Weakly nonlinear development of shock cells in imperfectly expanded supersonic jets

Abstract

Abstract Content of image described in text. A weakly nonlinear theory is developed for shock cells in an imperfectly expanded axisymmetric supersonic jet. By adopting a jet profile with a finite-thickness shear layer and a high-Reynolds-number asymptotic formalism, the shock cells are treated at leading order as a superposition of a countable number of standing-wave modes. The initial amplitude of each mode at the nozzle exit is proportional to normal upper Delta p Δp $\Delta p$ , the difference between the nozzle exit pressure and the ambient pressure. Effects of viscosity and non-parallelism, associated with the slow streamwise variation of the base flow, are accounted for. The nonlinear quadratic interactions among the leading-order modes generate sum and difference modes, as well as the harmonics and mean-flow distortion. These components interact with the leading modes at the cubic level, contributing usual nonlinear effects. In addition, we have identified for the first time resonant interactions among triplets and quartets of the shock-cell modes, which follow from a simple but hitherto unnoted fact that the eigenvalues form an arithmetic sequence to a remarkable degree of accuracy. The effects of these nonlinear interactions, resonant or non-resonant, are clarified through a weakly nonlinear analysis, which exhibits rather unique features and leads to a novel system of amplitude equations coupled with a set of parabolic partial differential equations governing the mean-flow distortion. The coefficients of all the nonlinear terms in the amplitude equations are purely imaginary, which means that the non-resonant nonlinear interactions modulate directly the phase of each mode, which then influences indirectly the amplitudes of other modes through resonant coupling. The resonant interactions directly facilitate the energy transfer among different modes, as a result of which a mode may be influenced and even excited by interactions of other modes, and certain modes may undergo amplification. Through modulating the phases and amplitudes of the constituting modes, the nonlinear interactions influence the overall shock-cell features and their spacing. The predicted shock-cell characteristics and spacing are found to be in favourable agreement with experimental data. The fundamental modes and nonlinearly generated sum/difference modes exhibit considerable magnitudes in the shear layer, the possible significance of which for the emission of shock-associated noise is discussed.