Benchmark Va Hl Davis 83 b

7/21/2019 Benchmark Va Hl Davis 83 b

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INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, VOL. 3 , 227-248 (1983)

NATURAL CONVECTION IN A SQUARE CAVITY:A COMPARISON EXERCISE

G. DE VAHL DAVIS

University of N.S. W., Kensington 2033, Australia

AND

I. P. JONES

A.E.R.E. Harwell OX1 ORA U. K .

SUMMARY

A number of contributed solutionsto t he p rob lemof laminar natural convection ina squa re cavity have

been com pared wi th what i s regarded as a so lu tionof high accuracy. Th e purposesof this exercise havebeen to conf i rm th e accuracyof th e bench mark solu t ion andto provide a basis for the assessmentofthe var ious method s and com puter codes used to ob ta in the contr ibuted solut ions .

KEY WORDS Comparison Natural ConvectionNumerical Methods Validation

INTRODUCTION

During the conference on Numerical Methods in Thermal Problems, which took place inSwansea in July 1979, Jones' proposed tha t buoyancy-driven flow in a square cavity withvertical sides which are differentially heated would be a suitable vehicle for testing andvalidating computer codes used for a wide variety of practical problems. Such problemsinclude reactor insulation, cooling of radioactive waste containers, ventilation of rooms, fire

prevention, solar energy collection, dispersion of waste heat in estuaries and crystal growthin liquids.

Following discussions at Swansea, we invitedz4 contributions t o th e solution of theproblem described below. A total of 37 contributions* from 30 contributors or groups ofcontributors in nine countries were received, and we are gratified by the world-wide interestexpressed in this project.

This paper summarizes and discusses the main features of the contributions, and providesa quantitative comparison between them and what we believe to be a high accuracy solutionsuitable for use as a bench mark. Full reprints of all the original contributions have beenpublished separately.sa A preliminary version of this paper was presented at a session of the2nd Conference on Numerical Methods in Thermal Problems.6 Since that preliminary pa perwas prepared, several contributors have taken the opportunity to provide improved results

and additional information. These further da ta a re given in full in Reference 5b and havealso been incorporated into the present paper.

Not c ounting multiple contributions, distinguished only by mesh size, from the same c ontributor(s).

027~-209~/83/03O227-22 O2.20983 by John Wiley Sons, Ltd.

Received 6 July 1982


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228 G . DE. VAHL DAVIS AND 1. P. JONES

D

THE COMPARISON PROBLEM2

Consider the two-dimensional flow of a Boussinesq fluid of Prandtl number 0 .71 in anupright square cavity described in non-dimensional terms by 0 == 1, 0 d d 1 with zvertically upwards. Assume that both components of the velocity are zero o n all the

boundaries, that the boundaries at z = 0 and 1 are insulated, aT/az 0, and that T 1 atx = O and T=O at x = l .Calculate the flow and thermal field for Rayleigh numbers, @ g AT D 3 / ~ v, f lo3, lo4, l o s

and lo6.

The following results should be supplied:

average Nusselt number, 6;;dzIx=. or 1

maximum and minimum local Nusselt numbers o n the hot wall, and their location;maximum vertical velocity o n the horizontal mid-plane and its location;maximum horizontal velocity o n the vertical mid-plane and its location;contour plots of th e velocity components and, if available, the stre am function, thepressure and the vorticity. These will be used only for a qualitative comparison and willnot be used to infer numerical values of these quantities.

In addition, contributions should include no more than two pages outlining the methodused and giving relevant computational details (grid, computer, C.P.U. ime, storage, etc.). Tofacilitate comparisons the non-dimensionalization used for presentation of results should usethe same reference scales as those described in Mallinson and de Vahl D a ~ i s . ~

T= T2

J=T~ -AT

1

T=Tl T= T2

J=T~ -AT

1

uxz O-DNototi%Non-dimensional temperature T =Non-dimensional coordinates x,z = x/D,L/D

Non-dimensional velocities u . w = uD/IC, wD/K

Thermal diffusivity K, kinematic vlscosrty v

Prandtl number Pr = v/K

Rayleigh number Ra =pgAT D3/hv

T- Tz

TI -T2

Figure 1. Notation

*A constraint few contributors found themselves able to satisfy


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NATURAL CONVECTION IN A SQUARE CAVITY 229

The notation, as specified in References 3 and 4, is summarized in Figure 1.After the submissions had been received and given preliminary consideration, we con-

tacted contributors with a request that they supply to us, for each of their solutions, the valueof the stream function at th e cavity mid-point and the value and location of the maximumstream function (where different from the mid-point value). We again sought information onthe C.P.U. ime used in obtaining the solutions.

BEST AVAILABLE SOLUTION

One of us (G de VD) used a finite difference (FTCS) method to solve the stream function-vorticity formulation of the equations o n successively finer meshes in an attempt to obtainthe right answers. Of course, the exact solutions were not found, and the uncertaintiesincrease with Rayleigh number. But the solutions obtained are probably the best currentlyavailable, and have been used as a bench mark to assess the contributions received. We areencouraged by the fact that at least two of the contributions agree well with the solutions weare presenting, and we feel that they have the same claim to be called the best.

Our opinion of the quality of the bench mark solution does not imply a judgement of thequality of the particular method employed to achieve it. It reflects the use of meshrefinement and Richardson extrapolation on a convergent method, together with a somewhatgenerous allocation of computer time.

The method, implemented in a program called FRECON, uses central differences on allspatial derivatives and forward differences on false transient time derivatives in the equationsfor all three variables (4, t and T . The finite difference approximations were solved byADI. The heat flux at the hot (or cold) wall was calculated by a three-point forward (orbackward) approximation to 8Tlax. The average value was found using Simpsons rule.

