Effect of Initial Temperature Profile on Recovery Factor Computations

Abstract

In many aeronautical applications the interaction between fluid flow and a high-speed vehicle is of crucial practical relevance, in view of vehicle performance as well as mechanical and thermal loading of the structure. Viscous flow effects, such as frictional drag and thermodynamic heating, usually play a dominant role in high-speed flight. Surface heating due to frictional effects becomes a design concern for flight Mach numbers of two and above. The elevated temperature that is reached under thermal equilibrium conditions (adiabatic wall temperature Taw) is described by the recovery factor r, as (1-3)Display FormulaTaw=Te+rue22cp (1) where ue and Te are the velocity and temperature outside the boundary layer and with the specific heat cp constant. In the context of the incompressible flat plate boundary layer the recovery effect was investigated already in 1921 by Pohlhausen (4), who showed that r≈Pr, where Pr is the Prandtl number. Several studies have revealed that this applies also under more general conditions, i.e., with arbitrary pressure gradient (1-3). A recent perturbation analysis by the present author confirmed that indeed the first-order pressure-gradient effect on the Prandtl number influence on r is absent, in the case of constant fluid properties (density, viscosity, and conductivity) (5).