Performance of two new block implicit procedures for the treatment of the velocitypressure coupling problem in incompressible fluid flow

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1 Performance of two new block implicit procedures for the treatment of the velocitypressure coupling problem in incompressible fluid flow Z. Mazhar Department of Mathematics, Eastern Mediterranean University, North Cyprus, Mersin 10, TURKEY Abstract The performance of two new non-segregated,block implicit procedures; BIP [l] and BIPEN [2] are compared with the well known SIMPLERprocedure, applied to benchmark problems of various sizes. It is demonstrated that BIP provides quicker convergence than SIMPLER by a factor of two, and BIPEN is %40 faster than BIP. In addition it is shown that both methods are extremely robust, even for large-sized problem, rendering them as a basis for a new generation of hybrid computation algorithms, for the efficient treatment of the velocity- pressure coupling problem. 1 Introduction Incompressible fluid flow is governed by the following nonlinear, coupled set of partial differential equations for mass and momentum conservation: The numerical solution of these equations poses a serious challenge to researchers due to two factors:

2 528 Advmces irl Fluid Mechunks W 1. Apart from the nonlinearities present in the momentum equations, there exists a strong coupling between the velocity and pressure components, imposed by the continuity equationn (3). 2. Despite this strong coupling, there exists no equation for pressure. Pressure components are embedded in the momentum equations as auxiliary components. To overcome this problem, various procedures have been proposed in the literature, like SIMPLE [3], PUMPIN [4], NEW PUMPIN [5],SIMPLER [6], SIMPLEC [7], FIMOSE [S], and PIS0 [9]. All of these procedures try to artificially extract a pressure equation from the momentum equations. The solution is then obtained through a so-called segregated-type procedure as follows: The discretized forms of the momentum equations are first solved using an assumed pressure field. Then the velocity field is corrected through a pressure correction field in such a way as to satisfy continuity. The pressure field is than updated according to this corrected velocity field hoping to satisfy the momentum equations. The procedure is repeated until all of the equations are satisfied, including the treatment of the nonlinearities as well. The solution of the linearized forms of the momentum equations poses no serious problem. Since the resulting matrices are diagonally dominant, they can be solved efficiently by an SOR- or ADI- type iterative procedure. The main drawback is in the solution of the derived pressure and pressure correction equations. These equations are of Poisson type, and therefore the resulting matrices are weakly diagonally dominant, suggesting that the above-type iterative solutions are very slow an costly. For this purpose, various solution procedures such as SIP [lo]and MSIP [ll]have been developed. Peconditioning [12] has also been considered to speed up the convergence of such algorithms. These procedures, however, necessitate the utilization of a relaxation factor in order to speed up convergence, optimum values of which should be selected very carefully. Weak diagonal dominance of the involved matrices enhances the importance of this process. The range of the relaxation factor giving fast and convergent results, however, is very restricted, which is a deficiency degrading the robustness of the overall procedure. In addition to the above, an additional relaxation factor, denoted by E in the formulation given below, need to be utilized. Altogether, up to five relaxation factors need to be optimized and utilized. It should be noted that, beyond other things, choice of the relaxation factors are decisive in the total solution cost of an algorithm. An alternative to the segregated process is to consider the system as a block and try to solve the discretized forms of all the equations at once. Such attempts have been successful, and it has been demonstrated in the literature [l3-16] that such procedures mybe considered. However, these type of procedures have not been utilized widely, due to relatively higher costs or higher storage requirements, or even due to the complexity of the implementation of the suggested algorithms themselves.

