Abstract
A high-order accurate reconstructed discontinuous Galerkin (rDG) method is developed for compressible inviscid and viscous flows in arbitrary Lagrangian-Eulerian (ALE) formulation on moving and deforming unstructured curved meshes. Taylor basis functions in the rDG method are defined on the time-dependent domain, where the integration and computations are performed. The Geometric Conservation Law (GCL) is satisfied by modifying the grid velocity terms on the right-hand side of the discretized equations at Gauss quadrature points. A radial basis function (RBF) interpolation method is used for propagating the mesh motion of the boundary nodes to the interior of the mesh. A third order Explicit first stage, Single Diagonal coefficient, diagonally Implicit Runge-Kutta scheme (ESDIRK3) is employed for the temporal integration. A number of benchmark test cases are conducted to assess the accuracy and robustness of the rDG-ALE method for moving and deforming boundary problems. The numerical experiments indicate that the developed rDG method can attain the designed spatial and temporal orders of accuracy, and the RBF method is effective and robust to avoid excessive distortion and invalid elements near moving boundaries.
Highlights
Many engineering problems require the solution on variable geometries, such as aeroelasticity, fluid-structure interaction, flapping flight and rotor-stator flows in turbine passage
How to satisfy the Geometric Conservation Law (GCL) is a critical issue for the arbitrary Lagrangian-Eulerian (ALE) formulation, especially for higher-order DG methods, where the basis functions being defined on the time-dependent or fixed reference domain/element will come into the picture, making the problem more complicated
7 Conclusions A reconstructed discontinuous Galerkin method has been presented for solving the unsteady compressible Navier-Stokes equations in an arbitrary Lagrangian-Eulerian (ALE) formulation on moving and deforming curved grids
Summary
Many engineering problems require the solution on variable geometries, such as aeroelasticity, fluid-structure interaction, flapping flight and rotor-stator flows in turbine passage. How to satisfy the Geometric Conservation Law (GCL) is a critical issue for the ALE formulation, especially for higher-order DG methods, where the basis functions being defined on the time-dependent or fixed reference domain/element will come into the picture, making the problem more complicated. A space-time type integration is used to obtain the discretized equations and the gas kinetic flux is computed to advance the solution This method is shown to preserve the uniform flow automatically, and applied to several moving boundary problems. By comparing each stage of the ESDIRK scheme with the space-time DG formulation, the following conditions could be obtained, such that each stage will satisfy the GCL Note that these conditions are required for each Gauss quadrature point. We can see that the designed 3rd-order of convergence has been achieved
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