Avoiding under-relaxations in SIMPLE algorithm

Abstract

The SIMPLE algorithm is devised by interpolating the mass continuity and non-advective momentum equations, provoking apparent simplicity and clarity in the formulation. The SIMPLE variant entitled as the SIMPLE-AC scheme convokes an artificial compressibility (AC) parameter to augment the diagonal dominance of discretized pressure-correction equation. Both methods are characteristically pressure-based, employing a cell-centered finite-volume Δ-formulation on a non-orthogonal collocated grid. A dual-dissipation scheme accompanied by limiting factors, repressing local extrema into the cell-face velocity and pressure, is used to unravel the issue of pressure-velocity decoupling; both SIMPLE and SIMPLE-AC schemes maintain an equivalent scaling (e.g., primary and auxiliary pseudo-time steps remain the same) with the cell-face dissipation and nodal influence coefficients. The phenomenal progress embedded in both contrivances facilitate an avoidance of the pervasive velocity/pressure under-relaxation. However, the SIMPLE-AC algorithm is benefited with using a higher CFL number, enhanced robustness, and convergence compared with the SIMPLE method.     

A COMPARATIVE ASSESSMENT OF THE PERFORMANCE OF MASS CONSERVATION-BASED ALGORITHMS FOR INCOMPRESSIBLE MULTIPHASE FLOWS

This work is concerned with the implementation and testing, within a structured collocated finite-volume framework, of seven incompressible-segregated multiphase flow algorithms that belong to the mass conservation-based algorithms (MCBA) group in which the pressure-correction equation is derived from overall mass conservation. The pressure-correction schemes in these algorithms are based on SIMPLE, SIMPLEC, SIMPLEX, SIMPLEM, SIMPLEST, PISO, and PRIME. The performance and accuracy of the multiphase algorithms are assessed by solving eight one-dimensional two-phase flow problems spanning the spectrum from dilute bubbly to dense gas-solid flows. The main outcome of this study is a clear demonstration of the capability of all MCBA algorithms to deal with multiphase flow situations. Moreover, results displayed in terms of convergence history plots and CPU times indicate that the performance of the MCBA versions of SIMPLE, SIMPLEC, and SIMPLEX are very close. In general, the performance of SIMPLEST approaches that of SIMPLE for diffusion-dominated flows. As expected, the PRIME algorithm is found to be the most expensive, due to its explicit treatment of the phasic momentum equations. The PISO algorithm is generally more expensive than SIMPLE, and its performance depends on the type of flow and solution method used. The behavior of SIMPLEM is consistent, and in terms of CPU effort it stands between PRIME and SIMPLE.     

A Fully Coupled Navier-Stokes Solver for Fluid Flow at All Speeds

This article deals with the formulation and testing of a newly developed, fully coupled, pressure-based algorithm for the solution of fluid flow at all speeds. The new algorithm is an extension into compressible flows of a fully coupled algorithm developed by the authors for laminar incompressible flows. The implicit velocity–pressure–density coupling is resolved by deriving a pressure equation following a procedure similar to a segregated SIMPLE algorithm using the Rhie-Chow interpolation technique. The coefficients of the momentum and continuity equations are assembled into one matrix and solved simultaneously, with their convergence accelerated via an algebraic multigrid method. The performance of the coupled solver is assessed by solving a number of two-dimensional problems in the subsonic, transsonic, supersonic, and hypersonic regimes over several grid systems of increasing sizes. For a desired level of convergence, results for each problem are presented in the form of convergence history plots, tabulated values of the maximum number of required iterations, the total CPU time, and the CPU time per control volume.     

