About the influences of compressibility, heat transfer and pressure gradients in compressible turbulent boundary layers

Abstract

This paper presents a comprehensive analysis of the momentum and energy transfer in compressible turbulent boundary layers based on integral identities. By considering data obtained from direct numerical simulations for a wide parameter range, the superordinate influences of compressibility, wall heat transfer and pressure gradient on the terms of the governing equations are identified and visualized. This allows us both to determine to what degree cases corresponding to different Mach number, heat transfer and pressure-gradient conditions have physically comparable behaviour and to design turbulent boundary-layer cases with specific sought-after behaviour. To this end, newly formulated identities for the skin-friction coefficient $c_f$ and the specific heat-transfer coefficient $c_h$ from wall-normal integrals based on the non-dimensional compressible momentum and total-enthalpy equations are derived and evaluated. As the individual terms of the resulting identities stay formally close to the terms of the governing equations, the integral analysis further allows the evaluation of common arguments derived from the ‘strong’ Reynolds analogy from an integral perspective. A particular formulation of the Eckert number $Ec$ is proposed as a similarity parameter, mapping cases with different Mach numbers and wall heat transfer conditions.

On the enhancement of boundary layer skin friction by turbulence: an angular momentum approach

“…As a result, the other terms cannot be interpreted as augmentations relative to the laminar case. A similar comment applies to the compressible flow relations of Xu et al (2021) and Wenzel et al (2022).…”

“…Xia, Zhang & Yang (2021) considered a similar double-integration procedure for predicting the wall shear stress from outer flow measurements. Double integral (or first moment) relations have also recently been introduced for high-speed boundary layers (Xu, Wang & Chen 2021; Wenzel, Gibis & Kloker 2022).…”

Skin-friction and heat-transfer decompositions in hypersonic transitional and turbulent boundary layers

“…the Eckert number in Wenzel et al. (2021 a ). In order to avoid the influence of the choice of the value of the adiabatic wall temperature on the conclusions obtained in this paper, only the variations along with the cooled wall temperature at the fixed Mach number are discussed, and the variations along with the cooled wall temperature are applicable in the Mach number range investigated in this paper.…”

“…2021; Wenzel et al. 2021 a ), mean kinetic energy (Renard & Deck 2016), mean energy (Arntz, Atinault & Merlen 2015), mean total enthalpy (Wenzel et al. 2021 a ) and mean internal energy (Zhang & Xia 2020).…”

“…According to the viewpoints of Zhang & Xia (2020) and Wenzel, Gibis & Kloker (2021 a ), the twofold repeated integration was better than the threefold repeated integration in the FIK identity. They showed that the decomposition based on the twofold repeated integration gets rid of the linearly weighted term on the Reynolds shear stress and has more intuitive physical interpretation of the underlying mechanisms of the generation of the skin-friction and heat-transfer coefficients.…”