Inferring Wall Shear Stress in Complex High-Speed Flows from the Integrated Favre-Averaged Streamwise Momentum Equation

This work is motivated by the necessity to study turbulent flow over blunt bodies with distributed surface roughness in hypersonic conditions, such as atmospheric entry capsules. We present a simple, integral formulation to compute the wall shear stress from the Favre-averaged streamwise momentum equation. The approach is general and applicable to flows with curvature, surface roughness, and pressure gradients. We build from existing integral formulations to demonstrate their capability to estimate wall shear stress in complex high-speed flows, such as those with surface roughness or surface curvature. Starting from the differential form of the Favre-averaged streamwise momentum equation, we integrate in the wall-normal direction only once to obtain a stress balance from which the wall shear stress can be inferred. The integral relation is then applied as a post-processing step for scale-resolving data, such as high-resolution Particle Image Velocimetry or numerical simulations. In the present work, eight demonstration cases are selected to illustrate the method's applicability for complex, high-speed, compressible flows involving pressure gradients, surface roughness, and surface curvature. We show the contributions of the various terms of the integral equality, the associated error of the estimate, and outline practical considerations when estimating the wall shear stress for complex flow conditions.