Asymptotic theory for calculating heat fluxes near the corner of a body

Abstract

A method is presented for calculating the distribution of the thermal fluxes, friction stresses, and pressure near the corner point of a body contour in whose vicinity the outer supersonic flow passes through an expansion wave. The method is based on a study of the asymptotic solutions of the Navier-Stokes equations as the Reynolds number R approaches infinity for the flow region in which the longitudinal gradients of the flow functions are large, invalidating conventional boundary layer theory. This problem was examined in part in [1], in which the distribution of the friction and pressure in a region with length on the order of a few thicknesses of the approaching boundary layer was obtained in the first approximation. The leading term of the expansion for the thermal flux to the surface of the body vanishes for a value of the Prandtl number equal to unity and for other values of the Prandtl number does not match directly with its value in the undisturbed boundary layer.

The thermal-flux distribution is obtained for values of the Prandtl number σ approaching unity. For this purpose it was necessary to consider a more general double passage to the limit as σ→1 and ε→0 for a finite value of the parameter B=[(σ−1)/σ] [−ln ε^1/4/ε]^1/4 characterizing the ratio of the effects of thermal conduction, viscous dissipation, and convection. The solution obtained previously [1] corresponds to the particular case ∥B∥ → ∞ and therefore for actual values of R=10⁴–10⁶, σ~ 0.7 overestimates considerably the effect of the dissipative term on heat transfer, although even in first approximation it describes the pressure distribution well and the friction distribution satisfactorily. For smooth matching of the solutions with the corresponding flow functions in the undisturbed boundary layer it was necessary to introduce a flow region with “free interaction” for the expansion flow. Equations and boundary conditions which describe the flow as a whole are presented. Examples are given of numerical calculations and comparison with experiment.