The Weakly Nonlinear Development of Shock Cells in Screeching Jets

The weakly nonlinear development of shock cells is described by an asymptotic theory in an axisymmetric supersonic jet. In this theory, shock cells are the superposition of all standing wave modes at leading order O(e_s), when nonlinearity is taken into account, the sum, difference and regeneration modes must be calculated as the same as the mean-flow distortion at higher orders. It shows that the nonlinear interaction between the O(e_s) and O(e_s^2) components will generate a real forcing term in the regenerated pressure equation, which balances the linear term iA'_n implying that the Landau coefficient is purely imaginary. It means nonlinearity does not affect the modulus of the amplitude function of shock cells, but only the phase of the amplitude function. The theoretical results clearly show the shock-cell spacings and the pressure distribution, and favourable agreement is obtained by comparing with previous experimental data. The present work is the first part of our asymptotic theory for the screech loop.