Role of various scale-similarity models in stabilized mixed subgrid-scale model
aa r X i v : . [ phy s i c s . f l u - dyn ] J un Role of various scale-similarity models in stabilized mixed subgrid-scale model
Kazuhiro Inagaki a) and Hiromichi Kobayashi Institute of Industrial Science, The University of Tokyo, 4-6-1 Komaba, Meguro-ku,Tokyo 153-8505, Japan Department of Physics & Research and Education Center for Natural Sciences,Hiyoshi Campus, Keio University, 4-1-1 Hiyoshi, Kohoku-ku, Yokohama 223-8521,Japan (Dated: 16 June 2020)
We investigate the physical role of various scale-similarity models in the stabilized mixedmodel [K. Abe, Int. J. Heat Fluid Flow, , 42 (2013); M. Inagaki and K. Abe, Int. J. HeatFluid Flow, , 137 (2017)] and evaluate their performance in turbulent channel flows.Among various models in the present study, the original model combined with the scale-similarity model for the subgrid-scale (SGS)-Reynolds term yields the best prediction forthe anisotropy of the grid-scale (GS) velocity fluctuations and the SGS stress, even incoarse grid resolutions. Moreover, it successfully predicts large intensities of the spec-tra close to the cut-off scale in accordance with the filtered direct numerical simulation,whereas other models predict a rapid decay of the spectra in the low-wavelength region.To investigate the behavior of the models close to the cut-off scale, we analyze the budgetequation for the GS Reynolds stress spectrum. The result shows that the scale-similaritymodel for the SGS-Reynolds term plays a role in the enhancement of the wall-normal ve-locity fluctuation close to the cut-off scale. Thereby, it activates turbulence close to thecut-off scale, leading to a reproduction of the proper streak structures observed in wall-bounded turbulent flows. The reproduction of velocity fluctuations close to the cut-offscale and turbulent structures is a key element for further development of SGS models. a) Electronic mail: [email protected] . INTRODUCTION Large-eddy simulation (LES) is an essential tool employed to predict high-Reynolds-numberturbulent flows. LES solves large-scale or grid-scale (GS, resolved scale, or super-filter scale) ed-dies in turbulent flows. Meanwhile, effects of subgrid-scale (SGS, unresolved, or sub-filter scale)turbulent eddies are modeled. This procedure is referred to as SGS modeling. SGS modeling wasstudied for half a century since the pioneering work by Smagorinsky . Numerous practical SGSmodels, including the Smagorinsky model , are based on the linear eddy-viscosity assumption,which models the effect of SGS eddies as an effective viscosity. To date, several eddy-viscositytype models have been proposed, e.g., dynamic models , one-equation models , and modifiedlocal eddy-viscosity models . Recently, LES approach is also applied to the lattice-Boltzmannmethod, employing the eddy-viscosity concept .Although the eddy-viscosity models are simple and handy, their physical reliability are notsufficient. Several studies showed that the principal axis of the exact SGS stress tensor does notgenerally align with that of the strain rate tensor . Hence, the eddy-viscosity models cannotreproduce the exact property of the SGS stress tensor; however, they are reasonable for estimationof the energy transfer rate from the GS to SGS fields . A traditional approach to improvethe SGS model is to employ scale-similarity models . Scale-similarity models were shown toyield a better correlation with the exact SGS stress than eddy-viscosity models . However,such scale-similarity models are not sufficiently dissipative to be employed by themselves for astable performance of the LES. Moreover, scale-similarity models cause backscatter or energytransfer from the SGS to GS fields , which can induce numerical instability. A remedyfor these difficulties is to combine scale-similarity models with the eddy-viscosity model. Theresulting model is referred to as the mixed model, which was first proposed by Bardina et al . .Several types of mixed models have been suggested to date (see, Refs. 13, 19–26). However,the backscatter caused by scale-similarity models still makes the mixed model difficult to applyit to engineering problems with complex geometries. Furthermore, Anderson and Domaradzki showed that the conventional scale-similarity model yields an excessive dissipation directly fromthe largest resolved scales, which is unphysical in the sense of localness in scale of energy transfer.In this sense, the physics of scale-similarity models itself also needs to be discussed in detail. Asanother branch of recent developments of SGS modeling, algebraic stress modeling approach,which was first developed in the Reynolds-averaged Navier–Stokes (RANS) modeling (see, e.g.,2efs. 28–31), was discussed and its performance was evaluated .For practical use of the SGS model, both applicability and physical consistency of the modelare required. As a unique approach to utilize the physics of scale-similarity models in a numer-ically stable manner, Abe proposed a new formalism of the mixed model, which is referred toas the stabilized mixed model (SMM). Further discussion on the modification or other modelingapproaches of the SMM are provided in Refs. 16, 36, and 37. Surprisingly, the SMM is signifi-cantly less sensitive to the grid resolution than conventional eddy-viscosity models . More-over, overestimation of the streamwise velocity fluctuation, which is often observed in the LES atcoarse grid resolutions, decreases significantly, yielding a better prediction of the anisotropy of theGS velocity fluctuations in comparison with the direct numerical simulation (DNS). However, thephysical mechanism that the SMM yields better results is still under discussion. Otsuka and Abe showed that the SMM maintains streamwise vortices even at coarse grid resolutions in turbulentchannel flows, hence reproducing the mean velocity profile. Abe showed that the non-eddy-viscosity term contributes significantly to the generation of the GS Reynolds shear stress in thechannel flow by performing an a priori test. These results suggest that a non-eddy-viscosity termis a key element to improve physical properties of SGS models.In the SMM, Abe adopted the scale-similarity model for the SGS-Reynolds term, followingthe suggestion of Horiuti , on the proper velocity scale for the SGS energy. However, previ-ous studies show that other scale-similarity models yield better correlations with the exact SGSstress . In this sense, physical consistency and reliability of the SMM are still unclear, albeitits attractive performance. If the SMM is used without understanding the physical properties, theresults of the SMM could not be evaluated exactly in its application to turbulent flows. Therefore,it is required to reveal physical properties of the SMM. To understand the physics of the SMMis helpful for further development of SGS modeling. Now, we pose a question regarding whichscale-similarity model exhibits the best performance in predicting turbulent flows. To investigatethe physics of SGS models, we should analyze their properties in their applications to numericalsimulations; namely, a posteriori tests. In the present study, we construct the SMM using variousscale-similarity models and evaluate their performance in turbulent channel flows.The rest of this paper is organized as follows. In Sec. II, we describe the modeling procedureof the SMM . In Sec. III, we construct various types of SMMs and evaluate their performancein turbulent channel flows. In this section, we find that the reproduction of the GS and SGSanisotropies significantly depends on SGS models, through the Lumley’s invariant map. Moreover,3e address that the critical difference among the models lies in the low-wavelength region of theGS Reynolds spectrum. To investigate the physical origin of the difference in the low-wavelengthregion, we discuss the budget for the GS Reynolds stress spectrum in Sec. IV. In this section,we also discuss the relation between the spectrum and the streak structures observed in wall-bounded turbulent flows. Conclusions are given in Sec V. II. STABILIZED MIXED MODEL WITH VARIOUS SCALE-SIMILARITY TERMSA. Governing equations and scale-similarity models
