Parallel implicit algorithms for direct numerical simulations of hypersonic boundary layer stability and transition

Due to the progress in computer technology in recent years, distributed memory parallel computer systems are rapidly gaining importance in direct numerical simulation (DNS) of the stability and transition of compressible boundary layers. In most works, explicit methods have mainly been used in such simulations to advance the compressible Navier-Stokes equations in time. However, the small wall-normal grid sizes for viscous flow simulations impose severe stability restriction on the allowable time steps in simulations using explicit method. This requires implicit treatment to the numerical methods. Although fully implicit methods are often used in steady-flow calculations to remove the stability restriction on time steps, they are seldom used in transient flow simulations because the time steps used in time-accurate calculations are often not large enough to offset high computational cost of using fully implicit methods. In this thesis, we present an efficient high-order semi-implicit method, which only treats the stiff terms implicitly, for the DNS study the hypersonic boundary-layer receptivity to freestream disturbances over blunt bodies. It is shown that the semi-implicit method can meet the requirements for both computational efficiency and numerical accuracy in the DNS studies. However, we can not implement our semi-implicit method on single computer to solve unsteady Navier-Stokes equations for the direct numerical simulation of supersonic and hypersonic boundary layer flows on parallel computers directly. The semi-implicit algorithm has to be modified to achieve the communications among processors in solving the global block linear systems. In this thesis, a divide and conquer (DAC) method is used to parallelly solve the block linear system from the semi-implicit method. A parallel Fourier collocation method is also implemented in the periodic spanwise direction. It is shown that by implementing the new parallel semi-implicit scheme the simulations of compressible transient flow can benefit greatly from parallel computer systems by increasing both simulation sizes and speed while maintaining high temporal accuracy.; To implement our new numerical methods on the numerical studies of compressible boundary layer stability and transitions, numerical simulations of the receptivity process of hypersonic boundary layer flows over 3-D blunt leading edges are chosen to be investigated because the receptivity phenomena are much more complex and currently not well understood. In this thesis, parametric simulations of receptivity freestream disturbances which includes fast acoustic waves, vorticity waves and entropy waves for Mach 15 flow over 3-D blunt leading edges have been carried out by using our new methods. The results show that initial transient growth generated and developed inside the hypersonic boundary layer near the leading edge can be observed in the receptivity of freestream standing vorticity or entropy waves, but not acoustic waves or traveling waves. It has been shown that this initial transient growth near the leading edge can be possibly explained by the transient growth theory. Additionally, cooling the surface will increase the growth. By adding inhomogeneous boundary conditions or random roughness on the surface can strongly increase the magnitude of growth.