%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % An example input file demonstrating the tcfd option of the SVJour % % document class for the journal: Theoret. Comput. Fluid Dynamics % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % \documentclass[tcfd]{svjour} \usepackage{graphicx} %%% \usepackage{times} %%% \usepackage{mathtime} \sloppy %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Several Macro's for this article programmed by the Author % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \input{example.sty} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{document} \title{Subgrid-Scale Models for Compressible Large-Eddy Simulations\thanks{The authors gratefully acknowledge the support from the Air Force Off\/ice of Scientific Research, under Grant Nos. AF/F49620-98-1-0035 (MPM and GVC) and AF/F49620-97-1-0244 (UP), monitored by D.L.~Sakell. This work was also sponsored by the Army High Performance Computing Research Center under the auspices of the Department of the Army, Army Research Laboratory cooperative agreement number DAAH04-95-2-0003/contract number DAAH04-95-C-0008, the content of which does not necessarily ref\/lect the position or the policy of the government, and no off\/icial endorsement should be inferred. A portion of the computer time was provided by the University of Minnesota Supercomputing Institute.}} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % You have to protect commands within the "Author"-macro by using the % \protect-command %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Author{M. Pino Mart\protect{\'\i}n} {Department of Aerospace Engineering and Mechanics, University of Minnesota,\\ 110 Union St. SE, Minneapolis, MN 55455, USA\\ pino@aem.umn.edu} \Author{Ugo Piomelli} {Department of Mechanical Engineering, University of Maryland,\\ College Park, MD 20742, USA\\ ugo@eng.umd.edu} \Author{Graham V. Candler} {Department of Aerospace Engineering and Mechanics, University of Minnesota,\\ 110 Union St. SE, Minneapolis, MN 55455, USA\\ candler@aem.umn.edu} \commun{Communicated by M.Y. Hussaini} \date{Received 12 March 1999 and accepted 11 August 1999} \abstract{An {\it a priori} study of subgrid-scale (SGS) models for the unclosed terms in the energy equation is carried out using the f\/low f\/ield obtained from the direct simulation of homogeneous isotropic turbulence. Scale-similar models involve multiple f\/iltering operations to identify the smallest resolved scales that have been shown to be the most active in the interaction with the unresolved SGSs. In the present study these models are found to give more accurate prediction of the SGS stresses and heat f\/luxes than eddy-viscosity and eddy-diffusivity models, as well as improved predictions of the SGS turbulent diffusion, SGS viscous dissipation, and SGS viscous diffusion.} \authorrunning{M.P. Mart\protect{\'\i}n, U. Piomelli, and G.V. Candler} \maketitle %%%%%%%%%%%%%%% %% Section 1 %% %%%%%%%%%%%%%%% \section{Introduction} \label{sec:1} Large-eddy simulation (LES) is a technique intermediate between the direct numerical simulation (DNS) of turbulent f\/lows and the solution of the Reynolds-averaged equations. In LES the contribution of the large, energy-carrying structures to momentum and energy transfer is computed exactly, and only the effect of the smallest scales of turbulence is modeled. Since the small scales tend to be more homogeneous and universal, and less affected by the boundary conditions, than the large ones, there is hope that their models can be simpler and require fewer adjustments when applied to different f\/lows than similar models for the Reynolds-averaged Navier--Stokes equations. While a substantial amount of research has been carried out into modeling for the LES of incompressible f\/lows, applications to compressible f\/lows have been signif\/icantly fewer, due to the increased complexity introduced by the need to solve an energy equation, which introduces extra unclosed terms in addition to the subgrid-scale (SGS) stresses that must be modeled in incompressible f\/lows. Furthermore, the form of the unclosed terms depends on the energy equation chosen (internal or total energy, total energy of the resolved f\/ield, or enthalpy). Early applications of LES to compressible f\/lows used a transport equation for the internal energy per unit mass $\varepsilon$ \citep{moi91,elh94} or for the enthalpy per unit mass $h$ \citep{spe88,erl92}. In these equations the SGS heat f\/lux was modeled in a manner similar to that used for the SGS stresses, while two additional terms, the SGS pressure-dilatation $\Pi_{\rm dil}$ and the SGS contribution to the viscous dissipation $\varepsilon_{\rm v}$, were neglected. \citet[b]{vre95a} performed {\it a priori} tests using DNS data obtained from the calculation of a mixing layer at Mach numbers from 0.2 to 0.6. They found that the SGS pressure-dilatation $\pi_{\rm dil}$ and SGS viscous dissipation $\varepsilon_{\rm v}$ are of the same order as the divergence of the SGS heat f\/lux $Q_j$, and that modeling $\varepsilon_{\rm v}$ improves the results, especially at moderate or high Mach numbers. They also proposed the use of a transport equation for the total energy of the f\/iltered f\/ield, rather than either the enthalpy or the internal energy equations; the same unclosed terms that appear in the internal energy and enthalpy equations are also present in this equation. Very few calculations have been carried out using the transport equation for the total energy, despite the desirable feature that it is a conserved quantity, and that all the SGS terms in this equation can be cast in conservative form. This equation has a different set of unclosed terms, whose modeling is not very advanced yet. \citet{nor92} performed calculations of a transitional boundary layer, and modeled only the SGS heat f\/lux, neglecting all the other terms. \citet{kni98} performed the LES of isotropic homogeneous turbulence on unstructured grids and compared the results obtained with the \citet{sma63} model with those obtained when the energy dissipation was provided only by the dissipation inherent in the numerical algorithm. They modeled the SGS heat f\/lux and an SGS turbulent diffusion term, and neglected the SGS viscous diffusion. \citet{com98} proposed the use of an eddy-diffusivity model for the sum of the SGS heat f\/lux and SGS turbulent diffusion, neglecting the SGS viscous diffusion. In this paper the f\/low f\/ield obtained from a DNS of homogeneous isotropic turbulence is used to compute the terms in the energy equations, and