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International Journal on Architectural Science, Volume 4, Number 1, p.24-35, 2003

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COMPARISON OF FOUR ALGORITHMS FOR SOLVING PRESSURE-VELOCITY LINKED EQUATIONS IN SIMULATING ATRIUM FIRE R. Yin and W.K. Chow Department of Building Service Engineering, The Hong Kong Polytechnic University, Hong Kong, China (Received 14 January 2003; Accepted 25 February 2003) ABSTRACT Four algorithms for solving the pressure-velocity linked equation were compared by simulating the pressure distribution in an atrium fire. The four algorithms studied were the SIMPLE, the SIMPLER, the SIMPLEC and the PISO. Four heat release rates of fire in an arbitrary atrium were considered. Different under-relaxation factors for velocity components were assigned. It is found that the flow variables predicted by the four algorithms are the same, though the pressure distributions are quite different. Two sets of predicted pressure were found. The SIMPLE/SIMPLEC/PISO resulted in one set and the SIMPLER resulted in another. Macroscopic parameters useful to the fire industry were also computed. 1. INTRODUCTION Studying pressure distribution induced by fire in an atrium is very important in understanding the smoke filling process [e.g. 1]. Accurate prediction on that enables the design of good smoke control systems as it helps in determining the dimensions and positions of the vents for natural ventilation, sizing the smoke extraction fans for mechanical ventilation, and estimating the pressure level required to pressurize the adjacent area. It is difficult and expensive to carry out full-scale burning tests. Fire zone models [e.g. 2,3], though a popular design tool, are not supposed to predict the pressure distribution in the compartment. In fact, constant compartment pressure was assumed in some fire zone models. Applying Computational Fluid Dynamics (CFD) (known as the CFD/Field model) is a possible solution to understand the pressure distribution in an atrium fire [e.g. 1]. The CFD/Field model [4] for fire studies (to some extent, indoor aerodynamics of buildings as well [5]) has three main characteristics: The k-ε model is commonly used to calculate

the average flow variables induced by a fire. The finite control volume method is used to

discretize the set of conservation equations. As there is no explicit equation on pressure,

algorithms for solving the pressure-velocity linked equation is required.

Questions are usually raised on the above three characteristics in fire studies. Preliminary comparison of turbulence models and algorithms for solving pressure-velocity linked equation have been made [6,7]. However, results are not yet convincing to recommend a suitable turbulence

model nor a numerical scheme for fire simulations. This paper compares different algorithms for solving the pressure-velocity linked equation in simulating fire-induced air flow. The findings of which would help to recommend a suitable algorithm for calculating the pressure distribution induced by an atrium fire using the CFD/Field model. The SIMPLE (Semi-Implicit Method for Pressure Linked Equations) type of algorithm [8] is commonly employed in CFD studies. The method was proposed years ago [8] but is still being used for simulating practical problems nowadays. In this paper, the SIMPLE, the SIMPLER (SIMPLE Revised) [9], the SIMPLEC (SIMPLE Consistent) [10] and the PISO (Pressure-Implicit with Splitting of Operators) [11] algorithms were tested for studying the fire-induced air flow in an atrium in order to get a better understanding of the pressure distribution. In the past two decades, there have been reports on comparing them on simulating different problems of fluid flow [e.g. 10,12], but not on computing fire-induced air flow. 2. GOVERNING EQUATIONS Steady-state two-dimensional simulations were performed to save CPU computing time. The equation describing the conservation of flow variables φ such as momentum and energy is:

φφφ +∂∂φ

Γ∂∂

+∂∂φ

Γ∂∂

=φρ∂∂

+φρ∂∂ S)

y(

y)

x(

x)v(

y)u(

x (1)

International Journal on Architectural Science

25

The finite difference control volume method with Centered Grid Nodes was used to discretize equation (1) to give:

φφφ +φ=φ ∑ Ui

iiPP Saa (2)

The coefficient φPa at node P (Fig. 1 refers) for φ is

expressed in terms of the coefficient at neighbourhood φ

ia (i = E, W, N, S) and φPS as:

φ

=

φφ −= ∑ PS,N,W,EiiP Saa (3)

Detailed derivation of the finite difference equation, application of boundary conditions and the treatment of non-uniform mesh have been discussed elsewhere [e.g. 4,13,14] and would not be repeated here. The CFD software concerned is also described in the literature [15]. An under-relaxation factor α was assigned to ensure convergence:

0PPU

iiiP

P a1Saa

φαα−

++φ=φα

φφφφ

∑ (4)

where 0Pφ is the value of φP from the previous step.

