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 viscous flow: Buckling of planar jets

The phenomenon of viscous fluid buckling has a long and distinguished history, dating back to Taylor (1968). This paper is concerned with demonstrating that a numerical method, GENSMAC, is capable of simulating this physical instability. A table of the parameter values (e.g. the Reynolds number, the Froude number, inlet width, inlet velocity and aspect ratio) is provided giving details of when buckling occurs and when it does not. This allows the deduction of a possible buckling condition in terms of the Reynolds number and the ratio of height of the jet to the inlet width, modifying a previous hypothesis. Visualization of jet buckling is provided. This work has been motivated by the need of industry to understand jet filling of containers; jet buckling can lead to air entrapment and this is undesirable. Copyright © 1999 John Wiley & Sons, Ltd.     

Numerical simulation of the vertical structure of discontinuous flows

A numerical method to solve the Reynolds‐averaged Navier–Stokes equations with the presence of discontinuities is outlined and discussed. The pressure is decomposed into the sum of a hydrostatic component and a hydrodynamic component. The numerical technique is based upon the classical staggered grids and semi‐implicit finite difference methods applied for quasi‐ and non‐hydrostatic flows. The advection terms in the momentum equations are approximated in order to conserve mass and momentum following the principles recently developed for the numerical simulation of shallow water flows with large gradients. Conservation of these properties is the most important aspect to represent near local discontinuities in the solution, following from sharp bottom gradients or hydraulic jumps. The model is applied to reproduce the flow over a step where a hydraulic jump forms downstream. The hydrostatic pressure assumption fails to represent this type of flow mainly because of the pressure deviation from the hydrostatic values downstream the step. Fairly accurate results are obtained from the numerical model compared with experimental data. Deviation from the data is found to be inherent to the standard k–ε model implemented. Copyright © 2001 John Wiley & Sons, Ltd.     

Numerical simulation of droplet ejection of thermal inkjet printheads

In this study, we present a method to predict the droplet ejection in thermal inkjet printheads including the growth and collapse of a vapor bubble and refill of the firing chamber. The three‐dimensional Navier–Stokes equations are solved using a finite‐volume approach with a fixed Cartesian mesh. The piecewise‐linear interface calculation‐based volume‐of‐fluid method is employed to track and reconstruct the ink–air interface. A geometrical computation based on Lagrangian advection is used to compute the mass flux and advance the interface. A simple and efficient model for the bubble dynamics is employed to model the effect of ink vapor on the adjacent ink liquid. To solve the surface tension‐dominated flow accurately, a hierarchical curvature‐estimation method is proposed to adapt to the local grid resolution. The numerical methods mentioned earlier have been implemented in an internal simulation code, CFD3. The numerical examples presented in the study show good performance of CFD3 in prediction of surface tension‐dominated free‐surface flows, for example, droplet ejection in thermal inkjet printing. Currently, CFD3 is used extensively for printhead development within Hewlett‐Packard. Copyright © 2015 John Wiley & Sons, Ltd.     

Numerical simulation of the flow in a conduit,in the presence of a confined air cushion

A rectangular conduit with a closed end has water flowing in/out at the other end. The water level at the open end has an imposed sinusoidal movement. When this level is higher than the ceiling of the conduit, a certain mass of air is trapped under the ceiling. In a previous article (T.D. Nguyen, La Houille Blanche, No. 2, 1990), it was supposed that this air is flowing out freely through the ceiling, so the relative pressure at the water surface is zero, and the water hammer at the dead end of the conduit was calculated when the conduit was thoroughly filled. In this article, it is supposed that the trapped air is compressed isothermally or adiabatically. The set of equations is resolved (water continuity and movement equations, air state equation) by supposing a regime of flow at each section (section submerged or not), a certain value for the air pressure and by using the sweep method to determine the water flow characteristics. The air volume calculated by iteration must converge, and the calculated regimes at each section (submerged or free) must agree with the supposed regimes. The simulation is performed first with a horizontal conduit then with an inclined conduit. As expected, adiabatic compression gives higher pressure than isothermal compression. The simulation shows also that when there is an air cushion, compared with the case when air is flowing out freely, the shock of the water hammer at the closed end of the conduit is significantly reduced. This method is aimed at calculating the flow with entrapped air in the inlet/outlet tunnel of a hydroelectric plant, or in sewer system pipe when a sudden discharge surge (due to turbin opening/closing or to urban storm) changes a previously free‐surface flow in a mostly full‐pipe flow, but with some air entrapped under the ceiling. Copyright © 1999 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.     

