A finite difference solution for the propagation of sound in near sonic flows

Abstract

SUMMARYAn explicit time/space finite difference procedure is used to model the propagation of sound in a quasi one-dimensional duct containing high Mach number subsonic flow. Nonlinear acoustic equations are derived by perturbing the time-dependent Euler equations about a steady, compressible mean flow. The governing difference relations are based on a fourth-order, two-step (predictor-corrector) MacCormack scheme. Difference equations for the source and termination boundary conditions are derived from the appropriate characteristic relationships. The solution algorithm functions by switching on a time harmonic source and allowing the difference equations to iterate to a steady state. A significant advantage with this approach is that the nonlinear terms can be retained and evaluated with only modest additional computer cost above that required for a linear model calculation.The principal effect of the nonlinearities was to shift acoustical energy to higher harmonics. With increased source strengths, wave steepening was observed. This phenomenon suggests that the acoustical response may approach a shock behavior at a higher sound pressure level as the throat Mach number approaches unity. Where applicable, comparisons were made with the calculations from a linear finite element algorithm. On a peak level basis, good agreement between the nonlinear finite difference and linear finite element solutions was observed, even though a peak sound pressure level of about 150 dB occurred in the throat region. Nonlinear steady state waveform solutions are shown to be in excellent agreement with a nonlinear asymptotic theory.

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