3. PRESSURE-VELOCITY LINKED

EQUATION The main objective of this paper is to compare four algorithms for solving the pressure-velocity linked equation. The four algorithms are the SIMPLE, the SIMPLER, the SIMPLEC and the PISO.

The u-momentum equation for the control volume centered at e as in Fig. 1 is:

eEPuUnbnbee A)PP(Suaua ⋅−++=∑ (5)

Pressure distribution P* was estimated from the velocities u* obtained by solving the u-momentum equation (5):

e*E

*P

uU

*nbnb

*ee A)PP(Suaua ⋅−++=∑ (6)

Velocity u given by equation (6) would not satisfy conservation of mass unless a correct pressure field is employed. Velocity and pressure fields are corrected by adding u′ and P′: P′ = P – P* (7) and u′ = u – u* (8) Substituting equation (6) into (5) would relate P′ to u′:

e'E

'P

'nbnb

'ee A)PP(uaua ⋅−+= ∑ (9)

The pressure P and velocity u satisfying both the mass and momentum constraints are u = u* + u′ (10) P = P* + P′ (11) Attention is now paid to the method for solving P′ using the four algorithms listed in Appendices A to D, with a flow chart shown in Fig. 2.

u

vv(i,j)

u(i,j)P(i,j)k(i,j)ε(i,j)T(i,j)

Controlvolume for v

Controlvolume for u

Grid pointat (i,j)

PW

S

E

N

we

s

n


26

Fig. 1: Control volume and staggered grid system

SIMPLEC Algorithm?

Yes

Start

Estimate P*, u*, v*, φ*

SIMPLER Algorithm?

Calculate pseudo-velocities using equation (B2)

Solve pressure P using equation (B4)

Solve momentum equation using equation (6)

No

Solve pressure correction using equation (A7)

Yes

Yes

Update pressure using equation (11)

Calculate coefficient of equations

Update velocity using equation (C2)

SIMPLER Algorithm?

Yes

Update velocity using equation (10)

PISO Algorithm?

YesSet P** = P, u** = u, v** = v

Solve second pressure correction using equation (D3)

Update pressure using equation (D2)

Solve other variables φ equations

No

No

No

Convergence?

End

Set u* = u v* = v P* = P φ* = φ

No


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Fig. 2: Summary of the four algorithms 4. NUMERICAL EXPERIMENTS Numerical experiments were performed in a section of an atrium of length 13.5 m and height 9 m as shown in Fig. 3. A heat source of length 1 m and height 0.5 m was located at the center of the atrium floor with fire of 4 heat release rates: 0.5 MWm-1, 1 MWm-1, 1.5 MWm-1 and 2 MWm-1. There was a 2 m high soffit at the atrium ceiling boundary. The atrium was divided into 32 parts along the x-direction and 32 parts along the y-direction with a non-uniform grid system. Under-relaxation factors used in solving the equations are listed in Table 1.

The discretized equations for momentum and other flow variables φ were solved by using the TDMA method. The number of iterations required to solve the equations was difficult to determine. There were some recommendations [10] for judging when to terminate the iteration process in solving problems in which momentum equations are not coupled to a scalar variable. However, those methods are not suitable for fire simulations because the air velocity depends on the local density and temperature. In this paper, the number of iterations was set to 5 for solving the corrected pressure equation and the temperature equation; and it was set to 1 for solving the other equations.

