A Coupled Finite Volume Solver for Incompressible Flows

F. Moukalled and M. Darwish

Department of Mechanical Engineering American University of Beirut

P.O.Box 11-0236 Riad El Solh, Beirut 1107 2020

Lebanon

Abstract. This paper reports on a pressure-based coupled algorithm for the solution of laminar incompressible flow problems. The implicit pressure-velocity coupling is accomplished by deriving a pressure equation in a way similar to a segregated SIMPLE algorithm with the extended set of equations solved simultaneously and having diagonally dominant coefficients. The superiority of the coupled approach over the segregated approach is demonstrated by solving the lid-driven flow in a square cavity problem using both methodologies and comparing their computational costs. Results indicate that the number of iterations needed by the coupled solver is grid independent. Moreover, recorded CPU time values reveal that the coupled approach substantially reduces the computational cost with the reduction rate for the problem solved increasing as the grid size increases and reaching a value as high as 115.

Keywords: Finite Volume Method, Pressure-Based Method, Coupled Solver. PACS: Replace this text with PACS numbers; choose from this list: http://www.aip.org/pacs/index.html

INTRODUCTION

The rate of convergence of pressure-based fluid-flow simulations is influenced greatly by the coupling between velocity and pressure. Effort to develop more robust and efficient velocity-pressure algorithms have resulted in solving several numerical problems affecting their performance [1-3]. However, the superiority of the coupled over the segregated approach (or vice versa) issue is still not settled. In the coupled approach, the conservation equations are discretized and solved as one system of equations as opposed to the segregated approach where the equation of each variable is solved separately using, previously computed, best estimate values of other dependent variables.

Pressure-based algorithms gained popularity through the development of the well-known segregated SIMPLE algorithm [4] for incompressible flows in the early 1970. The CFD research community widely adopted the SIMPLE algorithm leading to the development of a SIMPLE-like family of algorithms [5].

The first pressure-based coupled solver denoted by SIVA was developed by Carretto et al. [6] prior to the SIMPLE algorithm. However, in spite of its merits, the SIVA algorithm was quickly overshadowed by the SIMPLE algorithm that combined low memory requirement with coding simplicity, which were the two decisive factors given the state of computer technology at that time. Later work resulted in the development of several pressure-based coupled algorithms. These algorithms followed two approaches in their development. In the first, no pressure equation is introduced and the momentum and continuity equations are discretized in a straightforward manner [6- 8]. Since no pressure equation is derived, zeros are present in the main diagonal of the discretized continuity equation leading to an ill conditioned system of equations.

In the second approach a pressure equation is derived either through the addition of pseudo-velocities [9] as in the segregated SIMPLER algorithm [10] or without the addition of new variables as in the segregated SIMPLE algorithm [4].

In this paper the second approach is followed to develop a coupled solver on collocated grid systems. An algebraic multigrid solver is used to accelerate the solution of the extended system of equations. The coupled algorithm is assessed by comparing its performance with the segregated SIMPLE algorithm in a test problem.

FINITE VOLUME FORMULATION

The conservation equations governing steady, laminar incompressible Newtonian fluid flow are given by

(1)

(2)

Integrating the transport equations over a control volume, transforming the volume integrals of the diffusion and convection terms into surface integrals using the divergence theorem, and evaluating these integrals by representing the variables at the control volume faces in terms of nodal values, the discretized forms of the momentum and continuity equations are respectively given by

and (3)

The Collocated Simple Algorithm

In the segregated SIMPLE algorithm, the solution is obtained by iteratively solving the momentum equations and a pressure correction equation, derived from the continuity equation, while accounting for variations in the pressure field within the momentum equations by applying corrections to the velocity field. Denoting corrections with a prime, the corrected fields are written as

and (4)

Using the Rhie-Chow interpolation [25-27], the velocity correction along the control volume face, is written as

(5) By substituting from Eq. (5) into the continuity equation, the pressure correction equation is obtained as

(6)

The overall SIMPLE algorithm can be summarized as follows:

1. Solve the momentum equations implicitly for v using the available pressure field 2. Solve the pressure correction equation 3. Correct v and p 4. Return to the first step and repeat until convergence

The Coupled Algorithm

The low convergence rate of the SIMPLE algorithm is a due to the explicit treatment of the pressure gradient in the momentum equation and the velocity field in the continuity equation. The coupled algorithm overcomes this deficiency by treating both terms implicitly through coupling the momentum equation and the pressure equation form of the continuity equation through a set of coefficients that represent the mutual influence of the continuity and momentum equations on the pressure and the velocity fields. For that purpose the pressure gradient term in the momentum equations is integrated over the faces of the control volume and is evaluated implicitly by expressing the pressure at each face in term of the nodal values straddling the interface and resulting in the following discretized momentum equations:

