A priori tests of a. new dynamic subgrid-scale model for finite-difference

Transcription

1 A priori tests of a. new dynamic subgrid-scale model for finite-difference large-eddy simulations M. V. SalveGa) Dipartimento di Ingegneria Aerospaziale, Univemiti di Pisa, Piss, Italy S. Banerjee Chcrnical and Nuclear Engineering Department, University of California at Santa Barbara, Santa Barbara, Calalifornia (Received 6 December 1994; accepted 19 July 1994) This work focuses on subgrid-scale (SGS) modeling for finite-difference large-eddy simulations, employing filters in physical space. When a filter in physical space is used, an overlap is allowed between the unresolved and the resolved scales. For such a filter, all the three terms in the classical decomposition of the SGS stress tensor are present: the Leonard and cross-terms, due to the overlap between scales, and the true SGS Reynolds tensor, expressing the pure effect of the small scales. A dynamic subgrid-scale stress model is proposed, for finite-difference large-eddy simulation of incompressible and compressible flows in which the Leonard and cross-parts of the SGS stress tensor are assumed to he proportional to the resolved part (the modified Leonard term ), which is computed explicity. The SGS Reynolds stress is modeled by the eddy-viscosity Smagorinsky model. The two unknown parameters in this model are computed dynamically, as in German0 et al. [Phys. Fluids A 3, 1790 (1991)], but using a least squares technique. The model is tested using direct numerical simulation data for fully developed turbulent incompressible flows in presence of solid boundaries and free surfaces, and for compressible homogeneous turbulence. A box filter in physical space is used. Other SGS models are also tested, viz. the dynamic model of German0 et al. (DSM), and its compressible extension by Mom et al. [Phys. Fluids A 3, 2746 (1991)], and the dynamic mixed model in Zang et al. [Phys. Fluids A 5, 3186 (1993)] (DMM) and its compressible version developed here. Results on the behavior of the different models with regard to energy exchanges and correlation with the exact SGS stresses are presented for different filter widths. In particular high correlation is found between the modified Leonard and cross-terms thus justifying the basic assumption made in the model American Institute oj physics. I. INTRODUCTION The direct numerical simulation (DNS) of turbulent flows is restricted to low Reynolds numbers and simple geometries, because of the need to resolve all the spatial scales of turbulence. These restrictions are reduced in large-eddy simulation (LES), in which the filtered Navier-Stokes equations are solved. Only the large-scale field is directly resolved, while the effect of the unresolved small scales on the large-scale motion has to be modeled. The critical point in LES is to capture the effects of the subgrid scale (SGS) motion, using simple and universal models. In early LES calculations, the most widely used SGS model was due to Smagorinsky, which parametrized the subgrid-scale stresses in terms of an eddy viscosity and local strain rates, calculated from the resolved scales. This model however has some important drawbacks, amongst them an input model coefficient which is really how dependent, the incorrect prediction of limiting behavior near a wall and in laminar flow, and the lack of accounting for backscatter of energy from small scales to large scales. The dynamic subgrid stress model (DSM) proposed by German0 et a2. overcomes several of the limitations of the Smagorinsky model. In the DSM the Smagorinsky eddy- korresponding author: Maria Vittoria Salvetti, Dipartimento di Ingegneria Aerospaziale, Via Diotisalvi 2, PISA, Italy, tel.: ; fax: ; AERO@ICNUCEVM.CNUCE.CNR.I viscosity expression of the SGS stresses is the base model, in which the unknown coefficient is directiy computed using the information from the resolved scales. In this way, the model coefficient can be obtained as a function of space and time, in contrast to the original Smagorinsky model in which it is a constant given a priori. Moreover, this model has the correct asymptotic behavior near a wall and in laminar flow and permits energy backscatter from small scales to large scales. The DSM has been successfully used in several recent LES computations of both incompressible and compressible flows. -4 In spite of its many desirable features, the DSM has still some aspects to be unproved. The dynamically computed model coefficient in Refs. 2 and 3 is obtained by contracting a tensorial equation and is averaged either in homogeneous planes or in the global volume of the domain, to avoid possible singularities. Lilly proposed a least squares technique, which allows the model coefficient to be computed locally. Ghosal et al6 also proposed a method based on the idea of constrained optimization for the local dynamic computation of the unknown parameter in the Smagorinsky model. But in TBS computations in which the dynamic model coefficient is computed locally, large fluctuations of this coefficient occur, leading to numerical instabilities.7 Moreover, since the Smagorinsky model is used, the principal axes of the SGS stress tensor are assumed to be aligned with those of the resolved strain rate tensor. This is a consequence of the assumption of Phys. Fluids 7 (il), November /95/7(11)/2831/17/$ American Institute of Physics 2831

