z.g. wu, z.g. qu, ya-ling he and w.q. tao*nht.xjtu.edu.cn/paper/en/2008201.pdf · a comprehensive...

Progress in Computational Fluid Dynamics, Vol. 8, No. 5, 2008 233

Copyright © 2008 Inderscience Enterprises Ltd.

A comprehensive performance comparison for segregated algorithms of flow and heat transfer in complicated geometries

Z.G. Wu, Z.G. Qu, Ya-Ling He and W.Q. Tao* Stat Key Laboratory of Multiphase Flow and Heat Transfer, School of Energy and Power Engineering, Xi’an Jiaotong University, Xi’an, Shaanxi 710049, China E-mail: [email protected] E-mail: [email protected] E-mail: [email protected] E-mail: [email protected] *Corresponding author

Abstract: In this paper, a comprehensive performance comparison among CLEAR, SIMPLER and SIMPLEC algorithms at staggered grid system is performed for four incompressible fluid flow and heat transfer problems in complicated geometries. It is found that for most of the relaxation factor variation range CLEAR can appreciably enhance the convergence rate though it is slightly less robust than the other two algorithms. For the four problems tested, the ratio of iteration numbers of CLEAR over that of SIMPLER and SIMPLEC can reach as low as 0.18, 0.09 and the ratio of the CPU time as low as 0.23, 0.18, respectively.

Keywords: CLEAR; SIMPLER; SIMPLEC; fluid flow; heat transfer; numerical simulation; complex geometry.

Reference to this paper should be made as follows: Wu, Z.G., Qu, Z.G., He, Y-L. and Tao, W.Q. (2008) ‘A comprehensive performance comparison for segregated algorithms of flow and heat transfer in complicated geometries’, Progress in Computational Fluid Dynamics, Vol. 8, No. 5, pp.233–247.

Biographical notes: Zhi-Gen Wu is currently a Doctoral graduate student at the School of Energy and Power Engineering of Xi’an Jiaotong University, China. He received his Bachelor of Power Engineering in 2002 from XJTU. His research interests include numerical simulation and experimental study of fin-and-tube heat exchangers, and CO2 transcritical air condition system.

Zhi-Guo Qu is a Lecturer in the School of Energy and Power at Xi’an Jiaotong University of China. He received his PhD in Power Engineering and Engineering Thermophysics from Xi’an Jiatotong University, China. His research interests are enhanced heat transfer, computational heat transfer, heat exchangers with high efficiency and potential energy storage.

Ya-Ling He is a Professor of Xi’an Jiaotong University, China. She received her PhD of Engineering Thermophysics from Xi’an Jiaotong University, China. Her research interests include small-scale refrigerator and cryocooler in the high technology, computer simulation and optimisation of refrigeration system, heat exchangers with high efficiency; fluid flow and heat transfer in mirochannels. Currently, she is the Associate Editor of Applied Thermal Engineering.

Wen-Quan Tao is a Professor in the School of Energy and Power at Xi’an Jiaotong University of China. His research interests include enhanced heat transfer, numerical heat transfer, micro-scale heat transfer, and solar energy application. Currently, he is the Associate Editor of International Journal of Heat and Mass Transfer and International Communications in Heat and Mass Transfer.

1 Introduction There are two different strategies to solve the discretisation equations of the Navier–Stokes (NS) equations: the direct method and the segregated method (Tao, 2001; Shyy and Mittal, 1998; Braaten, 1985; Patankar, 1980). For the former, apart from computer memory problem, the non-linearity of the NS equations requires that the

resulting algebraic equations should be solved repeatedly with updated coefficients in the discretisation equations, leading to repeated direct solutions of the algebraic equations and, hence, a large consumption of the computational time. Therefore, for most engineering problems the segregated method, which is iterative in nature and needs less computer memory and time, has been more commonly applied. In the segregated method, the fluid

234 Z.G. Wu, Z.G. Qu, H. Ya-Ling and W.Q. Tao

pressure is usually solved as a primitive valuable, so it is referred to as the pressure-based approach. Although many numerical methods can be listed within the framework of the pressure-based approach, for example, the fractional step method (Kim and Moin, 1985), the artificial compressibility method (Lee and Tzong, 1992), the penalty method (Braaten and Shyy, 1986), and the pressure-correction method, according to the statistics of references published in the past three decades, the pressure-correction method is definitely the most widely used in the literature. The pressure-correction method solves the algebraic form of the momentum equations sequentially starting from a guessed pressure field or a field determined from a given velocity field. Through iteration, the pressure and velocity fields are gradually improved until they satisfy the momentum and the continuity equations.