Full details of the solution and its derivation are presented in a companion paper. The

Table I. The bench mark solution

Ra

10 lo4 los loh

1.174

3-6490.813

3.6970.178

1.118

1.118

1.117

1-5050-092

0.6921

5.071

16.1780-823

19.6170.119

2.243

2.243

2.238

35280.143

0-5861

9.111

9.6120.285, 0.601

34.73

68.59

0.855

0.066

4.5 19

4.5 19

4-509

7.7170.081

0.7291

16.32

16,7500-151, 0.547

64.630.850

0.0379

8.800

8.799

8.817

17.925

219.36

0-0378

0.9891


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230 G. DE VAHL DAVIS AND I. P. JONES

bench mark values of the items requested from contributors (but not the contour maps,which are in Reference 9) appea r in Table I. I t is believed that this solution is in erro r by nomore than 1 per cent at Ra = lo6, and probably by less than a tenth of that at the lowerRayleigh numbers. As discussed below, and also in Reference 9, Nu,,, at Ra = l o 6 may be inerror by a slightly greater amount.

Attention is drawn to the fact that the values given in Table I for urnax,wmax,Nu,,,, Nu,,and max are not necessarily the extreme mesh point values. They were computed bynumerical differentiation, using a fourth-order polynomial approximation.

THE CONTRIBUTIONS

Of the 37 contributions received, 21 used second-order finite difference methods (FDM) ofwhich one incorporated a fourth-order deferred correction step; ten used finite elementmethods (FEM); there were t hree variations of Hermitian methods in one contribution, oneadaptive finite difference method, one Galerkin method, a nd one use of spline approxima-tions.

Table I1 summarizes the essential features of the methods used in the various contribu-

tions.* Th e names and addresses of the contributors ar e given in the Appendix.All but one of t he FEM, and ten of the FDM, solved equations for the primitive variables(PV). Nine FD M and on e FEM solved equations for stream function and vorticity ( , sdid the Hermitian and spline methods. Two contributions considered the velocity-vorticityequations u 5 and one is based o n the biharmonic equat ion; each of these used FDM.

A variety of mesh sizes and element distributions, both uniform and non-uniform, wasused. Several contributors submitted results for two or more mesh (or element) sizes, enablingan indication of the residual truncation er rors to be obtained, bu t none attempted extrapola-tion to zero mesh/element size. One used an adaptive method based on approximations ofdifferent order to estimate the errors. In some contributions, advantage has been taken ofsymmetry to reduce storage and solution time. The equivalent full mesh has, however, beengiven in Table IT.

As mentioned, contributors were invited to report the amount of C.P.U. time taken toobtain their solutions. It is clear from the response to this request that it is not an easy matte rto compare C.P.U. imes. Most contributors emphasized that their methods were not optimaland that the times could have been substantially reduced if they had tuned their codes oradjusted their solution parameters. Some supplied the total C.P.U. time to solve for aparticular Rayleigh number from a cold start; others used a previous solution at a lowerRayleigh number or on a coarser mesh as an initial estimate, presumably reducing the totalC.P.U. time consumed and perhaps also easing difficulties due t o instability. Some con-tributors specified the C.P.U. ime per iteration, without giving the total number of iterationsrequired. Others gave the total time for all runs.

Schonauer (private communication) has pointed out that an adaptive method automaticallysupplies an error estimate. The time for such a method should therefore be compared to thetotal C.P.U. ime spent to obtain results in which the contributor has reasonable confidence,i.e. including runs o n coarser grids or with different locations of grid points, etc. Certainlynone of the contributors supplied this information.

'The contribution from Cooper and Pepper was, through no fault of either t hem or us, received after o ur originalclosing date and was not included in Reference 6. In order t hat th e numbering system of Reference 6 can beretained, the information about their contribution has been listed at the end of Table I1 instead of being insertedamong the other FDMs.


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NATURAL CONVECTION IN A SQUARE CAVITY 23 1

Other considerations which affect a comparison between C.P.U. times are the compilerused, the size of core available, the amount of I/O, and the computer used (speeds can evenvary between computers with the same name and number, depending upon some characteris-tics of the operating system).

It has not been possible to summarize concisely the information on C.P.U. imes which hasbeen supplied to us, nor to draw from it any meaningful conclusions. It is therefore notpresented here. Instead, t he information may be found in the original contributions and thesupplements to some of them.sd*sb

Tables I11 to VI contain the quantitative results requested.* The results have beentabulated to the precision given in the original contributions. In cases where results for twoor more mesh or element sizes have been submitted, only those for the finest are reportedhere.

Because the horizontal boundaries of the cavity are adiabatic it should not, in principle,matter where the overall Nusselt number Nu is calculated. In practice, it does seem tomatter, and as contributors presented results at various locations, a comparison betweenthem is affected. The original specification of the problem in References 3 and 4 did not fixwhere Nu was to be calculated; the choice was left to the contributors. Because of queriesreceived, and an editorial constraint imposed upon us by the Journal of luid Mechanics, thespecification there2 (which is the one presented here) required that Nu should be calculatedon one of the vertical walls (it should not matter which). It is clear, however, that theone-sided formula required for calculating a temperature gradient at a wall is less accuratethan the central formula of the same formal order of accuracy which may be used at themid-plane or on any internal vertical plane. It could be argued that the average of t he valuescalculated at all such vertical planes would be more accurate than that at any one plane; andit is suggested in Reference 9 that the mid-plane value Nu,,2 is better still. For this reason, ifsome of these various estimates for Nu have been given, that quoted here is either the meanNusselt number u or, if available, the mid-plane value

As has been noted, t he bench mark extreme values of N u and of velocity were obtained byinterpolation. Those given by contributors, on the other hand, were not all computed byinterpolation; nor is it always clear from the contributions when interpolation was used. Itwas found, in generating the bench mark solution, that the interpolated values differed by upto 1 per cent from the closest of the adjacent mesh point values; and in one case-that ofwmaXat Ra = l o 6where the profile is very sharp near the maximum-a discrepancy of 6 percent was encountered. This injects an uncertainty into the comparisons and emphasizes thepotential deficiency of using an extreme mesh point value as a function extreme value.

