large eddy simulation; see Leslie & Quarini (1979), Schumann (1995), Carati, Ghosal & Moin (1995), Kerr, Domaradzki & Barbier (1996), Domaradzki & Saiki (1991) and Piomelli et al. (1991, 1996) for the backward flux analysis and modelling. Let us note that this phenomenon leads to the opposite energy exchange between subgrid and resolved scales usually reproduced by the LES models which are based on the classical Richarson energy cascade. Both l c + and Ω B are shown in figure 2(a) as a function of the wall distance. 3. The Kolmogorov equation for the filtered velocity field In this section a scale-by-scale analysis similar to that just introduced will be applied to appropriately filtered DNS data. The turbulent fields are filtered with respect to the wall-parallel directions only, using a sharp cutoff filter in wavenumber space. Let us notice that such a procedure is quite established for the analysis of the physical behaviour of the subgrid scales. For example, with a similar approach Domaradzki et al. (1994) studied the instantaneous energy transfer mechanisms between welldefined spectral regions and Härtel et al. (1994) studied the single-point energy processes. In the present work we choose to analyse the results for filter lengths from l + F = 15 to 12 equal in the two directions parallel to the wall. This choice is aimed to study the effects of the subgrid scales as a function of a single filter parameter, varying with respect to the relevant scales of wall turbulence. The analysis presented in the following sections has been performed disregarding the filtering operation in the wall-normal direction. Indeed, in the near-wall resolved LES the required wall-normal resolution cannot be reduced such as in the homogeneous directions and, therefore, we expect a prominent role of the horizontal filter scale. In analogy with (2.2), an evolution equation for the filtered second-order structure function, δū 2 = δū i δū i, is derived and discussed for the first time. This generalized

8 Analysis of the Kolmogorov equation for filtered wall-turbulent flows 383 Kolmogorov equation specialized for the filtered velocity field in a turbulent channel flow reads ( ) δū 2 δū i + δū2 δu du +2 δūδ v + v δū 2 = 4 ɛ +2ν δū2 r i r x dy Y c r i r i 2 δ pδ v + ν 2 δū 2 4 ɛ 2 ρ Y c 2 sgs 4 τ ij δū i δτ i2δū i, (3.1) Y c r j Y c where τ ij = u i u j ū i ū j are the subgrid stresses, ɛ = ν( ū i / x j )( ū i / x j ) is the resolved viscous pseudo-dissipation, ɛ sgs = τ ij S ij is the subgrid dissipation with S ij =1/2( ū i / x j + ū j / x i ) the resolved strain-rate tensor and stands for average in the homogeneous directions. Let us point out that for this particular choice of the filter Ū = U, i.e. there is no alteration of the mean profile. Equation (3.1) allows us to analyse how the resolved processes change as a function of the filter scale l + F in different regions and scale range and to appreciate the effects of subgrid stresses both in physical and scale space. The three new terms represent the exchange of energy between grid and subgrid scales, a redistribution of the resolved scale energy in the spectrum of scales and in physical space, respectively. The r-averaged form of (3.1) is as follows: T r + T sgs r + Π + T c + T sgs c = Ē + D r + P + D c + E sgs, (3.2) where the effects of subgrid stresses appear together with the energy processes of (2.4), when evaluated with the filtered fields. Namely we can account for the redistribution of resolved energy in the space of scales and physical space: = 1 r 2 T sgs r Σ 4 τ ij δū i dr x dr z, Tc sgs = 1 δτ i2δū i dr r j r 2 x dr z (3.3) Σ Y c and the draining or sourcing of resolved energy: E sgs = 1 4 ɛ r sgs dr 2 x dr z. (3.4) Σ It is useful again to group together some terms in an effective resolved production Π e = Π + T c P and effective resolved dissipation Ē e = Ē + D r + D c, and the compact form of the r-averaged balance (3.2) is as follows: ( ) ( ) Π e + Tc sgs + T r + Tr sgs =(Ēe + E sgs ). (3.5) In the following, a detailed analysis of the scale-energy balance (3.2) is performed using the filtered DNS dataset. Energy transfer in the resolved motion will be discussed and emphasis will be put on the action of the subgrid stresses on the resolved motion in both physical and scale space, underlining the role of the filter scale compared to the cross-over scale l c + (Y c + ) and the region of reverse energy cascade Ω B (r +,Y c + ). Before discussing the scale-by-scale balances for the filtered Kolmogorov equation, we briefly discuss some results obtained with the resolved single-point turbulent kinetic energy q = ū i ū i /2. The evolution equation for q in a turbulent channel flow is d φ dy + d dy ū iτ i,2 = ū v du dy ɛ ɛ sgs, (3.6) where φ =( q v + p v /ρ νd q /dy). The analysis of (3.6) allows us to evaluate the subgrid stress effects and the resolved-energy processes in the physical space for

9 384 A. Cimarelli and E. De Angelis (a) (b) y + y Figure 3. (a) y-behaviour of production and dissipation. (b) Subgrid dissipation, ɛ sgs,and in the inset, [(π π) ( ɛ ɛ )]. DNS (circle), l F + = 2 (solid line), l+ F = 3 (dashed line) and = 6 (dash-dotted line). l + F various filter lengths, see also Härtel et al. (1994) for a discussion of the turbulent channel flow. In figure 3(a) the resolved-energy production by mean shear, ū v du/dy, and the resolved viscous dissipation, ɛ, are shown for three values of the filter length l + F. In the core flow, the former remains unaltered and a depletion of the latter at an increasing filter scale is observed. On the contrary, approaching the wall, both these quantities decrease with l + F. Indeed, while dissipation is always present at the small subgrid scales even in the centre of the channel, the production mechanism involves more and more subgrid scales moving towards the wall as the shear scale, L s ky +, diminishes. Figure 3(b) shows ɛ sgs at the same values of l + F. From the inspection of the plot, the role of the subgrid dissipation is deduced. The resolved turbulent kinetic energy is drained in the core flow and in the viscous sublayer, while in the buffer layer, for the larger filter scales, it becomes opposite in sign in an increasing region, implying that the subgrid scales feed the large scale of motion, i.e. a backward energy transfer occurs, see Härtel et al. (1994) for a similar discussion. The backward energy transfer can be related to the excess of turbulent energy in the subgrid scales due to the large subgrid turbulent production. To quantify this observation let us consider [(π π) ( ɛ ɛ )], where π = ū v du/dy and π = uv du/dy. This quantity represents somehow the net balance of energy production and dissipation acting in the subgrid scales. Indeed, the spectral cutoff filter used allows us to divide the unfiltered turbulent kinetic energy production and dissipation into the sum of two distinct contributions from resolved and subgrid scales. The same applies for the turbulent kinetic energy where u 2 = ū 2 + u 2 sgs with 2 ūu sgs =, see Sagaut (21). From the inset of figure 3(b), it is observed that dissipation is dominant in the subgrid scales in the core flow and the viscous sublayer where a draining of resolved energy occurs, ɛ sgs <, whereas in the subgrid scales of the buffer layer, production overcomes the viscous effects leading to a subgrid energy excess which feeds the resolved motion. Indeed, the correspondence between the region where [(π π) ( ɛ ɛ )] > and the region where ɛ sgs > is remarkable. It is worth anticipating that the different behaviours in the bulk, logarithmic and buffer regions can easily be related to the two scenarios that will be discussed in the next sections respectively.