基于表面力模型的液滴冲击弹性基底SPH模拟

采用光滑粒子动力学SPH方法建立液滴冲击弹性基底的流固耦合数值模型,给出描述粘性流体和弹性固体运动的SPH离散方程和数值处理格式,引入人工耗散项来抑制标准SPH方法的数值震荡。为模拟液滴的表面张力效应,通过精确检测边界粒子,采用拉格朗日插值方法计算表面法向量和曲率,结合界面理论中的连续表面力CSF方法,建立了适用于自由表面液滴的表面力模型,方形液滴变形的模拟结果与拉普拉斯理论解吻合较好。随后,采用SPH流固耦合模型模拟1.0 mm直径水滴以不同速度(0.2 m/s～3.0 m/s)冲击两种薄板型基底,分析了基底弹性变形对液滴铺展、收缩以及回弹行为的影响。     

不同界面计算方法对液滴在电场作用下变形数值模拟精度的影响

液滴在电场作用下的变形是电流体动力学的基础课题之一,表面张力的计算精度对液滴变形量的模拟结果有重要影响。本文以开源计算流体动力学平台OpenFOAM的VOF模型为框架,研究了MULES和isoAdvector两类界面更新算法与相分数梯度和RDF函数两类曲率算法对电场作用下液滴变形模拟精度的影响。研究表明,isoAdvector算法相比MULES算法对网格密度的要求更低,但其耦合相分数梯度算法计算表面张力的误差较高。isoAdvector算法耦合RDF函数算法计算误差较低,并且在使用轴对称网格时,只有该算法能够同时处理液滴平行于电场和垂直于电场方向的变形,得到的数值结果与解析解吻合较好。     

Numerical study of flows with multiple free surfaces

This paper demonstrates that a numerical method based on the generalized simplified marker and cell (GENSMAC) flow solver and Youngs' volume of fluid (Y‐VOF) surface‐tracking technique is an effective tool for studying the basic mechanics of hydraulic engineering problems with multiple free surfaces and non‐hydrostatic pressure distributions. Two‐dimensional flow equations in a vertical plane are solved numerically for this purpose. The numerical results are compared with experimental data and earlier numerical results based on a higher‐order depth‐averaged flow model available in the literature. Two classical problems, (i) flow in a free overfall and (ii) flow past a floor slot, are considered. The numerical results correspond very well with the experimental data for both sub‐critical and supercritical flows. Copyright © 2001 John Wiley & Sons, Ltd.     

过冷水滴撞击螺旋桨桨叶表面的数值模拟

本文介绍了某型螺旋桨在不同飞行状态下.求解桨叶水滴撞击特性的数值方法.该方法对桨叶运动模型进行简化,并在对绕桨叶运动的气流场计算的基础上,采用拉格朗日方法求解气流场中水滴运动方程,得到水滴运动轨迹.进而,确定了水滴对桨叶的撞击特性参数,为桨叶防冰系统设计提供条件.主要结论如下:(1)在巡航状态下,桨叶沿展向方向上总收集系数Em和局部收集系数β不断增大;(2)在爬升状态下,随着爬升高度H不断增大,飞行速度V0不断增大,水滴撞击在桨叶表面的范围有所增加,而且β随之增大;(3)随着水滴平均有效直径(MVD)的增大,水滴撞击在桨叶表面的范围明显增加同时,β在桨叶表面同一位置的值也随之增大.     

Incompressible SPH simulation of water entry of a free‐falling object

In this paper an incompressible smoothed particle hydrodynamics (Incom‐SPH) model is used to simulate the interactions between the free surface flow and a moving object. Incom‐SPH method is a two‐step semi‐implicit hydrodynamic formulation of the SPH algorithm and is capable of accurately treating the free surface deformations and impact forces during the solid–fluid interactions. For a free‐falling object, its motion is tracked by an additional Lagrangian algorithm based on Newton's law to couple with the Incom‐SPH program. The developed model is employed to investigate the water entry of a free‐falling wedge. The accuracy of the computations is validated by the good agreement in predicting the relevant hydrokinematic and hydrodynamic parameters. Finally, a numerical test is performed to study the influence of spatial resolution on the water entry features. The Incom‐SPH modeling coupled with the solid–fluid interaction algorithm could provide a promising computational tool to predict the slamming problems in coastal and offshore engineering. Copyright © 2008 John Wiley & Sons, Ltd.     

Numerical simulation of a single rising bubble by VOF with surface compression

The capability of the direct volume of fluid method for describing the surface dynamics of a free two‐dimensional rising bubble is evaluated using quantities of a recently published benchmark. The model equations are implemented in the open source computational fluid dynamics library OpenFOAM®. Here, a main ingredient of the numerical method is the so‐called surface compression that corrects the fluxes near the interface between two phases. The application of this method with respect to two test cases of a benchmark is considered in the main part. The test cases differ in physical properties, thus in different surface tension effects. The quantities centre of mass position, circularity and rise velocity are tracked over time and compared with the ones given in the benchmark. For test case one, where surface tension effects are more pre‐eminent, deviations from the benchmark results become more obvious. However, the flow features are still within reasonable range. Nevertheless, for test case two, which has higher density and viscosity ratios and above all a lower influence of the surface tension force, good agreement compared with the benchmark reference results is achieved. This paper demonstrates the good capabilities of the direct volume of fluid method with surface compression with regard to the preservation of sharp interfaces, boundedness, mass conservation and low computational time. Some limitation regarding the occurrence of parasitic currents, bad pressure jump prediction and bad grid convergence have been observed. With these restrictions in mind, the method is suitable for the simulation of similar two‐phase flow configurations. Copyright © 2012 John Wiley & Sons, Ltd.     

