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Numerical Heat Transfer, Part B:FundamentalsPublication details, including instructions for authors andsubscription information:http://www.tandfonline.com/loi/unhb20

Implementation of the IDEAL Algorithmon Nonorthogonal CurvilinearCoordinates for the Solution of 3-DIncompressible Fluid Flow and HeatTransfer ProblemsDong-Liang Son a b , Wen-Quan Tao b , Jin-Liang Xu a & Zhi-Guo Qu ba Beijing Key Laboratory of New and Renewable Energy, North ChinaElectric Power University, Beijing, People's Republic of Chinab State Key Laboratory of Multiphase Flow in Power Engineering,School of Energy and Power Engineering, Xi'an Jiaotong University,Xi'an, Shaanxi, People's Republic of China

Available online: 02 Feb 2011

To cite this article: Dong-Liang Son, Wen-Quan Tao, Jin-Liang Xu & Zhi-Guo Qu (2011):Implementation of the IDEAL Algorithm on Nonorthogonal Curvilinear Coordinates for the Solutionof 3-D Incompressible Fluid Flow and Heat Transfer Problems, Numerical Heat Transfer, Part B:Fundamentals, 59:2, 147-168

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IMPLEMENTATION OF THE IDEAL ALGORITHM ONNONORTHOGONAL CURVILINEAR COORDINATESFOR THE SOLUTION OF 3-D INCOMPRESSIBLE FLUIDFLOW AND HEAT TRANSFER PROBLEMS

Dong-Liang Son1,2, Wen-Quan Tao2, Jin-Liang Xu1, andZhi-Guo Qu21Beijing Key Laboratory of New and Renewable Energy, North ChinaElectric Power University, Beijing, People’s Republic of China2State Key Laboratory of Multiphase Flow in Power Engineering, School ofEnergy and Power Engineering, Xi’an Jiaotong University, Xi’an, Shaanxi,People’s Republic of China

Recently an efficient segregated algorithm for incompressible fluid flow and heat transfer

problems, called IDEAL (Inner Doubly Iterative Efficient Algorithm for Linked Equa-

tions), 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 stab-

ility 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 con-

travariant 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 compari-

son it can be concluded that the IDEAL algorithm is more robust and efficient than the

three other algorithms.

1. INTRODUCTION

Numerical approaches for solving the Navier-Stokes equations can be divided intotwo categories [1, 2]: the density-based and the pressure-based. The pressure-basedapproach, or the primitive-variables approach, was originally developed for solving

Received 28 July 2010; accepted 14 October 2010.

This work was supported by the Key Project of the National Natural Science Foundation of China

(50636050), the National Natural Science Foundation of China (50825603), the National Key Project for

R & D of China (2009CB219801) and the Fundamental Research Funds for the Central Universities

(09MG15).

Address correspondence to Wen-Quan Tao, State Key Laboratory of Multiphase Flow in Power

Engineering, School of Energy and Power Engineering, Xi’an Jiaotong University, 28 Xi’an Ning Road,

Xi’an, Shaanxi 710049, People’s Republic of China. E-mail: wqtao@mail.xjtu.edu.cn

Numerical Heat Transfer, Part B, 59: 147–168, 2011

Copyright # Taylor & Francis Group, LLC

ISSN: 1040-7790 print=1521-0626 online

DOI: 10.1080/10407790.2010.541357

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incompressible fluid flows and has been widely used in computational fluid flow and heattransfer problems. In recent years it has been successfully extended to compressibleflows. Among the pressure-based approaches, the pressure-correction method is themost widely used one because of its simplicity and straightforwardness in implemen-tation and clarity, and its physical reasonableness. The first pressure-correction methodwas the SIMPLE algorithm proposed by Patankar and Spalding in 1972 [3]. The majorapproximations made in the SIMPLE algorithm are [1, 2] that (1) the initial pressurefield and velocity field are assumed independently, thus the coupling between pressureand velocity is neglected, and (2) the effect of the velocity corrections of the neighboringgrids is not considered, to simplify the solution procedure, leading to the semi-implicitnature of the algorithm. The two approximations do not influence the final solutionsif the solution process converges [1, 4], but they do affect the convergence rate andthe stability of the solution [1]. Therefore, since the birth of the SIMPLE algorithm, anumber of modified methods, such as SIMPLER [5], SIMPLEC [6], SIMPLEX [7],PISO [8], CLEAR [9, 10], etc., have been proposed in order to overcome one or bothof the approximations. Unlike other algorithms, in order to obtain an incompressibleflow field which satisfies the mass conservation law, the CLEAR algorithm does notintroduce a pressure-correction term. The algorithm improves the intermediate velocityby solving a pressure equation tomake the algorithm fully implicit, since there no term isomitted in the derivation process. However, the robustness of the CLEAR algorithm issomewhat weakened by solving the pressure equation directly. To overcome this disad-vantage, an improved algorithm, called CLEARER, was proposed in [11]. CLEARERis a combination of CLEAR and SIMPLER, and it can somewhat improve the robust-ness of the algorithm but it makes the algorithm semi-implicit again. However, thesemi-implicit feature in turn may reduce the robustness of the algorithm. In order toovercome the disadvantage of CLEAR while still keep it fully implicit, the IDEALalgorithm (Inner Doubly iterative Efficient Algorithm for Linked equations) was pro-posed in [12, 13]. In that algorithm there exist inner doubly iterative processes for thepressure equation at each iteration level, which almost completely overcomes the twoapproximations in the SIMPLE algorithm. Thus the coupling between velocity andpressure is fully guaranteed, greatly enhancing the convergence rate and the stabilityof the solution process. Recently, the IDEAL algorithm was extended to the staggeredgrid system [14] and the collocated grid system in 3-D Cartesian coordinates [15].

