A finite element method for high Reynolds number viscous fluid flow using two step explicit scheme

This paper presents the finite element method for the analysis of unsteady viscous flow of fluid at high Reynolds numbers. The method is based on the explicit numerical integration scheme in time and uses three node triangular finite elements. For the convenience of the formulation, slight compressibility is considered. For the explicit scheme, the selective lumping two step scheme has been successfully employed. Vortex shedding behind a cylinder has been computed and compared with the conventional experimental results. The results agree favourably when both schemes are compared.     

DNS and Modelling of Passive Scalar Transport in Turbulent Channel Flow with a Focus on Scalar Dissipation Rate Modelling

A direct numerical simulation of turbulent channel flow with an imposed mean scalar gradient is analyzed with a focus on passive scalar flux modelling and in particular the treatment of the passive scalar dissipation equation. The Prandtl number is 0.71 and the Reynolds number based on the wall friction velocity and the channel half width is 265. Budgets are presented for the passive scalar variance and its dissipation rate, as well as for the individual scalar flux components. These form a basis for a discussion of modelling issues related to explicit algebraic scalar flux modelling. This revised version was published online in July 2006 with corrections to the Cover Date.     

A comparison of time integration methods in an unsteady low‐Reynolds‐number flow

This paper describes three different time integration methods for unsteady incompressible Navier–Stokes equations. Explicit Euler and fractional‐step Adams–Bashford methods are compared with an implicit three‐level method based on a steady‐state SIMPLE method. The implicit solver employs a dual time stepping and an iteration within the time step. The spatial discretization is based on a co‐located finite‐volume technique. The influence of the convergence limits and the time‐step size on the accuracy of the predictions are studied. The efficiency of the different solvers is compared in a vortex‐shedding flow over a cylinder in the Reynolds number range of 100–1600. A high‐Reynolds‐number flow over a biconvex airfoil profile is also computed. The computations are performed in two dimensions. At the low‐Reynolds‐number range the explicit methods appear to be faster by a factor from 5 to 10. In the high‐Reynolds‐number case, the explicit Adams–Bashford method and the implicit method appear to be approximately equally fast while yielding similar results. Copyright © 2002 John Wiley & Sons, Ltd.     

NUMERICAL STUDY OF HIGH-REYNOLDS-NUMBER FLOW PAST A BLUFF OBJECT

A semi-explicit finite difference scheme is proposed to study unsteady two-dimensional, incompressible flow past a bluff object at high Reynolds number. The bluff object comes from a class of elliptical cylinders in which the aspect ratio and the angle of attack are two controlled parameters. Associated with the streamfunction–vorticity formulation, the interior vorticity, streamfunction and wall vorticity are updated in turn for each time step. The streamfunction and wall vorticity are solved by means of a multigrid method and a projection method respectively. In regard to the vorticity transport equation, implicitness is merely associated with the diffusion operator, which can be made semi-explicit via approximate factorization. Low-diffusive upwinding is devised to handle the convection part. Numerical results are reported for Reynolds numbers up to 40,000. Comparisons with other numerical or physical experiments are included.     

Integral transform solution of developing laminar duct flow in Navier-Stokes formulation

The generalized integral transform technique is employed in the hybrid numerical-analytical solution of the Navier-Stokes equations in streamfunction-only formulation, which govern the incompressible laminar flow of a Newtonian fluid within a parallel plate channel. Owing to the analytic nature of this approach, the outflow boundary condition for an infinite duct is handled exactly, and the error involved in considering finite duct lengths is investigated. The present error-controlled solutions are used to inspect the relative accuracy of previously reported purely numerical schemes and to compare Navier-Stokes and boundary layer formulations for various combinations of inlet conditions and Reynolds number.     

ADAPTIVE LINEARIZATION AND GRID ITERATIONS WITH THE TRI-TREE MULTIGRID REFINEMENT–RECOARSEMENT ALGORITHM FOR THE NAVIER–STOKES EQUATIONS