evaluate eddy-viscosity and scale-similar models for their parametrization. We place emphasis on the total energy equation, both because of the lack of previous studies in the terms that appear in it, and because of the desirability of solving a transport equation for a conserved quantity. In the remainder of the paper the governing equations are presented and the unclosed terms are def\/ined. The DNS database used for the {\it a priori} tests is described. Finally, several models for the unclosed terms are presented and tested. %%%%%%%%%%%%%%% %% Section 2 %% %%%%%%%%%%%%%%% \section{Governing Equations} \label{sec:2} To obtain the equations governing the motion of the resolved eddies, we must separate the large from the small scales. LES is based on the def\/inition of a f\/iltering operation: a resolved variable, denoted by an overbar, is def\/ined as \citep{leo74} \begin{equation} \overline{f}({\bf x}) = \int_D f({\bf x'}) G({\bf x},{\bf x'};\delbar) \D{\bf x'}, \label{eq:filtering} \end{equation} where $D$ is the entire domain, $G$ is the f\/ilter function, and $\delbar$ is the f\/ilter-width associated with the wavelength of the smallest scale retained by the f\/iltering operation. Thus, the f\/ilter function determines the size and structure of the small scales. In compressible f\/lows it is convenient to use Favre-f\/iltering \citep[b]{fav65a} to avoid the introduction of SGS terms in the equation of conservation of mass. A Favre-f\/iltered variable is def\/ined as $\widetilde{f}=\overline{\rho f}/\overline{\rho}$. In addition to the mass and momentum equations, one can choose solving an equation for the internal energy, enthalpy, or total energy. Applying the Favre-f\/iltering operation, we obtain the resolved transport equations \begin{equation} \label{eq:mass-ff} \frac{\partial\rhob}{\partial t} + \frac{\partial}{\partial x_{j}} \left(\rhob\util_j\right) = 0 , \end{equation} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{equation} \label{eq:mom-ff} \frac{\partial\rhob\,\util_i}{\partial t} + \frac{\partial}{\partial x_{j}}\left(\rhob\util_i\util_j + \pb \delta_{ij} - \widetilde{\sigma}_{ji} \right) = - \frac{\partial\tau_{ji}}{\partial x_{j}} , \end{equation} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{equation} \label{eq:int-en-ff} \frac{\partial (\rhob\,\widetilde{\varepsilon}\,)}{\partial t} + \frac{\partial}{\partial x_{j}} \left(\rhob\util_j\widetilde\varepsilon\right) + \frac{\partial\widetilde{q}_j}{\partial x_j} + \pb\Stkk - \widetilde{\sigma}_{ji}\Stij = - C_{\rm v}\frac{\partial Q_j}{\partial x_{j}} - \Pi_{\rm dil} + \varepsilon_{\rm v} , \end{equation} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{equation} \label{eq:enth-ff} \frac{\partial (\rhob\,\widetilde{h})}{\partial t} + \frac{\partial}{\partial x_j}\left(\rhob\util_j\htil\right) + \frac{\partial\widetilde{q}_j}{\partial x_j} - \frac{\partial\pb}{\partial t} - \util_j\frac{\partial\pb}{\partial x_j} - \widetilde{\sigma}_{ji}\Stij = - C_{\rm v}\frac{\partial Q_j}{\partial x_j} - \Pi_{\rm dil} + \varepsilon_{\rm v} , \end{equation} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{equation} \label{eq:tot-en-ff} \frac{\partial}{\partial t} (\rhob\,\Etil) + \frac{\partial}{\partial x_j} \left[(\rhob\,\Etil + \pb)\util_j + \widetilde{q}_j - \widetilde{\sigma}_{ij}\util_i \right] = - \frac{\partial}{\partial x_j}\left(\gamma C_{\rm v}Q_j + {\textstyle\half}{\cal J}_j - {\cal D}_j \right) . \end{equation} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Here $\rho$ is the density, $u_j$ is the velocity in the $x_j$ direction, $p$ is the pressure, $\varepsilon=c_{\rm v} T$ is the internal energy per unit mass, $T$ is the temperature; $h=\varepsilon + p/\rho$ is the enthalpy per unit mass; $E=\varepsilon +u_iu_i/2$ is the total energy per unit mass, and the diffusive f\/luxes are given by \begin{equation} \label{eq:sig-hat} \widetilde{\sigma}_{ij} = 2\widetilde{\mu} \Stij - {\textstyle\frac{2}{3}}\widetilde{\mu} \delta_{ij}\Stkk, \qquad \widetilde{q}_j = - \widetilde{k}\frac{\partial\Ttil}{\partial x_j} , \end{equation} where $S_{ij}=\frac{1}{2} (\partial u_i/\partial x_i + \partial u_j/\partial x_i)$ is the strain rate tensor, and $\widetilde{\mu}$ and $\widetilde{k}$ are the viscosity and thermal conductivity corresponding to the f\/iltered temperature $\Ttil$. The effect of the SGSs appears on the right-hand side of the governing equations through the SGS stresses $\tij$; SGS heat f\/lux $Q_j$; SGS pressure-dilatation $\Pi_{\rm dil}$; SGS viscous dissipation $\varepsilon_{\rm v}$; SGS turbulent diffusion $\partial{\cal J}_j/\partial x_j$; and SGS viscous diffusion $\partial{\cal D}_j/\partial x_j$. These quantities are def\/ined as \begin{eqnarray} \label{eq:tauij} \tij & = & \rhob(\widetilde{u_iu_j}-\util_i\util_j), \\ \label{eq:qj} Q_j & = & \rhob \left(\widetilde{u_j T}-\util_j\widetilde{T}\right), \\ \label{eq:pdil} \Pi_{\rm dil} & = & \overline{p\Skk}-\pb\Stkk, \\ \label{eq:vdiss} \varepsilon_{\rm v} & = & \overline{\sigma_{ji}\Sij} - \widetilde{\sigma}_{ji}\Stij,\\ \label{eq:tdiff} {\cal J}_j & = & \rhob\left(\widetilde{u_ju_ku_k} - \util_j\widetilde{u_ku_k} \right), \\ \label{eq:vdiff} {\cal D}_j & = & \overline{\sigma_{ij}u_i} -\widetilde{\sigma}_{ij}\util_i . \end{eqnarray} The equation of state has been used to express pressure-gradient and pressure-diffusion correlations in terms of $Q_j$ and $\Pi_{\rm dil}$. It is also assumed that $\overline{\mu(T)\Sij} \simeq \mu(\Ttil)\Stij$, and that an equivalent equality involving the thermal conductivity applies. \citet{vre95b} performed {\it a priori} tests using DNS data obtained from the calculation of a mixing layer at Mach numbers in the range 0.2--0.6, and concluded that neglecting the nonlinearities of the diffusion terms in the momentum and energy equations is acceptable. %%%%%%%%%%%%%%% %% Section 3 %% %%%%%%%%%%%%%%% \section{{\em A priori} Test} \label{sec:3} One method to evaluate the performance of models for LES or RANS calculations is the {\it a priori} test, in which the velocity f\/ields obtained from a direct simulation are f\/iltered to yield the exact SGS terms, and the f\/iltered quantities are used to assess the accuracy of the parametrization. The database used in this study was obtained from the calculation of homogeneous isotropic turbulence decay. The Navier--Stokes equations were integrated in time using a fourth-order Runge--Kutta method.