Finite element methods for one-dimensional combustion problems

Three adaptive finite element methods based on equidistribution, elliptic grid generation and hybrid techniques are used to study a system of reaction–diffusion equations. It is shown that these techniques must employ sub-equidistributing meshes in order to avoid ill-conditioned matrices and ensure the convergence of the Newton method. It is also shown that elliptic grid generation methods require much longer computer times than hybrid and static rezoning procedures. The paper also includes characteristic, Petrov–Galerkin and flux-corrected transport algorithms which are used to study a linear convection–reaction–diffusion equation that has an analytical solution. The flux-corrected transport technique yields monotonic solutions in good agreement with the analytical solution, whereas the Petrov–Galerkin method with quadratic upstream-weighted functions results in very diffused temperature profiles. The characteristic finite element method which uses a Lagrangian–Eulerian formulation overpredicts the flame front location and exhibits overshoots and undershoots near the temperature discontinuity. These overshoots and undershoots are due to the interpolation of the results of the Lagrangian operator onto the fixed Eulerian grid used to solve the reaction–diffusion operator, and indicate that characteristic finite element methods are not able to eliminate numerical diffusion entirely.     

线性定常对流占优对流扩散问题的无网格解法

应用无网格Galerkin方法求解对流占优对流扩散问题时会出现非物理现象的数值伪振荡,本文将SUPG方法、GLS方法、SGS方法与无网格Galerkin方法相耦合,成功解决了对流扩散方程中对流项占优时的数值伪振荡问题。运用本文构造的方法,采用线性基和具有C2连续的权函数,应用移动最小二乘法可容易地构造高阶导数连续的形函数,从而避免了有限元方法中当采用线性元插值时,因忽略稳定项中二阶导数项而降低计算精度和稳定性的问题。数值实验表明:本文构造的方法具有计算精度高、稳定性好、计算算法实施简单、前后处理方便的优点,这些方法不仅能适用于对流项占优问题,而且也能很好地消除反应项占优时的数值伪振荡问题。     

A linearization technique for multi-species transport problems

We study a one-dimensional multi-species system of dispersive-advective contaminant transport equations coupled by nonlinear biological (kinetic reactions) and physical (adsorption) processes. To deal with the nonlinearities and the coupling, and to avoid additional computational costs, we propose a linearization technique based on first-order Taylor’s series expansions. A stabilized finite element in space, combined with an Euler implicit finite difference discretization in time, is used to approximate the dispersive-advective transport problem. Three computational tests are performed with different boundary conditions, retardation factors and kinetic parameters for a nonlinear reactive multi-species transport model. The proposed methodology is shown to be accurate and decrease computational costs in the numerical implementation of nonlinear reactive transport problems.     

Spectral element-FCT method for scalar hyperbolic conservation laws

A new algorithm based on spectral element discretizations and flux-corrected transport (FCT) ideas is developed for the solution of discontinuous hyperbolic problems. A conservative formulation is proposed, based on cell averaging and reconstruction procedures, that employs a staggered grid of Gauss–Chebyshev and Gauss–Lobatto–Chebyshev discretizations. In addition, high-order time-differencing schemes, a flux limiter and a general spectral filter are employed to improve the quality of the solution. It is demonstrated through model problems of linear advection and examples of one-dimensional shock formation that the proposed algorithm leads to stable, non-oscillatory solutions of high accuracy away from discontinuities. Typically, spectral or spectral element methods perform very poorly in the presence of even weak discontinuities, although they produce only exponentialy small errors for smooth solutions. Spectral element–FCT methods can provide spectral properties (i.e. minimum dispersion and diffusion errors) as well as great flexibility in the discretization, since a variable number of macroelements or collocation points per element can be employed to accommodate both accuracy and geometric requirements.     

RESEARCH DEVELOPMENTS OF SMOOTHED PARTICLE HYDRODYNAMICS METHOD AND ITS COUPLING WITH FINITE ELEMENT METHOD

拉格朗日型的有限元法和光滑粒子法在模拟材料大变形问题时各存优缺点, 而有限元与光滑粒子耦合算法实现了在小变形区域采用有限元法计算, 在局部的大变形区域采用光滑粒子法计算, 有效地综合了有限元法计算效率高和光滑粒子法能够自然地模拟材料大变形问题的特点.重点论述了有限元法、光滑粒子法以及有限元与光滑粒子耦合算法的研究现状及应用进展, 并讨论了各方法中需要进一步解决的问题.最后通过算例对3种方法的计算精度和计算效率进行了分析, 供研究人员参考.     

Comparisons of numerical methods with respect to convectively dominated problems

A series of numerical schemes: first‐order upstream, Lax–Friedrichs; second‐order upstream, central difference, Lax–Wendroff, Beam–Warming, Fromm; third‐order QUICK, QUICKEST and high resolution flux‐corrected transport and total variation diminishing (TVD) methods are compared for one‐dimensional convection–diffusion problems. Numerical results show that the modified TVD Lax–Friedrichs method is the most competent method for convectively dominated problems with a steep spatial gradient of the variables. Copyright © 2001 John Wiley & Sons, Ltd.     

An efficient 3D particle transport model for use in stratified flow

Random‐walk models are a versatile tool for modelling dispersion of both passive and active tracers in turbulent flow. The physical and mathematical foundations of stochastic Lagrangian models of turbulent diffusion have become more and more solid over the years. An important aspect of these types of models that has not received much attention is the behaviour of the particles near boundaries. Often, a simple stochastic, numerical scheme is used. Because turbulent mixing in the vertical direction is much more complicated than in the two horizontal directions, it is in the vertical direction that a simple numerical scheme, such as the Euler scheme, may cause problems. In this paper our main goal is the development of an efficient 3D particle transport model that can be used in stratified flow. For this type of situation the vertical direction is of special interest. First, a closer look is taken at some considerations that should be regarded when choosing a numerical scheme. Specifically schemes are investigated that can be used in the vertical direction, where the diffusion coefficient is varying in that direction. Experiments are performed regarding the accuracy of different numerical schemes in various situations. The behaviour of the particles near an impermeable layer interface is investigated. The stochastic Heun and Runge–Kutta schemes turn out to be very attractive for this type of model. For the simulation of the transport of various physical quantities, such as salinity, heat, silt, oxygen, or bacteria, different types of models are available. In this case we will take a closer look at the modelling of the transport of pollutants from point sources (either instantaneous or continuous transport). For this purpose a 3D particle transport model has been developed that is especially suited for stratified situations such as can be found in estuaries. The main idea is to use a simple numerical scheme for the horizontal directions and a higher‐order method for the vertical direction. The results play an important role in making specific choices for this type of particle transport model. Copyright © 2005 John Wiley & Sons, Ltd.     

CONSTRAINT VIOLATION STABILIZATION OF EULER-LAGRANGE EQUATIONS WITH NON-HOLONOMIC CONSTRAINTS

Two constraint violation stabilization methods are presented to solve the Euler Lagrange equations of motion of a multibody system with nonholonomic constraints. Compared to the previous works, the newly devised methods can deal with more complicated problems such as those with nonholonomic constraints or redundant constraints, and save the computation time. Finally a numerical simulation of a multibody system is conducted by using the methods given in this paper.     

Numerical solutions of incompressible Euler and Navier-Stokes equations by efficient discrete singular convolution method

An efficient discrete singular convolution (DSC) method is introduced to the numerical solutions of incompressible Euler and Navier-Stokes equations with periodic boundary conditions. Two numerical tests of two-dimensional Navier-Stokes equations with periodic boundary conditions and Euler equations for doubly periodic shear layer flows are carried out by using the DSC method for spatial derivatives and fourth-order Runge-Kutta method for time advancement, respectively. The computational results show that the DSC method is efficient and robust for solving the problems of incompressible flows, and has the potential of being extended to numerically solve much broader problems in fluid dynamics. The project supported by the National Natural Science Foundation of China (No.19902010).     

Positivity‐preserving,flux‐limited finite‐difference and finite‐element methods for reactive transport

A new class of positivity‐preserving, flux‐limited finite‐difference and Petrov–Galerkin (PG) finite‐element methods are devised for reactive transport problems.The methods are similar to classical TVD flux‐limited schemes with the main difference being that the flux‐limiter constraint is designed to preserve positivity for problems involving diffusion and reaction. In the finite‐element formulation, we also consider the effect of numerical quadrature in the lumped and consistent mass matrix forms on the positivity‐preserving property. Analysis of the latter scheme shows that positivity‐preserving solutions of the resulting difference equations can only be guaranteed if the flux‐limited scheme is both implicit and satisfies an additional lower‐bound condition on time‐step size. We show that this condition also applies to standard Galerkin linear finite‐element approximations to the linear diffusion equation. Numerical experiments are provided to demonstrate the behavior of the methods and confirm the theoretical conditions on time‐step size, mesh spacing, and flux limiting for transport problems with and without nonlinear reaction. Copyright © 2003 John Wiley & Sons, Ltd.     