Patankar and Spalding (1972) proposed the pressure-correction algorithm SIMPLE algorithm. As shown in the following section, it has two major approximations:

• the initial pressure field and the initial velocity fields are assumed independent; hence, the inherent interconnection between pressure and velocity is neglected, leading to some inconsistency between them

• the effects of the pressure corrections of the neighbouring nodes are arbitrarily dropped in order to simplify the solution procedure, thus making the algorithm semi-implicit.

These assumptions affect the convergence rate, though the final solution is not affected (Shyy and Mittal, 1998). At the same time, underrelaxation for the pressure-correction is needed to stabilise the iterative procedure. Therefore, how to overcome these two approximations is the objective of many evolutions of the SIMPLE algorithm (Patankar, 1981; van Doormaal and Raithby, 1984, 1985; Issa, 1985; Connell and Stow, 1986; Raithby and Schneider, 1988; Chatwani and Turan, 1991; Yen and Liu, 1993; Wen and Ingham, 1993; Sheng et al., 1998; Yu et al., 2001). In the following, a brief description of the evolution of SIMPLE-like algorithms is presented.

Patankar (1981) proposed the SIMPLER algorithm, which successfully overcomes the first approximation. Thus, the SIMPLER algorithm has been widely used in the CFD/NHT community since 1981. In the last two decades, the proposed algorithms were all focused on the second approximation. van Doormaal and Raithby (1984) proposed the SIMPLEC algorithm, in which, by improving the definition of the coefficients of the velocity-correction equation, the effects of this approximation are partially compensated. van Doormaal and Raithby (1985) and Raithby and Schneider (1988) proposed another SIMPLE-like algorithm, SIMPLEX, in which, through solving a set of algebraic equations for the coefficients in the velocity-correction equations, the effects of dropping the neighbouring grid node contributions are also taken into

account to some degree. Though there are more than ten variants of the SIMPLE-like algorithm, the second approximation has not been fully overcome. This is probably the reason why no particular SIMPLE-like algorithm has a predominant advantage and can replace the others, exhibiting the highly problem-dependent character of the algorithms. Generally speaking, in the SIMPLE family algorithms, the function of the pressure-correction term is to improve the current pressure and velocity fields by adding their corresponding corrections. Thus, the improved pressure and velocity can satisfy the mass conservation condition at each iteration level, as demonstrated in Blosch and Shyy (1993), which accelerates the iteration convergence. However, since in the derivation of the pressure-correction equation, the effects of the pressure-correction at the neighbouring points are neglected, the resulting pressure-correction is overestimated as far as the pressure itself is concerned and hence the total convergence rate is deteriorated somewhat.

Tao et al. (2004a, 2004b) introduced a novel segregated algorithm for incompressible fluid flow and heat transfer problems: Coupled and Linked Equations Algorithm Revised (CLEAR). It solves for the improved pressure directly and no term is dropped in the derivation of the pressure equation. As a result, the influence of the neighbouring points is fully taken into account. Thus, the second assumption of all SIMPLE-family algorithms is completely removed, greatly enhancing the convergence rate of the iteration process. In Tao et al. (2004a), comparisons between the CLEAR and SIMPLER algorithms for typical numerical examples have been presented and the results are promising. However, as described in van Doormaal and Raithby (1984) and Zeng and Tao (2003), SIMPLEC algorithm is better than SIMPLER in some cases. In addition, the examples presented in Tao et al. (2004a, 2004b) are for relatively simple flow geometry configurations. Therefore, it is important to test more complicated problems with the CLEAR, SIMPLER and SIMPLEC algorithms to investigate the performance of CLEAR more thoroughly.

