Numerical simulations of compressible mixing layers with a discontinuous Galerkin method

Discontinuous Galerkin（DG） method is known to have several advantages for flow simulations,in particular,in fiexible accuracy management and adaptability to mesh refinement. In the present work,the DG method is developed for numerical simulations of both temporally and spatially developing mixing layers. For the temporally developing mixing layer,both the instantaneous fiow field and time evolution of momentum thickness agree very well with the previous results. Shocklets are observed at higher convective Mach numbers and the vortex paring manner is changed for high compressibility. For the spatially developing mixing layer,large-scale coherent structures and self-similar behavior for mean profiles are investigated. The instantaneous fiow field for a three-dimensional compressible mixing layer is also reported,which shows the development of largescale coherent structures in the streamwise direction. All numerical results suggest that the DG method is effective in performing accurate numerical simulations for compressible shear fiows.

Wavy structures in compressible mixing layers

Semi-periodic structures namely inclined wavy structures （IWS） are experimentally observed in compressible mixing layers at two convective Mach numbers （Mc = 0.11 and 0.47）. Flow structures are visualized by the laserinduced planar laser Mie scattering （PLMS） technique. Two methods are developed to investigate the spatial distribu- tion and geometry of IWS： （1） the dominant mode extrac- tion （DME） method, to extract the dominant modes of IWS from the streamwise gray-level fluctuation, and （2） the phase tracking （PT） method, to identify the shape of IWS. The re- sults suggest that pressure perturbations account for the for- marion of IWS in the initial mixing region and the joint effect of dilatation and coherent vortices enhances IWS in the well- developed region. The large transverse （cross-flow） scale of the IWS and their relation to coherent vortices （CV） indicate that the disturbance originated from CV in the mixing center propagates far into the free streams. The DME and the PT method are shown to be the effective tools to study the geometrical features of wavy structures in compressible shear flows.

Vortex structure simulation for supersonic mixing layers using nonlinear PSE method

The method of nonlinear parabolized stability equations（PSE） is applied in the simulation of vortex structures in compressible mixing layer.The spatially-evolving unstable waves,which dominate the vortex structure,are investigated through spatial marching method.The instantaneous flow field is obtained by adding the harmonic waves to basic flow.The results show that T-S waves do not keep growing exponentially as the linear evolution,the energy transfer to high order harmonic modes,and that finally all harmonic modes get saturated due to nonlinear interaction.The mean flow distortion induced by the nonlinear interaction between the harmonic modes and their conjugate harmonic ones,makes great change of the average flow and increases the thickness of mixing layer. PSE methods can well capture the two- and three-dimensional large scale nonlinear vortex structures in mixing layers such as vortex roll-up,vortex pairing,and A vortex.

Modelling of pressure-strain correlation in compressible turbulent flow

Previous studies carried out in the early 1990s conjectured that the main compressible effects could be associated with the dilatational effects of velocity fluctuation. Later, it was shown that the main compressibility effect came from the reduced pressure-strain term due to reduced pressure fluctuations. Although better understanding of the compressible turbulence is generally achieved with the increased DNS and experimental research effort, there are still some discrepancies among these recent findings. Analysis of the DNS and experimental data suggests that some of the discrepancies are apparent if the compressible effect is related to the turbulent Mach number, Mt. From the comparison of two classes of compressible flow, homogenous shear flow and inhomogeneous shear flow （mixing layer）, we found that the effect of compressibility on both classes of shear flow can be characterized in three categories corresponding to three regions of turbulent Mach numbers： the low-Mr, the moderate-Mr and high-Mr regions. In these three regions the effect of compressibility on the growth rate of the turbulent mixing layer thickness is rather different. A simple approach to the reduced pressure-strain effect may not necessarily reduce the mixing-layer growth rate, and may even cause an increase in the growth rate. The present work develops a new second-moment model for the compressible turbulence through the introduction of some blending functions of Mt to account for the compressibility effects on the flow. The model has been successfully applied to the compressible mixing layers.