The governing equations in LES for an incompressible fluid are the filtered continuity andNavier–Stokes equations: ∂ u i ∂ x i = , (1) ∂ u i ∂ t = − ∂∂ x j ( u i u j + τ sgs i j ) − ∂ p ∂ x i + ∂∂ x j ( ν s i j ) , (2)where · denotes the filtering operation, u i is the GS velocity, τ sgs i j (= u i u j − u i u j ) is the SGSstress, p is the pressure divided by the density, ν is the kinematic viscosity, and s i j [= ( ∂ u i / ∂ x j + ∂ u j / ∂ x i ) / ] is the GS strain rate. To close the equations, we must model the SGS stress τ sgs i j . Aconventional model for τ sgs i j is an eddy-viscosity type model: τ sgs i j = τ sgs ℓℓ δ i j − ν sgs s i j , (3)where ν sgs denotes the SGS eddy viscosity. A widely accepted expression for ν sgs was proposedby Smagorinsky , which expresses ν sgs through the filter width ∆ and the GS strain rate s i j s i j . Thedynamic approach extends applicability of the Smagorinsky model to more complex turbulentflows. However, these models involve an intrinsic shortfall of negative viscosity, which causesexponential divergence . As a remedy, other local expressions for ν sgs have been proposed .Although the eddy-viscosity models are simple and handy, several studies showed that the strainrate tensor does not necessarily align with the exact SGS stress . An approach to improve thisproblem is to employ scale-similarity models . Conventionally, the SGS stress is decomposed4nto the following three terms : τ sgs i j = L i j + C i j + R i j , (4a) L i j = u i u j − u i u j , (4b) C i j = u i u ′′ j + u ′′ i u j , (4c) R i j = u ′′ i u ′′ j , (4d)where u ′′ i = u i − u i . We refer to L i j , C i j , and R i j as the Leonard, cross, and SGS-Reynolds terms,respectively. Note that the sum of the Leonard and cross terms satisfy the Galilean invariance,although the individual terms do not . The scale-similarity assumption gives the approximation u ′′ i u j ≃ u ′′ i u j . Under this scale-similarity assumption, the cross and SGS-Reynolds terms, Eqs. (4c)and (4d), yield C i j ≃ u i u ′′ j + u ′′ i u j = u i ( u j − u j ) + ( u i − u i ) u j , (5a) R i j ≃ u ′′ i u ′′ j = ( u i − u i )( u j − u j ) . (5b)The sum of L i j , C i j , and R i j under the scale-similarity assumption reads L i j + C i j + R i j ≃ L m i j = u i u j − u i u j , (6)where L m i j is referred to as the modified Leonard term, which is a redefined Leonard term em-ployed to satisfy the Galilean invariance . Employing the Gaussian or top-hat filter as the testfilter, the Taylor expansion of the filtered velocity yields b u i = u i + b ∆ ℓ ∂ u i ∂ x ℓ ∂ x ℓ + O ( b ∆ ) , (7)where the test filter operation b · is used to explicitly show which filter is expanded, and b ∆ i is thefilter width in the x i direction accompanied with b · . Hence, the modified Leonard term is expandedas L m i j = C i j − ∆ ℓ ∂ u i ∂ x ℓ ∂ x ℓ ∆ m ∂ u j ∂ x m ∂ x m + O ( ∆ )= C i j + O ( ∆ ) , (8) C i j = ∆ ℓ ∂ u i ∂ x ℓ ∂ u j ∂ x ℓ , (9)5here C i j is referred to as the Clark term. The second term on the first line of Eq. (8) correspondsto the Taylor expansion of the scale-similarity model for the SGS-Reynolds term R i j ; R i j ≃ ∆ ℓ ∂ u i ∂ x ℓ ∂ x ℓ ∆ m ∂ u j ∂ x m ∂ x m . (10)In this paper, we refer to terms expressed by Eqs. (5a), (5b), (6), and (9) as scale-similarity models. B. Stabilized mixed model
Abe proposed the following mixed model, referred to as the SMM: τ sgs i j = k sgs δ i j − ν sgs s i j + τ eat i j , τ eat i j = k sgs τ a i j | tl + ν a s i j τ a ℓℓ , ν a = − τ a i j | tl s i j s ℓ m s ℓ m , ν sgs = C sgs f ν ∆ √ k sgs , (11)Here, k sgs (= τ sgs ℓℓ / ) is the SGS kinetic energy, ν sgs is the SGS eddy viscosity, A i j | tl = A i j − A ℓℓ δ i j / τ a i j denotes an additional term, and C sgs is a constant. f ν is the wall damping functionbased on the Kolmogorov scale : f ν = − exp [ − ( d ε / A ) / ( + C ) ] , d ε = u ε n ν (cid:18) n ∆ (cid:19) C , u ε = ( νε sgs ) / , ε sgs = C ε ( k sgs ) / ∆ + ε wall , ε wall = ν k sgs n , (12)where n denotes the distance from the nearest solid wall, ε sgs denotes the SGS energy dissipationrate, and C ε , A , and C are constants. We refer to τ eat i j in Eq. (11) as the extra anisotropic term(EAT). In the original model , an unusual filter scale, ∆ = p max ( ∆ x ∆ y , ∆ y ∆ z , ∆ z ∆ x ) in Cartesiancoordinates, is adopted. Inagaki and Abe modified this unusual filter scale to the conventionalone, ∆ = ( ∆ x ∆ y ∆ z ) / in Cartesian coordinates, and set the model constants in C sgs = . C ε = . A =
13, and C = /
3. The SGS kinetic energy k sgs is obtained by numerically solving thefollowing model equation: ∂ k sgs ∂ t = − ∂∂ x j ( u j k sgs ) − τ sgs i j s i j − ε sgs + ∂∂ x j (cid:20)(cid:16) ν + C k f ν ∆ √ k sgs (cid:17) ∂ k sgs ∂ x j (cid:21) , (13)6here f ν and ε sgs are defined in Eq. (12) and C k is set as C k = . ν a related term is introduced to removethe backscatter and stabilize the model; namely, we have s i j τ eat i j = k sgs s i j τ a i j | tl + ν a s i j s i j τ a ℓℓ = , (14)which indicates that the last term on the first line of Eq. (11) does not contribute to the energytransfer between the GS and SGS fields. Thereby, the second term on the right-hand side ofEq. (13) yields − τ sgs i j s i j = ν sgs s i j s i j ≥ , (15)which indicates the absence of backscatter because ν sgs ≥
0. Hence, we calculate Eq. (13) in anumerically stable manner when utilizing scale-similarity models. Because Eq. (13) does not nec-essarily guarantee the positiveness of k sgs , we must clip the negative value of k sgs in the numericalsimulation. Otherwise, ν sgs , which is proportional to √ k sgs , cannot be calculated.Second, the EAT is expressed by a form of the normalized anisotropy tensor: τ eat i j = k sgs (cid:18) τ a i j + ν a s i j τ a ℓℓ + ν a s ℓℓ − δ i j (cid:19) = k sgs b a i j . (16)The advantage of this normalized form is that the model can predict the anisotropy of the SGS fieldeven when a component of τ a i j becomes small. This is because the denominator τ a ℓℓ decreases atthe same rate as τ a i j . Owing to these two features, we can make full use of scale-similarity modelsin a numerically stable manner.In this study, we note that Abe or Inagaki and Abe adopted τ a i j = ( u i − b u i )( u j − b u j ) , (17)for the EAT. Equation (17) corresponds to the scale-similarity model for the SGS-Reynolds term(5b), although the repeated filter operation of · is replaced with the test filter operation b · . Becausethe stabilization procedure of Abe is independent of the choice of τ a i j , we can construct varioustypes of SMMs, by for example using the modified Leonard term L m i j (6) or the Clark term C i j (9). Considering the results of numerous previous studies addressing the correlation betweenscale-similarity models and the exact SGS stress , we expect that the modified Leonard orClark terms yield a better performance in the reproduction of the physics of turbulent flows inthe LES than the scale-similarity model for the SGS-Reynolds term. In the following section,we construct the SMM using various scale-similarity models and evaluate their performance inturbulent channel flows as a typical case of wall-bounded turbulent shear flows.7 II. STATISTICS OF VARIOUS STABILIZED MIXED MODELS IN TURBULENTCHANNEL FLOWSA. Variety of stabilized mixed models
We construct various SMMs based on the model suggested by Inagaki and Abe (hereafterdenoted as IA), which was modified from the original model to enable the use of the conven-tional filter scale ∆ = ( ∆ x ∆ y ∆ z ) / in Cartesian coordinates. In the construction, we fix all modelconstants described in Sec. II B except C sgs for the cases excluding the EAT and only change τ a i j ,which is the scale-similarity model for the SGS-Reynolds term (17) in the original model. Theapplied models are listed in the followings:1. Original IA model (IA): τ a i j = ( u i − b u i )( u j − b u j ) ,
2. IA model with the Clark term (IA-CL): τ a i j = ∆ ℓ ∂ u i ∂ x ℓ ∂ u j ∂ x ℓ
3. IA model with the modified Leonard term (IA-ML): τ a i j = u i u j − u i u j
