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.     

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.     

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.     

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.     

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.     

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.     

Shallow water equations for power law and Bingham fluids

In this note,we provide a consistant thin layer theory for power law and Bingham incompressible fluids flowing down an inclined plane under the effect of gravity.The derivation of such equations is based on formal asymptotic expansions of solutions of Cauchy momentum equations in the shallow water scaling and in the neighbourhood of steady solutions so that we can close the average equations on the fluid height h and the total discharge rate q.     

Nonhydraulic Effects in Particle-Driven Gravity Currents in Deep Surroundings

In this article, we present an approach to modeling the flow of particle-driven gravity currents produced by the sudden release of well-mixed, fixed-volume suspensions into deep surroundings. Our model accounts for the initial turbulent energy of mixing in the release volume, characteristic of the classical lock–release experiments, as well as the spatiotemporal variability in the driving buoyancy forces attributable to particle settling. We show that, in contrast to compositionally driven flows, particle-driven flows cannot be described consistently in terms of shallow water theory. Specifically, we show that the presence of particles in the flow dynamics produces significant horizontal velocity shear, thereby changing the flow configuration in important ways from flows assumed to be governed by the shallow water equations. These new flow properties are calculated and contrasted with flow properties derived on the basis of the shallow water equations to show that the shallow water analysis misses dynamical features of the flow. We also show that our model provides significant improvement over the previous shallow water-based models in predicting the experimentally determined deposition patterns associated with the lock–release experiments.     

Shallow water model for lakes with friction and penetration

We deduce a shallow water model, describ‐ ing the motion of the fluid in a lake, assuming inflow–outflow effects across the bottom. This model arises from the asymptotic analysis of the 3D dimensional Navier–Stokes equations. We prove the global in time existence result for this model in a bounded domain taking the nonlinear slip/friction boundary conditions to describe the inflows and outflows of the porous coast and the rivers. The solvability is shown in the class of solutions with L_p‐bounded vorticity for any given p∈(1,∞]. Copyright © 2009 John Wiley & Sons, Ltd.     

A simple justification of the singular limit for equatorial shallow‐water dynamics

The equatorial shallow‐water equations at low Froude number form a symmetric hyperbolic system with large variable‐coefficient terms. Although such systems are not covered by the classical Klainerman‐Majda theory of singular limits, the first two authors recently proved that solutions exist uniformly and converge to the solutions of the long‐wave equations as the height and Froude number tend to 0. Their proof exploits the special structure of the equations by expanding solutions in series of parabolic cylinder functions. A simpler proof of a slight generalization is presented here in the spirit of the classical theory. © 2008 Wiley Periodicals, Inc.     

On the Integrability of a Generalized Variable‐Coefficient Forced Korteweg‐de Vries Equation in Fluids

With the inhomogeneities of media taken into account, under investigation is hereby a generalized variable‐coefficient forced Korteweg‐de Vries (vc‐fKdV) equation, which describes shallow‐water waves, internal gravity waves, etc. Under an integrable constraint condition on the variable coefficients, in this paper, the complete integrability of the generalized vc‐fKdV equation is proposed. By virtue of a generalization of Bells polynomials, we systematically present its bilinear representations, Bäcklund transformations, Lax pairs and Darboux covariant Lax pairs, which can be reduced to the ones of some integrable models, such as vcKdV model, cylindrical KdV equation, and an analytical model of tsunami generation. It is very interesting that its bilinear formulism is free for the integrable constraint condition. Besides, researching the Lax equations yield its infinitely conservation laws, all conserved densities and fluxes of them are obtained by explicit recursion formulas. Furthermore, by considering its bilinear formulism with an extra auxiliary variable, we present the soliton solutions and Riemann theta function periodic wave solutions of the equation. According to the constraint among the nonlinear, dispersive, and line‐damping coefficients, we further discuss the solitonic structures and interaction properties by some graphic analysis. Finally, the relationships between the periodic wave solutions and soliton solutions are presented in detail by a limiting procedure.     

A Least‐Squares Formulation for the Shallow Water Equations with Small Viscosity

A least‐squares finite element method for the shallow water equations with viscosity parameter μ > 0 is proposed and studied. The shallow water equations are reformulated as a first order system by adding a new variable for the velocity flux. The reformulated first order system is combined with a characteristic‐based time discretization and a least squares approach. For the correct boundary treatment in the limit case μ → 0, a trace theorem is presented. For the numerical computation of the velocity, the finite element spaces introduced recently by Mardal, Tai and Winther (SIAM Journal on Numerical Analysis 40, pp. 1605–1631) are used. The degrees of freedom in these spaces can be identified with the normal and tangential components, respectively. Numerical results for some test examples are shown. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

A new shallow water model with polynomial dependence on depth

In this paper, we study two‐dimensional Euler equations in a domain with small depth. With this aim, we introduce a small non‐dimensional parameter ε related to the depth and we use asymptotic analysis to study what happens when ε becomes small. We obtain a model for ε small that, after coming back to the original domain, gives us a shallow water model that considers the possibility of a non‐constant bottom, and the horizontal velocity has a dependence on z introduced by the vorticity when it is not zero. This represents an interesting novelty with respect to shallow water models found in the literature. We stand out that we do not need to make a priori assumptions about velocity or pressure behaviour to obtain the model. The new model is able to approximate the solutions to Euler equations with dependence on z (reobtaining the same velocities profile), whereas the classic model just obtains the average velocity. Copyright © 2007 John Wiley & Sons, Ltd.     

