Free Surface Flow Model Research Articles

Lattice Boltzmann Model for Free Surface Flow for Modeling Foaming

We present a 2D- and 3D-lattice Boltzmann model for the treatment of free surface flows including gas diffusion. Interface advection and related boundary conditions are based on the idea of the lattice Boltzmann equation. The fluid dynamic boundary conditions are approximated by using the mass and momentum fluxes across the interface, which do not require explicit calculation of gradients. A similar procedure is applied to fulfill the diffusion boundary condition. Simple verification tests demonstrate the correctness of the algorithms. 2D- and 3D-foam evolution examples demonstrate the potential of the method.

A 3‐D non‐hydrostatic pressure model for small amplitude free surface flows

AbstractA three‐dimensional, non‐hydrostatic pressure, numerical model with k–ε equations for small amplitude free surface flows is presented. By decomposing the pressure into hydrostatic and non‐hydrostatic parts, the numerical model uses an integrated time step with two fractional steps. In the first fractional step the momentum equations are solved without the non‐hydrostatic pressure term, using Newton's method in conjunction with the generalized minimal residual (GMRES) method so that most terms can be solved implicitly. This method only needs the product of a Jacobian matrix and a vector rather than the Jacobian matrix itself, limiting the amount of storage and significantly decreasing the overall computational time required. In the second step the pressure–Poisson equation is solved iteratively with a preconditioned linear GMRES method. It is shown that preconditioning reduces the central processing unit (CPU) time dramatically. In order to prevent pressure oscillations which may arise in collocated grid arrangements, transformed velocities are defined at cell faces by interpolating velocities at grid nodes. After the new pressure field is obtained, the intermediate velocities, which are calculated from the previous fractional step, are updated. The newly developed model is verified against analytical solutions, published results, and experimental data, with excellent agreement. Copyright © 2005 John Wiley & Sons, Ltd.

A Smart Bi-Directional Model of Heat Transfer and Free Surface Flow in Gas-Metal-Arc Fillet Welding for Practising Engineers

In recent years, numerical heat transfer and fluid flow models have provided significant insight about fusion welding processes and welded materials. A major problem in their practical use is that several input parameters cannot be easily prescribed from fundamental principles. Available inverse models of fusion welding for the determination of these unknown parameters have ignored important physical processes such as convection in the weld pool to make computational tasks tractable. As a result, these models are not very different from the neural network type models that are not required to obey any physical law. A smart, bi-directional, numerical model has been developed to determine three-dimensional temperature and velocity profiles, weld geometry and the shape of the solidified weld reinforcement surface during gas metal arc (GMA) welding of fillet joints. Apart from the transport of heat from the welding arc, additional heat from the metal droplets was also considered in the model. The model is capable of estimating unknown parameters such as the arc efficiency, effective thermal conductivity and effective viscosity from a limited number of data on weld geometry based on multivariable optimisation. Alternative strategies for the optimisation are examined. The calculated shape and size of the fusion zone, finger penetration characteristic of the GMA welds and the solidified free surface profile were in fair agreement with the experimental results for various welding conditions. In particular, the computed values of the leg length, the penetration depth and the actual throat agreed well with those measured experimentally for various heat inputs. The weld thermal cycles and the cooling rates were also in good agreement with the independent experimental data. The research presented here shows that advances in computational hardware and software have now made construction of smart, bi-directional, large transport phenomena based phenomenological models a useful undertaking.

Singularity of free-surface curvature in convergent flow: Cusp or corner?

The nature of singularities that arise in the mathematical modeling of free-surface flows and the ways of their analysis and regularization aimed at removing the physically unacceptable features is one of fundamental issues in theoretical fluid dynamics. The present work considers the type of the free-surface curvature singularity emerging in the steady two-dimensional convergent flow of a Newtonian fluid near a free boundary. The unphysical singularities in the flow field, unavoidable in the conventional model, are removed by describing this flow as a particular case of the interface formation/disappearance process in the framework of an earlier developed macroscopic theory of such processes which is applied without any ad hoc alterations. The near-field asymptotic analysis of the problem shows that at finite capillary numbers the singularity of the free-surface curvature is always a sharp corner, not a cusp.

