A geometry‐based level set method for curvilinear overset grids with application to ship hydrodynamics

In a previous work (Int. J. Numer. Meth. Fluids 2007; 55 :867–897), we presented a two‐phase level set method to simulate air/water turbulent flows using curvilinear body‐fitted grids for ship hydrodynamics problems. This two‐phase level set method explicitly enforces jump conditions across the interface, thus resulting in a fully coupled representation of the air/water flow. Though the method works well with multiblock curvilinear grids, severe robustness problems were found when attempting to use it with overset grids. The problem was tracked to small unphysical level set discontinuities across the overset grids with large differences in curvature. Though negligible for single‐phase approaches, the problem magnifies with large density differences between the phases, causing computation failures. In this paper, we present a geometry‐based level set method for curvilinear overset grids that overcomes these difficulties. The level set transport and reinitialization equations are not discretized along grid coordinates, but along the upwind streamline and level set gradient directions, respectively. The method is essentially an unstructured approach that is transparent to the differences between overset grids, but still the discretization is under the framework of a finite differences approach. As a result, significant improvements in robustness and to a less extent in accuracy are achieved for the level set function interpolation between overset grids, especially with big differences in grid curvature. Example tests are shown for the case of bow breaking waves around the surface combatant model David Taylor Model Basin (DTMB) 5415 and for the steady‐state ONR Tumblehome DTMB 5613 with superstructure. In the first case, the results are compared against experimental data available and in the second against results of a semi‐coupled method. Copyright © 2011 John Wiley & Sons, Ltd.     

Semi‐coupled air/water immersed boundary approach for curvilinear dynamic overset grids with application to ship hydrodynamics

For many problems in ship hydrodynamics, the effects of air flow on the water flow are negligible (the frequently called free surface conditions), but the air flow around the ship is still of interest. A method is presented where the water flow is decoupled from the air solution, but the air flow uses the unsteady water flow as a boundary condition. The authors call this a semi‐coupled air/water flow approach. The method can be divided into two steps. At each time step the free surface water flow is computed first with a single‐phase method assuming constant pressure and zero stress on the interface. The second step is to compute the air flow assuming the free surface as a moving immersed boundary (IB). The IB method developed for Cartesian grids (Annu. Rev. Fluid Mech. 2005; 37 :239–261) is extended to curvilinear grids, where no‐slip and continuity conditions are used to enforce velocity and pressure boundary conditions for the air flow. The forcing points close to the IB can be computed and corrected under a sharp interface condition, which makes the computation very stable. The overset implementation is similar to that of the single‐phase solver (Comput. Fluids 2007; 36 :1415–1433), with the difference that points in water are set as IB points even if they are fringe points. Pressure–velocity coupling through pressure implicit with splitting of operators or projection methods is used for water computations, and a projection method is used for the air. The method on each fluid is a single‐phase method, thus avoiding ill‐conditioned numerical systems caused by large differences of fluid properties between air and water. The computation is only slightly slower than the single‐phase version, with complete absence of spurious velocity oscillations near the free surface, frequently present in fully coupled approaches. Validations are performed for laminar Couette flow over a wavy boundary by comparing with the analytical solution, and for the surface combatant model David Taylor Model Basin (DTMB) 5512 by comparing with Experimental Fluid Dynamics (EFD) and the results of two‐phase level set computations. Complex flow computations are demonstrated for the ONR Tumblehome DTMB 5613 with superstructure subject to waves and wind, including 6DOF motions and broaching in SS7 irregular waves and wind. Copyright © 2008 John Wiley & Sons, Ltd.     