Table 1: The under-relaxation factors for flow variables

Flow variables

Turbulence kinetic energy αk

Turbulence energy

dissipation αε

TemperatureαT

Density αρ

Turbulence viscosity αη

Pressure αP

Under-relaxation

factor

0.7

0.7

0.5

1.0

0.6

0.7

9 parts2 parts9 parts 10 parts

5 parts

12 parts

4 parts

9 parts

9 m 2 m

2 m

9 m

4 m

13.5 m

Free boundary

Heat source

Fig. 3: Geometry for numerical experiments


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Even an Intel Pentium 166 MHz personal computer is capable of carrying out two-dimensional simulations. The associated CPU time required for simulation with different algorithms were recorded. The convergence criterion was 7 × 10-4 and the maximum sweep was 8000. That means computing would stop at 8000 sweeps even when the maximum residential errors do not satisfy the convergence criterion. Most cases satisfied the convergence criterion within 4000 sweeps. The additional 4000 sweeps were demonstrated to be useless. A summary of the number of steps and CPU time required for the four algorithms with different relaxation factors for velocities αu and αv is shown in Table 2. Results of velocity vectors, pressure and temperature contours are shown in Figs. 4 to 6. Predicted results on the flow variables using the four different schemes were basically the same except for the pressure field. The maximum deviation for velocity, temperature, turbulent kinetic energy k and turbulent energy dissipation ε computed from the four algorithms was less than 0.3%. However, a large deviation was found for the predicted pressure field. It is observed that using the schemes SIMPLE, SIMPLEC and PISO gave a set of similar results; whereas using the SIMPLER gave another set as shown in Figs. 5a and 5b. Predictions were repeated for heat release rate of 500 kWm-1 to check the convergence criteria and the number of steps in the TDMA methods. The

under-relaxation factors of the velocity component αu and αv were set to 0.5; the number of iterations in the TDMA method in calculating flow variables was set to 50; and the number of sweeps was set to 10000. The predicted results were roughly the same as those predicted from the above numerical conditions, with a maximum deviation less than 0.1%. Further, changing the number of iterations in the TDMA method would not decrease the total number of sweeps, but it would require longer CPU time and would even give divergent results. A possible reason is because it is a buoyancy-driven air flow problem. Setting the initial conditions to static situation by assigning all flow variables equal to the ambient conditions would not give converged results by increasing only the number of iterations in the TDMA without correcting the temperature field. 5. PRESSURE DISTRIBUTION Although the pressure Pij predicted at the node (i,j) from these four algorithms were quite different, the pressure differences ∆P between adjacent cells were quite the same. For the heat release rate of 1 MWm-1, values of the pressure Pi,j at nodes (10, 20), (10, 21) and (9, 20) computed by using the SIMPLE, SIMPLEC and PISO are: P10,20 = 11.94 Pa; P10,21 = 13.83 Pa; and P9,20 = 12.71 Pa

0 5 10 15x/m

0

3

6

9

y/m

Fig. 4 Velocity vectors(with heat release rate 1MWm

-1)

5 ms-1

Fig. 4: Velocity vectors (with heat release rate 1 MWm-1)

x / m

y / m

5 ms-1


29


30

Fig. 6: Temperature contours (with heat release rate 1 MWm-1) The pressure differences ∆P1 and ∆P2 among the nodes (10, 20) and (10, 21); (9, 20) and (10, 20) are: ∆P1 = P10,20 − P20,21 = −1.89 Pa ∆P2 = P9,20 − P10,20 = 0.77 Pa Using the SIMPLER, values of the pressure at nodes (10, 20), (10, 21) and (9, 20) are: P10,20 = 13.81 Pa; P10,21 = 16.53 Pa; and P9,20 = 14.56 Pa The corresponding pressure differences are: ∆P1 = P10,20 − P10,21 = −1.88 Pa ∆P2 = P9,20 − P10,20 = 0.75 Pa It is observed that the pressure differences ∆P1 and ∆P2 predicted from the four algorithms are similar. A possible reason is the method used in calculating the pressure field in the SIMPLER algorithm is different from the other three algorithms. For the SIMPLE/ SIMPLEC/PISO, the pressure field is estimated first, then corrected by the pressure-correction equations in the next iteration. Pressure-correction is not needed in SIMPLER since pressure is calculated directly from velocity field.

To solve the equations at the free boundary, instead of setting the pressure on the free boundary to certain values (such as setting directly to 0 Pa), coefficients Sp

p and Sup in the discretized pressure-

correction equations given by (2) and (3) are assigned as: Sp

p = −1030

Snp = 0.0

Pressure will not be corrected under free boundary conditions. If the velocity field is calculated from the mass conservation equation, results might not satisfy the momentum conservation equation. The pressure calculated at those points using the momentum conservation equation (for SIMPLER) would be different from the pressure estimated from the mass conservation equation (for SIMPLE/SIMPLEC/PISO). Inside the computing domain, velocities calculated by different methods are the same, giving similar pressure differences. 6. ERROR ANALYSIS Three kinds of errors should be considered [e.g. 16]: error in original data, truncation error and round-off error.

