The Dupuit Approximation for the Rectangular Dam Problem

A porous rectangular dam is above a horizontal impermeable base.There is a steady flow in which water seeps through the damfrom one reservoir (on the left)to a lower reservoir (on theright). Because of gravity, the water does not flow throughthe entire dam and the dam is dry near its upper-right corner.The interface separating the dry and wet regions of the damis a free boundary (in the hydrology literature called the phreaticsurface). Dupuit, in 1863, derived the formal approximationthat the phreatic surface was nearly a parabola. In recent years,mathematicians have obtained existence results. Here, relationsare established between the formal approximations and the solutionsobtained from the existence proofs.     

Conjugate gradient techniques for the optimal control evolution dam problem

This paper considers the numerical simulation of optimal control evolution dam problem by using conjugate gradient method.The paper considers the free boundary value problem related to time dependent fluid flow in a homogeneous earth rectangular dam.The dam is taken to be sufficiently long that the flow is considered to be two dimensional.On the left and right walls of the dam there is a reservoir of fluid at a level dependent on time.This problem can be transformed into a variational inequality on a fixed domain.The numerical techniques we use are based on a linear finite element method to approximate the state equations and a conjugate gradient algorithm to solve the discrete optimal control problem.This algorithm is based on Armijo's rule in the unconstrained optimization theory.The convergence of the discrete optimal solutions to the continuous optimal solutions,and the convergence of the conjugate gradient algorithm are proved.A numerical example is given to determine the location of the minimum surface     

The fluid flow through porous media. Regularity of the free surface

The fluid flow through an earth dam separating two water reservoirs of different levels gives rise to a free boundary problem. In [1] we have proved the existence of a solution to this problem. In this paper we show that the free boundary is regular.     

Kelvin–Helmholtz instability with a free surface

The well-posedness of the hydrostatic equations is linked to long wave stability criteria for parallel shear flows. We revisit the Kelvin--Helmholtz instability with a free surface. In the wall-bounded case, the flow is unstable to all wave lengths. Short wave instabilities are localized and independent of boundary conditions. On the other hand, long waves are shown to be stable if the upper boundary is a free surface and gravity is sufficiently small. We also consider smooth velocity profiles of the base flow rather than a velocity jump. We show that stability of long waves for small gravity generally holds for monotone profiles U(y). On the other hand, this need not be the case if U is not monotone.     

An optimal shape design formulation for inhomogeneous dam problems

In this paper, the flow problem of incompressible liquid through an inhomogeneous porous medium (say dam), with permeability allowing parametrization of the free boundary by a graph of continuous unidimensional function, is considered. We propose a new formulation on an optimal shape design problem. We show the existence of a solution of the optimal shape design problem. The finite element method is used to obtain numerical results which show the efficiency of the proposed approach. Copyright © 2002 John Wiley & Sons, Ltd.     

Non-linear seismic response of concrete gravity dams to near-fault ground motions including dam-water-sediment-foundation interaction

This study focuses on non-linear seismic response of a concrete gravity dam subjected to near-fault and far-fault ground motions including dam-water-sediment-foundation rock interaction. The elasto-plastic behavior of the dam concrete is idealized using Drucker–Prager yield criterion based on associated flow rule assumption. Water in the reservoir is represented by 9-noded isoparametric quadrilateral fluid finite elements while the dam, the foundation rock and the sediment layer are modeled by using 8-noded isoparametric quadrilateral solid finite elements. The program NONSAP modified for elasto-plastic analysis of fluid-structure systems using the Lagrangian fluid finite element is employed in the response calculations. The fluid element includes the effects of surface waves and sloshing behavior of fluids. Non-linear seismic analyses of the selected concrete dam subjected to both near-fault and far-fault ground motions are performed. The results obtained from linear and non-linear analyses are compared with each other.     

Application of Smoothed Particle Hydrodynamics for modelling gated spillway flows

Computational models of spillways are important for evaluating and improving dam safety, optimising spillway design and updating operating conditions. Traditionally, scaled down physical models have been used for validation and to collect hydraulic data. Computational fluid dynamics (CFD) models however provide advantages in time, cost and resource reduction. CFD models also provide greater efficiency when evaluating a range of spillway designs or operating conditions. Within the present literature, most studies of computational spillway models utilise a mesh-based method. In this work we use the particle based method of Smoothed Particle Hydrodynamics (SPH) to model weir flow through a four bay, gated, spillway system. Advantages of SPH for such modelling include automatic representation of the free surface flow behaviour due to the Lagrangian nature of the method, and the ability to incorporate complex and dynamic boundary objects such as gate structures or debris. To validate the SPH model, the reservoir water depth simulated is compared with a related physical study. The effect of SPH resolution on the predicted water depth is evaluated. The change in reservoir water level with discharge rates for weir flow conditions is also investigated, with the difference in simulated and experimental water depths found to range from 0.16% to 11.48%. These results are the first quantitative validation of the SPH method to capture spillway flow in three dimensions. The agreement achieved demonstrates the capability of the SPH method for modelling spillway flows.     

