Uncovering the forcing statistics in stochastic linear models for compressible wall-bounded turbulence

Modelling the nonlinear forcing is critical for linear models based on resolvent or input–output analyses. For compressible wall-bounded turbulence, little is known on what the real forcing looks like due to limited data, so the prediction agrees more qualitatively than quantitatively with direct numerical simulations (DNSs). Here, we present detailed forcing statistics of stochastic linear models, derived from elaborate DNS datasets for channel flows with bulk Mach number reaching 3. These statistics directly explain the success and failure of current models and provide guidance for further improvements. The benchmark linearised Navier–Stokes (LNS) and eLNS models are considered; the latter is assisted by eddy-viscosity-related terms. First, we prove the self-consistency of the models by using DNS-computed forcing as the input. Second, we present the spectral distributions of the forcing and its components. Third, we quantify the acoustic components, absent in incompressible cases, within the linear models. We reveal that the LNS forcing can exhibit relatively high coherence and low rank, very different from the modelled diagonal full-rank forcing. The eddy-viscosity-related term is not partial modelling of the LNS forcing; contrarily, the former is much larger than the latter, serving to disrupt the low-rank feature, enhance diagonal dominance and increase robustness across scales. The scales narrow in either horizontal direction are most susceptible to acoustic modes, while the others are little affected (2\, \% in energy). Furthermore, the extended strong Reynolds analogy is assessed in predicting the density and temperature components.