In the remainder of this paper, four fluid flow and heat transfer problems in complicated geometries are solved with the CLEAR algorithm. Comparisons are made with the solutions using the SIMPLER and SIMPLEC algorithm. In the following section, the solution procedures of four algorithms are briefly summarised. Then, the basis of comparison and the convergence criterion are described, followed by detailed presentations of the computational results of the four examples. Finally, some conclusions are drawn.

2 Brief description of the four algorithms The details of the four algorithms can be found in Tao (2001), Patankar (1980) and Tao et al. (2004a, 2004b). For the purpose of comparison, the solution procedure of the SIMPLE, SIMPLER, SIMPLEC and CLEAR algorithms are

A comprehensive performance comparison for segregated algorithms 235

described simply by the symbols: ‘⇒’ and ‘→’, which mean solving equation and calculating variables, respectively.

SIMPLE: 0 0

Assume 1

Assume 2

** *, , ,

* *, ,

* next iteration

u v p u v

p u u v v

p pα

⇒

′ ′ ′⇒ → + +

′+ ⋅ ⇒

Assumption 2:

e e nb nba u a u′ ′= ∑ ( )P E ep p A′ ′+ − (1)

* ( ).e e e P Eu u d p p′ ′= + − (2)

SIMPLEC: 0 0

Assume 1 Assume 2

* * **, , , ,

* *, next iteration

u v p u v p u u

v v p pα

′ ′⇒ ⇒ → +

′ ′+ + ⋅ ⇒

Assumption 2:

( ) ( )e nb e nb nb ea a u a u u′ ′ ′− ⋅ = −∑ ∑ ( )P E ep p A′ ′+ − (3)

* ( ).ee e P E

e nb

du u p p

a a′ ′= + −

−∑ (4)

SIMPLER: 0 0

Assume 2

* *ˆ ˆ, , ,* , next iteration

u v u v p uv p u v

→ ⇒ ⇒

′⇒ → ⇒

Assumption 2:

e e nb nba u a u′ ′= ∑ ( )P E ep p A′ ′+ − (5)

* ( ).e e e P Eu u d p p′ ′= + − (6)

CLEAR:

0 0 (1) 0 0

(2) (3)

0 0

* *, , ,

* * *, ,

, next iteration

u v u v p u

v u v p u

v u v

→ ⇒ ⇒

→ ⇒ →

⇒ ⇒

• 0 0

0 [(1 ) / ]/

nb nb u u e ee

e u

a u b a uu

aα α

α+ + −

= ∑ (7a)

0 00 [(1 ) / ]

/nb nb v v n n

nn v

a v b a vv

aα α

α+ + −

= ∑ (7b)

• * *

* [(1 ) / ]/

nb nb u u e ee

e u

a u b a uu

aβ β

β+ + −

= ∑ (8a)

* ** [(1 ) / ]

/nb nb v v n n

nn v

a v b a vv

aβ β

β+ + −

= ∑ (8b)

• * *

*

[(1 ) / ]/

( ) ( )

nb nb u u e ee

e u

e P E e e P E

a u b a uu

a

d p p u d p p

β ββ

+ + −=

+ − = + −

∑ (9a)

* *

*

[(1 ) / ]/

( ) ( ).

nb nb v v n nn

n v

n P N n n P N

a v b a vv

a

d p p v d p p

β ββ

+ + −=

+ − = + −

∑ (9b)

It can be seen from the above presentation that one of the evolutions of the SIMPLE algorithm, the SIMPLER algorithm, only overcomes Assumption 1, and that the SIMPLEC algorithm only alleviates the effects of Assumption 2. These two algorithms do not fully overcome Assumption 2 of the SIMPLE algorithm. However, it can be seen that no assumptions exist in CLEAR algorithm. In CLEAR, the first half of an iteration is the same as the SIMPLER algorithm. However, in the second half, the updated pseudo-velocities * *,u v are calculated instead of solving the pressure-correction equation, then the pressure equation is solved again to get the improved pressure, and subsequently the improved velocity is obtained from new expressions (9a) and (9b). So, in the entire solution process of one iteration, no terms are neglected in CLEAR, which makes the solution method fully implicit. This is the key difference between CLEAR and any SIMPLE-family algorithm.