Vertical slot fishways: Mathematical modeling and optimal management

Fishways are the main type of hydraulic devices currently used to facilitate migration of fish past obstructions (dams, waterfalls, rapids,…$\mathrm{rapids},\dots$) in rivers. In this paper we present a mathematical formulation of an optimal control problem related to the optimal management of a vertical slot fishway, where the state system is given by the shallow water equations, the control is the flux of inflow water, and the cost function reflects the need of rest areas for fish and of a water velocity suitable for fish leaping and swimming capabilities. We give a first-order optimality condition for characterizing the optimal solutions of this problem. From a numerical point of view, we use a characteristic-Galerkin method for solving the shallow water equations, and we use an optimization algorithm for the computation of the optimal control. Finally, we present numerical results obtained for the realistic case of a standard nine pools fishway.     

Interfacial long traveling waves in a two‐layer fluid with variable depth

Long wave propagation in a two‐layer fluid with variable depth is studied for specific bottom configurations, which allow waves to propagate over large distances. Such configurations are found within the linear shallow‐water theory and determined by a family of solutions of the second‐order ordinary differential equation (ODE) with three arbitrary constants. These solutions can be used to approximate the true bottom bathymetry. All such solutions represent smooth bottom profiles between two different singular points. The first singular point corresponds to the point where the two‐layer flow transforms into a uniform one. In the vicinity of this point nonlinear shallow‐water theory is used and the wave breaking criterion, which corresponds to the gradient catastrophe is found. The second bifurcation point corresponds to an infinite increase in water depth, which contradicts the shallow‐water assumption. This point is eliminated by matching the “nonreflecting” bottom profile with a flat bottom. The wave transformation at the matching point is described by the second‐order Fredholm equation and its approximated solution is then obtained. The results extend the theory of internal waves in inhomogeneous stratified fluids actively developed by Prof. Roger Grimshaw, to the new solutions types.     

Remarks on Leray's self‐similar solutions to the 3D magnetohydrodynamic equations

In this note, we examine small self‐similar solutions of the 3D incompressible magnetohydrodynamic equations in homogenous function spaces and show the uniqueness of self‐similar solutions when the velocity field is small in some homogeneous function spaces. This extend the recent results given by C. He and Z. Xin. “On the self‐similar solutions of the magneto‐hydro‐dynamic equations. Acta Mathematica Scientia, série B English Edition 2009; 29 (3): 583–598”. It is a natural way to extend the space widely and improve the previous results. Copyright © 2013 John Wiley & Sons, Ltd.     

Global solutions to the shallow water system with a method of an additional argument

The classical system of shallow water (Saint–Venant) equations describes long surface waves in an inviscid incompressible fluid of a variable depth. Although shock waves are expected in this quasi-linear hyperbolic system for a wide class of initial data, we find a sufficient condition on the initial data that guarantee existence of a global classical solution continued from a local solution. The sufficient conditions can be easily satisfied for the fluid flow propagating in one direction with two characteristic velocities of the same sign and two monotonically increasing Riemann invariants. We prove that these properties persist in the time evolution of the classical solutions to the shallow water equations and provide no shock wave singularities formed in a finite time over a half-line or an infinite line. On a technical side, we develop a novel method of an additional argument, which allows to obtain local and global solutions to the quasi-linear hyperbolic systems in physical rather than characteristic variables.     

An exact solution of start-up flow for the fractional generalized Burgers’ fluid in a porous half-space

Modified Darcy’s law for fractional generalized Burgers’ fluid in a porous medium is introduced. The flow near a wall suddenly set in motion for a fractional generalized Burgers’ fluid in a porous half-space is investigated. The velocity of the flow is described by fractional partial differential equations. By using the Fourier sine transform and the fractional Laplace transform, an exact solution of the velocity distribution is obtained. Some previous and classical results can be recovered from our results, such as the velocity solutions of the Stokes’ first problem for viscous Newtonian, second grade, Maxwell, Oldroyd-B or Burgers’ fluids.     

Mechanical Balance Laws for Boussinesq Models of Surface Water Waves

Depth-integrated long-wave models, such as the shallow-water and Boussinesq equations, are standard fare in the study of small amplitude surface waves in shallow water. While the shallow-water theory features conservation of mass, momentum and energy for smooth solutions, mechanical balance equations are not widely used in Boussinesq scaling, and it appears that the expressions for many of these quantities are not known. This work presents a systematic derivation of mass, momentum and energy densities and fluxes associated with a general family of Boussinesq systems. The derivation is based on a reconstruction of the velocity field and the pressure in the fluid column below the free surface, and the derivation of differential balance equations which are of the same asymptotic validity as the evolution equations. It is shown that all these mechanical quantities can be expressed in terms of the principal dependent variables of the Boussinesq system: the surface excursion ?? and the horizontal velocity w at a given level in the fluid.     

A new highly nonlinear shallow water wave equation

We derive a quasilinear shallow water equation directly from the governing equations for gravity water waves within a certain regime for large-amplitude waves which has not been studied so far. Furthermore, we demonstrate local well-posedness of the corresponding Cauchy problem and finally discuss some aspects of the blowup behavior of solutions.