Stratified Ambient Fluid Research Articles

Dynamics of sediment-laden plumes in the ocean

We present a series of experiments to illustrate the dynamics of positively or negatively buoyant particle-laden plumes in a cross-flow, with relevance for the discharge of sediment into the ocean during deep-sea mining operations. In an unstratified ambient fluid, our experiments identify three different regimes, corresponding to (i) a dense particle-laden plume, host to relatively dense saline fluid, in which the particles separate from the descending plume as the flow speed falls below the particle settling speed; (ii) a dense particle-laden plume, host to buoyant fluid, in which the fluid gradually rises from the sinking plume of particles, to form a second rising plume of source fluid; and (iii) a buoyant particle-laden plume, host to buoyant fluid, which rises from the discharge pipe, and from which particles gradually sediment. Classical models of single-phase plumes describe the initial motion of the plumes in cases (i) and (iii), but as the flow speed falls below the particle fall speed, sedimentation leads to a change in the averaged buoyancy, and, hence, the plume speed. Our data also suggest that the sedimentation leads to a reduction in the rate of entrainment of ambient fluid, compared with the classical single-phase plumes. We also show that with a density stratified ambient fluid, the stratification may arrest the plume prior to significant particle sedimentation, and in this case, the plume tends to spread downstream at the level of neutral buoyancy where particle sedimentation proceeds. The bulk density of the residual plume fluid may then remain intermediate between the density of the upper and lower layer fluid, or may become less dense than the upper layer fluid, in which case, following sedimentation, the plume fluid rises through the upper layer. While the dynamics of deep-sea mining plumes in the ocean are more complex, for example, owing to background turbulence and mixing, the results of our new laboratory experiments highlight the range of flow processes which may influence the initial dispersion and sedimentation of particles in such plumes following release into the water, depending on the initial conditions, the ambient density and the particle fall speed. We also discuss the relevance of our work in the context of ash dispersal by volcanic plumes.

Fire Plume in a Sharply Stratified Ambient Fluid

The present work analyses theoretically and numerically fire plumes evolving in a two-layer stratified environment. The ambient fluid consists of a lower cold (heavy) layer and an upper warm (light) layer with a sharp interface. The fire plume is controlled by the fire heat release rate Q. Depending on the density interface location, the temperatures of the layers and on the fire source conditions, the fire plume can rise indefinitely or can be captured by the density stratification. To determine a criterion between these two situations, a theoretical model is proposed. With this model, it is found that the trapping/escaping criterion of the fire plume is determined in terms of a stratification parameter \(\varLambda > 0.88\). This stratification parameter \(\varLambda\) is a function of the fire heat release, the density interface location and the ambient stratification. The model also allows the plume height to be determined analytically. The comparison with the numerical data shows an excellent agreement. Finally, for practical purposes, we propose a simplified correlation to estimate the height of the trapped plume as a function of the interface height, the temperatures of the layers and the fire heat release rate.

Intrusions and solitons: Propagation and collision dynamics

Triggering and evolution of internal solitary waves (ISWs) generated by intrusive gravity currents (IGCs) propagating into a stratified ambient fluid is analyzed by laboratory experiments. After the release of a fluid of uniform density, intermediate with respect to the upper (lower-density) and lower (higher-density) layers in the channel, the IGC develops and flows downstream, intruding into the pycnocline. Near the IGC leading front, the compression of the upper layer generates ISWs: they gradually separate from the current that propagates slower. Shoaling downstream over a uniform sloping boundary, solitons break and partially reflect. We investigate the dynamics of the interaction between the reflected ISWs and the incoming IGC. During the engage, an increase in the ISW celerity occurs, leading the celerity of the reflected waves to be even larger than the incident wave. Our analysis shows how both ISWs and IGCs can significantly change their features as they experience a change of the density structure in the water column. This is expected to occur, for example, in stratified small-scale basins, where river plumes intrude the seasonal thermocline. The radial ISWs, originated by IGCs, can then be reflected by the adjacent bottom bathymetry, spreading against the intrusive current from which they are generated.

