Quasilinear modelling of supersonic turbulent channel flow using incompressible flow data

This study extends the data-driven quasilinear approximation (DQLA) (Holford, Lee &amp; Hwang 2024 J. Fluid Mech. , vol. 980 , A12) to compressible turbulent channel flow. The DQLA employs the eddy viscosity enhanced linearised compressible Navier–Stokes operator (Chen et al. 2023 J. Fluid Mech. , vol. 962 , A7), driven by stochastic forcing whose streamwise weights are determined through self-similarity assimilated from an incompressible direct numerical simulation (DNS) database, and spanwise weights are obtained by minimising discrepancies in the Reynolds stresses between the mean and the fluctuation equations. Without any compressible DNS input, the extended DQLA reproduces turbulence intensities and energy spectra in close quantitative agreement with DNS up to bulk Mach number italic Ma Subscript b Baseline equals 1.5 <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content"> <mml:msub> <mml:mrow> <mml:mtext mathvariant="italic" class="MJX-tex-mathit">Ma</mml:mtext> </mml:mrow> <mml:mi>b</mml:mi> </mml:msub> <mml:mo>=</mml:mo> <mml:mn>1.5</mml:mn> </mml:math> $\textit{Ma}_b=1.5$ , and exhibits a consistent collapse of turbulence statistics across Mach numbers when expressed in semilocal units at the same centreline semilocal friction Reynolds number italic Re Subscript tau c Superscript asterisk <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content"> <mml:msubsup> <mml:mrow> <mml:mtext mathvariant="italic" class="MJX-tex-mathit">Re</mml:mtext> </mml:mrow> <mml:mrow> <mml:mi>τ</mml:mi> <mml:mi>c</mml:mi> </mml:mrow> <mml:mo>∗</mml:mo> </mml:msubsup> </mml:math> $\textit{Re}_{\tau c}^*$ . This collapse is largely inherited from the scaling properties of the mean flow and is preserved through DQLA, consistent with Morkovin’s hypothesis. At higher Mach number ( italic Ma Subscript b Baseline equals 3.0 <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content"> <mml:msub> <mml:mrow> <mml:mtext mathvariant="italic" class="MJX-tex-mathit">Ma</mml:mtext> </mml:mrow> <mml:mi>b</mml:mi> </mml:msub> <mml:mo>=</mml:mo> <mml:mn>3.0</mml:mn> </mml:math> $\textit{Ma}_b=3.0$ ), systematic deviations emerge, reflecting the limitations of the present modelling assumptions, particularly the neglect of nonlinear terms associated with density fluctuations. The results demonstrate the predictive capability of DQLA up to moderate Mach numbers, and establish its potential as a physically interpretable and computationally efficient framework for exploring compressible wall-bounded turbulence at high Reynolds numbers.