\break The spatial derivatives were computed using an eighth-order accurate central f\/inite-difference scheme. The\pagebreak\ results have been validated by comparison with the DNS data of \citet[1999]{mar98}. The simulations were performed on grids with 256$^3$ points. The computational domain is a periodic box with length 2$\pi$ in each dimension. The f\/luctuating f\/ields were initialized as in \citet{ris97}. The calculation was performed at a Reynolds number $Re_{\lambda}=u'\lambda/\nu=50$, where $\lambda=\langle u^2\rangle^{1/2}/$\break $\langle{(\partial u/\partial x)^2}\rangle^{1/2}$ is the Taylor microscale and $u'=(u_iu_i)^{1/2}$ is the turbulence intensity, and at a turbulent Mach number $M_t=u'/a=0.52$, where $a$ is the speed of sound. The initial f\/low f\/ield is allowed to evolve for four dimensionless time units $\tau_t=\lambda/u'$, so that the energy spectrum may develop an inertial range that decays as $k^{-5/3}$, where $k$ is the nondimensional wave number. The f\/iltered f\/ields were obtained using a top-hat f\/ilter, which is def\/ined in one dimension as \begin{equation} \label{eq:tophat} \overline{f}_i = \frac{1}{2n} \left( f_{i-{n}/{2}}+2\sum_{i-{n}/{2}+1}^{i+{n}/{2}-1}f_{i}+f_{i+{n}/{2}}\right) . \end{equation} Various f\/ilter-widths $\delbar=n\Delta$ (where $\Delta$ is the grid size and $n=4$, 8, 16, and 32) were used. Note that the grid resolution is high enough that $n=2$ would correspond to a DNS. The location of the various f\/ilter cutoffs along the energy spectrum at $t/\tau_t=6.5$ are shown in Figure~\ref{fig:fig01}; they cover the decaying range of the spectrum ($n=4$), the inertial range ($n=8$ and 16), and the energy-containing range ($n=32$). With the f\/ilters used, respectively, 5\%, 15\%, 40\%, and 70\% of the total turbulent kinetic energy resides in the SGSs. The two intermediate values are representative of actual LES calculations, in which the SGS kinetic energy\break is typically between 15\% and 30\% of the total energy. A higher percentage of SGS energy in general indicates an under-resolved calculation. In the following, results will be shown for $\delbar=8\Delta$, except when evaluating the effect of f\/ilter-width. The accuracy of a model is evaluated by computing the exact term $R$ and its model representation $M$ and comparing the two using the correlation coeff\/icient $C(R)$ and the root-mean-square (rms) amplitudes $\langle (R-\langle R\rangle)^2\rangle^{1/2}$ and $\langle(M-\langle M\rangle)^2\rangle^{1/2}$. The correlation coeff\/icient is given by \begin{equation} \label{eq:corr-coeff} C(R) = \frac{ \langle(R-\langle R \rangle) (M-\langle M \rangle) \rangle} {\left(\langle (R-\langle R \rangle)^2 \rangle \langle(M-\langle M \rangle)^2 \rangle\right)^{1/2}} , \end{equation} where the brackets $\langle\cdot\rangle$ denote averaging over the computational volume. A ``perfect'' model would give a correlation coeff\/icient of 1. In the following, the quantities plotted are made nondimensional using the initial values of $\rho$, $u'$, and $\lambda$. \begin{figure}[t] %\centering %\includegraphics[height=51mm]{01fig.eps} \vspace{51mm}% to simulate the figure \caption{Energy spectrum; \diam, location of the f\/ilter-widths used in the {\it a priori} test; \solid $k^{-5/3}$ slope; \dotted, DNS. $q^2 = u_iu_i$, and $\eta$ is the Kolmogorov length scale.} \label{fig:fig01} \end{figure} %%%%%%%%%%%%%%% %% Section 4 %% %%%%%%%%%%%%%%% \section{Models for the Momentum Equation} \label{sec:4} The SGS stresses (\ref{eq:tauij}) are the only unclosed term that appears in the momentum equation. Various types of models have been devised to represent the SGS stresses. Eddy-viscosity models try to reproduce the global exchange of energy between the resolved and unresolved stresses by mimicking the drain of energy associated with the turbulence energy cascade. Yoshizawa (1986) proposed an eddy-viscosity model for weakly compressible turbulent f\/lows using a multiscale direct-interaction approximation method. The anisotropic part of the SGS stresses is parametrized using the \citet{sma63} model, while the SGS energy $\tkk$ is modeled separately: \begin{equation} \label{eq:tauij-yoshi} \tij - \frac{\delta_{ij}}{3}\qsgs = - C_s^22\delbar^2\rhob|\St|\left(\Stij -\frac{\delta_{ij}}{3}\Stkk\right) = C_s^2\aij , \qquad \qsgs = C_I 2\rhob\delbar^2|\St|^2 = C_I \alpha, \end{equation} with $C_s=0.16$, $C_I=0.09$, and $|\St|=(2 \St_{ij}\St_{ij})^{1/2}$. \citet{moi91} proposed a modif\/ication of the eddy-viscosity model (\ref{eq:tauij-yoshi}) in which the two model coeff\/icients were determined dynamically, rather than input {\it a priori}, using the Germano identity $\Lij = T_{ij} - \widehat{\tau_{ij}}$ \citep{ger92}, which relates the SGS stresses $\tij$ to the ``resolved turbulent stresses'' $\Lij=\left(\widehat{\overline{\rho u_i}\,\overline{\rho u_j}/\rhob}\right) - \widehat{\overline{ \rho u_i }}\, \widehat{\overline{ \rho u_j}}/\rhobh$, and the subtest stresses $\Tij=\rhobh\breve{\widetilde{u_iu_j}} - \rhobh\utbi\utbj$ (where $\breve{\widetilde{f}}=\widehat{\overline{\rho f}}/\rhobh$, and the hat represents the application of the test f\/ilter $\widehat{{G}}$ of characteristic width $\delhat=2\delbar$) that appear if the f\/ilter $\widehat{{G}}$ is applied to (\ref{eq:mom-ff}). \citet{moi91} determined the model coeff\/icients by substituting (\ref{eq:tauij-yoshi}) into the Germano identity and contracting with $\Stij$. In the present paper the contraction proposed by \citet{lil92} to minimize the error in a least-squares sense are used instead. Accordingly, the two model coeff\/icients for the dynamic eddy-viscosity (DEV) model will be given by \begin{equation} \label{eq:coeff_dsm} C = C_s^2 = \frac{ \langle {\cal L}_{ij}M_{ij}\rangle } { \langle M_{kl}M_{kl}\rangle } , \qquad C_I = \frac{ \langle {\cal L}_{kk}\rangle } { \langle\beta-\widehat{\alpha}\rangle } , \end{equation} where $\beta_{ij} = -2\delhat^2\rhobh|\Stb| (\Stbij-\delta_{ij}\Stbkk/3)$, $\Mij=\bij-\widehat{\aij}$, and $\beta=2\delhat^2 \rhobh|\Stb\,|^2$. Scale-similar models are based on the assumption that the most active SGSs are those closer to the cutoff, and that the scales with which they interact are those immediately above the cutoff wave number \citep{bar80}. Thus, scale-similar models employ multiple operations to identify the smallest resolved scales and use the smallest ``resolved'' stresses to represent the SGS stresses. Although these models account for the local energy events, they underestimate the dissipation. \citet{spe88} proposed the addition of a scale-similar part to the eddy-viscosity model of \citet{yos86} introducing the mixed model. In this way, the eddy-viscosity contribution provides the dissipation that is