a comparison study of the convergence characteristics and ...nht.xjtu.edu.cn/paper/en/2003203.pdfa...

A comparison study of theconvergence characteristics

and robustness for fourvariants of SIMPLE-family

at regne gridsM Zeng and WQ Tao

School of Energy and Power Engineering Xian Jiaotong UniversityXirsquoan Shaanxi Peoplersquos Republic of China

Keywords Flow Heat transfer Numerical simulation Grids

Abstract A comparative study is performed to reveal the convergence characteristics and therobustness of four variants in the semi-implicit method for pressure-linked equations (SIMPLE)-family SIMPLE SIMPLE revised (SIMPLER) SIMPLE consistent (SIMPLEC) and SIMPLEextrapolation (SIMPLEX) The focus is concentrated in the solution at regne grid system Fourtypical macruid macrow and heat transfer problems are taken as the numerical examples (lid-driven cavitymacrow macrow in an axisymmetric sudden expansion macrow in an annulus with inner surface rotatingand the natural convection in a square enclosure) It is found that an appropriate convergencecondition should include both mass conservation and momentum conservation requirements Forthe four problems computed the SIMPLEX always requires the largest computational time theSIMPLER comes the next and the computational time of SIMPLE and SIMPLEC are the least Asfar as the robustness is concerned the SIMPLE algorithm is the worst the SIMPLER comes thenext and the robustness of SIMPLEX and SIMPLEC are superior to the others The SIMPLECalgorithm is then recommended especially for the computation at a regne grid system Briefdiscussion is provided to further reveal the reasons which may account for the difference of thefour algorithms

NomenclatureA = area of a cell facesb = source term in a discretized

equationd = ratio of area to momentum

coefregcientE = time step multiple

[= a=(1 2 a)]g = acceleration due to gravityD = characteristic length of cavityL = characteristic length of sudden

expansion

p = pressurep = intermediate pressurep0 = pressure correctionqm = mass macrow rater = radial coordinateRa = Rayleigh number

[= gb(TH 2 TC)D 3=(an)]Re = Reynolds numberRSMAX = the maximum residual of control-

volume mass macrow rateT = temperature

The Emerald Research Register for this journal is available at The current issue and full text archive of this journal is available at

httpwwwemeraldinsightcomresearchregister httpwwwemeraldinsightcom0264-4401htm

This work was supported by the National Key Project on Fundamental RampD of China (Grant NoG2000026303) and National Natural Science Foundation of China (Grant No 50076034)

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Received August 2002Revised November 2002Accepted November2002

Engineering ComputationsVol 20 No 3 2003pp 320-340

q MCB UP Limited0264-4401DOI 10110802644400310467234

u v = velocity components in the x andy direction

u v = intermediate velocitiesu 0 v 0 = velocity correctionx y = spatial coordinatesa = underrelaxation factorb = volumetric coefregcient of

expansionr = densitye1 e2 = pre-specireged small values to

control convergence

Superscripts = intermediate value

Subscriptse w n s = cell facesC = lowest temperatureH = highest temperatureIN = inlet diameterm = massnb = neighbor pointsOUT = out diameter

1 IntroductionSince the semi-implicit method for pressure-linked equations (SIMPLE)algorithm was proposed by Patankar and Spalding (1972) it has been widelyapplied to the regelds of computational macruid dynamics (CFD) and numerical heattransfer(NHT) Over the last three decades about ten variants (Acharya andMoukalled 1989 Date 1986 Gjesdal and Lossius 1997 Patankar 1980 1981Sheng et al 1998 Van Doormaal and Raithby 1984 1985 Yen and Liu 1993Yu et al 2001) were proposed to improve the convergence performance andthese algorithms consist the so-called SIMPLE-series or SIMPLE-familysolution algorithm Today the SIMPLE-family is probably the most popularalgorithm for solving incompressible Navier-Stokes equations with primitivevariables In the past decade the SIMPLE-series algorithms were alsosuccessfully extended to solving compressible macruid macrow (Demirdzic et al 1993Karki and Patankar 1989 Shyy et al 1992) Among the different variants themost often used algorithms are SIMPLE SIMPLE revised (SIMPLER) SIMPLEconsistent (SIMPLEC) and SIMPLE extrapolation (SIMPLEX) The SIMPLESTalgorithm (Spalding 1980) is essentially the same as SIMPLE with a differenceonly in the discretization scheme for the convection term By algorithm wemean the way to deal with the coupling between the velocity and pressure thuswe do not take it as a new variant of the SIMPLE-family

A number of comparisons between the different variants of the SIMPLE-family have been conducted (Barton 1998 Jang et al 1986 Latimer andPollard 1985 McCuirk and Palma 1993) The emphasis of these comparisonworks is often concentrated on the convergence rate for solving some typicalproblems and the grid number used is usually in the range of 10 pound 10 to30 pound 30 With the rapid advances in computer technologies and the constantly-reduced prices of computers nowadays even using a PC people can easily dealwith a network with 100 pound 100 grids Thus the convergence characteristics ofthese algorithms remain an interesting subject for further study To the authorsknowledge only Moukalled and Darwish (2000) and Van Doormaal andRaithby (1985) have clearly indicated the convergence characteristics of theabove algorithms at regne grids

A comparisonstudy for four

variants

321

When Van Doormaal and Raithby (1985) proposed the SIMPLEX algorithmthey regarded that the SIMPLEX experienced an optimization in convergencewith grid reregnement when compared to the SIMPLEC method RecentlyMoukalled and Darwish (2000) have made a unireged study about ten algorithmsbelonging to the SIMPLE-series In that article the authors indicated that in allSIMPLE-based methods no care is taken to ensure that the rate of convergencewill not degrade with grid reregnement This concern is addressed in SIMPLEXIt is emphasized that the SIMPLEX algorithm has a lower degradation in therate of convergence with grid reregnement as compared to other SIMPLE-likealgorithms And the following conclusion made in Van Doormaal and Raithby(1985) was recited in Moukalled and Darwish (2000) for sufregciently regne gridsSIMPLEX is more efregcient than SIMPLE SIMPLER and SIMPLEC Since themeaning of words ordfregneordm and ordfcoarseordm are only qualitative and relative it isnecessary to reveal speciregcally what is the regne grid used in Van Doormaal andRaithby (1985) It turns out that the so-called regne grid in Van Doormaal andRaithby (1985) is just 25 pound 25 for 2D case Deregnitely the grid system of(25 pound 25) could not be regarded as a regne one today So a question comes intobeing as whether the statement made in Van Doormaal and Raithby (1985) andMoukalled and Darwish (2000) is still applicable In the present study by regnegrid we mean a grid with grid number in the level of 100 pound 100 or so for 2Dcase We compare the convergence characteristics of the SIMPLE SIMPLECSIMPLER and SIMPLEX for four cases and get a conclusion different from thatof Van Doormaal and Raithby (1985)

In the following presentation the major features of the four algorithms willbe very briemacry reviewed at the staggered grid system and then convergencecomparison by using the four algorithms will be conducted for the four selectedproblems including the forced convection and natural convection in three 2Dcoordinates Two criterias will be selected to judge the convergence thecriterion of mass conservation and the criterion of both mass conservation andmomentum conservation The robustness of the four algorithms will also becompared Finally some conclusions will be drawn

2 Mathematical formulation of the four algorithms comparedThe problems we solve here are assumed to be at steady state with constantproperties Thus for the simplicity of presentation only the discretized massand momentum equations are dealt with The governing equation oftemperature does not effect the algorithm we compared here hence will beomitted Furthermore to show the major features of the different algorithmsthe pressure-correction equation is derived for a 2D incompressible macruid macrowproblem in Cartesian coordinates For the details of the derivation Patankar(1981) and Tao (2001) may be consulted The symbols used in Patankar (1981)are adopted here

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The discretized macrow governing equations are as followsMass conservation

