Compressible boundary layer velocity transformation based on a generalized form of the total stress

The effects of density and viscosity fluctuations on the total stress balance are identified and used to create a mean velocity transformation for compressible boundary layers. This work is enabled by an extensive database of direct numerical simulations that incorporate wall-cooling, semilocal Reynolds numbers ranging from 800 to 34 000, and Mach numbers up to 12. The role, significance, and physical mechanisms connecting density and viscosity fluctuations to the momentum balance and to the viscous, turbulent, and total stresses are presented, allowing the creation of generalized formulations. We identify the significant properties that thus far have been neglected in the derivation of velocity transformations: (1) the Mach invariance of the near-wall momentum balance for the generalized total stress and (2) the Mach invariance of the relative contributions from the generalized viscous and Reynolds stresses to the total stress. The proposed velocity transformation integrates both properties into a single transformation equation and successfully demonstrates a collapsing of all currently considered compressible cases onto the incompressible law of the wall, within the bounds of reported slope and intercept for incompressible data. Based on the physics embedded in the two scaling properties, the success of the proposed transformation is attributed to considering the effects of the viscous stress and turbulent stresses as well as mean and fluctuating density viscosity in a single transformation form.