The flow of high-Reynolds axisymmetric gravity currents of a stratified fluid into a stratified ambient: shallow-water and box model solutions

We consider the axisymmetric flow (in a full cylinder or a wedge) of high-Reynolds-number Boussinesq gravity currents and intrusions systems in which both the ambient and the propagating “current” are linearly stratified. The main focus is on a current of fixed volume released from a cylinder lock; the height ratio of the fluids H, and the stratification parameter of the ambient S, are quite general. We develop a one-layer shallow-water model. The internal stratification enters as a new dimensionless parameter, ${\sigma \in [0, 1]}$ . In general, the time-dependent motion is obtained by standard finite-difference solutions; a self-similar analytical solution exists for S?= 0. We show that, in general, the speed of propagation decreases when the internal stratification becomes more pronounced (σ increases). We also developed a box-model approximation, and show that the resulting radius of propagation is in good agreement with the more rigorous shallow-water prediction.     

Shoaling internal waves may reduce gravity current transport

Gravity currents descending along slopes have typically been studied in quiescent environments, despite the fact that in many geophysical settings there is significant externally driven motion. Here we investigate how the head of a gravity current is influenced by interfacial internal waves at the pycnocline of a two-layer ambient water column. Our experimental measurements show that larger amplitude internal waves, interacting with the gravity current, reduce both the mass transport by the gravity current and its thickness. These results suggest that the ambient internal wave field should be considered when estimating transport by gravity currents in geophysical settings with strong internal waves, such as lakes and the coastal ocean.     

Gravity currents and intrusions of stratified fluids into a stratified ambient

We consider high-Reynolds-number Boussinesq gravity currents and intrusions systems in which both the ambient and the propagating “current” are linearly stratified. The main focus is on a current of fixed volume released from a rectangular lock; the height ratio of the fluids H, and the stratification parameter of the ambient S, are quite general. We develop a one-layer shallow-water (SW) model which is an extension of previously used and tested formulations for currents and intrusions of constant density. The internal stratification enters as a new dimensionless parameter, s ? [0,1]{\sigma \in [0,1]} . Analytical results are obtained for the initial “slumping” stage during which the speed of propagation is constant, and finite-difference solutions are presented for the more general time-dependent motion. Overall, this is a versatile and robust self-contained prediction tool, which reduces smoothly to the classical case when σ = 0. We show that, in general, the speed of propagation decreases when the internal stratification becomes more pronounced (σ increases). An interesting non-expected behavior was detected: when the stratification of the ambient is weak and moderate then the height of the current decreases with σ, but the opposite occurs when the stratification of the ambient is strong (S ≈ 1, including the case of an intrusion). Moreover, when the stratification of the ambient is strong a current with internal stratification may “run out” of driving power. We also consider the Benjamin-type steady state current with internal linear stratification in a non-stratified ambient, and show that an analytical solution exists, and that the maximal thickness decreases to below half-channel depth when σ increases.     

Numerical simulations of intrusive gravity currents interacting with a bottom-mounted obstacle in a continuously stratified ambient

In this study, the flow dynamics of intrusive gravity currents past a bottom-mounted obstacle were investigated using highly resolved numerical simulations. The propagation dynamics of a classic intrusive gravity current was first simulated in order to validate the numerical model with previous laboratory experiments. A bottom-mounted obstacle with a varying non-dimensional height of \(\tilde{D}=D/H\), where D is the obstacle height and H is the total flow depth, was then added to the problem in order to study the downstream flow pattern of the intrusive gravity current. For short obstacles, the intrusion re-established itself downstream without much distortion. However, for tall obstacles, the downstream flow was found to be a joint effect of horizontal advection, overshoot-springback phenomenon, and associated Kelvin-Helmholtz instabilities. Analysis of the numerical results show that the relationship between the downstream propagation speed and the obstacle height can be subdivided into three regimes: (1) a retarding regime (\(\tilde{D}\) \(\approx \) 0–0.3) where a 30 % increase in obstacle height leads to a 20 % reduction in propagation speed, simply due to the obstacle’s retarding effect; (2) an impounding regime (\(\tilde{D}\) \(\approx \) 0.3–0.6) where the additional 30 % increase in obstacle height only leads to a further (negligible) 5 % reduction in propagation speed, due to the accelerating effect of upstream impoundment and downstream enhanced mixing; and (3) a choking regime (\(\tilde{D}\) \(\approx \) 0.6–1.0) where the propagation speed is dramatically reduced due to the dominance of the obstacle’s blocking effect. The obstacle thickness was found to be irrelevant in determining the downstream propagation speed at least for the parameter range explored in this study. The present work highlights the significance of topographic effects in stratified flows with horizontal pressure forcing.     