underestimated by purely scale-similar models. This mixed model was also used by \citet{erl92} and \citet{zan92}, and is given by \begin{equation} \label{eq:tauij-sezhu} \tij - \frac{\delta_{ij}}{3}\qsgs = C_s\aij + \Aij - \frac{\delta_{ij}}{3} \Akk , \qquad \qsgs = C_I \alpha + \Akk, \end{equation} where $\Aij=\rhob(\widetilde{\uti\util}_j-\utti\uttj)$. \citet{erl92} tested the constant coeff\/icient model {\it a priori} by comparing DNS and LES results of compressible isotropic turbulence and found good agreement in the dilatational statistics of the f\/low, as well as high correlation between the exact and the modeled stresses. \citet{zan92} compared the DNS and LES results of isotropic turbulence with various initial ratios of compressible to total kinetic energy. They obtained good agreement for the evolution of quantities such as compressible kinetic energy and f\/luctuations of the thermodynamic variables. Dynamic model adjustment can be also applied to the mixed model (\ref{eq:tauij-sezhu}), to yield the dynamic mixed model (DMM) \begin{equation} \label{eq:c-dmm} C = \frac{\langle\Lij\Mij\rangle - \langle\Nij\Mij\rangle} {\langle\Mlk\Mlk\rangle} , \qquad C_I = \frac{\langle\Lkk-\Nkk\rangle} {\langle \beta-\widehat{\alpha}\rangle}, \end{equation} with $\Bij=\rhobh(\breve{\widetilde{\utbi\utbj}} - \utbtbi\utbtbj)$, and $\Nij=\Bij-\widehat{\Aij}$. An issue that requires some attention is the necessity to model separately the trace of the SGS stresses $\qsgs$. \citet{yos86}, \citet{moi91}, and \citet{spe88} proposed a separate model for this term. \citet{erl92} conjectured that, for turbulent Mach numbers $M_t<0.4$ this term is negligible; their DNS of isotropic turbulence conf\/irm this conjecture. \citet{zan92} conf\/irmed these results {\it a posteriori}: they ran calculations with $0\leq C_I\leq0.066$ (the latter value is ten times higher than that predicted by the theory) and observed little difference in the results. \citet{com98} proposed incorporating this term into a modif\/ied pressure ${\cal P}$. This leads to the presence of an additional term in the equation of state, which takes the form \begin{equation} \label{eq:cl-state} {\cal P} = \rhob R\Ttil + \frac{3\gamma-5}{6}\qsgs ; \end{equation} for $\gamma=\frac{5}{3}$ the additional term is zero, and for $\gamma=\frac{7}{5}$ it might be negligible, unless $M_t$ is very large. This observation can be used to explain {\it a posteriori} the insensitivity of the LES results to the value of $C_I$ discussed by \citet{zan92}: the SGS stress trace can be approximately incorporated in the pressure with no modif\/ication to the equation of state. Another factor may be that both the calculations by \citet{erl92} and those by \citet{zan92} used mixed models, in which the scale-similar part gave a contribution to the normal SGS stresses. Thus, $\qsgs$ is taken into account, at least partially, by the scale-similar contribution. If the mixed model is used, the trace of the SGS stresses can be parameterized without requiring a separate term. A one-coeff\/icient dynamic mixed model (DMM-1) would be of the form \begin{equation} \label{eq:tauij-dmm-oc} \tij = C \aij + \Aij , \end{equation} with \begin{equation} \label{eq:c-dmm-oc} C = \frac{\langle\Lij\Mij\rangle - \langle\Nij\Mij\rangle} {\langle\Mlk\Mlk\rangle}. \end{equation} \begin{figure}[b] %\centering %\includegraphics[height=84mm]{02fig.eps} \vspace{84mm}% to simulate the figure \caption{{\it A priori} comparison of the normal SGS stresses $\tau_{11}$. (a) Correlation coeff\/icient and (b) nondimensional rms magnitude. \solid, Eddy-viscosity model DEV (\protect\ref{eq:tauij-yoshi})--(\protect\ref{eq:coeff_dsm}); \dashed, two-coeff\/icient mixed model DMM (\protect\ref{eq:tauij-sezhu})--(\protect\ref{eq:c-dmm}); \chndot, one-coeff\/icient mixed model DMM-1 (\protect\ref{eq:tauij-dmm-oc})--(\protect\ref{eq:c-dmm-oc}); \trian, DNS.} \label{fig:fig02} \end{figure} The models DEV (\ref{eq:tauij-yoshi})--(\ref{eq:coeff_dsm}), DMM (\ref{eq:tauij-sezhu})--(\ref{eq:c-dmm}), and DMM-1 (\ref{eq:tauij-dmm-oc})--(\ref{eq:c-dmm-oc}) are evaluated in Figures~\ref{fig:fig02}--\ref{fig:fig04}. Figure~\ref{fig:fig02}(a) shows that the DMM-1 model gives the highest correlation for the diagonal components of the SGS stress tensor; Figure~\ref{fig:fig02}(b) shows that neither the eddy-viscosity model nor the two-coeff\/icient mixed model DMM predict the rms of the SGS stresses accurately. The DMM-1 model gives the most accurate prediction among those tested. \begin{figure}[t] %\centering %\includegraphics[height=86mm]{03fig.eps} \vspace{86mm}% to simulate the figure \caption{{\it A priori} comparison of the off-diagonal SGS stresses $\tau_{12}$. (a) Correlation coeff\/icient and (b) nondimensional rms magnitude. \solid, Eddy-viscosity model DEV (\protect\ref{eq:tauij-yoshi})--(\protect\ref{eq:coeff_dsm}); \dashed, two-coeff\/icient mixed model DMM (\protect\ref{eq:tauij-sezhu})--(\protect\ref{eq:c-dmm}) and one-coeff\/icient mixed model DMM-1 (\protect\ref{eq:tauij-dmm-oc})--(\protect\ref{eq:c-dmm-oc}); \trian, DNS.} \label{fig:fig03} \end{figure} Figure~\ref{fig:fig03}(a) shows the correlation coeff\/icient for the off-diagonal components of the SGS stress. As in incompressible f\/lows, the eddy-viscosity model gives very poor correlation (near 0.2), while much improved results are obtained with the mixed models. Note that the correlation coeff\/icient for DMM and DMM-1 overlap in the f\/igure. Figure~\ref{fig:fig03}(b) shows the rms of $\tau_{12}$. DEV underpredicts the rms magnitude of the exact term, while DMM and DMM-1 slightly overpredict it. The coeff\/icient $C_s$ remained nearly constant at a value of 0.15 throughout the calculation, consistent with the theoretical arguments \citep{yos86}. The coeff\/icient of the SGS energy, $C_I$, on the other hand, has a value three times higher than predicted by the theory, consistent with the results of \citet{moi91}. \begin{figure}[t] %\centering %\includegraphics[height=46mm]{04fig.eps} \vspace{46mm}% to simulate the figure \caption{Nondimensional rms magnitude of $\tau_{11}$ versus f\/ilter-width at $t/\tau_t=6.5$. \solid, Eddy-viscosity model DEV (\protect\ref{eq:tauij-yoshi})--(\protect\ref{eq:coeff_dsm}); \dashed, two-coeff\/icient mixed model DMM (\protect\ref{eq:tauij-sezhu})--(\protect\ref{eq:c-dmm}); \chndot, one-coeff\/icient mixed model DMM-1 (\protect\ref{eq:tauij-dmm-oc})--(\protect\ref{eq:c-dmm-oc}); \trian, DNS.