Sediment Transportation Mechanics: Density Currents

Gravity currents modify their flow characteristics by entraining ambient fluid, which depends on a variety of governing parameters such as the initial density, Δρ, the total initial height of the fluid, H, and the slope of the terrain, α, from where it is released. It is imperative to study the entrainment dynamics of a gravity current in order to have a clear understanding of mixing transitions that govern the flow physics, the velocity mixing layer thickness, δu, and the density mixing layer thickness, δρ. Experiments were conducted in a lock-exchange facility in which the dense fluid was separated from the ambient lighter fluid using a gate. As the gate is released instantaneously, an energy conserving gravity current is formed, for which the only governing parameter is the Reynolds number defined as Re=Uhν, where U is the front velocity of the gravity current and h is the height of the current. In our study, the bulk Richardson number (inverse of Froude number, Fr), Rib = g′HUb2 = 1, takes a constant value for all the experiments, with Ub being the bulk velocity of the current defined as Ub = g′H. Simultaneous particle image velocimetry and planar laser induced fluorescence measurement techniques are employed to get the velocity and density statistics. Using the buoyancy conservation equation, a new flux-based method was formulated for calculating the entrainment coefficient, EF, near the front and head of the propagating gravity current for a Reynolds number range of Re ≈ 485-12 270 used in our experiments. At the head of the current, the results show a mixing transition at Re ≈ 2700 that is attributed to the flow transitioning from weak Holmboe waves to Kelvin-Helmholtz instabilities, in the form of Kelvin-Helmholtz vortex rolls. Following this mixing transition, the entrainment coefficient continued to increase with increasing Reynolds number owing to the occurrence of three-dimensional Kelvin-Helmholtz billows that promote further small-scale local mixing. Such a mixing transition indicates that a fully turbulent state is not reached even at Re = 12 270 and the amount of entrainment and ensuing mixing depends on the type of flow instability and presence of small-scale secondary structures. The entrainment dynamics were further substantiated using the ratio of δu and δρ. It was observed that δuδρ decreases with increasing Re and reaches a constant value of δuδρ≈ 1 at high values of Re. This trend is in contrast to the entrainment coefficient EF, which never reaches a constant value even at high enough Re. This disparity could be explained by the fact that EF accounts for small-scale scalar mixing, which is not captured by the ratio of mixing layer thicknesses. Experimentally, it was also observed that the EF value near the front of gravity current was 2-9 times higher than the head value depending on the value of the Reynolds numbers. At low Reynolds numbers, the entrainment near the front is an order of magnitude higher than the head and the value decreases with increasing Re. This could be attributed to different modes of entrainment near the front (dominated by vortical structures) and the head (dominated by turbulent flux exchange triggered by the nature of the flow instability). The results from this study improve our understanding of entrainment dynamics and would be useful in developing empirical parameterizations for mixing in stratified flows.

The lock-exchange problem is used extensively to study the flow dynamics of density-driven flows, such as gravity currents, and as a canonical problem to mixing in stratified flows. Opposite halves of a domain are filled with two fluids of different densities and held in place by a lock-gate. Upon release, the density difference drives the flow causing the fluids to slosh back and forth. In many scenarios, density stratification will also impose a viscosity stratification (e.g., if there are suspended sediments or the two fluids are distinct). However, numerical models often neglect variable viscosity. This paper characterizes the effect of both density and viscosity stratification in the lock-exchange configuration. The governing Navier–Stokes equations are solved using direct numerical simulation. Three regimes are identified in terms of the viscosity ratio μ2/μ1=(1+γ) between the dense and less dense fluids: when γ≪1, the flow dynamics are similar to the equal-viscosity case; for intermediate values ( γ∼1), viscosity inhibits interface-scale mixing leading to a global reduction in mixing and enhanced transfer between potential and kinetic energy. Increasing the excess viscosity ratio further ( γ≫1) results in significant viscous dissipation. Although many gravity or turbidity current models assume constant viscosity, our results demonstrate that viscosity stratification can only be neglected when γ≪1. The initial turbidity current composition could enhance its ability to become self-sustaining or accelerating at intermediate excess viscosity ratios. Currents with initially high excess viscosity ratio may be unable to dilute and propagate long distances because of the decreased mixing rates and increased dissipation.

Suspended particle transport is inherent in a wide range of industrial and environmental flows, from energy generation to manufacturing and from marine to atmospheric systems. Such flows are inherently stratified where varying concentration of particulates changes the mean density of the fluid; however, the role of stratification on particle suspension is poorly understood. Here the suspension of sediments in a channel (plane Poiseuille) flow is resolved using numerical simulation where the flow is stably thermally stratified and the particles passive. This set up enables quantification of stratification controls sediment transport processes. Sediment suspension is resolved for settling velocities of V s = 0.005 , V s = 0.01 , and V s = 0.02 , made dimensionless by the friction velocity u τ . Our results show that stratification generates flow-scale mixing processes, which invalidate extant models of sediment suspension that have been used for over 100 years. For stratified flows, the dimensionless collapse of flow scalars and turbulent statistics is poor near the core of the channel, where large concentration gradients coincide with large-scale mixing events. Here the differences between vertical turbulent diffusivities of sediment and temperature increase with increasing V s , reaching 15 % for V s = 0.02 . Discrepancies arise due to a breakdown of the linear gradient diffusion hypothesis. Predictions using the gradient diffusion hypothesis are expected to worsen with increasing settling velocity, indicating that classical models poorly predict dilute particle transport in strongly stratified flows, which are common in a range of environmental and industrial settings. Further, in stratified flows a feedback loop is established that generates strong buoyancy forces in the core of the duct, having the profound impact of building a self-reinforcing two-layer sediment concentration profile. These results highlight that classical sediment transport models may underpredict the amount of turbulent mixing, and therefore the capability of flows to suspend material, where flow-scale mixing events occur. They further reveal mechanisms for the formation of a barrier to mixing which significantly reduces suspended load capacity in flows with an internal velocity maximum, such as ducts, pipes, gravity currents, and jets.

Field observations imply that accumulation and advection of invertebrate larvae and other plankton by propagating fronts provides an efficient mechanism for the cross‐shelf transport and subsequent coastal settlement of larvae and plankton. Gravity currents propagating into a heavier, uniform‐density fluid are known to have near‐surface velocities u that exceed the propagation speed c of the gravity current nose. It is shown by simple dam‐break laboratory experiments and through corresponding numerical solutions of two‐dimensional stratified flow and Lagrangian particle advection that this characteristic of gravity currents can lead to accumulation in the gravity current nose of particles that originate well behind the gravity current head. However, efficient particle accumulation and transport with the speed c only occurs for particles with sufficient vertical buoyancy or vertical swimming capability. Weakly buoyant or neutral particles are swept down and back away from the gravity current head by the circulation in this region. A dam break into an ambient fluid with stratification (two‐layered) could result in the transport of some particles that are initially trapped in the forepart of the leading disturbance, an undular bore. However, this situation does not result in significant accumulation of particles at the leading edge of the disturbance. The numerical solutions show that particles that originate in the dense fluid ahead of the gravity current are much less likely to be accumulated at the gravity current nose. A simple scaling criterion for the minimum vertical buoyancy velocity for efficient accumulation and transport of buoyant particles based on the gravity current and particle characteristics is developed and compares favorably with the numerical results. The implications of these results for transport of larvae in coastal environments are discussed.

Internal waves, turbulence and mixing in stratified flows: a report on Euromech Colloquium 339

: The gravitational pattern in estuaries is often perturbed, at subtidal scales, by flows resulting from other processes. Wind forcing is the most familiar of these. Subtidal estuarine flow variability appears to be ubiquitous, but no predictive framework for these circulation patterns has yet been proposed. The estuarine-shelf exchanges driven at subtidal scales result in buoyant effluent plumes, which influence shelf chemistry and biology as well as physics. The dynamics of these plumes remains a fertile area of research, principally because of a lack of knowledge concerning mixing in stratified flows.

Several laboratory experiments have been used to study the vertical structure and the mixing efficiency across sharp and linear density interfaces. The two laboratory experiments described here exhibit no mean velocity shear; in the first one the turbulence is generated by a horizontally oscillating grid or rod system, in the other experiment a line of air bubbles injected in the base of a stratified tank produces localized mixing. We will consider here mixing in zero mean flows with a constant input of external kinetic energy, as well as the decay of the turbulence.

Computation of Mixing in Stratified Flow

Experimental investigation of density current patterns using dynamic fractal analysis

Shear-induced vertical mixing in a stratified flow is a key ingredient of thermohaline circulation. We experimentally determine the vertical flux of momentum and density of a forced gravity current using high-resolution velocity and density measurements. A constant eddy-viscosity model provides a poor description of the physics of mixing, but a Prandtl mixing length model relating momentum and density fluxes to mean velocity and density gradients works well. For the average gradient Richardson number Ri(g) approximately 0.08 and a Taylor Reynolds number Re(lambda) approximately 100, the mixing lengths are fairly constant, about the same magnitude, comparable to the turbulent shear length.