10 Analysis of the Kolmogorov equation for filtered wall-turbulent flows 385 (a) (b) r + r + Figure 4. Scale-energy balance in the logarithmic layer at Y c + = 16 for the unfiltered (lines) and filtered fields at l F + = 3 (symbols). (a) The terms in (3.5). Filtered field: Ē e,downward pointing triangles; Π e, upward pointing triangles; T r, circles, T sgs r, diamonds; T sgs c, stars; and Ē sgs, squares. Unfiltered field: E e, dashed line; Π e, solid line; and T r, dash-dotted line. (b) Contributions to the effective production. Filtered field: Π, upward pointing triangles; P, downward pointing triangles; T c, circles; and Tc sgs, stars. Unfiltered field: Π, solid line; P, dash-dotted line; and T c, dashed line. Inset: contribution to the effective dissipation. Filtered field: Ē, downward pointing triangles; E sgs,squares; D r, circles; and D c, upward pointing triangles. Unfiltered field: E, dashed line; D r, solid line; and D c, dash-dotted line. The vertical solid and dashed lines are l c + and l F + respectively Scale-by-scale budget in the logarithmic and bulk regions To ease the comprehension of the results in this region of the flow, let us recall that the logarithmic layer is characterized, see figure 2(a), by one relevant length, i.e. the cross-over scale l c +. Such a parameter represents somehow the border of the production-dominated range. In figure 4(a b), the scale-energy balance (3.2) in one representative plane for l + F = 3 is shown. At this location the filter scale is always smaller than the cross-over scale, l + F <l+ c and the subgrid scale dynamic reduces to the Richardson energy cascade. Such a condition leads to the preservation of the value for the statistical observables of the resolved fields, i.e. E e Ē e + E sgs, Π e Π e, T r T r. Indeed, for l + F <l+ c the energy production in the subgrid scales is negligible and the value obtained for the filtered field equals the unfiltered one, Π Π. Therefore, the rate of energy exchange between resolved and subgrid motions, E sgs, is exclusively determined by the viscous dissipation in the subgrid scales, implying that E Ē+E sgs, i.e. the sum of the subgrid and resolved viscous dissipation equals the unfiltered rate of energy dissipation. In these regions of the flow, the subgrid stress effects of resolved-scale-energy redistribution in physical and scale space, namely Tc sgs and Tr sgs, are both negligible as shown in figure 4(a). This is very important in the context of subgrid stress modelling. Indeed, the main goal of most LES models is the correct estimation of the energy exchange between grid/subgrid motions, E sgs, assuming that the resolved-energy redistribution due to subgrid stresses is negligible. The common interpretation of E sgs is a measure of the rate of energy transfer between scales larger and smaller than l + F. However, in a wall flow this rate of energy flux is predominantly determined by the balance between the processes of energy production and dissipation in the subgrid scales. Indeed, from figure 5 showing E sgs for different filter lengths and wall distances in the logarithmic and bulk regions, it is noticeable that as l + F approaches the Kolmogorov scale η+, E sgs decreases since a

11 386 A. Cimarelli and E. De Angelis Y + c = 25 + Y c = 15 + Y c = 8 Y + c = l F Figure 5. Scaling of the subgrid dissipation E sgs (l F + ) as a function of the filter scale l+ F different wall distances. The filled symbols correspond to l F + /l+ c =1. for larger fraction of energy dissipation occurs due to the resolved flow. When l + F /η+ is larger, E sgs increases monotonically accounting more and more for the energy dissipation, till it approaches the cross-over scale, l + F /l+ c 1, where a maximum is observed. For filter scales larger than the cross-over scale, l + F /l+ c > 1, E sgs decreases since production starts to be significant in the subgrid range and hence E sgs does not account exclusively for the energy dissipation but for the balance between the energy source and the sink processes occurring simultaneously in the subgrid scales. As a consequence Ē + E sgs E and Π Π. To ease the interpretation of the plot in figure 5, let us consider that in higher Reynolds number flows, when a large separation between l c + and viscous scales occurs in the logarithmic and bulk regions, a plateau is expected for η + l + F l+ c.inthis range the subgrid dissipation should equal the total amount of viscous dissipation, E sgs (l + F ) E, with a negligible contribution from the resolved scales, Ē(l+ F ), see Dunn & Morrison (23). The extension of this plateau represents a measure of the amplitude of the inertial range increasing with the distance from the wall. Since in such a range a Kolmogorov scaling sets in, δu 2 δu i r i /r ɛ, the behaviour of E sgs (l + F ) which is expected to be the same as that of the third-order structure function evaluated at the filter scale ( δu 2 δu i r i /r) l + F, see Cerutti & Meneveau (1998), should therefore equal E independently of the filter scale under consideration. These