Modified incompressible SPH method for simulating free surface problems

An incompressible smoothed particle hydrodynamics (I-SPH) formulation is presented to simulate free surface incompressible fluid problems. The governing equations are mass and momentum conservation that are solved in a Lagrangian form using a two-step fractional method. In the first step, velocity field is computed without enforcing incompressibility. In the second step, a Poisson equation of pressure is used to satisfy incompressibility condition. The source term in the Poisson equation for the pressure is approximated, based on the SPH continuity equation, by an interpolation summation involving the relative velocities between a reference particle and its neighboring particles. A new form of source term for the Poisson equation is proposed and also a modified Poisson equation of pressure is used to satisfy incompressibility condition of free surface particles. By employing these corrections, the stability and accuracy of SPH method are improved. In order to show the ability of SPH method to simulate fluid mechanical problems, this method is used to simulate four test problems such as 2-D dam-break and wave propagation.     

A coupled surface‐subsurface model for hydrostatic flows under saturated and variably saturated conditions

In this paper, the governing differential equations for hydrostatic surface‐subsurface flows are derived from the Richards and from the Navier‐Stokes equations. A vertically integrated continuity equation is formulated to account for both surface and subsurface flows under saturated and variable saturated conditions. Numerically, the horizontal domain is covered by an unstructured orthogonal grid that may include subgrid specifications. Along the vertical direction, a simple z‐layer discretization is adopted. Semi‐implicit finite difference equations for velocities, and a finite volume approximation for the vertically integrated continuity equation, are derived in such a fashion that, after simple manipulation, the resulting discrete pressure equation can be assembled into a single, two‐dimensional, mildly nonlinear system. This system is solved by a nested Newton‐type method, which yields simultaneously the (hydrostatic) pressure and a nonnegative fluid volume throughout the computational grid. The resulting algorithm is relatively simple, extremely efficient, and very accurate. Stability, convergence, and exact mass conservation are assured throughout also in presence of wetting and drying, in variable saturated conditions, and during flow transition through the soil interface. A few examples illustrate the model applicability and demonstrate the effectiveness of the proposed algorithm.     

Ensemble phase averaged equations for multiphase flows in porous media. Part 2: A general theory

Most models for multiphase flows in a porous medium are based on a straightforward extension of Darcy’s law, in which each fluid phase is driven by its own pressure gradient. The pressure difference between the phases is thought to be an effect of surface tension and is called capillary pressure. Independent of Darcy’s law, for liquid imbibition processes in a porous material, diffusion models are sometime used. In this paper, an ensemble phase averaging technique for continuous multiphase flows is applied to derive averaged equations and to examine the validity of the commonly used models. Closure for the averaged equations is quite complicated for general multiphase flows in a porous material. For flows with a small ratio of the characteristic length of the phase interfaces to the macroscopic length, the closure relations can be simplified significantly by an approximation with a second order error in this length ratio. This approximation reveals the information of the length scale separation obscured during an averaging process and leads to an equation system similar to Darcy’s law, but with additional terms. Based on interactions on phase interfaces, relations among closure quantities are studied.     

Three‐dimensional numerical modelling of free surface flows with non‐hydrostatic pressure

A three‐dimensional numerical model is developed for incompressible free surface flows. The model is based on the unsteady Reynolds‐averaged Navier–Stokes equations with a non‐hydrostatic pressure distribution being incorporated in the model. The governing equations are solved in the conventional sigma co‐ordinate system, with a semi‐implicit time discretization. A fractional step method is used to enable the pressure to be decomposed into its hydrostatic and hydrodynamic components. At every time step one five‐diagonal system of equations is solved to compute the water elevations and then the hydrodynamic pressure is determined from a pressure Poisson equation. The model is applied to three examples to simulate unsteady free surface flows where non‐hydrostatic pressures have a considerable effect on the velocity field. Emphasis is focused on applying the model to wave problems. Two of the examples are about modelling small amplitude waves where the hydrostatic approximation and long wave theory are not valid. The other example is the wind‐induced circulation in a closed basin. The numerical solutions are compared with the available analytical solutions for small amplitude wave theory and very good agreement is obtained. Copyright © 2002 John Wiley & Sons, Ltd.     

