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.     

Three‐dimensional finite element analysis of transient fluid flow with free‐surface using marker surface method and adaptive grid refinement

For three‐dimensional finite element analysis of transient fluid flow with free‐surface, a new marker surface method is proposed, in which the fluid flow is represented by the marker surface composed of marker elements instead of marker particles used in the marker particle method. This also involves an adaptive grid that is created under a new criterion of element categorization of filling states and the locations in the total region at each time step. The marker surface is used in order to represent the free‐surface accurately, as well as to decrease the memory and computation time, and to effectively display the predicted three‐dimensional free‐surface. By using the adaptive grid in which the elements, finer than those in internal and external regions, are distributed in the surface region through refinement and coarsening procedures, the analysis of three‐dimensional transient fluid flow with free‐surface is achieved more efficiently. Through three‐dimensional analysis of two kinds of problems using several grids, the efficiency of the proposed marker surface method and the adaptive grid are verified. Copyright © 1999 John Wiley & Sons, Ltd.     

Coupled lubrication and Stokes flow finite elements

A method is developed for performing a local reduction of the governing physics for fluid problems with domains that contain a combination of narrow and non‐narrow regions, and the computational accuracy and performance of the method are measured. In the narrow regions of the domain, where the fluid is assumed to have no inertia and the domain height and curvature are assumed small, lubrication, or Reynolds, theory is used locally to reduce the two‐dimensional Navier–Stokes equations to the one‐dimensional Reynolds equation while retaining a high degree of accuracy in the overall solution. The Reynolds equation is coupled to the governing momentum and mass equations of the non‐narrow region with boundary conditions on the mass and momentum flux. The localized reduction technique, termed ‘stitching,’ is demonstrated on Stokes flow for various geometries of the hydrodynamic journal bearing—a non‐trivial test problem for which a known analytical solution is available. The computational advantage of the coupled Stokes–Reynolds method is illustrated on an industrially applicable fully‐flooded deformable‐roll coating example. The examples in this paper are limited to two‐dimensional Stokes flow, but extension to three‐dimensional and Navier–Stokes flow is possible. Copyright © 2003 John Wiley & Sons, Ltd.     

Fringe element reconstruction for front tracking for three‐dimensional incompressible flow analysis

Fringe element reconstruction technique for tracking the free surface in three‐dimensional incompressible flow analysis was developed. The flow field was calculated by the mixed formulation based on a four‐node tetrahedral element with a bubble function at the centroid (P1+/P1). Since an Eulerian approach was employed in this study, the flow front interface was advected by the flow through a fixed mesh. For accurate modelling of interfacial movement, a fringe element reconstruction method developed can provide not only an accurate treatment of material discontinuity but also surface tension across the interface. The effect of surface tension was modelled by imposing tensile stress directly on the constructed surface elements at the flow front interface. To verify the numerical approach developed, the developed algorithm was applied to two examples whose solutions are available in references. Good agreement was obtained between the simulation results and these solutions. Copyright © 2005 John Wiley & Sons, Ltd.     

Numerical Solution of the Polymer System by Front Tracking

The paper describes the application of front tracking to the polymer system, an example of a nonstrictly hyperbolic system. Front tracking computes piecewise constant approximations based on approximate Riemann solutions and exact tracking of waves. It is well known that the front tracking method may introduce a blowup of the initial total variation for initial data along the curve where the two eigenvalues of the hyperbolic system are identical. It is demonstrated by numerical examples that the method converges to the correct solution after a finite time, and that this time decreases with the discretization parameter.For multidimensional problems, front tracking is combined with dimensional splitting, and numerical experiments indicate that large splitting steps can be used without loss of accuracy. Typical CFL numbers are in the range 10–20, and comparisons with Riemann free, high-resolution methods confirm the high efficiency of front tracking.The polymer system, coupled with an elliptic pressure equation, models two-phase, three-component polymer flooding in an oil reservoir. Two examples are presented, where this model is solved by a sequential time stepping procedure. Because of the approximate Riemann solver, the method is non-conservative and CFL numbers must be chosen only moderately larger than unity to avoid substantial material balance errors generated in near-well regions after water breakthrough. Moreover, it is demonstrated that dimensional splitting may introduce severe grid orientation effects for unstable displacements that are accentuated for decreasing discretization parameters.     

