The Thermodynamics of the Laminar Boundary Layer of a Heated Body in a High-Speed Gas Flow Field

Abstract

The thermal boundary-layer equations for a two-dimensional shockless gas flow are treated in a manner similar to Falkner and Skan's general solution of the velocity boundary-layer equations. When the local velocity outside the boundary layer increases as a ppwer of the distance along the surface, a particularly simple solution for the temperature field exists. A rather general differential equation defining the effect of the generation of heat by friction and shockless compressibility in the boundary-layer results. Some numerical integrations of this equation are presented, and it is found that the effect of the internal heat generation on the rate of heat transfer depends on the pressure gradient along the surface, as well as on the Prandtl Number of the gas when the latter differs from unity. In the absence of this internal heat generation, the difference between the gas stagnation temperature and the surface temperature is shown to be the temperature difference causing the flow of heat. The effect of the internal heat is only that of a correction factor on the stagnation temperature as the effective gas temperature. When the Prandtl Number of the gas is unity, no correction is required. An exact solution for the heat transfer from a body of arbitrary shape, obtainable through Taylor series expansions along the surface, is ^outlined. The tables required for its general application remain to be computed from the derived linear differential equations. Until these tables are completed, a first approximation may be based on solutions for velocity fields of the type: U — cx. I t is shown that these solutions indicate an analogy exists between the momentum loss in the boundary layer and the rate of heat transfer despite the presence of a pressure gradient. This finding is at variance with the common belief that skin friction and heat transfer are analogous quantities. The latter is true only in the absence of a pressure gradient. The analogy is applied to the calculation of the variation of the local heat transfer coefficient over the front half of a circular cylinder, and good agreement with experimental data is shown.