Instability and transition in finite-amplitude Kelvin–Helmholtz and Holmboe waves

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Abstract
We investigate the transition to turbulence in a free shear layer which contains a thin central region of stable density stratification. The fluid is assumed to possess Prandtl number significantly larger than unity, and the flow may exhibit either Holmboe or Kelvin–Helmholtz (KH) instability, depending upon the intensity of the stratification. A sequence of two-dimensional nonlinear numerical simulations of flows near the KH–Holmboe transition (i.e. having bulk Richardson numbers near 1/4) clearly illustrates the structural relationship between Holmboe and Kelvin–Helmholtz waves. The time-dependent nonlinear wave states delivered by the simulations are subjected to a three-dimensional normal-mode stability analysis in order to discover the physical processes that might drive the flow towards a turbulent state. Strong secondary instability is found to persist up to large spanwise wavenumbers, with no indication of a preferred lengthscale. These results indicate that secondary instability may lead the flow directly into the turbulent state.