Andrew J. Hogg

The effects of hydraulic resistance on dam-break and other shallow inertial flows

The effects of resistive forces on unsteady shallow flows over rigid horizontal boundaries are investigated theoretically. The dynamics of this type of motion are driven by the streamwise gradient of the hydrostatic pressure, which balances the inertia of the fluid and the basal resistance. Drag forces are often negligible provided the fluid is sufficiently deep. However, close to the front of some flows where the depth of the moving layer becomes small, it is possible for drag to substantially influence the motion. Here we consider three aspects of unsteady shallow flows. First we consider a regime in which the drag, inertia and buoyancy (pressure gradient) are formally of the same magnitude throughout the entire current and we construct a new class of similarity solutions for the motion. This reveals the range of solution types possible, which includes those with continuous profiles, those with discontinuous profiles and weak shocks and those which are continuous but have critical points of transition at which the gradients may be discontinuous. Next we analyse one-dimensional dam-break flow and calculate how drag slows the motion. There is always a region close to the front in which drag forces are not negligible. We employ matched asymptotic expansions to combine the flow at the front with the flow in the bulk of the domain and derive theoretical predictions that are compared to laboratory measurements of dam-break flows. Finally we investigate a modified form of dam-break flow in which the vertical profile of the horizontal velocity field is no longer assumed to be uniform. It is found that in the absence of drag it is no longer possible to find a kinematically consistent front of the fluid motion. However the inclusion of drag forces within the region close to the front resolves this difficulty. We calculate velocity and depth profiles within the drag-affected region, and obtain the leading-order expression for the rate at which the fluid propagates when the magnitude of the drag force is modelled using Chae#169;zy, Newtonian and power-law fluid closures; this compares well with experimental data and provide new insights into dam-break flows.

Morphological modelling of intertidal mudflats: the role of cross-shore tidal currents

Abstract We describe a mathematical model of the sediment transport resulting from cross-shore tidal currents on an intertidal mudflat. The model is integrated numerically to determine the long-term (“equilibrium”) behaviour of the morphodynamic system, and to investigate how the morphology of the flats depends on tidal range and sediment supply. Under a sinusoidal tide, the equilibrium flat is approximately linear below mean sea level (MSL) and convex above MSL, and advances seawards over long timescales. The cross-shore width of the flat is independent of tidal range, and increases with increasing sediment supply. Tidal asymmetry (flood- or ebb-dominance) leads to a steeper flat, and ebb-dominance can cause the flat to retreat landwards in the long term. Under a spring–neap tidal cycle, the shape of the equilibrium profile is very similar to that for a fixed tidal range, but the rate of accretion is significantly reduced.

Cross-shore sediment transport and the equilibrium morphology of mudflats under tidal currents

We describe a mathematical model of suspended sediment transport under cross-shore tidal currents on an intertidal mudflat. We employ a Lagrangian formulation to obtain periodic solutions for the sediment transport over idealized bathymetries and use these to investigate the process of settling lag. In deep water away from the shoreline the concentration of suspended sediment tends to a constant value, and we may invert the relation between bathymetry and offshore concentration to estimate how the gradient of a flat in a state of equilibrium varies with the sediment properties and supply. These analytical estimates are compared successfully with the numerical experiments of previous studies. We discuss the robustness of our modeling framework and demonstrate its application to different descriptions of sediment transport and tidal regimes.

On the slow draining of a gravity current moving through a layered permeable medium

We examine the gravitational dispersal of dense fluid through a horizontal permeable layer, which is separated from a second underlying layer by a narrow band of much lower permeability. We derive a series of analytical solutions which describe the propagation of the fluid through the upper layer and the draining of the fluid into the underlying region. The model predicts that the current initially spreads according to a self-similar solution. However, as the drainage becomes established, the spreading slows, and in fact the fluid only spreads a finite distance before it has fully drained into the underlying layer. We examine the sensitivity of the results to the initial conditions through numerical solution of the governing equations. We find that for sources of sufficiently large initial aspect ratio (defined as the ratio of height to length), the solution converges rapidly to the initially self-similar regime. For longer and shallower initial source conditions, this convergence does not occur, but we derive estimates for the run-out length of the current, which compare favourably with our numerical data. We also present some preliminary laboratory experiments, which support the model.

