Steepness expansion for free surface flows in coastal aquifers

Free surface flow of groundwater in aquifers has been studied since the early 1960s. Previous investigations have been based on the Boussinesq equation, derived from the non-linear kinematic boundary condition. In fact, the Boussinesq equation is the zeroth-order equation in the shallow-water expansion. A key assumption in this expansion is that the mean thickness of the aquifer is small compared with a reference length, normally taken to be the linear decay length. In this study, we re-examine the expansion scheme for free surface groundwater flows, and propose a new expansion wherein the shallow-water assumption is replaced by a steepness assumption. A comparison with experimental data shows that the new model provides a better prediction of water table levels than the conventional shallow-water expansion. The applicable ranges of the two expansions are exhibited. (c) 2004 Elsevier B.V. All rights reserved.

Deflection of jets discharged into a reservoir with a free surface

Deflections of jets discharged into a reservoir with a free surface are investigated numerically. The jets are known to deflect towards either side of the free surface or the bottom, whose direction is not determined uniquely in some experimental conditions, i.e. there are multiple stable states realizable in the same condition. The origin of the multiple stable states is explored by utilizing homotopy transformations in which the top boundary of the reservoir is transformed from a rigid to a free boundary and also the location of the outlet throat is continuously moved from mid-height to the top. We depicted bifurcation diagrams of the flow compiling the data of numerical simulations, from which we identified the origin as an imperfect pitchfork bifurcation, and obtained an insight into the mechanism for the direction to be determined. The parameter region where such multiple stable states are possible is also delimited. © 2011 The Japan Society of Fluid Mechanics and IOP Publishing Ltd.

The method of fundamental solutions for free surface Stefan problems

In this paper, free surface problems of Stefan-type for the parabolic heat equation are investigated using the method of fundamental solutions. The additional measurement necessary to determine the free surface could be a boundary temperature, a heat flux or an energy measurement. Both one- and two-phase flows are investigated. Numerical results are presented and discussed.

A New Method for Modeling Free Surface Flows and Fluid-structure Interaction with Ocean Applications

The computational modeling of ocean waves and ocean-faring devices poses numerous challenges. Among these are the need to stably and accurately represent both the fluid-fluid interface between water and air as well as the fluid-structure interfaces arising between solid devices and one or more fluids. As techniques are developed to stably and accurately balance the interactions between fluid and structural solvers at these boundaries, a similarly pressing challenge is the development of algorithms that are massively scalable and capable of performing large-scale three-dimensional simulations on reasonable time scales. This dissertation introduces two separate methods for approaching this problem, with the first focusing on the development of sophisticated fluid-fluid interface representations and the second focusing primarily on scalability and extensibility to higher-order methods.

We begin by introducing the narrow-band gradient-augmented level set method (GALSM) for incompressible multiphase Navier-Stokes flow. This is the first use of the high-order GALSM for a fluid flow application, and its reliability and accuracy in modeling ocean environments is tested extensively. The method demonstrates numerous advantages over the traditional level set method, among these a heightened conservation of fluid volume and the representation of subgrid structures.

Next, we present a finite-volume algorithm for solving the incompressible Euler equations in two and three dimensions in the presence of a flow-driven free surface and a dynamic rigid body. In this development, the chief concerns are efficiency, scalability, and extensibility (to higher-order and truly conservative methods). These priorities informed a number of important choices: The air phase is substituted by a pressure boundary condition in order to greatly reduce the size of the computational domain, a cut-cell finite-volume approach is chosen in order to minimize fluid volume loss and open the door to higher-order methods, and adaptive mesh refinement (AMR) is employed to focus computational effort and make large-scale 3D simulations possible. This algorithm is shown to produce robust and accurate results that are well-suited for the study of ocean waves and the development of wave energy conversion (WEC) devices.

Three-dimensional free surface modelling in an unstructured mesh environment for metal processing applications

In the casting of metals, tundish flow, welding, converters, and other metal processing applications, the behaviour of the fluid surface is important. In aluminium alloys, for example, oxides formed on the surface may be drawn into the body of the melt where they act as faults in the solidified product affecting cast quality. For this reason, accurate description of wave behaviour, air entrapment, and other effects need to be modelled, in the presence of heat transfer and possibly phase change. The authors have developed a single-phase algorithm for modelling this problem. The Scalar Equation Algorithm (SEA) (see Refs. 1 and 2), enables the transport of the property discontinuity representing the free surface through a fixed grid. An extension of this method to unstructured mesh codes is presented here, together with validation. The new method employs a TVD flux limiter in conjunction with a ray-tracing algorithm, to ensure a sharp bound interface. Applications of the method are in the filling and emptying of mould cavities, with heat transfer and phase change.