A front‐capture scheme for the simulation of homogeneous and particle‐laden gravity currents

This paper presents a new finite volume scheme to efficiently simulate gravity currents flowing over complex surfaces. The two‐dimensional shallow‐water equations, with terms to account for friction and particle transport, are solved using a non‐oscillatory technique. By applying a form drag at the front or head of the dense current, the scheme also implicitly captures the correct Froude number condition at the moving front as it intrudes into the less dense ambient fluid. The Froude number of the head region predicted by the numerical simulation is in good agreement with experimental results for a homogeneous current over a horizontal surface if a realistic profile drag coefficient is chosen. This new scheme avoids the development complexities of a general front‐tracking scheme (e.g., handling merging fronts and multiple currents) and the computational cost of solving the full three‐dimensional Euler equations while providing a fast, accurate simulation of gravity currents. Copyright © 2001 John Wiley & Sons, Ltd.     

Finite element analysis of filling stage in die‐casting process using marker surface method and adaptive grid refinement technique

The marker surface method and the adaptive grid refinement technique have been applied to the three‐dimensional (3‐D) finite element analysis of the filling stage in the die‐casting process. Especially, the marker surface plugging technique and the marker surface regeneration technique incorporated in the marker surface method have been proposed for the efficient analysis of 3‐D practical problems. Through the marker surface plugging technique, new parts of marker surface are effective lycreated in order to eliminate the gaps between the parts of marker surface or between the edge of marker surface and cavity wall. By using the marker surface regeneration technique, the marker surface including a great number of marker elements is recreated on the basis of its original shape in order to decrease the number of marker elements and computational time. A3‐D example used as the benchmark test and a typical industrial problem of the die‐casting process have been analysed. The numerical results have been in good agreement with the experimental results and the efficiency of the adaptive grid refinement technique has been verified. It has been shown that the proposed techniques incorporated in the marker surface method and the adaptive grid refinement technique can be effectively applied to general industrial problems. Copyright © 2004 John Wiley & Sons, Ltd.     

Numerical simulation of mold filling in foam reaction injection molding

A numerical simulation of reaction injection molding (RIM) of polymeric foam is developed, using a finite volume method (FVM). In this study we predict mold filling with a variable‐density fluid that fills a mold by self‐expansion. We deal with two‐dimensional, isothermal cases. With the assumptions of ideal mixing and rapid bubble nucleation, the foam is modelled as a continuum with a time‐dependent density. The continuum is assumed to be a Newtonian fluid. We develop a pressure‐based FVM for unstructured meshes that includes the SIMPLE algorithm with treatment of fluid compressibility. Cell‐based, co‐located storage is used for all physical variables. To treat the moving interface, an explicit high‐resolution interface capturing method is used. Foam flow in a slit is investigated, and the numerical calculations are in good agreement with an approximate analytic solution. For fountain flow in a rectangular cavity, the shape of the flow front is flatter and the traces of the particles are more complicated for an expanding foam than for a constant‐density fluid. An example of mold filling by an expanding foam demonstrates the geometric flexibility of the method. Copyright © 2003 John Wiley & Sons, Ltd.     

Three‐dimensional transient free‐surface flow of viscous fluids inside cavities of arbitrary shape

The three‐dimensional transient free‐surface flow inside cavities of arbitrary shape is examined in this study. An adaptive (Lagrangian) boundary‐element approach is proposed for the general three‐dimensional simulation of confined free‐surface flow of viscous incompressible fluids. The method is stable as it includes remeshing capabilities of the deforming free‐surface, and thus can handle large deformations. A simple algorithm is developed for mesh refinement of the deforming free‐surface mesh. Smooth transition between large and small elements is achieved without significant degradation of the aspect ratio of the elements in the mesh. The method is used to determine the flow field and free‐surface evolution inside cubic, rectangular and cylindrical containers. These problems illustrate the transient nature of the flow during the mixing process. Surface tension effects are also explored. Copyright © 2003 John Wiley & Sons, Ltd.     

Simulation of sharp gas–liquid interface using VOF method and adaptive grid local refinement around the interface

The volume of fluid (VOF) method is used to perform two‐phase simulations (gas–liquid). The governing Navier–Stokes conservation equations of the flow field are numerically solved on two‐dimensional axisymmetric or three‐dimensional unstructured grids, using Cartesian velocity components, following the finite volume approximation and a pressure correction method. A new method of adaptive grid local refinement is developed in order to enhance the accuracy of the predictions, to capture the sharp gas–liquid interface and to speed up the calculations. Results are compared with experimental measurements in order to assess the efficiency of the method. Copyright © 2004 John Wiley & Sons, Ltd.     