The tri-tree algorithm for refinements and recoarsements of finite element grids is explored. The refinement–recoarsement algorithm not only provides an accurate solution in certain parts of the grid but also has a major influence on the finite element equation system itself. The refinements of the grid lead to a more symmetric and linear equation matrix. The recoarsements will ensure that the grid is not finer than is necessary for preventing divergence in an iterative solution procedure. The refinement–recoarsement algorithm is a dynamic procedure and the grid is adapted to the instant solution. In the tri-tree multigrid algorithm the solution from a coarser grid is scaled relatively to the increase in velocity boundary condition for the finer grid. In order to have a good start vector for the solution of the finer grid, the global Reynolds number or velocity boundary condition should not be subject to large changes. For each grid and velocity solution the element Reynolds number is computed and used as the grid adaption indicator during the refinement–recoarsement procedure. The iterative tri-tree multigrid method includes iterations with respect to the grid. At each Reynolds number the same boundary condition s are applied and the grid is adapted to the solution iteratively until the number of unknowns and elements in the grid becomes constant. In the present paper the following properties of the tri-tree algorithm are explored: the influence of the increase in boundary velocities and the size of the grid adaption indicator on the amount of work for solving the equations, the number of linear iterations and the solution error estimate between grid levels. The present work indicates that in addition to the linear and non-linear iterations, attention should also be given to grid adaption iterations. © 1997 by John Wiley & Sons, Ltd.     

A volume-agglomeration multirate time advancing for high Reynolds number flow simulation

A frequent configuration in computational fluid mechanics combines an explicit time advancing scheme for accuracy purposes and a computational grid with a very small portion of much smaller elements than in the remaining mesh. Two examples of such situations are the travel of a discontinuity followed by a moving mesh, and the large eddy simulation of high Reynolds number flows around bluff bodies where together very thin boundary layers and vortices of much more important size need to be captured. For such configurations, multistage explicit time advancing schemes with global time stepping are very accurate but very CPU consuming. In order to reduce this problem, the multirate (MR) time stepping approach represents an interesting improvement. The objective of such schemes, which allow to use different time steps in the computational domain, is to avoid penalizing the computational cost of the time advancement of unsteady solutions that would become large due to the use of small global time steps imposed by the smallest elements such as those constituting the boundary layers. In the present work, a new MR scheme based on control volume agglomeration is proposed for the solution of the compressible Navier-Stokes equations equipped with turbulence models. The method relies on a prediction step where large time steps are performed with an evaluation of the fluxes on macrocells for the smaller elements for stability purpose and a correction step in which small time steps are employed. The accuracy and efficiency of the proposed method are evaluated on several benchmarks flows: the problem of a moving contact discontinuity (inviscid flow), the computation with a hybrid turbulence model of flows around bluff bodies like a flow around a space probe model at Reynolds number 10⁶, a circular cylinder at Reynolds number 8.4 × 10⁶, and two tandem cylinders at Reynolds number 1.66 × 10⁵ and 1.4 × 10⁵.     

高精度迎风紧致差分格式与热流计算研究的注记

利用高精度差分格式求解了可压缩 N-S方程球头热流问题。分析了不同差分格式在对球头粘性绕流热流计算中存在的问题 ,并分析了相应的网格雷诺数。在利用高精度迎风紧致 [1 ] 格式求解粘性绕流热流问题时 ,采用 Steger-Warming[2 ]的通量分裂技术将守恒型方程中的流通向量分裂成两部分 ,在此基础上据风向构造逼近于无粘项的高精度迎风格式。对方程中的粘性部分采用中心差分格式。数值结果表明 :高精度差分格式能在较大的网格雷诺数下较好地计算球头驻点热流     

Applications of URANS on predicting unsteady turbulent separated flows

Accurate prediction of unsteady separated turbulent flows remains one of the toughest tasks and a practi cal challenge for turbulence modeling. In this paper, a 2D flow past a circular cylinder at Reynolds number 3,900 is numerically investigated by using the technique of unsteady RANS （URANS）. Some typical linear and nonlinear eddy viscosity turbulence models （LEVM and NLEVM） and a quadratic explicit algebraic stress model （EASM） are evaluated. Numerical results have shown that a high-performance cubic NLEVM, such as CLS, are superior to the others in simulating turbulent separated flows with unsteady vortex shedding.     

A simple method to generate high-order accurate convection operators for explicit schemes based on linear finite elements

A simple method is proposed to generate high-order accurate convection operators for lumped-explicit schemes based on linear or multilinear finite elements. The basic idea is to reduce the truncation error on the first-order spatial derivatives by exploiting the consistent mass matrix of the finite element method in a purely explicit multistep procedure. The effectiveness of the method is demonstrated on pure convection problems in one and two dimensions.     