3 Numerical comparison conditions A comprehensive comparison between SIMPLER, SIMPLEC and CLEAR is presented in this paper. In order that the comparative results are meaningful, all numerical treatments should be the same except for the coupling algorithm between velocity and pressure. The major aspects of the comparison calculations include:

• The same discretisation scheme is adopted. In the present study, an absolutely stable scheme, the power-law scheme (Patankar, 1980) is used.

• The same grid system and distribution are used. The three algorithms are all implemented at staggered grid system. For each problem compared, a specific grid distribution will be presented.

• The same solution method for the algebraic equations is implemented. In this paper, the Alternative Direction Implicit method (ADI) incorporated in the block-correction technique (Prakash and Patankar, 1981) is used.

• The same values of the velocity underrelaxation factor α are used. For each problem, the values will be provided individually. For clarity of presentation, the time step multiple (van Doormaal and Raithby, 1985), E, instead of underrelaxation factor α, is used. The relation between α and E is defined by


(0 < < 1)1

E α αα

=−

(10)

Based on this practice, the second relaxation factor β of the CLEAR algorithm, which appears in Equations (9a) and (9b), is taken to be as follows:

0.5 0 0.51 1.5 0.5 < 1.β α

β α= < <

≤ ≤ ≤ (11)

For the SIMPLER and CLEAR algorithms, the pressure relaxation factor is also an important factor affecting the convergence rate. However, SIMPLEC needs not pressure relaxation (van Doormaal and Raithby, 1984).

• The convergence criterion is the same. In this study, the mass conservation condition is taken as the convergence criterion

* * * *

cv cvch

8

( ) ( ) ( ) ( )Rs MAXflow

5.0 10

w e s nu A u A v A v Aρ ρ ρ ρ

−

− + −=

≤ ×

(12)

and we also validated with Rscv ≤ 10–8 to obtain stricter conservation conditions in one case.

The SIMPLER, SIMPLEC and CLEAR algorithms are applied to four two-dimensional problems of fluid flow and heat transfer. They are:

• laminar fluid flow through a rectangular channel with a central constriction

• laminar fluid flow over roughness ribs in a rectangular channel

• laminar fluid flow in a serpentine rectangular channel

• natural convection heat transfer in a square cavity with an isolated horizontal plate.

The number of iterations for obtaining a converged solution, the CPU time, and the robustness of the algorithms are compared. All of the four problems are based on the following assumptions: laminar, incompressible, steady state, and constant fluid properties. The governing equations for such typical fluid flow and heat transfer are well documented in the literatures (Tao, 2001; Patankar, 1980) and, to save space, they are omitted here, while the boundary conditions will be described in detail for each problem studied.

4 Numerical experiments

Problem 1: Laminar fluid flow through rectangular channel with a central constriction.

The computational configuration is shown in Figure 1, where L1/H1 = 20, H2 = H3, L1/H2 = 60, L1/L3 = 15 and L1/L2 = 7/15. The Reynolds number is defined as

in 1Re / .u H ν= (13)

Figure 1 Laminar fluid flow through a rectangular channel with a central constriction

A grid system of 152 × 32 is adopted. The domain extension method is used to deal with the irregular domain: the protuberant solid regions are treated as a large viscosity fluid (Tao, 2001; Patankar, 1980). The inlet velocity distribution is assumed to be uniform and a fully developed boundary condition is assigned to the outflow boundary. A no-slip boundary condition is used for solid walls (the same treatment will be adopted for the solid walls of other three cases and will not be explicitly referred to again).