All of the extensions of the IDEAL algorithm mentioned above are implemen-ted in Cartesian coordinates. Thus, to simulate problems in irregular computationdomains, some approximations must be made. Usually, the domain extensionmethod [16] is applied for the irregular computation domain. The disadvantage ofthis method is that there are many stepwise approximations (90� sawteeth) on thecomputation boundary, which differs significantly from the real smoothed irregularboundary and may lead to large calculation errors. To overcome this disadvantage,nonorthogonal body-fitted coordinates are often adopted, in which the coordinatesand the real boundary overlap each other. In this case, collocated grid systems areusually applied for the convenience of code development. In this article, theimplementation method of the IDEAL algorithm will be extended to body-fittedcollocated grid systems. In the following, a brief review on the extension ofSIMPLE-like algorithms to collocated grid systems and related issues will bepresented.

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Implementation of the SIMPLE algorithm on body-fitted collocated gridsystems was proposed by Rhie and Chow [17] in 1983, which will be referred asthe extended SIMPLE algorithm hereafter. In the extended SIMPLE algorithm,the interfacial contravariant velocity is linearly interpolated from the values at themain nodes and an additional pressure gradient correction term is added to the inter-facial contravariant velocity to guarantee the coupling between velocity and press-ure. However, this additional term can demolish the local mass conservation.Since the decoupling issue is more severe than the problem of mass conservation,in the implementation of the SIMPLER algorithm on body-fitted collocated gridsystems [18], almost the same method is used to eliminate the decoupling betweenpressure and velocity, with the initial pressure being solved from a pressure equationaccording to the prespecified velocity field. Based on the extended SIMPLERalgorithm, Qu et al. proposed the SIMPLERM algorithm [19]. In this algorithmthe modified momentum interpolation method (MMIM) [20] is applied to calculatethe interfacial contravariant velocity and to eliminate the decoupling betweenpressure and velocity, which replaces the method used in the extended SIMPLEand SIMPLER algorithms and guarantees the mass conservation accurately. In2007, the CLEAR algorithm was implemented on body-fitted collocated girdsystems [21]. In the extended CLEAR algorithm, the interfacial contravariantvelocity is calculated by MMIM and is improved by solving the pressure equationdirectly. Thus the effects of neighboring velocity corrections and the cross-pressuregradient terms are all taken into account, so that the convergence rate is greatlyenhanced. However, it has been found by numerical experiments that the robustnessof the CLEAR algorithm is somewhat reduced by solving the pressure equationdirectly [11].

In this article, based on the above analysis, the IDEAL algorithm is extendedto body-fitted collocated grid systems in 3-D nonorthogonal curvilinear coordinates.The extended IDEAL algorithm adopts two successful methods: that the interfacialcontravariant velocity is calculated by the modified momentum interpolationmethod (MMIM); and that the interfacial contravariant velocity is improved bysolving the pressure equation directly. At the same time, for comparison, two newalgorithms, SIMPLEM and SIMPLECM, which are developed along the line ofSIMPLERM, are proposed by the present authors. Finally, three 3-D incompressiblefluid flow and heat transfer problems are provided to compare the convergence rateand robustness of IDEAL and three other algorithms (SIMPLEM, SIMPLERM,and SIMPLECM).

2. GOVERNING EQUATIONS

In order to simplify the formulation, an incompressible steady-state fluidflow and heat transfer in three dimensions is taken as example. For the 3-DCartesian coordinates, the governing equation in the physical space can beexpressed as

qðqu/Þqx

þ qðqv/Þqy

þ qðqw/Þqz

¼ qqx

C/q/qx

� �þ qqy

C/q/qy

� �þ qqz

C/q/qz

� �þ R/ ð1Þ

IDEAL ALGORITHM FOR 3-D FLUID FLOW AND HEAT TRANSFER 149

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where / is the general dependent variable, C/ is the nominal diffusion coefficient,and R/ is the source term (see Table 1).

The three curvilinear coordinates n, g, f are related with the three independentvariables of Cartesian coordinates as follows:

x ¼ xðn;g; fÞ ð2Þ

y ¼ yðn;g; fÞ ð3Þ

z ¼ zðn;g; fÞ ð4Þ

Based on Eqs. (2)–(4), the governing equation is converted from the physical space tothe computational space:

qqn

ðqU/Þ þ qqg

ðqV/Þ þ qqf

ðqW/Þ

¼ qqn

vJC/

q/qn

� �þ qqg

bJC/

q/qg

� �þ qqf

cJC/

q/qf

� �þ JS/ þ JS0

/

ð5Þ

In Eq. (5), U, V, and W are contravariant velocities and can be expressed as

U ¼ v1uþ v2vþ v3w ð6Þ

V ¼ b1uþ b2vþ b3w ð7Þ

W ¼ c1uþ c2vþ c3w ð8Þ

The source term S/ in Eq. (5) is converted from the source term R/ inCartesian coordinates, and its expression is as follows:

When

/ ¼ 1; S1 ¼ 0 ð9Þ

When

/ ¼ u; Su ¼ � v1J

qpqn

� b1J

qpqg

� c1J

qpqf

ð10Þ

Table 1. Values of /, C/, and R/

Equation / C/ R/

Continuity equation 1 0 0

u-Component momentum equation u g � qpqx

v-Component momentum equation v g � qpqy

w-Component momentum equation w g � qpqz

Energy equation T k=cp Depending on

individual problem

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When

/ ¼ v; Sv ¼ � v2J

qpqn

� b2J

qpqg

� c2J

qpqf

ð11Þ

When

/ ¼ w; Sw ¼ � v3J

qpqn

� b3J

qpqg

� c3J

qpqf

ð12Þ

In Eq. (5), the source term S0/ is introduced by the grid nonorthogonality and can be

expressed as

S0/ ¼ 1

J

�qqn

�u1

JC/

q/qg

�þ qqn

�u3

JC/

q/qf

�þ qqg

�u1

JC/

q/qn

�

þ qqg

�u2

JC/

q/qf

�þ qqf

�u3

JC/

q/qn

�þ qqf

�u2

JC/

q/qg

��ð13Þ

In Eqs. (5)–(13), parameters v1, v2, v3, b1, b2, b3, c1, c2, c3, u1, u2, u3, v, b, c, andJ are geometry factors. They are defined as

v1 ¼ ðygzf � yfzgÞ v2 ¼ ðxfzg � xgzfÞ v3 ¼ ðxgyf � xfygÞ ð14Þ

b1 ¼ ðyfzn � ynzfÞ b2 ¼ ðxnzf � xfznÞ b3 ¼ ðxfyn � xfyfÞ ð15Þ

c1 ¼ ðynzg � ygznÞ c2 ¼ ðxgzn � xnzgÞ c3 ¼ ðxnyg � xgynÞ ð16Þ

u1 ¼ v1b1 þ v2b2 þ v3b3 u2 ¼ b1c1 þ b2c2 þ b3c3 u3 ¼ v1c1 þ v2c2 þ v3c3ð17Þ

v ¼ v21 þ v22 þ v23 b ¼ b21 þ b22 þ b23 c ¼ c21 þ c22 þ c23 ð18Þ

J ¼ xnygzf þ xgznyf þ ynzgxf � xfygzn � xnyfzg � xgynzf ð19Þ

3. MODIFIED MOMENTUM INTERPOLATION METHOD

In this article, the interfacial contravariant velocity is calculated by the modi-fied momentum interpolation method (MMIM) [20], which replaces the method usedin the extended SIMPLE [17] and SIMPLER [18] algorithms and guarantees themass conservation accurately. For completeness of presentation, the MMIM forthe 3-D case is briefly introduced as follows.

The control volumes in the computational space are shown in Figure 1. (Forsimplicity of presentation, in the figure we give just the 2-D case; the extension tothe third coordinate is very straightforward.) In collocated grid system, the mainnodes are just grid points of control volumes where the variables to be solved arelocated. On the other hand, the contravariant velocity components are located inthe control-volume interfaces for calculating the discretized equation coefficients.

IDEAL ALGORITHM FOR 3-D FLUID FLOW AND HEAT TRANSFER 151

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In the collocated grid systems, the governing equation, Eq. (5), is discretized withthe finite-volume method (FVM) [16]. The final discretized results are expressed asfollows.

Discretized continuity equation:

ðqUÞe DgDf�ðqUÞw DgDfþ ðqVÞn DnDf� ðqVÞs DnDfþ ðqWÞtDnDg� ðqWÞb DnDg ¼ 0 ð20Þ

Discretized momentum equations:

auPau

uP ¼Xnb

aunbunb þ buP þ ð1� auÞauPau

u0P

� v1qpqn

� �P

DnDgDf � b1qpqg

� �P

DnDgDf� c1qpqf

� �P

DnDgDf ð21Þ

avPav

vP ¼Xnb

avnbvnb þ bvP þ ð1� avÞavPav

v0P

� v2qpqn

� �P

DnDgDf� b2qpqg

� �P

DnDgDf� c2qpqf

� �P

DnDgDf ð22Þ

awPaw

wP ¼Xnb

awnbwnb þ bwP þ ð1� awÞawPaw

w0P

� v3qpqn

� �P

DnDgDf� b3qpqg

� �P

DnDgDf� c3qpqf

� �P

DnDgDf ð23Þ

Figure 1. Geometry parameters in the computational space.

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where the superscript 0 represents the value of previous iteration and a is theunderrelaxation factor. The expressions of the coefficients a and sources b of thediscretized equations (21)–(23) depend on the discretized schemes. The detailedderivation process is given in [2, 16].

Equations (21)–(23) are rewritten in the explicit manner

uP ¼ �uup þ ð1� auÞu0P � duP v1

qpqn

� �P

� duP b1

qpqg

� �P

� duP c1

qpqf

� �P

ð24Þ

vP ¼ �vvP þ ð1� avÞv0P � dvP v2

qpqn

� �P

� dvP b2

qpqg

� �P

� dvP c2

qpqf

� �P

ð25Þ

wP ¼ �wwP þ ð1� awÞw0P � dw

P v3qpqn

� �P

� dwP b3

qpqg

� �P

� dwP c3

qpqf

� �P

ð26Þ

where

�uuP ¼

Pnb

aunbunb þ buP

auP=auduP ¼ DnDgDf

auP=auð27Þ

�vvP ¼

Pnb

avnbvnb þ bvP

avP=avdvP ¼ DnDgDf

avP=avð28Þ

�wwP ¼

Pnb

awnbwnb þ bwP

awP=awdwP ¼ DnDgDf

awP=awð29Þ

It should be noted that Eqs. (24)–(26) are of explicit type in that the velocitieson the right-hand side are all known: They take either the values of the previousiteration or the available values of the present iteration.