Dynamics of thin disk settling in two-layered fluid with density transition

Settling of solid particles in a stratified ambient fluid is a process widely encountered in geophysical flows. A set of experiments demonstrating the settling behaviour (the pattern of trajectory, variation of particle orientation, and settling velocity with depth) of thin disks descending through a nonlinear density transition was performed. The results showed complex hydrodynamic interactions between a particle and a liquid causing settling orientation instabilities and unsteady particle descent in low to moderate Reynolds number regime. Five phases of settling were observed: two phases with stable horizontal, one with stable vertical disk position, and two reorientation phases; moreover, two local minima of settling velocity were identified. It was demonstrated that thresholds for local minima and the first reorientation depend on the settling dynamics in an upper layer, stratification conditions, and disk geometry. The comparison of settling behaviour of thin disks varying in diameter revealed that settling dynamics is sensitive to particle geometry mainly in the upper part of density transition with a non-obvious result that the first minimum velocity is smaller for a disk with a larger diameter than for a disk with a smaller diameter. The analysis of settling trajectory showed that two reorientations are accompanied with a horizontal drift, which may be important in the context of interactions between particles settling in a group.

Bed shear stress and sediment entrainment potential for breaking of internal solitary waves

We investigate the interaction of strongly non linear internal solitary waves (ISWs) with boundaries having different slopes by means of high-resolution 3D large Eddy Simulations (LES). Releasing a volume of fresh water into a stratified ambient fluid, three different breaking mechanisms are produced: plunging, collapsing and surging breakers. The different shoaling dynamics affect the ISW evolution over the sloping boundary, inducing different effects on the bottom. In order to investigate the effects of the ISW breaking on the inclined surface, we calculate the bed shear stress and estimate the local flux of sediments entrained from the bed. We analyze the relationship between the breaking criteria and the related effects on the sloping surface. Although plunging breakers are expected to induce significant effects within the fluid, causing larger amount of mixing and fluid entrainment, the effects on the bottom are totally opposite. The collapsing breaker mechanism, indeed, generates boundary layer separation, which in turn induces whirling instabilities. Results show that the ISW interaction with the inclined surface occurs in its close proximity for collapsing breaker mechanism, which explains why the largest bed shear stresses and sediment re-suspension are predicted in the simulation where a collapsing breaker mechanism is observed.

Natural convection boundary-layer flow on an evenly heated vertical plate with time-varying heating flux in a stratified Pr < 1 fluid

The transient natural convection boundary-layer flow adjacent to a vertical plate heated with a time-dependent heat flux in an initially linearly stratified ambient fluid with Prandtl number (Pr) smaller than one was studied. Scaling analysis was used to obtain the scaling relations for the main parameters characterizing the flow behavior in terms of Pr, s, and fn at the four development stages, i.e., the startup, transitional, steady, and the decay stage, where s is the dimensionless temperature stratification parameter which represents the relative extent of the background stratification with respect to the temperature gradient across the plate and fn is the dimensionless natural frequency of the time-dependent heat flux applied to the plate. These parameters include the plate temperature, maximum vertical velocity, thermal and viscous boundary-layer thicknesses, and the corresponding time scales. The direct numerical simulation results over and show that these scaling relations provide a good representation of the flow behavior.

