Decomposition of the skin-friction coefficient of compressible boundary layers

Abstract

We derive an integral formula for the skin-friction coefficient of compressible boundary layers by extending the formula of Elnahhas and Johnson [“On the enhancement of boundary layer skin friction by turbulence: An angular momentum approach,” J. Fluid Mech. 940, A36 (2022)] for incompressible boundary layers. The skin-friction coefficient is decomposed into the sum of the contributions of the laminar coefficient, the change of the dynamic viscosity with the temperature, the Favre–Reynolds stresses, and the mean flow. This decomposition is applied to numerical data for laminar and turbulent boundary layers, and the role of each term on the wall-shear stress is quantified. We also show that the threefold integration identity of Gomez et al. [“Contribution of Reynolds stress distribution to the skin friction in compressible turbulent channel flows,” Phys. Rev. E 79(3), 035301 (2009)] and the twofold integration identities of Wenzel et al. [“About the influences of compressibility, heat transfer and pressure gradients in compressible turbulent boundary layers,” J. Fluid Mech. 930, A1 (2022)] and Xu et al. [“Skin-friction and heat-transfer decompositions in hypersonic transitional and turbulent boundary layers,” J. Fluid Mech. 941, A4 (2022)] for turbulent boundary layers all simplify to the compressible von Kármán momentum integral equation when the upper limit of integration is asymptotically large. The dependence of these identities on the upper integration bound is studied. By using asymptotic methods, we prove that the multiple-integration identity of Wenzel et al. [“About the influences of compressibility, heat transfer and pressure gradients in compressible turbulent boundary layers,” J. Fluid Mech. 930, A1 (2022)] degenerates to the definition of the skin-friction coefficient when the number of integrations is asymptotically large.