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.     

A stabilized finite element method for the incompressible Navier–Stokes equations using a hierarchical basis

Stabilized finite element methods have been shown to yield robust, accurate numerical solutions to both the compressible and incompressible Navier–Stokes equations for laminar and turbulent flows. The present work focuses on the application of higher‐order, hierarchical basis functions to the incompressible Navier–Stokes equations using a stabilized finite element method. It is shown on a variety of problems that the most cost‐effective simulations (in terms of CPU time, memory, and disk storage) can be obtained using higher‐order basis functions when compared with the traditional linear basis. In addition, algorithms will be presented for the efficient implementation of these methods within the traditional finite element data structures. Copyright © 2001 John Wiley & Sons, Ltd.     

A finite element model for free surface flows on fixed meshes

In this paper, we present a finite element model for free surface flows on fixed meshes. The main novelty of the approach, compared with typical fixed mesh finite element models for such flows, is that we take advantage of the particularities of free surface flow, instead of considering it a particular case of two‐phase flow. The fact that a given free surface implies a known boundary condition on the interface, allows us to solve the Navier–Stokes equations on the fluid domain uncoupled from the solution on the rest of the finite element mesh. This, together with the use of enhanced integration allows us to model low Froude number flows accurately, something that is not possible with typical two‐phase flow models applied to free surface flow. Copyright © 2007 John Wiley & Sons, Ltd.     

Periodic solution of turbulent oscillating channel flows

The time-dependent turbulent Navier–Stokes equations are solved numerically by a finite element method with an algebraic eddy viscosity model (Baldwin–Lomax formulation) for oscillating turbulent channel flows. The method of averaging is used to analyse the resulting periodic motion of the fluid. Numerical results are obtained for various Strouhal numbers and relative amplitudes. A comparison is made between the numerical and published experimental results. It appears that for low relative amplitudes in a certain range of frequencies the agreement is satisfactory.     

Parallel finite element computation of missile aerodynamics

A flow simulation tool, developed by the authors at the Army HPC Research Center, for compressible flows governed by the Navier–Stokes equations is used to study missile aerodynamics at supersonic speeds, high angles of attack and for large Reynolds numbers. The goal of this study is the evaluation of this Navier–Stokes computational technique for the prediction of separated flow fields around high-length-to-diameter (L/D) bodies. In particular, this paper addresses two issues: (i) turbulence modelling with a finite element computational technique and (ii) efficient performance of the computational technique on two different multiprocessor mainframes, the Thinking Machines CM-5 and CRAY T3D. The paper first provides a discussion of the Navier–Stokes computational technique and the algorithm issues for achieving efficient performance on the CM-5 and T3D. Next, comparisons are shown between the computation and experiment for supersonic ramp flow to evaluate the suitability of the turbulence model. Following that, results of the computations for missile flow fields are shown for laminar and turbulent viscous effects. © 1997 John Wiley & Sons, Ltd.     

Implicit unfactored implementation of two-equation turbulence models in compressible Navier–Stokes methods

An implicit unfactored method for the coupled solution of the compressible Navier–Stokes equations with two-equation turbulence models is presented. Both fluid-flow and turbulence transport equations are discretized by a characteristics-based scheme. The implicit unfactored method combines Newton subiterations and point-by-point Gauss–Seidel subrelaxation. Implicit-coupled and -decoupled strategies are compared for their efficiency in the solution of the Navier–Stokes equations in conjunction with low-Re two-equation turbulence models. Computations have been carried out for the flow over an axisymmetric bump using the k–ϵ and k–ω models. Comparisons have been obtained with experimental data and other numerical solutions. The present study reveals that the implicit unfactored implementation of the two-equation turbulence models reduces the computing time and improves the robustness of the CFD code in turbulent compressible flows. © 1998 John Wiley & Sons, Ltd.     

