Numerical Simulations of Physical Discontinuities in Single and Multi-Fluid Flows for Arbitrary Mach Numbers
Physical discontinuities, such as shocks and interfaces occur commonly in fluid mechanics. The emphasis of this dissertation lies on numerical simulations of shocked aerodynamic flows and two fluid flows with sharp interfaces in which Rayleigh-Taylor instability is present.
A numerical procedure, which can be classified as shock capturing strategy, based on implicit Pressure-Based finite control volume algorithm for the solution of compressible Euler/Navier-Stokes equation is developed. This method solves for the Cartesian mass flux component in a complex geometry domain. An implicit numerical dissipation model which includes second and fourth order difference terms expressed in pressure with a two-level smoothing function is adopted to create a dissipation mechanism based on pressure gradients to damp destabilizing numerical effects. % without smearing the physical discontinuity at shocks. In this method, retarded and semi-retarded density concepts are utilized to adjust the second-order dissipation in order to obtain a sharp shock capturing property. The simulation results have shown that for inviscid flow the proposed method has a comparable resolution of shocks to the advanced explicit time-marching method for aerodynamic flows, while the proposed method shows a better performance and obtains better results for turbulent flows.
A high-order scheme CHARM (Cubic-parabolic High Accuracy Resolution Method) for the convective modeling of discontinuities is designed for the further improvement of sharp shock capturing. The present scheme combines an algorithmic simplicity with the respective advantages of the higher-order accuracy and that of boundedness feature at discontinuities.
Another numerical procedure, which can be classified as front tracking strategy, based on an Eulerian level set technique has also been developed and tested in this work. This procedure is coupled with both an explicit Density-Based method and a Pressure-Based method to evaluate the motion of the interface separating two fluids with different densities. In this procedure, the dynamic interface is defined as a level set (or a level hypersurface) with a special value according to the initial condition. Then the solution of an equation describing the motion of the level sets embedded in the fluid flow can be advanced using the solution of main motion equations at each time step. This procedure has shown a remarkable ability to predict all main characteristics of Rayleigh-Taylor instabilities observed in experiments. The method is further developed to handle flame propagation problems subject to Landau-Darrieus and Rayleigh-Taylor instabilities. In spite of strong simplifications of the problem, the procedure gives results very close to experimental values.