2 equilibrium between dissipation and production, leading to the Smagorinsky type eddy-viscosity models, which is not generally correct. Following the classical decomposition, the SGS stress tensor can be expressed as the sum of three different terms: the Leonard term (which is known from the filtered velocity field), the cross-term and the true SGS Reynolds tensor. only the last term represents the pure effect of the subgrid scales, while the two others derive from the overlap between the resolved and the subgrid scales, provided by the filter. For instance, when a filter in physical space is employed, all the three aforementioned terms are present. More recently Germane proposed another decomposition of the SGS stress tensor, in which each term is separately Galilean invariant: the SGS stress tensor is expressed as the sum of the modifled Leonard tensor, represented only by resolved quantities, the modified cross-term and the modified SGS Reynolds tensor. Again, when a filter allowing the overlapping between resolved and unresolved scales is used, as for instance a filter in physical space, all the terms in the decomposition of German0 are present. From the perspective of large-eddy simulation of engineering flows, computations based on finite-difference formulations are certainly of great interest. Since finitedifference formulations most conveniently use filters in physical space, the problem of SGS modeling for these filters is not just academic. Recently Zang et al. modified the dynamic eddyviscosity model by employing the mixed model of Bardina et al. as the base model, in which the Leonard term is computed explicitly and the cross-term is modeled by the scalesimilarity assumption. It can easily be shown (see, also, Ref. 10) that the term resulting from the sum of the Leonard tensor and the scale similarity term is equivalent to the modified Leonard term in the decomposition proposed by German0 (see Sec. II). Thus, in the mixed model of Bardina, the modified Leonard term is explicitly computed and the true SGS Reynolds stress is modeled by a Smagorinsky model, while the other terms, such as the modified cross-terms, are neglected. The dynamic mixed model (DMM) retains the favorable features of the DSM, but it does not require the SGS stress tensor and the strain tensor to be aligned. The DMM has been successfully applied to the LES simulation of flows in a lid-driven cavity, using a finite-volume method and a box filter in physical space. The results show better agreement with experimental data than obtained using the DSM. Nevertheless, large tluctuations of the model coefficient still occur, even if in very localized regions. Indeed, spatial averaging over the test-filtering volume of the dynamically computed model parameter and the use of a cutoff for negative total viscosity are still needed, in order to prevent numerical instabilities due to too large negative values of this parameter. In this work, a new dynamic SGS model is presented, in which the modified cross-terms are assumed to be proportional to the resolved part (the modified Leonard term), which is computed explicitly. The true SGS Reynolds tensor is still expressed by a Smagorinsky-type model. The proposed model is thus characterized by two unknown param- eters, which are computed dynamically, following the procedure described in the following section. The model retains all the positive features of the DMM, but improves the modeling of the terms in the SGS stress tensor arising from the overlap between resolved and unresolved scales. Moreover, as discussed in Sec. III, since in the dynamic computation of the model parameters less burden is put on the eddyviscosity term, the model coefficient related to the eddyviscosity part exhibits fewer and smaller fluctuations than in previous models. Also, since it is anticipated that the scale similarity term (or equivalently the modified Leonard tensorj represents quite well energy backscatter, the present model is expected to improve the representation of energy backscatter, as, indeed, shown in Sec. III. These positive features of both DMM and the present dynamic two-parameter model (DTM) are significant only if a filter allowing overlapping between the resolved and unresolved scales is used. If a filter that does not permit this overlapping is employed, the model proposed here and the DMM reduce to the DSM, since the Leonard and cross-terms vanish. Clearly this is a situation that does not occur for finite-difference large-eddy simulations and, as discussed later, may not even be desirable from the view point of tluctuations in the model coefficient. Very recently Liu et al. l2 proposed a new similarity SGS model for incompressible flows, in which the SGS stress tensor is assumed to be proportional to the resolved stress tensor. The latter has the same form of the modified Leonard term, but it is obtained by filtering products of resolved velocities at a scale equal to twice the grid scale. The unknown parameter in this model is determined both from the energy dissipation balance and by a dynamic procedure, as described in Ref. 12. A priori tests have been carried out using experimental data in the far field of a round jet, at a relatively high Reynolds number, and a very good correlation is obtained between the SGS stresses computed with this model and those from experimental data, much higher than that obtained in the same tests with eddy-viscosity closures. Anyway, motivated by the fact that in earlier LES computations similarity models appeared to be not dissipative enough to ensure the stability of the calculations, Liu et al. also proposed a mixed model, in which an eddy-viscosity term is added to the similarity part, thus resulting in a modified model with two unknown coefficients. Although the model of the SGS stress tensor in the incompressible version of the DTM proposed here and the mixed model of Liu et az. were developed independently and on the basis of different considerations, they are somewhat similar. Nevertheless, substantial differences can be identified. First of all, the resolved stress tensor in the models of Liu et al. does not represent the modified Leonard term (the resolved part in DTM presented here). In fact, the resolved stress in Ref. 12 is obtained by filtering products of resolved velocities on a yvidth equal to twice the grid filter width. Conversely, to be consistent with the definition, in the modified Leonard term used in the DTM all the filtering operations have the same width (the width of the grid filter). Moreover, in the tests in Ref. 12 the coefficients in the mixed model are not computed dynamically but set to given values. In particular, the coeffi Phys. Fluids, Vol. 7, No. Ii, November 1995 M. V. Salvetti and S. Banerjee

