A computationally efficient 3D finite‐volume scheme for violent liquid–gas sloshing

We describe a semi‐implicit volume‐of‐fluid free‐surface‐modelling methodology for flow problems involving violent free‐surface motion. For efficient computation, a hybrid‐unstructured edge‐based vertex‐centred finite volume discretisation is employed, while the solution methodology is entirely matrix free. Pressures are solved using a matrix‐free preconditioned generalised minimum residual algorithm and explicit time‐stepping is employed for the momentum and interface‐tracking equations. The high resolution artificial compressive (HiRAC) volume‐of‐fluid method is used for accurate capturing of the free surface in violent flow regimes while allowing natural applicability to hybrid‐unstructured meshes. The code is parallelised for solution on distributed‐memory architectures and evaluated against 2D and 3D benchmark problems. Good parallel scaling is demonstrated, with almost linear speed‐up down to 6000 cells per core. Finally, the code is applied to an industrial‐type problem involving resonant excitation of a fuel tank, and a comparison with experimental results is made in this violent sloshing regime. Copyright © 2015 John Wiley & Sons, Ltd.     

A combined level set/ghost cell immersed boundary representation for floating body simulations

A six degrees of freedom (6DOF) algorithm is implemented in the open‐source CFD code REEF3D. The model solves the incompressible Navier–Stokes equations. Complex free surface dynamics are modeled with the level set method based on a two‐phase flow approach. The convection terms of the velocities and the level set method are treated with a high‐order weighted essentially non‐oscillatory discretization scheme. Together with the level set method for the free surface capturing, this algorithm can model the movement of rigid floating bodies and their interaction with the fluid. The 6DOF algorithm is implemented on a fixed grid. The solid‐fluid interface is represented with a combination of the level set method and ghost cell immersed boundary method. As a result, re‐meshing or overset grids are not necessary. The capability, accuracy, and numerical stability of the new algorithm is shown through benchmark applications for the fluid‐body interaction problem. Copyright © 2016 John Wiley & Sons, Ltd.     

An adaptive numerical scheme for solving incompressible 2‐phase and free‐surface flows

In this paper, we present a numerical scheme for solving 2‐phase or free‐surface flows. Here, the interface/free surface is modeled using the level‐set formulation, and the underlying mesh is adapted at each iteration of the flow solver. This adaptation allows us to obtain a precise approximation for the interface/free‐surface location. In addition, it enables us to solve the time‐discretized fluid equation only in the fluid domain in the case of free‐surface problems. Fluids here are considered incompressible. Therefore, their motion is described by the incompressible Navier‐Stokes equation, which is temporally discretized using the method of characteristics and is solved at each time iteration by a first‐order Lagrange‐Galerkin method. The level‐set function representing the interface/free surface satisfies an advection equation that is also solved using the method of characteristics. The algorithm is completed by some intermediate steps like the construction of a convenient initial level‐set function (redistancing) as well as the construction of a convenient flow for the level‐set advection equation. Numerical results are presented for both bifluid and free‐surface problems.     

Finite‐volume multi‐stage schemes for shallow‐water flow simulations

A finite‐volume multi‐stage (FMUSTA) scheme is proposed for simulating the free‐surface shallow‐water flows with the hydraulic shocks. On the basis of the multi‐stage (MUSTA) method, the original Riemann problem is transformed to an independent MUSTA mesh. The local Lax–Friedrichs scheme is then adopted for solving the solution of the Riemann problem at the cell interface on the MUSTA mesh. The resulting first‐order monotonic FMUSTA scheme, which does not require the use of the eigenstructure and the special treatment of entropy fixes, has the generality as well as simplicity. In order to achieve the high‐resolution property, the monotonic upstream schemes for conservation laws (MUSCL) method are used. For modeling shallow‐water flows with source terms, the surface gradient method (SGM) is adopted. The proposed schemes are verified using the simulations of six shallow‐water problems, including the 1D idealized dam breaking, the steady transcritical flow over a hump, the 2D oblique hydraulic jump, the circular dam breaking and two dam‐break experiments. The simulated results by the proposed schemes are in satisfactory agreement with the exact solutions and experimental data. It is demonstrated that the proposed FMUSTA schemes have superior overall numerical accuracy among the schemes tested such as the commonly adopted Roe and HLL schemes. Copyright © 2007 John Wiley & Sons, Ltd.     