A conservative semi‐implicit method for coupled surface–subsurface flows in regional scale

A semi‐implicit method for coupled surface–subsurface flows in regional scale is proposed and analyzed. The flow domain is assumed to have a small vertical scale as compared with the horizontal extents. Thus, after hydrostatic approximation, the simplified governing equations are derived from the Reynolds averaged Navier–Stokes equations for the surface flow and from the Darcy's law for the subsurface flow. A conservative free‐surface equation is derived from a vertical integral of the incompressibility condition and extends to the whole water column including both, the surface and the subsurface, wet domains. Numerically, the horizontal domain is covered by an unstructured orthogonal grid that may include subgrid specifications. Along the vertical direction a simple z‐layer discretization is adopted. Semi‐implicit finite difference equations for velocities and a finite volume approximation for the free‐surface equation are derived in such a fashion that, after simple manipulation, the resulting discrete free‐surface equation yields a single, well‐posed, mildly nonlinear system. This system is efficiently solved by a nested Newton‐type iterative method that yields simultaneously the pressure and a non‐negative fluid volume throughout the computational grid. The time‐step size is not restricted by stability conditions dictated by friction or surface wave speed. The resulting algorithm is simple, extremely efficient, and very accurate. Exact mass conservation is assured also in presence of wetting and drying dynamics, in pressurized flow conditions, and during free‐surface transition through the interface. A few examples illustrate the model applicability and demonstrate the effectiveness of the proposed algorithm. Copyright © 2015 John Wiley & Sons, Ltd.     

Discontinuous Galerkin finite element method applied to the coupled unsteady Stokes/Cahn-Hilliard equations

Two-phase flows driven by the interfacial dynamics are studied by tracking implicitly interfaces in the framework of the Cahn-Hilliard theory. The fluid dynamics is described by the Stokes equations with an additional source term in the momentum equation taking into account the capillary forces. A discontinuous Galerkin finite element method is used to solve the coupled Stokes/Cahn-Hilliard equations. The Cahn-Hilliard equation is treated as a system of two coupled equations corresponding to the advection-diffusion equation for the phase field and a nonlinear elliptic equation for the chemical potential. First, the variational formulation of the Cahn-Hilliard equation is presented. A numerical test is achieved showing the optimal order in error bounds. Second, the variational formulation in discontinuous Galerkin finite element approach of the Stokes equations is recalled, in which the same space of approximation is used for the velocity and the pressure with an adequate stabilization technique. The rates of convergence in space and time are evaluated leading to an optimal order in error bounds in space and a second order in time with a backward differentiation formula at the second order. Numerical tests devoted to two-phase flows are provided on ellipsoidal droplet retraction, on the capillary rising of a liquid in a tube, and on the wetting drop over a horizontal solid wall.     

Efficient simulation of free surface flows with discrete least-squares meshless method using a priori error estimator

In this article, a priori error estimate is employed to improve the efficiency of simulating free surface flows with discrete least-squares meshless (DLSM) method. DLSM is a fully least-squares approach in which both function approximation and the discretisation of the governing differential equations is carried out using a least-squares concept. The meshless shape functions are derived using the moving least-squares (MLS) method of function approximation. The discretised equations are obtained via a discrete least-squares method in which the sum of the squared residuals are minimised with respect to unknown nodal parameters. The governing equations of mass and momentum conservation are solved in a Lagrangian form using a pressure projection method. The proposed simulation strategy is composed of error estimation and a node moving refinement method. Since in free surface problems, the position of the free surface is of primary interest, a priori error estimate is used which automatically associates higher error to the nodes near the free surface. The node moving refinement method is used to construct a nodal configuration with dense nodal arrangement near the free surface. Four test problems namely dam break, evolution of a water bubble, solitary wave propagation and wave run-up on slope are investigated to test the ability and efficiency of the proposed efficient simulation method.     

