Jump Conditions for Hyperbolic Systems of Forced Conservation Laws with an Application to Gravity Currents

Weak solutions to systems of nonlinear hyperbolic conservation laws admit discontinuities that result from either an initial value or as part of the temporally developing solution itself. The propagation of such shocks or jumps is affected by forcing terms for the nonlinear system in a way that has not been investigated fully in standard references. Jump conditions for systems of conservation laws with discontinuous forcing terms are derived herein, following the method used to derive the Rankine–Hugoniot jump conditions, and the generalized results are illustrated for the one-dimensional inviscid Burger's equation with discontinuous forcing. The main application of this type of jump condition, and the primary motivation for its study, is its application to a shallow-water model of gravity currents previously described by the authors. Specifically, a new result relation between the front and height at a gravity current front is obtained by using the existing model. Front speeds for gravity currents resulting from instantaneous release are calculated numerically and used to determine the suitability of the jump conditions, which are then compared with existing theoretical expressions and experimental observations. New numerical results are portrayed for the gravity current model, suggesting that the standard method of modeling shallow-water gravity currents with a simple Froude number front condition may tend to suppress some of the finer details of the flow resolved by the numerical scheme used by the authors.     

Simulation of suspension gravity currents with different initial aspect ratio and layout of turbidity fence

A three-dimensional (3D) numerical model, using large eddy simulation (LES), is developed for simulating the motion of suspension gravity currents. The suitable values of model parameters are determined using the existing experimental data of a two-dimensional (2D) suspension (a mixture composed of water and glass bead particles) cloud. The simulated gravity current with different initial aspect ratio (length/breadth) of the suspension is compared with the reported data of 3D laboratory experiments to investigate the effect of initial aspect ratio on the flow characteristics and the diffusion of turbidity under the presence of a turbidity fence. The comparison of simulated results of such main flow characteristics as front height, front propagation velocity and particle deposition with the experimental data reveals that the model is capable of simulating the complex behavior of the 3D suspension gravity currents to a reasonably good accuracy under complex conditions.     

Lie group analysis of gravity currents

We consider shallow water theory to study the self-similar gravity currents that describe the motion of a heavy fluid flowing into another lighter ambient fluid. Gratton and Vigo investigated the shallow water theory representing the self-similar gravity currents by using dimensional analysis [J. Gratton, C. Vigo, Self-similarity gravity currents with variable inflow revisited: Plane currents, J. Fluid. Mech. 258 (1994) 77–104]. But in this study, the self-similarity solutions of the one-layer shallow-water equations representing gravity currents are investigated by using Lie group analysis and it is shown that Lie group analysis is the generalization of the dimensional analysis for investigating the self-similarity solutions of the one-layer shallow-water equations. Applying Lie group theory, reduced equations of the shallow water equations are found. Therefore, it becomes possible to obtain the similarity forms depending on the Lie group parameters and also the self-similarity solutions for the special values of these group parameters.     

Thermally Enhanced Motions of Variable-Inflow Surface Gravity-Driven Flows

In this paper, we report on theoretical and numerical studies of models for suddenly initiated variable-inflow surface gravity currents having temperature-dependent density functions when these currents are subjected to incoming radiation. This radiation leads to a heat source term that, owing to the spatial and temporal variation in surface layer thickness, is itself a function of space and time. This heat source term, in turn, produces a temperature field in the surface layer having nonzero horizontal spatial gradients. These gradients induce shear in the surface layer so that a depth-independent velocity field can no longer be assumed and the standard shallow-water theory must be extended to describe these flow scenarios. These variable-inflow currents are assumed to enter the flow regime from behind a partially opened lock gate with the lock containing a large volume of fluid whose surface is subjected to a variable pressure. Flow filament theory is used to arrive at expressions for the variable inflow velocity under the assumptions of an inviscid and incompressible fluid moving through a small opening under a lock gate at one end of a large rectangular tank containing the deep slightly more dense ambient fluid. Finding this time-dependent inflow velocity, which will then serve as a boundary condition for the solution of our two-layer system, involves solving a forced Riccati equation with time-dependent forcing arising from the surface pressure applied to the fluid in the lock.
The results presented here are, to the best of our knowledge, the first to involve variable-inflow surface gravity currents with or without thermal enhancement and they relate to a variety of phenomena from leaking shoreline oil containers to spring runoff where the variable inflow must be taken into account to predict correctly the ensuing evolution of the flow.     