A hybrid Lagrangian–Eulerian particle‐level set method for numerical simulations of two‐fluid turbulent flows

A coupled Lagrangian interface‐tracking and Eulerian level set (LS) method is developed and implemented for numerical simulations of two‐fluid flows. In this method, the interface is identified based on the locations of notional particles and the geometrical information concerning the interface and fluid properties, such as density and viscosity, are obtained from the LS function. The LS function maintains a signed distance function without an auxiliary equation via the particle‐based Lagrangian re‐initialization technique. To assess the new hybrid method, numerical simulations of several ‘standard interface‐moving’ problems and two‐fluid laminar and turbulent flows are conducted. The numerical results are evaluated by monitoring the mass conservation, the turbulence energy spectral density function and the consistency between Eulerian and Lagrangian components. The results of our analysis indicate that the hybrid particle‐level set method can handle interfaces with complex shape change, and can accurately predict the interface values without any significant (unphysical) mass loss or gain, even in a turbulent flow. The results obtained for isotropic turbulence by the new particle‐level set method are validated by comparison with those obtained by the ‘zero Mach number’, variable‐density method. For the cases with small thermal/mass diffusivity, both methods are found to generate similar results. Analysis of the vorticity and energy equations indicates that the destabilization effect of turbulence and the stability effect of surface tension on the interface motion are strongly dependent on the density and viscosity ratios of the fluids. Copyright © 2007 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.     

求解无粘可压Euler方程组的虚拟流方法

首先将三阶Godunov型半离散中心迎风格式推广到四阶,之后再将该新的四阶半离散中心迎风格式与Level Set方法以及虚拟流方法结合起来,成功地处理了非反应激波问题和多介质流中的爆轰间断问题。由于Level Set函数能隐式地追踪到界面的位置,而虚拟流的构造能隐式地捕捉到界面的边界条件,故而本文的方法可以很自然地推广到多维情况。     

A CWENO ghost fluid method for compressible multimaterial flow

In this work a new ghost fluid method (GFM) is introduced for multimaterial compressible flow with arbitrary equation of states. In previous researches, it has been shown that accurate wave decomposition at the interface by solving a Riemann problem alleviates the shortcomings of the standard GFM in dealing with the impingement of strong waves onto the interface but these Riemann‐based GFM are not consistent with the framework of the central WENO scheme in which the emphasis is to avoid solving Riemann problems at control volume faces and enjoy the black box property (being independent of equation of state). The aim of this work is to develop a new GFM that is completely consistent with the methodology behind central schemes; that is, it enjoys a black box property. The capabilities of the proposed GFM method is shown by solving various types of multimaterial compressible flows including gas–gas, gas–water and fluid–solid interfaces interacting with strong shock waves in one and two space dimensions. Copyright © 2013 John Wiley & Sons, Ltd.     

真实虚拟流方法在多介质可压缩流动模拟中的应用

为了克服原始虚拟流方法(ghost fluid method,GFM)在处理激波与大密度比流体-流体（气-水）界面相互作用时遇到的困难,采用真实虚拟流法(real ghost fluid method,RGFM)处理流体界面附近的虚拟点,结合HLLC(Harten-Lax-Van Leer with contact discontinuities)格式求解Euler方程,采用五阶WENO(weighted essentially nonoscillatory)格式求解level set输运方程。通过一维和二维算例的物质界面捕捉研究,证明RGFM在处理小密度比界面问题时优于GFM,同时RGFM还可用于求解激波与大密度比物质界面相互作用问题。计算表明,将RGFM引入到本文算法中,可精确捕捉到激波与界面（气-气、气-水界面）相互作用的变化细节,包括大密度比界面的剧烈变形和破碎,并具有较高的计算分辨率。     

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 geometric mass‐preserving redistancing scheme for the level set function

In this paper we describe and evaluate a geometric mass‐preserving redistancing procedure for the level set function on general structured grids. The proposed algorithm is adapted from a recent finite element‐based method and preserves the mass by means of a localized mass correction. A salient feature of the scheme is the absence of adjustable parameters. The algorithm is tested in two and three spatial dimensions and compared with the widely used partial differential equation (PDE)‐based redistancing method using structured Cartesian grids. Through the use of quantitative error measures of interest in level set methods, we show that the overall performance of the proposed geometric procedure is better than PDE‐based reinitialization schemes, since it is more robust with comparable accuracy. We also show that the algorithm is well‐suited for the highly stretched curvilinear grids used in CFD simulations. Copyright © 2010 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.     

