Two-equation (k, ϵ) turbulence computations by the use of a finite element model

A finite element technique is presented and applied to some one- and two-dimensional turbulent flow problems. The basic equations are the Reynolds averaged momentum equations in conjunction with a two-equation (k, ?) turbulence model. The equations are written in time-dependent form and stationary problems are solved by a time iteration procedure. The advection parts of the equations are treated by the use of a method of characteristics, while the continuity requirement is satisfied by a penalty function approach. The general numerical formulation is based on Galerkin's method. Computational results are presented for one-dimensional steady-state and oscillatory channel flow problems and for steady-state flow over a two-dimensional backward-facing step.     

Spectral/hp penalty least‐squares finite element formulation for unsteady incompressible flows

In this paper, we present spectral/hp penalty least‐squares finite element formulation for the numerical solution of unsteady incompressible Navier–Stokes equations. Pressure is eliminated from Navier–Stokes equations using penalty method, and finite element model is developed in terms of velocity, vorticity and dilatation. High‐order element expansions are used to construct discrete form. Unlike other penalty finite element formulations, equal‐order Gauss integration is used for both viscous and penalty terms of the coefficient matrix. For time integration, space–time decoupled schemes are implemented. Second‐order accuracy of the time integration scheme is established using the method of manufactured solution. Numerical results are presented for impulsively started lid‐driven cavity flow at Reynolds number of 5000 and transient flow over a backward‐facing step. The effect of penalty parameter on the accuracy is investigated thoroughly in this paper and results are presented for a range of penalty parameter. Present formulation produces very accurate results for even very low penalty parameters (10–50). Copyright © 2008 John Wiley & Sons, Ltd.     

A finite element analysis of the steady laminar entrance flow in a 90° curved tube

A standard Galerkin finite element penalty function method is used to approximate the solution of the three-dimensional Navier–Stokes equations for steady incompressible Newtonian entrance flow in a 90° curved tube (curvature ratio δ = 1/6) for a triple of Dean numbers (κ = 41, 122 and 204). The computational results for the intermediate Dean number (κ = 122) are compared with the results of laser–Doppler velocity measurements in an equivalent experimental model. For both the axial and secondary velocity components, fair agreement between the computational and experimental results is found.     

3D macrosegregation simulation with anisotropic remeshing

The article presents a three-dimensional coupled numerical solution of momentum, mass, energy and solute conservation equations, for binary alloy solidification. The interdendritic flow in the mushy zone is assumed to obey the Darcy's law. Microsegregation is governed by the lever rule, assuming local equilibrium at phase interfaces. The resulting energy and solute advection–diffusion equations are solved using the Streamline-Upwind/Petrov–Galerkin (SUPG) finite element method. A SUPG-PSPG velocity-pressure formulation is applied for the momentum equation. The full algorithm was implemented in the 3D code THERCAST, together with an anisotropic remeshing method. Two applications have been considered: a small ingot of Pb-48wt%Sn alloy and a large steel ingot. The numerical results of these two cases are presented with the evolution of temperature, liquid velocity, and solute concentration fields during solidification. To cite this article: S. Gouttebroze et al., C. R. Mecanique 335 (2007).     

On streamline diffusion arising in Galerkin FEM with predictor/multi‐corrector time integration

In the present paper, the author shows that the predictor/multi‐corrector (PMC) time integration for the advection–diffusion equations induces numerical diffusivity acting only in the streamline direction, even though the equations are spatially discretized by the conventional Galerkin finite element method (GFEM). The transient 2‐D and 3‐D advection problems are solved with the PMC scheme using both the GFEM and the streamline upwind/Petrov Galerkin (SUPG) as the spatial discretization methods for comparison. The solutions of the SUPG‐PMC turned out to be overly diffusive due to the additional PMC streamline diffusion, while the solutions of the GFEM‐PMC were comparatively accurate without significant damping and phase error. A similar tendency was seen also in the quasi‐steady solutions to the incompressible viscous flow problems: 2‐D driven cavity flow and natural convection in a square cavity. Copyright © 2002 John Wiley & Sons, Ltd.     

