A new method for accelerating the rate of convergence of the simple-like algorithm

A pressure correction formula is proposed for the SIMPLE-like algorithm in order to improve the rate of the convergence when solving laminar Navier–Stokes equations when there is rapidly varying pressure. Based on global mass conservation, a line average pressure correction is derived by integration of the momentum equation for approximate one-dimensional flow. The use of this formula with the SIMPLE-like algorithm can rapidly build up the pressure distribution in the region where the pressure undergoes a very large change, which normally causes the rate of convergence of the SIMPLE or the SIMPLEC schemes to be slow. In order to illustrate the technique, the performances of SIMPLE and of SIMPLEC with the average pressur correction are investigated for axisymmetric flow past and through a sampler. A comparison of these two techniques shows that the average pressure correction proposed in this paper significantly accelerates the rate of convergence.     

A NOTE ON THE APPLICATION OF THE AVERAGE CORRECTION TECHNIQUE

This note develops an average correction technique for accelerating the rate of convergence of the SIMPLE-like algorithm by implementing the average pressure correction method as proposed by Wen and Ingham (Int. j. numer. methods fluids, 17 , 385–400 (1993); 19 , 889–903 (1994)) with an average velocity correction. The technique is illustrated by considering the classical problem of fluid flow over a backward-facing step using (i) no average correction, (ii) an average velocity correction, (iii) an average pressure correction and (iv) both average velocity and pressure corrections. When both average velocity and pressure corrections are employed, it is found that the number of iterations required for convergence is almost independent of the initial guessed values of fluid velocity and pressure and the fastest rate of convergence may be achieved.     

On the improvement of the SIMPLE-like method for flows with complex geometry

For strongly nonorthogonal grids, the convergence difficulty is often encountered in the SIMPLE method unless very low relaxation factors are adopted. This is because that the cross derivatives of pressure corrections are usually neglected in deriving the pressure-correction equation. In this study, an approximate treatment is proposed to deal with the cross derivatives in the pressure-correction equation. Computations are performed for various laminar flows in a nonorthogonal cavity and the turbulent flows through the compressor cascades. It is found that the convergence difficulty encountered in the SIMPLE method for strongly nonorthogonal grids can be overcome when the SIMPLEC method is employed. In addition, when the present approximate treatment for the cross derivatives is adopted in the SIMPLEC method, the convergence rate can be speeded up in comparison with the original SIMPLEC method. In some cases, the CPU time can be decreased by more than 60%. The computational results are also compared with the numerical results of others and the available experimental data. Received on 30 November 1998     

Performance analysis of IDEAL algorithm for three‐dimensional incompressible fluid flow and heat transfer problems

Recently, an efficient segregated algorithm for incompressible fluid flow and heat transfer problems, called inner doubly iterative efficient algorithm for linked equations (IDEAL), has been proposed by the present authors. In the algorithm there exist inner doubly iterative processes for pressure equation at each iteration level, which almost completely overcome two approximations in SIMPLE algorithm. Thus, the coupling between velocity and pressure is fully guaranteed, greatly enhancing the convergence rate and stability of solution process. However, validations have only been conducted for two‐dimensional cases. In the present paper the performance of the IDEAL algorithm for three‐dimensional incompressible fluid flow and heat transfer problems is analyzed and a systemic comparison is made between the algorithm and three other most widely used algorithms (SIMPLER, SIMPLEC and PISO). By the comparison of five application examples, it is found that the IDEAL algorithm is the most robust and the most efficient one among the four algorithms compared. For the five three‐dimensional cases studied, when each algorithm works at its own optimal under‐relaxation factor, the IDEAL algorithm can reduce the computation time by 12.9–52.7% over SIMPLER algorithm, by 45.3–73.4% over SIMPLEC algorithm and by 10.7–53.1% over PISO algorithm. Copyright © 2009 John Wiley & Sons, Ltd.     

