Generalized Reynolds Analogy for Compressible Wall Turbulence: Unified Velocity--Temperature Relations from Mean to Fluctuating Field

Abstract

The Reynolds analogy between velocity and temperature fields is a central problem in the statistical theory of compressible wall turbulence. The generalized Reynolds analogy (GRA) established by the author describes the mean velocity--temperature relation accurately, but a self-consistent closure for the fluctuating field has remained elusive. Here, the instantaneous similarity relation that the generalized total enthalpy (minus its wall value) is proportional to the local velocity is shown to close both fields at once. The mean field reproduces the GRA solution, while the fluctuating field yields the closed relation $T'_{rms}/u'_{rms} = f(Re_τ,\Pr)\,\Pr_m^{-1/2}\,|\partial\bar{T}/\partial\bar{u}|$, where the prefactor $f$ is derived from the universal cascade dynamics of turbulence and the Obukhov--Corrsin theory, without adjustable parameters. The relation recovers the refined strong Reynolds analogy (RSRA) of Huang et al.~(2025), the most accurate benchmark to date, recasting it from an empirical relation into a consequence of universality principles and explaining the origin of its fitted prefactor $f=1.09$; and its ratio to boundary-layer and channel direct numerical simulation (DNS) data collapses onto unity across Mach numbers ($2.25$--$14$), Prandtl numbers ($0.025$--$1$) and wall thermal conditions, with an overall accuracy of about $5\%$.