4. IA model with the stress based on the Laplacian of velocity (IA-LV): τ a i j = ∆ ( ∇ u i )( ∇ u j )
5. IA model without the EAT (IA-LN): τ a i j = ( u i − b u i )( u j − b u j ) = b ∆ ℓ ∂ u i ∂ x ℓ ∂ x ℓ b ∆ m ∂ u i ∂ x m ∂ x m = γ ∆ ( ∇ u i )( ∇ u j ) , (18)8here γ = b ∆ α / ∆ α (= const. ) . Note that the SMM does not depend on the value of the coefficient γ / because τ a i j is normalized, as shown in Eqs. (11) or (16). For the numerical simulation ofturbulent channel flows, we often employ a rectangular grid, such that IA and IA-LV yield differentresults. B. Computational methods and numerical conditions
We employ a Cartesian coordinate grid and set the streamwise, wall-normal, and spanwisedirections as x (= x ) , y (= x ) , and z (= x ) , respectively. We use the staggered grid system andadopt the fully conservative central finite difference scheme for the x and z directions with fourth-order accuracy and the conservative central finite difference scheme on non-uniform grids forthe y direction with second-order accuracy, for both equations for the velocity and SGS kineticenergy k sgs . The boundary condition is periodic in the x and z directions and it is no slip in the y direction. The Poisson equation for pressure is solved using a fast Fourier transformation. Fortime marching in the velocity field, we adopt the second-order Adams–Bashforth method. For timemarching in k sgs , we adopt the explicit Euler method, except for the dissipation term ε sgs , which istreated implicitly. For the test filtering operation, we approximate it through the Taylor expansionas Eq. (7). The spatial derivative for each direction in Eq. (7) is discretized with second-orderaccuracy, i.e., we discretize b q ( I , J , K ) as b q ( I , J , K ) = q ( I , J , K ) + b ∆ x q ( I − , J , K ) − q ( I , J , K ) + q ( I + , J , K ) ∆ x + b ∆ y
24 1 ∆ y ( J ) " − − q ( I , J − , K ) + q ( I , J , K ) ∆ y ( J − / ) + − q ( I , J , K ) + q ( I , J + , K ) ∆ y ( J + / ) + b ∆ z q ( I , J , K − ) − q ( I , J , K ) + q ( I , J , K + ) ∆ z + O ( ∆ x ) + O ( ∆ y ) + O ( ∆ z ) , (19)where the superscript ( I , J , K ) denotes the grid point. Because b ∆ i ∝ ∆ x i , the test-filtered variablesretain fourth-order accuracy in the central finite difference scheme. We set ∆ i = ∆ x i , b ∆ i = ∆ i ,and b ∆ i = √ ∆ i to satisfy ∆ α + b ∆ α = b ∆ α , which is satisfied when the Gaussian filter is employedas the test filter . We perform two Reynolds number cases, Re τ =
180 and Re τ = τ (= u τ h / ν ) is the Reynolds number based on the channel half width h and the wall frictionvelocity u τ (= p | ∂ U x / ∂ | wall | ) , where U x (= h u x i ) is the streamwise mean velocity and h·i denotesthe statistical average. In the present simulation, we adopt the average over the x – z plane and timeto obtain the statistical average. For Re τ = τ = ∆ x + =
94 and ∆ z + =
47; medium resolution (MR), where ∆ x + =
47 and ∆ z + =
24; high resolution (HR), where ∆ x + =
94 and ∆ z + =
47, while ∆ y + is fixed. Values witha superscript ‘ + ’ denote those normalized by u τ and ν . Further, we perform the simulations withthe same resolution as LR, but in a large domain (LD), where L x = π h and L z = π h / τ = C sgs = . is too large to sustain the GS turbulent fluctuation. Hence, we establish areference linear eddy-viscosity model case with a smaller coefficient C sgs = .
042 (IA-LNcs42),which is optimized for an one-equation eddy-viscosity model . To evaluate the filtered values inthe a priori test through the DNS result, we adopt the sharp cut-off filter in the Fourier space inthe x and z directions, while no filtering operation is applied in the y direction. The filter scale isset in ∆ + x =
94 and ∆ + z =
47 to the same value as the LR cases in LES.To investigate a higher Reynolds number case, we perform LES at Re τ = ∆ x + =
65 and ∆ z + =
49, inwhich the spanwise resolution is almost the same as in LR at Re τ = ∆ x + =
130 and ∆ z + = C. Basic statistics
1. Mean velocity for basic models
Figure 1 shows the mean velocity profile for (a) DSM, (b) IA-LN, and (c) IA at various gridresolutions or domain sizes at Re τ = ABLE I. Flow cases and numerical parameters.Case Re τ L x × L y × L z N x × N y × N z ∆ x + ∆ y + ∆ z + C sgs IA180LR 180 4 π h × h × π h / × ×
16 94 1.1–11 47 0.075IA180MR 180 4 π h × h × π h / × ×
32 47 1.1–11 24 0.075IA180HR 180 4 π h × h × π h / × ×
64 24 1.1–11 12 0.075IA180LD 180 16 π h × h × π h / × ×
64 94 1.1–11 47 0.075IA-CL180LR 180 4 π h × h × π h / × ×
16 94 1.1–11 47 0.075IA-CL180LD 180 16 π h × h × π h / × ×
64 94 1.1–11 47 0.075IA-ML180LR 180 4 π h × h × π h / × ×
16 94 1.1–11 47 0.075IA-LV180LR 180 4 π h × h × π h / × ×
16 94 1.1–11 47 0.075IA-LN180LR 180 4 π h × h × π h / × ×
16 94 1.1–11 47 0.075IA-LNcs42-180LR 180 4 π h × h × π h / × ×
16 94 1.1–11 47 0.042IA-LNcs42-180MR 180 4 π h × h × π h / × ×
32 47 1.1–11 24 0.042IA-LNcs42-180HR 180 4 π h × h × π h / × ×
64 24 1.1–11 12 0.042IA-LNcs42-180LD 180 16 π h × h × π h / × ×
64 94 1.1–11 47 0.042DSM180LR 180 4 π h × h × π h / × ×
16 94 1.1–11 47 -DSM180MR 180 4 π h × h × π h / × ×
32 47 1.1–11 24 -DSM180HR 180 4 π h × h × π h / × ×
64 24 1.1–11 12 -no-model180LR 180 4 π h × h × π h / × ×
16 94 1.1–11 47 -no-model180MR 180 4 π h × h × π h / × ×
32 47 1.1–11 24 -no-model180HR 180 4 π h × h × π h / × ×
64 24 1.1–11 12 -DNS 180 4 π h × h × π h / × ×
256 8.8 0.23–6.8 2.9 -IA1000VLR 1000 2 π h × h × π h × ×
32 130 1.0–58 98 0.075IA1000LR 1000 2 π h × h × π h × ×
64 65 1.0–58 49 0.075IA-LN1000LR 1000 2 π h × h × π h × ×
64 65 1.0–58 49 0.075DSM1000LR 1000 2 π h × h × π h × ×
64 65 1.0–58 49 -no-model1000LR 1000 2 π h × h × π h × ×
64 65 1.0–58 49 - U x + y + (a) DSM180LRDSM180MRDSM180HRDNS 0 5 10 15 20 25 30 35 1 10 100 U x + y + (b) IA-LNcs42-180LRIA-LNcs42-180MRIA-LNcs42-180HRIA-LNcs42-180LDIA-LN180LRDNS 0 5 10 15 20 25 30 35 1 10 100 U x + y + (a) IA180LRIA180MRIA180HRIA180LDDNS 0 5 10 15 20 25 30 35 1 10 100 U x + y + (d) no-model180LRno-model180MRno-model180HRDNS FIG. 1. Mean velocity profile for (a) DSM, (b) IA-LN, (c) IA, and (d) no-model at various resolutions ordomain sizes for Re τ = lead to a laminar or parabolic profile because the SGS eddy viscosity ν sgs does not vanish dueto the mean value of k sgs . Overestimation of the mean velocity is also observed in other eddy-viscosity models . Hence, the EAT makes IA insensitive to the grid resolution. The results inLD overlap those in LR for IA and IA-LNcs42 almost perfectly, which suggests that the statisticsof the present simulation depend not on the domain size, but the grid resolution.Figure 2 shows the mean velocity profile for representative cases at Re τ = . IA yields a good prediction, while other cases result inoverestimation. Surprisingly, IA yields a reasonable prediction even in VLR, where the spanwisegrid size is ∆ z + =
98, which is close to the distance between the streak structure observed in thenear-wall region in wall-bounded turbulent flows . Notably, IA succeeds in predicting the meanvelocity profile in the near-wall region y + < U x + y + IA1000VLRIA1000LRIA-LN1000LRDSM1000LRno-model1000LRDNS
FIG. 2. Mean velocity profile for representative cases at Re τ = with respect to the spanwise grid size. The bulk mean velocity is defined by U m = Z h d y U x ( y ) , (20)where y = , h corresponds to the solid wall boundary. Figure 3 indicates that IA is the leastsensitive to the grid resolution, and the error is within 5% for both Reynolds numbers. Inter-estingly, the no-model overestimates the mean velocity in LR at Re τ = τ = τ = τ =
2. Mean velocity for various SMMs
Figure 4 shows the mean velocity profile for variable stabilized mixed models including the noEAT model (IA-LNcs42) in LR. The excessive overestimation of IA-ML results from the vanishingGS velocity fluctuations, similar to that shown with IA-LN180LR in Fig. 1(c). The reason behindthe failure of IA-ML may be attributed to the scale-similarity model for the SGS-Reynolds term13 U m / U m DN S ∆ z + IA180IA-LNcs42-180DSM180no-model180IA1000
