A complete Crocco integral for two-dimensional laminar boundary layer flow over an adiabatic wall for Prandtl numbers near unity

Abstract

The so-called Crocco integral establishes a relation between the velocity and temperature distributions in steady boundary layer flow. It corresponds to an exact solution of the flow equations in the case of unity Prandtl number and an adiabatic wall, where it reduces to the condition that the total enthalpy remains constant throughout the boundary layer, irrespective of pressure gradient and compressibility. The effect of Prandtl number is usually incorporated by assuming a constant recovery factor across the entire boundary layer. Strictly, however, this modification is in conflict with the conservation-of-energy principle. In search of a more complete expression for the Crocco integral the present study applies an asymptotic solution approach to the energy equation in constant-property flow. The analysis of self-similar boundary layer solutions results in a formulation of the Crocco integral which correctly incorporates the effect of Prandtl number to first order, and that is complete in the sense that it satisfies the energy conservation requirement. Furthermore, the result is found to be applicable not only to self-similar boundary layers, but also to provide a solution to the laminar flow equations in general as well. The effect of varying properties is considered with regard to the extension of the expression to more general flow conditions. In addition to the asymptotic expression for the Crocco integral, asymptotic solutions are also obtained for the recovery factor for various classes of flows.