A decoupling numerical method for fluid flow

A first-order non-conforming numerical methodology, Separation method, for fluid flow problems with a 3-point exponential interpolation scheme has been developed. The flow problem is decoupled into multiple one-dimensional subproblems and assembled to form the solutions. A fully staggered grid and a conservational domain centred at the node of interest make the decoupling scheme first-order-accurate. The discretization of each one-dimensional subproblem is based on a 3-point interpolation function and a conservational domain centred at the node of interest. The proposed scheme gives a guaranteed first-order accuracy. It is shown that the traditional upwind (or exponentially weighted upstream) scheme is less than first-order-accurate. The pressure is decoupled from the velocity field using the pressure correction method of SIMPLE. Thomas algorithm (tri-diagonal solver) is used to solve the algebraic equations iteratively. The numerical advantage of the proposed scheme is tested for laminar fluid flows in a torus and in a square-driven cavity. The convergence rates are compared with the traditional schemes for the square-driven cavity problem. Good behaviour of the proposed scheme is ascertained.     

Multigrid pressure correction techniques for the computation of quasi-incompressible internal flows

A study is reported on the possibility of improving the speed of convergence of existing numerical programmes for the simulation of flow in combustion chambers by applying the multigrid method to the pressure correction phase only. A version of the multigrid algorithm is introduced for this purpose which achieves a 1:10 residual reduction in a single V(1, 1) cycle. The overall decrease in computation time with respect to an industry-standard SIMPLE algorithm with single-grid pressure correction ranges from four to five times for SIMPLE itself and several other well-known algorithms to six times for a newly developed pressure correction strategy we call difference operator triangularization (DOT).     

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.     

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.     

连续分层海洋中内波传播的一种数值模式

基于Euler方程,使用有限体积法建立了一种密度为连续分层情况下、适应水深变化的水域中内波传播的数值模式.为了使计算格式能够达到二阶精度,对流项的处理使用了TVD (total variation diminishing)格式.将SIMPLE算法引入连续分层海洋中内波的数值计算,为了简化计算并方便地适应多种TVD格式,在计算预估速度场时采用了显式格式,而没有采用传统的隐式格式;鉴于在原始的SIMPLE算法中没有涉及到由于密度扰动而引起的静水压力场的改变问题,给出了该问题的计算方法.因此改进了SIMPLE算法.出流边界的处理采用阻尼消波和Sommerfeld辐射条件相结合的方式,以使内波得到有效的衰减和释放.将等水深水域的数值解和理论解进行了比较,两者吻合较好;并对存在潜堤时数值计算的不同时刻密度变化的空间分布进行了详细的定性分析.计算结果表明,所建立的数值模式能有效地模拟内波的传播和变形.      

A preconditioned alternating inner-outer iterative solution method for the mixed finite element formulation of the Navier-Stokes equations

In the present work a new iterative method for solving the Navier-Stokes equations is designed. In a previous paper a coupled node fill-in preconditioner for iterative solution of the Navier-Stokes equations proved to increase the convergence rate considerably compared with traditional preconditioners. The further development of the present iterative method is based on the same storage scheme for the equation matrix as for the coupled node fill-in preconditioner. This storage scheme separates the velocity, the pressure and the coupling of pressure and velocity coefficients in the equation matrix. The separation storage scheme allows for an ILU factorization of both the velocity and pressure unknowns. With the inner-outer solution scheme the velocity unknowns are eliminated before the resulting equation system for the pressures is solved iteratively. After the pressure unknown has been found, the pressures are substituted into the original equation system and the velocities are also found iteratively. The behaviour of the inner-outer iterative solution algorithm is investigated in order to find optimal convergence criteria for the inner iterations and compared with the solution algorithm for the original equation system. The results show that the coupled node fill-in preconditioner of the original equation system is more efficient than the coupled node fill-in preconditioner of the reduced equation system. However, the solution technique of the reduced equation system revals properties which may be advantageous in future solution algorithms.     