3 cient of the resolved part is chosen equal to 1. Furthermore, the tests in Ref. 12 are carried out for a flow having characteristics different from those studied in this work. In the incompressible tests presented here the behavior of the model in presence of different types of boundaries is studied, while the flow in Ref. 12 is an open flow. In addition, the numerical data used in the tests here are three dimensional, while the data used in Ref. 12 are two dimensional Finally, in this paper, tests are presented also for a compressible flow, giving an appraisal of the behavior of the compressible version of the model. In any case, the results of the a priori tests, carried out by Liu et al. for a flow different from the cases presented here, certainly support our general belief that a two-parameter model may behave well in LES computations, using filters in physical space. In the following section the mathematical formulation of the proposed model is described for both incompressible and compressible flows. The model is then tested using DNS data for both incompressible and compressible flows. The results are discussed in Sec. III and they are compared to those obtained for the same cases with the DSM2* and the DMM and its compressible version developed here. Concluding remarks are presented in Sec. IV. II. MATHEMATICAL FORMULATION In this section the mathematical formulation of the SGS model proposed here and the dynamical computation of the two unknown parameters are described in details for compressible tlows. The specialization of this formulation for incompressible flows is then presented briefly. Following the procedure of German0 et al., two filtering operators are introduced: a grid-scale filter, denoted by an overbar, and a te_st-scale filter, denoted by a hat. The width of the test filter (A) is assumed to be larger than that of the grid filter (8). The ratio between the width of test and grid filters is denoted by a. A. Modeling of the SGS stress tensor For compressible flows it is convenient to recast variables in terms of Favre-filtered quantities. A Favre-filtered variable is defined as: f = $16. Applying a Favre filter to the Navier-Stokes equations for compressible tlows, the effects of the unresolved subgrid scales in the momentum equations appear in the SGS stress term:3 13 rij=p(i+uiij), (1) where the superscript - denotes the Favre filter. By decomposing the velocity as Ui=ii+Uf and substituting in the tirst term of (l), the classical decomposition of the SGS stress tensor yields to rij = L ij $- C, -t R ij (3) where - L, are termed respectively the Leonard stress, cross-stress and the true subgrid-scale Reynolds stress tensors. In the Smagorinsky eddy-viscosity model, the anisotropic part of the SGS stress is assumed to be where C, is the unknown coefficient of the model, s ii= h<du ildxj-t au d&i) is the resolved strain rate tensor, and 131= (2s,,?,) j2. In the mixed model of Bardina et al.,l extended to compressible flows, the Leonard term is computed explicitly and the cross-term is modeled by the scale-similarity assumption. This leads to the following expression for 7ij, in which the constant of the scale-similarity term, originally proposed by Bardina as equal to 1.1, has been adjusted to 1 in order to satisfy Galilean invariance: l4 rij- J$ 7&= - 2c,ppjj&j+L;- % J&g ) (5) where Lc = P(Gj - u icgj) = Lij + Pij, pij being the scalesimilarity term. From Germano s redefinition of stresses, it is easy to show that the Lt term represents the resolved part of the SGS stress, the modified Leonard tensor. Indeed, decomposing the velocity, as in (2), and substituting in both terms of SGS stress tensor in (l), the following decomposition of the SGS stress tensor is obtained: where Tij=L$fCG+RGy are termed, respectively, modified Leonard tensor, modified cross-term, modified SGS Reynolds tensor. The modified Leonard tensor is represented only by resolved quantities. Incidentally we remark that each term in (6) is separately Galilean invariant. Thus in the perspective of this decomposition of the SGS stress tensor, the scale-similarity model of Bardina essentially calculates the modified Leonard term and models the first term in the modified SGS Reynolds tensor using an eddy-viscosity model. The modified cross-term and u;zzj are neglected. In the DSM and DMM, Eqs. (4) and (5) are respectively used as base SGS models, in which the unknown parameters are computed dynamically following the procedure suggested in Ref. 2. In the DTM proposed here, the SGS true Reynolds tensor [or equivalently the first term in Rz in (6)], expressing the pure effect of the unresolved scales, is still modeled by the eddy-viscosity Smagorinsky model. The modified cross- (6) Phys. Fluids, Vol. 7, No. 11, November 1995 M. V. Salvetti and S. Banerjee 2833

4 term is assumed to be proportional to the modified Leonard tensor, which is computed explicitly. This assumption is motivated by the fact that the modified Lenonard and crossterms all arise from the overlapping between scales provided by the filter used. To interpret this assumption in physical terms, note that the modified cross-term represents a stress which involves the resolved and unresolved fields. Using the usual scaling arguments, * this term is expected to be dominated by the smallest resolved scales and the largest unresolved eddies. From the concept of energy cascade, a similarity between the flow field in consecutive scales can be assumed, as shown also from a priori tests in Ref. 12. Hence, a good approximation of the modified cross-term seems to be the modified Leonard term, that is a stress involving only the resolved field and, following the same scaling arguments as previously, dominated by the smaller resolved eddies. Furthermore, this is confirmed by the analysis of the different terms of the SGS stress tensor obtained from DNS data in the a priori tests, as discussed in Sec. III. The term zz;cj has been found to be negligible in all the tests performed. Incidentally, in the perspective of the classical decomposition, it can be easily shown that the assumption made in the present model is equivalent to assuming the Leonard and cross-terms proportional to the modified Leonard tensor. Thus the anisotropic part of the SGS stress tensor is expressed as -2c,6 p]h?]( k?ij- 2 ski) rir 3 rkk= We remark that this model satisfies Galilean invariance of the SGS stress tensor, since the two terms on the right-hand side of (7) are separately Galilean invariant. The two unknown coefficients C, and K in (7) are computed dynamically in the way described below. Following the procedure of German0 et az., a test-filter is applied to the governing equations; the subtest scale stress tensor is then obtained: he ~ PUi PUj Tij=puiuj- ~. B It is modeled in the same manner as the SGS stress tensor: 03) The model coefficients C, and K are assumed to be the same at the grid and filter scales. Using the German0 identity,15 the sub-test-scale and SGS tensors are related as follows:3 where and - 1 he sij=tij-;jij=pu iuj- T (PUipUj). Using Eqs. j7), (9), and (ll), P we obtain Mij~aq;,(~ij- $j s,,i -p,s,( - t$ s,,i m 1 hh Hij=~UiUj- G (pusi/fiitj). P In obtaining Eq. (12) the unknown parameters of the model are assumed constant with respect to the test scale filtering. The limitations and inconsistencies of this assumption are discussed in Ref. 6, in which a new formulation is proposed. Nevertheless, in this paper, since the main interest is to compare the present model with the DSM and the DMM, this assumption, made in the computation of the model parameter in the DSM and in the DMM, is maintained. Anyway the technique proposed in Ref. 6 could well be used in the computations of the unknown parameters in the DTM. Furthermore, the fluctuations of the values of the model parameters obtained locally with the DTM are largely reduced, as discussed in Sec. III, showing that the inconsistency pointed out in Ref. 6 is not the only reason for the difficulties in the DSM model. A least squares approach is used here to compute the coefficients C, and K, as suggested by l3.ly.s Let Q be the square of the error in (12): 2. (10) (11) (13) The unknown coefficients are evaluated by setting aqlx,=o and c?q/c~k=o. Finally, we obtain K= -[m%ij-( sij/3)~kk][hij-(sij/3)hkk]+ba/d where t-2 Q gjz; g!+gj, i j ii $1 =(2&j&j) 2 and Lfi can be obtained by inserting the decomposition of the velocity field (2) in (8). This yields a2/d-[hij-( and UK-b C,=- 2A2d where Sii/3)HkJ2 (14) (15) 2834 Phys. Fluids, Vol. 7, No. 11, November 1995 M. V. Salvetti and S. Banerjee

