A Galerkin Method for the Simulation of Laminar Boundary Layers on Heated Walls

Abstract

This paper presents a new solution method for the calculation of laminar thermal boundary layers. The method consists of a coupling between a modal method (Galerkin method) in the direction normal to the wall and a finite volume method in the direction(s) tangential to the wall. It is similar to an integral method in the sense that only a surface mesh is required and that the unknowns are integral quantities (corresponding to the moments up to a fixed order of the temperature profile in the direction normal to the wall). A specificity of the Galerkin method used is that the domain over which the integrals are computed has a variable size that is also an unknown of the problem. Using a series of numerical tests (representative of situations that can be encountered in aeronautics in the case of a wing equipped with a thermal ice protection system), we show that the new method allows us to predict the quantities of interest with a maximum error of a few percent, while a usual integral method (with only one unknown per mesh cell) is unable to treat the case of boundary layers on heated walls with a strong longitudinal temperature gradient, as shown in the literature.