The effect of heat addition on slightly compressible flow: The example of vortex pair motion

The effect of a point heat source on the two-dimensional irrotational flow of a slightly compressible nonviscous and nondiffusive fluid is studied. Heat is added at a constant rate and with a prescribed spatial distribution. The Mach number M is assumed small and the governing equations are solved to order M 2 by a generalization of the Rayleigh-Janzen expansion. The interaction between entropy variations produced by the heat source and the pressure field generates a wake consisting of a vortex sheet and a dipole sheet extending downstream of the heat source, and this in turn perturbs the flow at order M*. A general result is that if heat is added where the gas is compressed by the flow, the impulse directed against the free stream is increased and vice versa. Cross-stream impulse is produced if the heat source is not on the axis of symmetry (if any) of the flow. This would result in a lifting force if a cylinder is providing the perturbation to the free stream. When heat is added on the downstream dividing streamline of a symmetrical flow the order M* velocity field at infinity is that due to an apparent mass source and it is shown that the required flux of mass and momentum is supplied from infinity via the wake. Detailed calculations are performed for the case of the self-propagation of a symmetrical vortex pair, where heat is added downstream of the rearward stagnation point. The resultant increase in the pair impulse is shown to lead to a slow increase in the vortex pair separation.