Finite element solution of an enclosed turbulent diffusion flame

A finite element formulation of enclosed turbulent diffusion flames is presented. A primitive variables approach is preferred in the analysis. A mixed interpolation is employed for the velocity and pressure. In the solution of the Navier-Stokes equations, a segregated formulation is adopted, where the pressure discretization equation is obtained directly from the discretized continuity equation, considering the velocity-pressure relationships in the discretized momentum equations. The state of turbulence is defined by a κ–? model. Near solid boundaries, a wall function approach is employed. The combustion rates are estimated using the eddy dissipation concept. The expensive direct treatment of the integrodifferential equations of radiation is avoided by employing the moment method, which allows the derivation of an approximate local field equation for the radiation intensity. The proposed finite element model is verified by investigating a technical turbulent diffusion flame of semi-industrial size, and comparing the results with experiments and finite difference predictions.     

剧变截面圆管内渗流的数值计算方法

对于剧变截面圆管的渗流问题写出不可压缩渗流的基本方程组,对直接求解原始变量(速度和压力)的数值计算方法作出改进。先由非主流方向的运动方程计算压力,后由主流方向的运动方程计算主流方向的速度分量,再由连续性方程计算非主流方向的速度分量。这样可以避免在一般的求解原始变量方法中由连续性方程计算压力时出现的困难和麻烦。根据本方法和剧变截面圆管的特点,采用半交错不等距非正交贴体混合网格系。本文详细写出差分方程和迭代计算公式,对剧变截面圆管内的渗流算例进行数值计算。本方法的优点是简单和实用,在工程上具有较大的应用价值。     

Dynamics of liquid membranes. II: Adaptive finite difference methods

Two domain-adaptive finite difference methods are presented and applied to study the dynamic response of incompressible, inviscid, axisymmetric liquid membranes subject to imposed sinusoidal pressure oscillations. Both finite difference methods map the time-dependent physical domain whose downstream boundary is unknown onto a fixed computational domain. The location of the unknown time-dependent downstream boundary of the physical domain is determined from the continuity equation and results in an integrodifferential equation which is non-linearly coupled with the partial differential equations which govern the conservation of mass and linear momentum and the radius of the liquid membrane. One of the finite difference methods solves the non-conservative form of the governing equations by means of a block implicit iterative method. This method possesses the property that the Jacobian matrix of the convection fluxes has an eigenvalue of algebraic multiplicity equal to four and of geometric multiplicity equal to one. The second finite difference procedure also uses a block implicit iterative method, but the governing equations are written in conservation law form and contain an axial velocity which is the difference between the physical axial velocity and the grid speed. It is shown that these methods yield almost identical results and are more accurate than the non-adaptive techniques presented in Part I. It is also shown that the actual value of the pressure coefficient determined from linear analyses can be exceeded without affecting the stability and convergence of liquid membranes if the liquid membranes are subjected to sinusoidal pressure variations of sufficiently high frequencies.     

A numerical‐variational procedure for laminar flow in curved square ducts

A new numerical method is presented for the solution of the Navier–Stokes and continuity equations governing the internal incompressible flows. The method denoted as the CVP method consists in the numerical solution of these equations in conjunction with three additional variational equations for the continuity, the vorticity and the pressure field, using a non‐staggered grid. The method is used for the study of the characteristics of the laminar fully developed flows in curved square ducts. Numerical results are presented for the effects of the flow parameters like the curvature, the Dean number and the stream pressure gradient on the velocity distributions, the friction factor and the appearance of a pair of vortices in addition to those of the familiar secondary flow. The accuracy of the method is discussed and the results are compared with those obtained by us, using a variation of the velocity–pressure linked equation methods denoted as the PLEM method and the results obtained by other methods. Copyright © 2004 John Wiley & Sons, Ltd.     

Modified incompressible SPH method for simulating free surface problems

An incompressible smoothed particle hydrodynamics (I-SPH) formulation is presented to simulate free surface incompressible fluid problems. The governing equations are mass and momentum conservation that are solved in a Lagrangian form using a two-step fractional method. In the first step, velocity field is computed without enforcing incompressibility. In the second step, a Poisson equation of pressure is used to satisfy incompressibility condition. The source term in the Poisson equation for the pressure is approximated, based on the SPH continuity equation, by an interpolation summation involving the relative velocities between a reference particle and its neighboring particles. A new form of source term for the Poisson equation is proposed and also a modified Poisson equation of pressure is used to satisfy incompressibility condition of free surface particles. By employing these corrections, the stability and accuracy of SPH method are improved. In order to show the ability of SPH method to simulate fluid mechanical problems, this method is used to simulate four test problems such as 2-D dam-break and wave propagation.     