FIG. 3. Bulk mean velocity normalized by DNS value for representative cases with respect to spanwise gridsize. entering the modified Leonard term involves with a negative coefficient, as seen in Eqs. (8) and(10). As both IA and IA-CL yield a reduction of the flow rate compared with other models, theClark and the SGS-Reynolds terms have similar effect. Moreover, ∆ i is large in LR. Thereby,the first and second terms on the first line of Eq. (8) are canceled each other in the modifiedLeonard term, yielding the weak contribution of the EAT to the SGS stress for IA-ML. Then, IA-ML leads to a laminarization due to the strong eddy viscosity, as shown in Fig. 1(b). IA-LNcs42and IA-LV overestimate the mean velocity, while IA-CL underestimates it. The prediction of themean velocity can be refined by tuning model parameters, e.g., C sgs . However, this is beyond ofscope of the present study. A notable point is that IA-CL and IA-LV succeed in sustaining theturbulence even in LR, where it leads to laminarization without the EAT. IA-LV cannot providea good prediction of the mean velocity, even though the rank of the spatial derivative is the sameas the scale-similarity model for the SGS-Reynolds term. Under the scale-similarity assumption, u i − b u i is evaluated as u ( I , J , K ) i − b u ( I , J , K ) i = h(cid:16) u ( I − , J , K ) i − u ( I , J , K ) i + u ( I + , J , K ) i (cid:17) + ∆ y ( J ) − − u ( I , J − , K ) i + u ( I , J , K ) i ∆ y ( J − / ) + − u ( I , J , K ) i + u ( I , J + , K ) i ∆ y ( J + / ) ! + (cid:16) u ( I , J , K − ) i − u ( I , J , K ) i + u ( I , J , K + ) i (cid:17)i , (21)14 U x + y + IA180LRIA-CL180LRIA-LNcs42-180LRIA-LV180LRIA-ML180LRDNS
FIG. 4. Mean velocity profile for various SMMs in LR at Re τ = while the Laplacian of the velocity for the finite difference with second-order accuracy yields ∇ u ( I , J , K ) i = ( ∆ y ( J ) ) "(cid:18) ∆ y ∆ x (cid:19) (cid:16) u ( I − , J , K ) i − u ( I , J , K ) i + u ( I + , J , K ) i (cid:17) + ∆ y ( J ) − − u ( I , J − , K ) i + u ( I , J , K ) i ∆ y ( J − / ) + − u ( I , J , K ) i + u ( I , J + , K ) i ∆ y ( J + / ) ! + (cid:18) ∆ y ∆ z (cid:19) (cid:16) u ( I , J , K − ) i − u ( I , J , K ) i + u ( I , J , K + ) i (cid:17) . (22)Note that the SMM does not depend on the coefficient 1 / ( ∆ y ( J ) ) , because the EAT is expressedby the form of the normalized tensor. In the present simulation, ∆ x and ∆ z are at least about 5 to 10times larger than ∆ y . Hence, the x - and z -derivative parts in Eq. (22) contributes little to the EAT.This represents a critical difference between IA and IA-LV. We confirm that the result does notchange for IA when the test filter is adopted only to the x and z directions (not shown). Namely, thetest filter in the x and z directions is essential in the scale-similarity model for the SGS-Reynoldsterm in turbulent channel flows. 15 R xx G S + y + (a) IA180LRIA-CL180LRIA-LNcs42-180LRIA-LV180LRDSM180LRf-DNS 0 2 4 6 8 10 12 14 1 10 100 R xx + , 〈 τ s g s xx 〉 + y + (b) IA180LRIA-CL180LRIA-LNcs42-180LRIA-LV180LRf-DNS 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1 10 100 R yy G S + y + (c) IA180LRIA-CL180LRIA-LNcs42-180LRIA-LV180LRDSM180LRf-DNS 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1 10 100 R yy + , 〈 τ s g s yy 〉 + y + (d) IA180LRIA-CL180LRIA-LNcs42-180LRIA-LV180LRf-DNS
FIG. 5. Profiles of the Reynolds stress of (a) R GS xx , (b) R xx and h τ sgs xx i , (c) R GS yy , and (d) R yy and h τ sgs yy i atRe τ =
3. Reynolds stress
In LES, the Reynolds stress R i j can be defined by R i j = R GS i j + h τ sgs i j i , R GS i j = h u ′ i u ′ j i , (23)where q ′ (= q − h q i ) is the fluctuation of q around the mean value. For the present SMM, we ex-plicitly solve the SGS kinetic energy k sgs . Thereby, we can calculate the SGS part of the Reynoldsstress, which involves k sgs as the isotropic part. Figure 5 shows the profile of the streamwise andwall-normal components of the Reynolds stress for various SMMs in LR at Re τ = R GS xx , which is twice as large as that of f-DNS. R GS xx contributes the most to the GS turbulentkinetic energy K GS (= R GS ii / ) in turbulent channel flows. Such overestimation of the GS turbu-16ent energy is often observed in the LES with coarse grid resolutions (see, e.g., Morinishi andVasilyev ). IA predicts R GS xx better than other models. Although IA-CL succeeds in reducing R GS xx ,it still results in overestimation. IA-LV overestimates R GS xx for y + >
20. Interestingly, IA-CL over-estimates R GS xx , while it underestimates the mean velocity, as shown in Fig. 4. This suggests thatthe overestimation of R GS xx is not necessarily caused by the overestimation of the velocity gradient.The overestimation of R GS xx leads to overestimation of the total Reynolds stress R xx . In Fig. 5(b), IAgives a reasonable prediction of the total value R xx , while all other models overestimate this value.IA and IA-CL provide a reasonable prediction of the SGS component h τ sgs xx i . IA-LNcs42 underes-timates h τ sgs xx i , whereas IA-LV overestimates it. Note that the eddy-viscosity part in the SGS stresscontributions little to the normal stress; namely, h τ eat αα i ≫ − h ν sgs s αα i ≃ R GS yy in Fig. 5(c), DSM seems to provide the best prediction; however, it overestimates R GS xx .Namely, DSM is more anisotropic than f-DNS. IA-LNcs42 is likewise excessively anisotropic. Inthe present cases, IA yields a good prediction of the wall-normal component of the GS Reynoldsstress R GS yy while also efficiently predicting the streamwise component R GS xx . In Fig. 5(d), IA-LNcs42 provides the profile of R yy or h τ sgs yy i far from f-DNS. As presented in Appendix A, theeddy-viscosity part in the SGS stress contributes little to the normal stress, such that h τ sgs yy i ≃ h k sgs i / h τ eat yy i <
0. Hence, these SMMs predict a smaller value of the wall-normal Reynoldsstress compared with the eddy-viscosity models. In Fig. 5(d), IA seems to yield the best prediction,although it underestimates the GS and the total value of the wall-normal stress.Figure 6 shows the profiles of the Reynolds shear stress for various cases in LR at Re τ = ν sgs is evaluated through ν sgs = − τ sgs i j s i j s ℓ m s ℓ m , (24)which is the same approach used in Abe . Note that ν sgs given by Eq. (24) is not necessarilypositive. Hence, it allows the backscatter through the eddy-viscosity term, while ν sgs in the presentLES is established to be positive. Although this is not a unique approach to determine ν sgs , wecan decompose the SGS stress into two parts through this procedure, where one plays the role of17nergy transfer between GS and SGS components through the eddy viscosity ν sgs , the other playsthe role of SGS forcing apart from the energy transfer. This decomposition is consistent with theconcept of the SMM described in Sec. II B. - R xy G S + y + (a) IA180LRIA-CL180LRIA-LNcs42-180LRIA-LV180LRDSM180LRf-DNS 0 0.2 0.4 0.6 0.8 1 1 10 100 - 〈 τ s g s xy 〉 + , 〈 ν s g s s - xy 〉 + y + (b) IA180LRIA-CL180LRIA-LNcs42-180LRIA-LV180LRDSM180LRf-DNS 0 0.2 0.4 0.6 0.8 1 1 10 100 - R xy + y + (c) IA180LRIA-CL180LRIA-LNcs42-180LRIA-LV180LRDSM180LRf-DNS
FIG. 6. Profiles of the Reynolds shear stress for (a) GS − R GS xy , (b) SGS −h τ sgs xy i , and (c) total components − R xy at Re τ = −h τ sgs xy i , while dashed lines depict the contributionfrom the eddy-viscosity term 2 h ν sgs s xy i . In Fig. 6(a), DSM overestimates the GS component R GS xy . IA-CL and IA-LNcs42 efficiently18redict R GS xy , whereas they overestimate the streamwise velocity fluctuation R GS xx in the same man-ner as DSM. This suggests that the statistical profiles of the mean velocity and the GS Reynoldsshear stress alone cannot account for the overestimation of R GS xx observed in Fig. 5(a), indicatingthe presence of some effect related to turbulent structures. We discuss this issue further in Sec. IV.IA and IA-LV underestimate R GS xy ; however, IA complements the underestimation of the GS valuewith the contribution from the SGS part as shown in Fig. 6(c). In Fig. 6(b), we find that the eddy-viscosity term contributes significantly to the SGS shear stress in f-DNS. IA and IA-CL reproducethis trend. In this sense, the value of the eddy-viscosity coefficient C sgs = .