A NON-OSCILLATORY NO-FREE-PARAMETER FINITE ELEMENT AND ITS APPLICATIONS IN CDF

A non-oscillatory no-free-parameter finite element method (NNFEM) is presented based on the consideration of wave propagation characteristic in different characteristic directions across a strong discontinuity through flux vector splitting in order to satisfy the increasing entropy condition. The algorithm is analysed in detail for the one-dimensional (1D) Euler equation and then extended to the 2D, axisymmetric and 3D Euler and Navier–Stokes equations. Its applications in various cases—in viscid oblique shock wave reflection, flow over a forward step, axisymmetric free jet flow, supersonic flows over 2D and 3D rectangular cavities—are given. These computational results show that the present NNFEM is efficient in practice and stable in operations and is especially capable of giving good resolution in simulating complicated separated and vortical flows interacting with shock waves. © 1997 by John Wiley & Sons, Ltd.     

An implicit mixed finite-volume–finite-element method for solving 3D turbulent compressible flows

The development of new aeronautic projects require accurate and efficient simulations of compressible flows in complex geometries. It is well known that most flows of interest are at least locally turbulent and that the modelling of this turbulence is critical for the reliability of the computations. A turbulence closure model which is both cheap and reasonably accurate is an essential part of a compressible code. An implicit algorithm to solve the 2D and 3D compressible Navier–Stokes equations on unstructured triangular/tetrahedral grids has been extended to turbulent flows. This numerical scheme is based on second-order finite element–finite volume discretization: the diffusive and source terms of the Navier–Stokes equations are computed using a finite element method, while the other terms are computed with a finite volume method. Finite volume cells are built around each node by means of the medians. The convective fluxes are evaluated with the approximate Riemann solver of Roe coupled with the van Albada limiter. The standard k–ϵ model has been introduced to take into account turbulence. Implicit integration schemes with efficient numerical methods (CGS, GMRES and various preconditioning techniques) have also been implemented. Our interest is to present the whole method and to demonstrate its limitations on some well-known test cases in three-dimensional geometries. © 1997 John Wiley & Sons, Ltd.     

Finite element and implicit Runge–Kutta implementation of an acoustics–convection upstream resolution algorithm for the time‐dependent two‐dimensional Euler equations

The second of a two‐paper series, this paper details a solver for the characteristics‐bias system from the acoustics–convection upstream resolution algorithm for the Euler and Navier–Stokes equations. An integral formulation leads to several surface integrals that allow effective enforcement of boundary conditions. Also presented is a new multi‐dimensional procedure to enforce a pressure boundary condition at a subsonic outlet, a procedure that remains accurate and stable. A classical finite element Galerkin discretization of the integral formulation on any prescribed grid directly yields an optimal discretely conservative upstream approximation for the Euler and Navier–Stokes equations, an approximation that remains multi‐dimensional independently of the orientation of the reference axes and computational cells. The time‐dependent discrete equations are then integrated in time via an implicit Runge–Kutta procedure that in this paper is proven to remain absolutely non‐linearly stable for the spatially‐discrete Euler and Navier–Stokes equations and shown to converge rapidly to steady states, with maximum Courant number exceeding 100 for the linearized version. Even on relatively coarse grids, the acoustics–convection upstream resolution algorithm generates essentially non‐oscillatory solutions for subsonic, transonic and supersonic flows, encompassing oblique‐ and interacting‐shock fields that converge within 40 time steps and reflect reference exact solutions. Copyright © 2005 John Wiley & Sons, Ltd.     

A semi‐implicit finite difference method for non‐hydrostatic,free‐surface flows

In this paper a semi‐implicit finite difference model for non‐hydrostatic, free‐surface flows is analyzed and discussed. It is shown that the present algorithm is generally more accurate than recently developed models for quasi‐hydrostatic flows. The governing equations are the free‐surface Navier–Stokes equations defined on a general, irregular domain of arbitrary scale. The momentum equations, the incompressibility condition and the equation for the free‐surface are integrated by a semi‐implicit algorithm in such a fashion that the resulting numerical solution is mass conservative and unconditionally stable with respect to the gravity wave speed, wind stress, vertical viscosity and bottom friction. Copyright © 1999 John Wiley & Sons, Ltd.     

An arbitrary Lagrangian–Eulerian finite element approach to non‐steady state turbulent fluid flow with application to mould filling in casting

This paper presents a two‐dimensional Lagrangian–Eulerian finite element approach of non‐steady state turbulent fluid flows with free surfaces. The proposed model is based on a velocity–pressure finite element Navier–Stokes solver, including an augmented Lagrangian technique and an iterative resolution of Uzawa type. Turbulent effects are taken into account with the k–ε two‐equation statistical model. Mesh updating is carried out through an arbitrary Lagrangian–Eulerian (ALE) method in order to describe properly the free surface evolution. Three comparisons between experimental and numerical results illustrate the efficiency of the method. The first one is turbulent flow in an academic geometry, the second one is a mould filling in effective casting conditions and the third one is a precise confrontation to a water model. Copyright © 2000 John Wiley & Sons, Ltd.     