3 Admcc~sill Fluid Mdxznics IV 529 The new approach proposed by the Author [ 1-21 addresses these problem in a very efficient way, paving the way to a new generation of fast, easier, cheaper and totally robust algorithms. In what follows, the new procedures are presented and tested on two benchmark problem most widely used in the literature for this purpose. Various grid resolutions were used m order to test their efficiency. Cost comparisons with the SIMPLER method, under similar conditions, are also presented. 2 The primitive finite difference equations The governing equations (1-3) are discretized by utilizing a conventional staggered grid [ 171 and upstream differencing [ 1S] and the E-factor formulation [4]. The following difference equations result ([191): Azus + XU,+A;u, + AiuE+AtuN+ &pp +A,Pp, = b; (4) AJv,+A,Ivw+A~v,+A,"v, +AivN+ A,Pp, +A,Pp, = b; (5) where &uw + A,"u,+ A,"v,+A,"., = b," (6) 3 The solution procedure After the boundary conditions are imposed, Eqs. 4-6 can be assembled mto a matrix form as shown on the next page, which can be denoted by AX=B (11) where X=(u,v,p). The main theme of the proposed solution procedure is to perform an incomplete decomposition of the matrix A of the form A=LD+C (12) The solution can than be accomplished by the successive iteration given by LDX("+l)= B-C$") (13) or, alternatively ~ ( n + l )= X ( " )+ ( ~ ~ 1 (B- - 1 A J I $ ~ ) ) (14) One iteration of the solution process, complemented with a relaxation parameter, is performed as follows:

4 530 Advmces irl Fluid MechmksW b" b" 8 I. Solve LY'*+" = B -AX'*' for ycn+l) 2. Solve DZ'~"' Y ( ~ + for I ~("+l) ) 3. Update X by&"+') = X'"' +CXZ("'", where cx parameter. is the relaxation In order that this process be successful and efficient, the following conditions must be satisfied: 1. The L and D matrices must be chosen in such a way that their structure allows a simple and economical direct solution of Y("+l) and Z("'l). 2. The condition P(LD)-' c < 1 must be met as a necessary and sufficient condition for the convergence of the algorithm. This is related to the effect of the "defect" matrix C. Therefore, the effect of the defect matrix C must be reduced as much as possible. For this the effect of the far diagonal coefficients in matrix A, denoted by A* and AC, should mainly be transferred to the L and D matrices, rather than the defect matrix. The procedures presented below satisfy both of the above conditions. 4 BIP (Block Implicit Procedure) In this procedure, the L, D and C matrices are taken as follows:

5 A d m c c ~ ill s Fluid Mdxznics IV 53 1 L= D= and FA;, 0 i 0 C= 0 A:: A&: A ; 0 \ 0 0 v The coefficients of the nine unknown vectors h,, A, ki, ni, g,, si, ti, qi, and y;: canbe written as follows: h,=& /A;, J;=A:/A;, i=l,n i=l,n-l ki=asp/a;, i=l,n ni=ap/a;,i=l,n-m i < gi=-(k,j-,+a:hi +As~ni-,+~ki), i=l,n si=-acj;/gi,i=l,n-l ei ti=-&h,-,/g,, i=2,n qi =-A;ki-,,,, i=m+l,n q =-Al,n,, i=l,n-m (15) (16) (17) (18) (19) (20) (21) (W (23)

6 532 Advmces irl Fluid Mechunks W where m and N are the number of grid points in one direction and the total number of grid points, respectively. 5 BIPEN (Block Implicit Procedure -Enhanced) This procedure is an enhancement of BIP in which the diagonal q in the matrix C is moved completely to the respective place in L, the cost of which returns to be the introduction of two auxiliary diagonals e and d, just below and above the original place of q in C.. The effect of these have been seen to be much less than that of q. In the new decomposition, D remains the same. L and C are redefined as follows: L= C= The coefficients of the 9 vectors h, k, n, g,s, t, q, and Y remain the same as above. The coefficients of the two extra vectors e and dare defined as follows: di= -qisi-,,, i = m +1,N (24) q = y i ti-,, i = m +2, N W 6 Test problems Two test problems are used. The first problem is a confined flow. Water is flowing into a closed square tank from one comer, and flows out from the opposing comer as shown in Fig. 1. For this problem, thee grid resolutions are used: 12xl2,22x22, and42x42. The second problem is the flow over a backward facing step as shown in Fig. 2. For this problem, three grid resolutions are used: 25x16, 50x32, and 64x40. For each problem and for each grid resolution, the problem was first solved to a tight convergence within machine accuracy limits. Then the velocity and pressure fields were set to zero, and the problem was solved again. The total number of multiplicative operations performed to reach a certain convergence level depending onthe maximum error in the pressure head of the entering fluid, was then recorded: E = 0.5pu~maxi I p y t-p,""" 1