An improved numerical algorithm for solution of convective heat transfer problems on nonstaggered grid system

In this paper, a SIMPLE-like algorithm on co-located grid system has been developed. The ability to suppress the spurious pressure field is achieved via introducing the pressure difference between adjacent two grid points into the convection-diffusion finite difference scheme, and the interfacial velocity is obtained by simple linear interpolation. The differencing scheme, discretization of governing equations and solution procedure of the algorithm are described in detail. In order to check the validity of the algorithm, several test cases which have analytical or benchmark solutions are presented. Good agreements are obtained between the numerical and the corresponding analytical or benchmark solutions. The ability of the improved algorithm to suppress the spurious pressure field is demonstrated via a 3-D example. Received on 11 March 1997     

An efficient parallel algebraic multigrid method for 3D injection moulding simulation based on finite volume method

Elapsed time is always one of the most important performance measures for polymer injection moulding simulation. Solving pressure correction equations is the most time-consuming part in the mould filling simulation using finite volume method with SIMPLE-like algorithms. Algebraic multigrid (AMG) is one of the most promising methods for this type of elliptic equations. It, thus, has better performance by contrast with some common one-level iterative methods, especially for large problems. And it is also suitable for parallel computing. However, AMG is not easy to be applied due to its complex theory and poor generality for the large range of computational fluid dynamics applications. This paper gives a robust and efficient parallel AMG solver, A1-pAMG, for 3D mould filling simulation of injection moulding. Numerical experiments demonstrate that, A1-pAMG has better parallel performance than the classical AMG, and also has algorithmic scalability in the context of 3D unstructured problems.     

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.     

Periodic flow and heat transfer using unstructured meshes

A numerical scheme has been developed for computing fluid flow and heat transfer in periodically repeating geometries. Unstructured solution-adaptive meshes are used in a cell-centred finite volume formulation. The SIMPLE algorithm is used for pressure‒velocity coupling. For periodic flows the static pressure is decomposed into a periodic component and one that varies linearly in the streamwise direction. The latter is computed from the imposition of overall mass balance at the periodic boundary. A subiteration between the periodic pressure correction equation and the correction to the linear component is used. For heat transfer a formulation using the physical rather than the scaled temperature is employed. The scheme is applied to both laminar and turbulent computations of periodic flow and heat transfer in a variety of heat exchanger geometries; comparison with published computations and experimental data is found to be satisfactory. © 1997 John Wiley & Sons, Ltd.     

Comparison of pressure correction smoothers for multigrid solution of incompressible flow

We compare the performance of different pressure correction algorithms used as basic solvers in a multigrid method for the solution of the incompressible Navier–Stokes equations on non-staggered grids. Numerical tests were performed on several cases of lid-driven cavity flow using four different pressure correction schemes, including the traditional SIMPLE and SIMPLEC methods as well as novel variants, and varying combinations of underrelaxation parameters. The results show that three of the four algorithms tested are robust smoothers for the multigrid solver and that one of the new methods converges fastest in most of the tests. © 1997 John Wiley & Sons, Ltd.     

Calculation of three‐dimensional turbulent flow with a finite volume multigrid method

A numerical method for the efficient calculation of three‐dimensional incompressible turbulent flow in curvilinear co‐ordinates is presented. The mathematical model consists of the Reynolds averaged Navier–Stokes equations and the k–ε turbulence model. The numerical method is based on the SIMPLE pressure‐correction algorithm with finite volume discretization in curvilinear co‐ordinates. To accelerate the convergence of the solution method a full approximation scheme‐full multigrid (FAS‐FMG) method is utilized. The solution of the k–ε transport equations is embedded in the multigrid iteration. The improved convergence characteristic of the multigrid method is demonstrated by means of several calculations of three‐dimensional flow cases. Copyright © 1999 John Wiley & Sons, Ltd.     

