Development of the spectral difference method and application in the numerical investigation of the separated and transitional flows over a low-Reynolds number airfoil

The development of the high-order accuracy spectral difference (SD) method on hexahedral mesh and its applications in aeroacoustic and aerodynamic problems are carried out in this work. Two absorbing boundary conditions, the absorbing sponge zone and the perfectly matched layer, are developed and implemented for the SD method discretizing the Euler and Navier-Stokes equations on unstructured grids. The performance of both boundary conditions is evaluated and compared with the characteristic boundary condition for a variety of benchmark problems including vortex and acoustic wave propagations. The applications of the perfectly matched layer technique in the numerical simulations of unsteady problems with complex geometries are also presented to demonstrate its capability.;Numerical simulations of the low-Reynolds number (Re = 104 ∼ 105 ) flows over a SD7003 airfoil at moderate incidences (<10°) are performed. A low-frequency convective instability is observed to dominate the spectrum near the leading edge and be responsible for the growth of the disturbance in the attached boundary layer. The characteristic frequency, the growth rate and the wave shape are investigated based on the numerical results. The growth of the low-frequency instability is not in agreement with parallel flow stability theory, nor with leading edge receptivity theory. And it has a higher growth rate than the Tollmien-Schlichting (T-S) wave. The effects of the angle-of-attack (AoA), the Reynolds number and the airfoil geometry on the low-frequency instability are investigated and discussed.;The mechanisms in the breakdown process are investigated and discussed. it is observed that the breakdown of the shedding vortices starts at approximately the location with the maximum negative streamwise flow velocity. And the reverse flow in the separation region directly triggers the generation of three dimensional disturbances and the streamwise vorticities. In addition, the secondary instability which initiates the breakdown process differs in cases at different AOAs. The elliptic and hyperbolic instabilities observed in bluff-body wakes are found to occur in the breakdown process of current cases. Furthermore, the sequence of breakdown states at various incidences is found to be similar to that of the bluff-body wakes at various Reynolds numbers.;A numerical investigation of passive LSB control techniques using roughness bumps on a low-Reynolds number wing is conducted as a further study. The previous case at Re = 6x104 and AoA = 4° is used as the baseline (uncontrolled) case. In the controlled cases, roughness bumps are strategically placed near the leading edge of the wing for the purpose of improving aerodynamic performance in terms of the lift to drag ratio. The location, bump size, the number of bumps and the AoA are varied to study the effects. The pressure drag forces in the controlled cases are found to be reduced significantly when the LSB are reduced or avoided, resulting in much improved lift over drag ratio.