By mimicking Eq. (24), the following equation can be written for the u velocityat the e interface:

ue ¼ �uue þ ð1� auÞu0e � due v1

qpqn

� �e

� due b1

qpqg

� �e

� due c1

qpqf

� �e

ð30Þ

where �uue; due are linearly interpolated from the corresponding values at neighboring

main nodes:

�uue ¼ f þe �uuP þ ð1� f þe Þ�uuE ð31Þ

due ¼ f þe du

P þ ð1� f þe ÞduE ð32Þ

f þe ¼ dneþdne

ð33Þ

IDEAL ALGORITHM FOR 3-D FLUID FLOW AND HEAT TRANSFER 153

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Similarly, the v and w velocities at the e interface can be written as

ve ¼ �vve þ ð1� avÞv0e � dve v2

qpqn

� �e

� dve b2

qpqg

� �e

� dve c2

qpqf

� �e

ð34Þ

we ¼ �wwe þ ð1� awÞw0e � dw

e v3qpqn

� �e

� dwe b3

qpqg

� �e

� dwe c3

qpqf

� �e

ð35Þ

Then, Eqs. (30), (34), and (35) are substituted into Eq. (6) for the interfacial contra-variant velocity Ue. Under the condition a¼ au¼ av¼ aw, the interfacial contravar-iant velocity Ue can be written as

Ue ¼ Ue þ ð1� aÞU0e � BU

e

qpqn

� �e

�CUe

qpqg

� �e

�DUe

qpqf

� �e

ð36Þ

where

Ue ¼ ðv1Þe�uue þ ðv2Þe�vve þ ðv3Þe�wwe ð37Þ

BUe ¼ ½du

e ðv1v1Þe þ dve ðv2v2Þe þ dw

e ðv3v3Þe� ð38Þ

CUe ¼ ½du

e ðb1v1Þe þ dve ðb2v2Þe þ dw

e ðb3v3Þe� ð39Þ

DUe ¼ ½du

e ðc1v1Þe þ dve ðc2v2Þe þ dw

e ðc3v3Þe� ð40Þ

Equation (36) can be rewritten as

Ue ¼ eUUe � BUe

qpqn

� �e

ð41Þ

where eUUe is the interfacial pseudo contravariant velocity,

eUUe ¼ Ue þ ð1� aÞU0e � CU

e

qp0

qg

� �e

�DUe

qp0

qf

� �e

ð42Þ

Equations (36) and (41) are derived for calculating the interfacial contravariantvelocity Ue by MMIM. Similarly, the interfacial contravariant velocities Vn and Wt

can be obtained by

Vn ¼ eVVn � CVn

qpqg

� �n

ð43Þ

Wt ¼ eWWt �DWt

qpqf

� �t

ð44Þ

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4. SIMPLEM AND SIMPLICM ALGORITHMS

In this article, in order to make algorithm comparison between IDEAL and theSIMPLE-like algorithms, the present authors extended the SIMPLE and SIMPLECalgorithms from Cartesian coordinates to body-fitted collocated grid systems bycalculating the interfacial contravariant velocity via MMIM. Such extension wasconducted in [19] for the SIMPLER algorithm, and the resulting algorithm is calledSIMPLERM. Thus the two newly extended algorithms are named SIMPLEM andSIMPLECM, respectively. Except for the process of calculating the interfacialcontravariant velocity, the solution procedures of SIMPLEM and SIMPLICMalgorithms are the same as for the SIMPLE [3] and SIMPLEC [6] algorithmsin Cartesian coordinates, respectively. Therefore, the solution procedures ofSIMPLEM and SIMPLECM algorithms are not introduced in this article.

5. SOLUTION PROCEDURE OF IDEAL ALGORITHM

In this article, the IDEAL algorithm is extended to body-fitted collocated gridsystems in 3-D nonorthogonal curvilinear coordinates. The extended IDEAL algor-ithm adopts two successful methods: The interfacial contravariant velocity is calcu-lated by the MMIM; the interfacial contravariant velocity is improved by solving thepressure equation directly. Figure 2 shows the framework of the solution process ofthe IDEAL algorithm in detail. The pressure-based solution method is iterative innature. In the following we will often use the term iteration level. By one iterationlevel we mean that all the computations are completed at the same values of the coef-ficients of the discretized momentum equations. In the IDEAL algorithm, at eachiteration level there exist two inner iteration processes, or inner doubly iterative pro-cesses, for pressure field solution. The first inner iteration process for the pressure

Figure 2. Framework of the solution process of the IDEAL algorithm.

IDEAL ALGORITHM FOR 3-D FLUID FLOW AND HEAT TRANSFER 155

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equation can almost completely overcome the first approximation in the SIMPLEalgorithm. The second inner iteration process can almost completely overcome thesecond approximation in the SIMPLE algorithm. The solution procedure of oneiteration level in the IDEAL algorithm is presented as follows.

