A hybrid finite-element–volume-of-fluid method for simulating free surface flows and interfaces

A numerical technique is developed for the simulation of free surface flows and interfaces. This technique combines the strength on the finite element method (FEM) in calculating the field variables for a deforming boundary and the versatility of the volume-of-fluid (VOF) technique in advection of the fluid interfaces. The advantage of the VOF technique is that it allows the simulation of interfaces with large deformations, including surface merging and breaking. However, its disadantage is that is solving the flow equations, it cannot resolve interfaces smaller than the cell size, since information on the subgrid scale is lost. Therefore the accuracy of the interface reconstruction and the treatment of the boundary conditions (i.e. viscous stresses and surface tension forces) become grid-size-dependent. On the other hand, the FEM with deforming interface mesh allows accurate implementation of the boundary conditions, but it cannot handle large surface deformations occurring in breaking and merging of liquid regions. Combining the two methods into a hybrid FEM-VOF method eliminates the major shortcomings of both. The outcome is a technique which can handle large surface deformations with accurate treatment of the boundary conditions. For illustration, two computational examples are presented, namely the instability and break-up of a capillary jet and the coalescence collision of two liquid drops.     

Application of a vortex method to free surface flows

Vortex methods have found wide applications in various practical problems. The use of vortex methods in free surface flow problems, however, is still very limited. This paper demonstrates a vortex method for practical computation of non-linear free surface flows produced by moving bodies. The method is a potential flow formulation which uses the exact non-linear free surface boundary condition at the exact location of the instantaneous free surface. The position of the free surface, on which vortices are distributed, is updated using a Lagrangian scheme following the fluid particles on the free surface. The vortex densities are updated by the non-linear dynamic boundary condition, derived from the Euler equations, with an iterative Lagrangian numerical scheme. The formulation is tested numerically for a submerged circular cylinder in unsteady translation. The iteration is shown to converge for all cases. The results of the unsteady simulations agree well with classical linearized solutions. The stability of the method is also discussed.     

An efficient three-dimensional semi-implicit finite element scheme for simulation of free surface flows

An efficient semi-implicit finite element model is proposed for the simulation of three-dimensional flows in stratified seas. The body of water is divided into a number of layers and the two horizontal momentum equations for each layer of water are first integrated vertically. Nine-node Lagrangian quadratic isoparametric elements are employed for spatial discretization in the horizontal domain. The time derivatives are approximated using a second-order-accurate semi-implicit time-stepping scheme. The distinguishing feature of the proposed numerical scheme is that only nodal values on the same vertical line are coupled. Two test cases for which analytic solutions are available are employed to test the proposed scheme. The test results show that the scheme is efficient and stable. A numerical experiment is also included to compare the proposed scheme with a finite difference scheme.     

Energy flux equalization: An open boundary condition for non-linear free surface flows

In the simulation of non-linear free surface flows in a finite domain a major concern is with the radiation condition to be applied at the ‘open’ boundary. No general theoretical radiation conditions are known to exist. In this paper a new open boundary condition is formulated based on energy flux equalization between the non-linear inner domain and a linear outer domain. The non-linear flow in the inner domain is solved numerically using a semi-Lagrangian procedure. The energy flux arriving at the open boundary is removed using a new open boundary condition which acts as a linear wave absorber. For the cases studied the performance of this boundary condition is found to be quite good.     

A Kantorovich computational method for free surface gravity flows

Computing free surface gravity flows involves basically two coupled problems, namely, the location of the free surface position and the determination of the internal flow field (for assumed values H₀ and Q of the total head and discharge, respectively). Solution techniques are invariably based on iterative procedures, but those that iterate between the two coupled problems may become unstable. In this paper we present a computational method in which the coupling is kept throughout the process of iteration. This is achieved by converting the coupled problems (by means of the Kantorovich method) into the single problem of finding a set of streamlines, including that of the free surface. These streamlines are moved (iteratively) to satisfy the stationary conditions of the governing variational principle. The algorithm is very stable and converges rapidly. It is also easy to implement to solve various types of steady flows with a free surface under gravity.     

