A Theoretical Investigation of Heat Transfer in the Laminar Flow Regions of Airfoils

Abstract

An analysis is presented of the thermal boundary layer about a certain infinite cylinder having a chordwise pressure gradient. An incompressible fluid of Prandtl Number equal to unity is considered. The thermal boundary-layer profiles are obtained at various chordwise stations and are shown to differ appreciably from the existing velocity boundary-layer profiles. The method of solution is analogous to tha t used by Sears in calculating the spanwise flow in the boundary layer of a yawed infinite cylinder. Although use of this method is not feasible for an arbitrary cylinder, it presents a case in which the results of more approximate methods may be evaluated. Consequently, the heat transfer from the cylinder is calculated by several commonly used methods and is compared to the more exact solution. The results indicate that , of several available approximate methods applicable to the estimation of heat transfer in this case, only the one proposed by Squire gives good agreement with the more exact results. I N T R O D U C T I O N T PROBLEM OF theoretically calculating the thermal boundary-layer profiles, about an arbitrary airfoil having a chordwise pressure gradient, is at present too complicated for a rigorous solution. Therefore, in order to solve the dependent problem of heat transfer from an airfoil, various simplifying assumptions have been made as to the shape of the thermal layer. Consequently, several approximate methods are now available for calculating the heat transfer from an arbitrary airfoil. It was noted that the equation for the spanwise velocity component in a steady, incompressible flow over a yawed infinite cylinder is identical to the equation for the temperature distribution in a steady two-dimensional laminar boundary-layer flow, provided that the Prandtl Number is unity and the viscous heating is neglected. Therefore, the distribution of spanwise velocity near a yawed cylinder is the same as the distribution of temperature when the cylinder surface is maintained at a uniform temperature different from the stream temperature. Sears presented a rigorous solution to the spanwise velocity gradient distribution Received December 7, 1949. * An abstract of a thesis submitted in partial fulfillment of the requirements for the degree of M.Aero.E. in the Graduate School of Aeronautical Engineering, Cornell University, 1948. t Now, Associate Research Engineer, Cornell Aeronautical Laboratory, Inc. about a certain infinite yawed cylinder. Consequently, this solution is used here to obtain the thermal boundary-layer profiles about the same cylinder. These profiles are then used as a basis to evaluate the various simplifying assumptions made in the above references.