Implicit Discontinuous Galerkin solution on unstructured mesh for turbine blade secondary flow

To enhance solution accuracy with high order methods, an implicit solver using the Discontinuous Galerkin (DG) discretization on unstructured meshes has been developed. The validity and effectiveness of the developed method have been examined in several test cases. The flows around a cylinder solved with up to 5th orders of accuracy on an extremely coarse mesh illustrate the need for high order geometrical representation in relation to the high-order flow field discretization. A calculated turbulent boundary layer on flat plate demonstrates the capability of the high order DG in capturing the laminar sub-layer in a coarse mesh (y+>20) without a wall function, in a clear contrast to a conventional 2nd order scheme. A main focal test case of the present work is a high pressure turbine cascade, where the flow losses are strongly influenced by the boundary layer transition on the blade surface and the secondary flow development in the endwall region. In this case the RANS solutions consistently over-predict the losses, prompting further modelling considerations. A higher order solution does indicate higher resolution for more detailed 3D secondary flow structures. It is however more remarkable that both overall and distributed losses of the unsteady laminar solutions on a coarse mesh are consistently better than the RANS solutions. The results indicate that for the turbine blade flows similar to the one presently tested, a direct unsteady laminar solution on a coarse mesh (far coarser than those typically required for LES) may be further explored