Compressible Boundary Layer Velocity Transformation Based on a Generalized Form of the Total Stress

TL;DR

This paper introduces a new velocity transformation for compressible boundary layers that accounts for density and viscosity fluctuations, successfully collapsing compressible data onto the incompressible law of the wall.

Contribution

It develops a generalized velocity transformation based on Mach-invariance properties of total stress, incorporating effects of density and viscosity fluctuations.

Findings

The transformation collapses all considered compressible cases onto the incompressible law of the wall.

It demonstrates the importance of Mach-invariance in the momentum balance.

The approach accounts for viscous and turbulent stresses along with density and viscosity effects.

Abstract

The effects of density and viscosity fluctuations on the total stress balance are identified and used to create a new mean velocity transformation for compressible boundary layers. This work is enabled by an extensive database of direct numerical simulations that incorporate wall-cooling, semi-local Reynolds numbers ranging from 800 to 34000, 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 Machinvariance 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 newly 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.