Cheese Wright Et Al 1988 Correlation of Experimental Velocity and Temprerature Etc

8/8/2019 Cheese Wright Et Al 1988 Correlation of Experimental Velocity and Temprerature Etc

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C140/88The correlation of experimental velocity an dtemperature d ata for a turbulent natural convectio nboundary laye rR CHEESEWR IGHT, BEng, PhD and M H MIRZAI, MS cDepartment of M echanical Engineering, Queen Mary C ollege, University of Londo n

It has been claimed that the wall shear stress isnot a relevant parameter for the correlation oftemperature and velocity data for turbulentnatural convection boundary layers. The pape rreports wall shear stress data of sufficientaccuracy for a check to be made of this claim .The correlation of the temperature data is shownto be insensitive to the wall shear stress overthe range of Grashof numbers covered but th evelocity data is correlated by splitting it into apart dependent on the shear stress and a partdirectly dependent on the temperature field .

1 INTRODUCTIONA heated vertical plate generates the simplest form of natural convectio nboundary layer and has been used by many investigators in the last twent yyears . In spite of numerous investigations in the turbulent region, only asmall amount of data are available on the velocity distribution . Althoughmuch more data for the corresponding temperature distribution exist, there i sno universally accepted basis to correlate either the temperature or the velocitydata. The problems of correlation are not assisted by the fact that many o fthe existing data are not entirely consistent, in that they do not satisfy anintegral ener balance .Fujii [1 ] reviewed all the experimental data available up to 1973 an dpointed out that there were deficiencies in measurement accuracy. Hconcluded that more experimental data were needed. George and Capp [2], intheir theoretical work mentioned that they had suffered from a lack ofaccurate data, especially velocity data very close to the wall . Hoogendoornand Euser [3] showed that although the data from different experiment sseemed to be consistent, they did not satisfy an integral energy balance an dthey suggested that the source of error lay in the mean velocity data. Thissuggestion was confirmed by the work of Cheesewright and Ierokipiotis [4] wh oused Laser Dopper Anemometry (LDA) to measure the velocity .In all the experiments to date the experimenter has been present in thelaboratory where the experiments were taking place and because of the verylow velocities which characterise the flow this may have introduced distur -

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bances. In order to ovecome this problem and to facilitate the acquisition o ftemperature and velocity data over a wide range of temperature differences ,the flat plate apparatus used by Ierokipiotis [7] was modernised to utilise afully automatic traversing mechanism. With the new arrangement the optica lhead of a fibre optic laser Doppler anemometer system and a thermocoupl eprobe could be traversed together across the boundary layer under compute rcontrol. In contrast to most previous experiments which had used only one ortwo temperature differences, usually of the order of 40 K, the present experi-ments included measurements at differences of approximately 10 K, 20 K, 30 K ,50 K and 60 KThe basis for the correlation of data suggested by George and Capp [2 ]depends on the assumption that the wall shear stress can be ignored in an ycorrelation. This view has been disputed by Cheesewright [8] and up t opresent there has not been any wall shear stress data available, in conjunctionwith temperature and velocity data, which could be used to check this point .With this in mind, considerable effort was expended in the presen texperimental work, in obtaining reliable wall shear stress data .

2 APPARATUS AND INSTRUMENTATIONThe vertical flat plate is 2.75 m high by 0.61 m wide. It consists of nineelectrical heating elements sandwiched between two aluminium plates . A stabi-lized mains supply is fed to nine variable transformers, each connected to on eof the heaters. By adjusting the current passing through the variable trans -formers, the temperature difference between the plate and the ambient can bekept uniform along the plate to an accuracy of 0 .5 K.A 25 um diameter, butt-joined chromel-alumel thermocouple was used t omeasure the temperature in the natural convection boundary layer . The signalfrom this thermocouple was amplified, low-pass filtered with a cut-off frequencyof 50 Hz, and then digitised and stored on a floppy disc of a PDP-11 min icomputer for further processing.For corresponding velocity measurements, a Dantec e 55X Fibre OpticLaser Doppler Anemometer (driven by a 35 mw laser) was used with aFrequency Shifter and Counter Processor. The signal from the photo-multiplie rdetector was band pass filtered in the range of 2-256 Hz before being passe dto the counter. In the absence of a direct digital connection between thecounter and the controlling computer, the analog output from the counter wa sdigitised by the same ADC which was used to convert the analog temperatur esignal and the data were stored on the same floppy disc .A 5 seconds burst of seeding (corn oil) was introduced every 200seconds from a position behind the hot plate. This enabled a uniform concen-tration of corn oil particles to be maintained during the five and half hour snecessary for the measurement of each profile .