COMPARISON OF THE CONTRIBUTIONS

It could be regarded as invalid or unfair to compare the different contributions with thebench mark solution for an assessment of their accuracy, since they were computed using a

variety of meshes. If a method is convergent, a more accurate solution can always beobtained by mesh refinement. Clearly, it could be argued, a 21 X 21 solution from contributorX cannot be expected to be as good as a 41 X 41 solution from contributor Y , particularly ifthe methods used are more or less the same.

And to a certain extent, that would be true. Accuracy should, perhaps, be compared on

*Not all contributors submitted all the information required.


6/22

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1 2 3 4 5 6 7 8 9 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3

B

e

C

e

dV

D

s

G

h

Gnh

I1

Gnh

I1

H Jo

I

Jo

I1

K

eI

Kbb

LQ

e

Lnh

P

e

Poa

Q R

na

R S Tee

Wo

B

s

D G

n

Hnc

L P Se

UoI

UoI1

Wine

K

eI

L

a

oux

R

I1

R

1

C

15

87

90

91

87

87

88

94

88

85

99

91

88

78

93

88

86

86

94

82

93

83

96

87

91

88

88

88

91

25

16

14

15

14

18

11

25

12

10

27

13

17

4 23

18

39

13

17

18

16

11

16

16

15

12

16

13

20

00

00

00

00

00

00

00

00

00

00

00

00

00

00

00

00

00

00

00

00

00

00

00

00

00

00

00

00

0

10

10

09

09

10

10

10

15

09

16

08

09

10

0 1 10

04

11

08

09

10

10

15

09

09

09

09

10

09

10

1 1 1 10

10

1 1 1 10

10

09

10

09

10

09

10

09

1 09

10

1 09

10

1 10

09

62

69

66

66

67

60

61

75

63

65

64

60

66

1 6 66

2

666

56

66

63

66

68

67

65

65

69

62

66

08

2

8

08

2

2

08

2

5

08

2

7

08

2

7

08

2

5

08

2

5

08

2

0

08

2

0

08

2

7

08

2

2

08

2

7

08

2

7

08

2

08

2

0

2

4

09

2

9

08

2

0

085

2

0

0

2

3

08

2

4

08

2

1

08

2

2

08

2

0

2

6

08

2

6

08

2

08

2

4

08

2

0

00

00

00

00

00

00

00

00

00

00

00

00

00

00

00

00

00

00

005

00

00

00

00

00

001

00

00

00

00

16

14

13

18

19

12

35

16

17

13

13

18

15

14

13

19

18

01

05

18

01

05

z

12

01

05

C?

13

01

05

c G

5 :

16

d 2

11

01

07

9

19

01

07

15

01

05

18

01

05

18

17

01

05

18

17

01

05

88

17

00

03

10

60

08

2

00

11


14/22

240 G. DE VAHL DAVIS AND I. P . JONES

the basis of equal C.P.U. cost, or storage-or even programming effort. However, the first twowere not generally reported, and the last is almost impossible to measure.

Having said that, it should be added that we acknowledge that not all contributors haveaccess to unlimited computing time. It is quite noticeable from the results and from thecomments of t he contributors that government laboratories generally seem to have thebetter computing facilities; and that many contributors took much time and trouble toparticipate in what was a peripheral activity for them. Indeed, several were under quitesevere pressure to complete other activities (such as doctoral theses) and were unable todevote as much effort to this project as they might have wished.

Quon* has gone to some trouble to suggest that a comparison of solutions on the basis ofgrid point maxima is inadequate and misleading, and that it is necessary to obtain thecorrect functional form of all field variables. We certainly agree that it would be best to beable to compare all features of t he contributions; but this is clearly not possible. We soughtto compare those aspects of a solution which authors tend to publish (because they havesome practical significance). Maxima are important, and are published. If contributors usedinterpolation to obtain their maxima, so much the better. If not, we could only compare gridpoint maxima.

Contributors were invited to submit solutions to a specified problem. No restrictions wereplaced on method, mesh or effort. It was left to contributors to determine these forthemselves in ways which would yield the solution. It was left to contributors, for example, todecide how to obtain Nu,,,: whether or not to use interpolation; if so, which scheme; etc.

It was, of course, clear that a comparison between contributions was to be made. We arethus entitled to consider that contributors have reached their own compromises betweenaccuracy and effort (or cost), and have sent us what they regard as the answer to theproblem.

In comparing the contributions, therefore, little consideration has been given to the factthat almost any of the methods would have yielded better solutions if only a finer mesh or amodified method had been used. We have compared what we have been given.