(ruDy)e 2 (ruDy)w + (rvDx)n 2 (rvDx)s = 0 (1)

Momentum conservation

aeue =P

aunbunb + bu + ( pP 2 pE )Ae

anvn =P

avnbvnb + b v + ( pP 2 pN )An

9=

(2)

The solution consequence is guess velocity regelds to evaluate the coefregcients ofthe momentum equations guess a pressure regeld p and solve the discretizedmomentum equations to obtain temporary solutions of velocity denoted by uv To improve u v such that the improved velocity satisfy the massconservation condition a pressure correction term p 0 and the correspondingvelocity correction terms u 0 v 0 should be added to their current values Then bysubtraction of the momentum equations for u v

aeue =

Pau

nbunb + bu + ( p

P 2 pE )Ae

anvn =

Pav

nbvnb + b v + ( p

P 2 pN )An

9=

(3)

from the momentum equations for u = u + u 0 v = v + v 0

ae(ue + u0

e) =P

aunb(u

nb + u 0

nb) + bu + [( pP + p 0

P ) 2 ( pE + p0

E )]Ae

an(vn + v 0

n) =P

avnb(v

nb + v 0

nb) + b v + [( pP + p0

P ) 2 ( pN + p 0

N )]An

9=

(4)

we can yield the following expressions

aeu0e =

Pau

nbu0nb + ( p 0

P 2 p 0E )Ae

anv 0n =

Pav

nbv0nb + ( p 0

P 2 p 0N )An

9=

(5)

Equation (5) tells us that the velocity correction consists of two parts One is thepressure correction difference between two adjacent points which are in thesame direction as the velocity and this part is the direct motive force bringingthe velocity correction The other part is caused by the neighborhood velocitycorrection which can be regarded as the indirect inmacruence of the pressurecorrections at nearby locations The main approximation made in the SIMPLEalgorithm is to neglect the inmacruence of these nearby velocity corrections Thishypothesis is equivalent to set coefregcients anb = 0 in equation

Pau

nbu0nb then

we can get the velocity correction equation


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323

u 0e = de( p 0

P 2 p 0E ) de = Ae

ae

v 0n = dn( p 0

P 2 p 0N ) dn = An

an

9gt=

gt(6)

Equation (6) is used to compute the velocity correction value in the SIMPLEalgorithm And the resulting velocity u = u + u 0 v = v + v 0 are taken asthe solution of this iteration level Substituting ue = u

e + de(p 0P 2 p 0

E ) andvn = v

n + dn( p0P 2 p 0

N ) into the continuity equation (1) we obtain the pressurecorrection equation

aPp 0P =

Xanbp

0nb + bP (7)

where aE = redeDy aN = rndnDx aP =P

anb and bP = (ruDy)we +

(rvDx)sn In the SIMPLE algorithm under-relaxation is needed for p 0 since

it is considered that the value of p0 is exaggerated because of the neglect of thenearby velocity corrections in equation (5)

The primary distinction between the SIMPLEX and SIMPLE and theSIMPLEC and SIMPLE is the determination of the coefregcient d in equation (6)Van Doormaal and Raithby (1984) take the following form for d

de =AeP

anb 2 ae

dn =AnP

anb 2 an(8)

The subtraction of the diagonal coefregcient from the summation of coefregcientsof the neighboring velocities in the denominator of equation (8) greatly alleviatethe inmacruence of neglecting the nearby velocity corrections in equation (5) Thisimprovement leads to the SIMPLEC algorithm and no underrelaxation isneeded for the pressure correction in the SIMPLEC algorithm

In another development Van Doormaal and Raithby (1985) extend theequation u 0

e = de( p 0P 2 p 0

E ) = deDp 0e to the computation of the neighborhood

velocity correction by supposing u 0nb = dnbDp 0

nb This practice is equivalent toextrapolate the expression of the velocity correction at location studied to allthe nearby locations The substitution of the above equation into the discretizedmomentum equation (3) results in the equation

aedeDp 0e =

XanbdnbDp 0

nb + AeDp 0e (9)

Further approximation is made in Van Doormaal and Raithby (1985) thatDp0

e = Dp 0nb Substitution of this approximation into equation (9) leads to the

following equation for de

aede =X

anbdnb + Ae (10)

The equation for dn ds and dw can be obtained similarly Because thecoefregcients of the discretized momentum equation for u and v are already

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calculated and available the values of de and dn can be determined by solvingthe algebraic equations on each iteration of the segregated solution procedureIt is considered that the de and dn such determined have considered theinmacruence of the nearby velocity corrections To identify the use of theextrapolation in determining d the character X is appended to SIMPLEleading to a new variant SIMPLEX (Moukalled and Darwish 2000) Obviouslythe pressure correction p0 in the SIMPLEX algorithm need not be underrelaxedeither The SIMPLEX algorithm is implanted by executing the followingsequence of steps

(1) guess a velocity regeld u0 and v0 to evaluate the coefregcients and theconstant of momentum equations

(2) guess a pressure regeld p

(3) compute the equations u de v and dn to get ue de v

n and dn equations

(4) solve the pressure correction equation to acquire p 0 according to thevalue of u

e de vn and dn

(5) improve the velocity regeld according to p0 de and dn

(6) using the improved velocity regeld (u + u0 v + v 0 ) and the pressureregeld ( p + p 0 ) return to step 3 Repeat this cycle until convergenceis achieved The velocity need to be underrelax ed except for thepressure

As far as the SIMPLER algorithm is concerned its major difference from theSIMPLE algorithm is that the pressure regeld is solved from the previousvelocity regeld rather than assumed And the pressure correction is only used tocorrect the velocity regelds not the pressure

The four algorithms discussed above have been implemented in this articleThe code developed is verireged through three benchmark problems (lid-drivencavity macrow in rectangular coordinates macrow in a 2D axisymmetric suddenexpansion and natural convection in a square cavity) The results agree wellwith benchmark solutions available in the literature Then the code is used toperform the comparison study for the convergence characteristics The resultsare presented in the following section

3 Results of comparison studyFour macrow and heat transfer problems (lid-driven cavity macrow macrow in a 2Daxisymmetric sudden expansion macrow in annulus with the inner wall rotatingabout the axis and natural convection in a square cavity) are used to comparethe convergence rate and robustness The four test cases are depictedschematically in Figure 1 The governing equations for the four problems arethose for 2D incompressible macruid For the natural convection we adopt theBoussinesq assumption All of these are well documented in Patankar (1980)and Tao (2001) and will not be restated here for simplicity


variants

325

The comparisons for the convergence characteristics of the four algorithms areconducted under two criteria for the iteration convergence The regrst one is themaximum relative residual of control-volume (SMAX) in the continuityequation which is less then a pre-specireged value

RSMAX =SMAX

qm emass (11)

where qm is the reference mass macrow rate For the open system such as macrow in a2D axisymmetric sudden expansion (Figure 1(b)) we take the inlet mass macrowrate as the referenced qm For the closed system for example lid-driven cavitymacrow (Figure 1(a)) and natural convection in a square cavity (Figure 1(c)) wemake a numerical integration for the macrow rate along any section in the regeld to

Figure 1Four problems forperformance study

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obtain the reference qm (Tao 2001) by adopting the absolute value of thevelocity The second criterion requires both the relative maximum massresidual and the relative residual module in the momentum equations are alsoless than pre-specireged small values Thus apart from equation (11) followingcondition is added

Aacute

node

X(

aeue 2

nb

Xanbunb + b + Ae( pP 2 pE )

)21=2

ru2m emom (12)

We take the inlet momentum as the referenced momentum in the aboveequation for the open system For the closed system regrst we get the numericalintegral of the momentum along any section then we adopt their average valueas the reference momentum