A conservative local interface sharpening scheme for the constrained interpolation profile method

A conservative local interface sharpening scheme has been developed for the constrained interpolation profile method with the conservative semi‐Lagrangian scheme, because the conservative semi‐Lagrangian scheme does not feature a mechanism to control the interface thickness, thus causing an increase of numerical error with the advance of the time step. The proposed sharpening scheme is based on the conservative level set method proposed by Olsson and Kreiss. However, because their method can cause excessive deformation of the free‐surface in certain circumstances, we propose an improvement of the method by developing a local sharpening technique. Several advection tests are presented to assess the correctness of the advection and the improved interface sharpening scheme. This is followed by the validations of dam‐breaking flow and the rising bubble flows. The mass of the fluid is exactly conserved and the computed terminal velocity of the rising bubble agrees well with the experiments compared with other numerical methods such as the volume of fluid method (VOF), the front tracking method, and the level set method. Copyright © 2011 John Wiley & Sons, Ltd.     

An efficient and accurate algebraic interface capturing method for unstructured grids in 2 and 3 dimensions: The THINC method with quadratic surface representation

We present in this paper an efficient and accurate volume of fluid (VOF) type scheme to compute moving interfaces on unstructured grids with arbitrary quadrilateral mesh elements in 2D and hexahedral elements in 3D. Being an extension of the multi‐dimensional tangent of hyperbola interface capturing (THINC) reconstruction proposed by the authors in Cartesian grid, an algebraic VOF scheme is devised for arbitrary quadrilateral and hexahedral elements. The interface is cell‐wisely approximated by a quadratic surface, which substantially improves the numerical accuracy. The same as the other THINC type schemes, the present method does not require the explicit geometric representation of the interface when computing numerical fluxes and thus is very computationally efficient and straightforward in implementation. The proposed scheme has been verified by benchmark tests, which reveal that this scheme is able to produce high‐quality numerical solutions of moving interfaces in unstructured grids and thus a practical method for interfacial multi‐phase flow simulations. Copyright © 2014 John Wiley & Sons, Ltd.     

A surface remeshing technique for a Lagrangian description of 3D two‐fluid flow problems

Classical Lagrangian schemes applied to update the front position between two immiscible incompressible fluids have been long recognized to provide a sharp representation of the interface. However, the main drawback of these approaches is the progressive distortion in the distribution of the markers used to identify the material front. To avoid this problem, a 3D interface remeshing algorithm is proposed in this work. In addition, the remeshed front is enforced to preserve the global volume. These aspects are incorporated in an existing fluid dynamics formulation for the analysis of two‐fluid flows problems. The resulting formulation, called as the 3D‐moving Lagrangian interface remeshing technique, is applied in the numerical analysis of two‐fluid flow problems. Copyright © 2009 John Wiley & Sons, Ltd.     

The intersection marker method for 3D interface tracking of deformable surfaces in finite volumes

Currently, the majority of computational fluid dynamics (CFD) codes use the finite volume method to spatially discretise the computational domain, sometimes as an array of cubic control volumes. The Finite volume method works well with single‐phase flow simulations, but two‐phase flow simulations are more challenging because of the need to track the surface interface traversing and deforming within the 3D grid. Surface area and volume fraction details of each interface cell must be accurately accounted for, in order to calculate for the momentum exchange and rates of heat and mass transfer across the interface. To attain a higher accuracy in two‐phase flow CFD calculations, the intersection marker (ISM) method is developed. The ISM method is a hybrid Lagrangian–Eulerian front‐tracking algorithm that can model an arbitrary 3D surface within an array of cubic control volumes. The ISM method has a cell‐by‐cell remeshing capability that is volume conservative and is suitable for the tracking of complex interface deformation in transient two‐phase CFD simulations. Copyright © 2015 John Wiley & Sons, Ltd.     