Efficient computation of two‐dimensional steady free‐surface flows

We consider a family of steady free‐surface flow problems in two dimensions, concentrating on the effect of nonlinearity on the train of gravity waves that appear downstream of a disturbance. By exploiting standard complex variable techniques, these problems are formulated in terms of a coupled system of Bernoulli equation and an integral equation. When applying a numerical collocation scheme, the Jacobian for the system is dense, as the integral equation forces each of the algebraic equations to depend on each of the unknowns. We present here a strategy for overcoming this challenge, which leads to a numerical scheme that is much more efficient than what is normally used for these types of problems, allowing for many more grid points over the free surface. In particular, we provide a simple recipe for constructing a sparse approximation to the Jacobian that is used as a preconditioner in a Jacobian‐free Newton‐Krylov method for solving the nonlinear system. We use this approach to compute numerical results for a variety of prototype problems including flows past pressure distributions, a surface‐piercing object and bottom topographies.     

Numerical solution of the incompressible Navier–Stokes equations with an upwind compact difference scheme

A new finite difference method for the discretization of the incompressible Navier–Stokes equations is presented. The scheme is constructed on a staggered‐mesh grid system. The convection terms are discretized with a fifth‐order‐accurate upwind compact difference approximation, the viscous terms are discretized with a sixth‐order symmetrical compact difference approximation, the continuity equation and the pressure gradient in the momentum equations are discretized with a fourth‐order difference approximation on a cell‐centered mesh. Time advancement uses a three‐stage Runge–Kutta method. The Poisson equation for computing the pressure is solved with preconditioning. Accuracy analysis shows that the new method has high resolving efficiency. Validation of the method by computation of Taylor's vortex array is presented. Copyright © 1999 John Wiley & Sons, Ltd.     

Numerical simulation of 3-D turbulent flows over dredged trenches

A 3-D free surface flow in open channels based on the Reynolds equations with thek-ε turbulence closure model is presented in this paper. Insted of the “rigid lid” approximation, the solution of the free surface equation is implemented in the velocity—pressure iterative procedure on the basis of the conventional SIMPLE method. This model was used to compute the flow in rectangular channels with trenches dredged across the bottom. The velocity, eddy viscosity coefficient, turbulent shear stress, turbulent kinetic energy and elevation of the free surface can be obtained. The computed results are in good agreement with previous experimental data.     

Entropy generation of radiation effect on laminar-mixed convection along a wavy surface

The effects of thermal radiation on laminar-forced and free convection along the wavy surface are studied. The optically thick limit approximation for the radiation flux is assumed. A modified form for the entropy generation equation is derived. The effect of geometry (e.g. flat surface, wavy surface), fluid friction and heat transfer (e.g. convection and radiation effects) are all included in the modified entropy generation form. Prandtl’s transposition theorem is used to stretch the ordinary coordinate system in certain directions. The wavy surface can be transformed into a calculable planar coordinate system. The governing equations are derived from the complete Navier–Stokes equations. A simple transformation is proposed to transform the governing equations into boundary layer equations for solution by the cubic spline collocation method.     

用Level Set方法求解具有自由面的流动问题

为采用Ｌｅｖｅｌ Ｓｅｔ方法来计算有自由的流动问题提出了一种方案,把自由水面视为水和空气的交界面,两种介质用统一的Ｎ－Ｓ方法求解,在自由面两侧分别采用各自的密度和粘性,并在自由面上给以适当的光滑;采用边界元法求解双调和方程来确定距离函数;Ｎ－Ｓ方程用投影法求解,文中给出了二维水池水面振荡和瞬时溃坝问题的算例,可以看出用ＬｅｖｅｌＳｅｔ方法求解有自由面流动问题是有效的。     

Numerical solution of the eXtended Pom-Pom model for viscoelastic free surface flows