(7)

The pressure equation is derived from the continuity equation by expressing the velocity at the control volume

face using the Rhie-Chow interpolation as

(8) Substituting given by Eq. (8) into the continuity equation, the algebraic form of the pressure equation is

obtained as

(9)

Combining the discretized momentum and continuity equations [Eqs. (7) and (9)], a system of equations

involving velocity components and pressure is obtained for each control volume and when expressed over the entire computational domain yields a system of equations of the form

(10)

where all variables (v, p) are now solved simultaneously. Note that the continuity equation is now written in

terms of pressure rather than pressure correction. The overall coupled algorithm can be summarized as follows:

1. Start with the nth iteration values 2. Assemble and solve the momentum and continuity equation for v*and p* 3. Assemble using the Rhie-Chow interpolation 4. Return to the first step and repeat until convergence

RESULTS AND DISCUSSION

The performance of the coupled algorithm is assessed in this section by presenting solutions to the lid-driven flow in a square cavity problem. The results are generated using both triangular and quadrilateral control volumes on six grid sizes with cell values of 104, 3x104, 5x104, 105, 2x105, and 3x105. The same initial guess was used for all grid sizes and for both coupled and segregated methods. The physical situation, which represents a square cavity of side L, along with illustrative portions of the quadrilateral and triangular meshes used are depicted in Figure 1(a).

(a) (b) (c)

FIGURE 1 (a) Physical domain and illustrative triangular and quadrilateral grids used for the driven flow in a square cavity, (b) streamlines, and (c) comparison of contours of constant u-velocity obtained using the coupled and segregated solvers.

Results are obtained for a value of Reynolds number (Re=ρUL/µ, U the velocity of the top horizontal wall) of 1000. The flow field in the cavity is shown by the streamlines presented in Figure 1(b). Figure 1(c) compares, on the densest grid employed (3x105 cells), contours of constant u-velocity component generated using both methods. As shown, contours are on top of each other and cannot be distinguished, validating both the segregated and coupled algorithms.

A summary of the number of iterations and CPU time needed by both segregated and coupled approaches using quadrilateral and triangular elements are presented for all grid sizes in Table 1. The number of iterations required by the coupled algorithm is almost grid independent for both quadrilateral and triangular elements while the iterations of the segregated approach increases with an increase of the grid size. For quadrilateral (triangular) elements the iteration ratio S/C increases from 45 (78) to 546 (642) as the grid size increases from 104 to 3x105 control volumes.

In terms of computational times, substantial savings are achieved using the coupled approach with the amount increasing with increases to the grid size. For quadrilateral (triangular) elements, the reduction factor increases from a value of 13 (13) on the coarsest grid (104 cells) to a value of 115 (104) on the densest grid (3x105 cells). This represents a significant decrease in computational time allowing solutions for the problem on the densest grid used to be obtained with the coupled solver in less than twenty minutes, while over 24 hours are required using the segregated approach.

TABLE 1. Comparison of the number of iterations and CPU time required by the segregated and coupled flow

solvers for the driven flow in a square cavity problem on meshes of different sizes. Quadrilateral Elements Triangular Elements # of Iterations CPU Time # of Iterations CPU Time

Size C S S/C C S S/C C S S/C C S S/C 10,000 17 768 45 26.52 351.23 13 18 1400 78 45.04 582.32 13 30,000 17 1653 97 86.88 2263.26 26 17 2208 130 130.16 3004.55 23 50,000 17 2444 144 150.53 5621.78 37 17 3412 201 205.89 6249.04 30 100,000 16 4475 280 346.55 24619.7 71 17 4418 260 426.78 14806.64 35 200,000 16 7012 438 698.64 67154.28 96 17 8453 497 815.51 67778.15 83 300,000 17 9280 546 1134.96 130890.4 115 17 10917 642 1211.53 125520.9 104

CLOSING REMARKS

A pressure-based fully coupled algorithm for the solution of laminar incompressible flow problems was presented. The performance of the method was assessed by comparing the number of iteration and CPU time required to produce a solution with those required using the segregated approach. It was found that the number of iterations needed by the coupled algorithm is grid independent. Moreover, results showed a substantial decrease in computational time using a coupled approach when compared to using the segregated method with the reduction rate increasing as the grid size increases.

ACKNOWLEDGMENTS

This work was partially supported by the URB of the AUB through Grant # 888322 and by the LNCSR trough Grant # 022142.

REFERENCES

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Numerical Heat Transfer, second edition, edited by W. J. Minkowycz, E. M. Sparrow, and J. Y. Murthy, Wiley, 2006, pp. 325-367.

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