Study of the effect of the non‐orthogonality for non‐staggered grids—The results

This article presents the effect of the grid skewness on the ranges of the underrelaxation factors for pressure and velocity. The effect is reflected by the relationship between the numbers of iterations required and the ranges of the underrelaxation factors for a converged solution. Four typical cavity flow problems are solved on non‐staggered grids for this purpose. Two momentum interpolation practices namely, practice A and practice B, together with SIMPLE, SIMPLEC and SIMPLER algorithms are employed. The results show that the ranges of the pressure underrelaxation factor values for convergence exist if the SIMPLE algorithm is used, while no restrictions are observed if the SIMPLEC algorithm is used. From the curves obtained using the SIMPLER algorithm, the ranges of those based on practice B are wider than those based on practice A. Copyright © 1999 John Wiley & Sons, Ltd.     

A numerical method for the calculation of incompressible,steady, separated flows around aerofoils

The present work deals with the numerical calculation of the incompressible turbulent flow around aerofoils. An orthogonal curvilinear grid of ‘C’ type is used for the solution of the time averaged equations and Reynolds stresses are modelled according to the κ-ε turbulence model. PISO and SIMPLE algorithms are used to solve the strongly coupled system of the derived finite volume equations and convergence is improved by applying the method of variable local underrelaxation factors. Comparisons between the calculated and measured pressure distributions are presented for NACA 0012 and NACA 4412 wing sections. The formation of separation bubbles according to calculations is also shown.     

Study of the effect of the non-orthogonality for non-staggered grids—the theory

An investigation has been conducted to determine the effect of the grid non-orthogonality on the convergence behavior of two-dimensional lid-driven cavity flows. The relevant theory is presented in this article. In the present work, the contravariant velocity fluxes are used as the dependent variables on non-orthogonal, non-staggered grids. The momentum equations retain a strongly conservative form. Two practices for treating the momentum interpolation method in general curvilinear co-ordinates are presented. In each practice, the momentum interpolation formulations with and without velocity underrelaxation factor are considered. The discretization equations are solved using the SIMPLE, SIMPLEC and SIMPLER algorithms. © 1998 John Wiley & Sons, Ltd.     

AN ALGEBRAIC MULTIGRID SOLVER FOR NAVIER–STOKES PROBLEMS IN DISCRETE SECOND–ORDER APPROXIMATION

An algebraic multigrid scheme is presented for solving the discrete Navier–Stokes equations to second-order accuracy using the defect correction method. Solutions for the driven cavity and asymmetric, sudden expansion test problems have been obtained for both structured and unstructured meshes, the resolution and resolution grading being controlled by global and local mesh refinements. The solver is efficient and robust to the extent that, for problems attempted so far, no underrelaxation of variables has been required to ensure convergence. Provided that the computational mesh can resolve the flow field, convergence characteristics are almost mesh-independent. Rates of convergence actually improve with refinement, asymptotically approaching mesh-independent values. For extremely coarse meshes, where dispersive truncation errors would be expected to prevent convergence (or even induce divergence), solutions can still be obtained by using explicit underrelaxation in the iterative cycle.     

基于非结构网格的高效求解方法研究

非结构网格的求解效率一直是计算流体力学工作者十分关注的问题。本文从一个新的角度分析了N-S(Euler/Navier-Stokes)方程求解效率的高低,表明计算效率不仅涉及时间离散的效率,空间离散和程序算法都与之息息相关。采用不同的计算状态,对目前非结构网格上广泛应用的LU-SGS、对称Gauss-Seidel和GMRES方法进行较详细地比较和分析,考查了空间离散的耗时对方程求解效率的影响。结果表明,LU-SGS方法的计算效率在所给的算例中均是最低的;在不考虑大量内存消耗时,GMRES算法求解Euler方程的效率较高,松耦合求解N-S方程时效率会有所降低;在大规模计算中,多次对称的Gauss-Seidel迭代方法应是较好的选择,特别是N-S方程的求解。     