075 may not be consid-ered an artificially large value, although it is too large to sustain the turbulent fluctuation withoutthe EAT as observed in Fig. 1(b). In Fig. 6(c), IA yields the best prediction of the total Reynoldsshear stress R xy , leading to the best prediction of the mean velocity, as shown in Figs. 1 and 4. ForIA-CL, the overestimation of − R xy in the near-wall region y + <
30 results in the underestimationof the mean velocity, as shown in Fig. 4.
4. Lumley’s invariant map
To quantitatively evaluate the anisotropy of the turbulent stress for various models, we inves-tigate Lumley’s invariant map (see, e.g., Hanjali´c and Launder ). We define the GS and SGSnormalized anisotropy tensor b GS i j and b SGS i j , respectively, by b GS i j = R GS i j R GS ℓℓ − δ i j , b SGS i j = h τ sgs i j ih τ sgs ℓℓ i − δ i j . (25)Their second and third invariants read II A b = b A i j b A i j / , III A b = b A i j b A j ℓ b A ℓ i / , (26)where A = GS , SGS. The realizability conditions for the Reynolds or SGS stress indicate thatthe invariants (26) lie in the following region : II A b ≤ III A b + , ( II A b ) ≥ ( III A b ) . (27)Figure 7 shows the invariant map for various cases at Re τ = y + <
20 are plottedwith symbols. In this map, the upper line, II A b = III A b + /
9, denotes the two-component turbu-lence. In turbulent channel flows, this corresponds to the condition that the wall-normal stress isnegligible relative to the streamwise and spanwise components. The upper right tip corresponds to19 II G S b III GS b (a) IA180LRIA-CL180LRIA-LNcs42-180LRIA-LV180LRDSM180LRf-DNS II S G S b III
SGS b (b) IA180LRIA-CL180LRIA-LNcs42-180LRIA-LV180LRf-DNS (b)
FIG. 7. Lumley’s invariant map for (a) GS and (b) SGS normalized anisotropy tensor at Re τ = y + <
20 are plotted with symbols. The inset shows the enlarged view in which IA turns around. Thechannel center corresponds to the end of lines closest to the origin. As it approaches the solid wall, f-DNS proceeds toward the right top, then turns around and asymptotically approaches the top line of thetwo-component turbulence, in both (a) and (b). the one-component turbulence (only the streamwise component), whereas the left tip correspondsto the two-component isotropic turbulence. The origin is the three-component isotropic condition.In Fig. 7(a), IA-LNcs42, IA-LV, and DSM approach the top right tip as they proceed toward thesolid wall, which indicates that the GS velocity fluctuation becomes nearly one component. Theprofiles for IA-CL and IA approach that for f-DNS; namely, they recover the asymptotic behaviorof the GS turbulence anisotropy in the vicinity of the wall. The anisotropy of the SGS componentexhibit more evident differences between the models than the GS component. In Fig. 7(b), itshould be noted that IA-LNcs42 lies at the origin, which indicates that the SGS stress is isotropic.This reflects the property of the eddy-viscosity model as the ‘isotropic’ model. IA-LV exhibitsstrange behavior, and it does not asymptotically approach the two-component turbulence in the20icinity of the wall. This is because IA-LV fails to reproduce the asymptotics of the wall-normalstress, h τ sgs yy i ∼ O ( y ) . IA-CL approaches the two-component turbulence in the vicinity of thewall; however it approaches the one-component turbulence tip and does not turn around. Only IAturns around in the near-wall region, such that it reproduces the asymptotic trend of the SGS stressin the vicinity of the wall, although it does not perfectly correspond to f-DNS. In summary, IAyields the best prediction among the presented models on the anisotropy of both the GS and SGSstress.
5. Reynolds stress spectrum
According to Abe , the SMM employing the scale-similarity model for the SGS-Reynoldsterm recovers the energy spectrum close to the cut-off scale. The energy spectrum reflects informa-tion concerning turbulent structures such as streak structures observed in wall-bounded turbulence.We define the spectrum of the GS Reynolds stress by E GS i j ( k x , y , k z ) = ℜ h ˜ u i ˜ u ∗ j i , (28) R GS i j ( y ) = k max x ∑ k x = k max z ∑ k z = E GS i j ( k x , y , k z ) ∆ k x ∆ k z , (29)where k max α = π N α / L α , ∆ k α = π / L α , α = x , z , the superscript ‘ ∗ ’ denotes the complex conjugate,and ˜ q is the Fourier coefficient of an instantaneous variable q , defined by˜ q ( J ) ( k x , k z ) = N x N z N x ∑ I = N z ∑ K = q ( I , J , K ) e − i ( k x IL x / N x + k z KL z / N z ) . (30)Hereafter, we focus on the streamwise spectrum E GS i j ( k x , y ) , which is defined by E GS i j ( k x , y ) = k max z ∑ k z = E GS i j ( k x , y , k z ) ∆ k z (31)Because the difference between the models is evident at y + =
20 for both the mean velocity inFig. 4 and the Reynolds stress in Figs. 5 and 6, we focus on the spectrum at that plane.Figure 8 shows the streamwise spectrum of the GS Reynolds stress for various cases in LRor LD at y + =
20 for Re τ = λ x (= π / k x ) instead of the wavenumber k x . First, the results in LD almost overlap21 -3 -2 -1 k x + E xx G S + λ x+ (a) IA180LRIA-CL180LRIA-LNcs42-180LRIA-LV180LRDSM180LRIA180LDIA-CL180LDIA-LNcs42-180LDf-DNSDNS10 -5 -4 -3 -2 -1 k x + E yy G S + λ x+ (b) IA180LRIA-CL180LRIA-LNcs42-180LRIA-LV180LRDSM180LRIA180LDIA-CL180LDIA-LNcs42-180LDf-DNSDNS10 -4 -3 -2 -1 - k x + E xy G S + λ x+ (c) IA180LRIA-CL180LRIA-LNcs42-180LRIA-LV180LRDSM180LRIA180LDIA-CL180LDIA-LNcs42-180LDf-DNSDNS
FIG. 8. Profiles of the GS Reynolds stress spectrum for (a) streamwise E GS xx , (b) wall-normal E GS yy , and (c)shear components − E GS xy at y + =