Numerical simulation of three-dimensional free surface flow in isopycnal co-ordinates

In this paper a semi-implicit method for three-dimensional circulation in isopycnal co-ordinates is derived and discussed. It is assumed that the flow is hydrostatic and characterized by isopycnal surfaces which can be represented by explicit, single-valued functions. The hydrostatic pressure is determined by using the conjugate gradient method to solve a block pentadiagonal linear system. The horizontal velocities are determined by solving a large set of tridiagonal systems. The stability of the resulting algorithm is shown to be independent of the surface and internal gravity wave speeds. © 1997 John Wiley & Sons, Ltd.     

Numerical simulation of viscous flows with free surface around realistic hull forms with transom

This paper describes a method for simulation of viscous flows with a free surface around realistic hull forms with a transom, which has been developed based on a FINFLO RANS solver with a moving mesh. A dry‐transom model is proposed and implemented for the treatment of flows off the transom. The bulk RANS flow with the artificial compressibility is solved by a cell‐centred finite volume multigrid scheme and the free surface deformed by wave motions is tracked by satisfying the kinematic and dynamic free‐surface boundary conditions on the actual location of the surface. The effects of turbulence on flows are evaluated with the Baldwin–Lomax turbulence model without a wall function. A test case is modern container ship model with a transom, the Hamburg Test Case. The calculated results are validated and they agree well with the measured results in terms of the free‐surface waves and the total resistance coefficient. Furthermore, the numerical solutions successfully captured many important features of the complicated interaction of the free surface with viscous flows around transom stern ships. In addition, the convergence performance and the grid refinement studies are also investigated. Copyright © 2001 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.     

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.     

Numerical prediction of droplet dynamics in turbulent flow,using the level set method

In the present article, the droplet dynamics in turbulent flow is numerically predicted. The modelling is based on an interfacial marker-level set (IMLS) method, coupled with the Reynolds-averaged Navier–Stokes (RANS) equations to predict the dynamics of turbulent two-phase flow. The governing equations for time-dependent, two-dimensional and incompressible two-phase flow are described in both phases and solved separately using a control volume approach on structured cell-centred collocated grids. The topological changes of the interface are predicted by applying the level set approach. The kinematic and dynamic conditions on the interface separating the two phases are satisfied. The numerical method proposed is validated against a well-known computational fluid dynamics problem. Further, the deformation and breakup of a single droplet either suddenly moved in air or exposed to turbulent stream are numerically investigated. In general, the developed numerical method demonstrates remarkable capability in predicting the characteristics of complex turbulent two-phase flows.     

Parallel finite element simulation of mooring forces on floating objects

The coupling between the equations governing the free‐surface flows, the six degrees of freedom non‐linear rigid body dynamics, the linear elasticity equations for mesh‐moving and the cables has resulted in a fluid‐structure interaction technology capable of simulating mooring forces on floating objects. The finite element solution strategy is based on a combination approach derived from fixed‐mesh and moving‐mesh techniques. Here, the free‐surface flow simulations are based on the Navier–Stokes equations written for two incompressible fluids where the impact of one fluid on the other one is extremely small. An interface function with two distinct values is used to locate the position of the free‐surface. The stabilized finite element formulations are written and integrated in an arbitrary Lagrangian–Eulerian domain. This allows us to handle the motion of the time dependent geometries. Forces and momentums exerted on the floating object by both water and hawsers are calculated and used to update the position of the floating object in time. In the mesh moving scheme, we assume that the computational domain is made of elastic materials. The linear elasticity equations are solved to obtain the displacements for each computational node. The non‐linear rigid body dynamics equations are coupled with the governing equations of fluid flow and are solved simultaneously to update the position of the floating object. The numerical examples includes a 3D simulation of water waves impacting on a moored floating box and a model boat and simulation of floating object under water constrained with a cable. Copyright © 2003 John Wiley & Sons, Ltd.     

气相中双液滴正碰过程数值模拟研究

液滴碰撞现象普遍存在于动力装置燃烧室喷嘴的下游区域,影响燃料的雾化性能。为了揭示相同直径的双液滴中心碰撞机理,求解了轴对称坐标系下的N-S方程,采用VOF(Volume of Fluid)方法捕捉液滴碰撞过程中气液自由表面的演化规律。利用Qian等提供的实验结果对计算模型进行数值校验,验证了模型的准确性。在此基础上,研究了环境压强对液滴碰撞反弹后不同结果(分离和融合)的影响,分析了环境压强和Weber数对液滴碰撞分离的影响。结果表明,液滴在碰撞反弹后的状态(分离或融合)是由液滴间气膜压强与环境气动阻力共同作用的结果,环境压强对液滴碰撞分离过程基本没有影响;Weber数越大,碰撞过程中变形的幅度越大。