ANALYSIS OF PRESSURE-VELOCITY COUPLING ON NONORTHOGONAL GRIDS

Extension of the SIMPLE pressure-velocity coupling algorithm to nonorthogonal grids results in a very complex pressure-correction equation (e.g., a 9-point computational molecule in a two-dimensional case, a 19-point.computational molecule in a three-dimensional case) The usual practice is therefore to further simplify this equation by neglecting the effect of nonorthogonality on the mass flux corrections, thus reducing the computational molecule to 5 or 7 points

The paper analyzes the performance of the simplified and full pressure-correction equations when the grid nonorthogonality becomes appreciable. It is demonstrated here that the efficiency of the simple coupling algorithm is not affected by the grid nonorthogonality, provided that no additional simplifications are introduced in the pressure-correction equation. However, the algorithm with the simplified equation becomes inefficient when the angle between grid lines approaches 45^° and it usually fails to converge for angles below 30^°. The problem of solving the full 9-point pressure-correction equation is best dealt with by employing the biconjugate gradient solver, which proved to be the most robust one in test calculations carried out in this study.     

Discretized pressure Poisson algorithm for the steady incompressible flow on a nonstaggered grid

In the aspect of numerical methods for incompressible flow problems, there are two different algorithms: semi-implicit method for pressure-linked equations (SIMPLE) series algorithms and the pressure Poisson algorithm. This paper introduced a new discretized pressure Poisson algorithm for the steady incompressible flow based on a nonstaggered grid. Compared with the SIMPLE series algorithms, this paper did not introduce three correction variables. So, there is no need to implement the guess-and-correct procedure for the calculation of pressure and velocity. Compared with the pressure Poisson algorithm, there is no need to calculate unsteady Navier–Stokes equations for steady problems in the new discretized pressure Poisson algorithm. Meanwhile, as the finite volume method and cell-centered grid are used, the governing equation for pressure is obtained from the continuity equation and the boundary conditions for pressure are easily obtained. This new discretized pressure Poisson algorithm was tested at the lid-driven cavity flow problem on a nonstaggered grid and the results are also reliable.     

Implementation of the IDEAL Algorithm on Nonorthogonal Curvilinear Coordinates for the Solution of 3-D Incompressible Fluid Flow and Heat Transfer Problems

Recently an efficient segregated algorithm for incompressible fluid flow and heat transfer problems, called IDEAL (Inner Doubly Iterative Efficient Algorithm for Linked Equations), has been proposed by the present authors. In the algorithm there exist inner doubly iterative processes for the pressure equation at each iteration level, which almost completely overcome the two approximations in the SIMPLE algorithm. Thus the coupling between velocity and pressure is fully guaranteed, greatly enhancing the convergence rate and stability of the solution process. In this article, the IDEAL algorithm is extended to the body-fitted collocated grid systems in 3-D nonorthogonal curvilinear coordinates. The extended IDEAL algorithm adopts two successful methods. One is that the interfacial contravariant velocity is calculated by the modified momentum interpolation method (MMIM); the other is that the interfacial contravariant velocity is improved by solving the pressure equation directly. Finally, three 3-D incompressible fluid flow and heat transfer problems are provided to compare the convergence rate and robustness between IDEAL and three other algorithms (SIMPLEM, SIMPLERM, and SIMPLECM). From the comparison it can be concluded that the IDEAL algorithm is more robust and efficient than the three other algorithms.     

HYBRID RELAXATION —A TECHNIQUE TO ENHANCE THE RATE OF CONVERGENCE OF ITERATIVE ALGORITHMS

Abstract

In solving multidimensional transient fluid flow and heat transfer problems, the strongly coupled conservation laws of mass, momentum, and energy require segregated iterative procedures. Derived from the SIMPLE algorithm, the fully explicit iteration scheme MAPLE (Modified Algorithm for Pressure-Linked Equations) for the pressure-velocity coupling is introduced here. A substantial speedup is gained in the iteration by utilizing hybrid relaxation, a combination of under- and overrelaxation, instead of the usual underrelaxation. Moreover, hybrid relaxation is not restricted to pressure—velocity algorithms only, but can be applied in more general iterations.     

A PRESSURE CORRECTION SCHEME USING COUPLED MOMENTUM EQUATIONS

Abstract

This article presents a method for the numerical solution of the incompressible 2D Navier-Stokes equations, based on the coupled solution of the momentum equations and a fully compatible pressure correction equation. With in each iteration, the linearized momentum equations are simultaneously, though inexactly, solved through a two-step, noniterative scheme. Their solution employs the appropriate factorization of the diagonal coefficient matrices into upper and lower triangular ones. The continuity equation is satisfied by means of the SIMPLE concept in a particular form that arises from the coupled solution of the momentum equations. The algorithm is more efficient in terms of convergence rates when compared to a segregated algorithm, given that identical discretization schemes are used.     