20 for Re τ = those in LR, in the same manner as the mean velocity in Fig. 1. Hence, the statistics in the presentLES do not depend on the domain size, but on the grid resolution. In Fig. 8(a), as shown by Abe ,IA recovers the intensity of the GS streamwise velocity spectrum close to the cut-off wavelengthscale. The spectrum of the GS streamwise velocity is related to the near-wall streak structure. Thespectrum accumulated in the high-wavelength region such as IA-LNcs42 or DSM, indicates that22he structure elongated in the streamwise direction is considerably more dominant than the shortscale structures. For IA-CL, the slope of the spectrum at the high-wavelength region is gentle. Thisindicates that IA-CL succeeds in reproducing the length of the streak structure to some extent. Thisis further discussed in Sec. IV B. Several studies discussed the relation between the breakdown ofthe streaks and the sustaining process of turbulent shear flows . In this sense, IA may succeed inreproducing the sustaining process of turbulent shear flow with the coarse grid by the enhancementof the turbulence close to the cut-off scale. As shown in Fig. 8(b), IA also predicts the GS wall-normal velocity spectrum in the entire wavelength range. Other models, including IA-CL, cannotreproduce the spectrum close to the cut-off wavelength scale. The difference between IA and IA-CL suggests that the EAT, based not on the first-order but on the second-order spatial derivative ofthe velocity field, is useful in restoring the behavior of the spectra close to the cut-off scale.For E GS xy in Fig. 8(c), the lines disappear at the low-wavelength region for some models. Thisis because − E GS xy is negative at wavelength regions below that scale. In contrast, IA predicts thepositive − E GS xy up to the cut-off wavelength as efficiently as f-DNS. Although IA-LV likewise pre-dicts a positive − E GS xy value in the entire wavelength region, the value is smaller than that obtainedby IA or f-DNS. Furthermore, IA-LV cannot predict E GS xx and E GS yy at the low-wavelength region.For IA in Fig. 8(c), the spectrum bends around λ + x = E GS yy , as described later in Sec. IV A 2. However, the success of IAsuggests that the reproduction of the GS Reynolds shear stress spectrum E GS xy close to the cut-offscale is an essence of further development of SGS models. IV. DISCUSSIONA. Budget equation for the GS Reynolds stress spectrum
As shown in Sec. III, the SGS models showed evident differences on anisotropy and structuresrepresented by the Reynolds stress spectrum, in the near-wall region. To investigate the effect ofthe SGS stress on both anisotropy and structures of turbulence in shear flows, it is useful to analyzethe budget equation for the Reynolds stress spectrum . In the use of SGS models, the budgetequation for the GS Reynolds stress spectrum yields ∂ E GS i j ∂ t + ∂∂ x ℓ ( U ℓ E GS i j ) ˇ P GS i j − ˇ ε GS i j + ˇ D t , GS i j + ˇ Φ GS i j + ˇ D p , GS i j + ˇ D v , GS i j + ˇ T GS i j + ˇ ξ SGS i j + ˇ D SGS i j . (32)Each term on the right-hand side is referred to as the production ˇ P GS i j , destruction ˇ ε GS i j , turbulentdiffusion ˇ D t , GS i j , pressure–strain correlation ˇ Φ GS i j , pressure diffusion ˇ D p , GS i j , viscous diffusion ˇ D v , GS i j ,inter-scale transfer ˇ T GS i j , SGS destruction ˇ ξ SGS i j , and SGS diffusion ˇ D SGS i j , respectively. When theturbulent field is inhomogeneous only in the y direction, they are defined as follows:ˇ P GS i j = − E GS iy ∂ U j ∂ y − E GS jy ∂ U i ∂ y , (33a)ˇ ε GS i j = νℜ D ˜ s ′ i ℓ ( ˜ ∂ ℓ ˜ u ′ j ) ∗ + ˜ s ′ j ℓ ( ˜ ∂ ℓ ˜ u ′ i ) ∗ E , (33b)ˇ D t , GS i j = − ∂∂ y ℜ D ˜ u ′ y ( g u ′ i u ′ j ) ∗ E , (33c)ˇ Φ GS i j = ℜ D ˜ p total ′ ˜ s ′ i j ∗ E , (33d)ˇ D p , GS i j = − ∂∂ y h ℜ D ˜ p total ′ ˜ u ′ i ∗ E δ jy + ℜ D ˜ p total ′ ˜ u ′ j ∗ E δ iy i , (33e)ˇ D v , GS i j = ν ∂∂ y ℜ D ˜ s ′ jy ˜ u ′ i ∗ + ˜ s ′ iy ˜ u ′ j ∗ E , (33f)ˇ T GS i j = ℜ D ˜ u ′ j ˜ N ∗ i + ˜ u ′ i ˜ N ∗ j E − ˇ P GS i j − ˇ D t , GS i j , (33g)ˇ ξ SGS i j = ℜ D ˜ τ sgs i ℓ ′ | tl ( ˜ ∂ ℓ ˜ u ′ j ) ∗ + ˜ τ sgs j ℓ ′ | tl ( ˜ ∂ ℓ ˜ u ′ i ) ∗ E , (33h)ˇ D SGS i j = − ∂∂ y ℜ D ˜ τ sgs jy ′ | tl ˜ u ′ i ∗ + ˜ τ sgs iy ′ | tl ˜ u ′ j ∗ E , (33i)where ˜ ∂ j = ( i k x , ∂ / ∂ y , i k z ) , and N i = − ∂ ( u i u j ) / ∂ x j . p total is the sum of p and the SGS dynamicpressure 2 k sgs /
3; namely, it is defined by p total = p + k sgs . (34)Furthermore, we decompose the SGS destruction term into two terms as follows:ˇ ξ SGS i j = − ˇ ε EV i j + ˇ ξ EAT i j , (35)where ˇ ε EV i j = ℜ D ^ ν sgs s i ℓ ( ˜ ∂ ℓ ˜ u ′ j ) ∗ + ^ ν sgs s j ℓ ( ˜ ∂ ℓ ˜ u ′ i ) ∗ E , (36a)ˇ ξ EAT i j = ℜ D ˜ τ eat i ℓ ′ ( ˜ ∂ ℓ ˜ u ′ j ) ∗ + ˜ τ eat j ℓ ′ ( ˜ ∂ ℓ ˜ u ′ i ) ∗ E , (36b)24 k x + E xy G S + e q . λ x+ (a) productiondestructiondiffusionspres.-straininter-scale transferev. destrictionani. redistribution -0.04-0.03-0.02-0.01 0 0.01 0.02 0.03 0.04 10 k x + E xy G S + e q . λ x+ (b) productiondestructiondiffusionspres.-straininter-scale transferev. destrictionani. redistribution-0.04-0.03-0.02-0.01 0 0.01 0.02 0.03 0.04 10 k x + E xy G S + e q . λ x+ (c) productiondestructiondiffusionspres.-straininter-scale transferev. destriction -0.04-0.03-0.02-0.01 0 0.01 0.02 0.03 0.04 10 k x + E xy G S + e q . λ x+ (d) productiondestructiondiffusionspres.-straininter-scale transferev. destrictionani. redistribution FIG. 9. Budget for the GS Reynolds shear stress spectrum E GS xy normalized by viscous unit for (a) IA180LD,(b) IA-CL180LD, (c) IA-LNcs42-180LD, and (d) f-DNS at y + =
20 for Re τ = where τ eat i j is defined in Eq. (11). We refer to ˇ ε EV i j and ˇ ξ EAT i j as the eddy-viscosity destruction andthe anisotropic redistribution terms, respectively. When the budget Eq. (32) is summed over thewavenumbers, it leads to the conventional budget for the GS Reynolds stress (see, Appendix B orAbe ): A GS i j = k max x ∑ k x = ˇ A GS i j ∆ k x , (37)where ˇ A GS i j corresponds to each term in Eq. (33a)–(33i), (36a), and (36b), while A GS i j correspondsto each term on the right-hand side of Eq. (B1).
1. Shear component
A critical feature of the IA model is that it reproduces the positive − E GS xy in the entire wave-length region, as shown in Fig. 8(c). Abe showed that the anisotropic redistribution term ξ EAT xy isindispensable to predict the profile of the SGS destruction term obtained from the filtered DNS in25he budget of the GS Reynolds shear stress R GS xy . Figure 9 shows the budget for the GS Reynoldsshear stress spectrum E GS xy for representative cases at y + =
20 for Re τ = Φ GS xy in Fig. 9(c). IA and IA-CL qualitatively reproduce theprofile of the pressure–strain correlation ˇ Φ GS xy and anisotropic redistribution ˇ ξ EAT xy obtained fromf-DNS. The difference between IA and IA-CL lies in the profile of the production term ˇ P GS xy . ˇ P GS xy in IA-CL in Fig. 9(b) disappears close to the cut-off wavelength scale. This arises from the profileof E GS yy , because ˇ P GS xy reads ˇ P GS xy = − E GS yy ∂ U x ∂ y . (38)Hence, the reproduction of ˇ P GS xy close to the cut-off wavelength scale relies on the reproduction of E GS yy .