The pressure relaxation factor for the CLEAR and SIMPLER algorithms is 1.0. The second relaxation factor β is 1.5 as the first relaxation factor α ≥ 0.5. The convergence criterion is Rscv ≤ 5 × 10–8. The effect of the convergence criterion on the convergence performance is also investigated with Rscv ≤ 10–8. The predicted flow fields for Re = 10 and 50 are shown in Figure 2.

Figure 2 Predicted stream function near the channel area in Problem 1: (a) Re = 10 and (b) Re = 50

(a)

(b)

The number of iterations and CPU time of the three algorithms are displayed in Figures 3 and 4 for two Reynolds numbers and in Figure 5 for the stricter convergence criterion. The ratios of iteration number and CPU time are presented in Figure 6. From the figures,


it can be observed that in the convergence range common to the three algorithms, CLEAR can save appreciable iterations and CPU time, up to about 70% saving in CPU time. Only for larger values of E (larger than 10–15, corresponding to α ≈ 1) is CLEAR perhaps inferior to the other two algorithms. For the whole range of comparison, the ratios are as follows:

For Re = 10, Rscv ≤ 5 × 10–8

The ratio of CLEAR over the SIMPLER: from 21% to 115% (ITER), from 29% to 156% (CPU time).

The ratio of CLEAR over the SIMPLEC: from 33% to 130% (ITER), from 50% to 187% (CPU time).

For Re = 50, Rscv ≤ 5 × 10–8



For Re = 10, Rscv ≤ 10–8



Figures 6(a) and (b) provide the numerical results obtained under more strict convergence conditions, and it can be observed that convergence condition is not the key factor affecting the ratio of the iteration number and CPU time. From Figures 3–5, it can be seen that CLEAR has the advantage in the practical range of E for engineering computations.

Figure 3 Comparison of iteration number and CPU time for Re = 10 in Problem 1: (a) Iteration number and (b) CPU time (continues on next column)

(a)

Figure 3 Comparison of iteration number and CPU time for Re = 10 in Problem 1: (a) Iteration number and (b) CPU time (continued)

(b)

Figure 4 Comparison of iteration number and CPU time for Re = 50 in Problem 1: (a) Iteration number and (b) CPU time

(a)

(b)


Figure 5 Comparison of iteration number and CPU time for Re = 10 with Rscv < 1 × 10–8 in Problem 1: (a) Iteration number and (b) CPU time

(a)

(b)

Figure 6 Ratios of iteration numbers and CPU time for Problem 1: (a) Re = 10, Rscv < 5 × 10–8, (b) Re = 10, Rscv < 10–8 and (c) Re = 50, Rscv < 5 × 10–8 (continues on next column)

(a)

Figure 6 Ratios of iteration numbers and CPU time for Problem 1: (a) Re = 10, Rscv < 5 × 10–8, (b) Re = 10, Rscv < 10–8 and (c) Re = 50, Rscv < 5 × 10–8 (continued)

(b)

(c)

As can be seen from Figures 3–5 and the above ratio list, the robustness of the three algorithms is different; SIMPLEC is best, followed by SIMPLER, and CLEAR. This is the weakness of CLEAR, i.e., for some cases, its robustness is not as strong as that of SIMPLEC. This weakness is exhibited in two ways: first, for CLEAR, the range of variation of the relaxation factor within which convergence can be obtained is less than that of SIMPLER and SIMPLEC; second, in the convergence range common to the three algorithms, at the extreme limit where the relaxation factor is large, the convergence rate of CLEAR may be less than that of the other two. In the above ratio list, for the large relaxation factor region, the ratio may be greater than 1, which is the result of this weakness.

Problem 2: Laminar fluid flow in a rib-roughened rectangular channel.

The problem is shown schematically in Figure 7, where L1/H1 = 15, H2 = H3, L1/H2 = 30, L1/L3 = 30 and L2/L3 = 6. The inlet velocity distribution is supposed to be uniform and the fully developed boundary condition is assigned to the outflow boundary. The Reynolds number is defined as

in 1Re / .u H ν= (14)


Figure 7 Laminar fluid flow in a rib-roughened rectangular channel

A grid system of 302 × 42 is adopted. The five central rectangular protuberant solid regions are treated as a large viscosity fluid. Pressure relaxation factor for CLEAR and SIMPLER algorithms is 1.0. Computations were conducted for Re = 50 and 150. The predicted flow fields are shown in Figure 8.

