From an alternative viewpoint, the results demonstrate how the effect of compressibility on laminar BLs can be used to gain insight into turbulent BLs. The AMI results quantify how the variation of mean density inside a turbulent BL impacts the ability of correlated velocity fluctuations to transport momentum and enhance the skin friction coefficient relative to the equivalent laminar BL. The derivation of the AMI equation itself suggests the use of shear stress-weighted average viscosity, which collapses the relative influence of turbulence on compressible BL skin friction over the wide range of edge Mach numbers and temperature boundary conditions considered. The DNS cases span a variety of wall-temperature boundary conditions ranging from nominally adiabatic to strongly cooled walls. Direct numerical simulations (DNSs) of zero pressure gradient isothermal flat plate BLs up to edge Mach number of 7 are used to demonstrate the utility of the AMI and MTEI equations. Similarly, a moment of total enthalpy integral (MTEI) equation is proposed to quantitatively map the effects of turbulence and other physical flow phenomena to the Stanton number relative to an equivalent laminar BL. By isolating the laminar BL friction at the same Reynolds number, the AMI equation obtains a straightforward interpretation for skin-friction alteration by other flow phenomena relative to an equivalent laminar BL. To accomplish such a mapping, an angular momentum integral (AMI) equation, originally derived for incompressible flows by Elnahhas and Johnson, is here introduced for compressible turbulent BL flows of a calorically perfect gas. An interpretable mapping of how various flow phenomena such as turbulence and the streamwise growth of the boundary layer (BL) thickness influence these key surface quantities is desirable for advancing our understanding of fundamental flow physics, as well as informing engineering design analysis. The enhancement of skin-friction drag and surface heat flux by the transition to turbulence is a crucial physical phenomenon for the design of high-speed vehicles.
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