Blockage of saline intrusions in restricted,two-layer exchange flows across a submerged sill obstruction

Results are presented from a series of large-scale experiments investigating the internal and near-bed dynamics of bi-directional stratified flows with a net-barotropic component across a submerged, trapezoidal, sill obstruction. High-resolution velocity and density profiles are obtained in the vicinity of the obstruction to observe internal-flow dynamics under a range of parametric forcing conditions (i.e. variable saline and fresh water volume fluxes; density differences; sill obstruction submergence depths). Detailed synoptic velocity fields are measured across the sill crest using 2D particle image velocimetry, while the density structure of the two-layer exchange flows is measured using micro-conductivity probes at several sill locations. These measurements are designed to aid qualitative and quantitative interpretation of the internal-flow processes associated with the lower saline intrusion layer blockage conditions, and indicate that the primary mechanism for this blockage is mass exchange from the saline intrusion layer due to significant interfacial mixing and entrainment under dominant, net-barotropic, flow conditions in the upper freshwater layer. This interfacial mixing is quantified by considering both the isopycnal separation of vertically-sorted density profiles across the sill, as well as calculation of corresponding Thorpe overturning length scales. Analysis of the synoptic velocity fields and density profiles also indicates that the net exchange flow conditions remain subcritical (G < 1) across the sill for all parametric conditions tested. An analytical two-layer exchange flow model is then developed to include frictional and entrainment effects, both of which are needed to account for turbulent stresses and saline entrainment into the upper freshwater layer. The experimental results are used to validate two key model parameters: (1) the internal-flow head loss associated with boundary friction and interfacial shear; and (2) the mass exchange from the lower saline layer into the upper fresh layer due to entrainment.     

On the fine structure of the thermal bar front

The thermal bar—a hydrodynamic phenomenon, arising in natural basins due to successive changes of the water temperature across the temperature of maximum density (T_m, which is close to 4°C)—has been studied in laboratory experiments and by numerical simulations. The experiments were performed in a rectangular tank with an inclined bottom, filled with water with initial temperature T₀ < T_m and then heated at the surface. During the heating a basin-wide circulation develops, consisting of down-slope cascades in regions where T < T_m, a subsurface off-shore jet in the region where T > T_m, and a compensating flow at intermediate depths towards the shallow part of the tank, supplying both off-shore flows with waters from deeper regions. Analysis of the water temperature and density fields as well as the currents has revealed that the location of the convergence zone of the surface current (when formed) does not coincide with that of the Tm-isotherm. The thermal bar front is typically understood as a convergence zone near the 4°C-isotherm, formed due to the effect of cabbeling. Our experiments demonstrate, however, that the front is associated with the leading edge of the subsurface current. The increasing distance between the 4°C-isotherm and the subsurface jet has been recorded in the laboratory experiments. Numerical simulation results corroborate the laboratory experiments. A scaling analysis predicts the speed of propagation of this frontal zone to be U ~ [g × Δρ/ρ × H]^1/2, where H is the depth (increasing with time) of the upper thermo-active layer, ρ₀ a reference density, and Δρ is the characteristic horizontal density difference across the front. A combined analysis of laboratory, field and numerical data has corroborated this law.     

