Numerical simulation of unsteady multidimensional free surface motions by level set method

This paper presents a numerical method that couples the incompressible Navier–Stokes equations with the level set method in a curvilinear co‐ordinate system for study of free surface flows. The finite volume method is used to discretize the governing equations on a non‐staggered grid with a four‐step fractional step method. The free surface flow problem is converted into a two‐phase flow system on a fixed grid in which the free surface is implicitly captured by the zero level set. We compare different numerical schemes for advection of the level set function in a generalized curvilinear format, including the third order quadratic upwind interpolation for convective kinematics (QUICK) scheme, and the second and third order essentially non‐oscillatory (ENO) schemes. The level set equations of evolution and reinitialization are validated with benchmark cases, e.g. a stationary circle, a rotating slotted disk and stretching of a circular fluid element. The coupled system is then applied to a travelling solitary wave, and two‐ and three‐dimensional dam breaking problems. Some interesting free surface phenomena are revealed by the computational results, such as, the large free surface vortices, air entrapment and splashing of the water surge front. The computational results are in excellent agreement with theoretical predictions and experimental data, where they are available. Copyright © 2003 John Wiley & Sons, Ltd.     

A level set technique applied to unsteady free surface flows

An unsteady Navier–Stokes solver for incompressible fluid is coupled with a level set approach to describe free surface motions. The two‐phase flow of air and water is approximated by the flow of a single fluid whose properties, such as density and viscosity, change across the interface. The free surface location is captured as the zero level of a distance function convected by the flow field. To validate the numerical procedure, two classical two‐dimensional free surface problems in hydrodynamics, namely the oscillating flow in a tank and the waves generated by the flow over a bottom bump, are studied in non‐breaking conditions, and the results are compared with those obtained with other numerical approaches. To check the capability of the method in dealing with complex free surface configurations, the breaking regime produced by the flow over a high bump is analyzed. The analysis covers the successive stages of the breaking phenomenon: the steep wave evolution, the falling jet, the splash‐up and the air entrainment. In all phases, numerical results qualitatively agree with the experimental observations. Finally, to investigate a flow in which viscous effects are relevant, the numerical scheme is applied to study the wavy flow past a submerged hydrofoil. Copyright © 2001 John Wiley & Sons, Ltd.     

A simulation of free surface waves for incompressible two‐phase flows using a curvilinear level set formulation

A level set formulation in a generalized curvilinear coordinate is developed to simulate the free surface waves generated by moving bodies or the sloshing of fluid in a container. The Reynolds‐averaged Navier–Stokes (RANS) equations are modified to account for variable density and viscosity in two‐phase (i.e. water–air) fluid flow systems. A local level set method is used to update the level set function and a least square technique adopted to re‐initialize it at each time step. To assess the developed algorithm and its versatility, a selection of different fluid–structure interaction problems are examined, i.e. an oscillating flow in a two‐dimensional square tank, a breaking dam involving different density fluids, sloshing in a two‐dimensional rectangular tank and a Wigley ship hull travelling in calm water. Copyright © 2005 John Wiley & Sons, Ltd.     

Numerical analysis of moving interfaces using a level set method coupled with adaptive mesh refinement

A novel numerical scheme is developed by coupling the level set method with the adaptive mesh refinement in order to analyse moving interfaces economically and accurately. The finite element method (FEM) is used to discretize the governing equations with the generalized simplified marker and cell (GSMAC) scheme, and the cubic interpolated pseudo‐particle (CIP) method is applied to the reinitialization of the level set function. The present adaptive mesh refinement is implemented in the quadrangular grid systems and easily embedded in the FEM‐based algorithm. For the judgement on renewal of mesh, the level set function is adopted as an indicator, and the threshold is set at the boundary of the smoothing band. With this criterion, the variation of physical properties and the jump quantity on the free surface can be calculated accurately enough, while the computation cost is largely reduced as a whole. In order to prove the validity of the present scheme, two‐dimensional numerical simulation is carried out in collapse of a water column, oscillation and movement of a drop under zero gravity. As a result, its effectiveness and usefulness are clearly shown qualitatively and quantitatively. Among them, the movement of a drop due to the Marangoni effect is first simulated efficiently with the present scheme. Copyright © 2004 John Wiley & Sons, Ltd.     

