A 2D modal integral boundary layer code for heat transfers computation was developed in order to allow accurate computation of non-uniform wall temperature flows. The model for this method as well as a previous thermal integral method are described in this paper. The model for both methods is based on solving an integrated form of the energy equation in the boundary layer. An unsteady Finite-Volume formulation is solved until convergence in order to compute the steady solution. This choice allows for an easy extension in three dimensions. Both methods are compared using an in-house code, CLICET, solving directly the Prandtl equations in the boundary layer through local meshing of the boundary layer and a marching algorithm, as a reference solution. Comparison are carried out on auto-similar wedge flows and wing profiles for both uniform and non-uniform imposed wall temperature.
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.
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