Solutions for Initial-Boundary-Value Problems Representing Gravity Currents Arising from Variable Inflow

In this article, we report on theoretical and numerical studies of models for suddenly initiated variable inflow gravity currents in rectangular geometry. These gravity currents enter a lighter, deep ambient fluid at rest at a time‐dependent rate from behind a partially opened lock gate and their subsequent dynamics is modeled in the buoyancy‐inertia regime using ½‐layer shallow water theory. The resistance to flow that is exerted by the ambient fluid on the gravity current is accounted for by a front condition which involves a non‐dimensional parameter that can be chosen in accordance with experimental observations. Flow filament theory is used to arrive at expressions for the variable inflow velocity under the assumptions of an inviscid and incompressible fluid moving through an opening of fixed area which is suddenly opened under a lock gate at one end of a large rectangular tank. The fluid in the lock is subjected to a (possibly) time varying pressure applied uniformly over its surface and the finite movement of the free surface is accounted for. Finding this time‐dependent inflow velocity, which will then serve as a boundary condition for the solution of the shallow‐water equations, involves solving forced non‐linear ordinary differential equations and the form of this velocity equation and its attendant solutions will, in general, rule out finding self‐similar solutions for the shallow‐water equations. The existence of self‐similar solutions requires that the gravity currents have volumes proportional to t ^α , where α≥ 0 and t is the time elapsed from initiation of the flow. This condition requires a point source of fluid with very special properties for which both the area of the gap and the inflow velocity must vary in a related and prescribed time‐dependent manner in order to preserve self‐similarity. These specialized self‐similar solutions are employed here as a check on our numerical approach. In the more natural cases that are treated here in which fluids flow through an opening of fixed dimensions in a container an extra dimensional parameter is introduced thereby ruling out self‐similarity of the solutions for the shallow‐water equations so that the previous analytical approaches to the variable inflow problem, involving the use of phase‐plane analysis, will be inapplicable. The models developed and analyzed here are expected to provide a first step in the study of situations in which a storage container is suddenly ruptured allowing a heavy fluid to debouch at a variable rate through a fixed opening over level terrain. They also can be adapted to the study of other situations where variable inflow gravity currents arise such as, for example, flows of fresh water from spring run‐off into lakes and fjords, flows from volcanoes and magma chambers, discharges from locks and flash floods.     

Nukerical anatomy of lock-release gravity current

Gravity currents created by releasing a fixed fluid into a less dense fluid are simulated numerically by using the renormalization group (RNG) κ- model for Reynolds-stress closure and a two-layer function as near wall treatment. A numerical anatomy of the gravity currents is made to characterize the concentration contour, velocity vector diagram, velocity profile, distribution of turbulent kinetic energy, streamlines and entrainment. Computed velocity structure of the current head explains why the front velocity decreases after the current advances about twelve lock lengths.     

Intrusive Gravity Currents

Intrusive gravity currents arise when a fluid of intermediate density intrudes into an ambient fluid. These intrusions may occur in both natural and human-made settings and may be the result of a sudden release of a fixed volume of fluid or the steady or time-dependent injection of such a fluid. In this article we analytically and numerically analyze intrusive gravity currents arising both from the sudden release of a fixed volume and the steady injection of fluid having a density that is intermediate between the densities of an upper layer bounded by a free surface and a heavier lower layer resting on a flat bottom. For the physical problems of interest we assume that the dynamics of the flow are dominated by a balance between inertial and buoyancy forces with viscous forces being negligible. The three-layer shallow-water equations used to model the two-dimensional flow regime include the effects of the surrounding fluid on the intrusive gravity current. These effects become more pronounced as the fraction of the total depth occupied by the intrusive current increases. To obtain some analytical information concerning the factors effecting bore formation we further reduce the complexity of our three-layer model by assuming small density differences among the different layers. This reduces the model equations from a 6×6 to a 4×4 system. The limit of applicability of this weakly stratified model for various ranges of density differences is examined numerically. Numerical results, in most instances, are obtained using MacCormack's method. It is found that the intrusive gravity current displays a wide range of flow behavior and that this behavior is a strong function of the fractional depth occupied by the release volume and any asymmetries in the density differences among the various layers. For example, in the initially symmetric sudden release problem it is found that an interior bore does not form when the fractional depth of the release volume is equal to or less than 50% of the total depth. The numerical simulations of fixed-volume releases of the intermediate layer for various density and initial depth ratios demonstrate that the intermediate layer quickly slumps from any isostatically uncompensated state to its Archimedean level thereby creating a wave of opposite sign ahead of the intrusion on the interface between the upper and lower layers. Similarity solutions are obtained for several cases that include both steady injection and sudden releases and these are in agreement with the numerical solutions of the shallow-water equations. The 4×4 weak stratification system is also subjected to a wavefront analysis to determine conditions for the initiation of leading-edge bores. These results also appear to be in agreement with numerical solutions of the shallow-water equations.     

