Wall-resolved LES of supersonic channel flows: Subgrid-scale closures for internal energy equation formulation

Subgrid-scale closures for wall-resolved explicit filtered LES of Supersonic Channel Flow with different thermal-wall boundary conditions are studied at bulk flow Mach number 1.5 and reference Reynolds number in wall units Re_tau = 190 and 340. Internal energy formulation is used to account for temperature in the system of equations, and three flow conditions : 1) two walls symmetrically isothermal, 2) symmetrically isothermal walls at different temperatures and 3) uniform property flow with viscous work subtracted from the energy equation, are taken up for study. In explicit filtered LES, the fields carry additional implicit filtering due to numerical error from truncation and the error is magnified due to nonlinearity of the viscous dissipation term containing velocity gradients. Mixed tensor-diffusivity model coupled with a dynamic Smagorinsky term is used for modeling SGS stress and SGS heat flux. The principle of partially reconstructing the resolved but unrepresented scales of the tensor-diffusivity model is extended to SGS viscous dissipation using a semi-empirical mixed model. The relative significance of the SGS terms are studied from a set of DNS data and a priori tests of the models are presented. Explicitly filtered LES is performed with three configurations of SGS viscous dissipation models on two levels of grid refinement. Arguments in favor of using a dynamically computed thermal diffusivity in place of extended mixing length analogy with a subgrid-scale Prandtl number, are made and validated. A new pair of two-term mixed models for viscous dissipation that accounts for the sub-filter scale stress work and global dissipation of the large-scales are proposed. From the large-eddy simulations of uniform property flows, the grid resolution was found to be a dominant factor influencing the accuracy of the simulations. In the variable property flows, the two term models are found to yield better prediction of dissipation and hence the mean temperature profiles. Comparing the LES on both the coarse and fine grid, the variable property simulation at Re_{\tau} = 340 yielded better results in terms of Reynolds stress correlation and {\it a posteriori} estimates of the SGS terms, indicating some favorable performance of the SGS models at a higher Reynolds number. The hot-wall half of the asymmetric channel simulations show a different behavior compared to the other flow cases in terms of SGS heat flux and viscous dissipation.