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.

A direct numerical simulation study of self-similar compressible flat-plate turbulent boundary layers (TBLs) with pressure gradients (PGs) has been performed for inflow Mach numbers of 0.5 and 2.0. All cases are computed with smooth PGs for both favourable and adverse PG distributions (FPG, APG) and thus are akin to experiments using a reflected-wave set-up. The equilibrium character allows for a systematic comparison between sub- and supersonic cases, enabling the isolation of pure PG effects from Mach-number effects and thus an investigation of the validity of common compressibility transformations for compressible PG TBLs. It turned out that the kinematic Rotta–Clauser parameter $\unicode[STIX]{x1D6FD}_{K}$ calculated using the incompressible form of the boundary-layer displacement thickness as length scale is the appropriate similarity parameter to compare both sub- and supersonic cases. Whereas the subsonic APG cases show trends known from incompressible flow, the interpretation of the supersonic PG cases is intricate. Both sub- and supersonic regions exist in the boundary layer, which counteract in their spatial evolution. The boundary-layer thickness $\unicode[STIX]{x1D6FF}_{99}$ and the skin-friction coefficient $c_{f}$, for instance, are therefore in a comparable range for all compressible APG cases. The evaluation of local non-dimensionalized total and turbulent shear stresses shows an almost identical behaviour for both sub- and supersonic cases characterized by similar $\unicode[STIX]{x1D6FD}_{K}$, which indicates the (approximate) validity of Morkovin’s scaling/hypothesis also for compressible PG TBLs. Likewise, the local non-dimensionalized distributions of the mean-flow pressure and the pressure fluctuations are virtually invariant to the local Mach number for same $\unicode[STIX]{x1D6FD}_{K}$-cases. In the inner layer, the van Driest transformation collapses compressible mean-flow data of the streamwise velocity component well into their nearly incompressible counterparts with the same $\unicode[STIX]{x1D6FD}_{K}$. However, noticeable differences can be observed in the wake region of the velocity profiles, depending on the strength of the PG. For both sub- and supersonic cases the recovery factor was found to be significantly decreased by APGs and increased by FPGs, but also to remain virtually constant in regions of approximated equilibrium.

A thorough self-similarity analysis is presented to investigate the properties of self-similarity for the outer layer of compressible turbulent boundary layers. The results are validated using the compressible and quasi-incompressible direct numerical simulation (DNS) data shown and discussed in the first part of this study; see Wenzel et al. (J. Fluid Mech., vol. 880, 2019, pp. 239–283). The analysis is carried out for a general set of characteristic scales, and conditions are derived which have to be fulfilled by these sets in case of self-similarity. To evaluate the main findings derived, four sets of characteristic scales are proposed and tested. These represent compressible extensions of the incompressible edge scaling, friction scaling, Zagarola–Smits scaling and a newly defined Rotta–Clauser scaling. Their scaling success is assessed by checking the collapse of flow-field profiles extracted at various streamwise positions, being normalized by the respective scales. For a good set of scales, most conditions derived in the analysis are fulfilled. As suggested by the data investigated, approximate self-similarity can be achieved for the mean-flow distributions of the velocity, mass flux and total enthalpy and the turbulent terms. Self-similarity thus can be stated to be achievable to a very high degree in the compressible regime. Revealed by the analysis and confirmed by the DNS data, this state is predicted by the compressible pressure-gradient boundary-layer growth parameter $\unicode[STIX]{x1D6EC}_{c}$, which is similar to the incompressible one found by related incompressible studies. Using appropriate adaption, $\unicode[STIX]{x1D6EC}_{c}$ values become comparable for compressible and incompressible pressure-gradient cases with similar wall-normal shear-stress distributions. The Rotta–Clauser parameter in its traditional form $\unicode[STIX]{x1D6FD}_{K}=(\unicode[STIX]{x1D6FF}_{K}^{\ast }/\bar{\unicode[STIX]{x1D70F}}_{w})(\text{d}p_{e}/\text{d}x)$ with the kinematic (incompressible) displacement thickness $\unicode[STIX]{x1D6FF}_{K}^{\ast }$ is shown to be a valid parameter of the form $\unicode[STIX]{x1D6EC}_{c}$ and hence still is a good indicator for equilibrium flow in the compressible regime at the finite Reynolds numbers considered. Furthermore, the analysis reveals that the often neglected derivative of the length scale, $\text{d}L_{0}/\text{d}x$, can be incorporated, which was found to have an important influence on the scaling success of common ‘low-Reynolds-number’ DNS data; this holds for both incompressible and compressible flow. Especially for the scaling of the $\bar{\unicode[STIX]{x1D70C}}\widetilde{u^{\prime \prime }v^{\prime \prime }}$ stress and thus also the wall shear stress $\bar{\unicode[STIX]{x1D70F}}_{w}$, the inclusion of $\text{d}L_{0}/\text{d}x$ leads to palpable improvements.

Abstract

This study extends the input-output framework for the receptivity analysis of an incompressible boundary layer introduced by Ran et al. (Stochastic receptivity analysis of boundary layer flow Phys. Rev. Fluids. 4, 093901, 2019) to the laminar adiabatic supersonic case.Spatially distributed in the wall-normal direction, a delta-correlated Gaussian noise is considered as input, both including the velocity and temperature fields. Similarly, components of the resulting velocity and/or temperature fields are chosen as outputs. To study effects on the boundary layer the measurements of the output are restricted within the δ 99 boundary layer thickness implying, however, that effects like acoustic radiation to the freestream are outside the scope of the present analysis. The main goal of the study is twofold: First, to