Step 1. Assume initial nodal velocities u0P; v0P; w0P and initial interfacial contra-

variant velocities U0e ; V 0

n ; W 0t .

Step 2. Calculate the coefficients a and source terms b of the discretized momen-tum Equations (21)–(23) for the u, v, w components by the initial velocity field.

First inner iteration process for pressure question.Step 3. Calculate the interfacial pseudo-contravariant velocities eUU0

e ;eVV 0

n, andeWW 0

t by the MMIM. Taking the calculation of eUU0

e as an example, the calculation pro-cess of the interfacial pseudo-contravariant velocity eUU0

e is as follows. First, �uu0p; �vv0p,

and �ww0p are calculated from Eqs. (27)–(29). Then, �uu0e is linearly interpolated from

the neighboring main nodes by Eq. (31). Similarly, �vv0e and �ww0e can be obtained. Then,

U0

e is calculated by substituting �uu0e ; �vv0e , and �ww0

e into Eq. (37). Finally, eUU0

e can beobtained by Eq. (42).

Step 4. Solve the pressure from the known velocities as follows [Eq. (45)], andobtain the temporary pressure pTemp.

apPap

pTempP ¼

Xapnbp

Tempnb þ bpP

apP ¼ a

pE þ a

pW þ a

pN þ a

pS þ a

pT þ a

pB

apE ¼ qeBUe ðDgDf=dneÞ apW ¼ qwB

Uw ðDgDf=dnwÞ

apN ¼ qnCVn ðDnDf=dgnÞ apS ¼ qsC

Vs ðDnDf=dgsÞ

apT ¼ qtD

Wt ðDnDg=dftÞ a

pB ¼ qbD

Wb ðDnDg=dfbÞ

bpP ¼ qw eUU0

wDgDf� qe eUU0

eDgDfþ qs eVV 0

sDnDf� qn eVV0

nDnDf

þ qb eWW 0

bDnDg� qt eWW 0

tDnDg þ ð1� apÞapPap

pPTempP ð45Þ

Equation (45) is obtained by substituting Eqs. (41), (43), and (44) into the dis-cretized continuity equation (20). In the first inner iteration process for the pressureequation, the pressure underrelaxation factor ap is incorporated into the pressureequation (45). The underrelaxation factor is used to make the solution process morestable for some very complicated cases. Generally speaking, the solution process ofthe IDEAL algorithm is stable enough, so for most cases the pressure in Eq. (45)needs not be underrelaxed and the pressure underrelaxation factor ap can be set as 1.

Step 5. Calculate the temporary nodal velocities uTempP ; v

TempP , and w

TempP from

the explicit discretized momentum equations (24)–(26) by the temporary pressurefinished and the next inner iteration step will be started.

Step 6. Regard uTempP ; vTemp

P ; wTempP , and pTemp calculated in Step 4 and Step 5

as the temporary nodal velocity and pressure of the previous inner iteration step,denoted by uPTemp

P ; vPTempP ; wPTemp

P , and pPTemp. Return to Step 3, and all the super-scripts 0 in Steps 3 and 4 are replaced by PTemp. Repeat the iteration process com-posed of Steps 3, 4, and 5 until the iteration times are equal to the prespecified times

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N1. After the first inner iteration process for the pressure equation is finished, thefinal temporary pressure PTemp is regarded as the initial pressure p�.

Step 7. Solve the discretized momentum equations (21)–(23) by the initial velo-city and pressure, and obtain the intermediate nodal velocities u�P; v�P; and w�

P.Second inner iteration process for pressure questionStep 8. Calculate the interfacial pseudo-contravariant velocities eUU�

e ;eVV �n; andeWW �

t by the MMIM. The calculation procedure is shown in Step 3 and is omitted herefor simplicity.

Step 9. Solve the pressure again by the following equation [Eq. (46)], andobtain the temporary pressure PTemp.

apPpTempP ¼

Xapnbp

Tempnb þ bpP

apP ¼ a

pE þ a

pW þ a

pN þ a

pS þ a

pT þ a

pB

apE ¼ qeBUe ðDgDf=dneÞ apW ¼ qwB

Uw ðDgDf=dnwÞ

apN ¼ qnCVn ðDnDf=dgnÞ apS ¼ qsC

Vs ðDnDf=dgsÞ

apT ¼ qtDWt ðDnDg=dftÞ apB ¼ qbD

Wb ðDnDg=dfbÞ

bpP ¼ qw eUU�wDgDf� qe eUU�

eDgDfþ qs eVV�sDnDf� qn eVV�

nDnDf

þ qb eWW �bDnDg� qt eWW �

tDnDg ð46Þ

Equation (46) is obtained by substituting Eqs. (41), (43), and (44) into thediscretized continuity equation (20). It should be noted that in the second inneriteration process for pressure, the pressure need not be underrelaxed.

Step 10. Calculate the temporary nodal velocities uTempP ; vTemp

P , and wTempP from

the explicit discretized momentum equations (24)–(26) by the temporary pressurepTemp. Then one inner iteration step is finished and the next inner iteration step willbe started.

