A three‐dimensional vortex particle‐in‐cell method for vortex motions in the vicinity of a wall

A new vortex particle‐in‐cell method for the simulation of three‐dimensional unsteady incompressible viscous flow is presented. The projection of the vortex strengths onto the mesh is based on volume interpolation. The convection of vorticity is treated as a Lagrangian move operation but one where the velocity of each particle is interpolated from an Eulerian mesh solution of velocity–Poisson equations. The change in vorticity due to diffusion is also computed on the Eulerian mesh and projected back to the particles. Where diffusive fluxes cause vorticity to enter a cell not already containing any particles new particles are created. The surface vorticity and the cancellation of tangential velocity at the plate are related by the Neumann conditions. The basic framework for implementation of the procedure is also introduced where the solution update comprises a sequence of two fractional steps. The method is applied to a problem where an unsteady boundary layer develops under the impact of a vortex ring and comparison is made with the experimental and numerical literature. Copyright © 2001 John Wiley & Sons, Ltd.     

Parallel three‐dimensional simulation of the injection molding process

This paper describes the development of a parallel three‐dimensional unstructured non‐isothermal flow solver for the simulation of the injection molding process. The numerical model accounts for multiphase flow in which the melt and air regions are considered to be a continuous incompressible fluid with distinct physical properties. This aspect avoids the complex reconstruction of the interface. A collocated finite volume method is employed, which can switch between first‐ and second‐order accuracy in both space and time. The pressure implicit with splitting of operators algorithm is used to compute the transient flow variables and couple velocity and pressure. The temperature equation is solved using a transport equation with convection and diffusion terms. An upwind differencing scheme is used for the discretization of the convection term to enforce a bounded solution. In order to capture the sharp interface, a bounded compressive high‐resolution scheme is employed. Parallelization of the code is achieved using the PETSc framework and a single program multiple data message passing model. Predicted numerical solutions for several example problems are considered. The first case validates the solution algorithm for moderate Reynolds number flows using a structured mesh. The second case employs an unstructured hybrid mesh showing the capability of the solver to describe highly viscous flows closer to realistic injection molding conditions. The final case presents the non‐isothermal filling of a thick cavity using three mesh sizes and up to 80 processors to assess parallel performance. The proposed algorithm is shown to have good accuracy and scalability. Copyright © 2008 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.     

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.     

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.     

Simulation of three‐dimensional flows over moving objects by an improved immersed boundary–lattice Boltzmann method

An improved immersed boundary–lattice Boltzmann method (IB–LBM) developed recently [28] was applied in this work to simulate three‐dimensional (3D) flows over moving objects. By enforcing the non‐slip boundary condition, the method could avoid any flow penetration to the wall. In the developed IB–LBM solver, the flow field is obtained on the non‐uniform mesh by the efficient LBM that is based on the second‐order one‐dimensional interpolation. As a consequence, its coefficients could be computed simply. By simulating flows over a stationary sphere and torus [28] accurately and efficiently, the proposed IB–LBM showed its ability to handle 3D flow problems with curved boundaries. In this paper, we further applied this method to simulate 3D flows around moving boundaries. As a first example, the flow over a rotating sphere was simulated. The obtained results agreed very well with the previous data in the literature. Then, simulation of flow over a rotating torus was conducted. The capability of the improved IB–LBM for solving 3D flows over moving objects with complex geometries was demonstrated via the simulations of fish swimming and dragonfly flight. The numerical results displayed quantitative and qualitative agreement with the date in the literature. Copyright © 2011 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.     

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.     

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.     

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.     

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.     

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.     

Large eddy simulation of turbulent incompressible flows by a three‐level finite element method

The variational multiscale method provides a methodical framework for large eddy simulation of turbulent flows. In this work, a particular implementation in the form of a three‐level finite element method separating large resolved, small resolved, and unresolved scales is proposed. Residual‐free bubbles are used for the numerical approximation of the small‐scale momentum equation. A stabilizing term is added, in order to take into account the effect of the small‐scale continuity equation. This implementation guarantees the stability of the method without further provisions and offers substantial computational savings on the small‐scale level. Furthermore, it is accounted for the unresolved scales by a specific dynamic modelling procedure. The method is tested for two different turbulent flow situations. Copyright © 2005 John Wiley & Sons, Ltd.