A kinetic theory solution method for the Navier–Stokes equations

The kinetic-theory-based solution methods for the Euler equations proposed by Pullin and Reitz are here extended to provide new finite volume numerical methods for the solution of the unsteady Navier–Stokes equations. Two approaches have been taken. In the first, the equilibrium interface method (EIM), the forward- and backward-flowing molecular fluxes between two cells are assumed to come into kinetic equilibrium at the interface between the cells. Once the resulting equilibrium states at all cell interfaces are known, the evaluation of the Navier–Stokes fluxes is straightforward. In the second method, standard kinetic theory is used to evaluate the artificial dissipation terms which appear in Pullin's Euler solver. These terms are subtracted from the fluxes and the Navier–Stokes dissipative fluxes are added in. The new methods have been tested in a 1D steady flow to yield a solution for the interior structure of a shock wave and in a 2D unsteady boundary layer flow. The 1D solutions are shown to be remarkably accurate for cell sizes large compared to the length scale of the gradients in the flow and to converge to the exact solutions as the cell size is decreased. The steady-state solutions obtained with EIM agree with those of other methods, yet require a considerably reduced computational effort.     

Using the Euler–Lagrange variational principle to obtain flow relations for generalized Newtonian fluids

The Euler–Lagrange variational principle is used to obtain analytical and numerical flow relations in cylindrical tubes. The method is based on minimizing the total stress in theflow duct using the fluid constitutive relation between stress and rate of strain. Newtonian and non-Newtonian fluid models, which include power law, Bingham, Herschel–Bulkley, Carreau, and Cross, are used for demonstration.     

Semi-implicit Skew Upwind Method for problems in environmental hydraulics

A new computational method is presented for reducing numerical diffusion in environmental fluid problems. This method, which is referred to as the Semi-Implicit Skew Upwind Method (SISUM), is a robust solution procedure for the conditional convergence of the discretized transport equations. The method retains the advantage of the low numerical diffusion of the conventional skew upwind schemes but does not suffer from over- or under-shooting often found in these methods due to the improved interpolation schemes. The effectiveness of SISUM is demonstrated in several examples. The comparison of the results of a hybrid scheme and SISUM with field observations of convection-dominated pollutant transport in strongly curvilinear river flow shows that SISUM successfully eliminates the high numerical diffusion produced by the hybrid scheme. The robustness of the method was tested by solving the hydrodynamics of a circular clarifier model with a large density gravity source term in the vertical-momentum equation.     

Euler–Lagrange simulation of high pressure shock waves

This paper presents the numerical simulation of overdriven detonation (or O.D.D.) that occurs when a high velocity object impacts an explosive. The pressure and the velocity at this state are higher than those of the Chapman–Jouguet (C–J) state. First, before the simulation of this event, a study of PBX air blast by using multi-material Eulerian method is presented. Pressure peaks are computed for several distances from the explosive. Second, the O.D.D. phenomenon is modeled by the Euler–Lagrange penalty coupling, which permits to couple a Lagrangian mesh of the flyer plate to multi-material Eulerian mesh of explosives and air. This coupling gives us the high detonation velocities in the acceptor explosive and demonstrates that it is able to handle shock–structure interaction problems.     

Effects of the Jacobian evaluation on Newton's solution of the Euler equations

Newton's method is developed for solving the 2‐D Euler equations. The Euler equations are discretized using a finite‐volume method with upwind flux splitting schemes. Both analytical and numerical methods are used for Jacobian calculations. Although the numerical method has the advantage of keeping the Jacobian consistent with the numerical residual vector and avoiding extremely complex analytical differentiations, it may have accuracy problems and need longer execution time. In order to improve the accuracy of numerical Jacobians, detailed error analyses are performed. Results show that the finite‐difference perturbation magnitude and computer precision are the most important parameters that affect the accuracy of numerical Jacobians. A method is developed for calculating an optimal perturbation magnitude that can minimize the error in numerical Jacobians. The accuracy of the numerical Jacobians is improved significantly by using the optimal perturbation magnitude. The effects of the accuracy of numerical Jacobians on the convergence of the flow solver are also investigated. In order to reduce the execution time for numerical Jacobian evaluation, flux vectors with perturbed flow variables are calculated only for neighbouring cells. A sparse matrix solver that is based on LU factorization is used. Effects of different flux splitting methods and higher‐order discretizations on the performance of the solver are analysed. Copyright © 2005 John Wiley & Sons, Ltd.     