Step 11. Regard uTempP ; vTemp

P ; wTempP ; and pTemp calculated in Step 9 and Step 10

as the temporary nodal velocity and pressure of the previous inner iteration step,denoted by u

PTempP ; v

PTempP ; w

PTempP ; and pPTemp. Return to Step 8, and all the super-

scripts � in Steps 8 and 9 are replaced by PTemp. Repeat the iteration process com-posed of Steps 8, 9, and 10 until the iteration times are equal to the prespecified timesN2. After the second inner iteration process for the pressure equation is finished, thefinal temporary nodal velocities are regarded as the final nodal velocities uP, vP, andwP of the current iteration level. Then, the final interfacial contravariant velocitiesUe, Vn, Wt can be calculated from Eqs. (41), (43), and (44) by the final nodalvelocities and pressure.

Step12. Solve the discretization equations of the other scalar variables if necessary.Step 13. Regard the final nodal velocities and the final interfacial contravariant

velocities as the initial nodal velocities u0P; v0P; w0P and the initial interfacial contra-

variant velocities U0e ; V0

n ; W 0t of the next iteration level, then return to Step 2.

Repeat the iterative procedure until convergence is reached.As indicated above, in one iteration level of the IDEAL algorithm, the values

with superscripts 0 in equations of Section 3 remain unchanged and the values with-out superscripts are changed with the inner iteration steps.

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It is interesting to note that in the IDEAL algorithm, the pressure field used tosolve the momentum equations, i.e., p�, is solved by the pressure equation. Since thealgebraic equation is solved iteratively, an initial pressure field is required, and thegoodness of this initial field has a profound effect on the solution convergence.The numerical practice provided in [22] revealed this important effect. Our numericalpractice shows that if the pressure results of the first inner iteration of the presentlevel are taken as the initial field for the next-level solution, the total solutionprocedure can be somewhat enhanced.

In the IDEAL algorithm, the first inner iteration times N1 and the second inneriteration times N2 (hereafter N1&N2) can be adjusted. N1&N2 should be increasedwith the increase of the velocity underrelaxation factor. At a larger velocity under-relaxation factor the solution process may become very unstable, and in such a casethe inner iteration times need to be increased to ensure the convergence of thesolution process and to enhance the robustness.

6. COMPARISON CONDITIONS AND CONVERGENCE CRITERION

Make meaningful comparisons of the four algorithms—SIMPLEM,SIMPLERM, SIMPLICM, and IDEAL—numerical comparison conditions andconvergence criterion should be specified. In our study, the comparison conditionsand convergence criterion include:

1. Discretization scheme. In order to guarantee the stability and accuracy of thenumerical solution, the SGSD scheme [23] is adopted, which is at least ofsecond-order accuracy and absolutely stable. For stability of the solution process,the deferred-correction method is adopted, which was proposed in [24].

2. Solution method of the algebraic equations. The algebraic equations are solvedby the alternative direction implicit method (ADI).

3. Under relaxation factor. In the SIMPLEM algorithm the pressure underrelaxa-tion factor is set as 0.4; in the SIMPLERM and IDEAL algorithms the pressureunderrelaxation factor is set as 1.0; in the SIMPLECM algorithm the pressureneeds not be underrelaxed at all [6]. In the comparison of the four algorithms,the pressure underrelaxation factor remains unchanged while the effects of thevelocity and temperature underrelaxation factors on the convergence are exam-ined. For the four algorithms, the same values are adopted for the velocity under-relaxation factors and also for the temperature underrelaxation factors.For convenience of presentation, the time step multiple E [7] is used in the follow-ing presentation, which relates to the underrelaxation factor a by Eq. (47):

E ¼ a1� a

ð0 < a < 1Þ ð47Þ

Some correspondence between a and E is presented in Table 2. It can be seen thatwith the time step multiple we have a much wider range to show the performanceof the algorithm in the high-value region of the underrelaxation factor.

4. Grid system. For each problem, the same grid system is used for the fouralgorithms. The details of each grid system will be presented individually.

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5. Convergence criterion. The adopted convergence criterion for the four algorithmsis the same and requires that both the relative maximum mass residual RsMass andthe relative maximum u, v, w component momentum residuals RsUMom, RsVMom,RsWMom are less than some prespecified small values [15].

7. NUMERICAL COMPARISONS

In the following presentation, comprehensive comparisons are made amongthe SIMPLEM, SIMPLERM, SIMPLECM, and IDEAL algorithms for three 3-Dfluid flow and heat transfer problems.

7.1. Problem 1: Lid-Driven Flow in an Inclined Cavity

The flow configuration is shown in Figure 3, where L¼H¼W¼ 1 and h is theinclination angle. Calculations are conducted for h¼ 5�, 25�, 45�, 80�, Re¼ 100, 500,grid number¼ 32� 32� 32, 52� 52� 52, and the allowed relative maximumresiduals RsMass, RsUMom, RsVMom, and RsWMom should be all less than 10�7. TheReynolds number is defined as

Re ¼ ulidH

nð48Þ

In the inclined cavity, the body-fitted grid system is generated by the transfiniteinterpolation method [2] (See Figure 4).

The location of central line CL is shown in Figure 3. In Figure 5 the u-velocityprofiles along the central line CL from the four algorithms are presented. The resultscalculated by the IDEAL algorithm are in excellent agreement with those calculated

Figure 3. Flow configuration of lid-driven flow in an inclined cavity (Problem 1).

