Dynamical instability of laminar axisymmetric flows of ideal incompressible fluid

Abstract

The instability of axisymmetric flows of inviscid compressible fluid with respect to two-dimensional infinitesimal perturbations with the nonconservation of angular momentum is investigated by numerically integrating the differential equations of hydrodynamics. The compressibility is taken into account for a homentropic flow with an adiabatic index varying over a wide range. The problem has been solved for two angular velocity profiles of an initial axisymmetric flow. In the first case, a power-law rotation profile with a finite enthalpy gradient at the flow edges has been specified. For this angular velocity profile, we show that the instability of sonic and surface gravity modes in a nearly Keplerian flow, when a radially variable vorticity exists in the main flow, can be explained by the combined action of the Landau mechanism and mode coupling. We also show that including a radially variable vorticity makes the limiting exponent in the rotation law at which the unstable surface gravity modes vanish dependent on the fluid compressibility. In the second case, a Keplerian rotation law with a quasi-sinusoidal deviation has been specified in such a way that the enthalpy gradient vanished at the flow edges. We have found than the sonic modes are then stabilized and the flow is unstable only with respect to the perturbations that also exist in an incompressible fluid.