Plumes in a rotating two-layer stratified fluid

Laboratory experiments are conducted to examine a descending plume in a rotating two-layer stratified ambient fluid such that the plume at the interface has moderate to large Rossby number. While the source fluid is more dense than the lower layer, the experiments are designed so that the mean density of the plume fluid impinging upon the interface is less than the lower layer density, as represented by a buoyancy parameter, $$\varLambda $$, being less than unity. In such cases, the discharged plume fluid spreads radially at the interface in the form of an intrusive gravity current at early times. At later times, this intrusion evolves to form an anticyclonic lens due to the influence of the Coriolis force. The measured radial position of the intrusion front, R(t), follows different power law relationships at early and late times during the spread of the intrusion: at early times when rotation does not play a significant role the power law exponent lies between 0.5 and 1.1; at late times when the intrusion acts as a rotationally influenced expanding lens the power law exponent ranges between 0.15 and 0.5, with generally smaller values for larger $$\varLambda $$. The plume fluid reaching the interface progressively increases in density due to re-entraining relatively dense fluid as the plume descends within the thickening lens. Consequently, the plume eventually penetrates through the interface and descends to the bottom of the tank. Faster rotation makes the lens thicker and hence increases the volume of the re-entrained lens-fluid, which decreases the time for the onset of penetration. The penetration time normalized by the rotation rate is found to hold a simple power law relationship with $$\varLambda $$.

Turbidity currents propagating down a slope into a stratified saline ambient fluid

Results are presented from highly resolved three-dimensional simulations of turbidity currents down a shallow slope into a stratified saline ambient, for various stratifications and particle settling velocity configurations. The front velocity is computed and the results are compared directly to experimental data by Snow and Sutherland (J Fluid Mech 755:251–273, 2014) and to an existing scaling law for gravity currents in stratified environments but on flat bottoms, with close agreement. The entrainment velocity of the ambient fluid into the current is computed from the DNS data and shows strong space time variability. The intrusion of the current into the ambient fluid is computed and compared to the experimental results of Snow and Sutherland (2014), an existing scaling law proposed by those authors, and a new model with increasing levels of complexity. The numerical results highlight the sensitivity of scaling tools to the choice of entrainment coefficient and their limitations in low entrainment but high settling rate scenarios. The new, simpler scaling relation for the intrusion depth is shown to be more robust and predictive in such cases. The energy budget of the flow is analysed in order to explain the governing processes in terms of energy transfers, with a focus on energy losses and potential to kinetic energy transfers. Particular attention is given to the Stokes losses with varying settling velocities, which validates the prediction of the concentration in the intrusion depth model introduced.

On the propagation of particulate gravity currents in circular and semi-circular channels partially filled with homogeneous or stratified ambient fluid

We present a combined theoretical-experimental investigation of particle-driven gravity currents advancing in circular cross section channels in the high-Reynolds number Boussinesq regime; the ambient fluid is either homogeneous or linearly stratified. The predictions of the theoretical model are compared with experiments performed in lock–release configuration; experiments were performed with conditions of both full-depth and partial-depth locks. Two different particles were used for the turbidity current, and the full range 0≤S≤1 of the stratification parameter was explored (S = 0 corresponds to the homogeneous case and S = 1 when the density of the ambient fluid and of the current are equal at the bottom). In addition, a few saline gravity currents were tested for comparison. The results show good agreement for the full-depth configuration, with the initial depth of the current in the lock being equal to the depth of the ambient fluid. The agreement is less good for the partial-depth cases and is improved by the introduction of a simple adjustment coefficient for the Froude number at the front of the current and accounting for dissipation. The general parameter dependencies and behaviour of the current, although influenced by many factors (e.g., mixing and internal waves), are well predicted by the relatively simple model.

Convection from a line-source into a two-layer stratified ambient fluid

We experimentally investigate the behaviour of a line-source plume falling through a finite two-layer stratified ambient where the depth of the upper ambient layer increases in time. Laboratory observations suggest one of two possible flow regimes depending on the value of $\unicode[STIX]{x1D706}$, which represents the relative loss of buoyancy experienced by the plume upon crossing the ambient interface. When $\unicode[STIX]{x1D706}&gt;1$, a classical filling-box-type flow is realized and plume fluid always reaches the bottom boundary. By contrast, when $\unicode[STIX]{x1D706}&lt;1$, we observe a transition by which an increasing fraction of plume fluid discharges along the interface. The approximate start time, $t_{v}$, and end time, $t_{t}$, of the transition process are well determined by $\unicode[STIX]{x1D706}$. After transition, the ambient density evolves to form a three-layer fluid with an intermediate layer that grows in time. Measured densities of the intermediate layer in experiments with $\unicode[STIX]{x1D706}&lt;1$ are well predicted using plume theory. We further characterize the horizontal speed of the intrusion that forms along the ambient interface, the mass of solute present in the intermediate layer at time $t_{t}$ and the rate of descent of the intrusion level for $t&gt;t_{t}$. The significance of our findings is discussed in the context of the ventilation of natural and hybrid ventilated buildings and of effluent discharge through marine outfall diffusers.