Figure 8 Predicted stream function around rib-roughness in Problem 2: (a) Re = 50 and (b) Re = 150

(a)

(b)

The numerical results shown in Figures 9–12 show the good performance of the CLEAR algorithm. For this rather complicated case, in the convergence region common to the three algorithms, the convergence performance of CLEAR is always superior, and the saving in iterations range from 19% to 75%, and that in CPU time from 18% to 70%. The detailed information of performance comparison is:

For Re = 50, Rscv ≤ 5 × 10–8

The ratio of CLEAR over SIMPLER: from 25% to 69% (ITER), from 30% to 82%(CPU time).

The ratio of CLEAR over SIMPLEC: from 25% to 80% (ITER), from 55% to 173% (CPU time).

For Re = 150, Rscv ≤ 5 × 10–8



For Re = 50, Rscv ≤ 10–8



The robustness of the three algorithms is in the same order as in Problem 1: i.e., SIMPLEC is the best, followed by SIMPLER, and CLEAR. For CLEAR the range of the relaxation factor within which convergence solution can be obtained is less than that of the two other algorithms.


(a)

(b)

Figure 10 Comparison of iteration number and CPU time for Re = 150 in Problem 2: (a) Iteration number and (b) CPU time (continues on next page)

(a)



(b)

Figure 11 Comparison of iteration number and CPU time for Re = 50 with Rscv < 1 × 10–8 in Problem 2: (a) Iteration number and (b) CPU time

(a)

(b)

Figure 12 Ratios of iteration numbers and CPU time for Problem 2: (a) Re = 50, Rscv < 5 × 10–8; (b) Re = 50, Rscv < 1 × 10–8 and (c) Re = 150, Rscv < 5 × 10–8

(a)

(b)

(c)

Problem 3: Laminar fluid flow in a serpentine rectangular channel.

The configuration is presented in Figure 13, where L1/H1 = 5, L1/L2 = 10, H1/H2 = 4/3 and H2 = H3. This example is calculated for Re = 50 and 100, which is defined as

in 1Re / .u H ν= (15)


Figure 13 Laminar fluid flow in a serpentine rectangular channel

The inlet velocity distribution is assumed to be uniform and the local one-way method is adopted to the outflow boundary treatment (Tao, 2001; Patankar, 1980).

Computations are performed on 202 × 42 grid system. The pressure relaxation factor is 1.0. The predicted streams are presented in Figure 14.

Figure 14 Predicted stream function around multi-cover plates in a rectangular area in Problem 3: (a) Re = 50 and (b) Re = 100

(a)

(b)

It can be seen from Figures 15–18 that the new algorithm performance is far superior to that of the SIMPLER algorithm and SIMPLEC algorithm in the range of convergence. The ratios are as follows:

For Re = 50, Rscv ≤ 10–7



For Re = 100, Rscv ≤ 10–7

The ratio of CLEAR over SIMPLER: from 26% to 83% (ITER), from 31% to 102% (CPU time).


For Re = 100, Rscv ≤ 10–8

The ratio of CLEAR over SIMPLER: from 25% to 80% (ITER), from 30% to 97% (CPU time).


The maximum saving in CPU time is up to 82% for SIMPLEC and 70% for SIMPLER, respectively.

As can be seen from Figures 15–18, for the problem studied, the ranking of robustness is different from Problems 1 and 2, and for this example, SIMPLER is the best, followed by SIMPLEC, and CLEAR.