5 and d=mijmij a )Mij, b=( ey?ij- ff Bkk)Mtj We also describe briefly the procedure for computing the model coefficient for the DMM: this is the extension to the compressible case of the formulation presented in Ref. 10 for incompressible flows. Using Eq. (5) and the German0 identity (ll), we obtain C,?, is determined using the least squares technique and is given by the following expression: c = -[~ij-(6ij/3)~kk-hij+ (~ijl3)hkjmij m 2~2MijMij (17) For the DSM, following the same procedure, the unknown model parameter is given by (18) We remark that in the compressible case the isotropic part of the SGS stress tensor must be modeled, in contrast to what happens for incompressible flows, where it is absorbed into the pressure term. In the present model, this part is given by B. SGS modeling in the energy equation In obtaining the filtered energy equation the same assumptions, as in Refs. 13 and 3, are made here. Thus only the SGS heat flux term must be modeled, to close the energy equation. The SGS heat llux vector has the following expression: - lp- Qk=PUkT- ; puk /IT, (22) where T is the temperature. In the present model the SGS heat flux vector is expressed as follows: (23) where vt= C,K21s l is the eddy viscosity with C, given by Eqs. (14) and (ls), and Pr, and h are the unknown parameters of the model. The term 4; is the resolved part of the SGS heat flux vector, obtained by decomposing the velocity field, as in (2), and the temperature as T=?+ T (24) and substituting in (22). This leads to the following decomposition of the SGS heat flux vector: where 4k=d+d!+d 7 (25) rkk=rkkfkl;k, (19) where R,, is the isotropic part of the true SGS Reynolds StXeSS. As a first approximation, we neglect here Rkk, as was done in Ref. 13. However, in Ref. 3, it has been shown that this term can be important in some cases. Thus it may be eventually modeled, as in Ref. 3. For incompressible flows, the SGS stress tensor has the following form: Tif=UiUj-UiUj * In the incompressible case the DTM writes: sij rfj- 3 rkk= -2C,z ]SIs,+K( L$- : LE), (21) ji& I_ -?+Z ( I I 1 is s resolved strain rate tensor, ISI =(2S,S,) 1 2, and Lb _ asr =UiUj-UiUj * The dynamical procedure for the determination of the unknown model parameters is completely analogous to that described for the compressible case and is omitted here for the sake of brevity. cw Expression (23) is obtained by assuming, analogously to what has been done for the SGS stress tensor, that the term PF may be expressed by an eddy-viscosity model and that the other terms are proportional to the resolved part &. The unknown parameters Pr, and h are determined dynamically, following the same procedure as for the SGS stress tensor. Applying the test filter to the energy equation, the sub-test-scale heat flux vector is obtained: - l-- ~ Qk=zT-; PGPT- P (261 It is modeled in the same way as the SGS heat flux vector: f&c=-- pr n ~ii2c$~ a? z+hqtv k (27) where Qi is the resolved part of Qk, that can be obtained by decomposing the velocity and temperature fields, as in (2) and (24), and substituting it in Eq. (27): =z 1-q Q;=jjikT- B (3) Phys. Fluids, Vol. 7, No. 11, November 1995 M. V. Salvetti and S. Banerjee 2835