It should be mentioned in this context that the fourth-order deferred correction method of

de Vahl Davis and Leong (number 3 in Table I1 etc.) benefited from inside knowledge: meshrefinement was not used to achieve convergence in isolation, but also to attempt toreproduce the known bench mark solution. It is therefore not entirely surprising that it hasperformed well. Moreover, the contributions of Jones, Thompson and Woodhouse I1(number 9), Linthorst and Schinkel (13), Phuoc and Tanner (27) and Kessler and Oer tel (32)were substantially improved in their later submission^.^^

The results of Rheinlander differ very considerably from the rest of the results presentedin Tables 111-VI. Th e reason for this is that he used a k--E turbulence model, his principalinterest being the simulation of time-dependent turbulent flow. He thus, in essence, solved adifferent problem. Since his results could be of value to others interested in his problem, theyhave been retained in the tables but have not been used in the comparison with the benchmark solution. It should be noted that his turbulence model was not disabled even at low Ra.

Tables VII-X contain the percentage relative differences between certain characteristics ofthe contributions and the corresponding characteristics of the bench mark solutions at thedifferent Rayleigh numbers. The positions of the various quantities have been omitted fromthis comparison, both in order to reduce the volume of data and because they are oftendifficult to determine accurately, particularly if the maximum is flat. Note that t he use ofrelative error magnifies the errors in Nu,,, in comparison with the errors in nd Nu,,,.Note also that the bench mark value of NuIR has been used for comparison with the


15/22

NATURAL CONVECTION IN A SQUARE CAVI'IY 24 1

Table VII. Percentage errors at Ra = 103

1

23456789

101112131415161819202122232425262728293031

323334353637

Bertell 0.2Cuvelier -0.5de Vahl Davis -0.2Gunther I 0.7Gunther I1 0.0Gunther I11 0.2Hackbusch 0.0Jones I -0.1Jones 11Kessler I 0.2Kublbeck 0.1Lequere -0.5Linthorst 0.2Portier 2.2Pro ahn 2.0

Quon 0-9Rue1 0.2Schonauer 0.0Thiele 0 - 1

Betts 0.2Donea -0-3

Heinrich -0.8Lava1 1-8Phuoc 1.1Stevens -0.2Upson I 0-0Upson I1 0.0

Wong 0.2

Gartling 0.0

Winters 0.2Kessler I1 0.0Lauriat 0.0Roux I 0.0Roux I1 -0.2Roux I11 0-6Cooper 0.2

1.0

-1.0-0.7

0.40.20.80.10.1

1.7-1.0-1.0

0.81.32.3

-0.80-30.10.4

-0.3-0.3-0.1

0.1-1.2

2.0-0.3-0 -5

0.10.10.30.10.70-3

-0.62.71.0

0.9-0.3

0.7-1.2-0.2-0.8

0.00.0

-0.35.51-2

-0.4-1.0-2.0

5.2-0.30.0

-0.4-0-1-0.3-0.6-0.1

0.8-0.2-1.7

-0.1-0.1-0.1-0.1-0.7-0.3

0.7-1.3-1.7

0.0-1.3

0.0-0.1

0.0-0.1

0.00.0

1-4-1.6-2.7-0.5

0.31.4

-1.6-9.60.00.0

-0.5-0.2

2.2-0.2-4.9-7-4

0-60.3

0.2-0.2

0.00-4

-0.1-0.6-0.7

2.8

0.4

0.60.0

-0.5-0.1-0.2-0.2

0.0

0.11.3

-1.5-0.5-0.2

2.0

-0.20.10.0

-0.2-0.5

0.11.60.0

-6,4-5-7

0.10.6

0-2-0.2

0.00.4

-0.2-1.2-1.3-1.8

1-4

0.0

0.90-4

-0.8

-2.6

-1.5-0.1

-0.30.00.50.0

0-5-12.3-0.3

0.10.10.1

-0.1-0.3-0.4-0,3

contributed average Nusselt numbers, since it is believed to be the most accurate estimate ofthe average.

The first general conclusion to be drawn is that although there are accurate contributionsusing both FEM and FDM, the former are by and large rather better. There is a lowertendency among the FEM entries towards a degradation of performance with increasing

Rayleigh number and a much lower number of contributions containing obvious majorerrors. The FEM was also better able to cope with the higher Rayleigh number, only one(27) failing to supply answers for R a = lo6 (the stated reason being cost).

It is tempting to attr ibute this superior performance to the more prevalent use of anon-uniform distribution of grid points in the FEM. However, those FDM which have used anon-uniform mesh (9, 12-16, 18 and 19) have not, on the whole, performed better thanthose which used only a uniform mesh. This conclusion was very surprising to us, since there


16/22


Table VIII. Percentage errors at Ra = lo

123456789

101112131415

161819202122232425262728293031323334353637

Bertelg

Cuvelierde Vahl DavisGunther IGunther I1Gunther 111HackbuschJones IJones 11Kessler IKublbeckLeyuereLinthorstPortierProjahnQuonRue1SchonauerThieleWongBettsDoneaGartlingHeinrichLava1PhuocStevensUpson IUpson 11Winters

