Reconstructed Discontinous Galerkin Method for the Compressible Navier-Stokes Equations in Arbitrary Langrangian and Eulerian Formulation

摘要

The discontinuous Galerkin (DG) methods are widely used in computational fluid dynamics (CFD) as higher-order schemes. However, the computational costs and storage requirements are expensive for DG methods. In order to reduce the high computing costs, the reconstructed discontinuous Galerkin (rDG) method is highly preferable. Although the rDG method has been studied frequently in the Eulerian formulation, its applications are relatively rare in the arbitrary Lagrangian- Eulerian (ALE) and the Lagrangian formulations. The objective of the effort presented in this PhD work is to investigate the performance of a Taylor-basis rDG method in all the three formulations (-- Eulerian, ALE and Lagrangian) for compressible Euler/Navier-Stokes equations.;In the Eulerian formulation, the focus is on a non-linear solver, called the non-linear Krylov acceleration (NKA). The implicit schemes due to the less demanding constraint on the CFL number, are frequently used in the rDG methods. In general, implicit methods require the solution of a nonlinear system of equations at each time step or stage. Among those nonlinear solvers Newton- GMRES is a popular one, in which the linear equations resulting from the Newton linearization procedure are solved by the generalized minimal residual method (GMRES). In this work, the NKA solver is incorporated into the rDG scheme, and used to solver the nonlinear system arising from the rDG formulation. A comparison study is performed between NKA and Newton-GMRES in terms of the CPU time cost, over a variety of steady and unsteady test cases. The results demonstrate that the NKA solver is comparable to its Newton-GMRES counterpart for steady problems, and is observed to be 2 to 5 times faster than Newton-GMRES for transient flow problems. Thus NKA provides an attractive alternative to solve systems of nonlinear equations in the context of the rDG formulation.;Many engineering problems requires the solution on variable geometries, such as aeroelasticity, fluid-structure interaction, flapping flight and rotor-stator flows in turbine passage. The arbitrary Lagrangian-Eulerian (ALE) formulation is often considered for solving such problems. Based on the success of the rDG method in Eulerian formulation, its extension to the ALE scheme is desirable. Thus, a rDG-ALE method is proposed in this work, to simulate flows over domains with moving and deforming grids. One critical issue in ALE methods is how to satisfy the geometric conservation law (GCL). In this work, we follow the idea from the literature and update the grid velocity terms at Gauss quadrature points on the right-hand side (RHS) of the semi-discrete equations, to enforce the GCL condition. For a typical moving boundary problem, the motion of the domain boundaries are usually given. To avoid excessive distortion and invalid elements, the motion of the boundary nodes can be propagated to the interior of the mesh by a smoothing procedure. In this work, the radial basis function (RBF) interpolation method is used for the mesh smoothing. Several numerical examples are set up to assess the performance of the proposed rDG-ALE method.Mesh and timestep refinement study shows that the designed spatial and temporal orders of accuracy are achieved. The results from the moving airfoil problems are compared with the experimental or numerical data in the literature, showing the capability of the rDG-ALE method in solving such problems.;Lagrangian method is particularly suitable for the evolution of flows undergoing large deformation due to strong compresssion or expansion. Since the mesh will follow the flow features, the method is naturally adaptive. And in this method, there's no mass flux across the boundary between cells, thus permitting accurately tracking of material interface. There're generally two approaches in Lagrangian method for the placement of the physical variables, the staggered-grid approach and the cell-centered approach. In cell-centered approach, the variables have consistent locations. However, how to determine the vertex velocity and guarantee the consistency between the mesh motion and the numerical flux is a challenge. One solution to this difficulty is to use a nodal Riemann solver, which provides a unique nodal velocity, and also the flux in momentum and total energy equations.;The Lagrangian formulation can be written in either total Lagrangian form or updated Lagrangian form. The total Lagrangian is usually discretized on the fixed initial mesh, while the updated Lagrangian on the time-dependent physical domain. In this work, we propose a cellcentered updated Lagrangian formulation using the rDG method, and take the advantage of the nodal Riemann solver. This method is a Lagrangian limit of the unsplit rDG-ALE formulation, with the conservative variables as the working variables. To suppress the oscillations near discontinuities, the vertex-based (VB) limiter by Kuzmin is preferred. The limiter on characteristic variables turns out to outperformits counterpart on the conservative physical variables, in terms of the monotonicity and symmetry preservation. The designed spatial order of accuracy is also achieved by setting up the smooth flow problems.;Overall, the rDG method using Taylor-basis has shown promising results in all the three formulations, and has a great potential to become an attractive and competitive method. In addition, the performance of the combination of the NKA nonlinear solver with rDG method is also quite impressive.