Numerical Study on the Influence of Particle Inertia in Particle-Driven Gravity Currents

We present two-dimensional numerical simulations of particle-driven gravity currents in a lock-exchange configuration. The fluid is described in an Eulerian framework, whereas the particles are tracked in a Lagrangian manner. The study is restricted to dilute suspensions, allowing to neglect particle-particle interactions. The particle forces considered are buoyancy and the Stokes drag. We study the influence of particle inertia on the flow evolution by performing simulations with different Stokes numbers. We also consider the case where particle inertia is neglected. Generally, we observe significant changes in the form and structure of the gravity current with increasing particle Stokes numbers. Particularly, the formation of Kelvin-Helmholtz vortices is more and more suppressed. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Sediment Transport and Deposition from a Two-layer Fluid Model of Gravity Currents on Sloping Bottoms

This article reports on a theoretical and numerical study of noneroding turbulent gravity currents moving down mildly inclined surfaces while depositing sediment. These flows are modeled by means of two-layer fluid systems appropriately modified to account for the presence of a sloping bottom and suspended sediment in the lower layer. A detailed scaling argument shows that when the density of the interstitial fluid is slightly greater than that of the ambient and the suspension is such that its volume fraction is of the order of the aspect ratio squared, for low aspect ratio flows a two-layer shallow-water theory is applicable. In this theory there is a decoupling of particle and flow dynamics. In contrast, however, when the densities of interstitial and ambient fluids are equal, so that it is the presence of the particles alone that drives the flow, we find that a consistent shallow-water theory is impossible no matter how small the aspect ratio or the initial volume fraction occupied by the particles. Our two-layer shallow-water formulation is employed to investigate the downstream evolution of flow and depositional characteristics for sloping bottoms. This investigation uncovers a new phenomenon in the formation of a rear compressive zone giving rise to shock formation in the post-end-wall-separation phase of the particle-bearing gravity flow. This separation of flow from the end wall in these fixed volume releases differs from what has been observed on horizontal surfaces where the flow always remains in contact with the end wall.     

Self-Similarity and the Evaluation of Long-Time Nonhydrostatic Effects in Compositionally Driven Gravity Flows in Deep Surroundings

In the study of compositionally driven gravity currents involving one or more homogeneous fluid layers, it has been customary to adopt the hydrostatic assumption for the pressure field in each layer which, in turn, leads to a depth‐independent horizontal velocity field in each of these layers and significant simplifications to the governing equations. Under this hydrostatic paradigm, each layer will then have its motion governed by the well‐known reduced dimension shallow‐water equations. For the so‐called ‐layer or reduced gravity shallow‐water equations, similarity solutions for fixed volume gravity currents released in rectangular geometry have been found. Very few attempts have been made to evaluate contributions arising from the possible loss of hydrostatic balance in the context of the problems treated using the classic shallow‐water approach. Where such attempts have been pursued, they have usually been carried out in a time‐independent context or using layer‐averaged equations and very small amplitude disturbances. The vast majority of these studies into nonhydrostatic effects do not include any relevant numerical work to assess these effects. In this paper, we develop an approach for evaluating nonhydrostatic contributions to the flow field for bottom gravity currents in deep surroundings and rectangular geometry. Our approach makes no assumptions on the amplitudes of the disturbances and does not depend on layer‐averaging in the governing equations. We seek asymptotic expansions of the solutions to the Euler equations for a shallow fluid by using the small parameter δ², where δ is the aspect ratio of the flow regime. At leading order the equations enforce hydrostatic balance while those obtained at first order retain certain nonhydrostatic effects which we evaluate. Our method for evaluation of these first‐order contributions employs the self‐similar nature of the solution to the leading‐order equations in the new first‐order equations without any vertical averaging procedures being employed.     