An unsteady single‐phase level set method for viscous free surface flows

The single‐phase level set method for unsteady viscous free surface flows is presented. In contrast to the standard level set method for incompressible flows, the single‐phase level set method is concerned with the solution of the flow field in the water (or the denser) phase only. Some of the advantages of such an approach are that the interface remains sharp, the computation is performed within a fluid with uniform properties and that only minor computations are needed in the air. The location of the interface is determined using a signed distance function, and appropriate interpolations at the fluid/fluid interface are used to enforce the jump conditions. A reinitialization procedure has been developed for non‐orthogonal grids with large aspect ratios. A convective extension is used to obtain the velocities at previous time steps for the grid points in air, which allows a good estimation of the total derivatives. The method was applied to three unsteady tests: a plane progressive wave, sloshing in a two‐dimensional tank, and the wave diffraction problem for a surface ship, and the results compared against analytical solutions or experimental data. The method can in principle be applied to any problem in which the standard level set method works, as long as the stress on the second phase can be specified (or neglected) and no bubbles appear in the flow during the computation. Copyright © 2006 John Wiley & Sons, Ltd.     

An accurate anisotropic adaptation method for solving the level set advection equation

In the present paper, a mesh adaptation process for solving the advection equation on a fully unstructured computational mesh is introduced, with a particular interest in the case it implicitly describes an evolving surface. This process mainly relies on a numerical scheme based on the method of characteristics. However, low order, this scheme lends itself to a thorough analysis on the theoretical side. It gives rise to an anisotropic error estimate which enjoys a very natural interpretation in terms of the Hausdorff distance between the exact and approximated surfaces. The computational mesh is then adapted according to the metric supplied by this estimate. The whole process enjoys a good accuracy as far as the interface resolution is concerned. Some numerical features are discussed and several classical examples are presented and commented in two or three dimensions. Copyright © 2011 John Wiley & Sons, Ltd.     

Numerical simulation of bubble and droplet deformation by a level set approach with surface tension in three dimensions

In this paper we present a three‐dimensional Navier–Stokes solver for incompressible two‐phase flow problems with surface tension and apply the proposed scheme to the simulation of bubble and droplet deformation. One of the main concerns of this study is the impact of surface tension and its discretization on the overall convergence behavior and conservation properties. Our approach employs a standard finite difference/finite volume discretization on uniform Cartesian staggered grids and uses Chorin's projection approach. The free surface between the two fluid phases is tracked with a level set (LS) technique. Here, the interface conditions are implicitly incorporated into the momentum equations by the continuum surface force method. Surface tension is evaluated using a smoothed delta function and a third‐order interpolation. The problem of mass conservation for the two phases is treated by a reinitialization of the LS function employing a regularized signum function and a global fixed point iteration. All convective terms are discretized by a WENO scheme of fifth order. Altogether, our approach exhibits a second‐order convergence away from the free surface. The discretization of surface tension requires a smoothing scheme near the free surface, which leads to a first‐order convergence in the smoothing region. We discuss the details of the proposed numerical scheme and present the results of several numerical experiments concerning mass conservation, convergence of curvature, and the application of our solver to the simulation of two rising bubble problems, one with small and one with large jumps in material parameters, and the simulation of a droplet deformation due to a shear flow in three space dimensions. Furthermore, we compare our three‐dimensional results with those of quasi‐two‐dimensional and two‐dimensional simulations. This comparison clearly shows the need for full three‐dimensional simulations of droplet and bubble deformation to capture the correct physical behavior. Copyright © 2009 John Wiley & Sons, Ltd.     

Numerical simulation of free‐surface flow using the level‐set method with global mass correction

A new numerical method that couples the incompressible Navier–Stokes equations with the global mass correction level‐set method for simulating fluid problems with free surfaces and interfaces is presented in this paper. The finite volume method is used to discretize Navier–Stokes equations with the two‐step projection method on a staggered Cartesian grid. The free‐surface flow problem is solved on a fixed grid in which the free surface is captured by the zero level set. Mass conservation is improved significantly by applying a global mass correction scheme, in a novel combination with third‐order essentially non‐oscillatory schemes and a five stage Runge–Kutta method, to accomplish advection and re‐distancing of the level‐set function. The coupled solver is applied to simulate interface change and flow field in four benchmark test cases: (1) shear flow; (2) dam break; (3) travelling and reflection of solitary wave and (4) solitary wave over a submerged object. The computational results are in excellent agreement with theoretical predictions, experimental data and previous numerical simulations using a RANS‐VOF method. The simulations reveal some interesting free‐surface phenomena such as the free‐surface vortices, air entrapment and wave deformation over a submerged object. Copyright © 2009 John Wiley & Sons, Ltd.     

