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.
Highlights
In compressible turbulent boundary layers (TBLs), the interaction of compressibility, wall heat transfer and pressure gradient results in a complex parameter space
From the current state of the literature, it can be concluded that the isolated influences of Mach number, heat transfer and, in some part, the pressure gradient on the momentum and energy transfer in TBLs are described to a reasonably good extent
Direct numerical simulations of both sub- and supersonic flat-plate TBLs are conducted at various wall-temperature conditions in both zero and adverse pressure-gradient conditions
Summary
In compressible turbulent boundary layers (TBLs), the interaction of compressibility, wall heat transfer and pressure gradient results in a complex parameter space. An intuitive statement about how the single terms of the governing equations behave for different conditions is difficult to formulate. Kloker aspects of our simplified understanding of the analogy between momentum and energy transfer in TBLs have to be modified for ‘strongly’ cooled cases; see e.g. Zhang, Duan & Choudhari (2018). If cases at various Mach numbers are considered, it is not at all obvious what ‘strongly’ means for each Mach-number case. As all the influences of compressibility, wall heat transfer and pressure gradient lead to a shift in the relevance of the single terms in the overall transfer processes with respect to each other, the question of comparability between certain cases always arises
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