A field-equation semi-local transformation for compressible wall turbulence

Abstract

Compressibility and wall heat transfer change the inner scaling of wall turbulence through the mean density and viscosity fields. Most semi-local transformations are applied after a wall-normal profile has been selected. Here the transformed coordinate and transformed velocity are defined by wall-anchored field equations before any profile is extracted. Wall density and friction velocity enter as boundary data for auxiliary field equations; the resulting reference density and friction velocity, together with the local density and viscosity, set the viscous scaling in the wall layer. The local logarithmic density contrast is introduced as a bounded correction to the coordinate-stretching factor. For weak density variation the corrected stretch follows the local inverse kinematic-viscosity variation, while the bounded form limits the influence of finite density contrasts. The transformed coordinate $Y^+$ and transformed velocity $\bm{U}^+$ are obtained from field equations using the corrected stretching and the mean viscous shear. In the constant-property limit the density contrast vanishes, $Y^+$ reduces to the ordinary wall coordinate and each transformed velocity component reduces to the conventional mean velocity component in wall units. For channel flows, the field equations are solved with wall boundary data and the extracted profiles retain their own wall origins. A cooled shock/boundary-layer interaction examines the response when the wall density and friction velocity vary in the streamwise direction. Across the cooled high-speed boundary layers considered here, the bounded density correction narrows the inner- and buffer-layer profile spread...