Numerical simulation of the target system of an ADSS

The study of thermal–hydraulic issues related to target module of an accelerator driven sub-critical nuclear reactor system (ADSS) play a crucial role in its design. For this, one needs to analyze laminar/turbulent flow and heat transfer characteristics of lead bismuth eutectic (LBE), target cum coolant liquid, at low Prandtl number in the target system of an ADSS. In this work, equations governing conservation of mass, momentum and energy together with equations for kinetic energy and its dissipation rate are solved numerically using streamline upwind Petrov-Galerkin (SUPG)-finite element (FE) method in a two-dimensional axisymmetric ADSS target system. The transfer of kinetic energy and its dissipation rate are modeled using standard k ? ε equations with the wall function approach. The complex target module comprises a 180° turn-around flow along a straight flow guide and a window interfacing the proton beam interaction with LBE. The principal purpose of the analysis is to trace the flow structure in the domain and the temperature distribution on the window. Increasing flow rate to turbulent regime is seen to minimize the number of re-circulation or stagnation zones that may lead to the development of hot spots in the flow domain.     

A momentum coupling method for the unsteady incompressible Navier–Stokes equations on the staggered grid

A new numerical method is developed to efficiently solve the unsteady incompressible Navier–Stokes equations with second-order accuracy in time and space. In contrast to the SIMPLE algorithms, the present formulation directly solves the discrete x- and y-momentum equations in a coupled form. It is found that the present implicit formulation retrieves some cross convection terms overlooked by the conventional iterative methods, which contribute to accuracy and fast convergence. The finite volume method is applied on the fully staggered grid to solve the vector-form momentum equations. The preconditioned conjugate gradient squared method (PCGS) has proved very efficient in solving the associate linearized large, sparse block-matrix system. Comparison with the SIMPLE algorithm has indicated that the present momentum coupling method is fast and robust in solving unsteady as well as steady viscous flow problems. © 1998 John Wiley & Sons, Ltd.     

On the numerical stability of mixed finite-element methods for viscoelastic flows governed by differential constitutive equations

Mixed finite-element methods for computation of viscoelastic flows governed by differential constitutive equations vary by the polynomial approximations used for the velocity, pressure, and stress fields, and by the weighted residual methods used to discretize the momentum, continuity, and constitutive equations. This paper focuses on computation of the linear stability of the planar Couette flow as a test of the numerical stability for solution of the upper-convected Maxwell model. Previous theoretical results prove this inertialess flow to be always stable, but that accurate calculation is difficult at high De because eigenvalues with fine spatial structure and high temporal frequency approach neutral stability. Computations with the much used biquadratic finite-element approximations for velocity and deviatoric stress and bilinear interpolation for pressure demonstrate numerical instability beyond a critical value of De for either the explicitly elliptic momentum equation (EEME) or elastic-viscous split-stress (EVSS) formulations, applying Galerkin's method for solution of the momentum and continuity equations, and using streamline upwind Petrov-Galerkin (SUPG) method for solution of the hyperbolic constitutive equation. The disturbance that causes the instability is concentrated near the stationary streamline of the base flow. The removal of this instability in a slightly modified form of the EEME formulation suggests that the instability results from coupling the approximations to the variables. A new mixed finite-element method, EVSS-G, is presented that includes smooth interpolation of the velocity gradients in the constitutive equation that is compatible with bilinear interpolation of the stress field. This formulation is tested with SUPG, streamline upwinding (SU), and Galerkin least squares (GLS) discretization of the constitutive equation. The EVSS-G/SUPG and EVSS-G/SU do not have the numerical instability described above; linear stability calculations for planar Couette flow are stable to values of De in excess of 50 and converge with mesh and time step. Calculations for the steady-state flow and its linear stability for a sphere falling in a tube demonstrate the appearance of linear instability to a time-periodic instability simultaneously with the apparent loss of existence of the steady-state solution. The instability appears as finely structured secondary cells that move from the front to the back of the sphere.Financial support for this research was given by the National Science Foundation, the Office of Naval Research, and the Defense Research Projects Agency. Computational resources were supplied by a grant from the Pittsburgh National Supercomputer Center and by the MIT Supercomputer Facility.     