Efficient discretization of pressure‐correction equations on non‐orthogonal grids

An alternative discretization of pressure‐correction equations within pressure‐correction schemes for the solution of the incompressible Navier–Stokes equations is introduced, which improves the convergence and robustness properties of such schemes for non‐orthogonal grids. As against standard approaches, where the non‐orthogonal terms usually are just neglected, the approach allows for a simplification of the pressure‐correction equation to correspond to 5‐point or 7‐point computational molecules in two or three dimensions, respectively, but still incorporates the effects of non‐orthogonality. As a result a wide range (including rather high values) of underrelaxation factors can be used, resulting in an increased overall performance of the underlying pressure‐correction schemes. Within this context, a second issue of the paper is the investigation of the accuracy to which the pressure‐correction equation should be solved in each pressure‐correction iteration. The scheme is investigated for standard test cases and, in order to show its applicability to practical flow problems, for a more complex configuration of a micro heat exchanger. Copyright © 2003 John Wiley & Sons, Ltd.     

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.     

On coupling the Reynolds‐averaged Navier–Stokes equations with two‐equation turbulence model equations

Two methods for coupling the Reynolds‐averaged Navier–Stokes equations with the q–ω turbulence model equations on structured grid systems have been studied; namely a loosely coupled method and a strongly coupled method. The loosely coupled method first solves the Navier–Stokes equations with the turbulent viscosity fixed. In a subsequent step, the turbulence model equations are solved with all flow quantities fixed. On the other hand, the strongly coupled method solves the Reynolds‐averaged Navier–Stokes equations and the turbulence model equations simultaneously. In this paper, numerical stabilities of both methods in conjunction with the approximated factorization‐alternative direction implicit method are analysed. The effect of the turbulent kinetic energy terms in the governing equations on the convergence characteristics is also studied. The performance of the two methods is compared for several two‐ and three‐dimensional problems. Copyright © 2005 John Wiley & Sons, Ltd.     

Simulation of unsteady turbulent flows around moving aerofoils using the pseudo‐time method

The pseudo‐time formulation of Jameson has facilitated the use of numerical methods for unsteady flows, these methods have proved successful for steady flows. The formulation uses iterations through pseudo‐time to arrive at the next real time approximation. This iteration can be used in a straightforward manner to remove sequencing errors introduced when solving mean flow equations together with another set of differential equations (e.g. two‐equation turbulence models or structural equations). The current paper discusses the accuracy and efficiency advantages of removing the sequencing error and the effect that building extra equations into the pseudo‐time iteration has on its convergence characteristics. Test cases used are for the turbulent flow around pitching and ramping aerofoils. The performance of an implicit method for solving the pseudo‐steady state problem is also assessed. Copyright © 2000 John Wiley & Sons, Ltd.     

Solving viscoelastic free surface flows of a second‐order fluid using a marker‐and‐cell approach

This work is concerned with the numerical simulation of two‐dimensional viscoelastic free surface flows of a second‐order fluid. The governing equations are solved by a finite difference technique based on the marker‐and‐cell philosophy. A staggered grid is employed and marker particles are used to represent the fluid free surface. Full details for the approximation of the free surface stress conditions are given. The resultant code is validated and convergence is demonstrated. Numerical simulations of the extrudate swell and flow through a planar 4:1 contraction for various values of the Deborah number are presented. Copyright © 2006 John Wiley & Sons, Ltd.     

Dynamics of fibres in a turbulent flow field – A particle-level simulation technique

A particle-level simulation technique has been developed for modelling the flow of fibres in a turbulent flow field. A single fibre is conceived here as a chain of segments, thus enabling the model fibre to have all the degrees of freedom (translation, rotation, bending and twisting) needed to realistically reproduce the dynamics of real fibres. Equations of motion are solved for each segment, accounting for the interaction forces with the fluid, the contact forces with other fibres and the forces that maintain integrity of the fibre.The motion of the fluid is resolved as a combination of 3D mean flow velocities obtained from a CFD code and fluctuating turbulent velocities derived from the Langevin equation. A case of homogeneous turbulence is treated in this paper.The results obtained show that fibre flocs in air-fibre flows can be created even when attractive forces are not present. In such a case, contacts between fibres, properties of an individual fibre (such as flexibility and equilibrium shapes) and properties of the flow of the carrying fluid are shown to govern the physics behind formation and breaking up of fibre flocs. Highly irregular fibre shapes and stiff fibres lead to strong flocculation.The modelling framework applied in this work aims at making possible a numerical model applicable for designing processes involving transport of fibres by air at industrial scale.     