In this paper we present a finite difference method for solving two-dimensional viscoelastic unsteady free surface flows governed by the single equation version of the eXtended Pom-Pom (XPP) model. The momentum equations are solved by a projection method which uncouples the velocity and pressure fields. We are interested in low Reynolds number flows and, to enhance the stability of the numerical method, an implicit technique for computing the pressure condition on the free surface is employed. This strategy is invoked to solve the governing equations within a Marker-and-Cell type approach while simultaneously calculating the correct normal stress condition on the free surface. The numerical code is validated by performing mesh refinement on a two-dimensional channel flow. Numerical results include an investigation of the influence of the parameters of the XPP equation on the extrudate swelling ratio and the simulation of the Barus effect for XPP fluids.     

Further experiences with computing non‐hydrostatic free‐surface flows involving water waves

A semi‐implicit, staggered finite volume technique for non‐hydrostatic, free‐surface flow governed by the incompressible Euler equations is presented that has a proper balance between accuracy, robustness and computing time. The procedure is intended to be used for predicting wave propagation in coastal areas. The splitting of the pressure into hydrostatic and non‐hydrostatic components is utilized. To ease the task of discretization and to enhance the accuracy of the scheme, a vertical boundary‐fitted co‐ordinate system is employed, permitting more resolution near the bottom as well as near the free surface. The issue of the implementation of boundary conditions is addressed. As recently proposed by the present authors, the Keller‐box scheme for accurate approximation of frequency wave dispersion requiring a limited vertical resolution is incorporated. The both locally and globally mass conserved solution is achieved with the aid of a projection method in the discrete sense. An efficient preconditioned Krylov subspace technique to solve the discretized Poisson equation for pressure correction with an unsymmetric matrix is treated. Some numerical experiments to show the accuracy, robustness and efficiency of the proposed method are presented. Copyright © 2004 John Wiley & Sons, Ltd.     

An asymptotic iteration method for the numerical analysis of near-critical free-surface flows

Satisfying the boundary conditions at the free surface may impose severe difficulties to the computation of turbulent open-channel flows with finite-volume or finite-element methods, in particular, when the flow conditions are nearly critical. It is proposed to apply an iteration procedure that is based on an asymptotic expansion for large Reynolds numbers and Froude numbers close to the critical value 1.The iteration procedure starts by prescribing a first approximation for the free surface as it is obtained from solving an ODE that has been derived previously by means of an asymptotic expansion (Grillhofer and Schneider, 2003). The numerical solution of the full equations of motion then gives a surface pressure distribution that differs from the constant value required by the dynamic boundary condition. To determine a correction to the elevation of the free surface we next solve an ODE that is obtained from the asymptotic analysis of the flow with a prescribed pressure disturbance at the free surface. The full equations of motion are then solved for the corrected surface, and the procedure is repeated until criteria of accuracy for surface elevation and surface pressure, respectively, are satisfied.The method is applied to an undular hydraulic jump as a test case.     

A finite volume approach for unsteady viscoelastic fluid flows

A finite volume, time‐marching for solving time‐dependent viscoelastic flow in two space dimensions for Oldroyd‐B and Phan Thien–Tanner fluids, is presented. A non‐uniform staggered grid system is used. The conservation and constitutive equations are solved using the finite volume method with an upwind scheme for the viscoelastic stresses and an hybrid scheme for the velocities. To calculate the pressure field, the semi‐implicit method for the pressure linked equation revised method is used. The discretized equations are solved sequentially, using the tridiagonal matrix algorithm solver with under‐relaxation. In both, the full approximation storage multigrid algorithm is used to speed up the convergence rate. Simulations of viscoelastic flows in four‐to‐one abrupt plane contraction are carried out. We will study the behaviour at the entrance corner of the four‐to‐one planar abrupt contraction. Using this solver, we show convergence up to a Weissenberg number We of 20 for the Oldroyd‐B model. No limiting Weissenberg number is observed even though a Phan Thien–Tanner model is used. Several numerical results are presented. Smooth and stable solutions are obtained for high Weissenberg number. Copyright © 2002 John Wiley & Sons, Ltd.     