Kessler I1LauriatRoux IRoux I1Roux 111Cooper

2.51.10.00.20.30.60.40.0

1.22.2

-0.50.65.61.2

1.00.80.20.8

-0.12-11.10.3

-2.110-3-1.0-0.1

0.20.10.30.10.20.30.1

0.8

5.4

0.1-0.6-0.1

0.80.91.10.6

4.03.50.61.23.73.2

-2.625.0

0.41-90.12.00.30-3

-3.06.2

-0.50.10.20.20.3

0.01.50.50.5

2.9

1.0

-7.80.5

-1.9-0.7

0.9-0.4

0.0

7.510.9

4.10.5

-1.40.7

16.022.9-0.3-1.9

0.50.7

-0.60.2

-0.49.8

-21.5

-0.1-0.2-0.2

0.0-1.4-0.6

0.5

-3.9

-0.2

-3.30.1

-0.2-0.4-0.5

6.10.3

-0.50.4

-3.9-0.9

0.10.1

-0.5-1.1

0.10.7

-0.5-0.5

3.20.0

-2.0-5.8

0.30.1

0.10.10.10.5

-0.1-2.0

-1.1

-0.2

-0.70.1

-0.3-0.5-0.5-0.4

0.1

-0.6-1.5-2.3-0.6-0.4-0.3

-0.32.0-0-4-1.0-0.6

1-52.10.11.61.8

-0.50.4

0.30.4

0.11.50.0

-1.9

-2.6

1.0

0.2

1.10.8

1.4

-2.4

-0.4

0.0

-0.20.02.50.1

0 7

2.00.4

0.00.1

0-1

0.0-0.7

0.4

is a popular theory (to which we subscribe) that a denser distribution of mesh points insuitably chosen locations will lead to improved accuracy-provided, of course, that theconsequent coarse distribution elsewhere does not introduce a countervailing contaminationof the solution.

Most of the non-uniform grid results, however, have been obtained using fewer grid

points than were used with the finest uniform grids. This presumably gives roughly the samenumber of grid points within the boundary layers for the higher Rayleigh number cases. Thismatter has also been discussed in detail by Quon'' in a comparison of his own results withsome of our contributions. It is clear from the present study and from Quon's paper thatthere is scope for more work on the use of co-ordinate stretching and selective meshrefinement at high Rayleigh numbers for both finite difference and finite element methods.

Some of the FEM tended to do well o n velocities, but not so well on heat transfer rates,


17/22


Table TX. Percentage errors at Ra = 10'

1

23456789

101112131415161819202122232425262728293031323334353637

Bertels

Cuvelierde Vahl DavisGunther IGiinther TIGunther TITHackbuschJones IJones TKessler IKublbeckLequereLi nthorstPortierProjahnQuonRue1SchbnauerThieleWongBettsDoneaGartlingHeinrichLavalPhuocStevensUpson IUpson ITWinters

Kessler TILauriatRoux IRoux TRoux T TCooper

8.4

6.20.01.21.00.41.80.1

2.6-1.1

0-47.73.10.42.00.2

-0.4-0.4

2.2-4.6

1.6-1.9

3-7-9.7-0.5

0-90.00.2

0.01-30.23.0

0.7

16.9

12.6-1.5

0.02.91.76.21.3

8.51.3

-0.18.74-8

-1.95.60.7

14.6-0.1

2.8-3.1

2.0-3.7

6.0-5.7

1.91.20.20.3

0.35.20.36.9

-0.3

-5.3

-46.51.4

-6.8-2.4

1.9-2.1

0.7

27.612.5

8.4-3.3

0-110.8

1.50.5

-12.44.50.10.11.10.69.5

-355

-0.1-0.2-0.3

0-8-1.5

1 o-0.2

-7.1

-0.2

-3.50.2

13.6-0-1

1.01.31.8

5.84.70.31.20.20.20.80.3

18-7-0.4

0.0-0.2

0.01.8

-0.42.1

-1.5-0.3-0.3

0.2

-0-11.9

-0.1-2.6

0.8

-2.9

-4.10.12.2

-1.2-1.0-0.9

0.1

-1.0-0.2

0.4-0.7

0.3-0.5

0.60.3

-3.8.6

-0.43.20.12.70.65.22.10.40-40.0

-0.32.1

-1.2-3.8

14.7

3-2

0.1

3.21.9

6.4

4.5

-0.1

0.4

0.4-0.3

1.6

0.4-3.1-0.1

0.10.0

3-0

0.1

4.02.0

5.3

-4.8

-0.7

0-3

0.10.4

-0.1

-0.2-1.2

0-30.10-10.3

0.1

0.0-0.9

4.0

particularly at the higher Rayleigh numbers, e.g. Betts and Lidder (22), Gartling (24), Doneaand Giuliani (23) and Laval (26). Th e consistent flux method of Upson et al. (30) gave moreaccurate mean Nusselt numbers than their more conventional Gauss point method (29). Thiswas also true for their values of Nu,,,, except at R a = lo6. As already noted, we are lessconfident of the accuracy of the bench mark value of Nu,,, at l o 6than we are of the othercharacteristics of the bench mark solution. And we note that two contributions-those ofUpson et al. (30) and Kessler and Oertel (32)-which otherwise agree well with the benchmark, have values which agree with each other (within 0.3 per cent), suggesting a value forNk , , of about 17.3, some 3.5 per cent below the bench mark value. However, two othersolutions-those of Quon (16) and Winters (3l)-which are almost as good, support a largervalue (say 17.7, or 1-3 per cent below the bench mark value). Stevens (28) and Winters (31)also consistently obtained accurate mean Nusselt numbers with their methods.


