Non-linear Instability of Compressible Mixing Layers

In this paper, we study the non-linear spatial stabi1it.y behavior of compressible mising layers, based on Landau’s pert,urbation series and Tollmien-Schlichting waves. It is wellknown from linear theory that, for spatially-developing perturbation waves, whether the disturbance is amplified, neutral, or damped is dependent on the imaginary part of the complex wavenumber, which could be negat.ive, zero, or positive. The non-linear theory goes further and predicts the possibilit,y for amplified wave to develop into supercritical equilibrium state and the possibility for damped waves to diverge into subcritical instability. The present study deals with the first of these possible situations since the non-linear behavior of the most amplified wave is being studied. It is found that the constants in Landau’s series determine the exist,ence of supercritical equilibrium state. If it exists, the constants will also determine the value of the equilibrium amplitude. If it does not exist, the constants will give the location of the singularity. We carry out a study of the influence of compressibility on the existence of supercritical equilibrium. The resutls show that supercritical equilibrium exists at relatively low Mach numbers, while singularity is more likely at high Mach numbers. The stability characteristics of compressible mixing layers are essential to efficient mixing bet,ween the fuel and OXidant in scramjet engine combustion. Stabi1it.y results also pro\-ide appropriate initial conditions for DYS. Comprehensive studies of this problem using linear theory were carried out by Jackson”2.3, h1ichalke4, Ragabs, Day6: Shin’, et.c. These st,udies contribute to the understanding of the phenomena and allow correct prediction of the first stage of the mixing process. In order to gain more knowledge of the evolution of the disturbance in mixing layers: we must rely on * Graduate student, Research .4ssistant t Associate Professor, Senior Member, AIA.4 f Professor 5 .41.4A member non-linear theory. Early non-linear studies were carried out by Landaus, Stuartg, Watsonl’,Reynoldsl’, Eagles12, et,c. Landau’s consideration lends to the well-known and most commonly used expansion for non-linear analysis which is referred to as the Landau’s series. This is also the basis for the present study. Using the Landau’s expansion, Stuart9 and Watsonl’ investigated the non-linear st,ability behavior of parallel flows. The procedure lends itself to the analysis of circular Couette flow and plane Poiseuille flow. Eagles12 studied the stability of Taylor vortices using the fifth-order amplitude expansion. STABILITY FORMULATIONS We define z, y and z as streamwise, normal and spanwise directions. The dependent variables of interest are p, U, ZI, w, T, and p, which are density, velocities in the z, y, z directions, temperature and pressure, respectively. For the convenience, we use variables, such as u, u, without any decorations to represent the total quantities. Paramet.ers for the base flow are denoted by small letters with subscript ‘00’ (e.g. uaa). Variables with subscript ‘10’ (e.g. 2~10) denot,e eigenfunctions of the linear part, which are equivalent to those with ‘ ’ on top (e.g. U) in most of the linear stability literature. Subscripts ‘Ol’, ‘20’, ‘11’ are used to denot.e the eigenfunctions for different non-linear parts. The reference quantities for nondimenionalization are those of the fast free stream, wherein the vorticity thickness 6, = Uc’n(+m)-uoo(-m) I* InlaX serves as the length scale. To simplify the stabilit.y analysis, we assume the following: