Self-similar compressible turbulent boundary layers with pressure gradients. Part 2. Self-similarity analysis of the outer layer

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 [STIX]x1D6ECc , which is similar to the incompressible one found by related incompressible studies. Using appropriate adaption, [STIX]x1D6ECc 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 [STIX]x1D6FDK=([STIX]x1D6FFK/[STIX]x1D70Fw)(dpₑ/dx) with the kinematic (incompressible) displacement thickness [STIX]x1D6FFK is shown to be a valid parameter of the form [STIX]x1D6ECc 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, dL₀/dx , 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 [STIX]x1D70Cuv stress and thus also the wall shear stress [STIX]x1D70Fw , the inclusion of dL₀/dx leads to palpable improvements.