Turbulent entrainment into fluid mud gravity currents

The entrainment of ambient water into non-Newtonian fluid mud gravity currents was investigated in this study. Constant volume release gravity currents were generated in a lock-exchange tank for a wide range of experimental conditions. A technique similar to the so-called light attenuation technique was used to find the boundary of the current, allowing for the calculation of both temporal and bulk entrainment parameters (in terms of the temporal and bulk entrainment velocities, respectively). It was found that the temporal entrainment velocity is dependent on different parameters in the different propagation phases. The slumping phase begins with an adjustment zone (henceforth, non-established zone) in which the temporal entrainment velocity is not a function of the current front velocity, followed by the established zone in which the temporal entrainment velocity is a function of the current front velocity. This dependence of the temporal entrainment velocity on the current front velocity carries through to the inertia-buoyancy phase. As expected, temporal entrainment velocity in the viscous-buoyancy phase was negligible in comparison to average entrainment velocity in the other phases. It is observed that the temporal entrainment characteristics in the non-established zone is governed by the competition between the entrainment-inhibiting density stratification effects and the entrainment-favouring effects of the Kelvin–Helmholtz billows that are quantified by the Richardson number and the Reynolds number of the gravity current, respectively. In the established zone, Reynolds number effects were observed to dominate over Richardson number effects in dictating temporal entrainment characteristics. A parameterization for the temporal entrainment velocity for non-Newtonian fluid mud gravity currents is developed based upon the experimental observations. This study also found that the bulk entrainment characteristics for the non-Newtonian fluid mud gravity currents can be parameterized by the Newtonian bulk entrainment parameterizations that rely solely on a bulk Richardson number. Interestingly, it was found that the non-Newtonian characteristics of the gravity current have little to no effect on the entrainment of the Newtonian ambient fluid.     

Gravity currents with tailwaters in Boussinesq and non-Boussinesq systems: two-layer shallow-water dam-break solutions and Navier–Stokes simulations

We consider the dam-break initial stage of propagation of a gravity current of density $\rho _{c}$ released from a lock (reservoir) of height $h_0$ in a channel of height $H$ . The channel contains two-layer stratified fluid. One layer, called the “tailwater,” is of the same density as the current and is of thickness $h_T (< h_0)$ , and the other layer, called the “ambient,” is of different density $\rho _{a}$ . Both Boussinesq ( $\rho _{c}/\rho _{a}\approx 1$ ) and non-Boussinesq systems are investigated. By assuming a large Reynolds number, we can model the flow with the two-layer shallow-water approximation. Due to the presence of the tailwater, the “jump conditions” at the front of the current are different from the classical Benjamin formula, and in some circumstances (clarified in the paper) the interface of the current matches smoothly with the horizontal interface of the tailwater. Using the method of characteristics, analytical solutions are derived for various combinations of the governing parameters. To corroborate the results, two-dimensional direct numerical Navier–Stokes simulations are used, and comparisons for about 80 combinations of parameters in the Boussinesq and non-Boussinesq domains are performed. The agreement of speed and height of the current is very close. We conclude that the model yields self-contained and fairly accurate analytical solutions for the dam-break problem under consideration. The results provide reliable insights into the influence of the tailwater on the propagation of the gravity current, for both heavy-into-light and light-into-heavy motions. This is a significant extension of the classical gravity-current theory from the particular $h_T=0$ point to the $h_T > 0$ domain.     

