Nonlinear three-mode interactions in a developing mixing layer

Abstract

A formulation based on the integral energy method is presented for the nonlinear interaction problem between three large-scale coherent modes in a developing, laminar, free shear layer. Both binary and three-mode nonlinear, modal energy and phase interactions are included. The modal evolution and the development of the mean flow are very sensitive to the initial energy contents and phases of the participating modes. A strong nonlinear coupling of the modal phase variations with the variations of the modal energies is shown to exist and is responsible for the rapid reversal of the intermodal energy interactions in favor of the occasionally declining mode. This nonlinear mechanism preserves the higher frequency modes far downstream and is shown to contribute to the appearance of mean flow contraction observed in experiments through local collective return of energy to the mean by all modes. Depending on initial conditions, our results support the arguments of R. A. Petersen and R. C. Clough [“The influence of higher harmonics on vortex pairing in an axisymmetric mixing layer,” J. Fluid Mech. 239, 81 (1992)] and M. R. Hajj, R. W. Miksad, and E. J. Powers [“Subharmonic growth by parametric resonance,” ibid. 236, 35 (1992)] according to which three-mode resonances are more efficient than the binary ones. Two cases relevant to situations where the first and second subharmonics of the most amplified mode are forced involving the 3/2 harmonic and 2/3 subharmonic, respectively, are examined. Comparisons of the calculations with experimental results indicate good qualitative agreement.