Effect of weak gravitation on the plane Poiseuille flow of a highly rarefied gas

Plane Poiseuille flow of a highly rarefied gas that flows horizontally in the presence of weak gravitation is studied based on the Boltzmann equation for a hard sphere molecular gas and the diffuse reflection boundary condition. The behavior of the solution in the regime of large mean free path and small strength of gravity is studied numerically based on the one-dimensional Boltzmann equation derived by means of the asymptotic analysis for a slow variation in the flow direction. It is clarified that the effect of weak gravity on the flow is not negligible when the gas is so rarefied that the mean free path is comparable to the maximum range that the molecules travel along the parabolic path within the channel. When the mean free path is much larger than this range, the effect of gravity that makes the molecules fall plays the dominant role in determining the distribution function, and thus the over-concentration in the distribution function as well as the flow velocity does not increase further even if the mean free path is increased. The upper bound of the flow velocity and the mass flow rate of the gas are obtained as a function of the gravitational acceleration.     

A Complementary Solution to the Dam Problem

This paper is concerned with the solution of the porous damproblem, as an example of an elliptic free boundary problem.We outline the main methods of solving this problem, togetherwith their drawbacks and then present a new method, a complementaryenergy approach based on a stream function formulation. Thisprovides a more satisfactory calculation of the unknown pointof contact of the free boundary with the dam wall and also adirect estimate of the "discharge" of the dam. Numerical resultsillustrating the accuracy of the method are given for homogeneousand inhomogeneous rectangular dams and an axisymmetric case.     

On a filtration problem through a porous medium

Summary We prove the existence of a unique solution for a free boundary problem relative to the stationary flow between two water reservoirs of different levels separated by a dam of a non-homogeneous porous medium. Entrata in Redazione il 5 maggio 1973.     

Stabilization of periodic Stokesian Hele–Shaw flows of ferrofluids

We consider an incompressible ferrofluid in a vertical Hele–Shaw cell and develop a proper analytic framework for the free interface and the velocity potential of the fluid in a periodic geometry. The flow is assumed to obey a non-Newtonian Darcy law. The forces influencing the fluid are gravity, surface tension and the response to a magnetic field induced by a current. In addition, the flow is stabilized at the lower boundary component by an external source b. We prove a well-posedness result for the flow near flat solutions. Moreover, we find conditions on the parameters and on the slope of b for the exponential stability and instability of flat interfaces. Furthermore, we identify values for the current's intensity ι where critical bifurcation of nontrivial finger-shaped solutions from the branch of trivial (flat) solutions takes place.     

Capillary driven Oscillations of a Free Liquid Interface under Non‐Isothermal Conditions

In this study the reorientation behavior of a free liquid interface in a partly filled right circular cylinder upon a step reduction in gravity is investigated experimentally. The experiments focus on the investigation of non‐isothermal boundary conditions on the liquid reorientation. The situation is similar to a spacecraft which enters a ballistic flight after the end of thrust. Heat flux between the liquid propellant and the tank wall occurs and influences the behavior of the liquid reorientation. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

On a free piston problem for potential ideal fluid flow

Our goal was to model and analyze a stationary and evolutionary potential ideal fluid flow through the junction of two pipes in the gravity field. Inside the ‘vertical’ pipe, there is a heavy piston that can freely move along the pipe. In the stationary case, we are interested in the equilibrium position of the piston in dependence on the geometry of junction, and in the evolutionary case, we study motion of the piston also in dependence on geometry. We formulate corresponding initial and boundary value problems and prove the existence results. The problem is nonlinear because the domain is unknown. Furthermore, we study some qualitative properties of the solutions and compare them with the qualitative properties of a free piston problem for Newtonian fluid flow. All theoretical results are illustrated with numerical experiments. Copyright © 2012 John Wiley & Sons, Ltd.     

Investigation of Flow Rate Limitation in Open Capillary Channel

A forced liquid flow through an open capillary channel with free liquid surfaces was investigated experimentally under low gravity. Since the free surfaces can only withstand a certain difference between the liquid pressure and the ambient pressure the flow rate in the channel is limited. The maximum flow rate is achieved when the surfaces collapse. The aim of the investigation is to determine the limitation of the flow rate and the corresponding critical flow velocity. The investigations were performed on board the sounding rocket mission TEXUS‐37 providing a reduced gravity environment of 10^‐4 g in all axis for 6 minutes.     