超音速粘性流动的SUPG有限元数值解法

本文构造了准简化 N-S 方程组的 SUPG(Streamline Upwind/Petrov-Galerk-in)加权剩余式,并利用该方法对 Burgers 方程、无粘性激波反射问题、以及超音速平板和压缩拐角的层流流动作了数值求解。计算结果表明,本文方法是精确、收敛和稳定的。     

Four-field Galerkin/least-squares formulation for viscoelastic fluids

A new Galerkin/Least-Squares (GLS) stabilized finite element method is presented for computing viscoelastic flows of complex fluids described by the conformation tensor; it extends the well-established GLS method for computing flows of incompressible Newtonian fluids. GLS methods are attractive for large-scale computations because they yield linear systems that can be solved easily with iterative solvers (e.g., the Generalized Minimum Residual method) and because they allow simple combinations of interpolation functions that can be conveniently and efficiently implemented on modern distributed-memory cache-based clusters.Like other state-of-the-art methods for computing viscoelastic flows (e.g., DEVSS-TG/SUPG), the new GLS method introduces a separate variable to represent the velocity gradient; with the aid of this variable, the conservation equations of mass, momentum, conformation, and the definition of velocity gradient are converted into a set of first-order partial differential equations in four unknown fields—pressure, velocity, conformation, and velocity gradient. The unknown fields are represented by low-order (continuous piecewise linear or bilinear) finite element basis functions.The method is applied to the Oldroyd-B constitutive equation and is tested in two benchmark problems—flow in a planar channel and flow past a cylinder in a channel. Results show that (1) the mesh-convergence rate of GLS is comparable to the DEVSS-TG/SUPG method; (2) the LS stabilization permits using equal-order basis functions for all fields; (3) GLS handles effectively the advective terms in the evolution equation of the conformation tensor; and (4) GLS yields accurate results at lower computational costs than DEVSS-type methods.     

Modelling and simulation of sprays in laminar flames

In the present paper we model and numerically simulate steady, laminar, premixed spray flames. The gasphase is described in Eulerian form by the equations governing the conservation of overall mass, momentum, energy and species mass. The liquid phase is described in Lagrangian form by the overall continuity equation, which reduces to an equation for the droplet size, the equations of motion, the energy equation and a droplet density function transport equation. The latter is the so-called spray equation, which, at any position in the chemically reacting flowfield, describes the joint distribution of droplet size, droplet velocity and droplet temperature. Herein the spray equation is solved using a Monte Carlo method. Detailed models of the exchange of mass, momentum and energy between the gaseous and the liquid phase are taken into account. The results presented in this paper are for an octane-air flame, where small amounts of liquid octane in form of a liquid spray are added to a fresh, unburnt gaseous octane-air mixture.Presented at Euromech Colloquium 324: The Combustion of Drops, Sprays and Aerosols, 25th–27th July 1994, Marseilles, France.     

Numerical modelling of shear-dependent mass transfer in large arteries

A numerical scheme for the simulation of blood flow and transport processes in large arteries is presented. Blood flow is described by the unsteady 3D incompressible Navier–Stokes equations for Newtonian fluids; solute transport is modelled by the advection–diffusion equation. The resistance of the arterial wall to transmural transport is described by a shear-dependent wall permeability model. The finite element formulation of the Navier–Stokes equations is based on an operator-splitting method and implicit time discretization. The streamline upwind/Petrov–Galerkin (SUPG) method is applied for stabilization of the advective terms in the transport equation and in the flow equations. A numerical simulation is carried out for pulsatile mass transport in a 3D arterial bend to demonstrate the influence of arterial flow patterns on wall permeability characteristics and transmural mass transfer. The main result is a substantial wall flux reduction at the inner side of the curved region. © 1997 John Wiley & Sons, Ltd.     

Least‐squares meshfree method for incompressible Navier–Stokes problems

A least‐squares meshfree method based on the first‐order velocity–pressure–vorticity formulation for two‐dimensional incompressible Navier–Stokes problem is presented. The convective term is linearized by successive substitution or Newton's method. The discretization of all governing equations is implemented by the least‐squares method. Equal‐order moving least‐squares approximation is employed with Gauss quadrature in the background cells. The boundary conditions are enforced by the penalty method. The matrix‐free element‐by‐element Jacobi preconditioned conjugate method is applied to solve the discretized linear systems. Cavity flow for steady Navier–Stokes problem and the flow over a square obstacle for time‐dependent Navier–Stokes problem are investigated for the presented least‐squares meshfree method. The effects of inaccurate integration on the accuracy of the solution are investigated. Copyright © 2004 John Wiley & Sons, Ltd.     