(a)

(b)

Figure 16 Comparison of iteration number and CPU time for Re = 100 in Problem 3: (a) Iteration number and (b) CPU time (continues on next page)

(a)



(b)

Figure 17 Comparison of iteration number and CPU time for Re = 100 with Rscv ≤ 1 × 10–8 in Problem 3: (a) Iteration number and (b) CPU time

(a)

(b)

Figure 18 Ratios of iteration numbers and CPU time for Problem 3: (a) Re = 50, Rscv < 1 × 10–7; (b) Re = 100, Rscv < 1 × 10–7 and (c) Re = 100, Rscv < 1 × 10–8

(a)

(b)

(c)


Problem 4: Natural convection heat transfer in a square cavity with isolated horizontal plate.

The geomerty is shown in Figure 19. The square cavity has two adiabatic walls (top and bottom), with its two vertical walls being maintained at constant hot temperature. A horizontal plate with cold temperature is placed in the upper part of the cavity. The geometric parameters are taken as: L1 = H1, L1/L2 = 10/3, L1/L3 = 5/2, H1/H2 = 10/7, H1/H3 = 10. Computations are performed for Ra = 104 and 106 based on the Boussinesq approximation. The Rayleigh number is defined by

3

Ra .g Tρ βδαµ

∆= (16)

Figure 19 Natural convection heat transfer in a square cavity with an isolated horizontal plate

Figure 20 Predicted streamlines in Problem 4: (a) Ra = 104 and (b) Ra = 106

(a)

(b)

A uniform grid of 102 × 102 is applied. The isolated rectangular cold region of the solid is treated as an isolated island by the same method referenced in Tao (2001). Pressure relaxation factor and temperature relaxation factor are 1.0 and 0.9, respectively.

Figures 20 and 21 show the numerical results including flow field and temperature field. Figures 22–25 show the comparison results of iteration number and CPU time.

Figure 21 Predicted isotherms in Problem 4: (a) Ra = 104 and (b) Ra = 106

(a)

(b)

Figure 22 Comparison of iteration number and CPU time for Ra = 104 in Problem 4: (a) Iteration number and (b) CPU time (continues on next page)

(a)


Figure 22 Comparison of iteration number and CPU time for Ra = 104 in Problem 4: (a) Iteration number and (b) CPU time (continued)

(b)

Figure 23 Comparison of iteration number and CPU time for Ra = 106 in Problem 4: (a) Iteration number and (b) CPU time

(a)

(b)

Figure 24 Comparison of iteration number and CPU time for Ra = 104 with Rscv ≤ 1 × 10–8 in Problem 4: (a) Iteration number and (b) CPU time

(a)

(b)

Figure 25 Ratios of iteration numbers and CPU time for Problem 4: (a) Ra = 104, Rscv < 5 × 10–8; (b) Ra = 104, Rscv < 10–8 and (c) Ra = 106, Rscv < 5 × 10–8 (continues on next page)

(a)


Figure 25 Ratios of iteration numbers and CPU time for Problem 4: (a) Ra = 104, Rscv < 5 × 10–8; (b) Ra = 104, Rscv < 10–8 and (c) Ra = 106, Rscv < 5 × 10–8 (continued)

(b)

(c)

Obviously, the performance of the CLEAR is much better than that of the SIMPLER and SIMPLEC in the region of lower time step multiple E (E ≤ 1), i.e., the lower underrelaxation factor α region α ≤ 0.5. Especially in the very low underrelaxation factor region, the convergence rate of CLEAR may be as large as five times than that of SIMPLER and SIMPLEC. These numerical results are consistent with that reported in Tao et al. (2004b), where it was also found that the CLEAR algorithm is more efficient for natural convection problems.

The comparison ratios are as follows:

For Ra = 104, Rscv ≤ 5 × 10–8

Comparing with the SIMPLER: from 25% to 88% (ITER), from 32% to 111% (CPU time).

Comparing with the SIMPLEC: from 23% to 71% (ITER), from 31% to 93% (CPU time).

For Ra = 106, Rscv ≤ 5 × 10–8



For Ra = 104, Rscv ≤ 10–8



The maximum saving in CPU time is up to 77% for SIMPLEC and 77% for SIMPLER, respectively.