31 Convergence comparison under the regrst criteriaUnder this condition we compare the four algorithms for the two problems lid-driven cavity macrow and macrow in annulus with the inner wall rotating about theaxis We adopted the underrelaxation factor of auv = 08 for all the algorithmcompared and ap = 03 for the SIMPLE only It was found that the SIMPLERalgorithm needs the least computation time but the results are inferior to allothers The behavior of SIMPLEX is the opposite The phenomenon becomessevere with grid reregnement The details are presented below

Problem 1 Lid-driven cavity macrow Figure 2 shows the velocity distributionsat the horizontal centerline together with the computational benchmarksolutions for Re = 1000 (Ghia et al 1982) In the abscissa of each reggure therequired CPU time of a PC with 128 M memory and 400 MHz frequency isindicated It can be observed that the solution differences among the fourschemes are insigniregcant when the grids are not regne (82 pound 82) but thedifferences increase with grid reregnement Because of the space limitation onlythe results of the regnest grid system (202 pound 202) are provided From the reggureit can be seen that the solution accuracy of SIMPLER is the worst and theresults of SIMPLEX is the best As far as the CPU time is concernedthe SIMPLER algorithm needs the least while the SIMPLEX the most Thephenomenon becomes more severe with grid reregnement The solutions ofSIMPLEC is superior to that of SIMPLE but inferior to that of SIMPLEX atregnest grid

Problem 2 Flow in annulus with the inner wall rotating about the axis Forthis case the analytical solution of the tangential velocity distribution along theradius is adopted as the benchmark solution (Bird et al 2002) The relativeresults are shown in Figure 3 The behavior of the four algorithms is the sameas that in problem 1


variants

327

Figure 2Comparison betweenfour algorithms underregrst convergencecriterion for lid-drivencavity macrow

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Figure 2


variants

329

32 Convergence comparison under the second criterionAs indicated above the second criterion includes the requirement for both massconservation and the momentum conservation The underrelaxation factorsused are auv = 08 aT = 08 and ap = 03 Computations are performed forfour problems mentioned above Since the qualitative results are more or lessthe same to save the space only the results for macrow in a 2D axisymmetricsudden expansion and natural convection in a square cavity are provided hereThese two problems cover the forced convection and natural convection andalso represent the Cartesian coordinates and cylindrical coordinates

Problem 3 Flow in a 2D axisymmetric sudden expansion The computationalresults are presented in Table I We can see there that the predicted value of therepresentative parameter Lr=Din are almost identical under the same griddensity for the four algorithms compared Such a uniformity in numericalresults are reasonable and expected Since it is generally considered that

Figure 2

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Figure 3Comparison between

four algorithms underregrst convergence

criterion for macrow in anannulus with inner

surface rotating


variants

331

Figure 3

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the accuracy of a numerical solution mainly depends on the discretizationscheme and the grid regneness while the iteration convergence rate depends onthe algorithm dealing with the coupling between velocity and pressure(Tao 2001) The solution uniformity of the four algorithms also implies that

Figure 3

LrDIN ( Re = 150) Time(s)

Grid SIMPLE SIMPLER SIMPLEC SIMPLEX SIMPLE SIMPLER SIMPLEC SIMPLEX

32 pound 12 6510505 6510505 6510505 6510505 0200288 0210302 0280403 052074852 pound 12 6288261 6288261 6288261 6288261 0350504 0420604 0450648 1321901102 pound 18 6709247 6709247 6709247 6709247 1682419 2453528 1542218 7801218152 pound 22 6633205 6633205 6633205 6633205 4816926 7240411 4696754 2611756202 pound 42 6628800 6628800 6628800 6628800 5188461 7719099 5236530 1421244Source Macagno and Hung (1967) 65

Table IThe predicted

values of Lr DIN andthe computation

time with differentgrid density


variants

333

the second criterion is more appropriate for the judging of convergenceAs far as the CPU time is concerned the time spent by SIMPLEX is the mostand secondly is the SIMPLER

Problem 4 Natural convection in a square cavity The results are listed inTable II Again we can see the uniformity of the four solutions The expense ofCPU time of SIMPLEX is the most and that of the SIMPLE is the least (exceptfor the 42 pound 42 case) while that of SIMPLEC is somewhere in between

33 Comparison of algorithm robustnessIt is generally considered that if an algorithm can lead to a convergencesolution within a wide rage of the relaxation factor the algorithm possessesgood robustness It is based on this understanding that we performed therobustness comparison In order to have a wide variation range the so-calledtime step multiple (Van Doormaal and Raithby 1984) is used instead of theunderrelaxation factor According to the above discussion the secondconvergence criterion is adopted here The results are presented in Figures 4-6In the reggures the X-coordinate stands for the time step multipleE (E = a=(1 2 a)) and the Y-coordinate is the computation time (TIME(s))The variation range of the time step multiple within which a convergencesolution can be acquired is regarded as the symbol of the robustness The widerthe range the better the robustness From these reggures the same conclusioncan be made the robustness of SIMPLE is always the worst whether the gridare coarse or not The SIMPLER behaves as the SIMPLEC and SIMPLEX incoarse grid and experiences a degradation in convergence with grid reregnementThe robustness of SIMPLEC and SIMPLEX are almost the same These regguresalso tells us that the SIMPLER and SIMPLEX need more computation timethan that of the SIMPLE and SIMPLEC

4 Further discussion on the difference between the four algorithmsWe have found that there are some differences among the four algorithmsFollowing discussion tries to further reveal the reasons that account forthe difference The main distinction among them is the determination of thecoefregcient de dn except that the SIMPLER needs to solve the pressure equationThe difference of d expression is listed in Table III From our numerical

Nu (Ra = 10000) Time(s)


22 pound 22 2299 2299 2299 2299 1452088 181261 1572261 2443514

42 pound 42 2258 2258 2258 2258 4296177 334481 4556552 6269014

82 pound 82 2248 2248 2248 2248 4331228 503721 4761847 6138827

102 pound 102 2247 2247 2247 2247 112672 132009 1218552 1657884

Source Yu et al (2001) 2245

Table IIThe results of theaverage Nusseltnumber and thecomputation timeunder different griddensity

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Figure 4Robustness comparison

for lid-driven cavity macrow


variants

335

Figure 5Robustness comparisonfor macrow in a 2Daxisymmetric suddenexpansion

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for natural convection ina square cavity


variants

337

practices we found that if we solve a problem under the same condition(the same grid density the same underrelaxation factors and the sameconvergence criterion etc) the values of de dn solved by SIMPLE andSIMPLER are close to each other when the convergence solution is approachedand the same applies to SIMPLEC and SIMPLEX Tables IV-V show the value ofde of two arbitrary points in two test cases We can see from the tables that thevalue of de solved by the SIMPLEC and SIMPLEX is about regve times larger thanthat of the SIMPLE and SIMPLER Whether the de is the factor to affect therobustness or not is open to discussion Now that the value of de solved by theSIMPLEC and SIMPLEX is nearly the same while the SIMPLEC need not solve

Method Approximation de

SIMPLEP

aunbu

0nb = 0 Aeae

SIMPLERP

aunbu

0nb = 0 Aeae

SIMPLECP

aunb(u

0nb 2 u 0

e) = 0 Ae=(ae 2P

anb)SIMPLEX Dp 0

e = Dp 0nb aede =

Panbdnb + Ae

Table IIIThe contrast of fourmethods inapproximation

22 pound 22 42 pound 42

SIMPLE SIMPLER SIMPLEC SIMPLEX SIMPLE SIMPLER SIMPLEC SIMPLEX

du (10 6) 1135 1135 5676 5634 05927 05927 2964 2928du (20 20) 1183 1183 5916 6060 05960 05960 2980 2979

82 pound 82

SIMPLE SIMPLER SIMPLEC SIMPLEX

du (10 6) 02981 02981 1490 1474du (20 20) 02975 02975 1488 1488

Table VThe value of de oftwo arbitrary points(natural convectionin a square cavity)



du (12 7) 2500 2500 1250 1252 1929 1930 9643 9525du (22 22) 2064 2064 1032 8593 1874 1873 9368 9265