Solving free surface fluid flow problems by the minimal kinetic energy functional

In this paper a more accurate minimization technique, namely the minimal kinetic energy method, is developed and used to investigate the free surface fluid flow caused by an obstacle on the bottom of a channel whose exact shape and location are unknown a priori. The fluid flow is assumed to be two‐dimensional, steady, inviscid, incompressible, irrotational and under the effect of the gravitational force. The minimization technique is based on the combination of the boundary integral method and the variational principle technique. This technique is extensively used in identifying unknown bottom surfaces. To illustrate this technique the free surface profile to be applied in the inverse analysis has been generated following a direct formulation when the solid bottom boundary possesses a double hump/double depression, a hump in front of a step, and a depression and a hump in front of a step. For all problems considered, the numerical results are in excellent agreement with the known analytical solution. In fact the computed profiles for both the bottom and free surfaces are graphically indistinguishable from the analytical results. Copyright © 2001 John Wiley & Sons, Ltd.     

High‐order time integrators for front‐tracking finite‐element analysis of viscous free‐surface flows

This paper proposes implicit Runge–Kutta (IRK) time integrators to improve the accuracy of a front‐tracking finite‐element method for viscous free‐surface flow predictions. In the front‐tracking approach, the modeling equations must be solved on a moving domain, which is usually performed using an arbitrary Lagrangian–Eulerian (ALE) frame of reference. One of the main difficulties associated with the ALE formulation is related to the accuracy of the time integration procedure. Indeed, most formulations reported in the literature are limited to second‐order accurate time integrators at best. In this paper, we present a finite‐element ALE formulation in which a consistent evaluation of the mesh velocity and its divergence guarantees satisfaction of the discrete geometrical conservation law. More importantly, it also ensures that the high‐order fixed mesh temporal accuracy of time integrators is preserved on deforming grids. It is combined with the use of a family of L‐stable IRK time integrators for the incompressible Navier–Stokes equations to yield high‐order time‐accurate free‐surface simulations. This is demonstrated in the paper using the method of manufactured solution in space and time as recommended in Verification and Validation. In particular, we report up to fifth‐order accuracy in time. The proposed free‐surface front‐tracking approach is then validated against cases of practical interest such as sloshing in a tank, solitary waves propagation, and coupled interaction between a wave and a submerged cylinder. Copyright © 2015 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.     

Numerical methods for conjunctive two‐dimensional surface and three‐dimensional sub‐surface flows

Sophisticated catchment runoff problems necessitate conjunctive modeling of overland flow and sub‐surface flow. In this paper, finite difference numerical methods are studied for simulation of catchment runoff of two‐dimensional surface flow interacting with three‐dimensional unsaturated and saturated sub‐surface flows. The equations representing the flows are mathematically classified as a type of heat diffusion equation. Therefore, two‐ and three‐dimensional numerical methods for heat diffusion equations were investigated for applications to the surface and sub‐surface flow sub‐models in terms of accuracy, stability, and calculation time. The methods are the purely explicit method, Saul'yev's methods, the alternating direction explicit (ADE) methods, and the alternating direction implicit (ADI) methods. The methods are first examined on surface and sub‐surface flows separately; subsequently, 12 selected combinations of methods were investigated for modeling the conjunctive flows. Saul'yev's downstream (S‐d) method was found to be the preferred method for two‐dimensional surface flow modeling, whereas the ADE method of Barakat and Clark is a less accurate, stable alternative. For the three‐dimensional sub‐surface flow model, the ADE method of Larkin (ADE‐L) and Brian's ADI method are unconditionally stable and more accurate than the other methods. The calculations of the conjunctive models utilizing the S‐d surface flow sub‐model give excellent results and confirm the expectation that the errors of the surface and sub‐surface sub‐models interact. The surface sub‐model dominates the accuracy and stability of the conjunctive model, whereas the sub‐surface sub‐model dominates the calculation time, suggesting the desirability of using a smaller time increment for the surface sub‐model. Copyright © 2000 John Wiley & Sons, Ltd.     