0 3 6 9 12 0

3

6

9

50oC

100oC

150oC

200oC

300oC

x / m

y / m


31

Error in original data arose because the initial conditions, particular values of k and ε have to be estimated. The solution would be changed depending on the sensitivity of the equation towards those parameters, and whether the errors can be reduced by iterative methods. If the equations are too sensitive to those parameters, divergent solution would be resulted. Under-relaxation method might improve the situation. Truncation error is due to truncating some terms in discretizing the equations. First order dis-cretization methods are applied in this study and so truncation errors will be expected. However, results are useful in estimating macroscopic parameters. Round-off error comes from floating-point arithmetic because of the fixed word length in the computer. A real number cannot be expressed exactly because of the hardware limitation and the associated errors are grouped under round-off error. The round-off error is very small initially, but it becomes bigger after several iterations. Finally, the result may be deviated from the expected values. This type of error is especially important when subtracting two quantities of similar magnitude. Decreasing the under-relaxation factors would improve the situation and so selecting suitable under-relaxation factors is very important. Overflow or ‘not converged’ results (labelled as NC) were encountered in cases listed in Table 2. By trial and error, 8000 sweeps was sufficient to reduce error to a value less than the defined residual errors, provided that the round-off errors were small enough. 7. COMPARISON OF THE FOUR

ALGORITHMS Different values of the under-relaxation factor for the momentum equation αu and αv were tested using the four algorithms for the four heat release rates, and are shown in Table 2. Longer CPU time was required for cases with smaller under-relaxation factors (up to 506 s for SIMPLER with αv and αv of 0.2). The followings are observed: Results predicted by using the SIMPLE

algorithm were unstable, especially for large value under-relaxation factors but it took roughly the same CPU time as the other three algorithms.

Using the SIMPLEC algorithm would not reduce the CPU time. This observation is different from the conclusions reported by Van Doormaal and Raithby [10]. Results predicted by this scheme were slightly more stable than those by using the SIMPLE.

Very stable results were predicted by using the SIMPLER algorithm, but simulations with this scheme took longer CPU time.

Results were also stable by using the PISO algorithm. In fact, this scheme is found to be much better than the other three and it took less CPU time.

The under-relaxation factors αu and αv cannot be too big nor too small. However, it is possible to use larger values of αu and αv for the schemes SIMPLER and PISO as shown in Table 2. 8. MACROSCOPIC PARAMETERS Macroscopic parameters useful to the fire industry were calculated from the predicted flow variables. These included the intake air flow rate Fin, hot air out flowing rate Fout, average air temperature of the atrium Tav and maximum height hmax of the neutral plane (0 Pa pressure with respect to the ambient). The air intake flow rate Fin was calculated from the horizontal incoming air speed ui

in and density ρi of the ith control volume at the opening (i.e. x = 9 m):

Fin = ∑ρi

iniiu (12)

The outflowing rate Fout of hot air was calculated in a similar manner using the outflowing air speed ui

out:

Fout = ∑ρi

outiiu (13)

The average temperature Tav was found from the temperature Ti and the area Ai of the ith control volume:

∑∑

=

ii

iii

av A

ATT (14)

The maximum neutral plane height hmax was judged from the predicted pressure distribution and measured from the atrium floor. The macroscopic parameters for the four heat release rates were calculated and shown in Table 3.


32

In general, the inflow rate and average temperature of the atrium increased as the heat release rate increased. However, the neutral plane height decreased to the floor level when the heat release

rate increased. This point is very important in understanding the smoke filling process in the atrium.

Table 3: Macroscopic parameters

Maximum Height of Neutral Plane hmax/m Heat release

rate /MWm-1

Inflow rate at opening

Fin/kgs-1m-1

Outflow rate at opening

Fout/kgs-1m-1

Average temperature

of atrium Tav/°C

SIMPLE/ SIMPLEC and

PISO SIMPLER

0.5 -6.88 6.89 96 1.58 2.50 1.0 -8.10 8.10 145 1.21 1.92 1.5 -8.73 8.72 205 1.06 1.60 2.0 -9.13 9.12 248 0.96 1.43

9. CONCLUSIONS Numerical experiments in an atrium with four heat release rates were performed to test four algorithms for solving the pressure-velocity linked equation. The four algorithms are the SIMPLE, the SIMPLER, the SIMPLEC and the PISO. The under-relaxation factors were tested by applying each scheme. A suitable algorithm is recommended for solving the pressure-velocity linked equation in atrium fire simulations. It is found that all the four schemes gave the same results on flow variables except for pressure. The SIMPLE, the SIMPLEC and the PISO gave similar pressure predictions but the SIMPLER gave another set of results. Further, the PISO gave more stable results and took less CPU time. Therefore, it can be concluded that both the PISO and the SIMPLER are suitable numerical schemes for solving the pressure-velocity linked equation in atrium fire simulation. ACKNOWLEDGMENT The paper is jointly funded by a PolyU-ASD account A-038 and B-Q063 from Research Grants Council of Hong Kong. REFERENCES 1. W.K. Chow, “Simulation of fire environment for

linear atria in Hong Kong”, ASCE Journal of Architectural Engineering, Vol. 3, No. 2, pp. 80-88 (1997).