} \label{fig:fig04} \end{figure} Figure~\ref{fig:fig04} shows the rms magnitude of $\tau_{11}$ versus the f\/ilter-width, at time $t/\tau_t=6.5$. For very small f\/ilter-widths ($\delbar/\Delta=4$), all the models are accurate, ref\/lecting the capability of dynamic models to turn off the model contribution when the grid becomes suff\/iciently f\/ine to resolve all the turbulent structures (models with constants assigned {\it a priori}, such as the \citet{sma63} model, do not have this characteristic). For $\delbar/\Delta=8$, consistent with the results shown above, the one-coeff\/icient mixed model DMM gives the most accurate predictions. For intermediate f\/ilter-widths, up to $\delbar/\Delta=16$, the best prediction is given by the DMM-1 model; when this f\/ilter-width is used the unresolved scales contain a considerable amount of energy, 40\%. For $\delbar/\Delta=32$, it appears that the DMM model predicts the rms magnitude accurately. However, since the DMM model overpredicts the rms signif\/icantly for $\delbar/\Delta=8$ and 16, the accurate prediction given by DMM for $\delbar/\Delta=32$ is a coincidence. When $\delbar/\Delta=32$ the SGSs contain a large contribution from the energy-containing eddies (70\% of the energy is in the SGS); since $\delbar/\Delta=32$ is not in the inertial range the assumptions on which LES modeling is based fail. The same results are found for $\tau_{12}$ (not shown). \begin{figure}[t] %\centering %\includegraphics[height=90mm]{05fig.eps} \vspace{90mm}% to simulate the figure \caption{Comparison of unclosed terms in the energy equations. (a) Nondimensional terms in the internal energy or enthalpy equations and (b) nondimensional terms in the total energy equation. \solid, Divergence of the SGS heat f\/lux, $C_{\rm v}\ \partial Q_j/\partial x_j$; \chndot, SGS viscous dissipation $\varepsilon_{\rm v}$; \dashed, pressure dilatation $\Pi_{\rm dil}$; \chndotdot, divergence of the SGS heat f\/lux, $\gamma C_{\rm v}\ \partial Q_j/\partial x_j$; \dotted, SGS turbulent diffusion $\partial{\cal J}_j/\partial x_j$; \ldash, SGS viscous diffusion $\partial{\cal D}_j/\partial x_j$.} \label{fig:fig05} \end{figure} %%%%%%%%%%%%%%% %% Section 5 %% %%%%%%%%%%%%%%% \section{Models for the Energy Equations} \label{sec:5} Figure~\ref{fig:fig05} compares the magnitude of the unclosed terms appearing in the internal-energy and enthalpy equations (\ref{eq:int-en-ff}) and (\ref{eq:enth-ff}), respectively (Figure~\ref{fig:fig05}(a)) and in the total energy equation (Figure~\ref{fig:fig05}(b)). Unlike in the mixing layer studied by \citet{vre95b}, in this f\/low the pressure dilatation $\Pi_{\rm dil}$ is negligible, and the viscous dissipation $\varepsilon_{\rm v}$ is one order of magnitude smaller than the divergence of the SGS heat f\/lux. In the total energy equation (\ref{eq:tot-en-ff}), the SGS turbulent diffusion $\partial{\cal J}_j/\partial x_j$ is comparable with the divergence of the SGS heat f\/lux and the SGS viscous diffusion is one order of magnitude smaller than the other terms. In this section several models for the more signif\/icant terms are examined. %%%%%%%%%%%%%%%%%%%% %% Subsection 5.1 %% %%%%%%%%%%%%%%%%%%%% \subsection{SGS Heat Flux} \label{sec:5.1} The simplest approach to modeling the SGS heat f\/lux $Q_j$ is to use an eddy-diffusivity model of the form \begin{equation} \label{eq:eddy-diff} Q_j = -\frac{\rhob\nu_{\rm T}}{\prt}\frac{\partial \Tt}{\partial x_j} = -C \frac{\delbar^2\rhob|\St|}{\prt}\frac{\partial \Tt}{\partial x_j}, \end{equation} where $C$ is the eddy-viscosity coeff\/icient that can be either assigned if a model of the form (\ref{eq:tauij-yoshi}) is used, or computed dynamically as in (\ref{eq:coeff_dsm}). The turbulent Prandtl number $\prt$ can be also f\/ixed or calculated dynamically according to\pagebreak \begin{equation} \label{eq:prt-dev} \prt = \frac{C\langle T_kT_k\rangle}{\langle\Kj T_j\rangle}, \end{equation} where \begin{equation} T_j= -\delhat^2\rhobh|\Stb|\frac{\partial\Ttb}{\partial x_j} +\delbar^2\widehat{\rhob|\St|\frac{\partial\Tt}{\partial x_j}} ,\qquad \Kj = \left(\frac{\widehat{\overline{\rho u_j} \,\overline{\rho T}}}{\rhob} \right) - \frac{\widehat{\overline{ \rho u_j }}\, \widehat{\overline{ \rho T }}}{\rhobh}. \end{equation} A mixed model of the form \begin{equation} \label{eq:qj-dmm} Q_j = -C\frac{\delbar^2\rhob|\St|}{\prt}\frac{\partial\Tt}{\partial x_j} +\rhob \left(\widetilde{\utj\Tt} - \uttj\Ttt\right) \end{equation} was proposed by \citet{spe88}. The model coeff\/icients $C$ and $\prt$ can again be assigned or adjusted dynamically according to (\ref{eq:c-dmm}) and \begin{equation} \label{eq:prt-dmm} \prt = C \frac{\langle T_kT_k\rangle} {\langle\Kj T_j\rangle-\langle V_jT_j\rangle } , \end{equation} with \begin{equation} V_j = \rhobh\left(\breve{\widetilde{\utbj\Ttb}} - \utbtbj\Ttbtb \right) - \widehat{\rhob \left(\widetilde{\utj\Tt} - \uttj\Ttt\right)}. \end{equation} \begin{figure}[t] %\centering %\includegraphics[height=85mm]{06fig.eps} \vspace{85mm}% to simulate the figure \caption{{\it A priori} comparison of the SGS heat f\/lux $Q_j$. (a) Correlation coeff\/icient and (b) nondimensional rms magnitude. \solid, Eddy-diffusivity model (\protect\ref{eq:eddy-diff}), $Pr_{\rm T}=0.7$; \dashed, eddy-diffusivity model (\protect\ref{eq:eddy-diff}), Prandtl number adjusted according to (\protect\ref{eq:prt-dev}); \chndot, mixed model (\protect\ref{eq:qj-dmm})--(\protect\ref{eq:prt-dmm}); \trian$\!$,\ $\,$ DNS.} \label{fig:fig06} \end{figure} Figure~\ref{fig:fig06}(a) shows the correlation coeff\/icient for the three models described above. Both eddy-viscosity models overlap on the plot giving a poor correlation factor, roughly 0.2, whereas the mixed model gives a correlation above 0.6. Both eddy viscosity models under-predict the rms of the exact $Q_j$, shown in Figure~\ref{fig:fig06}(b), while the mixed model is more accurate. The mixed model maintains accuracy for all f\/ilter-widths $\delbar/\Delta\leq16$ (Figure~\ref{fig:fig07}). \begin{figure}[t] %\centering %\includegraphics[height=45mm]{07fig.eps} \vspace{45mm}% to simulate the figure \caption{Nondimensional rms magnitude of $Q_j$ versus f\/ilter-width at $t/\tau_t=6.5$. \solid, eddy-diffusivity model (\protect\ref{eq:eddy-diff}), $Pr_{\rm T}=0.7$; \dashed, eddy-diffusivity model (\protect\ref{eq:eddy-diff}), Prandtl number adjusted according to (\protect\ref{eq:prt-dev}); \chndot, mixed model (\protect\ref{eq:qj-dmm})--(\protect\ref{eq:prt-dmm}); \trian, DNS.