Effects of external flow on compositional and particle gravity currents

The propagation at high Reynolds number of a heavy, two-dimensional gravity current of given initial volume at the base of a uniform flow is considered. An experimental setup is described for which a known volume of fluid is rapidly introduced halfway down a 9 m channel in which there is a uniform flow of water. The density excess of the released fluid is produced by either dissolving salt or suspending particles in water. The upstream and downstream propagation of the current was measured for dierent initial salt concentrations, particle sizes and concentrations. A simple box model for the motion of and deposit from the gravity current is constructed. The analytical results obtained compare well with our numerical solutions of one-layer and two-layer models incorporating the appropriate shallow-water equations. Both sets of results are in very good agreement with the experimental data.

Oblique shocks in rapid granular flows

The interaction between rapid, free-surface granular flows and deflecting dams is investigated by laboratory experimentation and by the formulation and analysis of a shallow-layer model of the motion. It is found that uniform, downslope flows of grains are deflected to flow parallel to the barrier and that upstream of the barrier, the flow state undergoes an abrupt transition whereby its depth, velocity, and direction of motion change. These oblique shocks are investigated for a range of Froude numbers and for a range of angles between the deflector and the direction of steepest descent. The experimental results are found to be in good agreement with predictions from the shallow-layer theory. Experiments were also conducted with rapid, free-surface flows of water. They reveal not only similarities between the steady deflection patterns of the water and grain flows, but also some differences in the nature of their initial interaction. A simple interpretation for this is given in terms of the relatively hig...

A three-phase mixture theory for particle size segregation in shallow granular free-surface flows

Particle-size segregation within granular materials is of great technological significance yet it is still very poorly understood. There are several causes of segregation, but this paper focuses on kinetic sieving which is the dominant mechanism in dense gravity-driven shallow free-surface flows, or, granular avalanches. The segregation model is derived from a three-phase mixture theory composed of large particles, small particles and a passive interstitial fluid. Steady-state solutions are constructed for a normally graded inflow in a steady uniform flow field. This problem is of fundamental interest, because it shows how an unstably stratified layer readjusts into a stable configuration. Expansion fans and concentration shocks are generated and sufficiently far downstream inversely graded segregated layers form, with the larger particles overlying the finer ones. This a good approximation for segregation in flows with weak diffusive remixing. The distance for complete segregation to occur is shown to increase with rising fluid density and tends to infinity as its density approaches that of the grains. If the particles are buoyant then the initial configuration is stable. An exact time-dependent two-dimensional solution is constructed for plug flow, which exploits the uncoupling of material columns of grains in the absence of shear. This yields insight into the nature of more complex numerical solutions for strong shear, which are computed with a high-resolution shock-capturing numerical scheme.

On sediment transport under dam-break flow

We present exact solutions for suspended sediment transport under one-dimensional dam-break flow, both over a dry bed and into a small depth of tail water. We explicitly calculate the suspended sediment concentration, including erosion and deposition, and investigate the effect of varying the erosional and depositional models employed. These solutions order insight into sediment transport processes under floods or sheet flow events, and we also discuss their application as test-bed solutions for the validation of numerical models.

Lock-release gravity currents and dam-break flows

Gravity current and dam-break flows, resulting from the instantaneous release of fluid initially at rest behind a lock gate, are modelled theoretically using the shallow water equations. By analysing the motion in the hodograph plane, the governing equations become linear and hence it is possible to integrate them analytically from lock-release initial conditions. This approach provides many advantages: not only are numerical computations obviated, but the analysis clearly reveals how the nature of the ensuing flow depends on the Froude number,

Occurrence and origin of submarine plunge pools at the base of the US continental slope

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