Implicit Discontinuous Galerkin Solution on Unstructured Mesh for Turbine Blade Secondary Flow

Abstract

To enhance solution accuracy with high-order methods, an implicit solver using the discontinuous Galerkin (DG) discretization on unstructured mesh has been developed. The DG scheme is favored chiefly due to its distinctive feature of achieving a higher-order accuracy by simple internal sub-divisions of a given mesh cell. It can thus relieve the burden of mesh generation for local refinement. The developed method has been assessed in several test cases. First, an examination on the mesh-dependency with different DG orders for discretization has demonstrated the expected mesh convergence order of accuracy correlation. The flow around a cylinder solved with up to the fifth-order accuracy demonstrates the need for a high-order geometrical representation corresponding to the high-order flow field discretization. A calculated turbulent flat plate boundary layer demonstrates the capability of the high-order DG in capturing the laminar sub-layer for a given coarse mesh (y + >20) without a wall function, in a clear contrast to a conventional second-order scheme. The focal test case of the present work is a high-pressure (HP) turbine cascade with the flow losses being strongly influenced by both the laminar separation induced transition on the blade suction surface and the secondary flow development in the endwall region. Fully turbulent RANS solutions consistently overpredict the losses, which can be significantly reduced by an empirically specified transition at a mid-chord point. Of particular interest is a contrasting behavior of pure laminar flow solutions. A steady laminar flow solution is difficult to converge, and even when converging, tends to give an unrealistically large separation. On the other hand when the laminar solution is run in a time-accurate unsteady mode, a qualitatively different flow pattern emerges. The separated shear layer is then shown to lead to coherently shed unsteady vortices with a net entrainment from the main stream to the near-wall region. The resultant time-averaged near-wall flow then effectively becomes a reattached boundary layer. Both overall and distributed losses computed from unsteady laminar solutions are shown to be consistently much better than the fully turbulent RANS solutions. It is remarkable and quite unexpected that for such a seemingly complex 3D flow problem, the straightforward unsteady laminar solutions with no empirical input seem to be comparable with (or slightly better than) the tripped RANS with empirical input.