Table 2. Some correspondence between a and E

a 0.1 0.5 0.9 0.95 0.96 0.97 0.98 0.99 1

E 0.111 1 9 19 24 32.3 49 99 Infinite

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by the other three algorithms. This comparison gives some support to the reliabilityof the proposed 3-D IDEAL algorithm and the developed code.

For the case of Re¼ 100 and grid number¼ 32� 32� 32, four different incli-nation angles (h¼ 5�, 25�, 45�, 80�) are simulated to compare the computation timeand robustness of the IDEAL, SIMPLEM, SIMPLERM, and SIMPLECM algo-rithms. The results are shown in Figure 6. The inner iteration times N1&N2 in theIDEAL algorithm are displayed at the top of these figures. For example, 2&2,4&4, and 5&5 at the top of Figure 6a show that in the three ranges of E the two inneriterative times are 2&2, 4&4, and 5&5, respectively. From the figure, the followingthree features may be noted. First, N1&N2 increase with the increase of time stepmultiple, i.e., with the underrelaxation factor. Second, among the four algorithmscompared, the IDEAL algorithm is far more robust than the SIMPLEM,SIMPLERM, and SIMPLECM algorithms, and it can converge almost at any timestep multiple even in the case of h¼ 5�. This implies that the IDEAL algorithm is still

Figure 5. Comparison of u-velocity profiles along the central line CL for the case of h¼ 45�, Re¼ 100, grid

number¼ 32� 32� 32 (Problem 1).

Figure 4. Body-fitted grid system in the inclined cavity (h¼ 45�) (Problem 1).

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robust, even when the grid nonorthogonality is severe. Third, the IDEAL algorithmconverges more quickly than the SIMPLEM, SIMPLERM, and SIMPLECM algo-rithms. For the case of h¼ 45�, Re¼ 500, and grid number¼ 52� 52� 52, the sameconclusions can be obtained, as shown in Figure 7.

Table 3 shows the reduced ratio of computation time of the IDEAL algorithmover the SIMPLEM, SIMPLERM, and SIMPLECM algorithms at their own optimaltime stepmultiples for different cases of Problem 1.When eachmethodworks at its ownoptimal time step multiple, the IDEAL algorithm can reduce the computation timeby 55.4–79.4% over the SIMPLEM algorithm, by 46.5–77.1% over the SIMPLERMalgorithm, and by 65.1–77.6% over the SIMPLECM algorithm for Problem 1.

7.2. Problem 2: Natural Convection in an Inclined Cavity

Natural convection in an inclined cavity is a velocity–temperature couplingproblem. The flow configuration of this problem is shown in Figure 8, where

Figure 6. Comparison of computation time and robustness of IDEAL, SIMPLEM, SIMPLERM, and

SIMPLECM algorithms (Problem 1, Re¼ 100, grid number¼ 32� 32� 32).

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L¼H¼W¼ 1 and the inclination angle h is 45�. The inclined cavity has four adia-batic walls, with two walls being maintained at constant but different temperatures.

Calculations are conducted for Ra¼ 104, 105 and grid numbers¼ 32� 32� 32,52� 52� 52 with the relative maximum residuals RsMass, RsUMom, RsVMom, andRsWMom being all less than 10�7. The Rayleigh number is defined as

Ra ¼ qgbH3ðTH � TCÞam

ð49Þ

The transfinite interpolation method [2] is applied to generate the body-fitted grid inthis problem.

The location of central section CS is shown in Figure 8. Figure 9 shows the fourtemperature fields on the central section CS, which are calculated by the SIMPLEM,SIMPLERM, SIMPLECM, and IDEAL algorithms, respectively. From the figurewe can see that the result calculated by the IDEAL algorithm agrees very well withthose calculated by the other three algorithms.

Figure 10 shows the computation time and robustness of the IDEAL,SIMPLEM, SIMPLERM, and SIMPLECM algorithms for different grid numbers

Table 3. Reduced ratio of computation time of IDEAL algorithm over SIMPLEM, SIMPLERM, and

SIMPLECM algorithms at their own optimal time step multiples in Problem 1

Grid number 32� 32� 32 52� 52� 52

Re 100 500

h 5� 25� 45� 80� 45�

Reduced ratio over SIMPLEM 66.7% 79.4% 78.1% 55.4% 73.9%

Reduced ratio over SIMPLERM 77.1% 76.2% 69.3% 46.5% 70.6%

Reduced ratio over SIMPLECM 74.5% 71.3% 77.6% 71.5% 65.1%

Figure 7. Comparison of computation time and robustness of IDEAL, SIMPLEM, SIMPLERM, and

SIMPLECM algorithms (Problem 1, h¼ 45�, Re¼ 500, grid number¼ 52� 52� 52).

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and different Rayleigh numbers. From this figure, we can see that the performance ofthe different algorithms in velocity–temperature coupling problems is almost thesame as that in decoupled fluid flow and heat transfer problems, for example,Problem 1. The IDEAL algorithm is the most robust and most efficient amongthe four algorithms compared.

Table 4 shows the reduced ratio of computation time of the IDEAL algorithmover the SIMPLEM, SIMPLERM, and SIMPLECM algorithms at their own opti-mal time step multiples for different cases of Problem 2. When each method uses its

Figure 9. Temperature fields on the central section CS for the case of Ra¼ 104, grid number¼ 32� 32� 32

(Problem 2).

Figure 8. Flow configuration of natural convection in an inclined cavity (Problem 2).