On surface tension modelling using the level set method

The paper describes and compares the performance of two options for numerically representing the surface tension force in combination with the level set interface‐tracking method. In both models, the surface tension is represented as a body force, concentrated near the interface, but the technical implementation is different: the first model is based on a traditional level set approach in which the force is distributed in a band around the interface using a regularized delta function, whereas in the second, the force is partly distributed in a band around the interface and partly localized to the actual computational cells containing the interface. A comparative study, involving analysis of several two‐phase flows with moving interfaces, shows that in general the two surface tension models produce results of similar accuracy. However, in the particular case of merging and pinching‐off of interfaces, the traditional level set model of surface tension produces an error that results in non‐converging solutions for film‐like interfaces (i.e. ones involving large contact areas). In contrast, the second model, based on the localized representation of the surface tension force, displays consistent first‐order convergence. Copyright © 2008 John Wiley & Sons, Ltd.     

Optimization of unsteady incompressible Navier–Stokes flows using variational level set method

This paper presents the optimization of unsteady Navier–Stokes flows using the variational level set method. The solid–liquid interface is expressed by the level set function implicitly, and the fluid velocity is constrained to be zero in the solid domain. An optimization problem, which is constrained by the Navier–Stokes equations and a fluid volume constraint, is analyzed by the Lagrangian multiplier based adjoint approach. The corresponding continuous adjoint equations and the shape sensitivity are derived. The level set function is evolved by solving the Hamilton–Jacobian equation with the upwind finite difference method. The optimization method can be used to design channels for flows with or without body forces. The numerical examples demonstrate the feasibility and robustness of this optimization method for unsteady Navier–Stokes flows.Copyright © 2012 John Wiley & Sons, Ltd.     

用隐式方法和WENO格式计算悬停旋翼跨声速无粘流场

寻找一种能够准确计算以涡为主要特征的复杂流场和克服尾迹耗散问题的数值方法,一直是旋翼空气动力学研究的热点和难点。本文发展了一种基于高阶迎风格式计算悬停旋翼无粘流场的隐式数值方法。无粘通量采用Roe通量差分分裂格式,为提高精度,使用五阶WENO格式进行左右状态插值,并与MUSCL插值进行比较。为提高收敛到定常解的效率,时间推进采用LU-SGS隐式方法。用该方法对一跨声速悬停旋翼无粘流场进行了数值计算,数值结果表明WENO-Roe的激波分辨率高于MUSCL-Roe,体现出了格式精度的提高对计算结果的改善,LU-SGS隐式方法的计算效率比5步Runge-Kutta显式方法的高。     

Convected level set method for the numerical simulation of fluid buckling

‘Fluid buckling’ is a phenomenon characterized mainly by the existence of fluid toroidal oscillations during flow. It appears when a high viscosity fluid flows vertically against a flat surface and may occur in industrial applications, as in injection molding of a propergol in complex‐shaped cavities. These coiling or folding oscillations appear during the mold filling stage, leading to air entrapment. To understand and to model this free surface flow problem, a convected level set method is proposed. First, a sinus filter is applied to the distance function to get a smooth truncation far from the interface. Second, the reinitialization is embedded in the transport equation model, avoiding it as a separate step during calculation. In order to validate the method, numerical results are presented on classical interface capturing benchmarks. Finally, results are shown on two‐dimensional and three‐dimensional viscous jet buckling problems. Copyright © 2010 John Wiley & Sons, Ltd.     

A simple finite element method for Stokes flows with surface tension using unfitted meshes

This paper presents a simple finite element method for Stokes flows with surface tension. The method uses an unfitted mesh that is independent of the interface. Due to the surface force, the pressure has a jump across the interface. Based on the properties of the level set function that implicitly represents the interface, the jump of the pressure is removed, and a new problem without discontinuities is formulated. Then, classical stable finite element methods are applied to solve the new problem. Some techniques are used to show that the method is equivalent to an easy‐to‐implement method that can be regarded as a traditional method with a modified pressure space. However, the matrix of the resulting linear system of equations is the same as that of the traditional method. Optimal error estimates are derived for the proposed method. Finally, some numerical tests are presented to confirm the theoretical results. Copyright © 2015 John Wiley & Sons, Ltd.     

Numerical simulation of orbitally shaken viscous fluids with free surface

Orbitally shaken bioreactors are an emerging alternative to stirred‐tank bioreactors for large‐scale mammalian cell culture, but their fluid dynamics is still not well defined. Among the theoretical and practical issues that remain to be resolved, the characterization of the liquid free surface during orbital shaking remains a major challenge because it is an essential aspect of gas transfer and mixing in these reactors. To simulate the fluid behavior and the free surface shape, we developed a numerical method based on the finite element framework. We found that the large density ratio between the liquid and the gas phases induced unphysical results for the free surface shape. We therefore devised a new pressure correction scheme to deal with large density ratios. The simulations operated with this new scheme gave values of wave amplitude similar to the ones measured experimentally. These simulations were used to calculate the shear stress and to study the mixing principle in orbitally shaken bioreactors. Copyright © 2012 John Wiley & Sons, Ltd.     