饱和黏弹性多孔介质中的平面波及能量耗散

研究了流体饱和不可压黏弹性多孔介质中的非均匀平面波及其能量流和能量耗散规律. 在流 相和固相物质微观不可压、固相骨架宏观服从积分型本构关系和小变形的假定下，利用 Helmholtz分解，得到了饱和黏弹性多孔介质中非均匀平面波的一般解以及纵波、横波相速 度和衰减率等的解析表达式，分析了平面波传播矢量和衰减矢量之间的关系. 数值结果表明 孔隙流体与固相骨架间的相互作用以及固相骨架的黏性对波的相速度、衰减率等有着显著的 影响. 同时，得到了饱和黏弹性多孔介质的能量方程，给出了能量流矢量和能量耗散率. 对 非均匀平面纵波和横波，推导了平均能量流矢量和平均能量耗散率的解析表达式.     

A high‐order accurate discontinuous Galerkin finite element method for laminar low Mach number flows

In this paper we present a discontinuous Galerkin (DG) method designed to improve the accuracy and efficiency of laminar flow simulations at low Mach numbers using an implicit scheme. The algorithm is based on the flux preconditioning approach, which modifies only the dissipative terms of the numerical flux. This formulation is quite simple to implement in existing implicit DG codes, it overcomes the time‐stepping restrictions of explicit multistage algorithms, is consistent in time and thus applicable to unsteady flows. The performance of the method is demonstrated by solving the flow around a NACA0012 airfoil and on a flat plate, at different low Mach numbers using various degrees of polynomial approximations. Computations with and without flux preconditioning are performed on different grid topologies to analyze the influence of the spatial discretization on the accuracy of the DG solutions at low Mach numbers. The time accurate solution of unsteady flow is also demonstrated by solving the vortex shedding behind a circular cylinder at the Reynolds number of 100. Copyright © 2012 John Wiley & Sons, Ltd.     

Heat transfer in a tapered passage

Heat transfer to laminar flow in tapered passages is studied for two types of thermal boundary conditions: prescribed heat flux on both walls, and on one wall with the other wall adiabatic. In the analysis, the flow is assumed to be purely radial. Temperature distributions and Nusselt number are obtained for the heat flux q ∞ r^δ. The Nusselt number depends on Reynolds number and taper angle. The fully developed Nusselt number decreases with increase in δ for converging flow and increases for diverging flow. Constant heat flux boundary conditions, δ = 0, for converging flow yield a reduction in Nusselt number when compared with the case of parallel channel flow.     

Implicit preconditioned high‐order compact scheme for the simulation of the three‐dimensional incompressible Navier–Stokes equations with pseudo‐compressibility method

This paper combines the pseudo‐compressibility procedure, the preconditioning technique for accelerating the time marching for stiff hyperbolic equations, and high‐order accurate central compact scheme to establish the code for efficiently and accurately solving incompressible flows numerically based on the finite difference discretization. The spatial scheme consists of the sixth‐order compact scheme and 10th‐order numerical filter operator for guaranteeing computational stability. The preconditioned pseudo‐compressible Navier–Stokes equations are marched temporally using the implicit lower–upper symmetric Gauss–Seidel time integration method, and the time accuracy is improved by the dual‐time step method for the unsteady problems. The efficiency and reliability of the present procedure are demonstrated by applications to Taylor decaying vortices phenomena, double periodic shear layer rolling‐up problem, laminar flow over a flat plate, low Reynolds number unsteady flow around a circular cylinder at Re = 200, high Reynolds number turbulence flow past the S809 airfoil, and the three‐dimensional flows through two 90°curved ducts of square and circular cross sections, respectively. It is found that the numerical results of the present algorithm are in good agreement with theoretical solutions or experimental data. Copyright © 2011 John Wiley & Sons, Ltd.     

Numerical experiments with several variant WENO schemes for the Euler equations

Numerical experiments with several variants of the original weighted essentially non‐oscillatory (WENO) schemes (J. Comput. Phys. 1996; 126 :202–228) including anti‐diffusive flux corrections, the mapped WENO scheme, and modified smoothness indicator are tested for the Euler equations. The TVD Runge–Kutta explicit time‐integrating scheme is adopted for unsteady flow computations and lower–upper symmetric‐Gauss–Seidel (LU‐SGS) implicit method is employed for the computation of steady‐state solutions. A numerical flux of the variant WENO scheme in flux limiter form is presented, which consists of first‐order and high‐order fluxes and allows for a more flexible choice of low‐order schemes. Computations of unsteady oblique shock wave diffraction over a wedge and steady transonic flows over NACA 0012 and RAE 2822 airfoils are presented to test and compare the methods. Various aspects of the variant WENO methods including contact discontinuity sharpening and steady‐state convergence rate are examined. By using the WENO scheme with anti‐diffusive flux corrections, the present solutions indicate that good convergence rate can be achieved and high‐order accuracy is maintained and contact discontinuities are sharpened markedly as compared with the original WENO schemes on the same meshes. Copyright © 2008 John Wiley & Sons, Ltd.     

