Development of a high-order implicit Discontinuous Galerkin method on unstructured meshes for turbomachinery flow
Keywords: turbomachinery blading design
high-order method
discontinuous finite element method
Computational fluid dynamics
secondary flow

There is a continued need to further capabilities of turbomachinery flow design and analysis method tools, both in terms of the flow modeling fidelity (e.g. moving towards DES/LES) and temporal and spatial accuracy of the numerical discretization methods. To enhance solution accuracy with high-order methods, an implicit solver using the Discontinuous Galerkin (DG) discretization on unstructured meshes has been developed. The DG scheme is favored chiefly due to its distinctive feature that a higher-order accuracy being achieved through simple internal sub-divisions of a given mesh cell. It thus can considerably relieve the burden of mesh generation for local refinement of complex configurations. The implicit formulation is well suited for an unstructured mesh solver, for which some well-established solution techniques for explicit methods (e.g. multigrid methods) are less effective.
The RANS equations in conjunction with the Sparlart-Allmaras one equation turbulence model are discretized using a high-order DG method on unstructured meshes. The all-speed AUSM+-up scheme is applied for the inviscid flux calculations and the BR2 scheme is implemented for the viscous flux discretization. An implicit time integration is adopted and the system linear equations are solved at each time step by using a preconditioned GMRES iterative algorithm with the ILU0. A modification of the turbulence model is also adopted to enhance the robustness for high-order turbulent flow simulations.
The validity and effectiveness of the present developed method have been examined in several test cases, for both validating the method implementation and illustrating salient features and characteristics of the methodology. The flows around a cylinder solved up to 5th-order 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 laminar boundary layer and the vortex shedding after a cylinder validate both the steady and unsteady modelling capabilities with the implicit time integration method. A calculated turbulent boundary layer on flat plate demonstrates the capability of a 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. The all-speed flows past an airfoil show the distinct capability of AUSM+-up scheme on solving a wide range of flow regimes.
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.