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Four problems in stratified flows


We extend the vorticity-based modeling approach of Borden & Meiburg (2013) to non-Boussinesq gravity currents and derive an analytical expression for the Froude number without the need for an energy closure. Via detailed comparisons with simulation results, we assess the validity of three key assumptions underlying both our as well as earlier models, viz. i) steady-state flow in the moving reference frame; ii) inviscid flow; and iii) horizontal flow sufficiently far in front of and behind the current. The current approach does not require an assumption of zero velocity in the current.

Double-diffusive lock-exchange gravity currents in the fingering regime are explored via two- and three-dimensional Navier-Stokes simulations in the Boussinesq limit. The front velocity of these currents exhibits a nonmonotonic dependence on the diffusivity ratio and the initial stability ratio due to the competing effects of increased buoyancy and increased drag. Scaling arguments based on the simulation results suggest that even low Reynolds number double-diffusive gravity currents are governed by a balance of buoyancy and turbulent drag.

The stability of an interface separating less dense, clear salt water above from more dense, sediment-laden fresh water below is explored via direct numerical simulations. We find that the destabilizing effects of double-diffusion and particle settling amplify each other above the diffusive interface, whereas they tend to cancel each other below. For large settling velocities, plume formation below the interface is suppressed. We identify the dimensionless parameter that determines in which regime a given flow takes place.

The effects of shear on double-diffusive fingering and on the settling-driven instability are assessed by means of a transient growth analysis. Shear is seen to dampen both instabilities, which is consistent with previous findings by other authors. The shear damping is more pronounced for parameter values that produce larger unsheared growth. These trends can be explained in terms of instantaneous linear stability results for the unsheared case. For both double-diffusive and settling-driven instabilities, low Pr-values result in less damping and an increased importance of the Orr mechanism, for which a quantitative scaling law is obtained.

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