Convergence analysis of trial free boundary methods for the two-dimensional filtration problem

Summary In this paper, we present a scheme of convergence analysis of trial free boundary methods for the two-dimensional filtration (or dam) problem. For the purpose we present a new variational principle of the filtration problem. This variational principle is defined on the set of admissible domains (candidates of the solution) in the dam. Under mild assumptions on the configuration of the dam, we may assume that all admissible domains are mapped from the unit disk by conformal mappings. Thus, proving convergence of trial free boundaries is reduced to proving convergence of the conformal mappings on the unit disk, and it is done using a method in the theory of minimal surfaces. Numerical examples are given.     

Solution to the initial boundary-value problem for the regularized Buckley-Leverett system

We solve the initial boundary-value problem for the regularized Buckley-Leverett system, which describes the flow of two immiscible incompressible fluids through a porous medium. This is the case of the flow of water and oil in an oil reservoir. The system is formed by a hyperbolic equation and an elliptic equation coupled by a vector field which represents the total velocity of the mixture. The regularization is done by means of a filter acting on the velocity field. We consider the critical situation in which we inject pure water into the reservoir. At this critical value for the water saturation, the spatial components of the characteristics of the hyperbolic equation vanish and this motivates the use of a new technique to prove the achievement of the boundary condition for the hyperbolic equation. We treat the case of a horizontal plane reservoir. We also prove that the time averages of the saturation component of the solution converge to one, as the time interval increases indefinitely, for almost all points of the reservoir, with a rate of convergence which depends only on the flux function.     

Smooth interface in a two-component Stokes flow

The problem considered is that of evolution of the free boundary Γ(t) separating two immiscible viscous fluids with different constant densities and viscosities. The motion is described by the Stokes equations driven by the gravity force. We prove the existence of classical solutions for small timet and establish that the free boundary Γ(t)∈C ^l+2 (l>0 is an arbitrary non-integer number)     

Numerical Solution of Non-Steady State Porous Flow Free Boundary Problems

The aim of this paper is the study of the convergence of a finite element approximation for a variational inequality related to free boundary problems in non-steady fluid flow through porous media. There have been many results in the stationary case, for example, the steady dam problems, the steady flow well problems, etc. In this paper we shall deal with the axisymmetric non-steady porous flow well problem. It is well know that by means of Torelli's transform this problem, similar to the non-steady rectangular dam problem, can be reduced a variational, inequality, and the existence, uniqueness and regularity of the solution can be obtained ([12, 7]). Now we study the numerical solution of this variational inequality. The main results are as follows: 1. We establish new regularity properties for the solution $W$ of the variation inequality. We prove that $W \in L^\infty(0, T; H^2(D))$, $γ_0W\in L^\infty(0, T; H^2(T_n))$ and $D_1γ_0W\in L^2(0, T; H^1(T_n))$ (see Theorem 2.5). Friedman and Torelli [7] obtained $W\in L^2(0, T; H^2(D))$. Our new regularity properties will be used for error estimation. 2. We prove that the error estimate for the finite element solution of the variational inequality is $$ ( \sum^N_{i=1}\| W^1 - W^1_h \|^2_{H^1(D)}\Delta t)^{1/2} = O(h+\Delta t^{1/2})$$ (see Theorem 3.4). In the stationary case the error estimate is $\|W-W_h\|_{H^1(D)} = O(k)$ ([3,6]). 3. We give a numerical example and compare the result with the corresponding result in the stationary case. The result of this paper are valid for the non-ready rectangular dam problem with stationary or quasi-stationary initial data (see [7], p.534).     

Probabilistic nonlinear analysis of CFR dams by MCS using Response Surface Method

This paper presents the probabilistic analysis of concrete-faced rockfill (CFR) dams according to the Monte Carlo Simulation (MCS) results which are obtained through the Response Surface Method (RSM). ANSYS finite element program is used to get displacement and principal stress components. First of all, some parametric studies are performed according to the simple and representative finite element model of dam body to obtain the optimum approximate model. Secondly, a sensitivity analysis is performed to get the most effective parameters on dam response. Then, RSM is used to obtain the approximate function through the selected parameters. After the performed analyses, star experimental design with quadratic function without mixed terms according to the k = 1 is determined as the most appropriate model. Finally, dam-foundation-reservoir interaction finite element model is constituted and probabilistic analyses are performed with MCS using the selected parameters, sampling method, function and arbitrary factor under gravity load for empty and full reservoir conditions. Geometrically and materially nonlinearity are considered in the analysis of dam-foundation-reservoir interaction system. Reservoir water is modeled by fluid finite elements based on the Lagrangian approach. Structural connections are modeled as welded contact and friction contact based on Coulomb’s friction law. Probabilistic displacements and stresses are presented and compared with deterministic results.