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Linear models, based on stochastically forced linearized equations, are deployed for spectral linear stochastic estimation (SLSE) of the velocity and temperature fluctuations in compressible turbulent channel flows with a bulk Mach number of 1.5. Through comparison with the direct numerical simulation (DNS) data, an eddy-viscosity-enhanced model (eLNS) outperforms the one not enhanced (LNS) in computing the coherence and amplitude ratio of streamwise velocity at different wall-normal heights, but they both largely deviate from DNS regarding the temperature prediction. For further investigation, the eigenspectra and pseudospectra of the linear operators are scrutinized. The eddy viscosity is shown to stabilize the eigenmodes and decrease the non-normality of the vortical modes. Consequently, the relative importance of acoustic and entropy modes increases, and they can contribute 20 % to 55 % of the response growth, which is not supported by DNS. Hence, it is an intrinsic defect of the eLNS model introduced by turbulence modelling. After a procedure of cospectrum decomposition, the contributions of acoustic and entropy components are filtered out. The resulting SLSE quantities for velocity, temperature and their coupling are basically agreeable with DNS, demonstrating that the coherent temperature fluctuation is dominated by advection and other vortical motions, instead of the compressibility effects. Moreover, a parameter study of Reynolds and Mach numbers (from 0.3 to 4) is conducted. The semi-local units are shown to well collapse the velocity SLSE quantities to the incompressible case for streamwise-elongated structures of high coherence.
Chen et al. (Fri,) studied this question.