Hydrostatic vs. non-hydrostatic modelling of density currents developing two dimensionally on steep and mild slopes

Turbidity currents in the ocean and lakes are driven by suspended sediment. The vertical profiles of velocity and excess density are shaped by the interaction between the current and the bed as well as between the current and the ambient water above. We present results of a set of 74 experiments that focus on the characteristics of velocity and fractional excess density profiles of saline density and turbidity currents flowing over a mobile bed. The gravity flows include saline density flows, hybrid saline/turbidity currents and a pure turbidity current. The use of dissolved salt is a surrogate for suspended mud that is so fine that it does not settle out readily. Thus, all the currents can be considered to be model turbidity currents. The data cover both Froude-subcritical and Froude-supercritical regimes. Depending on flow conditions, the bed remains flat or bed forms develop over time, which in turn affect vertical profiles. For plane bed experiments, subcritical flow profiles have velocity peaks located higher up in the flow, and display a sharper interface at the top of the current, than their supercritical counterparts. The latter have excess density profiles that decline exponentially upward from the bed, whereas subcritical flows show profiles with a region near the bed where excess density varies little. Wherever bed forms are present, they have a significant effect on the profiles. Especially for Froude-supercritical flow, bed forms push the location of peak velocity upward, and render the near-bed fractional excess density more uniform. In the case of subcritical flow, bed forms do not significantly affect fractional excess density profiles; velocity profiles are pushed farther upward from the bed than in the case of a plane bed, but to a lesser extent than for supercritical bed forms. Overall, the relative position of the velocity peak above the bed shows a dependence upon flow regime, being lowered for increasing Froude number Fd. Gradient Richardson numbers Rig in the near-bed region increase with increasing Fd, but are lower than the critical value of 0.25, indicating that near-bed turbulent structures are not notably suppressed. At the top interface, values of Rig are above the critical value for subcritical and mildly supercritical Fd, effectively damping turbulence. However as Fd increases, Rig goes below the critical value. Shape factors calculated from the profiles for use in the depth-averaged equation of motion are evaluated for different flow and bed conditions. Normalized experimental profiles for supercritical currents scale up well with observations of field-scale turbidity currents in the Monterey Canyon, and the range of average bed slopes and Froude numbers also compare favorably with estimated field-scale flow conditions for the Amazon canyon and fan. This suggests that the experimental results can be used to interpret the kinds of flows that are responsible for the shaping of major submarine canyon-fan systems.

When two liquid bodies with different density come in contact in non-equilibrium conditions, a flow is caused, known as gravity or density current. In the environment, as well as in the industrial framework, this kind of flow is very common and the scientifictechnical interest of the investigation on it is very high. The paper of Huppert (2006) and the book of Ungarish (2009b) give excellent reviews on the state of the art of the topic, while a huge collection of artificial, as well as natural, gravity currents and a qualitative description of their key features is given in the book of Simpson (1997). The investigation on gravity currents dates back to several decades ago (first important works are those of Von Karman, 1940; Yih, 1947; Prandtl, 1952 and Keulegan, 1957), nevertheless many aspects still need a better understanding. These aspects should be investigated in order to widen the knowledge on the considered phenomenon and are generally related to the geometry of the fluid domain and the use of particular fluids, like e.g. mixtures of liquid and sediments. Early studies on gravity currents were based on analytical and experimental methods and were concerned with 2D gravity currents: i.e. gravity currents whose description can be made in a vertical x-z plane. The seminal work of Benjamin (1968) formulates a fundamental theory, based on the perfect-fluid hypothesis and simple extensions of it (like the classical theory of hydraulic jumps), which gives a relationship between the thickness of the gravity current and the velocity of the front. The Benjamin’s theory is a milestone and analytical investigations on gravity currents, even the most recent (Shin et al., 2004; Lowe et al., 2005; Ungarish & Zemach, 2005; Ungarish, 2008; Ungarish, 2009) cannot disregard it. Laboratory gravity currents can be realized in very different ways (Simpson, 1997), depending on which features have to be investigated. The basic experimental setup, which permits to investigate the propagation’s features of the gravity current, is the lock exchange release experiment. This experiment consists in leaving two liquid bodies of different density in non-equilibrium condition, typically removing a sliding gate which originally separated them. The consequence is a flow of heavier liquid (the gravity current) under the

