Analytical equations for mean wall-normal velocity and momentum integral in compressible boundary-layer flows

Analytical expressions for the mean wall-normal velocity and wall shear stress in compressible boundary layers are derived by integrating the mean continuity and momentum equations. In the constant-density limit, the momentum integral formulation recovers the classical Kármán–Pohlhausen equation for incompressible boundary-layer flows. In compressible regimes, particularly under strong pressure gradients, streamwise density gradients are shown to play a crucial role in shaping boundary-layer dynamics. The derived analytical equations are validated against high-fidelity direct numerical simulation data, demonstrating both accuracy and robustness. Furthermore, the analytical equations offer insights into the physical mechanisms of compressible boundary layers, particularly the influence of density gradients. The effect of compressibility on the wall-normal velocity is explicitly demonstrated, highlighting the distinct behaviour of compressible boundary layers compared with incompressible flows. Finally, an analytical expression for the skin-friction coefficient is developed, revealing its close connection to the mean wall-normal velocity at the boundary-layer edge.