82 pound 82


du (12 7) 09999 09999 4999 4976du (22 22) 09612 09612 4803 4798

Table IVThe value of de oftwo arbitrary points(lid-driven cavitymacrow)

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the d equations the computation time by the SIMPLEC being less than thatof the SIMPLEX is the natural outcome From the four problems computed itmay be concluded that the SIMPLEC is superior to all others when regne gridsystem is used

5 ConclusionsIn this paper the convergence character and the robustness of four variants inthe SIMPLER-family are compared through four typical 2D macruid macrow and heattransfer problems at regne grids (grid number being in the order of 100 pound 100)The following conclusions can be made

(1) The criterion for judging the iteration convergence of macruid macrow and heattransfer problems is recommended to include both mass conservationand momentum conservation requirements The mass conservationcondition alone is not an adequate criterion in that different algorithmsmay lead to different numerical solutions with other conditions being thesame

(2) For the four problems computed with the appropriate convergencecriterion the SIMPLEX needs the largest CPU time the SIMPLER comesnext and the SIMPLE and SIMPLEC need the least computational time

(3) Under the same conditions the SIMPLE have the worst robustness andthe next is the SIMPLER The robustness of the SIMPLEC andSIMPLEX are almost the same and superior to that of the other twoalgorithms

(4) To sum up the SIMPLEC algorithm is recommended for the solution ofincompressible macruid macrow and heat transfer problems especially whenthe grid density is regne being in the order of 100 pound 100

References

Acharya S and Moukalled F (1989) ordfImprovements to incompressible macrow calculation on anon-staggered gridordm Numerical Heat Transfer Part B Vol 15 pp 131-52

Barton IE (1998) ordfComparison of SIMPLE- and PISO-type schemes for transient macrowsordm IntJ Numer Methods in Fluids Vol 26 pp 459-83

Bird RB Stewrat WE and Lightfoot EN (2002) Transport Phenomena Wiley New York

Date AW (1986) ordfNumerical prediction of natural convection heat transfer in horizontalannulusordm Int J Heat Mass Transfer Vol 29 pp 1457-64

Demirdzic I Lilek I and Peric M (1993) ordfA collocated regnite volume method for predictingmacrows at all speedsordm Int J Numer Methods Fluids Vol 16 pp 1029-50

Ghia U Ghia K and Kim CT (1982) ordfHigh-resolution for incompressible macrow using theNavier-Stokes equations and multigrid methodordm J Comp Phys Vol 48 p 387

Gjesdal T and Lossius MEH (1997) ordfComparison of pressure correction smoothers formultigrid solution of incompressible macrowordm Int J Numer Methods Fluids Vol 25pp 393-405


variants

339

Jang DC Jetli R and Acharya S (1986) ordfComparion of the PISO SIMPLER and SIMPLECalgorithms for the treatment of the pressure-velocity coupling in steady macrow problemsordmNumerical Heat Transfer Vol 10 pp 209-28

Karki KC and Patankar SV (1989) ordfPressure based calculation procedure for viscous macrows atall speedsordm AIAA J Vol 27 No 9 pp 1167-74

Latimer BR and Pollard A (1985) ordfComparison of pressure velocity coupling solutionalgorithmordm Numerical Heat Transfer Vol 8 pp 635-52

Macagno EO and Hung TK (1967) ordfComputational and experimental study of a captiveannular eddyordm J Fluid Mech Vol 28 pp 43-61

McCuirk JJ and Palma JMLM (1993) ordfThe efregciency of alternative pressure correctionformulation for incompressible turbulent macrow problemsordm Computers and Fluids Vol 22No 1 pp 249-72

Moukalled F and Darwish M (2000) ordfA unireged formulation of the segregated class ofalgorithms for macruid macrow at all speedsordm Numerical Heat Transfer Part B Vol 37pp 103-39

Patankar SV (1980) Numerical Heat Transfer Hemisphere Washington DC

Patankar SV (1981) ordfA calculation procedure for two-dimensional elliptic situationsordmNumerical Heat Transfer Vol 4 pp 409-25

Patankar SV and Spalding DB (1972) ordfA calculation procedure for heat mass and momentumtransfer in three-dimensional parabolic macrowsordm International Journal of Heat and MassTransfer Vol 15 pp 1787-806

Sheng Y Shoukri M Sheng G and Wood P (1998) ordfA modiregcation to the SIMPLE method forBouyancy driven macrowordm Numerical Heat transfer Part B Vol 33 pp 65-78

Shyy W Chen M-H and Sun C-S (1992) ordfPressure based multigrid algorithm for macrow at allspeedsordm AIAA J Vol 30 pp 2660-9

Spalding DB (1980) ordfMathematical modeling of macruid mechanics heat transfer and masstransfer processesordm Report HTS801 Mechanical Engineering Department ImperialCollege of Science Technology and Medicine London 1980

Tao WQ (2001) Numerical Heat Transfer 2nd ed Xirsquoan Jiaotong University Press Xirsquoanpp 220-1 238-43

Van Doormaal JP and Raithby GD (1984) ordfEnhancements of the SIMPLE method forpredicting incompressible macruid macrowsordm Numerical Heat Transfer Vol 7 pp 147-63

Van Doormaal JP and Raithby GD (1985) ordfAn evaluation of the segregated approach forpredicting incompressible macruid macrowsordm ASME Heat Transfer Conference August 1985Denver (Paper 85-HT-9)

Yen RH and Liu CH (1993) ordfEnhancement of the SIMPLE algorithm by an additional explicitcorrection stepordm Numerical Heat Transfer Part B Vol 40 No 24 pp 127-41

Yu B Ozoe H and Tao WQ (2001) ordfA modireged pressure correction scheme for the SIMPLERmethod MSIMPLERordm Numerical Heat Transfer Part B Vol 39 pp 435-49

Further reading

Barakos G and Mitsoulis E (1994) ordfNatural convection macrow in a square cavity revisitedlaminar and turbulent models with wall functionordm Int J Numer Methods Fluids Vol 18No 7 pp 695-719

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u v = velocity components in the x andy direction

u v = intermediate velocitiesu 0 v 0 = velocity correctionx y = spatial coordinatesa = underrelaxation factorb = volumetric coefregcient of

expansionr = densitye1 e2 = pre-specireged small values to

control convergence

Superscripts = intermediate value

Subscriptse w n s = cell facesC = lowest temperatureH = highest temperatureIN = inlet diameterm = massnb = neighbor pointsOUT = out diameter

1 IntroductionSince the semi-implicit method for pressure-linked equations (SIMPLE)algorithm was proposed by Patankar and Spalding (1972) it has been widelyapplied to the regelds of computational macruid dynamics (CFD) and numerical heattransfer(NHT) Over the last three decades about ten variants (Acharya andMoukalled 1989 Date 1986 Gjesdal and Lossius 1997 Patankar 1980 1981Sheng et al 1998 Van Doormaal and Raithby 1984 1985 Yen and Liu 1993Yu et al 2001) were proposed to improve the convergence performance andthese algorithms consist the so-called SIMPLE-series or SIMPLE-familysolution algorithm Today the SIMPLE-family is probably the most popularalgorithm for solving incompressible Navier-Stokes equations with primitivevariables In the past decade the SIMPLE-series algorithms were alsosuccessfully extended to solving compressible macruid macrow (Demirdzic et al 1993Karki and Patankar 1989 Shyy et al 1992) Among the different variants themost often used algorithms are SIMPLE SIMPLE revised (SIMPLER) SIMPLEconsistent (SIMPLEC) and SIMPLE extrapolation (SIMPLEX) The SIMPLESTalgorithm (Spalding 1980) is essentially the same as SIMPLE with a differenceonly in the discretization scheme for the convection term By algorithm wemean the way to deal with the coupling between the velocity and pressure thuswe do not take it as a new variant of the SIMPLE-family