results show the logarithmic layer as a region where the dominant energy processes can be reproduced with the resolved scales with the exception of a fraction of viscous dissipation which can be recovered with E sgs if l + F <l+ c. The subgrid scales are dominated by a classical Richardson energy cascade closed by viscous diffusion at small scales. Therefore, the main role of the subgrid scales is to drain the resolved energy, E sgs <, without redistribution effects, Tc sgs Tr sgs. This kind of phenomenology can be reproduced with good results with the commonly used purely dissipative eddy viscosity models as shown in 4. The present scenario should be substantially Reynolds number independent with an unaltered role of the filter position with respect to the cross-over scale. The only expected difference is quantitative, since the value of the cross-over scale could change due to the appearance of longer and wider structures in this region for a larger Reynolds number. Indeed, as reported in Saikrishnan et al. (211), an increase of l c + is observed when the same

12 Analysis of the Kolmogorov equation for filtered wall-turbulent flows 387 (a).2 (b) r + r + Figure 6. Effect of the filter scale at Y c + = 22 for l F + = 3 (filled symbols) and l+ F =6 (open symbols). (a) Π e, upward pointing triangles; Ē e downward pointing triangles; and T r, circles. (b) Tr sgs, diamonds; Tc sgs, circles; and E sgs squares. The vertical solid and dashed lines are l F + =3 and l+ F = 6 respectively. present approach is applied to datasets with larger values of the friction Reynolds number. In the bulk flow, the physics governing the subgrid motion is still the same as that reported for the log layer. Indeed, in this region the cross-over scale l c + is very large up to infinity in the core of the channel where production vanishes. Therefore, the filter scale l + F is always reasonably smaller than l+ c. In these conditions, the subgrid motion is characterized by a Richardson energy cascade and, therefore, drains the resolved energy, E sgs <, without resolved-energy redistribution effects, Tc sgs Tr sgs, as shown in figure 6(b) for two different filter scales. We would like to stress that even if in the bulk the subgrid stresses dynamics is the same as that of the log layer, there is a change in the dynamics of the resolved ones that should be considered with some care. In fact, as shown in figure 6(a), a depletion of the physics captured with the resolved motion is observed despite l + F <l+ c. The resolved effective production Π e is considerably smaller with respect to the unfiltered one (not reported), and a depletion is observed increasing the filter scale even if both the filters shown, l + F =3 and l + F = 6, are smaller than the cross-over scale. The observed depletion in the effective production is due to a poorly resolved spatial transfer rather than to an error in the evaluation of the production; hence, this could be regarded as a non-local effect potentially worsened by a not fully resolved physics in the buffer layer as will be shown in the next section Scale-by-scale budget in the buffer layer The physical scenario of the buffer layer is enriched by the presence of a range of scales where a reverse energy transfer occurs, namely the area Ω B in figure 2(a). In this region the cross-over scale is very small, the filter scale is always larger than l c + and the physics captured by the resolved field is largely affected by the position of the filter scale l + F with respect to the edge of the region Ω B, namely the scale l + B. The scale-energy budget in the buffer layer for l + F = 3 is reported in figure 7(a b) for Y c + = 12, corresponding to the peak of the turbulent production. At this location l + F >l+ B and it is shown that, since a non-negligible part of the production acts in the subgrid scales, Π Π, see figure 6(b). Approaching the buffer layer the cross-over scale l c + reaches the Kolmogorov scale η, asshowninfigure2(a), and the anisotropic action of turbulent production is relevant at all scales. In this region, the most

13 388 A. Cimarelli and E. De Angelis (a).8 (b) r + r + Figure 7. Scale-energy balance in the buffer layer for Y c + = 12. See the caption of figure 4 for more details. significant scales of motion are not those greater than l c + but those belonging to the reverse cascade region Ω B. To corroborate the role of Ω B, a comparison between the unfiltered and filtered mixed structure functions, S 12 (r + )= δūδ v, evaluated at Y c + = 25 and averaged on a square of dimension r +,isreportedinfigure8for different values of the filter length l + F.Ther+ -dependence of S 12 (r + ), that is the scaledependent component of the resolved turbulent production, recovers the unfiltered behaviour only for l + F <l+ B. Otherwise, the scale-energy production reproduced with the resolved scales is very poor, meaning that a large energy injection due to production occurs in the subgrid scales. In this region of the flow, the rate of energy exchange between grid/subgrid