6 Following Ref. 3, the heat flux vectors at the grid and test scales are related by the algebraic identity:.z$= Qk-&= jii& s (2% where.%5k can be computed explicitly. Using Eqs. (22), (26), and (29), we obtain where and c,zi2.%$= - Pr.,&~+k@~, n -1 -?G Qk= Q;-- 2=$&f- ; jik$t. P (30) The unknown coefficients Pr,, and h are obtained from Eq. (30) by the least squares approach. Let B be the square error in (30): g=(.%k+y&?k- /z.&)~, where y= C,L21Pr,, ; Pr, and h are obtained by setting &Z lay =0 and &%plah=o. Finally, we obtain and h=-.% & - a bld b ld - Qk/rk where a =.%?& s,, b =...&&, and d=~,.&&.&~. III. APPRAISAL OF THE MODELS USING DIRECT NUMERICAL SIMULATION DATA (32) The proposed model is tested using data from DNS for both incompressible and compressible flows. It is known *3 that a priori tests only tell part of the story in term of the performance of a SGS model in LES computations. However, they are an excellent l?rst step in the appraisal of SGS models, keeping in mind that a model that does badly in a priori tests, does not necessarily do badly in LES computations. Nevertheless, it will be shown that the DSM does worse than the DMM in these a priori tests, and this is consistent with the results of LES computations of Zhang et al. using these models for incompressible flows. On the other hand, a model showing also good behavior in a priori tests would be more desirable. In this work a box filter in physical space, in which integrals are computed by the trapezoidal rule and linear interpolation, is used both as grid and test filter. The box filter is applied in all directions and the filter width is defined as --- i3=a 1 A 2 A 37 A3=A1A2A3, where & and pi are, respectively, the width of the grid and test filter in direction i. In all cases the length of the test filter in each direction is twice that of the grid filter (a=2). This value has been chosen because it was found to be the optimal choice in a priori tests in Ref. 2. In order to obtain meaningful a priori tests, as pointed out in Ref. 12, the SGS stresses modeled are computed using solely the filtered velocity field, defined on the LES grid. In particular, twice filtered quantities are obtained at the grid and test scales using the expressions detailed in Ref. 10. A. Incompressible flows 1. Behavior of SGS stresses and model parameters For incompressible flows, tests are performed using the DNS data base of Lam and Banerjee16 for the flow between a no-slip wall and a surface with no shear at which free-slip conditions are imposed. The Reynolds number based on wall shear velocity and total depth is 171. In the simulation, 64 Fourier modes in streamwise and spanwise directions, and 65 Chebyshev polynomials in the normal direction were used. This corresponds to a 64X64X65 grid, non uniform in the normal direction. The data used here are the results of a direct numerical simulation performed by Lam and Banerjee for the same case as Ref. 16, but with 64 modes in streamwise direction instead of 32. Since, in Ref. 16, evidence is presented that 32X64 Fourier modes (in streamwise and spanwise directions) are enough to give an adequate resolution of the flow, this is true also for the DNS data used here. In a priori tests the grid filter width in each direction is taken to be equal to twice the grid spacing. Streamwise, spanwise, and normal directions are denoted by subscripts 1, 2, and 3, respectively. The SGS stresses computed by filtering the DNS data are compared to those obtained using the present model, the DSM and DMM. The SGS stresses are obtained using instantaneous DNS data. Tests for several time steps have been performed. Since the same remarks can be made for all the time steps, the results for a single time step only are presented. The model coefficients are computed locally using the procedure described in the previous section. In Fig. 1 the value of the 711 component of the trace-free SGS stress tensor obtained from DNS data, averaged on horizontal homogeneous planes, is shown as a function of the distance from the solid wall (in wall units). The values of the same component for the modified Leonard and cross terms and for the true SGS Reynolds tensor are also shown. Several features appear from this figure. First of all, the modified Leonard term represents the main part of T,~, especially near the solid wall, where r,, exhibits a large peak due mainly to the particular filter used and to the averaging in the shear normal direction. Incidentally, we remark that, when a sharp cutoff filter in the Fourier space is used, filtering Chebyshev expansions in the inhomogeneous direction is not trivial. However, filters in physical space can be easily applied to nonhomogeneous grids. Thus models which account for the modified Leonard.term, as DMM and DTM, are expected to give better agreement than the DSM, especially in high-shear regions. Moreover, the modified cross term also exhibits a peak in the near wall region, assuming values which are larger than the true SGS Reynolds stress. This 2836 Phys. Fluids, Vol. 7, No. 11, November 1995 M. V. Salvetti and S. Banerjee

7 * SGS stress tensor - modified Leonard tensor ---. modified cross term -. - true SGS Reynolds tensor FIG. 1. Decomposition of plane-averaged 711 component of the SGS tensor. result shows that proper modeling of these terms could improve noticeably the behavior of SGS models, especially in high-shear zones. It gives also support, a posteriori, together with the results obtained from the decomposition of the other components of the SGS stress tensor, to our reasons for modeling this term. A more extensive discussion of this point will be presented in the following section, from analysis of the correlation between the modified Leonard and cross-terms. As mentioned previously, the term ~2; ~2; has been found to be negligible in all the tests performed. In Fig. 2 we compare the values of rtl obtained from DNS with those computed respectively by DSM, DMM, and the present model. The value obtained with the DSM, compare very poorly with the value from DNS. Indeed, the DSM does not account for the modified Leonard and cross-terms that, for the filter used here (allowing the overlapping between resolved and unresolved scales), represent the main part of this component of the SGS stress tensor. Moreover, near the wail the main component of the strain rate tensor is the 13 component, due to the high shear in the normal direction; thus the DSM, in which the axes of the SGS stress tensor are assumed to be aligned to those of the strain rate tensor, is not able to reproduce the large peak in rll. The DMM, accounting for the modified Leonard part, gives better agreement. Nevertheless, the modified cross-term (neglected in the DMM) is also important. In the present model it is assumed to be proportional to the Leonard term: that allows much better correspondence to values of rll from DNS to be obtained, especially in the peak region near the wall. If the same comparisons are made for r13 (Fig. 3), the values from the DSM compare somewhat better with those from the DNS than for T,,, as expected on the basis of the previous discussion. However, even in this case, the best agreement is obtained with the present model. In Fig. 4 we compare the values of the Smagorinsky part =+ 160 FIG. 2. Plane-averaged q1 component of the SGS tensor. Phys. Fluids, Vol. 7,-No. 11, November 1995 M. V. Salvetti and S. Banerjee 2837