2. J. Quintiere, “Fundamentals of enclosure fire ‘zone’ models, Journal of Fire Protection Engineering”, Vol. 1, No. 2, pp. 99-119 (1989).

3. G.P. Forney and W.F. Moss, “Analyzing and exploiting numerical characteristics of zone fire models”, NISTIR 4763, National Institute of Standards and Technology, Gaithersburg, Maryland, U.S.A. (1992).

4. H.K. Versteeg and W. Malalasekera, An introduction to computational fluid dynamics - The finite volume method, Longman, Essex (1995).

5. D. Etheridge and M. Sandberg, Building ventilation: Theory and measurement, Wiley & Sons, New York (1996).

6. W.K. Chow and W.K. Mok, “On the simulation of forced-ventilation fires”, Numerical Heat Transfer, Part A: Applications, Vol. 28, pp. 321-338 (1995).

7. W.K. Chow and Y.L. Cheung, “Comparison of the algorithms PISO and SIMPLER for solving pressure-velocity linked equations in simulating compartmental fire”, Numerical Heat Transfer, Part A: Applications, Vol. 31, No. 1, pp. 87-112 (1997).

8. S.V. Patankar and D.B. Spalding, “A calculation procedure for heat, mass and momentum transfer in three-dimensional parabolic flows”, International Journal of Heat and Mass Transfer, Vol. 15, pp. 1787-1806 (1971).

9. S.V. Patankar, “A calculation procedure for two dimensional elliptic situations”, Numerical Heat Transfer, Vol. 14, pp. 409-425 (1984).

10. J.P. Van Doormaal and G.D. Raithby, “Enhancements of the SIMPLE method for predicting incompressible fluid flows”, Numerical Heat Transfer, Vol. 7, pp. 147-163 (1984).

11. R.I. Issa and A.D. Gosman and A.P. Watkins, “The computation of compressible and incompressible recirculating flows by a non-iterative implicit scheme”, Journal of Computational Physics, Vol. 62, pp. 66-82 (1986).

12. D.S. Jang, R. Jetli and S. Acharya, “Comparison of the PISO, SIMPLER, and SIMPLEC algorithms for the treatment of the pressure-velocity coupling


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in steady flow problems”, Numerical Heat Transfer, Vol. 10, pp. 209-228 (1986).

13. W.K. Chow and N.K. Fong, “Application of field modelling technique to simulate interaction of sprinkler and fire-induced smoke layer”, Combustion Science and Technology, Vol. 89, pp. 101-151 (1993).

14. W.K. Chow and W.M. Leung, “A short note on achieving convergent results in simulating building fire using the k-ε turbulent model”, Numerical Heat Transfer, Part A, Vol. 17, pp. 495-501 (1990).

15. R. Yin and W.K. Chow, “Studies on thermal responses of sprinkler heads in atrium buildings with fire field models”, Fire and Materials, Vol. 25, No. 1, pp. 13-19 (2001).

16. C.F. Gerald and P.O. Wheatley, Applied numerical analysis, 5th edition, Addison Wiley, New York (1994).

APPENDIX A: REVIEW OF THE SIMPLE ALGORITHM The exact equation for P′ derived from equation (9) is not suitable for quick calculation. In the SIMPLE procedure, eu′ is given by:

2e1ee uuu ′+′=′ (A1)

21 PPP ′+′=′ (A2) where

e

e1E1P1e a

A)PP(u ⋅′−′=′ (A3)

e

e2E2P

e

nbnb2e a

A)PP(

aua

u ⋅′−′+=′ ∑ (A4)

First order approximation would give zero value of

2eu ′ :

)PP(duu EPe*ee ′−′+= (A5)

where

e

ee a

Ad = (A6)

Putting equation (A5) into the continuity equation gives:

pUSSNNWWEEPP SPaPaPaPaPa +′+′+′+′=′ (A7)

where aE = (ρAd)E ; aW = (ρAd)W ; aS = (ρAd)S ; aN = (ρAd)N (A8) aP = aE + aW+ aN + aS (A9)

pUS = (ρu*A)w − (ρu*A)e + (ρv*A)s − (ρu*A)n (A10)

Values of P′ might be too large in this approximation, giving slow convergence rate or even divergence would be resulted during computation. An under-relaxation factor α is employed in the momentum and pressure equations to avoid that. Summary of the SIMPLE procedure: 1. Estimate a pressure field P*. 2. Evaluate the coefficients of the momentum

equations given by equation (6) and solve for u* and v*.