A marker-and-cell approach to viscoelastic free surface flows using the PTT model

This work is concerned with the development of a numerical method capable of simulating two-dimensional viscoelastic free surface flows governed by the non-linear constitutive equation PTT (Phan-Thien–Tanner). In particular, we are interested in flows possessing moving free surfaces. The fluid is modelled by a marker-and-cell type method and employs an accurate representation of the fluid surface. Boundary conditions are described in detail and the full free surface stress conditions are considered. The PTT equation is solved by a high order method which requires the calculation of the extra-stress tensor on the mesh contour. The equations describing the numerical technique are solved by the finite difference method on a staggered grid. In order to validate the numerical method fully developed flow in a two-dimensional channel was simulated and the numerical solutions were compared with known analytic solutions. Convergence results were obtained throughout by using mesh refinement. To demonstrate that complex free surface flows using the PTT model can be computed, extrudate swell and a jet flowing onto a rigid plate were simulated.     

Non-Newtonian viscous oscillating free surface jets, and a new strain-rate dependent viscosity form for flows experiencing low strain rates

A model for oscillating free surface jet flow of a fluid from an elliptical orifice, together with experimental measurements, can be exploited to characterize the elongational viscosity of non-Newtonian inelastic fluids. The oscillating jet flow is predominantly elongational, with a small strain that oscillates rapidly between large and zero strain rates. We find that to reproduce the experimentally observed steady oscillating jet flow in model simulations, the assumed form of the non-Newtonian viscosity as a function of strain rate must have zero gradient, i.e., be Newtonian, at zero strain rate (a behavior exhibited, in general, by real inelastic fluids). We demonstrate that the Cross, Carreau, Prandtl-Eyring, and Powell-Eyring forms, although they have finite viscosity at zero strain rate, have either nonzero or even unbounded gradient at zero, and hence are unable to model oscillating jet behavior. We propose a new non-Newtonian viscous form which has all of the desirable features of existing forms (high and low strain rate plateaus, with adjustable location and steepness of the transition) and the additional feature of Newtonian behavior at low strain rates. Received: 7 February 2000 Accepted: 31 October 2000     

A computational method for simulating transient motions of an incompressible inviscid fluid with a free surface

A new numerical method has been developed for the analysis of unsteady free surface flow problems. The problem under consideration is formulated mathematically as a two-dimensional non-linear initial boundary value problem with unknown quantities of a velocity potential and a free surface profile. The basic equations are discretized spacewise with a boundary element method and timewise with a truncated forward-time Taylor series. The key feature of the present paper lies in the method used to compute the time derivatives of the unknown quantities in the Taylor series. The use of the Taylor series expansion has enabled us to employ a variable time-stepping method. The size of time increment is determined at each time step so that the remainders of the truncated Taylor series should be equal to a given small error limit. Such a variable time-stepping technique has made a great contribution to numerically stable computations. A wave-making problem in a two-dimensional rectangular water tank has been analysed. The computational accuracy has been verified by comparing the present numerical results with available experimental data. Good agreement is obtained.     

A consistent boundary element method for free surface hydrodynamic calculations

Free surface phenomena are described by equations that exhibit two types of non-linearities. The first is inherent to the equations themselves and the second is caused by the application of boundary conditions at a free surface at an unknown location. Numerical calculations usually do not specifically recognize the second non-linearity, nor treat it in a fashion consistent with the more obvious non-linearities in the boundary conditions. A consistent formulation is introduced in the present paper. The field equation is integrated and the free surface boundary conditions are applied on the unknown geometry by means of appropriate series expansions. The consistent formulation introduces improvements in accuracy and computing speed. The method is demonstrated on several hydrodynamic free surface problems and an error analysis is included.     

A net inflow method for incompressible viscous flow with moving free surface

A new finite element procedure called the net inflow method has been developed to simulate time-dependent incompressible viscous flow including moving free surfaces and inertial effects. As a fixed mesh approach with triangular element, the net inflow method can be used to analyse the free surface flow in both regular and irregular domains. Most of the empty elements are excluded from the computational domain, which is adjusted successively to cover the entire region occupied by the liquid. The volume of liquid in a control volume is updated by integrating the net inflow of liquid during each iteration. No additional kinetic equation or material marker needs to be considered. The pressure on the free surface and in the liquid region can be solved explicitly with the continuity equation or implicitly by using the penalty function method. The radial planar free surface flow near a 2D point source and the dam-breaking problem on either a dry bed or a still liquid have been analysed and presented in this paper. The predictions agree very well with available analytical solutions, experimental measurements and/or other numerical results.     