18/22

244 G. DE VAHI. DAVIS A N D I. P. JONES

Table X. Percentage errors a t Ra = lo6

FirstNo. author N u Nu,,,, Nu,,, u r n a x Wmax 44. JI

1

23456789

101112131415161819202122232425262728293031

323334353637

Berteli 20.0

Cuvelierde Vahl Davis 0.2Gunther I 3.0Gunther I1 3-8Gunther 111 -0.5HackbuschJones I -0.2Jones I1 0.9Kessler 1Kublbec k 7.6Lequere 0.6Linthorst -3.2Portier 12.8Projahn 4.3Quon 0.8Ruel 5.8Schonauer 0.5Thiele -1.2

Betts 6.9Donea -6.4Gartling 6.6Heinrich -5.2Lava1 10-2PhuocStevens -0.4Upson I 4-2Upson I1 0.2Winters 0-4

Kessler I1 0-5Lauriat 4.3Roux IRoux I1Roux I11Cooper 0.9

Wong -1.2

14.7

4-08.59-22.8

-5.81.4

25.91.9

-5.215.8

8.0-0.913.2

-0.6123.0-3.2-6.4

-16.93.9

-15.69.6

-1.53 . 3

-3.5-1.8

-3.217.5

-1.3

10.7

7.7--8.9-8.7

2.7

2.21-9

55.70.1

69-3-15.9

-5.02.21.16.0

-53.518.3

-14.1-2.6

1.87.0

51.9

-0.5-0.9-1.4-0.3

4.9

-63.6

-0-6

0.50.0

-3-11-7

6.9-0.7

12.25.73.01.3

-1.01-62.11.5

300.1-3.1

-10.81.7

-0.40.005.0

3.2-0.1-0.1-1.1

0.97.8

0.6

10.3

0.9-0.8-2.1-0.3

0.10-6

0.3-2.9

1.5-1.0

0.20.22.1

-1.3-4.8-6.5-2.9

3.2-0.4

1.7-0.5

0.70-60.61.2

0.57.2

-3-8

7.8

0.6

6.43-2

10.2

1.7

-0.4

2.2

2.30.40-4

3.1

1.2

0.7

0.1

7.2

0.5

6.33.0

9.7

-2.4

-0.5

4.1

1.2-1.3

0.6

2.3

0.6-0.3-0.3

0.3

0-2

2.1

There is too much scatter in the finite difference results to enable any general conclusionsto be drawn. For example, Portier et al. (14) and de Vahl Davis and Leong 3 ) gave resultswhich have larger errors for Nu than for the velocities whereas th e results for Jones et al. (8,9) and Le Quere and Humphrey (12) generally had more accurate values for Nu.

Several contributions are, at least for some values of Ra, in general agreement with the

bench mark but contain one or two features which are significantly in error. These includeGunthers (4) value of u,,, at Ra = lo5; Ruel, Grand and Latrobes (18) value of u,,, at lo,and their Nu,,, and Numln at lo ; huoc and Tanners (27) value of at 10; and Quons(16) value of Nu,,,-a difficult quantity to obtain accurately-at each Ra except, curiously,at lo6, where his value is quite good.

Many contributions suffer from declining quality with increasing Rayleigh number, oftendespite the accompanying and counteracting use of mesh refinement. Particularly notable in


19/22


this respect are the contributions of BertelB l ) , Kublbeck and Straub ( l l ) , Thiele (20) andLauriat 33), although several others display this defect to a lesser extent. Thiele had thisdifficulty despite going to a 65 X6S mesh; he was also unable to obtain a steady solution atRa = lo6.

The non-Boussinesq ideal gas formulation of Le Quere and Humphrey (12) gave answerswhich were generally in good agreement with those for a Boussinesq fluid, particularly forthe overall Nusselt number. They used 293K and 273K as the hot and cold boundarytemperatures respectively; this temperature difference is below but close to the limit ofvalidity** of the Boussinesq approximation used in the bench mark and all other solutions.

Roux, Bo nt ow , Gilly and Grondin used three methods (34-36). The first, fully fourth-order, is the most accurate but was only employed at RaG 10'; it would be interesting to seeif its accuracy is maintained at R a = 10'. With a return to a second-order method for andthen also (at Ra= 103 only) for T, the quality of their results deteriorated.

The results of Hackbusch (7) are consistently fair, apart from umax t lo4 and Nu,,, at lo',which are high. His method-using the biharmonic equation for +--has the attraction ofavoiding the need for a vorticity boundary condition. However, it was unable to yield asolution at Ra = lo 6. It was also the only method that was noticeably very fast, e.g. 3.8 s on a

CDC Cyber 70/76 for a 33 X 33 grid at Ra= lo', including plotting time. Solution timescould have been reduced even further with the multi-grid method by requiring only twodigits of accuracy.

The method of Lava1 (26) achieved an accuracy which was more or less independent ofRayleigh number. The accuracy is not high-it is of the order of several per cent-but it isreasonable in the light of the coarse mesh used. Projahn and Rieger (15) also obtainedresults of fairly uniform (and somewhat better) quality. Gunther's third method (6) did evenbetter, although there appears to be a slight deteriorat ion of quality with increasing Ra. Thefourth-order deferred correction method of de Vahl Davis and Leong (3) did well, with thebenefit of knowledge of the bench mark solution as a goal.

Quon 16), as mentioned above, had difficulty with Nu,,,, but was otherwise within one ortwo per cent of the accurate solution, as were Jones t al. 8, 9), Stevens (28), Winters (31)

and Cooper and Pepper (37). Quon ' has also published additional results using a 60 x 6 0non-uniform grid for Ra = 10'. They are better than his values which are presented here andtheir accuracy is high. Gartling's velocities (24) were accurate but his Nusselt numbers werenot good at 10'.

Schonauer and Raith (19), and Wong and Raithby (21) each estimated the accuracy oftheir submissions to be about 1 per cent at Ra lo5;Schonauer and Raith also predicted a 4per cent error at 10'. If these two solutions are compared with the bench mark, the errorestimates will be seen to be quite reliable.

The best contributed results were achieved by the E M of Upson, Gresho and Lee(29 ,30) and the Galerkin method of Kessler and Oertel (32). In most cases their resultsagree with ours to better than 1 per cent. The average and extreme values of wall heat flux atthe higher Rayleigh numbers were the characteristics which agreed least well. It is perhaps

significant that Upson et al. took considerable care to provide a high density of grid points inthe wall and corner regions of the cavity.

