Mean flow effects on the linear stability of compressible planar jets

An analytical solution is derived for the two-dimensional, laminar, compressible, planar free jet. The solution assumes constant pressure, specific heats, and unity Prandtl number and accounts for the effects of heat conduction and viscous dissipation in a self-consistent fashion. Exact closed-form expressions are provided for the streamwise and transverse velocities, temperature, vorticity, and dilatation. Temporal instability analyses of these high Reynolds number mean flows indicate that jet-to-ambient temperature ratio exerts a far greater effect on instability growth rates than compressibility. Relative to isothermal conditions, a hot jet flowing into cold ambient fluid is an order of magnitude more unstable and is unstable over a far greater range of wavenumbers. For this hot jet both symmetric and antisymmetric modes are equally amplified whereas isothermal jets have relatively stronger amplification of their antisymmetric modes. A cold jet issuing into a hot fluid is very stable relative to isothermal conditions. Increasing compressibility suppresses instability growth rates for all temperature ratios. All modes found were two-dimensional. Comparison of the instability analysis and the mean vorticity transport equation indicates that the relative instability growth rates of these jets is qualitatively predicted by the mean inertial vorticity generation term, -omega theta. Jets exhibiting the greatest mean vorticity generation rate inside of the jet half-thickness are the most unstable.