Characteristic‐based inlet and outlet boundary conditions for incompressible flows

Determining boundary conditions (BCs) for incompressible flows is such a delicate matter that affects the accuracy of the results. In this research, a new characteristic‐based BC for incompressible Navier‐Stokes equations is introduced. Discretization of equations has been done via finite volume. Additionally, artificial compressibility correction has been employed to deal with equations. Ordinary extrapolation from inner cells of a domain was used as a traditional way to estimate pressure and velocities on solid wall and inlet/outlet boundaries. Here, this method was substituted by the newly proposed BCs based on the characteristics of artificial compressibility equations. To follow this purpose, a computer code has been developed to carry out series of numerical tests for a flow over a backward‐facing step and was applied to a wide range of Reynolds numbers and grid combinations. Calculation of convective and viscous fluxes was done using Jameson's averaging scheme. Employing the characteristic‐based method for determining BCs has shown an improved convergence rate and reduced calculation time comparing with those of traditional ones. Furthermore, with the reduction of domain and computational cells, a similar accuracy was achieved for the results in comparison with the ones obtained from the traditional extrapolation method, and these results were in good agreement with the ones in the literature.     

A segregated implicit solution algorithm for 2D and 3D laminar incompressible flows

A segregated algorithm for the solution of laminar incompressible, two- and three-dimensional flow problems is presented. This algorithm employs the successive solution of the momentum and continuity equations by means of a decoupled implicit solution method. The inversion of the coefficient matrix which is common for all momentum equations is carried out through an approximate factorization in upper and lower triangular matrices. The divergence-free velocity constraint is satisfied by formulating and solving a pressure correction equation. For the latter a combined application of a preconditioning technique and a Krylov subspace method is employed and proved more effecient than the approximate factorization method. The method exhibits a monotonic convergence, it is not costly in CPU time per iteration and provides accurate solutions which are independent of the underrelaxation parameter used in the momentum equations. Results are presented in two- and three-dimensional flow problems.     

NUMERICAL UNCERTAINTIES IN TRANSONIC FLOW CALCULATIONS FOR AEROFOILS WITH BLUNT TRAILING EDGES

Numerical uncertainties are quantified for calculations of transonic flow around a divergent trailing edge (DTE) supercritical aerofoil. The Reynolds-averaged Navier–Stokes equations are solved using a linearized block implicit solution procedure and mixing-length turbulence model. This procedure has reproduced measurements around supercritical aerofoils with blunt trailing edges that have shock, boundary layer and separated regions. The present effort quantifies numerical uncertainty in these calculations using grid convergence indices which are calculated from aerodynamic coefficients, shock location, dimensions of the recirculating region in the wake of the blunt trailing edge and distributions of surface pressure coefficients. The grid convergence index is almost uniform around the aerofoil, except in the shock region and at the point where turbulence transition was fixed. The grid convergence index indicates good convergence for lift but only fair convergence for moment and drag and also confirms that drag calculations are more sensitive to numerical error. © 1997 by John Wiley and 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.     