} \label{fig:fig07} \end{figure} %%%%%%%%%%%%%%%%%%%% %% Subsection 5.2 %% %%%%%%%%%%%%%%%%%%%% \subsection{SGS Viscous Dissipation} \label{sec:5.2} The other term in the enthalpy or internal energy equations that was found to be signif\/icant in the present f\/low is the viscous dissipation $\varepsilon_{\rm v}$. In this section the three models proposed by \citet{vre95b} are tested:\pagebreak \begin{eqnarray} \label{eq:ev-dss} \varepsilon_{\rm v}^{(1)} & = & C_{\varepsilon1} \left(\widetilde{\widetilde{\sigma}_{ji}\Stij} -\widetilde{\widetilde{\sigma}}_{ij}\Sttij \right) ;\\ \label{eq:ev-dtone} \varepsilon_{\rm v}^{(2)} & = & C_{\varepsilon2}\rhob \widetilde{q}^3/\delbar , \qquad \widetilde{q}^2 \sim\delbar^2|\St|^2 ; \\ \label{eq:ev-dttwo} \varepsilon_{\rm v}^{(3)} & = & C_{\varepsilon3}\rhob \widetilde{q}^3\delbar , \qquad \widetilde{q}^2 \sim \widetilde{\utk\util}_k-\uttk\uttk . \end{eqnarray} The f\/irst is a scale-similar model; the second and third represent the SGS dissipation as the ratio between the cube of the SGS velocity scale, $\widetilde{q}$, and the length scale. The velocity scale can be obtained using either the \citet{yos86} model, as in (\ref{eq:ev-dtone}), or the scale-similar model as in (\ref{eq:ev-dttwo}). \citet{vre95b} f\/ixed the values of the coeff\/icients by matching the rms magnitude of the modeled and exact terms obtained from the {\it a priori} test, and obtained $C_{\varepsilon1}=8$, $C_{\varepsilon2}=1.6$, and $C_{\varepsilon3}=0.6$. In the present study the dynamic procedure will be used instead to determine the coeff\/icients. The analog of the Germano identity for this term reads \begin{equation} \left\langle\widehat{\widetilde\sigma_{ji}\Stij} - \widehat{\overline{\rho\sigma_{ij}}}\, \widehat{\overline{\rho\Sij}}/\rhobh^2\right\rangle = \left\langle E_{\rm v}^{(n)}-\widehat{\varepsilon_{\rm v}^{(n)}}\right\rangle, \end{equation} and the modeled terms $\varepsilon_{\rm v}^{(n)}$ can be given respectively by (\ref{eq:ev-dss})--(\ref{eq:ev-dttwo}), while the $E_{\rm v}^{(n)}$ are \begin{eqnarray} E_{\rm v}^{(1)} & = & C_{\varepsilon1} \left(\breve{\widetilde{\breve{\widetilde{\sigma}}_{ji}\Stbij}} -\breve{\widetilde{\breve{\widetilde{\sigma}}}}_{ij}\Stbtbij \right) ;\\ E_{\rm v}^{(2)} & = & C_{\varepsilon2} \rhobh \breve{\widetilde{q}}^3/\delhat , \qquad \breve{\widetilde{q}}^2 \sim\delhat^2|\Stb|^2 ; \\ E_{\rm v}^{(3)} & = & C_{\varepsilon3} \rhobh \breve{\widetilde{q}}^3/\delhat , \qquad \breve{\widetilde{q}}^2 \sim \breve{\widetilde{\utbi\utbj}} - \utbtbi\utbtbj. \end{eqnarray} \begin{figure}[t] %\centering %\includegraphics[height=85mm]{08fig.eps} \vspace{85mm}% to simulate the figure \caption{{\it A priori} comparison of the SGS viscous dissipation $\varepsilon_{\rm v}$. (a) Correlation coeff\/icient and (b) nondimensional rms magnitude. \solid, scale-similar model (33); \dashed, dynamic model (34); \chndot, dynamic model (35); \trian, DNS.} \label{fig:fig08} \end{figure} Figure~\ref{fig:fig08}(a) shows the correlation coeff\/icient for the three models. The scale-similar model gives the highest correlation. The use of a velocity scale obtained from the scale-similar assumption, however, results in improved prediction of the rms magnitude; using $q\sim\delbar|\St|$ yields a signif\/icant overprediction of the rms. The values of the coeff\/icients obtained from the dynamic adjustment in this f\/low are signif\/icantly lower than those obtained in the mixing layer by \citet{vre95b}. For the particular f\/ilter-width shown, we obtained $C_{\varepsilon 1}=2.4$, and $C_{\varepsilon 2}=0.03$, while $C_{\varepsilon3}$ increased monotonically in time from 0.25 to 0.4. The fact that with these values the f\/irst and third models match the rms magnitude of the exact term indicates a lack of universality of these constants. Dynamic adjustment of the model coeff\/icient appears to be benef\/icial for this term. \begin{figure}[t] %\centering %\includegraphics[height=45mm]{09fig.eps} \vspace{45mm}% to simulate the figure \caption{Nondimensional rms magnitude of $\varepsilon_{\rm v}$ versus f\/ilter-width at $t/\tau_t=6.5$. \solid, scale-similar model (33); \dashed, dynamic model (34); \chndot, dynamic model (35); \trian, DNS.} \label{fig:fig09} \end{figure} The modeling of the viscous dissipation is more sensitive than the other terms to the f\/ilter-width. The prediction accuracy deteriorates with increasing f\/ilter-width, and in this case even for $\delbar/\Delta=16$ none of the models is particularly accurate (Figure~\ref{fig:fig09}). \pagebreak %%%%%%%%%%%%%%%%%%%% %% Subsection 5.3 %% %%%%%%%%%%%%%%%%%%%% \subsection{SGS Turbulent Diffusion} \label{sec:5.3} The SGS turbulent diffusion $\partial{\cal J}_j/\partial x_j$ appears in the total energy equation (\ref{eq:tot-en-ff}). Comte and Lesieur (1998) did not model this term explicitly, but added it to the SGS heat f\/lux by using an eddy-diffusivity model to parametrize \begin{equation} \label{eq:comtel-qj} \left( \widetilde{\rho Eu_j} +\widetilde{pu_j} \right) - \left(\rhob \widetilde{E}\util_j+\pb\util_j\right) = \gamma \rhob \left(\widetilde{u_j T}-\util_j\widetilde{T}\right) + {\cal J}_j \simeq -\frac{\nu_{\rm T}}{\prt} \frac{\partial\widetilde{T}}{\partial x_j}; \end{equation} with this model, however, the SGS turbulent diffusion ${\cal J}_j$, which depends mostly on the unresolved velocity f\/luctuations, is modeled in terms of the temperature gradient. In an isothermal f\/low, ${\cal J}_j$ may be nonzero, and, even if the temperature is not constant, there is no reason to couple a term due to mechanical energy gradients to the temperature. A model of the form (\ref{eq:comtel-qj}) effectively neglects ${\cal J}_j$. The only attempt to model the SGS turbulent diffusion was that by \citet{kni98}. They argue that $\uti\simeq\utti$ and propose a model of the form \begin{equation} \label{eq:tdif-knight} {\cal J}_j \simeq \utk\tau_{jk}. \end{equation} A dynamic scale-similar model can be obtained using the generalized central moments \citep{ger92} \begin{eqnarray} \label{eq:gcm1} \tau(u_i,u_j) & = & \rhob\left[\widetilde{u_iu_j} - \uti\utj\right], \\ \label{eq:gcm2} \tau(u_i,u_j,u_k) & = & \rhob\widetilde{u_iu_ju_k} - \uti\tau(u_j,u_k) - \utj\tau(u_i,u_k) - \utk\tau(u_i,u_j) - \rhob\uti\utj\utk. \end{eqnarray} Using this notation the turbulent diffusion term can be written as \begin{equation} \label{eq:tdiff2} 2 {\cal J}_j = \tau(u_j,u_k,u_k) + 2\utk\tau(u_j,u_k) , \end{equation} since $\tau(u_j,u_k)=\tau_{jk}$. Using this formalism, scale-similar models can be derived by approximating the quadratic terms using the f\/iltered velocities $\util_j$ to replace the velocities $u_j$; for instance, one can write \begin{equation} \label{eq:gcm_ss1} \tau(u_i,u_j) \sim \tau(\uti,\utj) \quad \Rightarrow \quad \rhob\left(\widetilde{u_iu_j}-\uti\utj\right) \sim \rhob\left(\widetilde{\uti\utj}-\utti\uttj\right) . \end{equation} If the proportionality constant in (\ref{eq:gcm_ss1}) is set to one, the scale-similar part of the mixed model (\ref{eq:tauij-sezhu}) is obtained. Analogously, the triple product can be written as \begin{eqnarray} \label{eq:tdiff_mod} 2{\cal J}_j & = & \tau(u_j,u_k,u_k) + 2\utk\tau(u_j,u_k) \nonumber \\ & \simeq & C_{J}\tau(\utj,\utk,\utk) + 2\utk\tau(u_j,u_k) \nonumber \\ & = & C_{J} \left[ \rhob\widetilde{\utj\utk\utk} - \rhob\uttj\uttk\uttk - \uttj\Akk - 2\uttk\Ajk \right] + 2\utk\tau_{jk} , \end{eqnarray} the last term is parametrized by the same model used in the momentum equation. The coeff\/icient $C_J$ can be set using the identity \begin{equation} \widehat{\rhob\utj\utk\utk} - \rhobh\utbj\utbk\utbk = 2 J_j - 2 \widehat{ {\cal J}_j} , \end{equation} where \begin{eqnarray} 2 J_j & = & C_{J} \left[ \rhobh\breve{\widetilde{\utbj\utbk\utbk}} - \rhobh\utbtbj\utbtbk\utbtbk - \utbtbj\Bkk - 2\utbtbk\Bjk \right] + 2\utbk T_{jk}, \nonumber \\ \Bjk & = & \rhobh\left(\breve{\widetilde{\utbj\utbk}} - \utbtbj\utbtbk\right), \end{eqnarray} to yield \begin{equation} C_J = \frac{\left\langle \left(\widehat{\rhob\utj\utk\utk} - \rhobh\utbj\utbk\utbk\right) {\cal P}_j -{\cal Q}_j{\cal P}_j\right\rangle} {\langle{\cal P}_k{\cal P}_k\rangle}, \end{equation} where \begin{eqnarray} {\cal P}_j & = & \left[ \rhobh\breve{\widetilde{\utbj\utbk\utbk}} - \rhobh\utbtbj\utbtbk\utbtbk - \utbtbj\Bkk - 2\utbtbk\Bjk \right] \nonumber \\ & - & \left[ \widehat{\rhob\widetilde{\utj\utk\utk}} -\widehat{\rhob\uttj\uttk\uttk} - \widehat{\uttj\Akk} - 2\widehat{\uttk\Ajk} \right], \\ {\cal Q}_j & = & 2 \left( \utbk T_{jk} - \widehat{\utk\tau_{jk}} \right) . \end{eqnarray} \begin{figure}[t] %\centering %\includegraphics[height=86mm]{10fig.eps} \vspace{86mm}% to simulate the figure \caption{{\it A priori} comparison of the SGS turbulent diffusion ${\cal J}_j$. (a) Correlation coeff\/icient and (b) nondimensional rms magnitude. \solid, knight \etal (1998); \dashed, scale-similar, one-coeff\/icient model; \trian, DNS.} \label{fig:fig10} \end{figure} Figure~\ref{fig:fig10}(a) shows the correlation coeff\/icient for the two models (\ref{eq:tdif-knight}) and (\ref{eq:tdiff_mod}) and using (21)--(22) to model $\tau_{jk}$. The correlation factor is greater than 0.7 for both models, and both models overpredict slightly the rms magnitude of ${\cal J}_j$ (Figure~\ref{fig:fig10}(b)). When the one-coeff\/icient, scale-similar model is used this overprediction is signif\/icantly reduced. Both models perform equally well for $\delbar/\Delta\leq16$, while neither is accurate for $\delbar/\Delta=32$. \pagebreak \begin{figure}[t] %\centering %\includegraphics[height=86mm]{11fig.eps} \vspace{86mm}% to simulate the figure \caption{{\it A priori} comparison of the SGS viscous diffusion ${\cal D}_j$. (a) Correlation coeff\/icient and (b) nondimensional rms magnitude. \solid, scale-similar model; \trian, DNS.} \label{fig:fig11} \end{figure} %%%%%%%%%%%%%%%%%%%% %% Subsection 5.4 %% %%%%%%%%%%%%%%%%%%%% \subsection{SGS Viscous Diffusion} \label{sec:5.4} The SGS viscous diffusion $\partial{\cal D}_j/\partial x_j$ is the smallest of the terms in the total energy equation, and is about 5\% of the divergence of $Q_j$. No model for this term has been proposed in the literature to date. One possibility is to parametrize it using a scale-similar model of the form\pagebreak \begin{equation} \label{eq:vdif} {\cal D}_j = C_D ( \widetilde{\widetilde{\sigma}_{ij}\uti} - \widetilde{\widetilde{\sigma}}_{ij}\utti ) , \end{equation} in which the coeff\/icient can be obtained from \begin{equation} \label{eq:cd} C_D = \frac{\left\langle\left[ \widehat{{\overline{\rho\sigma_{ij}}\,\overline{\rho u_i}}/ {\rhob^2}} - {\widehat{\overline{\rho\sigma_{ij}}}\, \widehat{\overline{\rho u_i}}}/ {\rhobh^2} \right] {\cal R}_j \right\rangle} { \left\langle{\cal R}_k{\cal R}_k\right\rangle } , \end{equation} where \begin{equation} {\cal R}_l = \left(\breve{\widetilde{ \breve{\widetilde{\sigma}}_{lk}\utbk}} -\breve{\widetilde{\breve{\widetilde{\sigma}}}}_{lk}\utbtbk \right) - \left(\widehat{\widetilde{\widetilde{\sigma}_{lk}\utk}} -\widehat{\widetilde{\widetilde{\sigma}}_{lk}\uttk} \right) . \end{equation} As can be seen from Figure~\ref{fig:fig11}, this model gives a poor correlation and poor agreement for the prediction of the rms magnitude. However, since the viscous diffusion is relatively small, its contribution to the total energy spectrum does not go to the inertial range, but rather to the decaying range. In this situation the accuracy of the model is degraded, as shown by \citet{men97}. Thus, the scale-similar approach may still give good predictions when this term is signif\/icant. In this particular f\/low, the error given by the model (or by not using a model) may be tolerable given the small contribution that the term gives to the energy budget. %%%%%%%%%%%%%%%%%%%% %% Subsection 5.5 %% %%%%%%%%%%%%%%%%%%%% \subsection{General Considerations} \label{sec:5.5} In addition to the term-by-term comparisons shown before, it is possible to evaluate the global accuracy of the models by comparing the sum of the exact SGS terms and the modeled quantity, namely, \begin{equation} \label{eq:global} E_{SGS}=\gamma C_{\rm v}Q_j + {\textstyle\half}{\cal J}_j - {\cal D}_j. \end{equation} \begin{figure}[t] %\centering %\includegraphics[height=85mm]{12fig.eps} \vspace{85mm}% to simulate the figure \caption{{\it A priori} comparison of the sum of the SGS terms in the total energy equation (\ref{eq:tot-en-ff}). (a) Correlation coeff\/icient and (b) nondimensional rms magnitude. \solid, Model; \trian, DNS.