Parallel‐in‐time simulation of the unsteady Navier–Stokes equations for incompressible flow

In this paper, we present an application of a parallel‐in‐time algorithm for the solution of the unsteady Navier–Stokes model equations that are of parabolic–elliptic type. This method is based on the alternated use of a coarse global sequential solver and a fine local parallel one. A standard finite volume/finite differences first‐order approach is used for discretization of the unsteady two‐dimensional Navier–Stokes equations. The Taylor vortex decay problem and the confined flow around a square cylinder were selected as unsteady flow examples to illustrate and analyse the properties of the parallel‐in‐time method through numerical experiments. The influence of several parameters on the computing time required to perform a parallel‐in‐time calculation on a PC cluster was verified. Among them we have analysed the influence of the number of processors, the number of iterations for convergence, the resolution of the spatial domain and the influence of the time‐step sizes ratio between the coarse and fine grids. Significant computer time saving was achieved when compared with the single processor computing time, particularly when the spatial dimension of the problem is low and the temporal scale is large. Copyright © 2004 John Wiley & Sons, Ltd.     

A fractional two‐step implicit algorithm using curvilinear collected grids

An efficient fractional two‐step implicit algorithm is reported to simulate incompressible fluid flows in a boundary‐fitted curvilinear collocated grid system. Using the finite volume method, the convection terms are discretized by the high‐accuracy Roe's scheme to minimize numerical diffusion. An implicitness coefficient Π is introduced to accelerate the rate of convergence. It is demonstrated that the proposed algorithm links the fractional step method to the pressure correction procedure, and the SIMPLEC method could be considered as a special case of the fractional two‐step implicit algorithm (when Π=1). The proposed algorithm is applicable to unsteady flows and steady flows. Three benchmark two‐dimensional laminar flows are tested to evaluate the performance of the proposed algorithm. Performance is measured by sensitivity analyses of the efficiency, accuracy, grid density, grid skewness and Reynolds number on the solutions. Results show that the model is efficient and robust. Copyright © 2006 John Wiley & Sons, Ltd.     

Accurate 3-D finite element simulation of elastic wave propagation with the combination of explicit and implicit time-integration methods

One of the big issues in finite element solutions of wave propagation problems is the presence of spurious high-frequency oscillations that may lead to divergent results at mesh refinement. The paper deals with the extension of the new two-stage time-integration technique developed in our previous papers to the solution of wave propagation problems with explicit time-integration methods.The explicit central difference method is used for accurate time-integration of the semi-discrete system of elastodynamics at the stage of basic computations and allows spurious high-frequency oscillations. To filter these oscillations, pre- or/and post-processing (the filtering stage) is applied using a few time increments of the implicit time-continuous Galerkin method with large numerical dissipation.A special calibration procedure is used for the selection of the minimum necessary amount of numerical dissipation (in terms of a time increment) at the filtering stage. In contrast to existing approaches that use a time-integration method with the same dissipation (or artificial viscosity) for all time increments, the new technique yields accurate and non-oscillatory results for wave propagation problems without interaction between user and computer code. The solutions of 3-D wave propagation and impact problems show the effectiveness of the new approach.     

Continuation methods applied to the 2D Navier–Stokes equations at high Reynolds numbers

Nonlinearities arise in aerodynamic flows as a function of various parameters, such as angle of attack, Mach number and Reynolds number. These nonlinearities can cause the change from steady to unsteady flow or give rise to static hysteresis. Understanding these nonlinearities is important for safety validation and performance enhancement of modern aircraft. A continuation method has been developed to study nonlinear steady state solutions with respect to changes in parameters for two‐dimensional compressible turbulent flows at high Reynolds numbers. This is the first time that such flows have been analysed with this approach. Continuation methods allow the stable and unstable solutions to be traced as flow parameters are changed. Continuation has been carried out on two‐dimensional aerofoils for several parameters: angle of attack, Mach number, Reynolds number, aerofoil thickness and turbulent inflow as well as levels of dissipation applied to the models. A range of results are presented. Copyright © 2012 John Wiley & Sons, Ltd.     

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.