From level set to volume of fluid and back again at second‐order accuracy

We present methods for computing either the level set function or volume fraction field from the other at second‐order accuracy. Both algorithms are optimal in that O(N) computations are needed for N total grid points and both algorithms are easily parallelized. This work includes a novel interface reconstruction algorithm in three dimensions that requires a smaller local block of volume fractions than existing algorithms. A compact local solver leads to better algorithm portability and efficiency: for example, fewer restrictions must be imposed on an adaptive mesh, and fewer grid cells must be communicated between processors in a parallel implementation. We also present a fast sweeping method for computing a unique approximation of the signed distance function to a piecewise linear interface. All of the numerical examples confirm second‐order accuracy on both uniform and tree‐based adaptive grids. Copyright © 2015 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.     

Spurious currents in finite element based level set methods for two‐phase flow

A study of spurious currents in continuous finite element based simulations of the incompressible Navier–Stokes equations for two‐phase flows is presented on the basis of computations on a circular drop in equilibrium. The conservative and the standard level set methods are used. It is shown that a sharp surface tension force, expressed as a line integral along the interface, can give rise to large spurious currents and oscillations in the pressure that do not decrease with mesh refinement. If instead a regularized surface tension representation is used, exact force balance at the interface is possible, both for a fully coupled discretization approach and for a fractional step projection method. However, the numerical curvature calculation introduces errors that cause spurious currents. Different ways to extend the curvature from the interface to the whole domain are discussed and investigated. The impact of using different finite element spaces and stabilization methods is also considered. Copyright © 2011 John Wiley & Sons, Ltd.     

A level set approach for computing solutions to inviscid compressible flow with moving solid boundary using fixed Cartesian grids

A level set approach for computing solutions to inviscid compressible flow with moving solid surface is presented. The solid surface is considered to be sharp and is described as the zero level set of a smooth explicit function of space and time. The finite volume TVD–MacCormack's two‐step procedure is used. The boundary conditions on the solid surface are easily implemented by defining the smooth level set function. The present treatment of the level set method allows the handling of fluid flows in the presence of irregularly shaped solid boundaries, escaping from the bookkeeping complexity in the so‐called ‘surface‐tracking’ method. Using the proposed numerical techniques, a two‐dimensional numerical simulation is made to investigate the aerodynamic phenomena induced by two high‐speed trains passing by each other in a tunnel. Copyright © 2001 John Wiley & Sons, Ltd.     

A discontinuous Galerkin‐based sharp‐interface method to simulate three‐dimensional compressible two‐phase flow

A numerical method for the simulation of compressible two‐phase flows is presented in this paper. The sharp‐interface approach consists of several components: a discontinuous Galerkin solver for compressible fluid flow, a level‐set tracking algorithm to follow the movement of the interface and a coupling of both by a ghost‐fluid approach with use of a local Riemann solver at the interface. There are several novel techniques used: the discontinuous Galerkin scheme allows locally a subcell resolution to enhance the interface resolution and an interior finite volume Total Variation Diminishing (TVD) approximation at the interface. The level‐set equation is solved by the same discontinuous Galerkin scheme. To obtain a very good approximation of the interface curvature, the accuracy of the level‐set field is improved and smoothed by an additional P_NP_M‐reconstruction. The capabilities of the method for the simulation of compressible two‐phase flow are demonstrated for a droplet at equilibrium, an oscillating ellipsoidal droplet, and a shock‐droplet interaction problem at Mach 3. Copyright © 2015 John Wiley & Sons, Ltd.     

Numerical approach for generic three-phase flow based on cut-cell and ghost fluid methods

In this paper, we introduce numerical methods that can simulate complex multiphase flows. The finite volume method, applying Cartesian cut-cell is used in the computational domain, containing fluid and solid, to conserve mass and momentum. With this method, flows in and around any geometry can be simulated without complex and time consuming meshing. For the fluid region, which involves liquid and gas, the ghost fluid method is employed to handle the stiffness of the interface discontinuity problem. The interaction between each phase is treated simply by wall function models or jump conditions of pressure, velocity and shear stress at the interface. The sharp interface method “coupled level set (LS) and volume of fluid (VOF)” is used to represent the interface between the two fluid phases. This approach will combine some advantages of both interface tracking/capturing methods, such as the excellent mass conservation from the VOF method and good accuracy of interface normal computation from the LS function. The first coupled LS and VOF will be generated to reconstruct the interface between solid and the other materials. The second will represent the interface between liquid and gas.     