Interface‐fitted moving mesh method for axisymmetric two‐phase flow in microchannels

A boundary‐fitted moving mesh scheme is presented for the simulation of two‐phase flow in two‐dimensional and axisymmetric geometries. The incompressible Navier‐Stokes equations are solved using the finite element method, and the mini element is used to satisfy the inf‐sup condition. The interface between the phases is represented explicitly by an interface adapted mesh, thus allowing a sharp transition of the fluid properties. Surface tension is modelled as a volume force and is discretized in a consistent manner, thus allowing to obtain exact equilibrium (up to rounding errors) with the pressure gradient. This is demonstrated for a spherical droplet moving in a constant flow field. The curvature of the interface, required for the surface tension term, is efficiently computed with simple but very accurate geometric formulas. An adaptive moving mesh technique, where smoothing mesh velocities and remeshing are used to preserve the mesh quality, is developed and presented. Mesh refinement strategies, allowing tailoring of the refinement of the computational mesh, are also discussed. Accuracy and robustness of the present method are demonstrated on several validation test cases. The method is developed with the prospect of being applied to microfluidic flows and the simulation of microchannel evaporators used for electronics cooling. Therefore, the simulation results for the flow of a bubble in a microchannel are presented and compared to experimental data.     

Lattice‐Boltzmann and finite element simulations of fluid flow in a SMRX Static Mixer Reactor

A detailed comparison between the finite element method (FEM) and the lattice‐Boltzmann method (LBM) is presented. As a realistic test case, three‐dimensional fluid flow simulations in an SMRX static mixer were performed. The SMRX static mixer is a piece of equipment with excellent mixing performance and it is used as a highly efficient chemical reactor for viscous systems like polymers. The complex geometry of this mixer makes such three‐dimensional simulations non‐trivial. An excellent agreement between the results of the two simulation methods was found. Furthermore, the numerical results for the pressure drop as a function of the flow rate were close to experimental measurements. Results show that the relatively simple LBM is a good alternative to traditional methods. Copyright © 1999 John Wiley & Sons, Ltd.     

Three‐dimensional boundary element analysis of drop deformation in confined flow for Newtonian and viscoelastic systems

An adaptive (Lagrangian) boundary element approach is proposed for the general three‐dimensional drop deformation in confined flow. The adaptive method is stable as it includes remeshing capabilities of the deforming interface between drop and suspending fluid, and thus can handle large deformations. Both drop and surrounding fluid are viscous incompressible and can be Newtonian or viscoelastic. A boundary‐only formulation is implemented for fluids obeying the linear Jeffrey's constitutive equation. Similarly to the formulation for two‐dimensional Newtonian fluids (Khayat RE, Luciani A, Utracki LA. Boundary element analysis of planar drop deformation in confined flow. Part I. Newtonian fluids. Engineering Analysis of Boundary Elements 1997; 19 : 279), the method requires the solution of two simultaneous integral equations on the interface between the two fluids and the confining solid boundary. Although the problem is formulated for any confining geometry, the method is illustrated for a deforming drop as it is driven by the ambient flow inside a cylindrical tube. The accuracy of the method is assessed by comparison with the analytical solution for two‐phase radial spherical flow, leading to good agreement. The influence of mesh refinement is examined for a drop in simple shear flow. Copyright © 2000 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.     

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速度场测量技术的研究在流体力学和空气动力学中具有极高的学术意义和实用价值．文中归纳介绍了近十多年来人们积极探索的各种全场测速新技术,特别是激光诱导荧光高速流场测试技术、二维粒子跟踪和粒子图象测试技术以及三维测速的最新进展      

Finite analytic boundary condition for a higher‐order finite analytic scheme

A higher‐order finite analytic scheme based on one‐dimensional finite analytic solutions is used to discretize three‐dimensional equations governing turbulent incompressible free surface flow. In order to preserve the accuracy of the numerical scheme, a new, finite analytic boundary condition is proposed for an accurate numerical solution of the partial differential equation. This condition has higher‐order accuracy. Thus, the same order of accuracy is used for the boundary. Boundary conditions were formulated and derived for fluid inflow, outflow, impermeable surfaces and symmetry planes. The derived boundary conditions are treated implicitly and updated with the solution of the problem. The basic idea for the derivation of boundary conditions was to use the discretized form of the governing equations for the fluid flow simplified on the boundaries and flow information. To illustrate the influence of the higher‐order effects at the boundaries, another, lower‐order finite analytic boundary condition, is suggested. The simulations are performed to demonstrate the validity of the present scheme and boundary conditions for a Wigley hull advancing in calm water. Copyright © 2005 John Wiley & Sons, Ltd.