Secondary Kelvin-Helmholtz instability in a 3D stably stratified temporal mixing layer by Direct Numerical Simulation

Abstract

The present work investigates the nature of the transition to turbulence in the stably stratified mixing layer, which is a complex process with great importance for geophysical and industrial flows. In the stably stratified mixing layer, the streamwise density gradient, which corresponds to the spanwise component of the baroclinic torque in the Boussinesq approximation, feeds the region between the Kelvin-Helmholtz (KH) vortices with vorticity and forms a thin vorticity layer, called baroclinic layer. The competition between buoyancy and inertial forces modifies the dynamics of this layer. As consequence, two different secondary instabilities are found to develop upon the baroclinic layer: one originated near the core region of the KH vortex, called near-core instability, that propagates towards the baroclinic layer and the other of Kelvin-Helmholtz type developed in the baroclinic layer itself. The development of these instabilities in the baroclinic layer depends on the Richardson number, the Reynolds number and the initial conditions. The main objective of this paper is to investigate the occurrence of secondary instabilities in the baroclinic layer of a three-dimensional stably stratified mixing layer using Direct Numerical Simulation (DNS). The development of streamwise vortices and its interactions with the secondary KH structures are focused. Typical Richardson numbers ranging from 0.07 to 0.167 are considered while the Reynolds number is kept constant ( 500 or 1000). White noise and forced perturbation are used as initial conditions. The Navier-Stokes equations, in the Boussinesq approximation, are solved numerically using a sixth-order compact finite difference scheme to compute the spatial derivatives, while the time integration is performed with a third-order low-storage Runge-Kutta method. The numerical results show the development of a jet in the baroclinic layer adjacent to vorticity layers of opposite signs. These layers are created baroclinically by convective motions inside the primary KH vortex and amplifies the near-core instability. It is shown that this instability appears due to the formation of a negative vorticity layer generated between two co-rotating positive vortices. The negative vorticity layer unstables the baroclinic layer and forms small vortices of the KH type. The intensity of the negative vorticity layer depends on the Richardson and Reynolds numbers and defines occurrence or not of secondary KH structures. Interactions between these secondary KH structures and streamwise vortices are also observed. They strongly depend on the initial conditions.