Contributions to the theory of heat transfer through a laminar boundary layer

Abstract

An approximation to the heat transfer rate across a laminar incompressible boundary layer, for arbitrary distribution of main stream velocity and of wall temperature, is obtained by using the energy equation in von Mises’s form, and approximating the coefficients in a manner which is most closely correct near the surface. The heat transfer rate to a portion of surface of length l (measured downstream from the start of the boundary layer) and unit breadth is given as -½ k /(⅓)! (3σρ/μ 2 ) ⅓ ∫ l 0 (∫ l x √{ T ( z )} dz ) ⅔ dT 0 ( x ), where k is the thermal conductivity of the fluid, σ its Prandtl number, ρ its density, μ its viscosity, T ( x ) is the skin friction, and T 0 ( x ) the excess of wall temperature over main stream temperature. A critical appraisement of the formula (§3) indicates that it should be very accurate for large σ, but that for σ of order 0.7 (i. e. for most gases) the constant ½3 ⅓ /(⅓) ! = 0.807 should be replaced by 0.73, when the error should not exceed 8 % for the laminar layers that occur in practical aerodynamics. This yields a formula Nu = 0.52σ ⅓ ( R √ C f ) ⅔ for Nusselt number in terms of the Reynolds number R and the mean square root of the skin friction coefficient C f , in the case of uniform wall temperature. However, for the boundary layer with uniform main stream, the original formula is accurate to within 3% even for σ = 0.7. By known transformations an expression is deduced for heat transfer to a surface, with arbitrary temperature distribution along it, and with a uniform stream outside it at arbitrary Mach number (equation (42)). From this, the temperature distribution along such a surface is deduced (§ 4) in the case (of importance at high Mach numbers) when heat transfer to it is balanced entirely by radiation from it. This calculation, which includes the solution of a non-linear integral equation, gives higher temperatures near the nose, and lower ones farther back (figure 2), than are found from a theory which assumes the wall temperature uniform and averages the heat transfer balance. This effect will be considerably mitigated for bodies of high thermal conductivity; the author is not in a position to say whether or not it will be appreciable for metal projectiles. But for stony meteorites at a certain stage of their flight through the atmosphere it indicates that melting at the nose and re-solidification farther back may occur, for which the shape and constitution of a few of them affords evidence. An appendix shows how the method for approximating and solving von Mises’s equation could be used to determine the skin friction as well as heat transfer rate, but this line seems to have no advantage over established approximate methods.