Numerical simulation method for viscoelastic flows with free surfaces—fringe element generation method

To simulate filling flow in injection moulding for viscoelastic fluids, a numerical method, based on a finite element method and a finite volume method, has been developed for incompressible isothermal viscoelastic flow with moving free surfaces. The advantages of this method are, first, good applicability to arbitrarily shaped mould geometries and, second, accurate treatment for boundary conditions on the free surface. Typical filling flows are simulated, namely filling flow into a 1:4 expansion cavity with and without an obstacle. Numerical results predict the position of weld lines and air-traps. The method also indicates the effects of elongational flow on molecular orientation.     

Modeling jet flows caused by the incidence of a shock wave on a profiled free surface

The problem of the incidence of a shock wave with a front-pressure amplitude of about 30 GPa at the profiled free surface of an aluminum sample is studied. It is shown that in the case of large perturbations (amplitude 1 mm and wavelength 10 mm), jet flows occur on the free surface. The data obtained are described using a kinetic fracture model that takes into account the damage initiation and growth in the material due to tensile stress and shear strain. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 48, No. 1, pp. 16–23, January–February, 2007.     

Development of new triangular elements for free surface flows

New finite elements have been developed to simulate steady and unsteady two-dimensional free surface flows. The depth-averaged velocity components with the free surface elevation have been used as independent variables in the model. The differences between the various elements presented lie in the choice of velocity approximation. The Newton–Raphson method has been used to solve the non-linear system of equations. Emphasis is put on bench-mark examples to assess the accuracy and efficiency of the elements. A simple stable new element tested herein shows promising advantages for industrial finite element codes.     

Multiscale simulation of viscoelastic free surface flows

A micro–macro approach based on combining the Brownian configuration fields (BCF) method [M.A. Hulsen, A.P.G. van Heel, B.H.A.A. van den Brule, Simulation of viscoelastic flow using Brownian configuration fields, J. Non-Newtonian Fluid Mech. 70 (1997) 79–101] with an Arbitrary Lagrangian–Eulerian (ALE) Galerkin finite element method, using elliptic mesh generation equations coupled with time-dependent conservation equations, is applied to study slot coating flows of polymer solutions. The polymer molecules are represented by dumbbells with both linear and non-linear springs; hydrodynamic interactions between beads are incorporated. Calculations with infinitely extensible (Hookean) and pre-averaged finitely extensible (FENE-P) dumbbell models are performed and compared with equivalent closed-form macroscopic models in a conformation tensor based formulation [M. Pasquali, L.E. Scriven, Free surface flows of polymer solutions with models based on the conformation tensor, J. Non-Newtonian Fluid Mech. 108 (2002) 363–409]. The BCF equation for linear dumbbell models is solved using a fully implicit time integration scheme which is found to be more stable than the explicit Euler scheme used previously to compute complex flows. We find excellent agreement between the results of the BCF based formulation and the macroscopic conformation tensor based formulation. The computations using the BCF approach are stable at much higher Weissenberg numbers, (where λ is the characteristic relaxation time of polymer, and is the characteristic rate of strain) compared to the purely macroscopic conformation tensor based approach, which fail beyond a maximum Wi. A novel computational algorithm is introduced to compute complex flows with non-linear microscopic constitutive models (i.e. non-linear FENE dumbbells and dumbbells with hydrodynamic interactions) for which no closed-form constitutive equations exist. This algorithm is fast and computationally efficient when compared to both an explicit scheme and a fully implicit scheme involving the solution of the non-linear equations with Newton’s method for each configuration field.     

A numerical investigation of 2d,steady free surface flows

Systematic tests have been performed to study the behaviour of a numerical method developed to calculate 2D, steady free surface flows. The Reynolds equations are solved in the physical space by employing a non–orthogonal staggered grid, while the k-ε model is adopted to approximate the Reynolds stresses. The free surface is calculated following an iterative procedure and various parameters that affect convergence and accuracy of the numerical solution have been examined. Calculated results are compared with measured data for two cases, i.e. the wave generation above a bottom topography at various Froude numbers and the free surface formation above a submerged hydrofoil. © 1997 John Wiley & Sons, Ltd.     