An efficient SIMPLER-revised algorithm for incompressible flow with unstructured grids

This paper proposes an efficient segregated solution procedure for the two-dimensional incompressible fluid flow on unstructured grids. The new algorithm is called SIMPLERR (SIMPLERR-revised algorithm). It includes an inner iterative process that consists of a pressure equation solution and an explicit velocity correction, and then, the pressure field is obtained by another solving process for the pressure equation. The features and advantages of the SIMPLERR algorithm are demonstrated by solving a benchmark flow problem, and the results indicate that the SIMPLERR algorithm can maintain a strong stability as the IDEAL algorithm and can converge faster than the SIMPLER algorithm or even than the IDEAL algorithm. The advantage of the SIMPLERR algorithm is especially evident for high-Re and fine-mesh fluid flow cases, where the SIMPLERR algorithm and the IDEAL algorithm can obtain a convergence result but the SIMPLER algorithm cannot. The SIMPLERR algorithm is about two times faster than the IDEAL algorithm.     

A time marching strategy for solving parabolic and elliptic equations with Neumann boundary conditions

Abstract

In the community of computational fluid dynamics, pressure Poisson equation with Neumann boundary condition is usually encountered when solving the incompressible Navier–Stokes equations in a segregated approach such as SIMPLE, PISO, and projection methods. To deal with Neumann boundary conditions more naturally and to retain high order spatial accuracy as well, a sixth-order accurate combined compact difference scheme developed on staggered grids (NSCCD6) is adopted to solve the parabolic and elliptic equations subject to Neumann boundary conditions. The staggered grid system is usually used when solving the incompressible Navier–Stokes equations. By adopting the combined compact difference concept, there is no need to discretize Neumann boundary conditions with one-sided discretization scheme which is of lower accuracy order. The conventional Crank–Nicolson scheme is applied in this study for temporal discretization. For two-dimensional cases, D’yakonov alternating direction implicit scheme is adopted. A newly proposed time step changing strategy is adopted to improve convergence rate when solving the steady state solutions of the parabolic equation. High accuracy order of the currently proposed NSCCD6 scheme for one- and two-dimensional cases are shown in this article.     

Higher-Order Numerical Scheme for the Fractional Heat Equation with Dirichlet and Neumann Boundary Conditions

In this article, we consider a higher-order numerical scheme for the fractional heat equation with Dirichlet and Neumann boundary conditions. By using a fourth-order compact finite-difference scheme for the spatial variable, we transform the fractional heat equation into a system of ordinary fractional differential equations which can be expressed in integral form. Further, the integral equation is transformed into a difference equation by a modified trapezoidal rule. Numerical results are provided to verify the accuracy and efficiency of the proposed algorithm.     

A Coupled Pressure-Based Finite-Volume Solver for Incompressible Two-Phase Flow

This article reports on the formulation and testing of a coupled pressure-based algorithm for the solution of steady incompressible disperse two-phase-flow problems. The method is formulated within a Eulerian-Eulerian framework in the context of a collocated finite-volume scheme. An equation for pressure is derived from overall mass conservation following the segregated mass conservation–based algorithm (MCBA) approach and using an extended two-phase flow form of the Rhie-Chow interpolation technique. The newly developed pressure-based coupled solver differs from pressure-based segregated solvers in that it accounts implicitly for the pressure–velocity and the interphase drag couplings that are present in disperse multiphase flows to yield a system of coupled equations linking the velocity and pressure fields. The performance and accuracy of the coupled multiphase algorithm are assessed by solving eight one-dimensional two-phase flow problems spanning the spectrum from dilute bubbly to dense gas–solid flows. Each problem is solved over three grid systems with sizes of 10,000, 30,000, and 50,000 control volumes, respectively. Results are compared in terms of iterations and CPU time with similar ones generated using the segregated MCBA-SIMPLE algorithm. The newly developed coupled solver is shown to yield substantial decrease in the required number of iterations and CPU time, with the rate of solution acceleration varying between 1.3 and 4.6.     

