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.

Internal bores, also know as internal hydraulic jumps, can develop from phenomena in both oceanic and atmospheric situations. Classical approaches handle these bores in cases when density differences between the two layers are large, and more sophisticated approaches can now predict the bore height and propagation velocity in certain cases when the two layers have similar densities. These two-layer models, which conserve mass separately in each layer while conserving momentum across both layers, can generate reasonable predictions for bore velocity if the up and downstream layer heights are known. Traditionally, these models have needed to make assumptions about restricting the energy loss to either the upper or lower layer, but these assumptions are made unnecessary by utilizing conservation of vorticity. Within this work we propose utilizing vorticity conservation to first close the system of equations; after doing so, the energy drop across the bore can be calculated analytically. If we then enforce conservation of energy a predicted downstream layer height for bores can be generated that fits our analytical assumptions. By using this method we compare these model predictions to two-dimensional direct numerical simulations and find that it is possible to predict bore velocity, geometry, and to a lesser extent downstream behavior based on initial conditions only.

Dense particle-laden flows play an important role in many environmental processes, including the shaping of rivers and the formation of landslides. Despite decades of study, researchers have not been able to accurately predict the onset of erosion and the amount of sediment transported by flows, due in part to the difficulty in measuring dense particle-laden flows. Highly-resolved numerical simulations, on the other hand, allow us to study the physics of particle-fluid and particle-particle interactions in much more detail.

We develop a code to accurately simulate dense, polydisperse, particle-laden flows as well as methods by which to analyze them. The code solves the Navier-Stokes equations for the fluid phase and resolves the flow around each individual particle using an immersed boundary method. We also develop a collision model to accurately resolve particle-particle interactions within the fluid. We then perform simulations of a pressure-driven flow over a bed of spherical particles that agree with experimental results for particle velocities and flow rates. Using a control volume momentum balance, we analyze fluid and particle stresses within the simulations, which reveal the mechanisms by which the particle bed expands and contracts during changes in flow rates. These same stresses also allow us to measure the rheology of the particle-laden flows, where we find some agreement with existing constitutive models but also reveal the need to develop these models further.

Many fundamental processes in oceanic transport and limnology occur in geophysical flows that are both local in space and transient in time, and that require equally space and time-resolved methods of analysis. The importance of providing physics-based, quantitative modeling of such flows has driven the development of numerical methods for geophysical fluid dynamics for over three decades. Here, we use direct numerical simulations to investigate a range of stratified, particle-laden flows that are accurately described by the three-dimensional Navier-Stokes equations for an incompressible flow in the Boussinesq limit. We firstly investigate the propagation, transport and mixing dynamics of density-driven gravity currents moving in stratified environments. We propose new models for the intrusion of a turbidity current into a linearly stratified ambient based on three-dimensional simulations. We then describe the interaction between a gravity-current and an internal wave and characterize a phenomenological change in the long-term effect of the interaction at a critical wave height. We then quantify the role of double-diffusive processes in the Dead Sea in Summer and their role in the seasonality of salt crystallization and deposition. We also describe large-scale double-diffusive instabilities that arise in high-Prandtl sedimentary double-diffusive systems such as linearly stratified particle-laden salt water. Finally, we quantify mixing induced by a swarm of small-scale self-propelled organisms migrating in a stratified ambient fluid. We compare the relative contribution to mixing by individual swimmers within the swarm to that of the large-scale motion produced by the collective motion of the swarm.

Within the present investigation, the broad span of applications of the vorticity-based modeling concept for stratified flows, based on the simultaneous use of horizontal and vertical momenta equations in the form of vorticity balance principle, is studied in detail. Towards this objective, this approach, originally introduced by Borden and Meiburg [Z. Borden and E. Meiburg, Phys. Fluids 25 (10), 101301 (2013); Z. Borden and E. Meiburg, J. Fluid Mech. 726, R1 (2013)], for gravity currents propagating into unstratified ambients and internal bores traveling at the interface of two-layer fluids, respectively, is extended to various well known stratified flow problems, in the following. These flows normally involve several fronts which can be analyzed according to the quasisteady conservation laws of mass and momentum by appropriate shift in the reference frame, or possibly unsteady sections for which the flow cannot be rendered quasisteady by any finite number of changes in the reference frames. The analyses of various flow components are then superimposed and matched to obtain the whole flow field. It is also demonstrated that under certain conditions the propagation of gravity currents (or intrusions) can lead to the formation of interfacial perturbations in the form of rarefaction waves or internal bores, which are a source of unsteadiness, and can substantially impact the flow dynamics as well as its energy budget.

Enforcing the conservation laws for horizontal and vertical momenta concurrently, enables us to avoid employing energy-based closure assumptions invoked by previous peer models. Consequently, the assessment of flow energetics becomes plausible, which can be utilized to investigate the validity of the energy-related arguments made by other authors. Furthermore, the predictions of the current study obtained by detailed parametric studies are compared to the results of our two-dimensional direct numerical simulations as well as the theoretical and experimental findings of earlier investigations, where very good agreement is observed with regard to all flow properties.