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diff --git a/macros/latex/contrib/springer/svjour/tcfd/example/example.tex b/macros/latex/contrib/springer/svjour/tcfd/example/example.tex new file mode 100644 index 0000000000..3095bdb40c --- /dev/null +++ b/macros/latex/contrib/springer/svjour/tcfd/example/example.tex @@ -0,0 +1,1172 @@ +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% 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} |