Velocity-scalar filtered density function for large eddy simulation of ...

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2334 Phys. Fluids, Vol. 15, No. 8, August 2003 Sheikhi et al.FIG. 14. Cross-stream variations of some of the components of at t60.FIG. 12. Cross-stream variations of the Reynolds-averaged values of thefiltered scalar field at t80.low values of C , C are not recommended as they wouldresult in too much SGS energy in comparison to the resolvedenergy.The computational cost of VSFDF simulations relativeto those required by DNS and by the Smagorinsky model isthe same as that reported previously. 16 The typical ratios ofthe normalized Smagorinsky–VSFDF–DNS run times are1O30O200.VI. SUMMARY AND CONCLUDING REMARKSThe filtered density function FDF methodology hasproven effective for LES of turbulent reactive flows. In previousinvestigations, either the marginal FDF of the scalar, orthat of the velocity were considered. The objective of presentwork is to develop the joint velocity-scalar FDF methodology.For this purpose, the exact transport equation governingthe evolution of VSFDF is derived. It is shown that effects ofthe SGS convection and chemical reaction appear in a closedform. The unclosed terms are modeled in a fashion similar tothose typically followed in PDF methods. The modeledVSFDF transport equation is solved numerically via a LagrangianMonte Carlo MC scheme via consideration of asystem of equivalent stochastic differential equationsFIG. 13. Temporal variations of the scalar thickness.FIG. 15. Cross-stream variations of some of the components of at t80.Downloaded 22 Sep 2004 to 140.121.120.39. Redistribution subject to AIP license or copyright, see http://pof.aip.org/pof/copyright.jsp
Phys. Fluids, Vol. 15, No. 8, August 2003Velocity-scalar filtered density function2335FIG. 16. Cross-stream variations of some of the components of R¯ at t60.FIG. 18. Cross-stream variations of r¯ at t60.SDEs. These SDEs are discretized via the Euler–Maruyamma approximation. The consistency of the VSFDFmethod and the convergence of its MC solutions are assessedin LES of a two-dimensional 2D temporally developingmixing layer. This assessment is done by comparing the resultsobtained by the MC procedure with those of the finitedifferencescheme LES–FD for the solution of the transportequations of the first two moments of VSFDF. Byincluding the third moments from the VSFDF into the LES–FD, the consistency and convergence of the MC solution aredemonstrated by good agreements of the first two SGS momentswith those obtained by LES–FD.The VSFDF predictions are compared with LES resultswith the Smagorinsky 18 SGS model. All of these results arealso compared with direct numerical simulation DNS dataof a three-dimensional, temporally developing mixing layer.It is shown that the VSFDF performs well in predicting someof the phenomena pertaining to the SGS transport. Most ofthe overall flow features, including the mean field, the resolvedand total stresses as predicted by VSFDF are in goodagreement with DNS data. However, the model does requirethe input of three empirical constants. Also, the numericalimplementation of VSFDF is more expensive than the traditionalmodels. It may be possible to improve the predictivecapabilities of the VSFDF by two ways: 1 development ofa dynamic procedure to determine the model coefficients,and/or 2 implementation of higher order closures for thegeneralized Langevin model parameter G ij . 50 Future work isrecommended for development and application of the jointfiltered velocity-scalar mass density function VSFMDF toallow for LES of variable density flows with/or without thepresence of chemical reaction.FIG. 17. Cross-stream variations of some of the components of R¯ at t80.FIG. 19. Cross-stream variations of r¯ at t80.Downloaded 22 Sep 2004 to 140.121.120.39. Redistribution subject to AIP license or copyright, see http://pof.aip.org/pof/copyright.jsp
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