From Figures 22–25, it can be observed that the worse robustness of CLEAR limits its application in the higher underrelaxation region. For the problem studied, the convergence rate and robustness of SIMPLER is better than SIMPLEC.

In the above four examples, it can be found that, generally speaking, the CLEAR algorithm can greatly improve the convergence rate of the iterative process when compared with the SIMPLER and SIMPLEC algorithm for fluid flow and heat transfer problems in complicated geometries. It should be noted that, compared with the relatively simple examples shown in Tao et al. (2004a, 2004b), the weakness of CLEAR in its robustness is revealed more clearly for complicated geometries. Generally speaking, this will not affect the application of the CLEAR algorithm. The first reason for this is that in the CLEAR algorithm, a larger second relaxation factor β could improve the convergence characteristics as indicated in Tao et al. (2004a, 2004b). The second reason is that the convergence rate with CLEAR is not as sensitive to the relaxation factor α as the SIMPLEC and SIMPLER algorithms. The third reason is that a smaller relaxation factor α is often adopted to ensure iteration convergence in complex fluid flow and heat transfer problems, and the working range of the underrelaxation factor of CLEAR still covers the major part used in engineering computations. In addition, a stricter convergence condition is not the key factor affecting the advantage of CLEAR algorithm. For the four tested problems, the minimal iteration numbers for CLEAR are about 18% of the SIMPLER algorithm and 17% of the SIMPLEC algorithm; the minimal ratio of CPU time is about 23% in both cases. It should be noted that during the execution of the present project, the authors conducted numerical comparisons for more examples than presented in this paper. Since the conclusions are more or less the same, for the simplicity of presentation, the other examples are not shown in this paper.

However, the reduced robustness of CLEAR is still an unpleasant feature, and it is useful to understand the reason for this. The present authors consider that there are maybe two basic reasons. One comes from its fully implicit character. The fully implicit feature makes a larger variation


between two successive solutions in the iteration procedure, and for the iterative solution procedure of non-linear problems, too much variation between the successive solutions may lead to divergence. This characteristic was known to the authors of Tao et al. (2004a, 2004b) and the second underrelaxation factor was therefore proposed. By changing the second relaxation factor, the variation range of the (first) relaxation factor for the convergence solution can be extended to some extent. The second reason also comes from its fully implicit character, but exhibited in a different manner. In the CLEAR algorithm, in each iteration, the improved velocity is directly solved from the improved pressure, which in turn is solved from the intermediate velocity of each iteration. In the SIMPLER and SIMPLEC algorithms, the improved velocity is obtained by adding a small amount of correction. Therefore, the effect of any numerical fluctuation (or noise) on the improved velocity in CLEAR will be much more severe than that in SIMPLER or SIMPLEC. The accumulation of such numerical fluctuation may lead to inferior robustness. Therefore, the present authors believe that it is valuable to pay more attention to this problem for further improvement of CLEAR.

It is worth noting that the extension of CLEAR algorithm to the collocated grid system and Qu et al. (2005) completed the comparisons between different algorithms, and similar conclusions were obtained.

5 Conclusion In this paper, comprehensive numerical experiments with complex geometry have been conducted for the CLEAR, SIMPLEC, and SIMPLER algorithms. Numerical experiments generally confirm that the CLEAR algorithm can significantly enhance the convergence rate of the iteration process compared with the SIMPLE family algorithms. For the four complex problems tested, compared with SIMPLEC and SIMPLER algorithm, the CLEAR algorithm can reduce the number of iterations to converge by typically 82%, and the CPU time by 77%. Although the robustness of the CLEAR algorithm is worse than that of SIMPLER and SIMPLEC, the working region of the underrelaxation factor of CLEAR still covers most of the range used in engineering computations. Therefore, it is considered that this will not affect its engineering application.

The full coupling between velocity and pressure of CLEAR results in its weaker robustness, and a discussion of this issue is presented in the paper. Further improvement and the extension of the CLEAR algorithm are now underway in the authors’ group.

Acknowledgement This study was supported by the National Natural Science Foundation of China (Grant Nos. 50656030, 50425620).

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