Using a segregated finite element scheme to solve the incompressible Navier-Stokes equations

In this paper, a segregated finite element scheme for the solution of the incompressible Navier-Stokes equations is proposed which is simpler in form than previously reported formulations. A pressure correction equation is derived from the momentum and continuity equations, and equal-order interpolation is used for both the velocity components and pressure. Algorithms such as this have been known to lead to checkerboard pressure oscillations; however, the pressure correction equation of this scheme should not produce these oscillations. The method is applied to several laminar flow situations, and details of the methods used to achieve converged solutions are given.     

Wet/dry areas method for interfacial (free surface) flow

In this paper, a new numerical method is developed for two‐dimensional interfacial (free surface) flows, based on the control volume method and conservative integral form of the Navier–Stokes equations with a standard staggered grid. The new method deploys two continuity equations, the continuity equation of the mass conservation for better convergence of the implicit scheme and the continuity equation of the volume conservation for the equation of pressure correction. The convection terms (the total momentum flux) on the surfaces of control volume are accurately calculated from the wet area exposed to the water, and the dry area exposed to the air. The numerical results produced by the new numerical method agree very well with the analytical solution, experimental images and experimentally measured velocity. Copyright © 2012 John Wiley & Sons, Ltd.     

A face-based monolithic approach for the incompressible magnetohydrodynamics equations

A novel numerical algorithm has been developed to solve the incompressible resistive magnetohydrodynamics equations in a fully coupled form. The numerical method is based on the face-centered unstructured finite volume approximation, where the velocity and magnetic field vector components are defined at the center of edges/faces; meanwhile, the pressure term is defined at element centroid. In order to enforce a divergence-free magnetic field, the gradient of a scalar Lagrange multiplier is introduced into the induction equation. A special attention will be given to satisfy the continuity equation and the Gauss' law for magnetism within each element and the summation of the equations can be exactly reduced to the domain boundary. The first modification to the original algorithm involves the evaluation of the convective fluxes over the two neighboring elements, where the discrete continuity equations are exactly satisfied. The second modification is based on the neglecting electric field term from the Lorentz force in two dimensions. The resulting large-scale algebraic linear equations are solved in a fully coupled manner using the one- and two-level restricted additive Schwarz preconditioners to avoid any time step restrictions forced by stability requirements. The spatial convergence of the algorithm is confirmed by solving the Hartmann flow, and then the algorithm is applied to the classical lid-driven cavity and backward facing step benchmark problems in two and three dimensions. The lid-driven cavity flow calculations at relatively high Stuart numbers indicate the perfect braking effect of the magnetic field in two dimensions.     

A stabilized finite element method for the Stokes problem based on polynomial pressure projections

A new stabilized finite element method for the Stokes problem is presented. The method is obtained by modification of the mixed variational equation by using local L² polynomial pressure projections. Our stabilization approach is motivated by the inherent inconsistency of equal‐order approximations for the Stokes equations, which leads to an unstable mixed finite element method. Application of pressure projections in conjunction with minimization of the pressure–velocity mismatch eliminates this inconsistency and leads to a stable variational formulation. Unlike other stabilization methods, the present approach does not require specification of a stabilization parameter or calculation of higher‐order derivatives, and always leads to a symmetric linear system. The new method can be implemented at the element level and for affine families of finite elements on simplicial grids it reduces to a simple modification of the weak continuity equation. Numerical results are presented for a variety of equal‐order continuous velocity and pressure elements in two and three dimensions. Copyright © 2004 John Wiley & Sons, Ltd.     

A Helmholtz pressure equation method for the calculation of unsteady incompressibl eviscous flows

A time-implicit numerical method for solving unsteady incompressible viscous flow problems is introduced. The method is based on introducing intermediate compressibility into a projection scheme to obtain a Helmholtz equation for a pressure-type variable. The intermediate compressibility increases the diagonal dominance of the discretized pressure equation so that the Helmholtz pressure equation is relatively easy to solve numerically. The Helmholtz pressure equation provides an iterative method for satisfying the continuity equation for time-implicit Navier–Stokes algorithms. An iterative scheme is used to simultaneously satisfy, within a given tolerance, the velocity divergence-free condition and momentum equations at each time step. Collocated primitive variables on a non-staggered finite difference mesh are used. The method is applied to an unsteady Taylor problem and unsteady laminar flow past a circular cylinder.     