CONCLUSIONS

We have presented here a summary of the results of a comparison exercise which is intendedto provide a basis for the assessment of numerical methods for the solution of problems of


20/22

246 G. DE VAHL DAVIS AND 1. P. JONES

buoyancy-driven flow. It is extremely gratifying that so many of the numerical results aresubstantially in agreement with each other, and that those which one would expect to givemore accurate results (e.g. higher order methods and those with more grid points) do so.This, in itself, has enabled the principal aim of our exercise to be met: the generation of ahigh accuracy solution which has been validated by several independent calculations. We feel

that this agreement between solutions allows us to assert with some confidence that thebench mark solution, and of course those which agree with it closely, represent an accuratesolution of the problem. It is to be hoped that those wishing in the future to verify theiralgorithms and programs will compute results for the same standard set of parameters toenable an objective assessment to be obtained. It is also to be hoped that our contributors,and others interested in problems of this type, will now set out to improve their existingalgorithms and throw light on many of the interesting features that have emerged from thisstudy. Certainly very few of us have any cause to be complacent.

It is invidious to seek best methods or winners from an exercise such as this; and it was notour aim to do so. It is, however, pleasurable to repor t tha t th e three most accurate results forall the parameter values were provided by a finite element method, a Galerkin method and afinite difference method, viz. the contributions of Upson, Gresho and Lee (29, 30) and of

Kessler and Oerte l (32) together with the bench mark itself. Many others were in closepursuit but, because of the scatter in the errors, it is not possible to separate them intodifferent categories.

Finally, we would again like to express our sincere thanks to the contributors and t o manyothers for their active encouragement, interest and patience throughout the course of thisexercise.

APPENDIX

Following are the names and addresses of all contributors, in alphabetical order of firstcontributor but numbered in accordance with Table TI;1

22

37

2

3

23

24

4-6

7

2s

M. Bertelri, UniversitA di Firenze, Facolta di Tngegneria, Istituto di Energetica,Firenze, Italy.

P. L. Betts and J. S. Lidder, Department of Mechanical Engineering, UMIST,Manchester M60 1QD, U.K.R. E. Cooper and D. W. Pepper, Environmental Transport Division, E.I. du Pont deNemours and Co., Savannah River Laboratory, Aiken, SC, 29801, U.S.A.C . Cuvelier, Department of Mathematics, University of Delft, Delft, The Nether-lands.G. de Vahl Davis and S . S . Leong, School of Mechanical and Industrial Engineering,University of New South Wales, Kensington, 2033, Australia.J. Donea and S. Giuliani, Applied Mathematics Division, Commission of theEuropean Communities, Joint Research Centre, Ispra Establishment, Italy.D. K. Gartling, Fluid Mechanics and Heat Transfer Division, Sandia Laboratories,Albuquerque, NM, 87185, U.S.A.

C. Giinther, Kernforschungszentrum Karlsruhe, Institut fur Reaktorbauelemente,Postfach 3640, 7500 Karlsruhe 1, West Germany.W. Hackbusch, Ruhr-Universitat Bochum, Mathematisches Tnstitut, Postfach102148, 4630 Bochum 1, West Germ any.J. C. Heinrich, Department of Aerospace and Mechanical Engineering, Universityof Arizona, Tuscon, AZ, 85721, U.S.A.M. Strada, Facolta di Fisica Tecnica, Via Marzolo 9, 35100 Padova, Italy.


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N AT U R A L C O N V E C T I O N IN ASQUARE CAVITY 247

0 C. Zienkiewicz, Department of Civil Engineering, University of Swansea, Swansea,SA2 8PP, Wales, U.K.I. P. Jones and C. P. Thompson, Computer Science and Systems Division, AtomicEnergy Research Establishment, Harwell, OX1 1 ORA, U.K.E. A. Woodhouse, F International Ltd., The Bury, Church Street, Chesham,Buckinghamshire, U.K.

10,32 R. Kessler and H. Oertel Jr., Institut fur Stromungslehre und Stromungsmaschinen,