Sea and land breeze circulations driven by surface temperature differences between land and sea often evolve into gravity currents with sharp fronts. Along narrow peninsulas, islands and enclosed seas, sea/land breeze fronts from opposing shorelines may converge and collide and may initiate deep convection and heavy precipitation. Here we investigate the collision of two sea breeze gravity current fronts in an analogue laboratory setting. We examine these collisions by means of ‘lock‐exchange’ experiments in a rectangular channel. The effects of differences in gravity current density and height are studied. Upon collision, a sharp front separating the two currents develops. For symmetric collisions (the same current densities and heights) this front is vertical and stationary. For asymmetric collisions (density differences, similar heights) the front is tilted, changes shape in time and propagates in the same direction as the heavier current before the collision. Both symmetric and asymmetric collisions lead to upward displacement of fluid from the gravity currents and mixing along the plane of contact. The amount of mixing along the collision front decreases with asymmetry. Height differences impact post‐collision horizontal propagation: there is significant propagation in the same direction as the higher current before collision, independent of density differences. Collisions of two gravity current fronts force sustained ascending motions which increase the potential for deep convection. From our experiments we conclude that this potential is larger in stationary collision fronts from symmetric sea breeze collisions than in propagating collision fronts from asymmetric sea breeze collisions.

Interactions between gravity waves and cold air outflows in a stably stratified uniform flow forced by various combinations of prescribed heat sinks and sources are studied using a hydrostatic two-dimensional nonlinear numerical model. It is found that the formation time for the development of a stagnation point or reversed flow at the surface is not always directly proportional to the Froude number when wave reflections exist from upper levels. It is shown that a density current is able to form by the wave-outflow interaction, even though the Froude number is greater than a critical value. This is the result of the wave-outflow interaction shifting the flow response to a different location in the characteristic parameter space. A density current is able to form or be destroyed due to the wave-outflow interaction between a traveling gravity wave and a cold air outflow. This is proved by performing experiments with a steady-state heat sink and an additional transient heat source. In a quiescent fluid, a region of cold air, convergence, and upward motion is formed after the collision between two outflows produced by two prescribed heat sinks. After the collision, the individual cold air outflows lose their own identity and merge into a single, stationary, cold air outflow region. Gravity waves tend to suppress this new stationary cold air outflow after the collision. The region of upward motion associated with the collision is confined to a very shallow layer. In a moving airstream, a density current produced by a heat sink may be suppressed or enhanced nonlinearly by an adjacent heat sink due to the wave-outflow interaction.

Recent studies have modelled the flow of particle–driven gravity currents over horizontal boundaries using either shallow–water equations or simple ‘box’ (integral) models. The shallow–water equations are typically integrated numerically, whereas box models admit analytical solutions. However, the theoretical validity of the latter models has not been fully established. In this paper a novel mathematical technique is developed which permits the derivation of analytical solutions to the shallow–water model for gravity current motion. These solutions, confirmed by comparison with the results of numerical integration, are in good agreement with experimental observations. They also indicate why the simplified box models have been so successful. Moreover, they reveal how the internal dynamics of particle–driven flows are different from gravity currents arising solely due to compositional density differences. While compositionally driven gravity currents, which have a fixed density difference between the intruding and ambient fluids, may be modelled using similarity solutions to the governing equations, particle–driven gravity currents do not possess such solutions because their density is progressively reduced by particle sedimentation. Instead the new analysis determines how their behaviour progressively diverges from the similarity solution. By a change of independent variables, it is possible to develop convergent series expansions for each of the dependent variables which characterize the motion. It is suggested that this approach may find application to a number of other problems in which the dynamics are initially governed by a simple dynamical balance which is progressively lost as extra physical effects begin to influence the motion.

The evolution of a mono-disperse gravity current in the lock-exchange configuration is investigated by means of direct numerical simulations for various Reynolds numbers and settling velocities for the deposition. We limit our investigations to gravity currents over a flat bed in which density differences are small enough for the Boussinesq approximation to be valid. The concentration of particles is described in an Eulerian fashion by using a transport equation combined with the incompressible Navier-Stokes equations. The most interesting results can be summarized as follows: (i) the settling velocity is affecting the streamwise vortices at the head of the current with a substantial reduction of their size when the settling velocity is increased; (ii) when the Reynolds number is increased the lobe-and-cleft structures are merging more frequently and earlier in time, suggesting a strong Reynolds number dependence for the spatio-temporal evolution of the head of the current; (iii) the temporal imprint of the lobe-and-cleft structures can be recovered from the deposition map, suggesting that the deposition pattern is defined purely and exclusively by the structures at the front of the current.

A simple model is developed of a double-diffusive gravity current flowing without friction on a horizontal surface. This model captures the effect of the increase in density difference between the gravity current and the environment which is caused by the double-diffusive convection. The double-diffusive flux of buoyancy into the flowing current decreases with time (i.e. with distance from the source) and eventually approaches zero. Steady solutions can then be found which are a function only of distance down-stream from the source. Conservation of momentum shows that entrainment of fluid into the density current is necessary in order to match the imposed condition of a fixed Richardson number at the nose to the arbitrary boundary conditions at the source. When double-diffusive convection and entrainment are both parameterized a priori, the model is overconstrained in the sense that the downstream Richardson number condition at the nose can be satisfied only by a particular set of inlet conditions. This leads us to the conclusion that entrainment and double-diffusive convection must not act independently of each other, but that the presence of the entrainment process must reduce the effectiveness of double-diffusive convection.