7 Ad~mcc~s ill Fluid MdxznicsIV 533 u=o v=0 J- ( U=0 / v=0,"p ' L x " & L O L=004 m vout= qn = 0.7 mls Figure 1: Square tank geometry. H=0.04 m yn=o.1 m/s Figure 2: Backward facing step geometry. The SIMPLER, BIP andbipenwere applied using various values of the E factor used in the discretization, and for various values of tie a parameter used in BIP and BIPEN, and for the same parameter used in the solution of velocities, pressure correction and pressure through the MSIP algorithm in SIMPLER. 7 Results and comparison In Fig. 3, the solution costs of each procedure for the square tank problem, is shown. Fig. 4 shows the solution costs of the three procedures for the backward facing step problem.

8 534 Advmces irl Fluid Mechunks W Cost (xioonoperations) 100 m m X ~ 1.0 a Cost (xioon operations) SIMPLER( (L =O 65) Cost W O O Noperations) c) 42x42 L c a. Cost (x1000n operations) 600 (c)42x oKBlP(; =0.7) 50 BIPEN a =l.0 ) BIPEN(lX =OS( E+1 E+2 E+3 E*4 E+5 E ;~~ ~ E+1 E+2 E+3 E+4 E+5 E -L E+1 E+2 E+3 E+4 E+5 E Figure 3: Solution cost of SIMPLER,BIPand BIPEN for the a) 12x12, b) 22x22 and 42x42 square tank problem. 4) i 3) M O 400 m 204 m 440 m 3w WENlE=20) x x W E N [EZZOJ l.0 a a a B I P W W I W E+0 E d E+2 E+3 E+4 E+5 E+0 E+1 E+2 E+3 E+4 E+5 E+0 Et1 E+2 E+3 E+4 Et5 E E E Figure 4: Solution cost of SIMPLER, BIP and BIPEN for the a) 25x16, 50x32 and 64x40 back steo problem.

9 Ad~mcc~s ill Fluid Mdxznics IV 535 The main observations that can be dram from these figures can be listed as follows: 1. BIP is faster than SIMPLER by a factor of two, under the condition that optimum relaxation parameters are used. 2. BIPEN is faster than BIP by a factor of 1.5, when optimum relaxation parameters are used. 3. BIP converges for a wider range of the relaxation parameter a than SIMPLER. BIPEN converges for any relaxation parameter a. SIMPLER converges in a very narrow range of the E factor (E<8),but BIP and BIPEN converge for any E value. 8 Conclusion It has been been demonstrated, for the test problem chosen, that BIP is a clear cut alternative to SIMPLER in handling the strong coupling of the velocity and pressure fields. BIPEN, furthermore, proved to be convergent for virtually any relaxation parameters. In fact, BIPEN is seen to converge faster for very large E values, suggesting that the E-factor formulation can be abandoned at all. This means that the time independent formulation of the governing equations can be used for steady state problems. On the other hand, however, although BIPEN converges regardless of the a parameter used, there still exist an optimum value at which the solution can be obtained in a lowest computational cost. It should also be noted that the formulations of BIP and BIPEN are very simple; even much simpler than that of SIMPLER. Furthermore, not too much extra storage is needed than SIMPLER. Lastly, SIMPLER requires the use of five optimized relaxation parameters. BIP requires two, and BIPEN requires only one. With the above conclusions, BIPEN seem to be a very attractive alternative procedure for the treatment of the velocity-pressure coupling problem, therefore deserving further attention of researchers in this field. It is also open to further enhancement, for example, to the application of preconditioning. References [l] Mazhar, Z., A procedure for the treatment of the velocity-pressure coupling problem in incompressible fluid flow, Numerical Heat Transfer, Part B, 39, pp ,2001. [2] Mazhar, Z., An enhancement to the block implicit procedure for the treatment of the velocity-pressure coupling problem in incompressible fluid flow, Numerical Heat Transfer, Part B,41, pp. 1-8, 2002 [3] Patankar, S. V. & Spalding, D. B., A Calculation Procedure for Heat, Mass and MomentumTransfer in Three-Dimensional Parabolic Flows, M J. Heat Mass Transfer, 2, pp , 1972.

