Numerical investigation of the formation of three-dimensional structures in stably-stratified mixing layers

Abstract

A temporally-growing mixing layer has been directly simulated with a pseudospectral technique, for initial bulk Richardson numbers from 0.0 to 0.2 and for Prandtl numbers from 0.00535 to 2.2. Several different initial conditions for the velocity fluctuations were imposed. For the two-dimensional (2-D) case only purely-deterministic conditions were used, whereas purely-deterministic, combined deterministic-random, or purely-random conditions were imposed in the three-dimensional (3-D) cases. The numerical procedure allowed fields with very different characteristic lengths to be resolved, with spectral accuracy maintained. The evolution of the velocity, active (temperature), and passive scalar fields were followed independently by adaptively redistributing collocation points in the regions of high shear and rapid scalar variations. The vertical boundary conditions were imposed at infinity to eliminate any boundary-layer effects and an exponential mapping was used to translate infinite physical space into finite computational space. The birth and time evolution of the longitudinal structures have been investigated. Variations in the initial modal forcing are reflected in different outcomes from the competition between core- and braid-centered instabilities in unstratified flow. For relatively strong fundamental forcing (compared to the 3-D forcing) the most unstable mode is braid-centered, whereas if the fundamental forcing is weak the most-unstable, core-centered mode determines the overall three-dimensionalization of the flow by generating large deformations of the main vortex cores. In subcritically-stratified air and water flow, instead, the braid-centered instability supersedes the core-centered instability, whatever the initial forcing. In stratified liquid sodium the shear-aligned convective instabilities observed in air and water are not excited. The conversion of potential into kinetic energy by convective overturning in the braid region does not occur because the high thermal conduction precludes the existence of the unstably-stratified regions necessary to drive the instabilities. The initial conditions, the Richardson, and the Prandtl numbers accordingly play a significant role in the free-shear-layer evolution and need to be explicitly considered for modeling purposes.