Gravity currents in a linearly stratified ambient fluid created by lock release and influx in semi-circular and rectangular channels

We present an experimental investigation, supported by a theoretical model, of the motion of lock-release, constant inflow, and time varying inflow gravity currents (GCs) into a linearly stratified ambient fluid at large Reynolds number. The aim is the experimental validation of a simple model able to predict the slumping phase front speed and the asymptotic self-similar front speed for rectangular and circular cross section channels. The first investigated system is of Boussinesq type with the dense current (salt water dyed with aniline) released in a circular channel of 19 cm diameter and 400 cm long (605 cm in the inflow experiments), half-filled of linearly stratified ambient fluid (salt water with varying salt concentration). The second system has the same components but with a channel of rectangular cross section of 14 cm width, 11 cm ambient fluid depth, and 504 cm length. The density stratification of the ambient fluid was obtained with a computer controlled set of pumps and of mixing tanks. For the experiments with inflow, a multi-pipes drainage system was set at the opposite end with respect to the inflow section, computer controlled to avoid the selective withdrawal. The numerous experiments (28 for circular cross section, lock release; 26 for circular and 14 for rectangular cross section, constant inflow (fluid volume ∝tα, with α = 1); 6 for circular cross section, linearly increasing inflow (α = 2)), with several combination of the stratification parameter (0 &amp;lt; S &amp;lt; 1) confirm the theory within ≈30% (≈40% for a single series of experiments), which is considered a good result in view of the various underlying simplifications and approximations. The results on the front speed of the GCs are discussed in the presence of the internal waves, which have a celerity given by a theoretical and experimentally tested model for the rectangular but not for the circular cross section. The theoretical analysis of internal waves in circular cross sections has been extended and experimentally validated.

Sustained axisymmetric intrusions in a rotating system

We analyse the effects of rotation on the propagation of an axisymmetric intrusion through a linearly stratified ambient fluid, arising from a sustained source at the level of neutral buoyancy. This scenario occurs during the horizontal spreading of a volcanic ash cloud, which occurs after the plume has risen to its neutral buoyancy level. A simple and well-accepted approximation for the flow at late times is that inertial effects are negligible. This leads to a lens-shaped intrusion governed by a balance between Coriolis accelerations and horizontal pressure gradients, with the radius scaling with time as rN∼t1/3. However, we show using a shallow-layer model that inertial forces cannot be neglected until significant times after the beginning of the influx. These inertial forces result in the flow forming two distinct domains, separated by a moving hydraulic jump: an outer ‘head’ region in which the radial velocity and thickness vary with time, and a thinner ‘tail’ region in which the flow is steady. Initially, the flow expands rapidly and this tail region occupies most of the flow. After about one half-revolution of the system, Coriolis accelerations halt the advance of the front, and the hydraulic jump separating the two regions propagates back towards the source of the intrusion. Only after approximately one and a half rotations of the system does inertia become insignificant and the Coriolis lens solution, with rN∼t1/3, become established. Importantly, this means that neither inertia nor Coriolis accelerations can be neglected when modelling intrusions from volcanic eruptions. We exploit the two-region flow structure to construct a new hybrid model, comprising just two ordinary differential equations for the intrusion radius and location of the hydraulic jump. This hybrid model is much simpler than the shallow-layer model, but nonetheless accurately predicts flow properties such as the intrusion radius at all stages of motion, without requiring fitted or adjustable parameters.