2. Wall-normal component
Figure 10 shows the budget for the wall-normal component of the GS Reynolds stress spectrum E GS yy for representative cases at y + =
20 for Re τ = ξ EAT yy obtained from f-DNS. However, they provide areasonable prediction of the sum of the pressure–strain correlation and the anisotropic redistribu-tion terms ˇ Φ GS yy + ˇ ξ EAT yy . For IA in Fig. 10(a), ˇ Φ GS yy + ˇ ξ EAT yy is positive close to the cut-off wavelengthscale, as is the case with f-DNS in Fig. 10(d). Hence, IA succeeds in reproducing the large inten-sity of the wall-normal component of the Reynolds stress spectrum E GS yy , as shown in Fig. 8(b).For IA in Fig. 10(a), the wavelength in which the anisotropic redistribution term ˇ ξ EAT yy exhibits adent corresponds to that in which − E GS xy is bent in Fig. 8(c). This suggests that the reproduction ofthe spectrum close to the cut-off scale for the shear component − E GS xy is realized by the increaseof the wall-normal velocity fluctuation at that scale attributed to the EAT through the anisotropicredistribution term ˇ ξ EAT yy .Figure 11 shows the contour map of the sum of the pressure–strain correlation and theanisotropic redistribution terms ˇ Φ GS yy + ˇ ξ EAT yy in the budget equation for the wall-normal componentof the GS Reynolds stress spectrum E GS yy , for representative cases at y + =
20 for Re τ = k x + E yy G S + e q . λ x+ (a) destructiondiffusionspres.-straininter-scale transferev. destrictionani. redistributionpres.-strain + ani. redistribution -0.01-0.005 0 0.005 0.01 10 k x + E yy G S + e q . λ x+ (b) destructiondiffusionspres.-straininter-scale transferev. destrictionani. redistributionpres.-strain + ani. redistribution-0.01-0.005 0 0.005 0.01 10 k x + E yy G S + e q . λ x+ (c) destructiondiffusionspres.-straininter-scale transferev. destriction -0.01-0.005 0 0.005 0.01 10 k x + E yy G S + e q . λ x+ (d) destructiondiffusionspres.-straininter-scale transferev. destrictionani. redistributionpres.-strain + ani. redistribution FIG. 10. Budget for the wall-normal component of the GS Reynolds spectrum E GS yy normalized by viscousunit for (a) IA180LD, (b) IA-CL180LD, (c) IA-LNcs42-180LD, and (d) f-DNS at y + =
20 for Re τ = a comparable value to the middle wavelength scale, as is the case with f-DNS. The negativecontribution for y + <
10 corresponds to the ‘splatting’ effect , leading to the two-componentturbulence, as observed in Fig. 7(a). The positive contribution of ˇ Φ GS yy + ˇ ξ EAT yy in the region y + > λ + x <
300 for IA in Fig. 11(a) leads to the reproduction of E GS yy close to the cut-off scale, asshown in Fig. 8(b). The reproduction of E GS yy close to the cut-off scale results in feedback to thebudget of E GS xy as a source term through the production term ˇ P GS xy .
3. Streamwise component
Figure 12 shows the budget for the streamwise component of the GS Reynolds stress spec-trum E GS xx for representative cases at y + =
20 for Re τ = P GS xx for IA inFig. 12(a) provides a positive value close to the cut-off wavelength scale owing to the reproduction27 a) λ x+ y + -0.008-0.006-0.004-0.002 0 0.002 0.004 0.006 0.008 (b) λ x+ y + -0.008-0.006-0.004-0.002 0 0.002 0.004 0.006 0.008 (c) λ x+ y + -0.008-0.006-0.004-0.002 0 0.002 0.004 0.006 0.008 FIG. 11. Contour map of sum of the pressure–strain correlation and the anisotropic redistribution termsˇ Φ GS yy + ˇ ξ EAT yy in budget for the wall-normal component of the GS Reynolds stress spectrum E GS yy normalizedby viscous unit for (a) IA180LD, (b) IA-CL180LD, and (c) f-DNS at Re τ = of E GS xy , because ˇ P GS xx reads ˇ P GS xx = − E GS xy ∂ U x ∂ y . (39)Moreover, the anisotropic redistribution term ˇ ξ EAT xx is likewise positive in the entire wavelengthrange for IA. Although the profiles of each term are different between IA and f-DNS, IA succeedsin enhancing the GS streamwise velocity fluctuation in the low-wavelength region close to thecut-off scale. In contrast, in IA-CL, the anisotropic redistribution term ˇ ξ EAT xx contributes little to28 k x + E xx G S + e q . λ x+ (a) productiondestructiondiffusionspres.-straininter-scale transferev. destrictionani. redistribution-0.08-0.06-0.04-0.02 0 0.02 0.04 0.06 0.08 10 k x + E xx G S + e q . λ x+ (b) productiondestructiondiffusionspres.-straininter-scale transferev. destrictionani. redistribution-0.08-0.06-0.04-0.02 0 0.02 0.04 0.06 0.08 10 k x + E xx G S + e q . λ x+ (c) productiondestructiondiffusionspres.-straininter-scale transferev. destrictionani. redistribution FIG. 12. Budget for the streamwise component of the GS Reynolds spectrum E GS xx normalized by viscousunit for (a) IA180LD, (b) IA-CL180LD, and (c) f-DNS at y + =
20 for Re τ = the budget in the entire wavelength range, and the production becomes negative close the cut-offwavelength scale. This leads to the rapid decay of the spectrum in the low-wavelength region, asshown in Fig. 8(a). Furthermore, the increase in the velocity fluctuation in the low-wavelengthregion leads to the enhancement of the dissipation. In fact, the eddy-viscosity destruction for IAin Fig. 12(a) is larger than that for IA-CL in Fig. 12(b). This accounts for the difference between29he profiles of the GS streamwise velocity fluctuation for IA and IA-CL in Fig. 5(a). Hence, theoverestimation of the GS streamwise velocity fluctuation can be restored by the increase of theturbulence close the cut-off wavelength scale.