Deepwater pipelines and flowlines that are routed across areas of steep slopes have the potential for seabed instability. They are at risk of being impacted by geohazards in the form of mass gravity flows (e.g. mud flows, debris flows etc.) and turbidity current flows. This paper discusses the impact of accidental environmental/geohazard loads on oil and gas pipelines routed across an area of seabed with a steep slope. The vertical gradient along the route ranges between 5° and 15° with a maximum local gradient in excess of 36°. The main design issues that needed to be resolved were the pipelines' global stability while resting on the steep seabed slope and their capacity to resist environmental loads (geohazards). Due to the nature of the problem that encompasses several unknown factors, conventional pipeline/flowline engineering was not sufficient to adequately define the response of the overall system. To address this issue, a series of fit-for-purpose full 3-Dimensional finite element models were completed to assess the global stability of the system and subsequently assess the impact of geohazard loads. With the help of extensive 3D finite element modeling, the impact of geohazard loads on the pipelines was assessed and the feasibility of the system was qualified through application of Accidental Limit State (ALS) design. Introduction Routing a pipeline or flowline across a steep slope in deepwater is not considered an uncommon design practice. There are several existing pipeline systems routed across aggressive seabed features. Some of these features include steep seabed gradients, areas of seafloor faults, canyons etc. However, there are some locations where an additional design challenge is introduced due to the potential for seabed slope instability. If locations of seabed slope instability are close to the pipeline, it adds further complexity due to the risk of the geohazard potentially inducing large displacements of the pipeline system. Additionally, uncertainties in the historical and geotechnical/geophysical data also contribute to the risk associated with routing a pipeline across a steep slope. This paper addresses design issues related to global axial stability of oil and gas pipelines that is routed across an area of steep seabed slope. In addition, the paper discusses the structural integrity of these pipelines when subjected to geohazard events such as debris flows, turbidity currents and subsequent condition of the pipelines following these events. A comparison between the results from FEA simulations and the field data obtained during as-laid and post-hydrotest survey has also been discussed. Pipeline Routes The oil and gas pipelines (D/t ~20, Steel Grade = X65) considered for this case study are located in the northern Gulf of Mexico and traverse the Outer Continental Shelf, an area of steep seabed slope and Upper Continental Rise (see Figure 1). As the routes ascend the slope the seabed shoals steeply with gradients generally ranging between 5° and 15°, but in some areas it is in excess of 20° to 36°. Figure 2 shows the histogram of the seabed vertical gradients along the sloped section of the pipeline route. The average gradient within the slope section lies between 6° and 8°. The seabed slopes close to the pipeline routes were investigated to identify potential areas of instability that could impact the feasibility of the pipeline system. These areas and soil provinces were then used as input parameters for debris flow/turbidity current simulation modeling. The results of the debris flow/turbidity current modeling were subsequently used to assess their impact on the pipeline stability and integrity.

Water stratification commonly exists in nature, such as thermocline in lakes and oceans and halocline in estuaries and oceans (He et al. 2017). Turbidity currents in estuary often encounter stratified sea water, which may significantly influence their propagation and deposition. This study presents high-resolution numerical simulations of lock-exchange gravity and turbidity currents in linear stratifications on a flat bed. Laboratory experiments are conducted to validate the numerical model and good agreements between numerical results and measurements are found. The evolution process, front velocity, internal wave, and entrainment ratio are analyzed based on the numerical results. For a gravity current in a strong stratification, its front velocity can be maintained as a near constant state for a long time after an initial acceleration period because of interactions between the current and internal waves. However, sedimentation of suspended particles due to the damping effect of ambient stratification on turbulence makes a turbidity current quickly lose its structure so the maintaining effect of the internal waves on its front velocity is quite weak. During the evolution process of a turbidity current, the ambient stratification is found to damp the turbulent structures, and front velocity. Stratification can also decrease the entrainment ratios between a gravity current and ambient water after the initial period, but it has an insignificant influence on the entrainment ratios of a turbidity current. This study provides a better understanding of gravity and turbidity currents in estuary stratifications.

Breaking nonlinear internal waves (NLIWs) of depression on boundary slopes drives mixing in the coastal ocean. Of the different breaker types, fission is most commonly observed on mild slopes of continental margins. However, fission on mild slopes has rarely been investigated in the laboratory owing to limitations on flume length. In the present work, a train of NLIWs of depression is generated in an 18.2 m wave flume and shoaled upon a mild uniform slope. During fission, each NLIW of depression scatters into one or two NLIWs of elevation, which transforms into a bolus at the bolus birth point, where shear instability occurs through the pycnocline. The bolus propagates upslope, decreasing in size until it degenerates by shear and lobe-cleft instability, while losing volume to a return flow along the bed. The location of the bolus birth point, bolus propagation length scale, initial size and the number of boluses from each incident wave are parameterized from the wave half-width and the wave Froude number associated with the incident NLIW. These are compared with the characteristics of boluses generated by other breaking mechanisms on steeper slopes. Some bolus characteristics (height to length ratio, change in size and velocity field) are similar for boluses generated by fission, collapsing sinusoidal waves and internal solitary waves of elevation; however, the number of boluses, their birth point and initial height differ. The boluses formed by fission have more initial energy and no reflection. Further research is required to better quantify bolus-driven mixing on continental margins.

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 Co/Ci by the gravity current varies as a function of the particle diameter ds. We classify the transport pattern by PDGC into three regimes: (1) a suspended regime (ds 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 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.