8-9

11

33

26

12

13

27

14

15

16

17

34-36

18

19

28

2029-30

31

21

Universitat Karlsruhe, Germany.K. Kublbeck and J. Straub, Lehrstuhl A fur Thermodynamik, TechnischeUniversitat Munchen, Postfach 202420, 8000 Munchen 2, Germany.G. Lauriat, Laboratoire de Thermique, CNAM, 292 rue Saint Martin, 75141 ParisCedex 03, France.H. Laval, Applied Mechanics Division, Commission of the European Communities,Joint Research Centre, Ispra Establishment, Italy.P. Le Quere and J. A. C. Humphrey, Department of Mechanical Engineering,University of California, Berkeley, CA, 94720, U.S.A.S . J. M. Linthorst and W. M. M. Schinkel, Department of Applied Physics, DelftUniversity of Technology, Delft, The Netherlands.H. B. Phuoc and R. I. Tanner, Department of Mechanical Engineering, Universityof Sydney, Sydney, 2006, Australia.J. J. Portier and M. P. Fraikin, Institute of Thermodynamics, State University ofLiege, Liege, Bel, wum.0 A. Arnas, Department of Mechanical Engineering, Louisiana State University,Baton Rouge, LA, U.S.A.U. Projahn and H. Rieger, Technische Hochschule Darmstadt, Institiit furTechnische Thermodynamik, Petersenstrasse 30, 6 100 Darmstadt, Germany.C. Quon, Ocean Circulation Division, Atlantic Oceanographic Laboratory, BedfordInstitute of Oceanography, P.O. Box 1006, Dartmouth, NS, B2Y 4A2, Canada.J. Rheinliinder, Technische Universitat Berlin, Fachbereich Univelttecknik (FB21),Hermann-Rietschel-Institiit fur Heizurps und Klimstechnik, Sekr. HL45, March-strasse 4, D-1000 Berlin 10, Germany.B. Roux, P. Bontoux, B. Gilly and J. C. Grondin, Institut de Mkcanique des Fluidesde Marseille, 1 rue Honnorat, 13003 Marseille, France.F. Ruel, I . Grand and A. Latrobe, Centre dEtudes Nuclkaires de Grenoble,Services des Transferts Thermiques, 85X, 38041 Grenoble, Cedex, France.W. Schonauer and K. Raith, Rechenzentrum der Universitat, 0- 750 0 Karlsruhe 1,West Germany.W. N. R. Stevens, Department of Electrical and Electronic Engineering, Universityof Nottingham, Nottingham, U.K.F. Thiele, Technische Universitiit Berlin, Sekr. H F 1, D-1000 Berlin 12, Germany.C. D. Upson, P. M. Gresho and R. L. Lee, Lawrence Livermore Laboratory,

University of California, Livermore, CA , 94550, U.S.A.K. H. Winters, Theoretical Physics Division, Atomic Energy Research Establish-ment, Harwell, OX11 ORA, U.K.H. H. Wong, Analysis and Modelling, Contract Engineering, Atomic Power Divi-sion, Westinghouse Canada Ltd., Hamilton, Ontario, Canada L8N 3K2.G. D. Raithby, Thermal Engineering Group, Department of Mechanical Engineer-ing, University of Waterloo, Waterloo, Ontario, Canada, N2L 3G1.


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REFERENCES

1. 1. P. Jones, A comparison problem for numerical methods in fluid dynamics: the double-glazing problem,Numerical Methods in Thermal Problems, R. W. Lewis and K. Morgan (eds.), Pineridge Press, Swansea, U.K.,

J . Fluid Mech., 95 4, inside back cover (1979).Computers and Fluids, 7 , 315 (1979).Numerical Heat Transfer, 2 , 395 (1979).

an Enclosed Cavity, AERE-R9955, HMSO, 1981.

G . de Vahl Davis and 1. P. Jone s, Natural convection in a sauare cavitv-a comuarison exercise. Numerical

1979, pp. 338-348.2.3 .4.5a. I. P. Jones and C. P. Thompson (eds), Numerical Solutions for a Comparison Problem on Natural Con vection in

Sb. I. P. Jones and C. P. Thompson, ibid., Supplement l(1981).

Methods in Thermal Problems, vol. TI, R. W. Lewis, K. Morgan and B: A. Schrefler (eds.), Pineridge Press,Swansea, U.K., 1981, pp. 552-572.G. D. Mallinson and G . de Vahl Davis, Three-dimensional natural convection in a box: a numerical study, 1Fluid Mech., 83, 1-31 (1977).G. de Vahi Davis, Program FRECON for the numerical solution of free convection in a rectangular cavity,Report 1976/FMT/1 Univ. N.S.W., School of Mech. and Indnst. Eng. (1976).G. de Vahl Davis, Natural convection of air in a square cavity: a bench mark solution, Int. j numer. methods

C. Quon, Effects of grid distribution on the computation of high Rayleigh number convection in adifferentially heated cavity, R o c . 2nd Nat. Conf. Numerical Meth. in Heat Transfer, Hemisphere PublishingCorp., Washington, D.C., U.S.A. (1981).E. Leonardi and J . A. Reizes, Natural convection in a compressible fluid with variable properties, NumericalMethods in Thermal Problems, R. W. Lewis and K. Morgan (eds.), Pineridge Press, Swansea, U.K., 1979. pp.

fluids, 3, 249-264 (1983).

297-306.

NOTE AD DE D IN PROOF:

Since the submission of this paper, two additional computations have come to o ur att ention.1) P. Le Quere and T. Alziary de Roquefort C . R . Acad . Sc. Paris, 294 Series 11,

941-944, 1982 ) solved th e PV equations using Chebychev polynomials with a semi-implicitspectral method. With 17 x 17 nodes, their results were good: errors of less than 0 .3 per centin the variables of Tables VII-X at R aS 1 0 5 , although up t o several per cent at lo6; with33 X 33 nodes they were excellent: less than 0 .2 per cent for R a S o 5 and (except for Nu,,,),less than 1 per cent at lo6. Their value of Nu,,, was 17.553 (with 33x33 nodes),somewhere between th e bench mark value and th e values computed by Upson et al. (30) andKessler et al. (32).

(2) N. C. Markato s and K. A. Pericleous Report PDRICHAM UKl16, Cham Ltd,London, 1982) also solved the PV equations using an upwind finite domain code im-plemented in a commercially available program called PHOENICS. The grids used were30 x 30 at 10; 40 x 40 at lo4 and lo5;and 80 x 80 at lo6;all non-uniform. The results areonly fair with errors in velocities of 5 per cent at R a = lo, up to 1 2 per cent in u at 10(but only 0 - 2 per cent in w t 10). Errors in NU^,^ and Nu,,, range from 1 to 7 per cent;they appear to have been obtained by graphical, rather than numerical, differentiation.Errors in Nu,,,, not surprisingly under those circumstances, are up to 23 per cent.

Benchmark Va Hl Davis 83 b

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