} \label{fig:fig12} \end{figure} The mixed model (26)--(27) was used for the SGS heat f\/lux, the scale-similar model (44)--(45) for the SGS turbulent diffusion, and the SGS viscous diffusion has been neglected. Figure~\ref{fig:fig12}(a) shows the correlation coeff\/icient for the exact and modeled quantities. While the individual correlations were roughly 0.6 and 0.7 for the SGS heat f\/lux model and the SGS turbulent diffusion, respectively, the global correlation drops just below 0.6 when considering the sum of the terms. Figure~\ref{fig:fig12}(b) shows the rms for both quantities. The agreement between the exact and modeled quantities is slightly less\pagebreak\ accurate than for the SGS heat f\/lux alone, Figure~\ref{fig:fig06}(b), but more accurate than for the SGS turbulent diffusion alone, Figure~\ref{fig:fig10}(b). Figure~\ref{fig:fig12} shows that the overall performance is very good. %%%%%%%%%%%%%%% %% Section 6 %% %%%%%%%%%%%%%%% \section{Conclusions} \label{sec:6} Several mixed and eddy-viscosity models for the momentum and energy equations have been tested. The velocity, pressure, density, and temperature f\/ields obtained from the DNS of homogeneous isotropic turbulence at $Re_\lambda=50$, $M_t=0.52$ were f\/iltered and the unclosed terms in the momentum, internal energy, and total energy equations were computed. In the momentum equation, mixed models were found to give better prediction, in terms of both correlation and {\rm rms} amplitude, than the pure eddy-viscosity models. The dynamic adjustment of the model coeff\/icient was benef\/icial, as already observed by \citet{moi91}. In the internal energy and enthalpy equations only the divergence of the SGS heat f\/lux was signif\/icant in this f\/low; the SGS pressure dilatation $\Pi_{\rm dil}$ and viscous dissipation $\varepsilon_{\rm v}$, which were signif\/icant in the mixing layer studied by \citet{vre95b}, were found to be negligible here. Once again, mixed dynamic models gave the most accurate results. In the total energy equation two additional terms are present, one of which, the turbulent diffusion $\partial{\cal J}_j/\partial x_j$, is signif\/icant. The model proposed by \citet{kni98} and a new scale-similar model proposed here correlate well with the actual SGS turbulent diffusion, and predict the correct {\rm rms} amplitude. However, the new scale-similar model was found to be more accurate. A mixed model for the SGS viscous diffusion was also proposed and tested, although this term is much smaller than the others. The accuracy of the models for the sum of the terms was also evaluated, and it was found that the models proposed still predict nearly the correct {\rm rms} amplitude, and an acceptable value of the correlation coeff\/icient. The results obtained in this investigation are promising and indicate that it is possible to model accurately the terms in the energy equations. Further work may extend these results to cases in which the pressure-dilatation is signif\/icant, as well as to inhomogeneous f\/lows, and evaluate these models {\it a posteriori}. %%%%%%%%%%%%%%%% %% References %% %%%%%%%%%%%%%%%% \begin{thebibliography}{} \bibitem[Bardina \etal(1980)]{bar80} Bardina, J., Ferziger, J.H., and Reynolds, W.C. (1980). Improved subgrid-scale models for large eddy simulation. AIAA Paper 80-1357. \bibitem[Comte and Lesieur(1998)]{com98} Comte, P., and Lesieur, M. (1998). Large-eddy simulation of compressible turbulent f\/lows. In: {\it Advances in Turbulence Modeling}, edited by D.~Olivari. Von Karman Institute for Fluid Dynamics, Rhode-Ste-Gen\`ese, 4:1--4:133. \bibitem[El-Hady \etal(1994)]{elh94} El-Hady, N., Zang, T.A. , and Piomelli, U. (1994). Application of the dynamic subgrid-scale model to axisymmetric transitional boundary layer at high speed. \pofa{6}, 1299--1309. \bibitem[Erlebacher \etal(1992)]{erl92} Erlebacher, G., Hussaini, M.Y., Speziale, C.G., and Zang, T.A. (1992). Toward the large-eddy simulation of compressible turbulent f\/lows. \jfm{238}, 155--185. \bibitem[Favre(1965a)]{fav65a} Favre, A. (1965a). \'{E}quations des gaz turbulents compressible. I. Formes g\'{e}n\'{e}rales. {\it J. M\'ec.}, {\bf 4}, 361--390. \bibitem[Favre(1965b)]{fav65b} Favre, A. (1965b). \'{E}quations des gaz turbulents compressible. II. M\'{e}thode des vitesses moyennes; m\'{e}thode des vitesses macroscopiques pond\'{e}r\'{e}es par la masse volumique. {\it J. M\'ec.}, {\bf 4}, 391--421. \bibitem[Germano(1992)]{ger92} Germano, M. (1992). Turbulence: the f\/iltering approach. \jfm{238}, 325--336. \bibitem[Knight \etal(1998)]{kni98} Knight, D., Zhou, G., Okong'o, N., and Shukla, V. (1998). Compressible large eddy simulation using unstructured grids. AIAA Paper 98-0535. \bibitem[Leonard(1974)]{leo74} Leonard, A. (1974). Energy cascade in large-eddy simulations of turbulent f\/luid f\/lows. {\it Adv. Geophys.}, {\bf 18A}, 237--248. \bibitem[Lilly(1992)]{lil92} Lilly, D.K. (1992). A proposed modif\/ication of the Germano subgrid-scale closure method. \pofa{4}, 633--635. \bibitem[Mart\protect{\'\i}n and Candler(1998)]{mar98} Mart\'{\i}n, M.P., and Candler, G.V. (1998). Effect of chemical reactions on decaying isotropic turbulence. {\it Phys.\ Fluids}, {\bf 10}, 1715--1724. \bibitem[Mart\protect{\'\i}n and Candler(1999)]{mar99} Mart\'{\i}n, M.P., and Candler, G.V. (1999). Subgrid-scale model for the temperature f\/luctuations in reacting hypersonic turbulent f\/lows. {\it Phys.\ Fluids}, {\bf 11}, 2765--2771. \bibitem[Meneveau and Lund(1997)]{men97} Meneveau, C., and Lund, T.S. (1997). The dynamic Smagorinsky model and scale-dependent coeff\/icients in the viscous range of turbulence. {\it Phys.\ Fluids}, {\bf 9}, 3932--3934. \bibitem[Moin \etal(1991)]{moi91} Moin, P., Squires, K.D., Cabot, W.H., and Lee, S. (1991). A dynamic subgrid-scale model for compressible turbulence and scalar transport. \pofa{3}, 2746--2757. \bibitem[Normand and Lesieur(1992)]{nor92} Normand, X., and Lesieur, M. (1992). Direct and large-eddy simulation of laminar breakdown in high-speed axisymmetric boundary layers. {\it Theoret.\ Comput.\ Fluid Dynamics}, {\bf 3}, 231--252. \bibitem[Ristorcelli and Blaisdell(1997)]{ris97} Ristorcelli, J.R., and Blaisdell, G.A. (1997). Consistent initial conditions for the DNS of compressible turbulence. {\it Phys.~Fluids}, {\bf 9}, 4--6. \bibitem[Smagorinsky(1963)]{sma63} Smagorinsky, J. (1963). General circulation experiments with the primitive equations. I.~The basic experiment. {\it Mon. Weather Rev.} {\bf 91}, 99--164. \bibitem[Speciale \etal(1988)]{spe88} Speziale, C.G., Erlebacher, G., Zang, T.A., and Hussaini, M.Y. (1988) The subgrid-scale modeling of compressible turbulence. \pofa{31}, 940--942. \bibitem[Vreman \etal(1995a)]{vre95a} Vreman, B., Geurts, B., and Kuerten, H. (1995a). A priori tests of large eddy simulation of the compressible mixing layer. {\it J. Engrg.\ Math.}, {\bf 29}, 299--327. \bibitem[Vreman \etal(1995b)]{vre95b} Vreman, B., Geurts, B., and Kuerten, H. (1995b). Subgrid-modeling in LES of compressible f\/low. {\it Appl. Sci.\ Res.}, {\bf 54}, 191--203. \bibitem[Yoshizawa(1986)]{yos86} Yoshizawa, A. (1986). Statistical theory for compressible turbulent shear f\/lows, with the application to subgrid modeling. \pofa{29}, 2152--2164. \bibitem[Zang \etal(1992)]{zan92} Zang, T.A., Dahlburg, R.B., and Dahlburg, J.P. (1992). Direct and large-eddy simulations of three-dimensional compressible Navier--Stokes turbulence. \pofa{4}, 127--140. \end{thebibliography} \end{document}