A NON-LINEAR ADAPTIVE FULL TRI-TREE MULTIGRID METHOD FOR THE MIXED FINITE ELEMENT FORMULATION OF THE NAVIER–STOKES EQUATIONS

The full adaptive multigrid method is based on the tri-tree grid generator. The solution of the Navier–Stokes equations is first found for a low Reynolds number. The velocity boundary conditions are then increased and the grid is adapted to the scaled solution. The scaled solution is then used as a start vector for the multigrid iterations. During the multigrid iterations the grid is first recoarsed a specified number of grid levels. The solution of the Navier–Stokes equations with the multigrid residual as right-hand side is smoothed in a fixed number of Newton iterations. The linear equation system in the Newton algorithm is solved iteratively by CGSTAB preconditioned by ILU factorization with coupled node fill-in. The full adaptive multigrid algorithm is demonstr ated for cavity flow. © 1997 by John Wiley & Sons, Ltd. Int. j. numer. methods fluids 24: 1037ndash;1047, 1997.     

可压缩多介质粘性流体和湍流的大涡模拟

在可压缩多介质粘性流体动力学高精度计算方法MVPPM(multi-viscous-fluid piecewise parabolicmethod)基础上,引入Smagorinsky和Vreman亚格子湍流模型,采用大涡数值模拟方法求解可压缩粘性流体NS(Navier-Stokes)方程,给出适用于可压缩多介质流体界面不稳定性发展演化至湍流阶段的计算方法和二维计算程序MVFT(multi-viscosity-fluid and turbulence)。在2种亚格子湍流模型下计算了LANL(Los Ala-mos National Laboratory)激波管单气柱RM不稳定性实验,分析了气柱的形状、流场速度以及涡的特征,通过与LANL实验和计算结果的比较可知,Vreman模型略优于Smagorinsky模型,MVFT方法和计算程序可用于对界面不稳定性发展演化至湍流阶段的数值模拟。     

A boundary element method for steady viscous fluid flow using penalty function formulation

A new boundary element method is presented for steady incompressible flow at moderate and high Reynolds numbers. The whole domain is discretized into a number of eight-noded cells, for each of which the governing boundary integral equation is formulated exclusively in terms of velocities and tractions. The kernels used in this paper are the fundamental solutions of the linearized Navier–Stokes equations with artificial compressibility. Significant attention is given to the numerical evaluation of the integrals over quadratic boundary elements as well as over quadratic quadrilateral volume cells in order to ensure a high accuracy level at high Reynolds numbers. As an illustration, square driven cavity flows are considered for Reynolds numbers up to 1000. Numerical results demonstrate both the high convergence rate, even when using simple (direct) iterations, and the appropriate level of accuracy of the proposed method. Although the method yields a high level of accuracy in the primary vortex region, the secondary vortices are not properly resolved. © 1997 John Wiley & Sons, Ltd.     

Convergence properties of high-Reynolds-number separated flow calculations

The convergence properties of an iterative solution technique for the Reduced Navier–Stokes equations are examined for two-dimensional steady subsonic flow over bump and trough geometries. Techniques for decreasing the sensitivity to the initial pressure approximation, for fine meshes in particular, are investigated. They are shown to improve the robustness of the relaxation process and to decrease the computational work required to obtain a converged solution. A semi-coarsening multigrid technique that has previously been found to be particularly advantageous for high-Reynolds-number (Re) flows with flow separation and with highly stretched surface-normal grids is applied herein to further accelerate convergence. Solutions are obtained for the laminar flow over a trough that is more severe than has been considered to date. Sufficient axial grid refinement in this case leads to a shock-like reattachment and, for sufficiently large Re, to a local ‘divergence’ of the numerical computations. This ‘laminar flow breakdown’ appears to be related to an instability associated with high-frequency fine-grid modes that are not resolvable with the present modelling. This behaviour may be indicative of dynamic stall or of incipient transition. The breakdown or instability is shown to be controllable by suitable introduction of transition turbulence models or by laminar flow control, i.e. small amounts of wall suction. This lends further support to the hypothesis that the instability is of a physical rather than numerical character and suggests that full three-dimensional analysis is required to properly capture the flow behaviour. Another inference drawn from this investigation is that there is a need for careful grid refinement studies in high-Re flow computations, since coarser grids may yield oscillation-free solutions that cannot be obtained on finer grids.