Development of a 3D non-hydrostatic pressure model for free surface flows

A three-dimensional, non-hydrostatic pressure, numerical model for free surface flows is presented. By decomposing the pressure term into hydrostatic and non-hydrostatic parts, the numerical model uses an integrated time step with two fractional steps. In the first fractional step, the momentum equations are solved without the hydrostatic pressure term using Newton's method in conjunction with the generalised minimal residual (GMRES) method. This combined method does not require the determination of a Jacobian matrix explicitly but simply the product of the Jacobian and a vector, thereby reducing the amount of storage required and significantly decreasing the overall computational time required. By using Newton's method, the numerical model can handle implicitly almost all variables, unlike many other numerical models. Hence numerical stability is achieved effectively. In the second fractional step, the pressure-Poisson equation is solved iteratively with a preconditioned linear GMRES method. It is shown that preconditioning reduces the processing time dramatically. After the new pressure field is obtained the intermediate velocities, which are calculated from the previous fractional step, are updated and then these updated velocities preserve the local mass conservation. The newly developed model is verified against analytical solutions, with good agreement.

Asymptotic model for free surface flow of an electrically conducting fluid in a high-frequency magnetic field

We study the behaviour of a layer of an electrically conducting inviscid incompressible fluid in a high-frequency alternating magnetic field. We derive nonlinear asymptotic equations governing the evolution of the fluid layer in the high-frequency limit. As a test for the model, we consider the linearised stability problem for an infinite planar free surface of a layer of finite depth.

Simulation of free-surface flow in a tank using the Navier-Stokes model and unstructured finite volume method

Modelling free-surface flow has very important applications in many engineering areas such as oil transportation and offshore structures. Current research focuses on the modelling of free surface flow in a tank by solving the Navier-Stokes equation. An unstructured finite volume method is used to discretize the governing equations. The free surface is tracked by dynamically adapting the mesh and making it always surface conforming. A mesh-smoothing scheme based on the spring analogy is also implemented to ensure mesh quality throughout the computaiton. Studies are performed on the sloshing response of a liquid in an elastic container subjected to various excitation frequencies. Further investigations are also carried out on the critical frequency that leads to large deformation of the tank walls. Another numerical simulation involves the free-surface flow past as submerged obstacle placed in the tank to show the flow separation and vortices. All these cases demonstrate the capability of this numerical method in modelling complicated practical problems.

A study of time integration schemes for the numerical modelling of free surface flows

A semi-discrete finite element methodology for the modelling of transient free surface flows in the context of Eulerian interface capturing is proposed. The focus of this study is put on the choice of an appropriate time integration strategy for the accurate modelling of the dynamics of free surfaces and of interfacial physics. It is composed of an adaptive time integration scheme for the Navier–Stokes equations, and of the implicit midpoint rule for the transport equation of the Eulerian marker variable. The adaptive scheme allows the automatic determination of a time-step size that follows the physics of the problem under study, which facilitates the accurate modelling of stiff free surface flows. It is shown that the implicit midpoint rule reduces mass loss for each fluid. Various free surface flow problems are studied to verify and validate the proposed time integration strategy. Copyright © 2005 John Wiley & Sons, Ltd.