Numerical investigation of fluid flow and heat transfer around a solid circular cylinder utilizing nanofluid

In this study, flow-field and heat transfer through a copper–water nanofluid around circular cylinder has been numerically investigated. Governing equations containing continuity, N–S equation and energy equation have been developed in polar coordinate system. The equations have been numerically solved using a finite volume method over a staggered grid system. SIMPLE algorithm has been applied for solving the pressure linked equations. Reynolds and Peclet numbers (based on the cylinder diameter and the velocity of free stream) are within the range of 1 to 40. Furthermore, volume fraction of nanoparticles (φ) varies within the range of 0 to 0.05. Effective thermal conductivity and effective viscosity of nanofluid have been estimated by Hamilton–Crosser and Brinkman models, respectively. The effect of volume fraction of nanoparticles on the fluid flow and heat transfer characteristics are investigated. It is found that the vorticity, pressure coefficient, recirculation length are increased by the addition of nanoparticles into clear fluid. Moreover, the local and mean Nusselt numbers are enhanced due to adding nanoparticles into base fluid.     

Keller-Box shooting method and its application to nanofluid flow over convectively heated sheet with stability and convergence

Abstract

The main aim of the present contribution is to present a coupling of Keller-Box method using Jacobi and Gauss–Seidel iterative methods with shooting approach and also this method is implemented on nanofluid flow problem. Present method of Keller-Box shooting method can be considered as an explicit approach whereas the standard Keller-Box method is an implicit method. Previously constructed nanofluid flow models are extended with exothermic/endothermic chemical reactions and the flow is considered over stretching, rotating, porous disk which is convectively heated from its bottom with the hot liquid. Similarity transformations are utilized to reduce the set of nonlinear partial differential equations into nonlinear ordinary differential equations with an assumption. Presently modified Keller-Box shooting method using Jacobi and Gauss–Seidel iterative methods is applied to investigate MHD nanofluid flow problem subject to Dirichlet, Neumann, and Robin’s boundary conditions. Von Neumann stability criterion is adopted to check the stability of the present method using Gauss–Seidel iterative method. The bounds for maximum errors are found for Jacobi and Gauss–Seidel iterative methods and convergence conditions are given for the case of Gauss–Seidel iterative method. The obtained results from presently developed modified Keller-Box shooting method are in good agreement with those obtained by Matlab built in solver “bvp4c” in case of wall temperature gradient and obtained results are in good agreement with those obtained by Runge–Kutta (4, 5) shooting method in case of skin friction coefficient.     

A Modified Pressure-Correction Scheme for the SIMPLER Method,MSIMPLER

A new method is proposed to accelerate the convergence rate for the SIMPLER algorithm by artificially changing the underrelaxation term to match the dependent variable to be solved. Based on this idea, a new pressure-correction equation is derived, and the modified algorithm is named MSIMPLER. Five numerical experiments show that the MSIMPLER algorithm can appreciably enhance the convergence rate for cases of low and moderate underrelaxation factors with good robustness.     

A fitting algorithm for solving inverse problems of heat conduction

The paper presents an algorithm for solving inverse problems of heat transfer. The method is based on iterative solving of direct and adjoint model equations with the aim to minimize a fitting functional. An optimal choice of the step length along the descent direction is proposed. The algorithm has been used for the treatment of a steady-state problem of heat transfer in a region with holes. The temperature and the heat flux density were known on the outer boundary of the region, whereas these values on the boundaries of the holes are to be determined. The idea of the algorithm consist in solving of Neumann problems where the heat flux on the outer boundary is prescribed, whereas the heat flux on the inner boundary is guessed. The guess is being improved iteratively to minimize the mean quadratic deviation of the solution on the outer boundary from the given distribution.The results obtained show that the algorithm provides smooth, non-oscillating, and stable solutions to inverse problems of heat transfer, that is, it avoids disadvantages inherent in other computational methods for such problems. The ill-conditioning of inverse problems in the Hadamard sense is exhibited in that a very quick convergence of the fitting functional to its minimum does not imply a comparable rate of convergence of the recovered temperature on the inner boundary to the true distribution.The considered method can easily be extended to nonlinear problems.Numerical calculation has been carried out with the FE program Felics developed at the Chair of Mathematical Modelling of the Technical University of Munich.     