An upstream flux‐splitting finite‐volume scheme for 2D shallow water equations

An upstream flux‐splitting finite‐volume (UFF) scheme is proposed for the solutions of the 2D shallow water equations. In the framework of the finite‐volume method, the artificially upstream flux vector splitting method is employed to establish the numerical flux function for the local Riemann problem. Based on this algorithm, an UFF scheme without Jacobian matrix operation is developed. The proposed scheme satisfying entropy condition is extended to be second‐order‐accurate using the MUSCL approach. The proposed UFF scheme and its second‐order extension are verified through the simulations of four shallow water problems, including the 1D idealized dam breaking, the oblique hydraulic jump, the circular dam breaking, and the dam‐break experiment with 45° bend channel. Meanwhile, the numerical performance of the UFF scheme is compared with those of three well‐known upwind schemes, namely the Osher, Roe, and HLL schemes. It is demonstrated that the proposed scheme performs remarkably well for shallow water flows. The simulated results also show that the UFF scheme has superior overall numerical performances among the schemes tested. Copyright © 2005 John Wiley & Sons, Ltd.     

An efficient method for capturing free boundaries in multi‐fluid simulations

An easy‐to‐use front capturing method is devised by directly solving the transport equation for a volume of fluid (VOF) function. The key to this method is a semi‐Lagrangian conservative scheme, namely CIP_CSL3, recently proposed by the author. In the CIP_CSL3 scheme, the first‐order derivative of the interpolation polynomial at each cell centre is used to control the shape of the reconstructed profile. We show in the present paper that the first‐order derivative, which plays a crucial role in reconstructing the interpolation profile, can also be used to eliminate numerical diffusion. The resulting algorithm can be directly used to compute the VOF‐like function and retain the compact thickness of the moving interface in multi‐fluid simulations. No surface reconstruction based on the value of VOF function is required in the method, which makes it quite economical and easy to use. The presented method has been tested with various interfacial flows including pure rotation, vortex shearing, multi‐vortex deformation and the moving boundaries in real fluid as well. The method gives promising results to all computed problems. Copyright © 2003 John Wiley & Sons, Ltd.     

Improvement of moving particle semi‐implicit method for simulation of progressive water waves

Precise simulation of the propagation of surface water waves, especially when involving breaking wave, takes a significant place in computational fluid dynamics. Because of the strong nonlinear properties, the treatment of large surface deformation of free surface flow has always been a challenging work in the development of numerical models. In this paper, the moving particle semi‐implicit (MPS) method, an entirely Lagrangian method, is modified to simulate wave motion in a 2‐D numerical wave flume preferably. In terms of consecutive pressure distribution, a new and simple free surface detection criterion is proposed to enhance the free surface recognition in the MPS method. In addition, a revised gradient model is deduced to diminish the effect of nonuniform particle distribution and then to reduce the numerical wave attenuation occurring in the original MPS model. The applicability and stability of the improved MPS method are firstly demonstrated by the calculation of hydrostatic problem. It is revealed that these modifications are effective to suppress the pressure oscillation, weaken the local particle clustering, and boost the stability of numerical algorithm. It is then applied to investigate the propagation of progressive waves on a flat bed and the wave breaking on a mild slope. Comparisons with the analytical solutions and experimental results indicate that the improved MPS model can give better results about the profiles and heights of surface waves in contrast with the previous MPS models. Copyright © 2017 John Wiley & Sons, Ltd.     

A meshless method for numerical simulation of depth‐averaged turbulence flows using a k‐ϵ model

A three‐dimensional numerical model is developed to analyze free surface flows and water impact problems. The flow of an incompressible viscous fluid is solved using the unsteady Navier–Stokes equations. Pseudo‐time derivatives are introduced into the equations to improve computational efficiency. The interface between the two phases is tracked using a volume‐of‐fluid interface tracking algorithm developed in a generalized curvilinear coordinate system. The accuracy of the volume‐of‐fluid method is first evaluated by the multiple numerical benchmark tests, including two‐dimensional and three‐dimensional deformation cases on curvilinear grids. The performance and capability of the numerical model for water impact problems are demonstrated by simulations of water entries of the free‐falling hemisphere and cone, based on comparisons of water impact loadings, velocities, and penetrations of the body with experimental data. For further validation, computations of the dam‐break flows are presented, based on an analysis of the wave front propagation, water level, and the dynamic pressure impact of the waves on the downstream walls, on a specific container, and on a tall structure. Extensive comparisons between the obtained solutions, the experimental data, and the results of other numerical simulations in the literature are presented and show a good agreement. Copyright © 2015 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.     

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.     

