The effect of small viscosity and diffusivity on the marginal stability of stably stratified shear flows

The effect of non-zero, but small, viscosity and diffusivity on the marginal stability of a stably stratified shear flow is examined by making perturbations around the neutral solution for an inviscid and non-diffusive flow. The results apply to turbulent flows in which horizontal and vertical turbulent transports of momentum and buoyancy are represented by eddy coefficients of viscosity and diffusivity that vary in the vertical ($z$) direction. General expressions are derived for the modified phase speed and the growth rate of small disturbances as a function of wavenumber. To first order in their coefficients, the effect on the phase speed of adding viscosity and diffusivity is zero. Growth rates are found for two mean flows when the horizontal or vertical coefficients of viscosity and diffusivity vary in $z$ in such a way that the rates can be found analytically. The first flow, denoted as a ‘Holmboe flow’, has a velocity and density interface: the mean horizontal velocity and the density are both proportional to $\tanh az$, where $a$ is proportional to the inverse of the interface thickness. The second, ‘Drazin flow’, has a similar velocity variation in $z$ but uniform density gradient. The analytical results compare favourably with numerical calculations. Small horizontal coefficients of viscosity and diffusivity may affect disturbances to the flow in opposite ways. Although the effect of uniform vertical coefficients of viscosity is to decrease the growth rates, and uniform vertical coefficients of diffusivity increase them, cases are found in which, with suitably chosen $z$ dependence, vertical coefficients of viscosity (or diffusivity) may cause a previously neutral disturbance to grow (or to diminish); viscosity may destabilize a stably stratified shear flow. The introduction of viscosity and diffusivity may consequently increase the critical Richardson number to a value exceeding $1/ 4$. While some patterns of behaviour are apparent, no simple rule appears to hold about whether flows that are neutral in the absence of these effects (viscosity or diffusivity) will be stabilized or destabilized when they are added. One such rule, namely our conjecture that viscosity is always stabilizing and that diffusivity is destabilizing, is explicitly refuted.

Entrainment Correlations Based on a Fuel/Water Stratified Shear Flow

The purpose of this work was twofold: first, to develop correlations for the entrainment of small fuel droplets into water in a stratified fuel/water shear flow; second, to implement the correlations in a CFD code and validate it with experimental effluent fuel concentration data. It is assumed that the droplets act as passive scalars and are advected far from their generation regions where they may cause fuel contamination problems far down-stream. This work relied upon extensive experimental data obtained from a stably stratified shear flow: droplet number, droplet PDF, fluid fraction and velocity field data. The droplet data was expressed as a nondimensional entrainment velocity (E) for the volume flux of fuel due to small droplets. The fluid fraction and velocity fields at the interface were expressed in terms of Richardson numbers (Ri). It was found that E = CeRi−n where n = 1 and Ce is a constant, gives a good fit for the two experimental velocity cases. The best correlation was implemented in a computational simulation of the stably stratified shear flow, and the results show that the simulation can predict the entrainment quite well. A second simulation was performed for a flow with energetic vertical buoyant jets (“buoyant flow events”) and stably stratified shear flows with very large Richardson numbers. In this case, the simulations underpredicted effluent fuel concentrations by two orders of magnitude. Ad hoc corrections to the entrainment correlations show marked improvements.

Large-eddy simulation of homogeneous turbulence and diffusion in stably stratified shear flow

By means of large-eddy simulation, homogeneous turbulence is simulated for neutrally and stably stratified shear flow at gradient-Richardson numbers between zero and one. We investigate the turbulent transport of three passive species which have uniform gradients in either the vertical, downstream or cross-stream direction. The results are compared with previous measurements in the laboratory and in the stable atmospheric boundary layer, as well as with results from direct numerical simulations. The computed and measured flow properties agree with each other generally within the scatter of the measurements. At strong stratification, the Froude number becomes the relevant flow-controlling parameter. Stable stratification suppresses vertical overturning and mixing when the inverse Froude number based on a turn-over timescale exceeds a critical value of about 3. The turbulent diffusivity tensor is strongly anisotropic and asymmetric. However, only the vertical and the cross-stream diagonal components are of practical importance in shear flows. The vertical diffusion coefficient is much smaller than the cross-stream one at strong stratification. This anisotropy is stronger than predicted by second-order closure models. Turbulence fluxes in downstream and cross-stream directions follow classical mixing-length models.