A number of comparisons between the different variants of the SIMPLE-family have been conducted (Barton 1998 Jang et al 1986 Latimer andPollard 1985 McCuirk and Palma 1993) The emphasis of these comparisonworks is often concentrated on the convergence rate for solving some typicalproblems and the grid number used is usually in the range of 10 pound 10 to30 pound 30 With the rapid advances in computer technologies and the constantly-reduced prices of computers nowadays even using a PC people can easily dealwith a network with 100 pound 100 grids Thus the convergence characteristics ofthese algorithms remain an interesting subject for further study To the authorsknowledge only Moukalled and Darwish (2000) and Van Doormaal andRaithby (1985) have clearly indicated the convergence characteristics of theabove algorithms at regne grids


variants

321




EC203

322




aeue =P


anvn =P


9=

(2)


aeue =

Pau

nbunb + bu + ( p

P 2 pE )Ae

anvn =

Pav

nbvnb + b v + ( p

P 2 pN )An

9=

(3)


ae(ue + u0

e) =P

aunb(u

nb + u 0

nb) + bu + [( pP + p 0

P ) 2 ( pE + p0

E )]Ae

an(vn + v 0

n) =P

avnb(v

nb + v 0

nb) + b v + [( pP + p0

P ) 2 ( pN + p 0

N )]An

9=

(4)


aeu0e =

Pau

nbu0nb + ( p 0

P 2 p 0E )Ae

anv 0n =

Pav

nbv0nb + ( p 0

P 2 p 0N )An

9=

(5)


Pau

nbu0nb then



variants

323

u 0e = de( p 0

P 2 p 0E ) de = Ae

ae

v 0n = dn( p 0

P 2 p 0N ) dn = An

an

9gt=

gt(6)


e + de(p 0P 2 p 0

E ) andvn = v

n + dn( p0P 2 p 0


aPp 0P =

Xanbp

0nb + bP (7)






de =AeP

anb 2 ae

dn =AnP

anb 2 an(8)



e = de( p 0P 2 p 0




aedeDp 0e =

XanbdnbDp 0

nb + AeDp 0e (9)




aede =X

anbdnb + Ae (10)


EC203

324





n and dn equations


e de vn and dn







variants

325


RSMAX =SMAX

qm emass (11)



EC203

326


Aacute

node

X(

aeue 2

nb


)21=2

ru2m emom (12)






variants

327


EC203

328

Figure 2


variants

329



Figure 2

EC203

330




surface rotating


variants

331

Figure 3

EC203

332


Figure 3








variants

333





Nu (Ra = 10000) Time(s)


22 pound 22 2299 2299 2299 2299 1452088 181261 1572261 2443514

42 pound 42 2258 2258 2258 2258 4296177 334481 4556552 6269014

82 pound 82 2248 2248 2248 2248 4331228 503721 4761847 6138827

102 pound 102 2247 2247 2247 2247 112672 132009 1218552 1657884



EC203

334




variants

335


EC203

336




variants

337



SIMPLEP

aunbu

0nb = 0 Aeae

SIMPLERP

aunbu

0nb = 0 Aeae

SIMPLECP

aunb(u

0nb 2 u 0

e) = 0 Ae=(ae 2P

anb)SIMPLEX Dp 0

e = Dp 0nb aede =

Panbdnb + Ae




du (10 6) 1135 1135 5676 5634 05927 05927 2964 2928du (20 20) 1183 1183 5916 6060 05960 05960 2980 2979

82 pound 82


du (10 6) 02981 02981 1490 1474du (20 20) 02975 02975 1488 1488




du (12 7) 2500 2500 1250 1252 1929 1930 9643 9525du (22 22) 2064 2064 1032 8593 1874 1873 9368 9265

82 pound 82


du (12 7) 09999 09999 4999 4976du (22 22) 09612 09612 4803 4798


EC203

338







References









variants

339


















Further reading


EC203

340




EC203

322




aeue =P


anvn =P


9=

(2)


aeue =

Pau

nbunb + bu + ( p

P 2 pE )Ae

anvn =

Pav

nbvnb + b v + ( p

P 2 pN )An

9=

(3)


ae(ue + u0

e) =P

aunb(u

nb + u 0

nb) + bu + [( pP + p 0

P ) 2 ( pE + p0

E )]Ae

an(vn + v 0

n) =P

avnb(v

nb + v 0

nb) + b v + [( pP + p0

P ) 2 ( pN + p 0

N )]An

9=

(4)


aeu0e =

Pau

nbu0nb + ( p 0

P 2 p 0E )Ae

anv 0n =

Pav

nbv0nb + ( p 0

P 2 p 0N )An

9=

(5)


Pau

nbu0nb then



variants

323

u 0e = de( p 0

P 2 p 0E ) de = Ae

ae

v 0n = dn( p 0

P 2 p 0N ) dn = An

an

9gt=

gt(6)


e + de(p 0P 2 p 0

E ) andvn = v

n + dn( p0P 2 p 0


aPp 0P =

Xanbp

0nb + bP (7)






de =AeP

anb 2 ae

dn =AnP

anb 2 an(8)



e = de( p 0P 2 p 0




aedeDp 0e =

XanbdnbDp 0

nb + AeDp 0e (9)




aede =X

anbdnb + Ae (10)


EC203

324





n and dn equations


e de vn and dn







variants

325


RSMAX =SMAX

qm emass (11)



EC203

326


Aacute

node

X(

aeue 2

nb


)21=2

ru2m emom (12)






variants

327


EC203

328

Figure 2


variants

329



Figure 2

EC203

330




surface rotating


variants

331

Figure 3

EC203

332


Figure 3








variants

333





Nu (Ra = 10000) Time(s)


22 pound 22 2299 2299 2299 2299 1452088 181261 1572261 2443514

42 pound 42 2258 2258 2258 2258 4296177 334481 4556552 6269014

82 pound 82 2248 2248 2248 2248 4331228 503721 4761847 6138827

102 pound 102 2247 2247 2247 2247 112672 132009 1218552 1657884



EC203

334




variants

335


EC203

336




variants

337



SIMPLEP

aunbu

0nb = 0 Aeae

SIMPLERP

aunbu

0nb = 0 Aeae

SIMPLECP

aunb(u

0nb 2 u 0

e) = 0 Ae=(ae 2P

anb)SIMPLEX Dp 0

e = Dp 0nb aede =

Panbdnb + Ae




du (10 6) 1135 1135 5676 5634 05927 05927 2964 2928du (20 20) 1183 1183 5916 6060 05960 05960 2980 2979

82 pound 82


du (10 6) 02981 02981 1490 1474du (20 20) 02975 02975 1488 1488




du (12 7) 2500 2500 1250 1252 1929 1930 9643 9525du (22 22) 2064 2064 1032 8593 1874 1873 9368 9265

82 pound 82


du (12 7) 09999 09999 4999 4976du (22 22) 09612 09612 4803 4798


EC203

338







References









variants

339


















Further reading


EC203

340




aeue =P


anvn =P


9=

(2)


aeue =

Pau

nbunb + bu + ( p

P 2 pE )Ae

anvn =

Pav

nbvnb + b v + ( p

P 2 pN )An

9=

(3)


ae(ue + u0

e) =P

aunb(u

nb + u 0

nb) + bu + [( pP + p 0

P ) 2 ( pE + p0

E )]Ae

an(vn + v 0

n) =P

avnb(v

nb + v 0

nb) + b v + [( pP + p0

P ) 2 ( pN + p 0

N )]An

9=

(4)


aeu0e =

Pau

nbu0nb + ( p 0

P 2 p 0E )Ae

anv 0n =

Pav

nbv0nb + ( p 0

P 2 p 0N )An

9=

(5)


