Solution of unsteady Euler equations: Gust–cascade interaction tones

The problem of interaction of a vortical gust with a two-dimensional cascade is considered. Full nonlinear time dependent Euler equations governing the flow are solved employing a 6th-order accurate spatial differencing scheme and a 4th-order accurate time marching technique. The vortical gust is represented by a Fourier series which includes three harmonics. The acoustic response of the cascade for single and multi frequency (vortical) excitations are calculated. The solutions show the generation and propagation of modes that are expected from the theory. It is demonstrated that at low amplitudes of excitation, the time domain analysis produces characteristics of the propagating modes such as the complex mode amplitudes, phase variations, axial waveforms, and tangential waveforms that are in very good agreement with those expected from the linear theory. The exponential decay of the cutoff modes of the first harmonic is also clearly observed. The sound pressure levels of the propagating modes obtained from the present nonlinear time domain analysis are compared with the results of a linearized Navier–Stokes solution and a linearized Euler solution (frequency domain analyses) and good agreement between the results is observed for all the propagating modes.