Incompressible Roe‐like scheme for turbulent flows

In the current study, numerical investigation of incompressible turbulent flow is presented. By the artificial compressibility method, momentum and continuity equations are coupled. Considering Reynolds averaged Navier–Stokes equations, the Spalart–Allmaras turbulence model, which has accurate results in two‐dimensional problems, is used to calculate Reynolds stresses. For convective fluxes a Roe‐like scheme is proposed for the steady Reynolds averaged Navier–Stokes equations. Also, Jameson averaging method was implemented. In comparison, the proposed characteristics‐based upwind incompressible turbulent Roe‐like scheme, demonstrated very accurate results, high stability, and fast convergence. The fifth‐order Runge–Kutta scheme is used for time discretization. The local time stepping and implicit residual smoothing were applied as the convergence acceleration techniques. Suitable boundary conditions have been implemented considering flow behavior. The problem has been studied at high Reynolds numbers for cross flow around the horizontal circular cylinder and NACA0012 hydrofoil. Results were compared with those of others and a good agreement has been observed. Copyright © 2012 John Wiley & Sons, Ltd.     

Numerical Simulation of Forced Convective Heat Transfer Past a Square Diamond-Shaped Porous Cylinder

Fluid flow and heat transfer around and through a porous cylinder is an important issue in engineering applications. In this paper a numerical study is carried out for simulating the fluid flow and forced convection heat transfer around and through a square diamond-shaped porous cylinder. The flow is two-dimensional, steady, and laminar. Conservation laws of mass, momentum, and heat transport equations are applied in the clear region and Darcy–Brinkman–Forchheimer model for simulating the flow in the porous medium has been used. Equations with the relevant boundary conditions are numerically solved using a finite volume approach. In this study, Reynolds and Darcy numbers are varied within the ranges of $1<Re<45$ and $10^{-6}<Da<10^{- 2}$ , respectively. The porosity $(\varepsilon )$ is 0.5. This paper presents the effect of Reynolds and Darcy numbers on the flow structure and heat transfer characteristics. Finally, these parameters are compared among solid and porous cylinder. It was found that the drag coefficient decreases and flow separation from the cylinder is delayed with increasing Darcy number. Also the size of the thermal plume decreases by decreasing Darcy number.     

Normal Mode Analysis of the High Speed Channel Flow in a Bidisperse Porous Medium

The linear Darcy–Brinkman model of the high speed flow in a bidisperse porous medium proposed by Nield and Kuznetsov (Transport Phenomena in Porous Media, 2005) is revisited in this paper. For the steady unidirectional flow in a parallel plane channel the exact analytical solutions for the fluid velocities are worked out by the normal-mode reduction of the governing equations. The limiting cases of the weak and strong momentum transfer between the flows in the fracture and porous phases are discussed in detail. A comparison to the nonlinear Forchheimer extension of the model proposed recently by Nield and Kuznetsov (Transport Porous Media, 2013) shows that, in the considered parameter range, the nonlinear effect of the Forchheimer drag is negligibly small. Even the simplest zero-momentum transfer solution yields an acceptable approximation.     

Analysis of flow in the plate‐spiral of a reaction turbine using a streamline upwind Petrov–Galerkin method

The prediction of the flow field in a novel spiral casing has been accomplished. Hydraulic turbine manufacturers are considering the potential of using a special type of spiral casing because of the easier manufacturing process involved in its fabrication. These special spiral casings are known as plate‐spirals. Numerical simulation of complex three‐dimensional flow through such spiral casings has been accomplished using a finite element method (FEM). An explicit Eulerian velocity correction scheme has been deployed to solve the Reynolds‐average Navier–Stokes equations. The simulation has been performed to describe the flow in high Reynolds number (10⁶) regimes. For spatial discretization, a streamline upwind Petrov–Galerkin (SUPG) technique has been used. The velocity field and the pressure distribution inside the spiral casing reveal meaningful results. Copyright © 2000 John Wiley & Sons, Ltd.     

A Bubble Element for the Compressible Euler Equations

In this paper, we present a new Galerkin finite element method with bubble function for the compressible Euler equations. This method is derived from the scaled bubble element for the advection-diffusion problems developed by Simo and his colleagues, which is based on the equivalence between the Galerkin method employing piecewise linear interpolation with bubble functions and the Streamline-Upwind/Petrov Galerkin (SUPG) finite element method using P₁ approximation in the steady advection-diffusion problem. Simo and this author have applied this approach to transient advection-diffusion problems by using a special scaled bubble function called P-scaled bubble, which is designed to work in the transient advection-diffusion problems for any Peclet number from 0 to ∞. The method presented in this paper is an application of this p-scaled bubble element to a pure hyperbolic system.