Intrusion-generated waves in a linearly stratified fluid

AbstractWe present an experimental and numerical study of the upstream internal wavefield in a channel generated by constant density intrusions propagating into a linearly stratified ambient fluid during the initial phase of translation. Using synthetic schlieren imaging and two-dimensional direct numerical simulations, we quantify this wave motion within the ambient stratified fluid ahead of the advancing front. We show that the height of the neutral buoyancy surface in the ambient fluid determines the vertical modal response with the predominant waves being mode 2 for intrusions near the mid-depth of the channel and mode 1 waves being produced by intrusions nearer the top or bottom of the domain. All higher vertical modes travel slower than the intrusion and so do not appear upstream ahead of the intrusion front. We find the energy flux into this upstream wavefield to be approximately constant, and to be between 10 and 30 % of the rate of available potential energy transfer into the flow.

Radial intrusions from turbulent plumes in uniform stratification

Laboratory experiments investigate the radial spread of an intrusion created by a turbulent forced plume in uniformly stratified ambient fluid. The flow evolution is determined as it depends upon the ambient buoyancy frequency, N, and the source momentum and buoyancy fluxes, M0 and F0, respectively. The plume reaches its maximum vertical extent, Zm, collapses back upon itself as a fountain and then spreads radially outwards at its neutral buoyancy depth, Zs, where the intrusion has the same density as the ambient. Through theory and experiments we determine that Zs = f(σ)Hp, in which Hp = M03/4F0−1/2, σ = (M0N/F0)2, and f(σ) ∝ σ−3/8 for σ ≲ 50 and f(σ) ∝ σ−1/4 for σ ≳ 50. In the inertia-buoyancy regime the intrusion front advances in time approximately as Rf ∝ t3/4, consistent with models assuming a constant buoyancy flux into the intrusion. Where the intrusion first forms, at radius R1, its thickness h1 is approximately constant in time. The thickness of the intrusion as a whole, h(r, t), adopts a self-similar shape of the form h/h1 ≃ [(Rf − r)/(Rf − R1)]p, with p ≃ 0.55 ± 0.03. The comparison of these results to large volcanic plumes penetrating into and spreading in the stratosphere is discussed.

Naturally bounded plumes

AbstractThis paper investigates theoretically the vertical evolution of a turbulent plume into a linearly stratified ambient fluid, by regarding it as composed of two distinct regions. In the first region, called the positive buoyant region, the plume buoyancy and the plume momentum act in the same upward direction, whereas in the second region, called the negative buoyant region, they act in opposite directions. In a first step, analytical expressions for the plume variables at the transition height (i.e. between the two regions) are obtained from one-dimensional conservation equations, using the plume entrainment model and under the Boussinesq approximation. In a second step, these variables are used in order to determine analytically the buoyancy and volume fluxes as well as the density deficit of the plume at its top. In this investigation, the transition height (denoted ${z}_{t} $) and the total plume height (denoted ${z}_{p} $) are obtained in the form of two integrals. These integrals are evaluated asymptotically in three different cases associated with particular flow regimes. Finally, the limit of the Boussinesq assumption for such flows is discussed.

The flow of an axisymmetric stratified gravity current into a stratified ambient in a rotating system

The effect of rotation on the behaviour of an inviscid axisymmetric stratified gravity current released from a lock and propagating into a stratified ambient fluid is considered. It is assumed that both the ambient and the current are linearly stratified. We develop a one-layer shallow-water model and show that the flow is governed by four dimensionless parameters which represent the stratifications in the current and ambient, the ratio of Coriolis to inertia effects, and the height ratio of the ambient to the lock. The solution, obtained by a finite-difference method, provides the shape of the interface, radial and azimuthal speeds, and radius of propagation, as functions of time. The propagation reaches a maximum radius, dependent on the input parameters; the increase of the internal stratification reduces the radius. The system admits steady-state lens structures for which simple approximate solutions are also presented. In general, the internal stratification of the current reduces the velocity of propagation, and enhances the Coriolis effects in a rotating system.