Multigrid solution of the incompressible Navier–Stokes equations for three‐dimensional recirculating flow: coupled and decoupled smoothers compared

A comparison of multigrid methods for solving the incompressible Navier–Stokes equations in three dimensions is presented. The continuous equations are discretised on staggered grids using a second‐order monotonic scheme for the convective terms and implemented in defect correction form. The convergence characteristics of a decoupled method (SIMPLE) are compared with those of the cellwise coupled method (SCGS). The convergence rates obtained for computations of the three‐dimensional lid‐driven cavity problem are found to be very similar to those obtained for computations of the corresponding two‐dimensional problem with comparable grid density. Although the convergence rate of SCGS is thus superior to that of SIMPLE, the decoupled method is found to be more efficient computationally and requires less computing time for a given level of convergence. The linewise implementation of the coupled method (CLGS) is also investigated and shown to be more efficient than SCGS, although the convergence rate and computing time required per cycle are both found to depend on the direction of sweep. The optimal implementation of CLGS is found to be only marginally more effective than SIMPLE, but a change to the structure of the data storage would increase the advantage. Copyright © 1999 John Wiley & Sons, Ltd.     

Comparison of using Cartesian and covariant velocity components on non-orthogonal collocated grids

In this paper, the Cartesian velocity components and the covariant velocity components are adopted respectively as the main variables in solving the momentum equations in the SIMPLE-like method to calculate a lid-driven cavity flow on non-orthogonal collocated grids. In total, more than 400 computer runs are carried out for a two-dimensional problem. The accuracy and convergence performance of using Cartesian and covariant velocity components are compared in detail. Comparisons show that both the Cartesian and covariant velocity methods have the same numerical accuracy. The convergence rate of the covariant velocity method can be faster than that of the Cartesian velocity method if the relaxation factor for pressure is small enough. However, the convergence range of the relaxation factor for pressure in the covariant velocity method is quite narrow. When the cross-derivatives in the pressure-correction equation are retained approximately, its convergence performance can be greatly improved. Copyright © 1999 John Wiley & Sons, Ltd.     

Finite volume multigrid method of the planar contraction flow of a viscoelastic fluid

This paper reports on a numerical algorithm for the steady flow of viscoelastic fluid. The conservative and constitutive equations are solved using the finite volume method (FVM) with a hybrid scheme for the velocities and first‐order upwind approximation for the viscoelastic stress. A non‐uniform staggered grid system is used. The iterative SIMPLE algorithm is employed to relax the coupled momentum and continuity equations. The non‐linear algebraic equations over the flow domain are solved iteratively by the symmetrical coupled Gauss–Seidel (SCGS) method. In both, the full approximation storage (FAS) multigrid algorithm is used. An Oldroyd‐B fluid model was selected for the calculation. Results are reported for planar 4:1 abrupt contraction at various Weissenberg numbers. The solutions are found to be stable and smooth. The solutions show that at high Weissenberg number the domain must be long enough. The convergence of the method has been verified with grid refinement. All the calculations have been performed on a PC equipped with a Pentium III processor at 550 MHz. Copyright © 2001 John Wiley & Sons, Ltd.     

An extension of the SIMPLE based discontinuous Galerkin solver to unsteady incompressible flows

In this paper, we present a SIMPLE based algorithm in the context of the discontinuous Galerkin method for unsteady incompressible flows. Time discretization is done fully implicit using backward differentiation formulae (BDF) of varying order from 1 to 4. We show that the original equation for the pressure correction can be modified by using an equivalent operator stemming from the symmetric interior penalty (SIP) method leading to a reduced stencil size. To assess the accuracy as well as the stability and the performance of the scheme, three different test cases are carried out: the Taylor vortex flow, the Orr‐Sommerfeld stability problem for plane Poiseuille flow and the flow past a square cylinder. (1) Simulating the Taylor vortex flow, we verify the temporal accuracy for the different BDF schemes. Using the mixed‐order formulation, a spatial convergence study yields convergence rates of k + 1 and k in the L²‐norm for velocity and pressure, respectively. For the equal‐order formulation, we obtain approximately the same convergence rates, while the absolute error is smaller. (2) The stability of our method is examined by simulating the Orr–Sommerfeld stability problem. Using the mixed‐order formulation and adjusting the penalty parameter of the symmetric interior penalty method for the discretization of the viscous part, we can demonstrate the long‐term stability of the algorithm. Using pressure stabilization the equal‐order formulation is stable without changing the penalty parameter. (3) Finally, the results for the flow past a square cylinder show excellent agreement with numerical reference solutions as well as experiments. Copyright © 2015 John Wiley & Sons, Ltd.