motions, E sgs, is not determined by the viscous dissipation, but by the balance between production, which is never negligible (l c + η + ), and dissipation in the subgrid scales; therefore, Ē + E sgs E, see figure 7(a b). The resolved field cannot reproduce the unfiltered physics because most of the energy processes act in the subgrid scales and theroleofe sgs is strongly modified up to a backward energy transfer, E sgs > when l + F >l+ B, leading to an energy flux from subgrid to resolved scales, as for the location Y c + = 12. As shown in figure 3(b), the intensity of the backward energy transfer and the region of the flow involved increase with the filter scale l + F. Indeed, as shown in figure 2(b), representing T r (r +,Y c + ), for large scales r + corresponding to the filter lengths considered, the region of the reverse energy cascade and the magnitude of this process increase. Furthermore, for this choice of parameters, the nonlinear interactions in the resolved scales do not reproduce the energy flux across scales, T r, and the spatial flux T c,as showninfigure7(a b), while, on the other hand, the subgrid stresses significantly act in the energy redistribution with Tr sgs and Tc sgs, meaning that most of the processes depend on the scales below l + F that, in the wall region, are those responsible for the coherent structures. In order to better exploit the role of the filter length, in figure 9(a b) a comparison between two balances at a different value of l + F, smaller and larger than l + B, is reported. It is interesting to observe that the contribution of Tr sgs and Tc sgs in the case when l + F = 6, the one larger than l+ B, is fundamental in reproducing the energy redistribution mechanisms, while for l + F = 3 it is negligible. The effects of nonlinear interactions, Tr sgs, as a function of the wall distance and evaluated at a scale r + = l + F = 3, are shown in figure 1(a) where T r and T r also appear. In the bulk of the channel the transfer is only due to the nonlinear interactions in the resolved scales, T r, while Tr sgs is negligible. Approaching the wall,

14 Analysis of the Kolmogorov equation for filtered wall-turbulent flows l + F = 15 l + F = 2 l + F = 3 l + F = 4 l + F = 5 l + F = r + 13 Figure 8. Scaling of the averaged filtered mixed structure function, S 12 = δūδ v, for different filter scales l F + in the buffer layer Y c + = 25. Comparison with the unfiltered mixed structure function S 12 = δuδv (solid line). The vertical line represents the scale l B + at Y c + =25. the subgrid interaction term, Tr sgs, becomes more and more significant highlighting a strong coupling between subgrid and resolved motions occurring in the buffer layer. This statistical occurrence can presumably be attributed to the fact that when l + F >l+ B both subgrid and resolved scales belong to the same coherent structures. The same phenomenology is also observed for other scales and filters considered, not shown here, and for the spatial fluxes T c, T c and Tc sgs shown in figure 1(b) under the same conditions. In general, it is very important to estimate the subgrid nonlinear energy distribution effects in the buffer layer for l + F >l+ B in order to correctly reproduce the near-wall regeneration mechanisms. As also suggested by Domaradzki et al. (1994), in the buffer layer an intermediate range of scales exists which loses energy to feed both larger (reverse energy cascade) and smaller (forward energy cascade) scales in the same physical location and other ranges of scales of the adjacent regions of the flow via the spatial flux. In the present framework, the scale l + B (Y c + ) can be thought as the centre of this range of scales. This range, responsible for the backward energy transfer, is related to the dynamics of the coherent structures producing turbulent fluctuations, see Piomelli et al. (1996). When l + F >l+ B, a large fraction of this range belongs to the subgrid scales and therefore both Tr sgs and Tc sgs account for a significant part of the energy cascade and spatial fluxes in the resolved scales and the resolved physics is very poor. Furthermore, under these conditions a large energy-excess due to turbulent production occurs in the subgrid scales as highlighted in figure 8, feeding the backward energy transfer, E sgs >. In a context of LES of wall flows, it is very important to capture the physics of the buffer layer because this region is responsible for the sustainment of turbulence. However, according to the present results, the filtered velocity field is able to capture the turbulent dynamics only when the filter scale allows for the resolution of the reverse energy cascade region Ω B. Otherwise, the resolved physics is very poor; see figure 9(a), where Π e, T r and Ē e recover the unfiltered physics (not shown) only for l + F = 3, which is outside Ω B, and are underestimated otherwise. Furthermore, if l + F >l+ B, LES models should be able to reproduce a backward energy transfer and the complex but significant energy redistribution due to subgrid stresses.

