Structure and stability of the compressible Stuart vortex

Abstract

The structure and two- and three-dimensional stability properties of a linear array of compressible Stuart vortices (CSV; Stuart 1967; Meiron et al. 2000) are investigated both analytically and numerically. The CSV is a family of steady, homentropic, two-dimensional solutions to the compressible Euler equations, parameterized by the free-stream Mach number $M_{\infty}$, and the mass flux $\epsilon$ inside a single vortex core. Known solutions have $0 \,{<}\,M_{\infty}\,{<}\,1$. To investigate the normal-mode stability of the generally spatially non-uniform CSV solutions, the linear partial-differential equations describing the time evolution of small perturbations to the CSV base state are solved numerically using a normal-mode analysis in conjunction with a spectral method. The effect of increasing $M_{\infty}$ on the two main classes of instabilities found by Pierrehumbert & Widnall (1982) for the incompressible limit $M_{\infty} \,{\rightarrow}\, 0$ is studied. It is found that both two- and three-dimensional subharmonic instabilities cease to promote pairing events even at moderate $M_{\infty}$. The fundamental mode becomes dominant at higher Mach numbers, although it ceases to peak strongly at a single spanwise wavenumber. We also find, over the range of $\epsilon$ investigated, a new instability corresponding to an instability on a parallel shear layer. The significance of these instabilities to experimental observations of growth in the compressible mixing layer is discussed. In an Appendix, we study the CSV equations when $\epsilon$ is small and $M_{\infty}$ is finite using a perturbation expansion in powers of $\epsilon$. An eigenvalue determining the structure of the perturbed vorticity and density fields is obtained from a singular Sturm–Liouville problem for the stream-function perturbation at $O(\epsilon)$. The resulting small-amplitude steady CSV solutions are shown to represent a bifurcation from the neutral point in the stability of a parallel shear layer with a tanh-velocity profile in a compressible inviscid perfect gas at uniform temperature.