A high‐order alternating direction implicit method for the unsteady convection‐dominated diffusion problem

A high‐order alternating direction implicit (ADI) method for solving the unsteady convection‐dominated diffusion equation is developed. The fourth‐order Padé scheme is used for the discretization of the convection terms, while the second‐order Padé scheme is used for the diffusion terms. The Crank–Nicolson scheme and ADI factorization are applied for time integration. After ADI factorization, the two‐dimensional problem becomes a sequence of one‐dimensional problems. The solution procedure consists of multiple use of a one‐dimensional tridiagonal matrix algorithm that produces a computationally cost‐effective solver. Von Neumann stability analysis is performed to show that the method is unconditionally stable. An unsteady two‐dimensional problem concerning convection‐dominated propagation of a Gaussian pulse is studied to test its numerical accuracy and compare it to other high‐order ADI methods. The results show that the overall numerical accuracy can reach third or fourth order for the convection‐dominated diffusion equation depending on the magnitude of diffusivity, while the computational cost is much lower than other high‐order numerical methods. Copyright © 2011 John Wiley & Sons, Ltd.     

Adaptive Euler simulations of airfoil–vortex interaction

A grid redistribution method is used together with an improved spatially third‐order accurate Euler solver to improve the accuracy of direct Euler simulations of airfoil–vortex interaction. The presented numerical results of two airfoil–vortex interaction cases indicate that with combination of the two methods, the numerical diffusion of vorticity inherent in the direct Euler simulations is drastically reduced without increasing the number of grid points. With some extra works due to grid redistribution, the predicted vortex structure is well preserved after a long convection and much sharper acoustic wave front resulting from airfoil–vortex interaction is captured. Copyright © 2006 John Wiley & Sons, Ltd.     

Numerical simulation of a dense solid particle flow inside a cyclone separator using the hybrid Euler–Lagrange approach

This paper presents a numerical simulation of the flow inside a cyclone separator at high particle loads. The gas and gas–particle flows were analyzed using a commercial computational fluid dynamics code. The turbulence effects inside the separator were modeled using the Reynolds stress model. The two phase gas–solid particles flow was modeled using a hybrid Euler–Lagrange approach, which accounts for the four-way coupling between phases. The simulations were performed for three inlet velocities of the gaseous phase and several cyclone mass particle loadings. Moreover, the influences of several submodel parameters on the calculated results were investigated. The obtained results were compared against experimental data collected at the in-house experimental rig. The cyclone pressure drop evaluated numerically underpredicts the measured values. The possible reason of this discrepancies was disused.     

Numerical simulation of vortical ideal fluid flow through curved channel

A numerical algorithm to study the boundary‐value problem in which the governing equations are the steady Euler equations and the vorticity is given on the inflow parts of the domain boundary is developed. The Euler equations are implemented in terms of the stream function and vorticity. An irregular physical domain is transformed into a rectangle in the computational domain and the Euler equations are rewritten with respect to a curvilinear co‐ordinate system. The convergence of the finite‐difference equations to the exact solution is shown experimentally for the test problems by comparing the computational results with the exact solutions on the sequence of grids. To find the pressure from the known vorticity and stream function, the Euler equations are utilized in the Gromeka–Lamb form. The numerical algorithm is illustrated with several examples of steady flow through a two‐dimensional channel with curved walls. The analysis of calculations shows strong dependence of the pressure field on the vorticity given at the inflow parts of the boundary. Plots of the flow structure and isobars, for different geometries of channel and for different values of vorticity on entrance, are also presented. Copyright © 2003 John Wiley & Sons, Ltd.     

A hybrid approach for nonlinear computational aeroacoustics predictions

In many aeroacoustics applications involving nonlinear waves and obstructions in the far-field, approaches based on the classical acoustic analogy theory or the linearised Euler equations are unable to fully characterise the acoustic field. Therefore, computational aeroacoustics hybrid methods that incorporate nonlinear wave propagation have to be constructed. In this study, a hybrid approach coupling Navier–Stokes equations in the acoustic source region with nonlinear Euler equations in the acoustic propagation region is introduced and tested. The full Navier–Stokes equations are solved in the source region to identify the acoustic sources. The flow variables of interest are then transferred from the source region to the acoustic propagation region, where the full nonlinear Euler equations with source terms are solved. The transition between the two regions is made through a buffer zone where the flow variables are penalised via a source term added to the Euler equations. Tests were conducted on simple acoustic and vorticity disturbances, two-dimensional jets (Mach 0.9 and 2), and a three-dimensional jet (Mach 1.5), impinging on a wall. The method is proven to be effective and accurate in predicting sound pressure levels associated with the propagation of linear and nonlinear waves in the near- and far-field regions.