A hybrid vortex method for the simulation of three‐dimensional flows

This paper presents an integral vorticity method for solving three‐dimensional Navier–Stokes equations. A finite volume scheme is implemented to solve the vorticity transport equation, which is discretized on a structured hexahedral mesh. A vortex sheet algorithm is used to enforce the no‐slip boundary condition through a vorticity flux at the boundary. The Biot–Savart integral is evaluated to compute the velocity field, in conjunction with a fast algorithm based on multipole expansion. This method is applied to the simulation of uniform flow past a sphere. Copyright © 2007 John Wiley & Sons, Ltd.     

Finite element solution of three‐dimensional turbulent flows applied to mold‐filling problems

This paper presents a finite element solution algorithm for three‐dimensional isothermal turbulent flows for mold‐filling applications. The problems of interest present unusual challenges for both the physical modelling and the solution algorithm. High‐Reynolds number transient turbulent flows with free surfaces have to be computed on complex three‐dimensional geometries. In this work, a segregated algorithm is used to solve the Navier–Stokes, turbulence and front‐tracking equations. The streamline–upwind/Petrov–Galerkin method is used to obtain stable solutions to convection‐dominated problems. Turbulence is modelled using either a one‐equation turbulence model or the κ–ε two‐equation model with wall functions. Turbulence equations are solved for the natural logarithm of the turbulence variables. The change of dependent variables allows for a robust solution algorithm and good predictions even on coarse meshes. This is very important in the case of large three‐dimensional applications for which highly refined meshes result in untreatable large numbers of elements. The position of the flow front in the mold cavity is computed using a level set approach. Finally, equations are integrated in time using an implicit Euler scheme. The methodology presents the robustness and cost effectiveness needed to tackle complex industrial applications. Copyright © 2000 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.     

An unstructured grid,three‐dimensional model based on the shallow water equations

A semi‐implicit finite difference model based on the three‐dimensional shallow water equations is modified to use unstructured grids. There are obvious advantages in using unstructured grids in problems with a complicated geometry. In this development, the concept of unstructured orthogonal grids is introduced and applied to this model. The governing differential equations are discretized by means of a semi‐implicit algorithm that is robust, stable and very efficient. The resulting model is relatively simple, conserves mass, can fit complicated boundaries and yet is sufficiently flexible to permit local mesh refinements in areas of interest. Moreover, the simulation of the flooding and drying is included in a natural and straightforward manner. These features are illustrated by a test case for studies of convergence rates and by examples of flooding on a river plain and flow in a shallow estuary. Copyright © 2000 John Wiley & Sons, Ltd.     

CFD simulation of two‐ and three‐dimensional free‐surface flow

The paper describes the implementation of moving‐mesh and free‐surface capabilities within a 3‐d finite‐volume Reynolds‐averaged‐Navier–Stokes solver, using surface‐conforming multi‐block structured meshes. The free‐surface kinematic condition can be applied in two ways: enforcing zero net mass flux or solving the kinematic equation by a finite‐difference method. The free surface is best defined by intermediate control points rather than the mesh vertices. Application of the dynamic boundary condition to the piezometric pressure at these points provides a hydrostatic restoring force which helps to eliminate any unnatural free‐surface undulations. The implementation of time‐marching methods on moving grids are described in some detail and it is shown that a second‐order scheme must be applied in both scalar‐transport and free‐surface equations if flows driven by free‐surface height variations are to be computed without significant wave attenuation using a modest number of time steps. Computations of five flows of theoretical and practical interest—forced motion in a pump, linear waves in a tank, quasi‐1d flow over a ramp, solitary wave interaction with a submerged obstacle and 3‐d flow about a surface‐penetrating cylinder—are described to illustrate the capabilities of our code and methods. Copyright © 2003 John Wiley & Sons, Ltd.     

A first‐order volume of fluid convection model in three‐dimensional space

To improve the numerical analysis of free surface convections and reconstruction in a three‐dimensional space, a first‐order algorithm is developed based on the volume of fluid (VOF) theory. The methodology applied to the first‐order method (FOM) is to define a first‐order surface as near to the horizontal as possible while satisfying the defined volume fraction of a cell. The developed method is compared against the donor cell method of zeroth‐order through simulation of the transitional and rotational convection of liquid spheres. Although the donor cell method shows relatively good predictions for the sphere of a large diameter, it shows poor performance of large distortions for a sphere of a relatively small diameter. However, the FOM developed in this study always shows quite satisfactory prediction results for free surface convection. Copyright © 2001 John Wiley & Sons, Ltd.     

Calculation of three‐dimensional turbulent flow with a finite volume multigrid method

A numerical method for the efficient calculation of three‐dimensional incompressible turbulent flow in curvilinear co‐ordinates is presented. The mathematical model consists of the Reynolds averaged Navier–Stokes equations and the k–ε turbulence model. The numerical method is based on the SIMPLE pressure‐correction algorithm with finite volume discretization in curvilinear co‐ordinates. To accelerate the convergence of the solution method a full approximation scheme‐full multigrid (FAS‐FMG) method is utilized. The solution of the k–ε transport equations is embedded in the multigrid iteration. The improved convergence characteristic of the multigrid method is demonstrated by means of several calculations of three‐dimensional flow cases. Copyright © 1999 John Wiley & Sons, Ltd.     