Free surface and mesh control for the FV simulation of semi-levitation melting

Abstract not available

On nonlinear dynamics in a free fluid surface of a tank excited by a non-ideal power source

We investigate the nonlinear oscillations in a free surface of a fluid in a cylinder tank excited by non-ideal power source, an electric motor with limited power supply. We study the possibility of parametric resonance in this system, showing that the excitation mechanism can generate chaotic response. Additionally, the dynamics of parametrically excited surface waves in the tank can reveal new characteristics of the system. The fluid-dynamic system is modeled in such way as to obtain a nonlinear differential equation system. Numerical experiments are carried out to find the regions of chaotic solutions. Simulation results are presented as phase-portrait diagrams characterizing the resonant vibrations of free fluid surface and the existence of several types of regular and chaotic attractors. We also describe the energy transfer in the interaction process between the hydrodynamic system and the electric motor. Copyright © 2011 by ASME.

Jacobian-free Newton-Krylov methods with GPU acceleration for computing nonlinear ship wave patterns

The nonlinear problem of steady free-surface flow past a submerged source is considered as a case study for three-dimensional ship wave problems. Of particular interest is the distinctive wedge-shaped wave pattern that forms on the surface of the fluid. By reformulating the governing equations with a standard boundary-integral method, we derive a system of nonlinear algebraic equations that enforce a singular integro-differential equation at each midpoint on a two-dimensional mesh. Our contribution is to solve the system of equations with a Jacobian-free Newton-Krylov method together with a banded preconditioner that is carefully constructed with entries taken from the Jacobian of the linearised problem. Further, we are able to utilise graphics processing unit acceleration to significantly increase the grid refinement and decrease the run-time of our solutions in comparison to schemes that are presently employed in the literature. Our approach provides opportunities to explore the nonlinear features of three-dimensional ship wave patterns, such as the shape of steep waves close to their limiting configuration, in a manner that has been possible in the two-dimensional analogue for some time.

Wake angle for surface gravity waves on a finite depth fluid

Linear water wave theory suggests that wave patterns caused by a steadily moving disturbance are contained within a wedge whose half-angle depends on the depth-based Froude number $F_H$. For the problem of flow past an axisymmetric pressure distribution in a finite-depth channel, we report on the apparent angle of the wake, which is the angle of maximum peaks. For moderately deep channels, the dependence of the apparent wake angle on the Froude number is very different to the wedge angle, and varies smoothly as $F_H$ passes through the critical value $F_H=1$. For shallow water, the two angles tend to follow each other more closely, which leads to very large apparent wake angles for certain regimes.

Finite Element Solution Of Surface-Tension Driven Flows In Laser Surface-Melting

Lasers are very efficient in heating localized regions and hence they find a wide application in surface treatment processes. The surface of a material can be selectively modified to give superior wear and corrosion resistance. In laser surface-melting and welding problems, the high temperature gradient prevailing in the free surface induces a surface-tension gradient which is the dominant driving force for convection (known as thermo-capillary or Marangoni convection). It has been reported that the surface-tension driven convection plays a dominant role in determining the melt pool shape. In most of the earlier works on laser-melting and related problems, the finite difference method (FDM) has been used to solve the Navier Stokes equations [1]. Since the Reynolds number is quite high in these cases, upwinding has been used. Though upwinding gives physically realistic solutions even on a coarse grid, the results are inaccurate. McLay and Carey have solved the thermo-capillary flow in welding problems by an implicit finite element method [2]. They used the conventional Galerkin finite element method (FEM) which requires that the pressure be interpolated by one order lower than velocity (mixed interpolation). This restricts the choice of elements to certain higher order elements which need numerical integration for evaluation of element matrices. The implicit algorithm yields a system of nonlinear, unsymmetric equations which are not positive definite. Computations would be possible only with large mainframe computers.Sluzalec [3] has modeled the pulsed laser-melting problem by an explicit method (FEM). He has used the six-node triangular element with mixed interpolation. Since he has considered the buoyancy induced flow only, the velocity values are small. In the present work, an equal order explicit FEM is used to compute the thermo-capillary flow in the laser surface-melting problem. As this method permits equal order interpolation, there is no restriction in the choice of elements. Even linear elements such as the three-node triangular elements can be used. As the governing equations are solved in a sequential manner, the computer memory requirement is less. The finite element formulation is discussed in this paper along with typical numerical results.

Nonlinear surface acoustic waves on an isotropic solid

A systematic derivation of the approximate coupled amplitude equations governing the propagation of a quasi-monochromatic Rayleigh surface wave on an isotropic solid is presented, starting from the non-linear governing differential equations and the non-linear free-surface boundary conditions, using the method of mulitple scales. An explicit solution of these equations for a signalling problem is obtained in terms of hyperbolic functions. In the case of monochromatic excitation, it is shown that the second harmonic amplitude grows initially at the expense of the fundamental and that the amplitudes of the fundamental and second harmonic remain bounded for all time.