Design sensitivity and finite element analysis of free surface flows with application to optimal design of casting rigging systems

A novel, fully-analytical design sensitivity formulation for transient, turbulent, free surface flows is derived and implemented in the context of finite element analysis. The time-averaged, turbulent form of the Navier–Stokes equations are solved using a mixing length model, in conjunction with the volume of fluid (VOF) method to model the free surface movement. The design derivatives of these governing equations are computed and solved to find the analytical sensitivities of the fluid position, velocity and pressure fields with respect to shape design variables. The computational efficiency produced by evaluating the sensitivities analytically is demonstrated. The design of the runner and gating system of a simple block casting is presented as an example application for using sensitivity information in design. The analytical sensitivity routine is coupled to a numerical optimizer to yield an automated method for optimal design of the casting rigging system. The results produce runner shapes which eliminate mold-gas aspiration. © 1998 John Wiley & Sons, Ltd.     

Retracted and replaced: A flow‐condition‐based interpolation finite element procedure for triangular grids

A flow‐condition‐based interpolation finite element scheme is presented for use of triangular grids in the solution of the incompressible Navier–Stokes equations. The method provides spatially isotropic discretizations for low and high Reynolds number flows. Various example solutions are given to illustrate the capabilities of the procedure. This article and been retracted and replaced. See retraction and replacement notice DOI: 10.1002/fld.1247 . Copyright © 2005 John Wiley & Sons, Ltd.     

A two‐dimensional vertical non‐hydrostatic σ model with an implicit method for free‐surface flows

An implicit finite difference model in the σ co‐ordinate system is developed for non‐hydrostatic, two‐dimensional vertical plane free‐surface flows. To accurately simulate interaction of free‐surface flows with uneven bottoms, the unsteady Navier–Stokes equations and the free‐surface boundary condition are solved simultaneously in a regular transformed σ domain using a fully implicit method in two steps. First, the vertical velocity and pressure are expressed as functions of horizontal velocity. Second, substituting these relationship into the horizontal momentum equation provides a block tri‐diagonal matrix system with the unknown of horizontal velocity, which can be solved by a direct matrix solver without iteration. A new treatment of non‐hydrostatic pressure condition at the top‐layer cell is developed and found to be important for resolving the phase of wave propagation. Additional terms introduced by the σ co‐ordinate transformation are discretized appropriately in order to obtain accurate and stable numerical results. The developed model has been validated by several tests involving free‐surface flows with strong vertical accelerations and non‐linear waves interacting with uneven bottoms. Comparisons among numerical results, analytical solutions and experimental data show the capability of the model to simulate free‐surface flow problems. Copyright © 2004 John Wiley & Sons, Ltd.     

Numerical computations of internal flows for axisymmetric and two-dimensional nozzles

MacCormack's explicit time-marching scheme is used to solve the full Navier–Stokes unsteady, compressible equations for internal flows. The requirement of a very fine grid to capture shock as well as separated flows is circumvented by employing grid clustering. The numerical scheme is applied for axisymmetric as well as two-dimensional flows. Numerical predictions are compared with experimental data and the qualitative as well as the quantitative agreement is found to be quite satisfactory. © 1997 John Wiley & Sons, Ltd.     

Numerical simulation of turbulent free surface flow with two‐equation k–ε eddy‐viscosity models

This paper presents a finite difference technique for solving incompressible turbulent free surface fluid flow problems. The closure of the time‐averaged Navier–Stokes equations is achieved by using the two‐equation eddy‐viscosity model: the high‐Reynolds k–ε (standard) model, with a time scale proposed by Durbin; and a low‐Reynolds number form of the standard k–ε model, similar to that proposed by Yang and Shih. In order to achieve an accurate discretization of the non‐linear terms, a second/third‐order upwinding technique is adopted. The computational method is validated by applying it to the flat plate boundary layer problem and to impinging jet flows. The method is then applied to a turbulent planar jet flow beneath and parallel to a free surface. Computations show that the high‐Reynolds k–ε model yields favourable predictions both of the zero‐pressure‐gradient turbulent boundary layer on a flat plate and jet impingement flows. However, the results using the low‐Reynolds number form of the k–ε model are somewhat unsatisfactory. Copyright © 2004 John Wiley & Sons, Ltd.     