A direct reinitialization approach of level‐set/splitting finite element method for simulating incompressible two‐phase flows

Computation of a moving interface by the level‐set (LS) method typically requires reinitialization of LS function. An inaccurate execution of reinitialization results in incorrect free surface capturing and thus errors such as mass gain/loss so that an accurate and robust reinitialization process in the LS method is essential for the simulation of free surface flows. In the present study, we pursue further development of the reinitialization process, which directly corrects the LS function after advection is carried out by using the normal vector to the interface instead of solving the reinitialization equation of hyperbolic type. The Taylor–Galerkin method is adopted to discretize the advection equation of the LS function and the P1P1 splitting finite element method is applied to solve the Navier–Stokes equation. The proposed algorithm is validated with the well‐known benchmark problems, i.e. stretching of a circular fluid element, time‐reversed single‐vortex, solitary wave propagation, broken dam flow and filling of a container. The simulation results of these flows are in good agreement with previously existing experimental and numerical results. Copyright © 2010 John Wiley & Sons, Ltd.     

Finite‐element/level‐set/operator‐splitting (FELSOS) approach for computing two‐fluid unsteady flows with free moving interfaces

The present work is devoted to the study on unsteady flows of two immiscible viscous fluids separated by free moving interface. Our goal is to elaborate a unified strategy for numerical modelling of two‐fluid interfacial flows, having in mind possible interface topology changes (like merger or break‐up) and realistically wide ranges for physical parameters of the problem. The proposed computational approach essentially relies on three basic components: the finite element method for spatial approximation, the operator‐splitting for temporal discretization and the level‐set method for interface representation. We show that the finite element implementation of the level‐set approach brings some additional benefits as compared to the standard, finite difference level‐set realizations. In particular, the use of finite elements permits to localize the interface precisely, without introducing any artificial parameters like the interface thickness; it also allows to maintain the second‐order accuracy of the interface normal, curvature and mass conservation. The operator‐splitting makes it possible to separate all major difficulties of the problem and enables us to implement the equal‐order interpolation for the velocity and pressure. Diverse numerical examples including simulations of bubble dynamics, bifurcating jet flow and Rayleigh–Taylor instability are presented to validate the computational method. Copyright © 2004 John Wiley & Sons, Ltd.     

Simulation of two‐phase flow–body interaction problems using direct forcing/fictitious domain–level set method

In the present paper, a direct forcing/fictitious domain (DF/FD)–level set method is proposed to simulate the twophase flow–body interaction. The DF/FD does not sacrifice accuracy and robustness by employing a discrete δ (Dirac delta) function to transfer quantities between the Eulerian nodes and Lagrangian points explicitly as the immersed boundary method. The advantages of this approach are the simple concept, the easy implementation and the utilization of original governing equation without modification. The main idea is to combine DF/FD method with the level set method in the Cartesian coordinates. We present the results of a number of test cases to illustrate the effectiveness of the proposed method for single‐phase flow–body interaction problem and the two‐phase flows with a stationary body. Eventually, the simulations of various water entry problems have been conducted to validate the capability and the accuracy of the present method on solving the twophase flow–body interaction. Consequently, the present results are found to be in good agreement with those of previous studies. Copyright © 2013 John Wiley & Sons, Ltd.     

Enablers for high‐order level set methods in fluid mechanics

In the context of numerical simulations of multiphysics flows, accurate tracking of an interface and consistent computation of its geometric properties are crucial. In this paper, we investigate a level set technique that satisfies these requirements and ensures local third‐order accuracy on the level set function (near the interface) and first‐order accuracy on the curvature, even for long‐time computations. The method is developed in a finite differences framework on Cartesian grids. As in usual level set strategies, reinitialization steps are involved. Several reinitialization algorithms are reviewed and mixed to design an accurate and fast reinitialization procedure. When coupled with a time evolution of the interface, the reinitialization procedure is performed only when there are too large deformations of the isocontours. This strategy limits the number of reinitialization steps and shows a good balance between accuracy and computational cost. Numerical results compare well with usual level set strategies and confirm the necessity of the reinitialization procedure, together with a limited number of reinitialization steps. Copyright © 2015 John Wiley & Sons, Ltd.