Effect of density gradients in confined supersonic shear layers. I. Two-dimensional disturbances

Abstract

The effect of density gradients on two-dimensional (2-D) supersonic wall modes (acoustic modes) of a 2-D confined compressible shear layer are investigated using linear analysis. Due to the inadequacies of the hyperbolic tangent velocity profile, the boundary layer basic flow profiles are used. First, a test case is taken with the same parameters as in a previous analysis by Tam and Hu [J. Fluid Mech. 203, 51 (1989)], who used a hyperbolic tangent profile. For the boundary layer profiles, three generalized inflection points are found giving rise to three modes. The first two show similar properties to those found by Tam and Hu, whereas the third is a new mode that can have a higher growth rate than the others. As the density ratio is increased above that of the test case, the smallest of the three neutral phase speeds tends toward the speed of the lower-velocity stream, and the other two eventually coalesce and then disappear. These two effects lead to a linear resonance between some of the modes that increases the cutoff frequency and growth rate of the lowest mode. In fact, growth rates of two to four times the test case were found as the density ratio was increased to 7. A similar trend (linear resonance) is observed as the density ratio is decreased from the test case, but the growth rate is only slightly changed from the test case.