Simulation of three‐dimensional bubbles using desingularized boundary integral method

A very simple model based on the three‐dimensional desingularized boundary integral method is applied to study the evolution of bubble(s) with or without the presence of solid structures. The choice of the desingularization parameters, which is crucial to the success of the method, is studied in the context of bubble dynamics. With the proper choice of parameters, the new model is far more efficient than previous models with virtually the same level of accuracy being achieved. This is largely attributed to the simplicity of the desingularization method. Furthermore, the new model offers a simple and attractive way for mesh refinement. Although it has limitations in the sense that, with this model the time stepping tends to slow down as two surfaces approach each other, this can be easily rectified by switching over to a direct method so that the two surfaces can be drawn closer as required in the context of jet impact. After this the new model can be reinstated to treat the complicated doubly connected geometry involving toroidal bubbles that would otherwise be very difficult to deal with. Copyright © 1999 John Wiley & Sons, Ltd.     

Three‐dimensional modeling of non‐hydrostatic free‐surface flows on unstructured grids

A three‐dimensional numerical model is presented for the simulation of unsteady non‐hydrostatic shallow water flows on unstructured grids using the finite volume method. The free surface variations are modeled by a characteristics‐based scheme, which simulates sub‐critical and super‐critical flows. Three‐dimensional velocity components are considered in a collocated arrangement with a σ‐coordinate system. A special treatment of the pressure term is developed to avoid the water surface oscillations. Convective and diffusive terms are approximated explicitly, and an implicit discretization is used for the pressure term to ensure exact mass conservation. The unstructured grid in the horizontal direction and the σ coordinate in the vertical direction facilitate the use of the model in complicated geometries. Solution of the non‐hydrostatic equations enables the model to simulate short‐period waves and vertically circulating flows. Copyright © 2015 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.     

Bounds on outputs of the exact weak solution of the three‐dimensional Stokes problem

A method for obtaining rigorous upper and lower bounds on an output of the exact weak solution of the three‐dimensional Stokes problem is described. Recently bounds for the exact outputs of interest have been obtained for both the Poisson equation and the advection‐diffusion‐reaction equation. In this work, we extend this approach to the Stokes problem where a novel formulation of the method also leads to a simpler flux calculation based on the directly equilibrated flux method. To illustrate this technique, bounds on the flowrate are calculated for an incompressible creeping flow driven by a pressure gradient in an endless square channel with an array of rectangular obstacles in the center. Copyright © 2009 John Wiley & Sons, Ltd.     

注塑成型流动诱导应力的数值计算

根据注塑成型的特点引入合理的假设,简化了粘性、可压缩、非等温塑料熔体流动的控制方程及基于PTT(Phan Thien Tanner)模型的本构方程,用分部积分法推导了关于压力场的拟Poisson方程,用待定系数法导出了流动应力的解析表达式.用有限元法求解压力场,有限差分及"上风"法离散求解温度场,根据解析表达式计算速度场及应力场,再进行应力-速度迭代求出非线性问题的最终解.比较了PC材料的模拟结果与光弹实验结果,模拟结果与实验结果基本一致.     

An adaptive front tracking technique for three‐dimensional transient flows

An adaptive technique, based on both surface stretching and surface curvature analysis for tracking strongly deforming fluid volumes in three‐dimensional flows is presented. The efficiency and accuracy of the technique are demonstrated for two‐ and three‐dimensional flow simulations. For the two‐dimensional test example, the results are compared with results obtained using a different tracking approach based on the advection of a passive scalar. Although for both techniques roughly the same structures are found, the resolution for the front tracking technique is much higher. In the three‐dimensional test example, a spherical blob is tracked in a chaotic mixing flow. For this problem, the accuracy of the adaptive tracking is demonstrated by the volume conservation for the advected blob. Adaptive front tracking is suitable for simulation of the initial stages of fluid mixing, where the interfacial area can grow exponentially with time. The efficiency of the algorithm significantly benefits from parallelization of the code. Copyright © 2000 John Wiley & Sons, Ltd.     

A three‐dimensional hydrodynamic model for shallow waters using unstructured Cartesian grids

This paper presents a three‐dimensional unstructured Cartesian grid model for simulating shallow water hydrodynamics in lakes, rivers, estuaries, and coastal waters. It is a flux‐based finite difference model that uses a cut‐cell approach to fit the bottom topography and shorelines and, at the same time, has the flexibility of discretizing complex geometries with Cartesian grids that can be arbitrarily downsized in the two horizontal directions simultaneously. Because of the use of Cartesian grids, the grid generation is very simple and does not suffer the grid generation headache often seen in many other unstructured models, as the unstructured Cartesian grid model does not have any requirements on the orthogonality of the grids. The newly developed unstructured Cartesian grid model was validated against analytical solutions for a three‐dimensional seiching case in a rectangular basin, before it was compared with another three‐dimensional model named LESS3D for circulations and salinity transport processes in an idealized embayment that is driven by tides and freshwater inflows. Model tests show that the numerical procedure used in the unstructured Cartesian grid model is robust. Similar to other unstructured models, a variable grid size has resulted in a smaller number of grids required for a reasonable model simulation, which in turn reduces the CPU time used in the model run. Copyright © 2010 John Wiley & Sons, Ltd.     

注塑成型三维流动的数值模拟

建立了非等温、粘性、不可压缩、非牛顿流体流动的控制方程。为了避免同时求解耦合的压力场、速度场,本文通过修改Galerkin方法的变分方程,导出了关于压力场的拟Poisson方程,用迭代法独立地求解连续性方程、动量方程,并进行速度一粘度迭代求出最终的压力场、速度场。由于直接使用Galerkin方法求解能量方程容易引起温度场的振荡,本文采用隐式格式及“上风”法离散能量方程,用超松驰迭代法求解温度场的代数方程组。比较了模拟结果与等温管道流动的解析解及法兰的实际注射结果,算例表明本文方法可以预测注射成型流动过程中的一些重要特征。与传统Galerkin方法相比,本文方法可以减少内存,提高数值方法的稳定性。