8 DNS - DSM ---. DMM D-,-M -.-.._ =+ 160 FIG. 3. Plane-averaged 713 component of the SGS tensor. coefficient (C), averaged on horizontal planes, obtained respectively with DSM, DMM, and the present model as a function of the wall coordinate. For all the models, as expected, the coefficient vanishes at the wall. But for the present model the values of C are lower and fluctuate less than for the other models. This is particularly important near the free surface, where the DSM and the DMM show a large peak, almost nonexistent for the DTM. This result is not surprising. In fact, since in the DSM (as also in the DMM) the only adjustable parameter is the Smagorinsky part coefficient, in the dynamical procedure this coefficient is computed from not only the true SGS Reynolds tensor, but from the whole SGS stress tensor (or the SGS stress tensor minus the modified Leonard term in the DMM). Thus in the DSM (and to a lesser extent in the DMM) a considerable burden is put on the coefficient related to the Smagorinsky part. In Fig. 5 the same plot is shown for the second parameter in the present model (K) averaged on horizontal planes. The value of K increases near the wall to reach an almost constant value of about 1.3, then drops slightly near the free surface. Since one of the points of main interest in the tests presented here is to study the behavior of the different models near boundaries, and, in particular, near free surfaces, tests have been performed using the data from a direct numerical simulation of the near interface turbulence in gas-liquid coupled flows. The computational domain is divided in two subdomains, in which gas and liquid flows between two freeslip external boundaries are simulated. At the interface continuity of shear stress and velocity is assumed. Thus the interface may be considered a free-surface under shear, while no shear is present at the external surfaces. The Reynolds number, equal to 171, is based on the shear velocity and the depth of each subdomain and defines the extent of each subdomain in shear-based nondimensional units. In each subdomain 64 Fourier modes were used in streamwise and spanwise directions and 65 Chebyshev polynomials in the normal DSM - DMM ---. D-,-f,,, FIG. 4. Plane-averaged Smagorinsky part coefficient Phys. Fluids, Vol. 7, No. 11, November 1995 M. V. Salvetii and S. Banerjee

9 z+ 160 FIG. 5. Plane-averaged parameter K in the present model. direction. As in the previous case, the grid filter width in each direction is taken to be twice the grid spacing and the SGS stresses are computed using the instantaneous values from DNS for several time steps. In Fig. 6 we compare the values of the horizontal-plane average of 711 obtained, respectively, from DNS, DSM, DMM, and the present model for the liquid subdomain. They are plotted as a function of the nondimensional distance from the sheared interface. It is interesting to remark that near the sheared surface a large peak of ri1 occurs, as was found near the solid wall in the previous tests. As was pointed out previously, this peak is an effect of the particular filter used. Moreover, the behavior of the models near the sheared interface is found to be similar to that observed near the wall in the previous tests. Indeed, for this case also, for the reasons explained earlier. the DSM correlates poorly with the DNS data, while the DMM, accounting for the Leonard term, gives better agreement. The best agreement is again obtained with the present model. In Fig. 7 the same plots are shown for T,~; remarks similar to those for the previous tests can be made. In Fig. 8 the values of the horizontal-plane average of the Smagorinsky part coefficient (C) obtained respectively by DSM, DMM, and the present model (DTM) are plotted as a function of the distance from the interface in wall units. The same plot for the second parameter (K) in the present model is shown in Fig. 9. As in the previous case, and in accord with the previous considerations, the coefficient C obtained with the present model takes values which are lower than those of the other models. Moreover, the large fluctuations, characterizing this coefficient for the DSM and the DMM, especially near the free-surface, are largely reduced in the present model. The second parameter K, except for a slight decrease near the free surfaces, is almost constant at the same value of the previous tests. Since the model parameters are computed locally, it is interesting to Iook not DNS - DSM - DMM , [ 1,! i!!i :i!! : &._- ~ 0 FIG. 6. Plane-averaged 711 component of the SGS tensor of the liquid subdomain. Phys. Fluids, Vol. 7, No. 11, November i 995 M. V. Salvetti and S. Banerjee 2839

10 DMM :, I z+ 160 FIG. 7. Plane-averaged r13 component of the SGS tensor for the liquid subdomain I DSM ~ DMM --- DTM _m-- *-* _ / _ _ I-.--._., _+... _---i, = _ 8 :. :,\ ; FIG. 8. Plane-averaged Smagorinsky part coefficient for the liquid subdomain z+ 160 FIG. 9. Plane-averaged parameter K in the present model for the liquid subdomain Phys. Fluids, Vol. 7, No. 11, November 1995 M. V. Salvetti and S. Banerjee