A stencil-like volume of fluid （VOF） method for tracking free interface

A stencil-like volume of fluid （VOF） method is proposed for tracking free interface. A stencil on a grid cell is worked out according to the normal direction of the interface, in which only three interface positions are possible in 2D cases, and the interface can be reconstructed by only requiring the known local volume fraction information. On the other hand, the fluid-occupying-length is defined on each side of the stencil, through which a unified fluid-occupying volume model and a unified algorithm can be obtained to solve the interface advection equation. The method is suitable for the arbitrary geometry of the grid cell, and is extendible to 3D cases. Typical numerical examples show that the current method can give ＂sharp＂ results for tracking free interface.     

3D free‐surface flow computation using a RANSE/Fourier–Kochin coupling

A coupling method for numerical calculations of steady free‐surface flows around a body is presented. The fluid domain in the neighbourhood of the hull is divided into two overlapping zones. Viscous effects are taken in account near the hull using Reynolds‐averaged Navier–Stokes equations (RANSE), whereas potential flow provides the flow away from the hull. In the internal domain, RANSE are solved by a fully coupled velocity, pressure and free‐surface elevation method. In the external domain, potential‐flow theory with linearized free‐surface condition is used to provide boundary conditions to the RANSE solver. The Fourier–Kochin method based on the Fourier–Kochin formulation, which defines the velocity field in a potential‐flow region in terms of the velocity distribution at a boundary surface, is used for that purpose. Moreover, the free‐surface Green function satisfying this linearized free‐surface condition is used. Calculations have been successfully performed for steady ship‐waves past a serie 60 and then have demonstrated abilities of the present coupling algorithm. Copyright © 2003 John Wiley & Sons, Ltd.     

An implicit three‐dimensional fully non‐hydrostatic model for free‐surface flows

An implicit method is developed for solving the complete three‐dimensional (3D) Navier–Stokes equations. The algorithm is based upon a staggered finite difference Crank‐Nicholson scheme on a Cartesian grid. A new top‐layer pressure treatment and a partial cell bottom treatment are introduced so that the 3D model is fully non‐hydrostatic and is free of any hydrostatic assumption. A domain decomposition method is used to segregate the resulting 3D matrix system into a series of two‐dimensional vertical plane problems, for each of which a block tri‐diagonal system can be directly solved for the unknown horizontal velocity. Numerical tests including linear standing waves, nonlinear sloshing motions, and progressive wave interactions with uneven bottoms are performed. It is found that the model is capable to simulate accurately a range of free‐surface flow problems using a very small number of vertical layers (e.g. two–four layers). The developed model is second‐order accuracy in time and space and is unconditionally stable; and it can be effectively used to model 3D surface wave motions. Copyright © 2004 John Wiley & Sons, Ltd.     

A decoupling numerical method for fluid flow

A first-order non-conforming numerical methodology, Separation method, for fluid flow problems with a 3-point exponential interpolation scheme has been developed. The flow problem is decoupled into multiple one-dimensional subproblems and assembled to form the solutions. A fully staggered grid and a conservational domain centred at the node of interest make the decoupling scheme first-order-accurate. The discretization of each one-dimensional subproblem is based on a 3-point interpolation function and a conservational domain centred at the node of interest. The proposed scheme gives a guaranteed first-order accuracy. It is shown that the traditional upwind (or exponentially weighted upstream) scheme is less than first-order-accurate. The pressure is decoupled from the velocity field using the pressure correction method of SIMPLE. Thomas algorithm (tri-diagonal solver) is used to solve the algebraic equations iteratively. The numerical advantage of the proposed scheme is tested for laminar fluid flows in a torus and in a square-driven cavity. The convergence rates are compared with the traditional schemes for the square-driven cavity problem. Good behaviour of the proposed scheme is ascertained.     

An implicit numerical algorithm for solving non‐hydrostatic free‐surface flow problems

Details are given of the development of a two‐dimensional vertical numerical model for simulating unsteady free‐surface flows, using a non‐hydrostatic pressure distribution. In this model, the Reynolds equations and the kinematic free‐surface boundary condition are solved simultaneously, so that the water surface elevation can be integrated into the solution and solved for, together with the velocity and pressure fields. An efficient numerical algorithm has been developed, deploying implicit parameters similar to those used in the Crank–Nicholson method, and generating a block tri‐diagonal algebraic system of equations. The model has been applied to simulate a range of unsteady flow problems involving relatively strong vertical accelerations. The results show that the numerical algorithm described is able to produce accurate predictions and is also easy to apply. Copyright © 2001 John Wiley & Sons, Ltd.