Finite element modelling of free-surface flows with non-hydrostatic pressure andk-ɛ turbulence model

Validation of 3D finite element model for free-surface flow is conducted using a high quality and high spatial resolution data set. The commonly numerical models with the conventional hydrostatic pressure still remain the most widely used approach for the solution of practical engineering problems. However, when a 3D description of the velocity field is required, it is useful to resort to a more accurate model in which the hydrostatic assumption is removed. The present research finds its motivation in the increasing need for efficient management of geophysical flows such as estuaries (multiphase fluid flow) or natural rivers with the presence of short waves and/or strong bathymetry gradient, and/or strong channel curvature. A numerical solution is based on the unsteady Reynolds-averaged Navier-Stokes equations on the unstructured grid. The eddy viscosity is calculated from the efficient k-e turbulence model. The model uses implicit fractional step time stepping, and the characteristics method is used to compute the convection terms in the multi-layers system (suitable for the vertical stratified fluid flow), in which the vertical grid is located at predefined heights and the number of elements in the water column depends on water depth. The bottommost and topmost elements of variable height allow a faithful representation of the bed and the time-varying free-surface, respectively. The model is applied to the 3D open channel flows of various complexity

Analysis of a Local Hydrodynamic Model with Marangoni Effect

We study a mathematical model treating of the dynamic contact line problem, supposed to describe the main features of the advancing triple line (rolling motion and variable contact angle) and to remove the singularity. The model is composed by a macroscopic hydrodynamic free surface flow model (HFSM) (Navier–Stokes) coupled with a mesoscopic local surface model (LSM). Detailed mathematical and numerical analysis of the 1D steady-state local surface model are done existence and uniqueness of the exact and numerical solutions, extra properties of the derivatives, and convergence of finite element schemes. Some numerical results of the two models treated separately are presented for a 2D plunging tape configuration.

A Numerical Technique for Solving the Maxwell Model for Free Surface Flows

This work is concerned with the development of a numerical technique for solving free surface flows of a Maxwell fluid. The governing equations for the flow of a Maxwell type fluid together with appropriate boundary conditions are given. The free surface stress conditions are treated in details. A novel formulation for calculating the extra stress components on rigid boundaries is given. The numerical technique presented in this work employs the finite difference method on a staggered grid and employs the ideas of the MAC (Marker-and-Cell) method. Numerical results demonstrating that this numerical technique can solve viscoelastic flows governed by the Maxwell model are presented. Moreover, validation results are presented.

A free surface updating methodology for marker function‐based Eulerian free surface capturing techniques on unstructured meshes

AbstractThe numerical modelling of non‐miscible free surface flows using an interface capturing strategy, where a marker function is used to identify each fluid, is popular in the computational science literature. As the marker function is advected by the fluid flow, the region of transition of the marker gets deformed, this being caused by the finite element discretization of the transport equation, the time‐integration scheme and the type of mesh used. The challenge is to update this region of transition at regular time intervals, without artificially creating or losing mass. A methodology is proposed to update the region of transition of the marker variable on unstructured meshes, using a local least‐squares‐based strategy which reduces numerical mass loss. It is shown that this technique proves to be an effective strategy for the updating of an advected marker variable. Copyright © 2004 John Wiley & Sons, Ltd.

A Cartesian method for fitting the bathymetry and tracking the dynamic position of the shoreline in a three-dimensional, hydrodynamic model

This paper presents a Cartesian method for the simultaneous fitting of the bathymetry and shorelines in a three-dimensional, hydrodynamic model for free-surface flows. The model, named LESS3D (Lake & Estuarine Simulation System in Three Dimensions), solves flux-based finite difference equations in the Cartesian-coordinate system ( x, y, z). It uses a bilinear bottom to fit the bottom topography and keeps track the dynamic position of the shoreline. The resulting computational cells are hybrid: interior cells are regular Cartesian grid cells with six rectangular faces, and boundary/bottom cells (at least one face is the water–solid interface) are unstructured cells whose faces are generally not rectangular. With the bilinear interpolation, the shape of a boundary/bottom cell can be determined at each time step. This allows the Cartesian coordinate model to accurately track the dynamic position of the shorelines. The method was tested with a laboratory experiment of a Tsunami runup case on a circular island. It was also tested for an estuary in Florida, USA. Both model applications demonstrated that the Cartesian method is quite robust. Because the present method does not require any coordinate transformation, it can be an attractive alternative to curvilinear grid model.