Pau

nbu0nb then



variants

323

u 0e = de( p 0

P 2 p 0E ) de = Ae

ae

v 0n = dn( p 0

P 2 p 0N ) dn = An

an

9gt=

gt(6)


e + de(p 0P 2 p 0

E ) andvn = v

n + dn( p0P 2 p 0


aPp 0P =

Xanbp

0nb + bP (7)






de =AeP

anb 2 ae

dn =AnP

anb 2 an(8)



e = de( p 0P 2 p 0




aedeDp 0e =

XanbdnbDp 0

nb + AeDp 0e (9)




aede =X

anbdnb + Ae (10)


EC203

324





n and dn equations


e de vn and dn







variants

325


RSMAX =SMAX

qm emass (11)



EC203

326


Aacute

node

X(

aeue 2

nb


)21=2

ru2m emom (12)






variants

327


EC203

328

Figure 2


variants

329



Figure 2

EC203

330




surface rotating


variants

331

Figure 3

EC203

332


Figure 3








variants

333





Nu (Ra = 10000) Time(s)


22 pound 22 2299 2299 2299 2299 1452088 181261 1572261 2443514

42 pound 42 2258 2258 2258 2258 4296177 334481 4556552 6269014

82 pound 82 2248 2248 2248 2248 4331228 503721 4761847 6138827

102 pound 102 2247 2247 2247 2247 112672 132009 1218552 1657884



EC203

334




variants

335


EC203

336




variants

337



SIMPLEP

aunbu

0nb = 0 Aeae

SIMPLERP

aunbu

0nb = 0 Aeae

SIMPLECP

aunb(u

0nb 2 u 0

e) = 0 Ae=(ae 2P

anb)SIMPLEX Dp 0

e = Dp 0nb aede =

Panbdnb + Ae




du (10 6) 1135 1135 5676 5634 05927 05927 2964 2928du (20 20) 1183 1183 5916 6060 05960 05960 2980 2979

82 pound 82


du (10 6) 02981 02981 1490 1474du (20 20) 02975 02975 1488 1488




du (12 7) 2500 2500 1250 1252 1929 1930 9643 9525du (22 22) 2064 2064 1032 8593 1874 1873 9368 9265

82 pound 82


du (12 7) 09999 09999 4999 4976du (22 22) 09612 09612 4803 4798


EC203

338







References









variants

339


















Further reading


EC203

340

u 0e = de( p 0

P 2 p 0E ) de = Ae

ae

v 0n = dn( p 0

P 2 p 0N ) dn = An

an

9gt=

gt(6)


e + de(p 0P 2 p 0

E ) andvn = v

n + dn( p0P 2 p 0


aPp 0P =

Xanbp

0nb + bP (7)






de =AeP

anb 2 ae

dn =AnP

anb 2 an(8)



e = de( p 0P 2 p 0




aedeDp 0e =

XanbdnbDp 0

nb + AeDp 0e (9)




aede =X

anbdnb + Ae (10)


EC203

324





n and dn equations


e de vn and dn







variants

325


RSMAX =SMAX

qm emass (11)



EC203

326


Aacute

node

X(

aeue 2

nb


)21=2

ru2m emom (12)






variants

327


EC203

328

Figure 2


variants

329



Figure 2

EC203

330




surface rotating


variants

331

Figure 3

EC203

332


Figure 3








variants

333





Nu (Ra = 10000) Time(s)


22 pound 22 2299 2299 2299 2299 1452088 181261 1572261 2443514

42 pound 42 2258 2258 2258 2258 4296177 334481 4556552 6269014

82 pound 82 2248 2248 2248 2248 4331228 503721 4761847 6138827

102 pound 102 2247 2247 2247 2247 112672 132009 1218552 1657884



EC203

334




variants

335


EC203

336




variants

337



SIMPLEP

aunbu

0nb = 0 Aeae

SIMPLERP

aunbu

0nb = 0 Aeae

SIMPLECP

aunb(u

0nb 2 u 0

e) = 0 Ae=(ae 2P

anb)SIMPLEX Dp 0

e = Dp 0nb aede =

Panbdnb + Ae




du (10 6) 1135 1135 5676 5634 05927 05927 2964 2928du (20 20) 1183 1183 5916 6060 05960 05960 2980 2979

82 pound 82


du (10 6) 02981 02981 1490 1474du (20 20) 02975 02975 1488 1488




du (12 7) 2500 2500 1250 1252 1929 1930 9643 9525du (22 22) 2064 2064 1032 8593 1874 1873 9368 9265

82 pound 82


du (12 7) 09999 09999 4999 4976du (22 22) 09612 09612 4803 4798


EC203

338







References









variants

339


















Further reading


EC203

340





n and dn equations


e de vn and dn







variants

325


RSMAX =SMAX

qm emass (11)



EC203

326


Aacute

node

X(

aeue 2

nb


)21=2

ru2m emom (12)






variants

327


EC203

328

Figure 2


variants

329



Figure 2

EC203

330




surface rotating


variants

331

Figure 3

EC203

332


Figure 3








variants

333





Nu (Ra = 10000) Time(s)


22 pound 22 2299 2299 2299 2299 1452088 181261 1572261 2443514

42 pound 42 2258 2258 2258 2258 4296177 334481 4556552 6269014

82 pound 82 2248 2248 2248 2248 4331228 503721 4761847 6138827

102 pound 102 2247 2247 2247 2247 112672 132009 1218552 1657884



EC203

334




variants

335


EC203

336




variants

337



SIMPLEP

aunbu

0nb = 0 Aeae

SIMPLERP

aunbu

0nb = 0 Aeae

SIMPLECP

aunb(u

0nb 2 u 0

e) = 0 Ae=(ae 2P

anb)SIMPLEX Dp 0

e = Dp 0nb aede =

Panbdnb + Ae




du (10 6) 1135 1135 5676 5634 05927 05927 2964 2928du (20 20) 1183 1183 5916 6060 05960 05960 2980 2979

82 pound 82


du (10 6) 02981 02981 1490 1474du (20 20) 02975 02975 1488 1488




du (12 7) 2500 2500 1250 1252 1929 1930 9643 9525du (22 22) 2064 2064 1032 8593 1874 1873 9368 9265

82 pound 82


du (12 7) 09999 09999 4999 4976du (22 22) 09612 09612 4803 4798


EC203

338







References









variants

339


















Further reading


EC203

340


RSMAX =SMAX

qm emass (11)



EC203

326


Aacute

node

X(

aeue 2

nb


)21=2

ru2m emom (12)






variants

327


EC203

328

Figure 2


variants

329



Figure 2

EC203

330




surface rotating


variants

331

Figure 3

EC203

332


Figure 3








variants

333





Nu (Ra = 10000) Time(s)


22 pound 22 2299 2299 2299 2299 1452088 181261 1572261 2443514

42 pound 42 2258 2258 2258 2258 4296177 334481 4556552 6269014

82 pound 82 2248 2248 2248 2248 4331228 503721 4761847 6138827

102 pound 102 2247 2247 2247 2247 112672 132009 1218552 1657884



EC203

334




variants

335


EC203

336




variants

337



SIMPLEP

aunbu

0nb = 0 Aeae

SIMPLERP

aunbu

0nb = 0 Aeae

SIMPLECP

aunb(u

0nb 2 u 0

e) = 0 Ae=(ae 2P

anb)SIMPLEX Dp 0

e = Dp 0nb aede =

Panbdnb + Ae




du (10 6) 1135 1135 5676 5634 05927 05927 2964 2928du (20 20) 1183 1183 5916 6060 05960 05960 2980 2979

82 pound 82


du (10 6) 02981 02981 1490 1474du (20 20) 02975 02975 1488 1488




du (12 7) 2500 2500 1250 1252 1929 1930 9643 9525du (22 22) 2064 2064 1032 8593 1874 1873 9368 9265

