We use the method developed by Elnahhas and Johnson ["On the enhancement of boundary layer skin friction by turbulence: An angular momentum approach," J. Fluid Mech. 940, A36 (2022)] and Xu et al. ["Decomposition of the skin-friction coefficient of compressible boundary layers," Phys. Fluids 35, 035107 (2023)] for the decomposition of the skin-friction coefficient to integrate the mean temperature equation for high-Reynolds-number compressible boundary layers and arrive at an identity for the decomposition of the wall-heat flux. The physical interpretation of the identity and the limitations of this approach are discussed. We perform an integration on the mean temperature equation to obtain an identity that is the heat-transfer analog to the compressible von Kármán momentum integral equation for the skin-friction coefficient. This identity is applied to numerical data for laminar and turbulent compressible boundary layers, revealing that the mean-flow dissipation and production of turbulent kinetic energy given by the Favre-Reynolds stresses dominate the thermal-energy balance. The term related to the growth of the turbulent boundary layer opposes the wall cooling. Other identities for the wall-heat flux, inspired by the method of Fukagata et al. ["Contribution of Reynolds stress distribution to the skin friction in wall-bounded flows," Phys. Fluids 14(11), L73-L76 (2002)], are studied numerically and by asymptotic methods. The terms of these identities depend spuriously on the upper integration bound because this bound is a mathematical quantity used in the derivation. When the bound is asymptotically large, the integral identities simplify to the heat-transfer analog to the von Kármán momentum equation. We also prove that an existing multiple-integration identity reduces to the definition of the wall-heat flux when the number of integrations is asymptotically large. No information about the wall-heat transfer is extracted because the impact of the integration number is nonphysical.
D. Xu et al., "Decomposition Of The Wall-heat Flux Of Compressible Boundary Layers," Physics of Fluids, vol. 35, no. 6, article no. 065117, American Institute of Physics, Jun 2023.
The definitive version is available at https://doi.org/10.1063/5.0150696
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01 Jun 2023