A multiple-scales model of the shock-cell structure of imperfectly expanded supersonic jets

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Abstract
A linear solution modelling the shock-cell structure of an axisymmetric supersonic jet operated at slightly off-design conditions is developed by the method of multiple scales. The model solution takes into account the gradual spatial change of the mean flow in the downstream direction. Turbulence in the mixing layer of the jet has the tendency to smooth out the sharp velocity and density gradients induced by the shocks. To simulate this effect, eddy-viscosity terms are incorporated in the model. Extensive comparisons between the numerical results of the present model and experimental measurements gathered at the NASA Langley Research Center over the Mach number range of |Mj2 − Md2| [les ] 1.0 for underexpanded and overexpanded supersonic jets are carried out. Here Mj is the fully expanded jet Mach number and Md is the design Mach number of the convergent–divergent nozzle. Very favourable agreement is found. This is especially true for the gross features of the shock cells, including the shock-cell spacings and the pressure amplitudes associated with the shocks. The measured data show that the pressure distributions over the first three or four shock cells usually are rich in fine structures. These fine structures are reproduced by the calculated results. Beyond the first few shock cells the model predicts that the shock-cell structure can be represented by a single Fourier mode of the mean flow. This is confirmed by a careful examination of the experimental data. The appropriate turbulent Reynolds number for shock-cell structure calculation is investigated. It is shown that the best choice is the same as the value found to give the best results for jet mean-flow calculation. The present model is used to explain some of the observed characteristics of broadband shock-associated noise.