82 pound 82


du (12 7) 09999 09999 4999 4976du (22 22) 09612 09612 4803 4798


EC203

338







References









variants

339


















Further reading


EC203

340


Aacute

node

X(

aeue 2

nb


)21=2

ru2m emom (12)






variants

327


EC203

328

Figure 2


variants

329



Figure 2

EC203

330




surface rotating


variants

331

Figure 3

EC203

332


Figure 3








variants

333





Nu (Ra = 10000) Time(s)


22 pound 22 2299 2299 2299 2299 1452088 181261 1572261 2443514

42 pound 42 2258 2258 2258 2258 4296177 334481 4556552 6269014

82 pound 82 2248 2248 2248 2248 4331228 503721 4761847 6138827

102 pound 102 2247 2247 2247 2247 112672 132009 1218552 1657884



EC203

334




variants

335


EC203

336




variants

337



SIMPLEP

aunbu

0nb = 0 Aeae

SIMPLERP

aunbu

0nb = 0 Aeae

SIMPLECP

aunb(u

0nb 2 u 0

e) = 0 Ae=(ae 2P

anb)SIMPLEX Dp 0

e = Dp 0nb aede =

Panbdnb + Ae




du (10 6) 1135 1135 5676 5634 05927 05927 2964 2928du (20 20) 1183 1183 5916 6060 05960 05960 2980 2979

82 pound 82


du (10 6) 02981 02981 1490 1474du (20 20) 02975 02975 1488 1488




du (12 7) 2500 2500 1250 1252 1929 1930 9643 9525du (22 22) 2064 2064 1032 8593 1874 1873 9368 9265

82 pound 82


du (12 7) 09999 09999 4999 4976du (22 22) 09612 09612 4803 4798


EC203

338







References









variants

339


















Further reading


EC203

340


EC203

328

Figure 2


variants

329



Figure 2

EC203

330




surface rotating


variants

331

Figure 3

EC203

332


Figure 3








variants

333





Nu (Ra = 10000) Time(s)


22 pound 22 2299 2299 2299 2299 1452088 181261 1572261 2443514

42 pound 42 2258 2258 2258 2258 4296177 334481 4556552 6269014

82 pound 82 2248 2248 2248 2248 4331228 503721 4761847 6138827

102 pound 102 2247 2247 2247 2247 112672 132009 1218552 1657884



EC203

334




variants

335


EC203

336




variants

337



SIMPLEP

aunbu

0nb = 0 Aeae

SIMPLERP

aunbu

0nb = 0 Aeae

SIMPLECP

aunb(u

0nb 2 u 0

e) = 0 Ae=(ae 2P

anb)SIMPLEX Dp 0

e = Dp 0nb aede =

Panbdnb + Ae




du (10 6) 1135 1135 5676 5634 05927 05927 2964 2928du (20 20) 1183 1183 5916 6060 05960 05960 2980 2979

82 pound 82


du (10 6) 02981 02981 1490 1474du (20 20) 02975 02975 1488 1488




du (12 7) 2500 2500 1250 1252 1929 1930 9643 9525du (22 22) 2064 2064 1032 8593 1874 1873 9368 9265

82 pound 82


du (12 7) 09999 09999 4999 4976du (22 22) 09612 09612 4803 4798


EC203

338







References









variants

339


















Further reading


EC203

340

Figure 2


variants

329



Figure 2

EC203

330




surface rotating


variants

331

Figure 3

EC203

332


Figure 3








variants

333





Nu (Ra = 10000) Time(s)


22 pound 22 2299 2299 2299 2299 1452088 181261 1572261 2443514

42 pound 42 2258 2258 2258 2258 4296177 334481 4556552 6269014

82 pound 82 2248 2248 2248 2248 4331228 503721 4761847 6138827

102 pound 102 2247 2247 2247 2247 112672 132009 1218552 1657884



EC203

334




variants

335


EC203

336




variants

337



SIMPLEP

aunbu

0nb = 0 Aeae

SIMPLERP

aunbu

0nb = 0 Aeae

SIMPLECP

aunb(u

0nb 2 u 0

e) = 0 Ae=(ae 2P

anb)SIMPLEX Dp 0

e = Dp 0nb aede =

Panbdnb + Ae




du (10 6) 1135 1135 5676 5634 05927 05927 2964 2928du (20 20) 1183 1183 5916 6060 05960 05960 2980 2979

82 pound 82


du (10 6) 02981 02981 1490 1474du (20 20) 02975 02975 1488 1488




du (12 7) 2500 2500 1250 1252 1929 1930 9643 9525du (22 22) 2064 2064 1032 8593 1874 1873 9368 9265

82 pound 82


du (12 7) 09999 09999 4999 4976du (22 22) 09612 09612 4803 4798


EC203

338







References









variants

339


















Further reading


EC203

340



Figure 2

EC203

330




surface rotating


variants

331

Figure 3

EC203

332


Figure 3








variants

333





Nu (Ra = 10000) Time(s)


22 pound 22 2299 2299 2299 2299 1452088 181261 1572261 2443514

42 pound 42 2258 2258 2258 2258 4296177 334481 4556552 6269014

82 pound 82 2248 2248 2248 2248 4331228 503721 4761847 6138827

102 pound 102 2247 2247 2247 2247 112672 132009 1218552 1657884



EC203

334




variants

335


EC203

336




variants

337



SIMPLEP

aunbu

0nb = 0 Aeae

SIMPLERP

aunbu

0nb = 0 Aeae

SIMPLECP

aunb(u

0nb 2 u 0

e) = 0 Ae=(ae 2P

anb)SIMPLEX Dp 0

e = Dp 0nb aede =

Panbdnb + Ae




du (10 6) 1135 1135 5676 5634 05927 05927 2964 2928du (20 20) 1183 1183 5916 6060 05960 05960 2980 2979

82 pound 82


du (10 6) 02981 02981 1490 1474du (20 20) 02975 02975 1488 1488




du (12 7) 2500 2500 1250 1252 1929 1930 9643 9525du (22 22) 2064 2064 1032 8593 1874 1873 9368 9265

82 pound 82


du (12 7) 09999 09999 4999 4976du (22 22) 09612 09612 4803 4798


EC203

338







References









variants

339


















Further reading


EC203

340




surface rotating


variants

331

Figure 3

EC203

332


Figure 3








variants

333





Nu (Ra = 10000) Time(s)


22 pound 22 2299 2299 2299 2299 1452088 181261 1572261 2443514

42 pound 42 2258 2258 2258 2258 4296177 334481 4556552 6269014

82 pound 82 2248 2248 2248 2248 4331228 503721 4761847 6138827

102 pound 102 2247 2247 2247 2247 112672 132009 1218552 1657884



EC203

334




variants

335


EC203

336




variants

337



SIMPLEP

aunbu

0nb = 0 Aeae

SIMPLERP

aunbu

0nb = 0 Aeae

SIMPLECP

aunb(u

0nb 2 u 0

e) = 0 Ae=(ae 2P

anb)SIMPLEX Dp 0

e = Dp 0nb aede =

Panbdnb + Ae




du (10 6) 1135 1135 5676 5634 05927 05927 2964 2928du (20 20) 1183 1183 5916 6060 05960 05960 2980 2979

82 pound 82


du (10 6) 02981 02981 1490 1474du (20 20) 02975 02975 1488 1488




du (12 7) 2500 2500 1250 1252 1929 1930 9643 9525du (22 22) 2064 2064 1032 8593 1874 1873 9368 9265

82 pound 82


du (12 7) 09999 09999 4999 4976du (22 22) 09612 09612 4803 4798


EC203

338







References









variants

339


















Further reading


EC203

340

Figure 3

EC203

332


Figure 3








variants

333





Nu (Ra = 10000) Time(s)