Numerical simulation of particle-driven gravity currents

Particle-driven gravity currents frequently occur in nature, for instance as turbidity currents in reservoirs. They are produced by the buoyant forces between fluids of different density and can introduce sediments and pollutants into water bodies. In this study, the propagation dynamics of gravity currents is investigated using the FLOW-3D computational fluid dynamics code. The performance of the numerical model using two different turbulence closure schemes namely the renormalization group (RNG) ${k-\epsilon}$ scheme in a Reynold-averaged Navier-Stokes framework (RANS) and the large-eddy simulation (LES) technique using the Smagorinsky scheme, were compared with laboratory experiments. The numerical simulations focus on two different types of density flows from laboratory experiments namely: Intrusive Gravity Currents (IGC) and Particle-Driven Gravity Currents (PDGC). The simulated evolution profiles and propagation speeds are compared with laboratory experiments and analytical solutions. The numerical model shows good quantitative agreement for predicting the temporal and spatial evolution of intrusive gravity currents. In particular, the simulated propagation speeds are in excellent agreement with experimental results. The simulation results do not show any considerable discrepancies between RNG ${k-\epsilon}$ and LES closure schemes. The FLOW-3D model coupled with a particle dynamics algorithm successfully captured the decreasing propagation speeds of PDGC due to settling of sediment particles. The simulation results show that the ratio of transported to initial concentration C _o /C _i by the gravity current varies as a function of the particle diameter d _s . We classify the transport pattern by PDGC into three regimes: (1) a suspended regime (d _s is less than about 16 μm) where the effect of particle deposition rate on the propagation dynamics of gravity currents is negligible i.e. such flows behave like homogeneous fluids (IGC); (2) a mixed regime (16 μm < d _s <40 μm) where deposition rates significantly change the flow dynamics; and (3) a deposition regime (d _s ?> 40 μm) where the PDGC rapidly loses its forward momentum due to fast deposition. The present work highlights the potential of the RANS simulation technique using the RNG ${k-\epsilon}$ turbulence closure scheme for field scale investigation of particle-driven gravity currents.     

Effect of initial excess density and discharge on constant flux gravity currents propagating on a slope

The effect of the upstream conditions on propagation of gravity current over a slope is investigated using three-dimensional numerical simulations. The current produced by constant buoyancy flux, is simulated using a large eddy simulation solver. The dense saline solution used at the inlet is the driving force of the flow. Higher replenishment of the current is possible either by a high inflow discharge or high initial fractional density excess. In the simulations, it is observed that these two parameters affect the flow in different ways. Results show that the front speed of the descending current is proportional to the cube root of buoyancy flux, $(g_o^{\prime } Q)^{1/3}$ , which agrees with the previous experimental and numerical observations. The height of the tail of the current grows linearly in the streamwise direction. Formation of a strong shear layer at the boundary of mixed upper layer and dense lower layer is observed within the body and the tail of the current. Over the tail of the current far enough from the inlet, the vertical velocity and density profiles are compared to the ones from an experimental study. Distance from the bed to the point of maximum velocity increases with an increase in inflow discharge, while it remains practically unchanged with increasing initial fractional excess density in the simulations. Even though the velocity profiles are in good agreement, some discrepancies are observed in fractional excess density profiles among experimental and numerical results. Possible reasons for these discrepancies are discussed. Generally, gravity current type of flows could be expressed in layer-integrated formulation of governing equations. However, layer integration introduces several constants, commonly known as shape factors, to the equations of motion. The values of these shape factors are calculated based on simulation results and compared to the values from experiments and to the favorably used ‘top hat’ assumption.     

Temperature and salinity tolerances of embryos and larvae of the deep-sea mytilid mussel ��Bathymodiolus�� childressi

We examined temperature and salinity tolerances of early embryonic and larval stages of the deep-sea, cold-seep mussel ??Bathymodiolus?? childressi to determine whether they may control the dispersal depth of larvae. Salinity and temperature tolerances increased with developmental stage, but tolerance ranges were not as wide for the larvae of ??B.?? childressi as for the larvae of the related shallow-water mussel Mytilus trossulus. Normal development occurred in ??B.?? childressi from 7 to 15°C and at salinities of 35 and 45. Greater tolerance of ??B.?? childressi embryos to high than low salinities may aid development of negatively buoyant early embryos at brine seeps. Although there was a decreasing trend in survival of ??B.?? childressi larvae with increasing temperature, survival of ??B.?? childressi trochophores was not significantly different at 20°C than at the adults?? ambient temperature. Since larvae tolerate increasing temperatures as they age and seawater temperatures at 100?m depth do not exceed 20°C in months following the mussels?? spawning season, we suggest that temperature would not limit vertical migration of the veliger larvae of ??B.?? childressi into even the uppermost layer of the water column above the cold seeps in the Gulf of Mexico.     