3. Evaluate the source term given by equation (A10) and solve for P′ by equation (A7).

4. Correct the velocity field given by equation (10) and pressure field given by equation (11).

5. Solve for other φ equations and update the coefficients.

6. Take the corrected P as new P*, repeat step (2) to step (6) until convergence.

APPENDIX B: REVIEW OF THE SIMPLER ALGORITHM Using the SIMPLER algorithm, the pressure field is calculated from the velocity field which is also corrected. The momentum equation is expressed as:

)PP(aA

a

Suau EP

e

e

e

uUnbnb

e −++

= ∑ (B1)

The pseudo-velocity eu~ is defined as:

e

uUnbnb

e aSua

u~ ∑ += (B2)

Putting equation (B2) into equation (B1) gives:

)PP(du~u EWeee −+= (B3)

(A8)


34

Substituting the above equation into the continuity equation, the pressure equation can be derived from equation (A7):

PUSSNNWWEEPP SPaPaPaPaPa ++++= (B4)

where aP = (ρAd)E ; aW = (ρAd)W ; aS = (ρAd)S ; aN = (ρAd)N (B5) aP = aE + aW + aN + aS (B6)

nsewPU )Av~()Av~()Au~()Au~(S ρ−ρ+ρ−ρ= (B7)

The coefficients of equation (B5) are the same as those in the pressure correction equation (A7). However, the source term is evaluated using the pseudo-velocities. The SIMPLER procedure is summarized as: 1. Evaluate the coefficients of the momentum

equations given by equation (6). 2. Calculate pseudo-velocities using equation

(B2). 3. Solve for pressure field P* given by equation

(B4). 4. Solve for u* and v*. 5. Evaluate the source term in equation (B7) and

solve for P′ given by equation (A7). 6. Correct the velocity field given by equation

(10). 7. Solve for other φ equations; update the

coefficients. 8. Repeat step (1) to step (8) until convergence. APPENDIX C: REVIEW OF THE SIMPLEC ALGORITHM The term ∑ ′nbnbua in equation (9) is neglected in the SIMPLE algorithm. To introduce a “consistent” approximation, the term ∑ ′enbua is subtracted from both sides of the equation to give:

e'E

'P

'e

'nbnb

'enbe A)PP()uu(au)aa( ⋅−+−=− ∑∑

(C1) In the SIMPLEC approximation, the term

)uu(a enbnb∑ ′−′ in equation (C1) is neglected. The velocity correction equations is changed to:

)PP(duu EPe*ee ′−′+= (C2)

where

∑−=

nbe

ee aa

Ad (C3)

However, P′ should not be under-relaxed. APPENDIX D: REVIEW OF THE PISO ALGORITHM The major difference between the SIMPLE and the PISO is on the pressure field correction. In the SIMPLE, only one correction is used to update the pressure field. However, two corrections are introduced in the PISO. After correcting the velocity and pressure fields, equations (10) and (11) become:

2**

21** uuuuuuuu ′+=′+′+=′+= (D1)

2

**21

** PPPPPPPP ′+=′+′+=′+= (D2) Substituting the above equations into the discretized momentum equations and the mass equation. The equation for the second pressure correction is:

bPaPaPaPaPa SSNNWWEEPP +′′+′′+′′+′′=′′ (D3) where aE = (ρAd)E ; aW = (ρAd)W ; aS = (ρAd)S ; aN = (ρAd)N (D4) aP = aE + aW + aN + aS (D5)

∑∑

∑∑′ρ

−′ρ

+′ρ−′ρ

=

nbnbNnbnbS

nbnbEnbnbW

va)aA(va)

aA(

ua)aA(ua)

aA(b

(D6)

Applying the above method, the second correction pressure field can be calculated from equation (D2).

(B5)

(D4)


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