The effects of Prandtl number on the nonlinear dynamics of Kelvin–Helmholtz instability in two dimensions. (11th March 2021) - Parker, J.P.; Caulfield, C.P.; Kerswell, R.R.
Abstract: Abstract : It is known that the pitchfork bifurcation of Kelvin–Helmholtz instability occurring at minimum gradient Richardson number $Ri_m \simeq 1/4$ in viscous stratified shear flows can be subcritical or supercritical depending on the value of the Prandtl number, $Pr$ . Here, we study stratified shear flow restricted to two dimensions at finite Reynolds number, continuously forced to have a constant background density gradient and a hyperbolic tangent shear profile, corresponding to the 'Drazin model' base flow. Bifurcation diagrams are produced for fluids with $Pr=0.7$ (typical for air), 3 and $7$ (typical for water). For $Pr=3$ and $7$, steady billow-like solutions are found to exist for strongly stable stratification of $Ri_m$ beyond $1/2$ . Interestingly, these solutions are not a direct product of a Kelvin–Helmholtz instability, having half the wavelength of the linear instability, and arising through a superharmonic bifurcation. These short-wavelength states can be tracked down to at least $Pr ≈ 2.3$ and act as instigators of complex dynamics, even in strongly stratified flows. Direct numerical simulations of forced and unforced two-dimensional flows are performed, which support the results of the bifurcation analyses. Perturbations are observed to grow approximately exponentially from random initial conditions where no modal instability is predicted by a linear stability analysis.