Inside pyroclastic density currents – uncovering the enigmatic flow structure and transport behaviour in large-scale experiments

The propagation dynamics of the turbid density currents of the Imha Reservoir is investigated using the FLOW-3D computational fluid dynamics code. The renormalization group (RNG) κ-e turbulence scheme in a Reynolds-averaged Navier–Stokes (RANS) framework was applied for the field investigation. A new particle dynamics algorithm was developed and coupled with the FLOW-3D model to simulate the settling of sediment particles, and the model was tested with field measurements during Typhoon Ewiniar. The coupled model correctly predicted the spatial and temporal evolution of intrusive density currents with plunging flows at a depth of 20 m and interflows propagating downstream at about 0.2 m/s. The simulation results with sediment settling demonstrated that the ratio of transported to initial sediment concentration (C0/Ci) varies as a function of particle size ds and sediment concentration. Density currents can be classified into three regimes: (1) a suspended regime (ds<10 μm), where interflows will ...

Chapter 6 Behavior of the turbidity plume relating to a density current in a tidal river

Direct Numerical Simulation of Three-dimensional Gravity Current on a Uniform Slope

Experimental study of long wave generation on sloping bottoms

Density currents occur when fluid of one density propagates along a horizontal boundary into fluid of a different density. In dam reservoirs, density currents are the main transport mechanism for the incoming sediments and they play an important role in redistribution of existing sediments. This paper aims to investigate velocity structure in the body of density currents. To this end, laboratory experiments were performed on density currents having various initial conditions and bottom slopes. Then, vertical velocity profiles were recorded in the body of density currents. The velocity structure of the currents was investigated by fitting equations to the wall and jet regions of the measured profiles, and the constants of the equations were yielded with R2 more than 0.80. Temporal and spatial evolution of density currents were also analysed to study the dynamics of the frontal region of the currents. It was observed that the currents having more bottom slope travel at a further distance. It was also found that 400% increase in the initial concentration of the currents can increase their frontal velocity up to 97%.

We report on the dynamics of circular finite-release Boussinesq gravity currents on a uniform slope. The study comprises a series of highly resolved direct numerical simulations for a range of slope angles between $5^{\circ }$ and $20^{\circ }$. The simulations were fixed at Reynolds number $Re=5000$ for all slopes considered. The temporal evolution of the front is compared to available experimental data. One of the interesting aspects of this study is the detection of a converging flow towards the centre of the gravity current. This converging flow is a result of the finite volume of the release coupled with the presence of a sloping boundary, which results in a second acceleration phase in the front velocity of the current. The details of the dynamics of this second acceleration and the redistribution of material in the current leading to its development will be discussed. These finite-release currents are invariably dominated by the head where most of the mixing and ambient entrainment occurs. We propose a simple method for defining the head of the current from which we extract various properties including the front Froude number and entrainment coefficient. The Froude number is seen to increase with steeper slopes, whereas the entrainment coefficient is observed to be weakly dependent on the bottom slope.

Particle-driven gravity currents, as exemplified by either turbidity currents in the ocean or ignimbrite flows in the atmosphere, are buoyancy-driven flows due to a suspension of dense particles in an ambient fluid. We present a theoretical study on the dynamics of and deposition from a turbulent current flowing down a uniform planar slope from a constant-flux point source of particle-laden fluid. The flow is modelled using the shallow-water equations, including the effects of bottom friction and entrainment of ambient fluid, coupled to an equation for the transport and settling of the particles. Two flow regimes are identified. Near the source and for mild slopes, the flow is dominated by a balance between buoyancy and bottom friction. Further downstream and for steeper slopes, entrainment also affects the behaviour of the current. Similarity solutions are also developed for the simple cases of homogeneous gravity currents with no settling of particles in the friction-dominated and entrainment-dominated regimes. Estimates of the width and length of the deposit from a monodisperse particle-driven gravity current with settling are derived from scaling analysis for each regime, and the contours of the depositional patterns are determined from numerical solution of the governing equations.

Turbidity currents act to sculpt the submarine environment through sediment erosion and deposition. A sufficiently swift turbidity current on a steep slope can be expected to be supercritical in the sense of the bulk Richardson number; a sufficiently tranquil turbidity current on a mild slope can be expected to be subcritical. The transition from supercritical to subcritical flow is accomplished through an internal hydraulic jump. Consider a steady turbidity current flowing from a steep canyon onto a milder fan, and then exiting the fan down another steep canyon. The flow might be expected to undergo a hydraulic jump to subcritical flow near the canyon–fan break, and then accelerate again to critical flow at the fan–canyon break downstream. The problem of locating the hydraulic jump is here termed the ‘jump problem’. Experiments with fine-grained sediment have confirmed the expected behaviour outlined above. Similar experiments with coarse-grained sediment suggest that if the deposition rate is sufficiently high, this ‘jump problem’ may have no solution with the expected behaviour, and in particular no solution with a hydraulic jump. In such cases, the flow either transits the length of the low-slope fan as a supercritical flow and shoots off the fan–canyon break without responding to it, or dissipates as a supercritical flow before exiting the fan. The analysis presented below confirms the existence of a range associated with rapid sediment deposition where no solution to the ‘jump problem’ can be found. The criterion for this range is stated in terms of an order-one dimensionless parameter involving the fall velocity of the sediment. The criterion is tested and confirmed against the experiments mentioned above. A sample field application is presented.