10 536 Advmces irl Fluid Mechunks W [4] Raithby, G. D. & Schneider, G. E., Numerical Solution of Problem in Incompressible Fluid Flow: Treatment of the Velocity-Pressure Coupling, Numer. Heat Transfer, 2, pp , [5] Mazhar, Z. & Raithby, G. D., A Refined PUMPIN (Pressure Update by Multiple Path INtegration) Method for Updating Pressures in the Numerical Solution of the IncompressibleFluid Flow Equations, Proc. 2nd Int. Con$ on Numerical Methods in Laminar and Turbulent Flow, Pineridge Press, pp , [6] Patankar, S. V., A Calculation Procedure for Two-Dimensional Elliptic Situations, Numer. Heat Transfer, 4, pp , [7] Van Doormaal, J. P. & Raithby, G. D., Enhancements of the SIMPLE Method for Predicting IncompressibleFluid Flow, Numer. Heat Transfer, 7, pp , [S] Latimer, B. R. & Pollard, A., Comparison of Pressure-Velocity Coupling Solution Algorithms, Numer. Heat Transfer, 8, pp , [9] Issa, R. I., Solution of Implicitly Discretized Fluid Flow equations by Operator-Splitting, Comput. Phys, 62, pp , [lo] Stone, H. L., Iterative solution of implicit approximations of multidimensional partial differential equations, SIAMJ. Numer. Anal., 5, pp , [l l]zedan, M. G., A Modified Strongly Implicit Procedure for the Numerical Solution of Field Problems, Master's Thesis, University of Waterlo, Waterloo, Ontario, Canada, Evans, D. J., The use of pre-conditioning in iterative methods for solving linear equations with symmetric positive definite matrix, J. Inst. Maths. Applics., 4,pp , 1967 [13] Vanka, S. P., Block-Implicit Calculation of Steady, Turbulent, Recirculating Flows, Int. J. HeatMass Transfer, 28(11), pp , [14] Hanby, R. F., Silvester, D. J. &L Chow, J. W., A Comparison of Coupled and segregated Iterative Solution Techniques for Incompressible Swirling Flow, Int. J. Numer. Mtds. in Fluids, 22, pp , [151 Galpin, P. F., Solution of mass and momemtum equations by application of a coupled equation line solver, Int. J. Numer. Meth. Fluids, 5,pp , [16] Gungormus, T., A Block Implicit Method for Incompressible Fluid Flow Problems,Master's Thesis, Eastern MediterraneanUniversity, [17] Harlow, F. H. &L Welch, J. E., Numerical Calculation of Time-dependent Viscous Incompressible Flow of Fluid with Free Surface, Phys. Fluids, 8(12),pp , [l81 Raithby, G. D. & Torrance, K. E., Upstream-Weighted Differencing Schemes and Their Application to Elliptic Problems Involving Fluid Flow, Comput. Fluids, 2, pp ,1965. [19] Mazhar, Z., An Evaluation of the segregated Solution Procedures for the Solution of Incompressible Fluid Flow Problem, PhD Thesis, University of Waterloo, 1982.