Fourier Analysis of the Effect of Grid Non-Orthogonality on the SIMPLE Algorithm

In this article, the error amplification matrix is developed for the SIMPLE algorithm formulated on the nonorthogonal grid by Fourier decomposition. The effect of grid skewness on the convergence properties of the SIMPLE algorithm is investigated in terms of robustness and convergence rate. As the grid nonorthogonality increases, the robustness of the SIMPLE algorithm, characterized by the convergence range of the pressure relaxation factor for a given velocity relaxation factor, weakens, and an effective remedy is to take the cross pressure correction terms into account rather than omit them, which is in agreement with the conclusions reached by Peric. In the case of convergence, the convergence rate seems to be strongly dependent on flow field, and the optimal rate can be reached when the grid direction coincides well with the local flow direction. The analysis also suggests that a collocated layout should be adopted in preference to a staggered layout, due to its better robustness on a nonorthogonal grid.     

FINITE-ELEMENT ANALYSIS OF HEAT CONDUCTION PROBLEMS WITH IRREGULAR REGION BOUNDARY USING MULTILEVEL TECHNIQUES

The main goal of this article is to introduce a numerical procedure for the calculation of solidification and melting using the PISO algorithm. Since coupling among velocity, pressure, temperature, and liquid fraction is important in phase-change problems, an improved PISO algorithm is required to couple these variables properly. To achieve this goal, the present study proposes the use of the enthalpy method for the inner iterations of the matrix solver, in order to predict more accurate temperature and liquid fraction that satisfy the energy balance criterion simultaneously. Another difference of the present method from other conventional coupling methods is in updating the permeability term of the momentum equation with liquid fractions predicted in the first correction step. This may couple the velocities with the liquid fractions more closely. A comparison of the proposed method with a standard iterative method is made for three different cases, showing that improved predictions are obtained in terms of computing speed and solution robustness.     

Numerical investigation of laminar forced convection of water upwards in a narrow annulus at supercritical pressure

In the present work, convection heat transfer of water at supercritical pressure in a narrow annulus at low Reynolds numbers (less than 1500) has been investigated numerically. The continuity, momentum and energy equations have been solved simultaneously using computational fluid dynamics techniques with the inlet Reynolds number ranging from 250 to 1000, Grashof number from 2.5 × 10⁵ to 1 × 10⁶ and the inlet fluid temperature from 360 °C to 380 °C. In all of the case studies, a sub-cooled water flow at supercritical pressure (25 MPa) and a temperature close to the pseudo-critical point enters the annular channel with constant heat flux at inner wall surface and insulated at outer wall. To calculate the velocity and temperature distributions of the flow, discretized form of the governing equations in the cylindrical coordinate system are obtained by the finite volume method and solved by the SIMPLE algorithm. It has been shown that the effect of buoyancy is strong and causes extensive increase in velocity near the inner wall, and consequently an increase in the convective heat transfer, which is desirable. Besides, the effects of inlet Reynolds number, Grashof number and inlet temperature on the velocity distribution and also on the heat transfer have been investigated.     

Equivalence of Artificial Compressibility Method and Penalty-Function Method

Abstract

Due to an assumption made on the pressure-velocity coupling for the SIMPLE algorithm and its variants, the corrected velocity can be obtained from the corrected pressure. However, substituting these quantities into the momentum equations may result in failure to satisfy the momentum equations. Therefore, the equations should be solved iterativety to obtain better velocities, thus giving a more satisfactory solution to the equations. In this article an explicit corrector step is proposed that is imposed on the first corrected velocities, which are obtained from the existing algorithms. This new corrector step has been tested by three flow problems, driven cavity flow, backward-facing step flow, and rectangular tank flow, with different Reynolds numbers. With this additional corrector step imposed on the SIMPLEC and PISO algorithms, the results show that the number of iterations can be reduced drastically due to the much better satisfaction of the momentum equations. Considerable savings in computing effort can be gained.