Two-layer Gravity Currents with Topography

Two-dimensional and time-dependent gravity currents involving the initial release of a fixed volume of heavy fluid over a gradually sloping bottom and underlying a layer of lighter fluid are considered. The equations which describe the resulting two-layer flow are derived from the Navier–Stokes equations for a constant density, inviscid, nonrotating fluid, neglecting kinematic viscosity, surface tension, and entrainment between the layers. A new addition to the theory is introduced in the form of a forcing term in the lower layer horizontal momentum equation which is incorporated to produce the characteristic structure typical of such gravity currents in the laboratory. This delaying term is restricted to the front of the gravity current, and as such is shown to be valid under conventional shallow-water scaling assumptions. The hyperbolic character of the equations of motion is shown, a simple numerical test for hyperbolicity is derived from theoretical considerations, and these results are related to the stability Froude number of the flow. Well-posedness of the initial boundary value problem is proven via localization of the equations, and the discussion is extended to a two-point boundary value problem with examples of steady-state and traveling wave solutions given for a bottom surface of constant slope. Numerical results are obtained by using a recently developed finite-difference relaxation scheme for conservation laws, sufficiently modified herein to include spatial variability and forcing terms, which approximates the material interface at the front of the lower fluid layer as a shock. The effects of slope and the delaying force are investigated numerically to determine their theoretical importance, and the range of expected values is compared to published experimental results. Some calculations for the temporal evolution of the flow are produced that display the phenomenon of rear wall separation for nonzero slopes.     

Direct numerical simulation of bi-disperse particle-laden gravity currents in the channel configuration

We present a numerical investigation of bi-disperse particle-laden gravity currents in the lock-exchange configuration. Previous results, based on numerical simulation and laboratory experiments, are used to establish comparisons. Our discussion focuses on explaining how the presence of more than one particle diameter influences the main features of the flow, such as deposit profile, the evolution of the front location and suspended mass. We develop the complete energy budget equation for bi-disperse flows. A set of two and three-dimensional direct numerical simulations (DNS), with different initial compositions of coarse and fine particles, are carried out for Reynolds number equal to 4000. Such simulations show that the energy terms are strongly affected by varying the initial particle fractions. The addition of a small amount of fine particles into a current predominantly composed of coarse particles increases its run-out distance. In particular, it is shown that higher amounts of coarse particles have a dumping effect on the current development. Comparisons show that the two-dimensional simulation does not reproduce the intense turbulence generated in 3D cases accurately, which results in a significant difference in the suspended mass, front position as well as the dissipation term due to the advective motion.     

PIV observation of instantaneous velocity structure of lock release gravity currents in the slumping phase

The experiments of lock release gravity currents were performed in a rectangular channel with salt water and fresh water. The spreading law of the salt water is validated by using a digital video to record the progress of the current. Detailed instantaneous velocity structure of lock release gravity currents in the slumping phase was studied experimentally with particle image velocimetry (PIV). The time variation of the spatial distribution of velocity and vorticity is obtained, which shows some qualitative characters of the current as well as the effects of the bottom boundary layer and upper mixing layer.     

Steady Deep-Water Waves on a Linear Shear Current

The properties of steady, periodic, deep-water gravity waves on a linear shear current are investigated. Numerical solutions for all waveheights, up to and including the limiting ones, are computed from a formulation which involves only the wave profile (parametrized in a natural way) and some constants of the motion. It is found that for some shear currents the highest waves are not necessarily those waves with sharp crests known as extreme waves. Furthermore a certain nonuniqueness in the sense of a fold is shown to exist, and a new type of limiting wave is discovered. For both small-amplitude waves and extreme waves the numerical results are compared with theoretical predictions.     

Random Dynamics of the Boussinesq System with Dynamical Boundary Conditions

Abstract

A coupled system of the two-dimensional Navier–Stokes equations and the salinity transport equation with spatially correlated white noise on the boundary as well as in fluid is investigated. The noise affects the system through a dynamical boundary condition. This system may be considered as a model for gravity currents in oceanic fluids. The noise is due to uncertainty in salinity flux on fluid boundary. After transforming this system into a random dynamical system, we first obtain asymptotic estimates on system evolution, and then show that the long time dynamics is captured by a random attractor.     

Euclidean and Lorentzian quantum gravity—lessons from two dimensions

No theory of four-dimensional quantum gravity exists as yet. In this situation the two-dimensional theory, which can be analyzed by conventional field-theoretical methods, can serve as a toy model for studying some aspects of quantum gravity. It represents one of the rare settings in a quantum-gravitational context where one can calculate quantities truly independent of any background geometry.We review recent progress in our understanding of 2d quantum gravity, and in particular the relation between the Euclidean and Lorentzian sectors of the quantum theory. We show that conventional 2d Euclidean quantum gravity can be obtained from Lorentzian quantum gravity by an analytic continuation only if we allow for spatial topology changes in the latter. Once this is done, one obtains a theory of quantum gravity where space-time is fractal: the intrinsic Hausdorff dimension of usual 2d Euclidean quantum gravity is four, and not two. However, certain aspects of quantum space-time remain two-dimensional, exemplified by the fact that its so-called spectral dimension is equal to two.     