4. Summary of analysis through budget for the GS Reynolds stress spectrum
We summarize the physical role of the EAT in the SMM in turbulent channel flows. The EAT inIA model enhances the wall-normal component of the velocity fluctuation through the anisotropicredistribution term. This feeds the streamwise and shear components of the GS Reynolds stress inthe low-wavelength region close to the cut-off scale, through the production term. Consequently,IA succeeds in activating turbulence in the entire wavelength range. To achieve this activation,the scale-similarity model for the SGS-Reynolds term is more appropriate than the Clark term, asthe former is based on a higher-order spatial derivative than the latter. Hence, the scale-similaritymodel for the SGS-Reynolds term is efficient in enhancing the turbulence close to the cut-off scale.Anderson and Domaradzki discussed the physics of inter-scale energy transfer among thelargest, small resolved, and sub-grid scales, through scale-similarity models. They showed thatthe modified Leonard-like term yields an excessive dissipation directly from the largest resolvedscales, which is unphysical in the concept of localness in scale of energy transfer. Such a non-localenergy transfer does not occur in the SMM because the energy transfer between the GS and SGSdue to the EAT is prohibited, although the EAT can redistribute energy between mean and the GSturbulent kinetic energies, as shown in Appendix B. A failure of IA-CL may be partly attribute tothe similar mechanism suggested by Anderson and Domaradzki , because the Clark term retainsonly the leading term of the scale-similarity model for the SGS stress, as shown in Eq. (6) and(8). Namely, the Clark term may be dominant in the largest resolved scale, although the statisticalcontribution of ˇ ξ EAT xx is negligible, as shown in Figs. 12(b). In contrast, the scale-similarity modelfor the SGS-Reynolds term (17) is composed only by the small resolved scale velocity, whichmore correlates to the small resolved scale than the largest scale. This is observed from ˇ ξ EAT xx inFigs. 12(a). Thereby, IA does not suffer from the non-local property of the scale-similarity modelsuggested by Anderson and Domaradzki .We also investigate the model where the EAT enters with a negative coefficient, τ sgs i j | tl = − ν sgs s i j − τ eat i j , through the scale-similarity model for the SGS-Reynolds term, because this termyields a negative correlation on the energy transfer between the GS and SGS fields around an30lliptic Burgers vortex . However, this model excessively overestimates the mean velocity (notshown). Moreover, it violates the realizability conditions for the SGS stress, as h τ sgs xx i < τ a xx / τ a ℓℓ > / h τ sgs xx i ≃ h k sgs [ / − ( τ a xx / τ a ℓℓ − / )] i <
0. Furthermore, the form of the EAT providedin Eq. (16) is likewise a key element. The present simulation of IA shows that h k sgs / τ a ℓℓ i ≫ B. Streak structures in stabilized mixed models
As shown in Fig. 8(a), all SGS models except for IA result in the rapid decay of the spectrum ofthe streamwise turbulent fluctuation in the low-wavelength region. The spectrum accumulated inthe high-wavelength region corresponds to the flow structure excessively elongated in the stream-wise direction. Figure 13 shows the contour map of the instantaneous streamwise velocity fluctu-ation u x − h u x i x – z -plane at y + =
20 for Re τ = C GS xx defined by C GS xx ( r x , y ) = h u ′ x ( x + r x e x ) u ′ x ( x ) ih u ′ x i , (40)where e x denotes the unit vector in the x direction. Figure 14 shows the streamwise velocitycorrelation at y + =
20 for Re τ = a)
0 1000 2000 3000 4000 5000 6000 7000 8000 9000 x + z + -10-5 0 5 10 (b)
0 1000 2000 3000 4000 5000 6000 7000 8000 9000 x + z + -10-5 0 5 10 (c)
0 1000 2000 3000 4000 5000 6000 7000 8000 9000 x + z + -10-5 0 5 10 FIG. 13. Contour map of instantaneous streamwise velocity fluctuation u x − h u x i x – z -plane for (a) IA180LD,(b) IA-CL180LD, (c) IA-LNcs42-180LD at y + =
20 for Re τ = in the wall-bounded turbulent shear flow. V. CONCLUSIONS
We investigated the physical role of various scale-similarity models in the SMM . TheSMM proposed by Abe adopted the scale-similarity model for the SGS-Reynolds term although,other scale-similarity models yield a better correlation with the exact SGS stress. In the present32 C xx G S + r x+ IA180LDIA-CL180LDIA-LNcs42-180LDf-DNSDNS -0.2 0 0.2 0.4 0.6 0.8 1 0 200 400 600 800 1000IA180LRIA-CL180LRIA-LNcs42-180LRDSM180LR
FIG. 14. Profile of streamwise velocity correlation C GS xx at y + =
20 for Re τ = study, we applied various scale-similarity models to the SMM and evaluated their performancein turbulent channel flows. As previously shown in previous studies , the SMM usingthe scale-similarity model for the SGS-Reynolds term is less sensitive to the grid resolution thanthe conventional eddy-viscosity models in the prediction of the mean velocity. In particular, itcan predict the near-wall mean velocity profile even in coarse grid resolutions at both low- andhigh-Reynolds numbers. In various SMMs, the original model using the SGS-Reynolds termprovides the best prediction of the Reynolds stress, whereas other models overestimate the GSstreamwise velocity fluctuation. We also investigated Lumley’s invariant map to quantitativelyevaluate the anisotropy of the GS and SGS turbulent stress. The result indicates that the originalmodel predicts a similar near-wall behavior as the filtered DNS. The GS velocity fluctuations forthe eddy-viscosity models result in a nearly one-component turbulence in the vicinity of the solidwall, instead of the conventional two-component state. Moreover, the eddy-viscosity model cannotpredict the anisotropy of the SGS stress, which reflects the isotropic property of the eddy-viscositymodel. A critical difference between various scale-similarity models is found in the spectra of theGS Reynolds stress close to the cut-off scale. The original SMM using the scale-similarity modelfor the SGS-Reynolds term succeeds in predicting the large intensities of the spectra close to thecut-off scale in accordance with the filtered DNS, whereas other models predict a rapid decay ofthe spectra in the low-wavelength region. The success of the scale-similarity model for the SGS-Reynolds term relies on the property that it is expressed by the higher-order spatial derivative,unlike other scale-similarity models. 33o investigate the behavior of the models close to the cut-off scale, we analyzed the budgetequation for the GS Reynolds stress spectrum . As a result, it was shown that the scale-similarity model for the SGS-Reynolds term plays a role in enhancing the wall-normal componentof the GS velocity fluctuation close to the cut-off scale. This leads to the enhancement of thestreamwise and shear components of the GS Reynolds stress in that scale through the productionterm. Hence, the activation of turbulence close to the cut-off scale is achieved. Owing to theseproperties, the streak structures observed in wall-bounded turbulent flows are successfully repro-duced. Although the SMM employing the scale-similarity model for the SGS-Reynolds term doesnot predict the overall profiles of the budget of the GS Reynolds stress spectrum obtained fromthe filtered DNS, it predicts both the statistics and structures in the wall-bounded turbulent flow atthe coarse grid resolution. For further development of SGS models, one should consider how toreproduce the turbulence structures including the low-wavelength region close to the cut-off scale. ACKNOWLEDGMENTS
The authors would like to acknowledge Prof. K. Abe for valuable discussions. K. I. is gratefulto Prof. F. Hamba for fruitful comments and discussions. The work of H. K. is supported in part byKeio Gijuku Academic Development Funds. We would also like to thank the referees for valuablecomments for improvement of this paper.
Appendix A: Contribution of the eddy-viscosity term in SGS normal stress
Figure 15 shows the profile of the eddy-viscosity term and the EAT in the normal componentof the SGS stress for IA180LR. The eddy-viscosity term is negligible compared with the EAT. IA-CL and f-DNS exhibit a similar result (not shown). The eddy-viscosity term in IA-LNcs42 is alsonegligible compared with the SGS kinetic energy k sgs , which can be found in the Lumley invariantmap in Fig. 7(b). Hence, the eddy-viscosity model cannot predict the anisotropy in turbulentchannel flows. 34 〈 τ ea t αα 〉 + , - 〈 ν s g s s - αα 〉 + y + IA180LR EV xxIA180LR EV yyIA180LR EV zzIA180LR EAT xxIA180LR EAT yyIA180LR EAT zz
FIG. 15. Profile of the eddy-viscosity term and EAT in the normal components of SGS stress for IA180LR.Solid lines denote the eddy-viscosity term, while dashed lines denote the EAT.
Appendix B: Budget for the GS Reynolds stress
The budget equation for the GS Reynolds stress yields ∂ R GS i j ∂ t + ∂∂ x ℓ ( U ℓ R GS i j )= P GS i j − ε GS i j + D t , GS i j + Φ GS i j + D p , GS i j + D v , GS i j − ε EV i j + ξ EAT i j + D SGS i j . (B1)Terms on the right-hand side are similar to those in Eq. (32). Note that the inter-scale transfer termvanishes when it is summed over the wavenumbers: k max x ∑ k x = ˇ T GS i j ∆ k x = . (B2)The trace of the pressure–strain correlation Φ GS i j should disappear due to incompressibility: Φ GS ii = . (B3)Therefore, it is sometimes referred to as the redistribution term, which plays a role of the redis-tribution of intensities among normal stress components. In contrast, the trace of the anisotropicredistribution term does not vanish, ξ EAT ii =
0, even though τ eat i j does not exchange energy betweenthe GS and SGS fields, as shown in Eq. (14). This is because ξ eat ii = (cid:10) τ eat i j s i j (cid:11) − (cid:10) τ eat i j (cid:11) S i j = − (cid:10) τ eat i j (cid:11) S i j = . (B4)35 - ε ii E V + / , ξ ii E A T + / y + IA180LRIA-CL180LRIA-LNcs42-180LRf-DNS
FIG. 16. Profile of trace part of the eddy-viscosity destruction − ε EV ii / ξ EAT ii / τ = The mean kinetic energy equation reads ∂∂ t (cid:18) U i U i (cid:19) = − (cid:10) τ eat i j (cid:11) S i j + · · · . (B5)Therefore, h τ eat i j i S i j is interpreted as the energy transfer between the mean and SGS kinetic en-ergies, while ξ EAT ii / ξ EAT ii / ξ EAT i j the anisotropic redistribution term.Figure 16 shows the profile of the trace part of the eddy-viscosity destruction − ε EV ii / ξ EAT ii / τ = ξ EAT ii is relatively small compared with ε EV ii . However, IA provides a positive ξ EAT ii at y + =
20 in the same manner as f-DNS, supporting the increase in the GS velocity fluctu-ations. The success of IA may partly lie in the property of EAT, which enhances the GS velocityfluctuations in the buffer layer 10 < y + < DATA AVAILABILITY
The data that support the findings of this study are available from the corresponding authorupon reasonable request. 36
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