Lattice Boltzmann model for thermal free surface flows with liquid-solid phase transition

Purpose

The main objective of this work is to develop an algorithm to use the Lattice Boltzmann method for solving free surface thermal flow problems with solid/liquid phase changes.

Approach

A multi-distribution function model is applied to simulate hydrodynamic flow and the coupled thermal diffusion-convection problem.

Findings

The free surface problem, i.e. the reconstruction of the missing distribution functions at the interface, can be solved by applying a physical transparent momentum and heat flux based methodology. The developed method is subsequently applied to some test cases in order to assess its computational potentials.

Practical implications

Many industrial processes involve problems where non-isothermal motion and simultaneous solidification of fluids with free surface is important. Examples are all castings processes and especially foaming processes which are characterized by a huge and strongly changing surface.

Value

A reconstruction algorithm to treat a thermal hydrodynamic problem with free surfaces is presented which is physically transparent and easy to implement.     

Validation of a CIP-based tank for numerical simulation of free surface flows

A constrained interpolation profile CIP-based numerical tank is developed to simulate violent free surface flows.The numerical simulation is performed by the CIP-based Cartesian grid method,which is described in the present paper.The tangent of hyperbola for interface capturing(THINC) scheme is applied for capturing complex free surfaces.The new model is capable of simulating a flow with violently varied free surface.A series of computations are conducted to assess the developed algorithm and its versatility.These tests include the collapse of water column with and without an obstacle,sloshing in a fixed tank,the generation of regular waves in a tank,the generation of extreme waves in a tank.Excellent agreements are obtained when numerical results are compared with available analytical,experimental,and other numerical results.     

A multigrid pseudo-spectral method for incompressible Navier–Stokes flows

A pseudo-spectral solver with multigrid acceleration for the numerical prediction of incompressible non-isothermal flows is presented. The spatial discretization is based on a Chebyshev collocation method on Gauss–Lobatto points and for the discretization in time the second-order backward differencing scheme (BDF2) is employed. The multigrid method is invoked at the level of algebraic system solving within a pressure-correction method. The approach combines the high accuracy of spectral methods with efficient solver properties of multigrid methods. The capabilities of the proposed scheme are illustrated by a buoyancy driven cavity flow as a standard benchmark case. To cite this article: K. Krastev, M. Schäfer, C. R. Mecanique 333 (2005).     

A boundary-integral equation method for free surface viscous flows

Summary A boundary integral equation method is proposed for the solution of viscous recirculating flows with free surfaces. In particular the method is applied to thermocapillary convection and to drop formation, both in micro-gravity conditions, the latter to test its capability to handle real unsteady problems.The presence of non linear terms in Navier-Stokes equations leads to a volume integral, which has to be approximated by a linearization procedure.Several numerical results for thermocapillary flows, both with fixed and moving free surface, are discussed in comparison with previously obtained finite difference solutions. Some preliminary results, and in particular the time evolution of the free surface shape, are also presented for the drop formation problem. Only plane two dimensional fields are considered for both problems.

---

Sommario Si propone un metodo basato sulla soluzione di equazioni integrali di contorno per flussi viscosi con superficie libera. Tale metodo è applicato allo studio della convezione termocapillare ed al processo di formazione di una goccia, entrambi in condizioni di microgravità. La presenza dei termini non lineari nell'equazione di Navier-Stokes comporta un integrale di volume che viene approssimato mediante un processo di linearizzazione.Risultati numerici per flussi termocapillari con superficie libera sia fissa che mobile sono confrontati con altri ottenuti in precedenza con un metodo alle differenze finite. Si presentano inoltre alcuni risultati preliminari sul problema della formazione della goccia ed in particolare l'evoluzione nel tempo della configurazione geometrica della superficie libera. Nei due casi si analizzano solo campi bidimensionali.

---

---

Presented at the VII National Conference AIDAA, Naples, September 1983.

---

In leave of absence from Tianjin University, China.