Propagation and stability of vorticity–entropy waves in a non-uniform flow

Abstract

The evolution of disturbances in an annular duct with a non-isentropic radially varying mean flow is studied. Linear and nonlinear analyses are carried out to examine how the mean velocity and density gradients affect the stability and coupling between the disturbances. To isolate the effect of the mean-velocity gradients from that of the mean-density gradients two mean flows are considered, one with a Gaussian density profile and a uniform axial velocity and the other with Gaussian density and Gaussian axial-velocity distributions. For small-amplitude disturbances with the former mean flow profile, the vortical disturbances convect with the mean flow and density fluctuations grow linearly in space as a result of the interaction of the mean-density gradient with the disturbance radial velocity. Eigenmode analysis of the latter profile shows that unstable modes with exponential growth occur owing to the inflection point in the mean-velocity profile. These modes are almost independent of the mean-density profile and are most unstable for low azimuthal wavenumbers. Nonlinear solutions support the linear results and show an algebraic growth of the density for a range of azimuthal wavenumbers and both uniform and non-uniform mean-velocity profiles. The growth of the velocity fluctuations, however, is strongly dependent on the azimuthal wavenumber of the incident disturbance and the mean-velocity profile. The largest growth in the disturbance is observed at radial locations where the largest mean-flow gradients exist. Owing to the growth of the density fluctuations, coupled vorticity–entropy waves are observed downstream of a forced harmonic excitation in a non-isentropic flow. The forcing amplitudes of the incident waves were varied to see how the solutions change with amplitude. As the amplitude is increased, the waves continue to grow and a steepening of the gradients is observed as they propagate downstream until eventually very sharp density and velocity fronts form. These results show that the mean-flow and density profiles play an important role in the evolution of low-azimuthal-wavenumber disturbances which can couple strongly to the duct acoustic modes during combustion instabilities.