A study of time integration schemes for the numerical modelling of free surface flows

A semi‐discrete finite element methodology for the modelling of transient free surface flows in the context of Eulerian interface capturing is proposed. The focus of this study is put on the choice of an appropriate time integration strategy for the accurate modelling of the dynamics of free surfaces and of interfacial physics. It is composed of an adaptive time integration scheme for the Navier–Stokes equations, and of the implicit midpoint rule for the transport equation of the Eulerian marker variable. The adaptive scheme allows the automatic determination of a time‐step size that follows the physics of the problem under study, which facilitates the accurate modelling of stiff free surface flows. It is shown that the implicit midpoint rule reduces mass loss for each fluid. Various free surface flow problems are studied to verify and validate the proposed time integration strategy. Copyright © 2005 John Wiley & Sons, Ltd.     

A marker‐and‐cell approach to free surface 2‐D multiphase flows

This work describes a methodology to simulate free surface incompressible multiphase flows. This novel methodology allows the simulation of multiphase flows with an arbitrary number of phases, each of them having different densities and viscosities. Surface and interfacial tension effects are also included. The numerical technique is based on the GENSMAC front‐tracking method. The velocity field is computed using a finite‐difference discretization of a modification of the Navier–Stokes equations. These equations together with the continuity equation are solved for the two‐dimensional multiphase flows, with different densities and viscosities in the different phases. The governing equations are solved on a regular Eulerian grid, and a Lagrangian mesh is employed to track free surfaces and interfaces. The method is validated by comparing numerical with analytic results for a number of simple problems; it was also employed to simulate complex problems for which no analytic solutions are available. The method presented in this paper has been shown to be robust and computationally efficient. Copyright © 2012 John Wiley & Sons, Ltd.     

SINGLE-LAYER FINITE ELEMENT APPROXIMATION OF 3D NAVIER–STOKES EQUATIONS WITH FREE BOUNDARIES FOR SHALLOW WATER FLOWS

In order to simulate flows in the shallow water limit, the full incompressible Navier–Stokes equations with free boundaries are solved using a single layer of finite elements. This implies a polynomial approximation of the velocity profile in the vertical direction, which in turn distorts the wave speed. This fact is verified by numerical results: the wave speed depends on the vertical discretization. When at least two layers of finite elements are used, the boundary layer at the bottom can be simulated and the correct solution for the shallow water limit is recovered. Then this algorithm is applied to the prediction of Tsunami event.     

Implementation of high‐order compact finite‐difference method to parabolized Navier–Stokes schemes

The numerical solution to the parabolized Navier–Stokes (PNS) and globally iterated PNS (IPNS) equations for accurate computation of hypersonic axisymmetric flowfields is obtained by using the fourth‐order compact finite‐difference method. The PNS and IPNS equations in the general curvilinear coordinates are solved by using the implicit finite‐difference algorithm of Beam and Warming type with a high‐order compact accuracy. A shock‐fitting procedure is utilized in both compact PNS and IPNS schemes to obtain accurate solutions in the vicinity of the shock. The main advantage of the present formulation is that the basic flow variables and their first and second derivatives are simultaneously computed with the fourth‐order accuracy. The computations are carried out for a benchmark case: hypersonic axisymmetric flow over a blunt cone at Mach 8. A sensitivity study is performed for the basic flowfield, including profiles and their derivatives obtained from the fourth‐order compact PNS and IPNS solutions, and the effects of grid size and numerical dissipation term used are discussed. The present results for the flowfield variables and also their derivatives are compared with those of other basic flow models to demonstrate the accuracy and efficiency of the proposed method. The present work represents the first known application of a high‐order compact finite‐difference method to the PNS schemes, which are computationally more efficient than Navier–Stokes solutions. Copyright © 2008 John Wiley & Sons, Ltd.