Extended Mild-Slope Equation

ＩｎｔｒｏｄｕｃｔｉｏｎＡｃｃｕｒａｔｅｍｏｄｅｌｌｉｎｇｏｆｓｕｒｆａｃｅｗａｖｅｄｙｎａｍｉｃｓｉｎｃｏａｓｔａｌｒｅｇｉｏｎｓｈａｓｂｅｅｎｔｈｅｇｏａｌｏｆｍｕｃｈｒｅｃｅｎｔｒｅｓｅａｒｃｈ ,ｗｈｉｃｈｈａｓｂｅｅｎｓｕｍｍａｒｉｚｅｄｉｎｓｕｒｖｅｙｓｂｙＤｉｎｇｅｍａｎｓ( 1 997) [1]ａｎｄＫｉｒｂｙ( 1 997) [2 ].Ｔｈｅｒｉｃｈｎｅｓｓｏｆｃｏａｓｔａｌｗａｖｅｄｙｎａｍｉｃｓａｒｉｓｅｓｆｒｏｍｔｈｅｓｔｒｏｎｇａｍｂｉｅｎｔｃｕｒｒｅｎｔｓａｎｄｔｈｅｗｉｄｅｖａｒｉａｔｉｏｎｓ…     

Analyzing the 2D fracture problems via the enriched boundary element-free method

In this paper, the enriched boundary element-free method for two-dimensional fracture problems is presented. An improved moving least-squares (IMLS) approximation, in which the orthogonal function system with a weight function is used as the basis function, is used to obtain the shape functions. The IMLS approximation has greater computational efficiency and precision than the existing moving least-squares (MLS) approximation, and does not lead to an ill-conditioned system of equations. Combining the boundary integral equation (BIE) method and the IMLS approximation, a boundary element-free method (BEFM), for two-dimensional fracture problems is obtained. For two-dimensional fracture problems, the enriched basis function is used at the tip of the crack, and then the enriched BEFM is presented. In comparison with other existing meshless boundary integral equation methods, the BEFM is a direct numerical method in which the basic unknown quantity is the real solution of the nodal variables, and the boundary conditions can be implemented easily, which leads to a greater computational precision. When the enriched BEFM is used, the singularity of the stresses at the tip of the crack can be shown better than that in the BEFM. For the purposes of demonstration, some selected numerical examples are solved using the enriched BEFM.     

General steady-state solutions for transversely isotropic thermoporoelastic media in three dimensions and its application

This paper presents a set of 3D general solutions for thermoporoelastic media for the steady-state problem. By introducing two displacement functions, the equations governing the elastic, pressure and temperature fields are simplified. The operator theory and superposition principle are then employed to express all the physical quantities in terms of two functions, one of which satisfies a quasi–Laplace equation and the other satisfies a differential equation of the eighth order. The generalized Almansi's theorem is used to derive the displacements, pressure and temperature in terms of five quasi-harmonic functions for various cases of material characteristic roots. To show its practical significance, an infinite medium containing a penny-shaped crack subjected to mechanical, pressure and temperature loads on the crack surface is given as an example. A potential theory method is employed to solve the problem. One integro-differential equation and two integral equations are derived, which bear the same structures to those reported in literature. For a penny-shaped crack subjected to uniformly distributed loads, exact and complete solutions in terms of elementary functions are obtained, which can serve as a benchmark for various kinds of numerical codes and approximate solutions.     

The use of a reformulation of the hydrostatic approximation Navier-Stokes equations for geophysical fluid problems

In order to simulate geophysical general circulation processes, to simplify the governing equations of motion, often the vertical momentum equation of the Navier-Stokes equations is replaced by the hydrostatic approximation equation. The resulting equations are reformulated and a variational formulation of the linearized problem is derived. Iteration schemes are presented to solve this problem. A finite element method is discussed, as well as a finite difference method which is based on a grid that is often used in geophysical general circulation models. The schemes are extended to the non-linear case. Numerical examples are presented to demonstrate the performance of the derived iteration schemes.