Instability Properties Of Compressible Mixing Layers To Long-wavelength Perturbations

Recent direct numerical simulations of instability in supersonic mixing layer indicate that shocklets appear during the nonlinear evolution process of instability waves. As an effort to seek a mathematical theory that might explain the formation of shocklets, the linear and nonlinear nature of long-wave disturbances in compressible mixing layer is studied in this thesis. For the sake of simplicity, a temporal-mode formulation is adopted in this paper.In the linear stability analysis, dispersive relation is derived from the Rayleigh equation, and an approximate analytical expression for the growth rateωi is obtain up to O (α³), whereα<<1 represents wave number. Neutral curves and curves showing growth rateω_i as a function ofαare obtained for both cases of constant and non-constant mean temperature. These results reveal the effect of the temperature on three-dimensional disturbances. The following conclusions can be dawn:1. Three-dimensional disturbances are less unstable than two-dimensional disturbance in a mixing layer. Disturbances are unstable for all values of Mach number, and as the Mach number increases, the maximum growth rate first increases and then decreases significantly.2. The instability is very sensitive to the temperature ratioβ_T. It is less unstable whenβ_T= 1 than the case whenβ_Tdeviates appreciably from unity (e.g.β_T= 0.5orβ_T= 2.0). In the latter case, disturbances become very unstable at low Mach number ( 2^1/2 < M< 2), but as the Mach number increases further, the maximum growth rate then decreases significantly.3. The O (α³) term in the growth rate ofωi has stabilizing role, leading to a finite neutral wave number of instability wave.An important fact about long-wave modes in mixing layer is that they are non-dispersive waves approximately. The non-dispersive nature means that a resonant interaction can take place among a continuous band of modes, which might lead to formation of shocklets. Based on this consideration, we analyzed the nonlinear evolution of long waves. The evolution system governing the disturbance is derived by analyzing the dynamics within the critical layer. It is further regularized by using the results of the linear analysis. The regularized evolution system may have shocklet solutions. It is hoped that the system may prove useful in elucidating the mechanism of shocklet formation.