A 2-D model of Rayleigh instability in capillary tubes??surfactant effects

The Rayleigh instability of stagnant liquid films lining the interior of capillary tubes is analyzed with the aid of a 2-D free surface flow model; this axisymmetric model is previously validated using already published theoretical and experimental results. The Galerkin-finite element method is used to transform the complete set of governing equations and boundary conditions into a discrete set, which is then simultaneously solved at each time step by Newton's method. Predictions of well known simplified models represented by nonlinear evolution equations derived on the one-dimensional flow assumption are compared with those obtained from the present one. The comparisons are made for pure liquids and also for liquids contaminated with insoluble surfactants; they show that the simpler models represent the free surface evolution reasonable well. However, the 1-D models generally underestimate the time needed to complete the unstable process that ends––if the film is thick enough––when the inner gas phase becomes disconnected due to the formation of liquid lenses regularly spaced; these discrepancies become larger when surface active agents are present. Surfactant effects and the wealth of information produced by the 2-D model are both evidenced through sample results presented at the end of the paper.

A 2-D model of Rayleigh instability in capillary tubes––surfactant effects

The Rayleigh instability of stagnant liquid films lining the interior of capillary tubes is analyzed with the aid of a 2-D free surface flow model; this axisymmetric model is previously validated using already published theoretical and experimental results. The Galerkin-finite element method is used to transform the complete set of governing equations and boundary conditions into a discrete set, which is then simultaneously solved at each time step by Newton's method. Predictions of well known simplified models represented by nonlinear evolution equations derived on the one-dimensional flow assumption are compared with those obtained from the present one. The comparisons are made for pure liquids and also for liquids contaminated with insoluble surfactants; they show that the simpler models represent the free surface evolution reasonable well. However, the 1-D models generally underestimate the time needed to complete the unstable process that ends––if the film is thick enough––when the inner gas phase becomes disconnected due to the formation of liquid lenses regularly spaced; these discrepancies become larger when surface active agents are present. Surfactant effects and the wealth of information produced by the 2-D model are both evidenced through sample results presented at the end of the paper.

Using a piecewise linear bottom to fit the bed variation in a laterally averaged, z‐co‐ordinate hydrodynamic model

AbstractIn developing a 3D or laterally averaged 2D model for free‐surface flows using the finite difference method, the water depth is generally discretized either with the z‐co‐ordinate (z‐levels) or a transformed co‐ordinate (e.g. the so‐called σ‐co‐ordinate or σ‐levels). In a z‐level model, the water depth is discretized without any transformation, while in a σ‐level model, the water depth is discretized after a so‐called σ‐transformation that converts the water column to a unit, so that the free surface will be 0 (or 1) and the bottom will be ‐1 (or 0) in the stretched co‐ordinate system. Both discretization methods have their own advantages and drawbacks. It is generally not conclusive that one discretization method always works better than the other. The biggest problem for the z‐level model normally stems from the fact that it cannot fit the topography properly, while a σ‐level model does not have this kind of a topography‐fitting problem.To solve the topography‐fitting problem in a laterally averaged, 2D model using z‐levels, a piecewise linear bottom is proposed in this paper. Since the resulting computational cells are not necessarily rectangular looking at the x–z plane, flux‐based finite difference equations are used in the model to solve the governing equations. In addition to the piecewise linear bottom, the model can also be run with full cells or partial cells (both full cell and partial cell options yield a staircase bottom that does not fit the real bottom topography). Two frictionless wave cases were chosen to evaluate the responses of the model to different treatments of the topography. One wave case is a boundary value problem, while the other is an initial value problem. To verify that the piecewise linear bottom does not cause increased diffusions for areas with steep bottom slopes, a barotropic case in a symmetric triangular basin was tested. The model was also applied to a real estuary using various topography treatments. The model results demonstrate that fitting the topography is important for the initial value problem. For the boundary value problem, topography‐fitting may not be very critical if the vertical spacing is appropriate. Copyright © 2004 John Wiley & Sons, Ltd.