Method for taking into account gravity in free-surface flow simulation

A numerical algorithm that correctly takes into account the force of gravity in the presence of density discontinuities is constructed using unstructured collocated grids and splitting algorithms based on SIMPLE-type methods. A correct hydrostatic pressure field is obtained by explicitly extracting the gravity force contribution to the pressure equation and computing it using the solution of the gravity equilibrium problem for a two-phase medium. To ensure that the force of gravity is balanced by the pressure gradient in the case of a medium at rest, an algorithm is proposed according to which the pressure gradient in the equations of motion is replaced by a modification allowing for the force of gravity. Well-known free-surface problems are used to show that, in contrast to previously known algorithms, the proposed ones on unstructured meshes correctly predict hydrostatic pressure fields and do not yield velocity oscillations or free-surface distortions.     

The similarity forms and invariant solutions of two-layer shallow-water equations

In this study symmetry group properties and general similarity forms of the two-layer shallow-water equations are discussed by Lie group theory. We represent that Lie group theory can be used as an effective approach for investigation of the self-similar solutions for the shallow-water equations with variable inflow as the generalization of dimensional analysis that was used so far for a regular approach in the literature. We also represent that the results obtained by dimensional analysis are just a special case of the results obtained by Lie group theory and it is possible to obtain the new similarity forms and the new variable inflow functions for the study of gravity currents in two-layer flow under shallow-water approximations based on Lie group theory. The symmetry groups of the system of nonlinear partial differential equations are found and the corresponding similarity and reduced forms are obtained. Some similarity solutions of the reduced equations are investigated. It is shown that reduced equations and similarity forms of the system depend on the group parameters. We show that an analytic similarity solution for the system of equations can be found for some special values of them. For other values of the group parameters, the similarity solutions of the two-layer shallow-water equations representing the gravity currents with a variable inflow are found by the numeric integration.     

Third order approximation to capillary gravity short crested waves with uniform currents

A third-order analytical solution for the capillary gravity short crested waves with uniform current (the main current direction is along the vertical wall) in front of a vertical wall is derived through a perturbation expanding technique. The validity and advantage of the new solution were proved by comparing the results of wave profiles and wave pressures with those of Huang and Jia [H. Huang, F. Jia, The patterns of surface capillary gravity short-crested waves with uniform current fields in coastal waters, Acta Mech. Sinica 22 (2006) 433–441] and Hsu [J.R.C. Hsu, Y. Tsuchiya, R. Silvester, Third-order approximation to short-crested waves, J. Fluid Mech 90 (1979) 179–196]. The important influences of currents on the wave profiles, wave frequency, the ratio of maximum crest height to the total wave height, and wave pressure are investigated for both small-scale (for example, the incident wave wavelength is 9.35 cm) and larger-scale (for example, the incident wave wavelength is 5 m) short crested wave. By numerical computation, we find wave frequency of short crested wave system is greatly affected by incident wave amplitude, incident angle, water depth, surface tension coefficient and the strength of the currents for small-scale incident wave. Furthermore, for the larger-scale short crested wave system, the higher-order solution with uniform current is particularly important for the prediction of wave profile and wave pressure for different water depth and incident angle. Computational results show that with the increase of the current speed, the crests of wave profile and wave pressure become more and more steep. In some cases, the crest value of wave pressure with strong current would be larger about six times than that of no current. Therefore, ocean engineers should consider the short crested wave-current load on marine constructs carefully.     

The well‐posedness of the electric field integral equation for transient scattering from a perfectly conducting body

We consider the scattering of a transient electromagnetic field incident on a body with a smooth, perfectly conducting surface. A standard numerical method for calculating the scattered field is to use a time dependent, surface integral equation (called the electric field integral equation) to calculate the surface currents and charges induced by the incident field—these currents and charges then yield the scattered fields by means of standard integral representations (vector and scalar potentials). In this paper we show that the time‐dependent electric field integral equation is well‐posed in a suitable function space setting. We also investigate the behaviour of the solutions at large time. Copyright © 1999 John Wiley & Sons, Ltd.