On thermo-acoustic acoustic-vortical-entropy waves and flow stability

The stability of combustion in jet and rocket engines is related to thermoacoustic effects and also to swirling flow that enhances chemical reactions. This problem is addressed by studying acoustic- vortical-entropy (AVE) waves that are considered as axisymmetric linear non-dissipative perturbations of a compressible, non-isentropic, swirling mean flow, with constant axial velocity and constant angular velocity. The axisymmetric AVE wave equation is obtained for the radial velocity perturbation, specifying its radial dependence for a given frequency and axial wavenumber. The exact solutions are obtained for small and large radius, respectively as ascending and descending series valid inside and outside a critical radius, where the isothermal Mach number for the swirl velocity is unity. Thus they specify exactly the perturbations of: (i,ii) the radial and azimuthal velocity; (iii,iv) pressure and mass density; (v,vi) entropy and temperature. It is shown that there are: (a) oscillatory solutions with decaying amplitude, corresponding to a stable mean flow; (b) monotonic solutions with increasing amplitude, corresponding to instability of the mean flow. The instability condition is that the frequency times a function of the adiabatic exponent is less than the vorticity (or twice the angular velocity); in this case, the instability typical of vortical flows dominates the stability typical of potential flows. This suggests a condition for stable combustion in a confined space: the peak vorticity (multiplied by a factor of order unity dependent on the adiabatic exponent) should be less than the lowest or fundamental frequency of the cavity.