An arbitrary Lagrangian Eulerian (ALE) formulation for free surface flows using the characteristic‐based split (CBS) scheme

An arbitrary Lagrangian Eulerian (ALE) method for non‐breaking free surface flow problems is presented. The characteristic‐based split (CBS) scheme has been employed to solve the ALE equations. A simple mesh smoothing procedure based on coordinate averaging (Laplacian smoothing) is employed in the calculations. The mesh velocity is calculated at each time step and incorporated as part of the scheme. Results presented show an excellent agreement with the available experimental data. Copyright © 2005 John Wiley & Sons, Ltd.     

A new implicit surface tension implementation for interfacial flows

A new implementation of surface tension effects in interfacial flow codes is proposed which is both fully implicit in space, that is the interface never has to be reconstructed, and also semi‐implicit in time, with semi‐implicit referring to the time integration of the surface tension forces. The main idea is to combine two previously separate techniques to yield a new expression for the capillary forces. The first is the continuum surface force (CSF) method, which is used to regularize the discontinuous surface tension force term. The regularization can be elegantly implemented with the use of distance functions, which makes the level set method a suitable choice for the interface‐tracking algorithm. The second is to use a finite element discretization together with the Laplace–Beltrami operator, which enables simple reformulation of the surface tension term into its semi‐implicit equivalent. The performance of the new method is benchmarked against standard explicit methods, where it is shown that the new method is significantly more robust for the chosen test problems when the time steps exceed the numerical capillary time step restriction. Some improvements are also found in the average number of nonlinear iterations and linear multigrid steps taken while solving the momentum equations. Copyright © 2005 John Wiley & Sons, Ltd.     

A Q2Q1 finite element/level‐set method for simulating two‐phase flows with surface tension

A Q2Q1 (quadratic velocity/linear pressure) finite element/level‐set method was proposed for simulating incompressible two‐phase flows with surface tension. The Navier–Stokes equations were solved using the Q2Q1 integrated FEM, and the level‐set variable was linearly interpolated using a ‘pseudo’ Q2Q1 finite element when calculating the density and viscosity of a fluid to avoid an unbounded density/viscosity. The advection of the level‐set function was calculated through the Taylor–Galerkin method, and the direct approach method is employed for reinitialization. The proposed method was tested by solving several benchmark problems including rising bubbles exhibiting a large density difference and the surface tension effect. The numerical results of the rising bubbles were compared with the existing results to validate the benchmark quantities such as the centroid, circularity, and rising velocity. Furthermore, we focused our attention mainly on mass conservation and time‐step. We observed that the present method represented a convergence rate between 1.0 and 1.5 orders in terms of mass conservation and provided more stable solutions even when using a larger time‐step than the critical time‐step that was imposed because of the explicit treatment of surface tension. Copyright © 2011 John Wiley & Sons, Ltd.     

Time-dependent liquid metal flows with free convection and a deformable free surface

The finite element method is employed to investigate time-dependent liquid metal flows with free convection, free surfaces and Marangoni effects. The liquid circulates in a two-dimensional shallow trough with differentially heated vertical walls. The spatial formulation incorporates mixed Lagrangian approximations to the velocity, pressure, temperature and free surface position. The time integration is performed with the backward Euler and trapezoid rule methods with step size control. The Galerkin method is used to reduce the problem to a set of non-linear equations which are solved with the Newton–Raphson method. Calculations are performed for conditions relevant to the electron beam vaporization of refractory metals. The Prandtl number is 0·015 and Grashof number are in the transition range between laminar and turbulent flow. The results reveal the effects of flow intensity, surface tension gradients, mesh refinement and time integration strategy.     

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.     

A spectral method for free surface flows of inviscid fluids

In this paper, an efficient numerical method for unsteady free surface motions, with simple geometries, has been devised. Under the potential flow assumption, the governing equation of free surface flows becomes a Laplace equation, which is treated here by means of a series expansions of the velocity potential. The free surface is represented with a height function. The present method is applied to surface gravity waves to test the stability and accuracy of the method. To show the versatility of the method, a model for a dip formation is considered. © 1998 John Wiley & Sons, Ltd.