11 6 6 Y Y FIG. 10. Fluctuations of the Smagorinsky part coefficient on a horizontal plane in the middle of the liquid subdomain for DSM. The plane average value is ~*. FIG. 12. Fluctuations of the Smagotinsky part coefficient on a horizontal plane in the middle of the Liquid subdomain for the present model. The plane average value is lo-. only at averaged values but also at the variation of these parameters on horizontal planes. In Figs. lo-12 the fluctuations (difference between the local and the plane-averaged values) of the Smagorinsky part coefficient are plotted for a C horizontal plane in the center of the liquid subdomain, respectively, for DSM, DMM, and the present model. The model coefficient for the DSM shows large fluctuation peaks, both positive and negative, about an order of magnitude 0.: larger than the averaged value. The number and the intensity of these peaks decrease significantly for DMM and in even larger measure, for the present model. This behavior, which is due to the reasons explained before for Fig. 4, is a desirable feature in actual LES computations to avoid numerical instabilities. The second parameter in the present model also varies on horizontal planes (Fig. 13) but as a small percentx age of its average value, without exhibiting large isolated peaks. The values of these fluctuations relative to the average RG. 13. Fluctuations of the coefficient K for the present model on a horivalue are much smaller than those observed for the Smago- zontal plane in the middle of the liquid subdomain. The plane average value rinsky part coefficient in the DSM. The same remarks can be is 1.3. made for the fluctuations of the parameters C on a horizontal plane at the free surface (Figs ). FIG. 11. Fluctuations of the Smagorinsky part coefficient on a horizontal plane in the middle of the liquid subdomain for DMM. The plane average is z. FIG. 14. Fluctuations of the Smagorinsky part coefficient on a horizontal plane near the free surface in the liquid subdomain for the DSM. The plane average value is w3. Phys. Fluids, Vol. 7, No. 11, November 1995 M. V. Salvetti and S. Banerjee 2841

12 ' ;*;; 1 5 0: ' FIG. 15. Fluctuations of the Smagorinsky part coefficient on a horizontal plane near the free-surface in the liquid subdomain for the DMhC The plane average value is Oe3. In order to investigate if the good agreement between the exact SGS stress tensor components and those obtained by the present model, indicated by plane-averaged values, is achieved also locally, we compare in Figs the isocontours of rrl on the same horizontal plane as in Fig. 14, obtained respectively from DNS, by the DSM, the DMM and present model. As for the plane-averaged values, the DSM agrees very poorly with the values from DNS. The isocontours for the DMM have a shape very similar to those from DNS, but not very satisfactory quantitative agreement. Conversely, excellent agreement between the isocontours from the present model and those from DNS is obtained, both quantitatively and qualitatively. FIG. 17. Isocontours of rtr on a horizontal plane near the free surface in the liquid subdomain from DNS. 5 'O x y FIG. 18. Isocontours of rt, on a horizontal plane near the free surface in the liquid subdomain for DSM Correlation with exact SGS stresses and energy For plane-averaged and local SGS stresses, the present model shows better agreement with DNS data than the other models. For the purpose of better understanding the reasons of this behavior, the correlation coefficients between some components of the SGS stress tensor and the same components, respectively, of the SGS strain rate tensor and the modified Leonard term are shown in Table I. The correlation coefficients between the modified Leonard tensor and the 5 lo x 0.48 C Y FIG. 19. Isocontours of rlr on a horizontal plane near the free surface in the liquid subdomain for DMhK X 5 '0 x FIG. 16. Fluctuations of the Smagorinsky partcoefficient on a horizontal plane near the free surface in the liquid subdomain for the present model. The plane average value is m3. FIG. 20. Isocontours of rtr on a horizontal plane near the free surface in the liquid subdomain for the present model Phys. Fluids, Vol. 7, No. 11, November 1995 M. V. Salvetti and S. Banerjee