A steady-state model for asymmetric intrusive gravity currents in a linearly stratified ambient

The behavior of the steady intrusive gravity current of thickness h and density ρ _c which propagates with speed U at the neutral buoyancy level of a long horizontal channel of height H into a stratified ambient fluid whose density increases linearly from ρ _o to ρ _b is investigated. The intrusive and the ambient fluids are assumed to be asymmetric with respect to the neutral-buoyancy level. The Boussinesq, high-Reynolds number two-dimensional configuration is considered. Long’s model combined with the flow-force balance over the width of the channel and the pressure balances over a density current are used to obtain the desired results. It is shown that the intrusion velocity decreases with decreasing the asymmetry of the system and approaches its minimum for the symmetric configuration (however, the difference of speed between asymmetric and symmetric configurations shows no significant differences).     

Front velocity and structure of bottom gravity currents with a low volume of release propagating in a porous medium

The paper reports results of large eddy simulations of lock exchange compositional gravity currents with a low volume of release advancing in a horizontal, long channel. The channel contains an array of spanwise-oriented square cylinders. The cylinders are uniformly distributed within the whole channel. The flow past the individual cylinders is resolved by the numerical simulation. The paper discusses how the structure and evolution of the current change with the main geometrical parameters of the flow (e.g., solid volume fraction, ratio between the initial height of the region containing lock fluid and the channel depth, ratio between the initial length and height of the region containing lock fluid) and the Reynolds number. Though in all cases with a sufficiently large solid volume fraction the current transitions to a drag-dominated regime, the value of the power law coefficient, α, describing the front position’s variation with time (x _f  ~ t ^α , where t is the time measured from the removal of the lock gate) is different between full depth cases and partial depth cases. The paper also discusses how large eddy simulation (LES) results compare with findings based on shallow-water equations. In particular, LES results show that the values of α are not always equal to values predicted by shallow water theory for the limiting cases where the current height is comparable, or much smaller, than the channel depth.     

Reductive dechlorination of chlorobenzene in supercritical water catalyzed by Fe/ZrO<Subscript>2</Subscript>

Reductive dechlorination of chlorobenzene by Fe/ZrO₂ in supercritical water was investigated. The effects of the operation conditions were studied. The dechlorination of chlorobenzene obeyed pseudo-first-order kinetics models. In supercritical water, the rate constant increases more rapidly with temperature than those in subcritical water. The results showed that the rate-determining step of dechlorination in subcritical water is diffusion; whereas the rate-determining of dechlorination in supercritical water is chemical reaction. The reaction mechanism in subcritical water might involve with an ionic mechanism; whereas the reaction mechanism in supercritical water might involve with a homolytic reaction.     

Flow and drag force around a free surface piercing cylinder for environmental applications

This paper investigates flows around a free surface piercing cylinder with Froude number F > 0.5 and Reynolds number around Re = 50,000. The aim of this work is to gain a better understanding of the flow behaviour in environmental systems such as fishways. The advances are based upon experimental and numerical results. Several flow discharges and slopes are tested to obtain both subcritical and supercritical flows. The drag force exerted on the cylinder is measured with the help of a torque gauge while the velocity field is obtained using particle velocimetry. For the numerical part, two URANS turbulence models are tested, the k-\(\omega\) SST and the RNG k-\(\varepsilon\) models using the OpenFOAM software suite for subcritical cases, and then compared with the corresponding experimental results. With fishways applications in mind, the changes in drag coefficient \(C_d\) versus Froude number and water depth are studied and experimental correlations proposed. We conclude that the most suitable URANS turbulence model for reproducing this kind of flow is the k-\(\omega\) SST model.     

