The GRP Scheme For Computing Compressible Fluid Flows And Its Applications

The generalized Rieamnn problem(GRP) scheme is a second order conservative finite volume scheme.According to the method of resolving the rarefaction waves,the GRP scheme has two different versions:the central GRP scheme and the direct Eulerian GRP scheme.The main feature of the centeral GRP scheme is the analytic resolution of curved rarefaction waves,based on the Lagrangian formation.This approach has the advantage that the contact discontinuity in each local wave pattern is always fixed with speed zero and the rarefaction waves and shock waves are located on either side.However,the passage from the Lagrangian to the Eulerian version is sometimes quite delicate,particularly for sonic cases.In contrast,the direct Eulerian GRP scheme directly resolves rarefaction waves in the Eulerian system by using Riemarm invariants and characteristic coordinates,avoiding the transformation between the Eulerian system and the Lagrangian system.In this thesis,we take the idea of the direct Eulerian GRP scheme and consider the following three issues.In the first part,we use the regularity of Riemann invariants and characteristic coordinate transform to resolve the rarefaction waves in the Lagrangian coordinate system directly and devise the Lagrangian GRP scheme.In the second part,we consider the radially symmetric compressible fluid flows.Spherical explosion and implosion in air,water and other media are well-known problems in application.Typical difficulties of these radially symmetric compressible fluid flows lie in the treatment of singularity in the geometrical source and the imposition of boundary conditious at the center,in addition to the resolution of classical discontinuities(shocks and contact discontinuities).This part presents the implementation of direct GRP scheme to resolve this issue.The scheme is directly obtained by the time integration of the fluid flows. The new contribution of this part is to show analytically that the singularity is removable and the momentum vanishes at the center.We then derive the updating formulae of mass and energy at the center,which is incorporated into numerical boundary conditions.The main ingredient is the passage from the Cartesian coordinates to the radially symmetric coordinates.In the third part,we deal with the overheating phenomenon by tracking the contact discontinuity. Conventional shock-capturing schemes accurately predict pressure and velocity, but calculate temperature excessively and calculate density less than exact solution near the contact discontinuity.This overheating phenomenon results from the generation of spurious entropy,being caused by cell-averaging of conserved variables.By tracking the contact discontinuity in the Eulerian coordinates,we avoid the average of conserved variables across it as far as possible.Thus the overheating phenomenon is weakened by the GRP scheme for tracking the contact discontinuity.