A study of nonlinear waves and resonance in intrusion flows

Abstract

A stratified intrusion flow is considered in which there are three moving (horizontal) fluid layers and two interfaces. The top and bottom layers move with different speeds and may even move in opposite directions, producing an exchange flow. The middle layer is in motion relative to the outer two, and possesses shear so that the speed in the three-fluid system is continuous when the interfaces are both unperturbed. The flow configuration supports the propagation of periodic waves. A linearized analysis for small wave amplitudes is presented. This is compared to some nonlinear periodic solutions found numerically using a Fourier technique. Such solutions permit nonlinear resonances between the various solution modes and these have been computed extensively. References

• P. O. Rusas and J. Grue. Solitary waves and conjugate flows in a three-layer fluid. European Journal of Mechanics, B/Fluids, 21:185--206, 2002. doi:10.1016/S0997-7546(01)01163-3.
• L. K. Forbes, G. C. Hocking, and D. E. Farrow. An intrusion layer in stationary incompressible fluids: Part 1: Periodic waves. Euro. Jnl of Applied Mathematics, 17:557--575, 2006. doi:10.1017/S0956792506006711.
• G. C. Hocking and L. K. Forbes. A note on the flow of a homogeneous intrusion into a two-layer fluid. Euro. Jnl of Applied Mathematics, 18:181--193, 2007. doi:10.1017/S0956792507006924.
• H. Michallet and F. Dias. Non-linear resonance between short and long waves. Proc. of the 9th International Offshore and Polar Engineering Conference, pages 193--198, 1999. http://cat.inist.fr/?aModele=afficheN&cpsidt=1174804.
• D. I. Pullin and R. H. J. Grimshaw. Interfacial progressive gravity waves in a two-layer shear flow. Phys. Fluids, 26:1731--1739, 1983. doi:10.1063/1.864372.