Experiments on gravity currents propagating on unbounded uniform slopes

Gravity currents propagating on \(12^\circ \), \(9^\circ \), \(6^\circ \), \(3^\circ \) unbounded uniform slopes and on an unbounded horizontal boundary are reported. Results show that there are two stages of the deceleration phase. In the early stage of the deceleration phase, the front location history follows \({(x_f+x_0)}^2 = {(K_I B)}^{1/2} (t+t_{I})\), where \((x_f+x_0)\) is the front location measured from the virtual origin, \(K_I\) an experimental constant, B the total buoyancy, t time and \(t_I\) the t-intercept. In the late stage of the deceleration phase for the gravity currents on \(12^\circ \), \(9^\circ \), \(6^\circ \) unbounded uniform slopes, the front location history follows \({(x_f+x_0)}^{8/3} = K_{VS} {{B}^{2/3} V^{2/9}_0 }{\nu }^{-1/3} ({t+t_{VS}})\), where \(K_{VS}\) is an experimental constant, \(V_0\) the initial volume of heavy fluid, \(\nu \) the kinematic viscosity and \(t_{VS}\) the t-intercept. In the late stage of the deceleration phase for the gravity currents on a \(3^\circ \) unbounded uniform slope and on an unbounded horizontal boundary, the front location history follows \({(x_f+x_0)}^{4} = K_{VM} {{B}^{2/3} V^{2/3}_0 }{\nu }^{-1/3} ({t+t_{VM}})\), where \(K_{VM}\) is an experimental constant and \(t_{VM}\) the t-intercept. Two qualitatively different flow morphologies are identified in the late stage of the deceleration phase. For the gravity currents on \(12^\circ \), \(9^\circ \), \(6^\circ \) unbounded uniform slopes, an ‘active’ head separates from the body of the current. For the gravity currents on a \(3^\circ \) unbounded uniform slope and on an unbounded horizontal boundary, the gravity currents maintain an integrated shape throughout the motion. Results indicate two possible routes to the final stage of the gravity currents on unbounded uniform slopes.

     

A general solution of Benjamin-type gravity current in a channel of non-rectangular cross-section

We consider the steady-state propagation of a high-Reynolds-number gravity current in a horizontal channel along the horizontal coordinate x. The bottom and top of the channel are at z =?0, H, and the cross-section is given by the quite general form ?f ₁(z) ≤?y ≤?f ₂(z) for 0 ≤?z ≤?H, where f _1,2 are piecewise continuous functions and f ₁ +?f ₂ >?0 for ${z \in(0,H)}$ . The interface of the current is horizontal, the (maximum) thickness is h, its density is ρ _c . The reduced gravity g′ =?|ρ _c /ρ _a ? 1|g (where ${- g\hat{z}}$ is the gravity acceleration and ρ _a the density of the ambient) drives the current with speed U into the stationary ambient fluid. We show that the dimensionless Fr =?U/(g′ h)^1/2, the rate of energy dissipation (scaled with the rate of pressure work), and the dimensionless head-loss Δ/h, can be expressed by compact formulas which involve three integrals over the cross-section areas of the current and ambient. By some standard manipulations these integrals are simplified into quite simple line-integrals of the shape-function of the channel, f(z) =?f ₁(z) +?f ₂(z), and of z f(z). This theory applies to Boussinesq and non-Boussinesq currents of “heavy” (bottom) and “light” (top) type. The classical results of Benjamin (J Fluid Mech 31:209–248, 1968) for a rectangular channel are fully recovered. We also recover the Fr results of Marino and Thomas (J Fluid Eng 131(5):051201, 2009) for channels of shape y =?±b z ^α (where b, α are positive constants); in addition, we consider the energy dissipation of these flows. The results provide insights into the effect of the cross-section shape on the behavior of the steady-state current, in quite general cases, for both heavy-into-light and light-into-heavy fluid systems, Boussinesq and non-Boussinesq. In particular, we show that a very deep current displays ${Fr = \sqrt{2}}$ , and is dissipative; the value of Fr and rate of dissipation (absolute value) decrease when the thickness of the current increases. However, in general, energy considerations restrict the thickness of the current by a clear-cut condition of the form h/H ≤?a _max ?< 1.