A promising boundary element formulation for three‐dimensional viscous flow

In this paper, a new set of boundary‐domain integral equations is derived from the continuity and momentum equations for three‐dimensional viscous flows. The primary variables involved in these integral equations are velocity, traction, and pressure. The final system of equations entering the iteration procedure only involves velocities and tractions as unknowns. In the use of the continuity equation, a complex‐variable technique is used to compute the divergence of velocity for internal points, while the traction‐recovery method is adopted for boundary points. Although the derived equations are valid for steady, unsteady, compressible, and incompressible problems, the numerical implementation is only focused on steady incompressible flows. Two commonly cited numerical examples and one practical pipe flow problem are presented to validate the derived equations. Copyright © 2004 John Wiley & Sons, Ltd.     

Prediction of incompressible flow separation with the approximate factorization technique

This paper describes one application of the approximate factorization technique to the solution of incompressible steady viscous flow problems in two dimensions. The velocity-pressure formulation of the Navier-Stokes equations written in curvilinear non-orthogonal co-ordinates is adopted. The continuity equation is replaced with one equation for the pressure by means of the artificial compressibility concept to obtain a system parabolic in time. The resulting equations are discretized in space with centred finite differences, and the steady state solution obtained by a time-marching ADI method requiring to solve 3 x 3 block tridiagonal linear systems. An optimized fourth-order artificial dissipation is introduced to damp the numerical instabilities of the artificial compressibility equation and ensure convergence. The resulting solver is applied to the prediction of a wide variety of internal flows, including both streamlined boundaries and sharp corners, and fast convergence and good results obtained for all the configurations investigated.     

A semi‐implicit scheme for large Eddy simulation of piston engine flow and combustion

A semi‐implicit scheme is presented for large eddy simulation of turbulent reactive flow and combustion in reciprocating piston engines. First, the governing equations in a deforming coordinate system are formulated to accommodate the moving piston. The numerical scheme is made up of a fourth‐order central difference for the diffusion terms in the transport equations and a fifth‐order weighted essentially nonoscillatory (WENO) scheme for the convective terms. A second‐ order Adams–Bashforth scheme is used for time integration. For higher density ratios, it is combined with a predictor–corrector scheme. The numerical scheme is explicit for time integration of the transport equations, except for the continuity equation which is used together with the momentum equation to determine the pressure field and velocity field by using a Poisson equation for the pressure correction field. The scheme is aimed at the simulation of low Mach number flows typically found in piston engines. An efficient multigrid method that can handle high grid aspect ratio is presented for solving the pressure correction equation. The numerical scheme is evaluated on two test engines, a laboratory four‐stroke engine with rectangular‐shaped engine geometry where detailed velocity measurements are available, and a modified truck engine with practical cylinder geometry where lean ethanol/air mixture is combusted under a homogeneous charge compression ignition (HCCI) condition. Copyright © 2012 John Wiley & Sons, Ltd.     

Numerical solution of three‐dimensional velocity–vorticity Navier–Stokes equations by finite difference method

This paper describes the finite difference numerical procedure for solving velocity–vorticity form of the Navier–Stokes equations in three dimensions. The velocity Poisson equations are made parabolic using the false‐transient technique and are solved along with the vorticity transport equations. The parabolic velocity Poisson equations are advanced in time using the alternating direction implicit (ADI) procedure and are solved along with the continuity equation for velocities, thus ensuring a divergence‐free velocity field. The vorticity transport equations in conservative form are solved using the second‐order accurate Adams–Bashforth central difference scheme in order to assure divergence‐free vorticity field in three dimensions. The velocity and vorticity Cartesian components are discretized using a central difference scheme on a staggered grid for accuracy reasons. The application of the ADI procedure for the parabolic velocity Poisson equations along with the continuity equation results in diagonally dominant tri‐diagonal matrix equations. Thus the explicit method for the vorticity equations and the tri‐diagonal matrix algorithm for the Poisson equations combine to give a simplified numerical scheme for solving three‐dimensional problems, which otherwise requires enormous computational effort. For three‐dimensional‐driven cavity flow predictions, the present method is found to be efficient and accurate for the Reynolds number range 100?Re?2000. Copyright © 2004 John Wiley & Sons, Ltd.     

Unstructured pressure‐correction solver based on a consistent discretization of the Poisson equation

A finite volume incompressible flow solver is presented for three‐dimensional unsteady flows based on an unstructured tetrahedral mesh, with collocation of the flow variables at the cell vertices. The solver is based on the pressure‐correction method, with an explicit prediction step of the momentum equations followed by a Poisson equation for the correction step to enforce continuity. A consistent discretization of the Poisson equation was found to be essential in obtaining a solution. The correction step was solved with the biconjugate gradient stabilized (Bi‐CGSTAB) algorithm coupled with incomplete lower–upper (ILU) preconditioning. Artificial dissipation is used to prevent the formation of instabilities. Flow solutions are presented for a stalling airfoil, vortex shedding past a bridge deck and flow in model alveoli. Copyright © 2000 John Wiley & Sons, Ltd.     

Calculation of vertical velocity in three-dimensional,shallow water equation,finite element models

Computation of vertical velocity within the confines of a three-dimensional, finite element model is a difficult but important task. This paper examines four approaches to the solution of the overdetermined system of equations arising when the first-order continuity equation is solved in conjunction with two boundary conditions. The traditional (TRAD) method neglects one boundary condition, solving the continuity equation with the remaining boundary condition. The vertical derivative of continuity (VDC) method involves solution of the second-order equation obtained by differentiation of the continuity equation with respect to the vertical co-ordinate. The least squares (LS) method minimizes the residuals of the continuity equation (in discrete form) and the two boundary conditions. The adjoint (ADJ) method minimizes the residuals of the continuity equation (in continuous form) and the two boundary conditions. Two domains are considered: a quarter-annular harbour and the southwest coast of Vancouver Island. Results indicate that the highest-quality solution is obtained with both LS and ADJ. Furthermore, ADJ requires less CPU and memory than LS. Therefore the optimal method for computation of vertical velocity in a three-dimensional finite element model is the adjoint (ADJ) method. © 1997 John Wiley & Sons, Ltd.     

An algorithm that accelerates convergence rates for incompressible Navier-Stokes problems

An algorithm, called the Algebraic Continuity Equations Solver (ACES), is developed based on the concept that two algebraic equations (three for 3D problems) can be generated from rearranging the discretized continuity equations. These rearranged equations are used to re-compute the two velocity components (three for 3D problems), whose values are already obtained from solving the momentum equations. When written in a Navier-Stokes computer code, this algorithm is equivalent to a fairly concise set of statements and can be implemented immediately after the computation of the continuity equation. In our analysis, ACES is used in conjunction with a grid having nodal velocity components at the vertices and the nodal pressure at the centre of each computational cell. With the aid of ACES, correction of velocity components during the iteration can be inexpensively made, leading to faster convergence rates or rendering otherwise divergent computations convergent. Test problems include benchmark problems such as lid-driven cavity flows and buoyancy-driven cavity flows of various parametric values and grid sizes. A 3D time-dependent flow in an irregular geometry is also investigated. Discussions are presented to clarify some relevant issues. A possible reason why we think ACES is capable of improving the convergence rates is also given.     

Similarity solutions of vertical plane wall plume based on finite analytic method

The turbulent flow of vertical plane wall plume with concentration variation was studied with the finite analytical method. The k-epsilon model with the effect of buoyancy on turbulent kinetic energy and its dissipation rate was adopted. There were similarity solutions in the uniform environment for the system of equations including the equation of continuity, the equation of momentum along the flow direction and concentration, and equations of k, epsilon. The finite analytic method was applied to obtain the similarity solution. The calculated data of velocity, relative density difference, the kinetic energy of turbulence and its dissipation rate distribution for vertical plane plumes are in good agreement with the experimental data at the turbulent Schmidt number equal to 1.0. The variations of their maximum value along the direction of main flow were also given. It shows that the present model is good, i.e., the effect of buoyancy on turbulent kinetic energy and its dissipation rate should be taken into account, and the finite analytic method is effective.     

Coupled Groundwater Flow and Transport in Porous Media. A Conservative or Non-conservative Form?

A new formulation for the modeling of density coupled flow and transport in porous media is presented. This formulation is based on the development of the mass balance equation by using the conservative form. The system of equations obtained by coupling the flow and transport equations using a state equation is solved by a combination of the mixed hybrid finite element method (MHFEM) and the discontinuous finite element method (DFEM). The former is applied in order to solve the flow equation and the dispersive part of the transport equation, whilst the latter is used to solve the advective part of the transport equation. Although the advantages of the MHFEM are known (efficiency calculation of velocity field and continuity of fluxes from one element to an adjacent one), its application in a classical development form (volumetric fluxes as unknowns) leads to the non-conservative version of the mass balance equation. The associated matrix of the system of equations obtained by hybridization is positive definite but non-symmetrical. By using a new approach (mass fluxes as unknowns) the conservative form of the continuity equation is preserved and the associated matrix of the system of equations obtained by hybridization becomes symmetrical. When applied to Elder's problem involving a strong density contrast, this new approach, with a lower calculation cost, leads to similar or identical results to those found in the specialized literature. The comparison between the conservative and non-conservative formulations solved with the same MHFEM and DFEM combination emphasizes the rigor and the pertinence of this new approach. Furthermore, we show the existence of a limit refinement defining the stability of the numerical solution for Elder's problem.