Analytical Analysis of Indirect Combustion Noise in Subcritical Nozzles

This article revisits the problem of indirect combustion noise in nozzles of finite length. The analytical model proposed by Moase for indirect combustion noise is rederived and applied to subcritical nozzles having shapes of increasing complexity. This model is based on the equations formulated by Marble and Candel for which an explicit solution is obtained in the subsonic framework. The discretization of the nozzle into $n$ elementary units of finite length implies the determination of $2n$ integration constants for which a set of linear equations is provided in this article. The analytical method is applied to configurations of increasing complexity. Analytical solutions are compared to numerical results obtained using SUNDAY (a 1D nonlinear Euler solver in temporal space) and CEDRE (3D Navier–Stokes flow solver). Excellent agreement is found for all configurations thereby showing that acceleration discontinuities at the boundaries between adjacent elements do not influence the actual acoustic transfer functions. The issue of nozzle compactness is addressed. It is found that in the subcritical domain, spectral results should be nondimensionalized using the flow-through-time of the entire nozzle. Doing so, transfer functions of nozzles of different lengths are successfully compared and a compactness criterion is proposed that writes $ω*∫0Ldζ/u(ζ)<1$ where $L$ is the axial length of the nozzle. Finally, the entropy wave generator (EWG) experimental setup is considered. Analytical results are compared to the results reported by Howe. Both models give similar trends and show the important role of the rising time of the fluctuating temperature front on the amplitude of the indirect acoustic emission. The experimental temperature profile and the impedance coefficients at the inlet and outlet are introduced into the analytical formulation. Results show that the indirect combustion noise mechanism is not alone responsible for the acoustic emission in the subcritical case.