22 pound 22 2299 2299 2299 2299 1452088 181261 1572261 2443514

42 pound 42 2258 2258 2258 2258 4296177 334481 4556552 6269014

82 pound 82 2248 2248 2248 2248 4331228 503721 4761847 6138827

102 pound 102 2247 2247 2247 2247 112672 132009 1218552 1657884



EC203

334




variants

335


EC203

336




variants

337



SIMPLEP

aunbu

0nb = 0 Aeae

SIMPLERP

aunbu

0nb = 0 Aeae

SIMPLECP

aunb(u

0nb 2 u 0

e) = 0 Ae=(ae 2P

anb)SIMPLEX Dp 0

e = Dp 0nb aede =

Panbdnb + Ae




du (10 6) 1135 1135 5676 5634 05927 05927 2964 2928du (20 20) 1183 1183 5916 6060 05960 05960 2980 2979

82 pound 82


du (10 6) 02981 02981 1490 1474du (20 20) 02975 02975 1488 1488




du (12 7) 2500 2500 1250 1252 1929 1930 9643 9525du (22 22) 2064 2064 1032 8593 1874 1873 9368 9265

82 pound 82


du (12 7) 09999 09999 4999 4976du (22 22) 09612 09612 4803 4798


EC203

338







References









variants

339


















Further reading


EC203

340


Figure 3








variants

333





Nu (Ra = 10000) Time(s)


22 pound 22 2299 2299 2299 2299 1452088 181261 1572261 2443514

42 pound 42 2258 2258 2258 2258 4296177 334481 4556552 6269014

82 pound 82 2248 2248 2248 2248 4331228 503721 4761847 6138827

102 pound 102 2247 2247 2247 2247 112672 132009 1218552 1657884



EC203

334




variants

335


EC203

336




variants

337



SIMPLEP

aunbu

0nb = 0 Aeae

SIMPLERP

aunbu

0nb = 0 Aeae

SIMPLECP

aunb(u

0nb 2 u 0

e) = 0 Ae=(ae 2P

anb)SIMPLEX Dp 0

e = Dp 0nb aede =

Panbdnb + Ae




du (10 6) 1135 1135 5676 5634 05927 05927 2964 2928du (20 20) 1183 1183 5916 6060 05960 05960 2980 2979

82 pound 82


du (10 6) 02981 02981 1490 1474du (20 20) 02975 02975 1488 1488




du (12 7) 2500 2500 1250 1252 1929 1930 9643 9525du (22 22) 2064 2064 1032 8593 1874 1873 9368 9265

82 pound 82


du (12 7) 09999 09999 4999 4976du (22 22) 09612 09612 4803 4798


EC203

338







References









variants

339


















Further reading


EC203

340





Nu (Ra = 10000) Time(s)


22 pound 22 2299 2299 2299 2299 1452088 181261 1572261 2443514

42 pound 42 2258 2258 2258 2258 4296177 334481 4556552 6269014

82 pound 82 2248 2248 2248 2248 4331228 503721 4761847 6138827

102 pound 102 2247 2247 2247 2247 112672 132009 1218552 1657884



EC203

334




variants

335


EC203

336




variants

337



SIMPLEP

aunbu

0nb = 0 Aeae

SIMPLERP

aunbu

0nb = 0 Aeae

SIMPLECP

aunb(u

0nb 2 u 0

e) = 0 Ae=(ae 2P

anb)SIMPLEX Dp 0

e = Dp 0nb aede =

Panbdnb + Ae




du (10 6) 1135 1135 5676 5634 05927 05927 2964 2928du (20 20) 1183 1183 5916 6060 05960 05960 2980 2979

82 pound 82


du (10 6) 02981 02981 1490 1474du (20 20) 02975 02975 1488 1488




du (12 7) 2500 2500 1250 1252 1929 1930 9643 9525du (22 22) 2064 2064 1032 8593 1874 1873 9368 9265

82 pound 82


du (12 7) 09999 09999 4999 4976du (22 22) 09612 09612 4803 4798


EC203

338







References









variants

339


















Further reading


EC203

340




variants

335


EC203

336




variants

337



SIMPLEP

aunbu

0nb = 0 Aeae

SIMPLERP

aunbu

0nb = 0 Aeae

SIMPLECP

aunb(u

0nb 2 u 0

e) = 0 Ae=(ae 2P

anb)SIMPLEX Dp 0

e = Dp 0nb aede =

Panbdnb + Ae




du (10 6) 1135 1135 5676 5634 05927 05927 2964 2928du (20 20) 1183 1183 5916 6060 05960 05960 2980 2979

82 pound 82


du (10 6) 02981 02981 1490 1474du (20 20) 02975 02975 1488 1488




du (12 7) 2500 2500 1250 1252 1929 1930 9643 9525du (22 22) 2064 2064 1032 8593 1874 1873 9368 9265

82 pound 82


du (12 7) 09999 09999 4999 4976du (22 22) 09612 09612 4803 4798


EC203

338







References









variants

339


















Further reading


EC203

340


EC203

336




variants

337



SIMPLEP

aunbu

0nb = 0 Aeae

SIMPLERP

aunbu

0nb = 0 Aeae

SIMPLECP

aunb(u

0nb 2 u 0

e) = 0 Ae=(ae 2P

anb)SIMPLEX Dp 0

e = Dp 0nb aede =

Panbdnb + Ae




du (10 6) 1135 1135 5676 5634 05927 05927 2964 2928du (20 20) 1183 1183 5916 6060 05960 05960 2980 2979

82 pound 82


du (10 6) 02981 02981 1490 1474du (20 20) 02975 02975 1488 1488




du (12 7) 2500 2500 1250 1252 1929 1930 9643 9525du (22 22) 2064 2064 1032 8593 1874 1873 9368 9265

82 pound 82


du (12 7) 09999 09999 4999 4976du (22 22) 09612 09612 4803 4798


EC203

338







References









variants

339


















Further reading


EC203

340




variants

337



SIMPLEP

aunbu

0nb = 0 Aeae

SIMPLERP

aunbu

0nb = 0 Aeae

SIMPLECP

aunb(u

0nb 2 u 0

e) = 0 Ae=(ae 2P

anb)SIMPLEX Dp 0

e = Dp 0nb aede =

Panbdnb + Ae




du (10 6) 1135 1135 5676 5634 05927 05927 2964 2928du (20 20) 1183 1183 5916 6060 05960 05960 2980 2979

82 pound 82


du (10 6) 02981 02981 1490 1474du (20 20) 02975 02975 1488 1488




du (12 7) 2500 2500 1250 1252 1929 1930 9643 9525du (22 22) 2064 2064 1032 8593 1874 1873 9368 9265

82 pound 82


du (12 7) 09999 09999 4999 4976du (22 22) 09612 09612 4803 4798


EC203

338







References









variants

339


















Further reading


EC203

340



SIMPLEP

aunbu

0nb = 0 Aeae

SIMPLERP

aunbu

0nb = 0 Aeae

SIMPLECP

aunb(u

0nb 2 u 0

e) = 0 Ae=(ae 2P

anb)SIMPLEX Dp 0

e = Dp 0nb aede =

Panbdnb + Ae




du (10 6) 1135 1135 5676 5634 05927 05927 2964 2928du (20 20) 1183 1183 5916 6060 05960 05960 2980 2979

82 pound 82


du (10 6) 02981 02981 1490 1474du (20 20) 02975 02975 1488 1488




du (12 7) 2500 2500 1250 1252 1929 1930 9643 9525du (22 22) 2064 2064 1032 8593 1874 1873 9368 9265

82 pound 82


du (12 7) 09999 09999 4999 4976du (22 22) 09612 09612 4803 4798


EC203

338







References









variants

339


















Further reading


EC203

340







References









variants

339


















Further reading


EC203

340


















Further reading


EC203

340

a comparison study of the convergence characteristics and ...nht.xjtu.edu.cn/paper/en/2003203.pdfa...

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