13 TABLE I. Correlation coefficients between SGS stresses and respectively the strain rate and the modified Leonard term. Correlation coefficients between the modified Leonard and cross-terms. (a) LES grid: 32X32X33; (b) LES grid: 16X16X33; (c) LES grid: 16X16x17; (d) LES grid: 8X8X (4 '$1 (4 ( TABLE II. Correlation coefficients between the exact SGS stresses and those obtained by the different models. (a) LES grid: 32X32X33; (b) LES grid: 16X16x33; (c) LES grid: 16X16X17; (d) LES grid: 8X8X9. DSM DMM DTM ( I Cc) modified cross-term are also shown for the same components. The sensitivity to the filter width is also investigated and the correlations are presented also for tests performed with LES grids of respectively 16X 16X33, 16x 16X 17, and 8 X 8 X 9 points. Incidentally, we remark that, in LES of high Reynolds number flows, the grid used must be much coarser than the grid needed in direct simulation of such flows, because of computer resource limitations. For this reason we explore a range of LES filters down to quite coarse nodalizations and the performance of the various models in the very coarse nodalization range is of particular interest. The correlation coefficient between two variables x and y is defined as usual: The symbol (.) indicates the averaging over a volume and several time steps. The data used are those from the direct numerical simulation of the near interface turbulence in gasliquid coupled flows, presented in the previous section. The correlation coefficients between the exact SGS stresses and those obtained with the different models are shown in Table II. The correlations between the SGS stress and the strain rate tensor are low for all components and all filter widths. This is in agreement with numerous previous results of u pl-ioc tests from DNS data and also with the results of a p&ri analysis of high Re experimental results in Ref. 12. The DSM gives a much higher correlation with the exact SGS stress tensor for the 13 component than that between r13 and Isl,?,,. This is due to the fact that in the dynamic procedure the model parameter is obtained by contracting the term Zlj in the German0 identity by the difference between the strain rate tensors respectively at the grid and test scales [see Eq. (1811. S ince the 13 component of the strain rate tensor, both at grid and test scales, is much larger than the others, the model parameter is computed optimizing the agreement between the 13 component of the exact and modeled SGS stress tensors. These results are consistent with the considerations made previously from the plane-averaged stresses. Nevertheless, as the filter width increases, this mechanism becomes less efficient; for instance, the correlation obtained with a 8X8 X9 LES grid for the 13 component is lower than for the 11 component. The correlation between the modified Leonard term and the SGS stress tensor is high for all the components and all the filter widths, as already shown by Bardina in Ref. 11 and in other previous works. The correlation between the exact SGS stresses and those obtained with DMM is practically the same than that between the modified Leonard term and the SGS stresses, except for the 13 component, for which it is noticeably lower especially when the filter width is larger. This can be explained by the fact that, when the filter width increases, the contribution of the modified Leonard term to the SGS stress decreases noticeably for the 13 component, while for others components it remains almost constant. As an example, the averaged values of the modified Leonard and cross-terms normalized by the averaged SGS stress are presented for the 11 and 13 components in Table III. At the same time for the 13 component the correlation of the Smagorinsky part decreases noticeably as the filter width increases, as discussed previously. It should be also stressed that the contribution of the modified cross-term to the SGS stress tensor is not negligible in all cases and can be higher than that of the modified Le- onard term, as, for instance, for the 13 component for a LES grid of 8X8X9 points. Phys. Fluids, Vol. 7, No. 11, November 1995 M.V. Salvetti and S. Banerjee 2843

14 TABLE III. Averaged values of the modified Leonard and cross-terms normalized by the averaged SGS stress. (a) LES grid: 32X32X33; (bj LES grid: 8X8X9. ( Y FIG. 22. Isocontours of (7;; on a horizontal plane in the middle of the liquid subdomain. The most interesting feature appearing from Table I is the good correlation (4BO.6) between the modified Leonard and cross-terms obtained for all the components and all the filter widths. This result corroborates the assumption of similarity between the modified cross- and Leonard terms made in the present model. This similarity is observed also locally, as can be seen, for instance, from the isocontours of the 13 component of the modified Leonard and cross-terms on a horizontal plane at z+ = 19.4 presented in Figs. 21 and 22, for the 32X32X33 LES grid. Indeed, a quite good correspondence of high gradient regions is found. According to these considerations, the correlations between the exact stresses and those modeled by the DTM are higher than those obtained with the other models, for all the components and all the filter widths. Moreover, the gain in comparison with the DMM is more important when the contribution of the modified cross term to the SGS stress is higher, as, for instance, for the 13 component when the filter width is larger. Previous results show that the assumption of proportionality between the modified cross- and Leonard terms made in the present model is justified and that it improves the global and local agreement with the exact SGS stresses. We wish to examine now how the different parts of the SGS stress contribute to the SGS energy dissipation Z?=Q~~~. For this purpose, the correlation coefficients - - between the SGS dissipation and, respectively, -S,S, and ~=I*~j;ij are presented in Table IV for different filter widths. The represents the SGS energy dissipation due to the modified Leonard term. The correlations are not reported for the 8X8 X9 LES grid, since the number of points available for the averaging was not enough to obtain statistical conver- gence. In fact, when computing g, we are dealing with a third-order moment of the velocity and such moments are known to converge - - slowly. As expected, the correlation between B and -SijSii is low, indicating that Smagorinskytype models could not give a satisfactory approximation of the SGS energy. Again the SGS dissipation due to the modified Leonard term correlates well with the whole SGS dissipation. However, it must be remarked that previous LES calculations with scale similarity models encountered instability problems because such models are not dissipative enough; hence, and eddy-viscosity term is usually added, yielding to the mixed models. On the other hand, it is anticipated that the scale similarity model gives a good representation of energy backscatter from unresolved to resolved scales, corresponding to positive values of cy. This is confirmed by the high correlation between the whole SGS energy and the part due to the modified Leonard term, obtained by averaging only in regions where backscatter occurs (Table IV). As previously for SGS stresses, the most interesting feature appearing from Table IV is the high correlation existing between the SGS energy due to the modified Leonard tensor and that due to the modified CEsij, meaning that the assumption of similarity between the modified Leonard and cross-terms made in the present model is justified also from the perspective of energy exchanges between resolved and unresolved scales. In the first line of Table V the correlation coefficients between the SGS energy from DNS and that from the different models, obtained by averaging over the whole computational domain, are shown. The values in the second line represent the correlation coefficients obtained averaging in regions where backscatter occurs. It is interesting to remark FIG. 21. Isocontours of LI; subdomain. 5 '0 on a horizontal plane in the middle of the liquid x Y TABLE IV. Correlation coefficients between: the SGS dissipation and ?,s,, the SGS dissipation and the SGS dissipation due to the modified 4 Leonard term, the backscatter and backscatter due to the modified Leonard 3 term, the SGS dissipation, due to the modified Leonard term and that from the modified cross-term. 2 1 $NK-s,js,j) wp;l) +w+ 7%) d@m 32X32X X16X x 16x Phys. Fluids, Vol. 7, No. 11, November 1995 M. V. Salvetti and S. Banerjee