New Mathematical Framework for Analyzing Compressible Wall Turbulence Using Elliptic Equations

A preprint submitted to arXiv on June 8, 2026 introduces a semi-local transformation framework for compressible wall-bounded turbulence that departs from conventional approaches by defining the transformed coordinate and velocity through elliptic equations before any wall-normal profiles are extracted. Compressibility and wall heat transfer complicate turbulence scaling by altering mean density and viscosity fields, and existing semi-local transformations typically address this only after profiles are selected. The new method constructs a semi-local wall coordinate from local density and viscosity scaling, then applies a Helmholtz-type elliptic equation to generate a bounded density-induced correction to wall-normal coordinate stretching; projection equations then yield the transformed coordinate Y⁺ and projected velocity vector U⁺. In the limit of uniform density and viscosity, the formulation recovers standard wall coordinates and conventional mean velocity components, confirming consistency with classical theory. A single set of wall-layer constants is calibrated using two canonical cooled flat-plate datasets and subsequently applied without re-tuning to zero-pressure-gradient boundary layers, isothermal channels, and mixed thermal-wall channels. Results show improved placement of the inner and buffer layers in cooled high-speed boundary layers, with asymmetric behavior in mixed thermal-wall channels where the isothermal side responds more strongly than the adiabatic side. The work represents a methodological advance in turbulence coordinate transformations relevant to high-speed aerodynamics and thermal management engineering.