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own optimal time stepmultiple, the IDEAL algorithm can reduce the computation timeby 45.4–52.6% over the SIMPLEM algorithm, by 38.2–52.2% over the SIMPLERMalgorithm, and by 13.4–45.2% over the SIMPLECM algorithm for Problem 2.

7.3. Problem 3: Laminar Fluid Flow through a Twisted-Tape-InsertedTube

Figure 11 shows the flow configuration of laminar fluid flow through atwisted-tape-inserted tube, where d is the tape thickness, D is the tube outer diam-eter, and H is the axial distance of tape rotating through 180� in the axial direction.The twist ratio y is defined as

y ¼ H

Dð50Þ

In this article, the configuration of y¼ 5 is adopted to compare the computationtime and robustness of the IDEAL, SIMPLEM, SIMPLERM, and SIMPLECMalgorithms.

Figure 10. Comparison of computation time and robustness of IDEAL, SIMPLEM, SIMPLERM, and

SIMPLECM algorithms (Problem 2).

Table 4. Reduced ratio of computation time of IDEAL algorithm over SIMPLEM, SIMPLERM, and

SIMPLECM algorithms at their own optimal time step multiples in Problem 2

Reducing ratio over

SIMPLEM (%)

Reducing ratio over

SIMPLERM (%)

Reducing ratio over

SIMPLECM (%)

Ra¼ 104 Grid

number¼ 32� 32� 32

45.4 52.2 13.4

Ra¼ 105 Grid

number¼ 52� 52� 52

52.6 38.2 45.2

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Figure 12 shows the computation domain of Problem 3, which is the volume ofthe shadowed region (see Figure 11) rotating through 360� in the axial direction. Inthe computation domain, the periodic boundary condition [2] is adopted at the inletand outlet cross sections, and the no-slip boundary condition [2] is used at the tubewall and the twisted tape. The body-fitted grid system is generated by the differentialequation method [2].

Calculations are conducted for Re¼ 100, 500, grid number¼ 276� 30� 30,and the allowed relative maximum residuals RsMass, RsUMom, RsVMom, and RsWMom

are all set to be less than 10�6. The Reynolds number is defined as

Re ¼ uinD

nð51Þ

where uin is the mean velocity at the inlet cross section.Table 5 shows the predicted friction factors f calculated by the four different

algorithms. f is defined as

f ¼ ðpin � poutÞqu2in=2

D

2Hð52Þ

Figure 11. Flow configuration of laminar fluid flow through a twisted-tape-inserted tube (Problem 3).

Figure 12. Computation region and body-fitted grid system (Problem 3).

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where pin and pout represent the mean pressures at the inlet and outlet cross sections,respectively. From the table we can see that the results computed by the IDEALalgorithm are almost the same as those by the other three algorithms.

Figure 13 shows the computation time and robustness of the IDEAL,SIMPLEM, SIMPLERM, and SIMPLECM algorithms for different Reynolds num-bers. From the figure, we can see that the SIMPLEM and SIMPLERM algorithmsare the least robust and least efficient and can obtain convergence results just in therange of very small time step multiples, the performance of SIMPLECM is in themiddle, and the IDEAL algorithm is the most robust and efficient.

Table 6 shows the reduced ratio of computation time of the IDEAL algorithmover the SIMPLEM, SIMPLERM, and SIMPLECM algorithms at their own opti-mal time step multiples for different cases of Problem 3. When each method uses itsown optimal time step multiple, the IDEAL algorithm can reduce the computation

Table 5. Predicted friction factors for different Re (Problem 3)

SIMPLEM SIMPLERM SIMPLECM IDEAL

Re¼ 100 1.7731 1.7707 1.7715 1.7719

Re¼ 500 0.4147 0.4157 0.4151 0.4150

Figure 13. Comparison of computation time and robustness of IDEAL, SIMPLEM, SIMPLERM, and

SIMPLECM algorithms (Problem 3).

Table 6. Reduced ratio of computation time of IDEAL algorithm over SIMPLEM, SIMPLERM, and

SIMPLECM algorithms at their own optimal time step multiples in Problem 3

Reduced ratio over

SIMPLEM (%)

Reduced ratio over

SIMPLERM (%)

Reduced ratio

over SIMPLECM (%)

Re¼ 100 59.6 60.0 23.2

Re¼ 500 71.0 67.5 38.0

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time by 59.6–71.0% over the SIMPLEM algorithm, by 60.0–67.5% over theSIMPLERM algorithm, and by 23.2–38.0% over the SIMPLECM algorithm forProblem 3.

8. CONCLUSION

In the present article, the IDEAL algorithm has been extended to body-fittedcollocated grid systems in 3-D nonorthogonal curvilinear coordinates, and the per-formance of the IDEAL algorithm has been analyzed by a systematic comparisonwith three other algorithms (SIMPLEM, SIMPLERM, SIMPLECM). The mainconclusions are as follows.

1. The IDEAL algorithm is the most robust and efficient among the four algorithmscompared.

2. The IDEAL algorithm can converge at almost any time step multiple for the threeproblems studied, and it is robust even when the grid nonorthogonality is veryserious.

3. When each algorithm works at its own optimal time step multiple, the IDEALalgorithm can reduce the computation time by 45.4–79.4% over the SIMPLEMalgorithm, by 38.2–77.1% over the SIMPLERM algorithm, and by 13.4–77.6%over the SIMPLECM algorithm.

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