15 39 A. Cimarelli and E. De Angelis (a).6 (b) r + r + Figure 9. Effects of the filter scale in the buffer layer at Y + c = 25. See the caption of figure 6 for more details. The incorrect resolution of the leading physical processes of the buffer layer affects the flow, especially in the core region, via spatial energy fluxes. In figure 11(a b) the Y c + -behaviour of the resolved spatial flux φ c = v δū 2 +2 δ pδ v /ρ νd δū 2 /2dY c is shown in comparison with the unfiltered one Φ c for large and small scales and different filter lengths. It is also shown that φ c+sgs = φ c + δτ i2 δū i, which accounts for the energy redistribution in physical space due to subgrid stresses. When the filter length allows us to capture the whole region Ω B,asforl + F =15 shown in figure 11(a), the amount of resolved energy which leaves the buffer layer to feed the viscous sublayer and the core flow recovers the unfiltered process, φ c Φ c, for both large and small scales. The subgrid stress effects are negligible, φ c φ c+sgs. Otherwise, a depletion of the resolved spatial flux is observed as for l + F = 3 shown in figure 11(b). In this case the filter scale that does not allow us to resolve the region Ω B and the subgrid stresses accounts for a significant part of the resolved-energy redistribution in physical space. The reduction in the resolved spatial energy fluxes strongly affects the dynamics of the turbulent fluctuations in the core flow. Indeed, the turbulence in this region is sustained by the energy incoming from the wall since production goes to zero. The resolved physics in this region is strongly altered even if the local main processes are captured, i.e. if l + F <l+ c. As shown in figure 6(a) fory c + = 22, a depletion of the resolved statistical observable occurs moving from l + F = 3 to 6 even if both the filter lengths are smaller than l c +. Indeed, for l + F = 6 a larger fraction of the region Ω B is not resolved leading to a stronger decrease of the spatial fluxes which feed this region. It should be pointed out that under the same conditions the dynamics of the logarithmic region remains unaffected. Indeed in the log layer these effects are negligible even if a large decrease of the resolved spatial flux occurs as shown in figure 11(a b). In fact the spatial flux being almost constant, the amount of energy carried out by it is not released in this region. Therefore, despite the large value of the spatial flux in the log layer, its divergence is sufficiently small in comparison with the other terms of the scale-by-scale budget (2.2) and (3.1) to be dynamically ineffective. The inhomogeneity plays only a minor role here which is also expected to vanish for a larger Reynolds number. Therefore, the condition l + F <l+ c presented in the previous section is still sufficient to ensure that the main physical processes of the log layer are well reproduced with the resolved scales.

16 Analysis of the Kolmogorov equation for filtered wall-turbulent flows 391 (a).8 (b) Y + c Y + c Figure 1. (a) Y c -behaviour of T r (solid line), T r (dotted line) and Tr sgs (dashed line) for r + =3 and l F + = 3. (b) Y c-behaviour of T c (solid line), T c (dotted line) and Tc sgs (dashed line) for r + =3 and l F + = Assessment of the LES models In this section, a methodology for using the filtered Kolmogorov equation (3.1) as a new tool for the assessment of LES models will be briefly outlined. An a priori analysis of the most common and widely used eddy viscosity models will be shown as an example, but the same procedure could also be performed for the a posteriori analysis. According to the previous analysis, LES models should reproduce a backward energy transfer in the wall region if l + F >l+ B ; however, it will be shown that in this condition dissipative models do not capture the correct subgrid stress effects. According to the idea of isotropic recovery at small scales of all types of flow for a sufficiently large Reynolds number, the eddy viscosity models assume that the energy exchange between resolved and subgrid scales is similar to the viscous dissipation, leading to subgrid stresses in the form τ ij 1/3τ kk δ ij = 2ν T S ij.infact,whenisotropic conditions are recovered, (2.2) leads to an energy exchange across the filter scale, δu 2 δu i r i = 4ν S ij S ij, (4.1) lf which is similar to E sgs evaluated with eddy viscosity, i.e. E sgs = 8ν T S ij S ij.letus note that in expression (4.1) the pseudo-dissipation ɛ is replaced with the proper dissipation ν S ij S ij since in homogeneous conditions these terms are coincident. The simplest and most commonly used eddy viscosity model is the one developed in Smagorinsky (1963), where it is assumed that ν T =(C s l F ) 2 S. This model is known to give good results in homogeneous and isotropic turbulence, but it is too dissipative in the near-wall turbulence. To alleviate this deficiency, for wall-bounded flows the Smagorinsky constant C s is often multiplied by a damping factor depending on the wall-normal distance, see Van Driest (1956). Instead, in other models like the Smagorinsky shear-improved model (Lévêque et al. 27), the eddy viscosity takes into account mean-shear effects and, therefore, the non-isotropic nature of near-wall turbulence, and naturally decreases to zero at the wall, i.e. ν T =(C s l F ) 2 ( S S ). More in general, the Smagorisky approach can be improved to a large degree via a dynamic procedure (Germano et al. 1991) where the model coefficient is evaluated from the resolved motion and it is applied to the subgrid scales assuming the scale invariance suggested by relation (4.1). Formally, this methodology is based on