3-D Modeling of unsteady free-surface flow in open channel

We present the hydrodynamic part of a three-dimensional model for pollutant dispersion in turbulent free-surface flow in open channels. It does not require any strong assumption such as hydrostatic pressure and is able to simulate an unsteady free-surface flow as flood waves or discharge over a weir. Our main objective was to model the moving free-surface and the boundary conditions under two constraints. The first constraint was to keep a point of view as close as possible to that of the 1-D modeling generally used in river engineering, especially relative to the upstream and downstream boundary conditions. The second one was that our model had to be sufficiently flexible to be implemented in a general computational fluid dynamics package. After describing our model, we present some numerical tests and comparisons with 1-D simulations from a hydrodynamic point of view.

A numerical analysis of free-surface flow in curved open channel with velocity?pressure-free-surface correction

This paper presents a numerical study onfree-surface flow in curved open channel. An improved SIMPLEC algorithm with velocity-pressure-free-surface coupled correction is developed and validated. Such algorithm differs from the traditional SIMPLEC algorithm and includes three correction equations which are named as the velocity correction equation, the free-surface correction equation derived from the continuity equation with the kinematic boundary conditions on the free-surface and the bottom bed, and the pressure correction equation taking the same formulation as the traditional SIMPLEC algorithm does. In this study, the improved method is used to solve the incompressible, three-dimensional, Reynolds-averaged Navier–Stokes equation set combined with the standard k–ɛ model and/or the low Reynolds number k–ɛ model for free-surface viscous flow in curved open channels. The power law scheme (POW) is used to discretize the convection terms in these equations with a finite-volume method. The practical cases studied are free-surface flow through the 180° curved open channel with different hydraulic discharge rates. The comparisons between computations and experiments reveal that the model is capable of predicting the detailed velocity field, including changes in secondary motion, the distribution of bed shear, and the variations of flow depth in both the transverse and the longitudinal directions. In summary, the improved SIMPLEC algorithm is feasible and effective for numerical study of free-surface viscous flows in curved open channels.

Arbitrary Lagrangian–Eulerian finite element analysis of free surface flow using a velocity–vorticity formulation

This paper describes the application of velocity–vorticity formulation of the Navier–Stokes equations for two-dimensional free surface flow using an arbitrary Lagrangian–Eulerian method. The velocity Poisson equations and the vorticity transport equations are solved using a finite element method to obtain the velocity and the vorticity fields in the interior region of the computational domain. The boundary-fitted coordinates system is adopted to solve the boundary equations for kinematic and dynamic conditions at the free surface using a finite difference method. The numerical model for the velocity–vorticity formulation is validated for a square cavity flow at Re=400 and 1000. The solitary wave reflected from a vertical wall is chosen as a test case for comparison and validation of the free surface flow model. Then the proposed numerical model is used to obtain flow results for the following free surface flow cases: (i) interaction between two opposite solitary waves, (ii) seiche phenomenon in a rectangular reservoir, and (iii) solitary wave through a submerged rectangular structure in a viscous fluid. The efficiency of the present numerical model for numerical treatment of free surface flows is discussed. Furthermore the advantage of this formulation with respect to primitive variables formulation is addressed from the computational point of view.

Three-dimensional hydrodynamic model for free surface flow

A three-dimensional mathematical model for free surface flow based on the Reynolds-averaged Navier-Stokes equations is developed in a sigma co-ordinate system. The time-splitting method is used to separate advection and diffusion terms from the pressure terms in the governing equations. The pressure variable is further separated into hydrostatic and hydrodynamic pressures so that the computer rounding errors can be largely avoided. The resulting hydrodynamic pressure equation is solved by a multigrid method, while the hydrostatic pressure equations are solved very efficiently by an Alternating Direction Implicit (ADI) scheme. The convection terms are discretized by the Roe's scheme of second-order accuracy. A staggered mesh is used. The model is tested against available analytical solutions and experimental data. Good agreement has been achieved.