A general description of a gravity current front propagating in a two-layer stratified fluid

AbstractThe behaviour of a gravity current propagating into a two-layer stratified ambient fluid is described in detail. A comprehensive description is given of the different flow regimes, with particular emphasis on the front condition linking the thickness of the gravity current to its speed of propagation and the transfer of energy to upstream disturbances in the form of internal bores and nonlinear solitary waves. Hydraulic theory analogous to that of two-layer flow over topography (Baines, J. Fluid Mech., vol. 146, 1984, pp. 127–167) is extended to the gravity current problem to classify frontal behaviour into the following regimes: Type I, subcritical currents; Type II, currents that generate upstream undular bores; Type III, currents that generate an upstream monotonic bore connected by a rarefaction; Type IV, supercritical fronts with a large-amplitude trapped solitary-wave-like disturbance; and Type V, supercritical gravity currents. Over 200 two-dimensional Boussinesq–Euler simulations spanning a range of gravity current properties demonstrate good agreement, for both the behavioural regime and the front condition ${U}_{o} (h)$, with hydraulic theory that extends original work by Rottman &amp; Simpson (Q. J. R. Meteorol. Soc., vol. 115, 1989, pp. 941–963) to arbitrary ambient layer thickness, and uses an improved closure for the upstream bore that correctly predicts the behaviour in the limit of large bore amplitude. In addition, the energy balance is analysed, and it is shown that the energy transfer from the gravity current to upstream disturbances is significant, and consistent with the hydraulic theory. The results demonstrate a clear connection to the problem of upstream resonance in two-layer flow over topography, and have significant implications for interpreting field observations of nonlinear internal waves generated by atmospheric density currents and coastal river plumes.

Modeling glacial meltwater plume dynamics and sedimentation in high-latitude fjords

[1] A model, SedPlume, has been developed to simulate marine sediment deposited by glacial meltwater plumes emerging from tidewater glaciers. Turbid meltwater emerging from beneath a glacier into a fjord rises as a buoyant forced plume due to density contrasts with the ambient fjord water. SedPlume assumes that meltwater discharge flows at a constant rate for long enough periods that the plume reaches a steady state. Entrainment of ambient fluid into the turbulent plume is assumed to occur at a rate proportional to the local velocity of the plume. Plume motion is considered in two dimensions: one horizontal dimension (perpendicular to the glacier front) and the vertical dimension. An integral model is formulated for the conservation equations of volume, momentum, buoyancy, and sediment flux along the path of a turbulent plume injected into stably stratified ambient fluid. Sedimentation occurs from the plume when the radial component of the sediment fall velocity exceeds the entrainment velocity. When the plume reaches the surface, it is treated as a radially spreading surface gravity current, for which exact solutions exist for the sediment deposition rate. Flocculation of silt and clay particles is modeled using empirical measurements of particle settling velocities in fjords to adjust the settling velocity of fine-grained sediments. SedPlume has been applied to McBride Inlet, Alaska, a temperate glaciated fjord where the majority of sedimentation originates from meltwater sources. SedPlume produces rates and patterns of sedimentation in good agreement with observations, with calculated peak ice-proximal annual sedimentation rates of approximately 22 m yr−1.

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 ρc 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 ρo to ρb 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).

On axisymmetric intrusive gravity currents: The approach to self-similarity solutions of the shallow-water equations in a stratified ambient

The axisymmetric intrusion of a fixed volume of fluid, which is released from rest and then propagates radially at the neutral buoyancy level in a deep linearly stratified ambient fluid is investigated. The SW equations representing the high-Reynolds number motion are used. For the long-time motion an analytical similarity solution indicates propagation with t 1 / 3 , but the shape is peculiar: the intrusion propagates like a ring and the inner domain contains a thin tail of clear ambient fluid. To avoid accumulation of numerical errors the problem was reformulated in terms of new variables and solved by finite-difference scheme. It is shown that the initial-value problem tends to the similarity prediction. Comparison with the non-stratified case is presented. It was found that for the non-stratified case there is a similar “tail-ring” stage of propagation, however this stage is only a transient to a different self-similar shape.