3 RESULTSThe low frequencies which are characteristic of turbulence in natural convectionflows necessitated the use of an averaging period of 6 minutes, which with adigitisation rate of 100 Hz gave 36 000 samples per point .Each profile comprised 50 points, ofwhich 24 were taken within thefirst 10 mm away from the wall. This concentration of points near the wal lwas necessary to enable the wall temperature to be obtained by extrapolatio nof the temperature profile and the local heat transfer rate and the wall shear

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stress to be obtained from the temperature and velocity gradients respectively .The distance between points on the profiles was known to an accurac yof 0.5 mm. This degree of uncertainty was not acceptable and so it wa snecessary to determine the exact position of the wall from the measure dprofiles . In the case of the temperature profile previous workers [5[, [7] haveshown that profiles of both mean temperature and the intensity of th etemperature fluctuations are linear over the first 2 mm away from the wal land that the latter profile goes to zero at the wall . Thus a least squares fi tof the fluctuation profile was used to determine the position of the wal lrelative to the thermocouple probe . It is estimated that this was accurate to 0.05 mmThe determination of the wall position for the velocity profile was mor edifficult . The measuring volume of the fibre optic LDA probe was 0 .16 mmdiameter by 2.4 mm long, and was situated approximately 55 mm from th eprobe head. The length meant that the axis of the probe could not b eperpendicular to the wall . Equally, the diameter of the probe head (16 mm) ,meant that the axis could not be parallel to the wall . Thus it had to beinclined at 9 to the wall . The polished surface of the wall caused any partof the measuring volume which was actually incident on the wall to b ereflected back into the flow thus giving rise to erroneously large estimates ofthe veloci ty . All the measured profiles have at lease one point very close tothe wall which is in error in this way. Typical examples of the profiles ar eshown in Fig. 1.The velocity profile for that part of the flow which is in both th econductive sublayer and the viscous sublayer is known to be given by :Tw gL(Tw-Tm)

g R=Yo+3pv 2v 6 v where Y is the true distance from the wall.Two slightly different methods of analysing the velocity profile data wer eused, both based on equation 1 . In the first method the least squarestechnique was used to fit data for the first 2.5 mm away from the wall to anequation of the formU=Co +C Y m +C 2 Ym +C3 Yin ( 2)where Ym is the apparent or measured distance from the wall .The fit was repeated with the data point closest to the wall being pro-gressively dropped from the analysis until consistent values of the coefficient swere obtained. Cwas then the difference between the measured and thetrue distance from the wall and the wall shear stress could be obtained fromdU/dY at Y = O. By trial and error it was found that data out to 5 mmfrom the wall could be included in the analysis without significantly alteringthe estimated wall shear stress .The incentive to try a slightly different analysis came from the obser-vation that the fitted coefficients in Eqn. 2 did not agree well with what coul dbe estimated from the quadratic and cubic terms in Eqn. 1, using the valuesof Tw an d Qo obtained from the temperature profile. The analysis started

with the estimation of the difference between the true and the measure ddistance from the wall from a graph of measured velocity against measure ddistance from the wall . The corrected distance from the wall was then used ,

( 1 )

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together with the values of Tw and Qo from the temperature profile t oconstruct a graph of :g BT Tm) g S QU +2 6v3 against Y (3 )Fig. 2 shows an example of such a graph and it will be seen that the dat aform a good straight line. The slope of the straight line gives (dU/dY)Y=0for the actual velocity profile. Values of the wall shear stress obtained by th e

two methods were quite close but over the whole set of profile measurement sit seemed that the second method gave the more consistent results .The estimates of the wall shear stress for all the profiles measured, ar eplotted in non dimensional form against the Grashof Number in Fig. 3 and aselection of the temperature and velocity profiles are plotted in the mannersuggested by George and Capp [2] in Figs. 4 and 5. Data obtained byChokouhmand [6] from experiments in water is also included in Fig . 44 DISCUSSION

It can be seen that the approach of George and Capp [2] gives a reasonabl ecorrelation of the temperature data for the region away from the wall but thatit does not correlate the water data in the conductive sublayer very near th ewall. This failure in the conductive sublayer is not unexpected since weknow that the temperature profile in this region is given by :lb = C (4 )

and if we accept * that the wall heat transfer data correlates as :Nux = Grx i/3 F(Pr)

= T Pr2/3 F(Pr )and the profile has the form :

0 = 1 -YPr 2/3 F(Pr) = 1 = Y1 (Pr)nT nTand if we omit the function of Pr we shall not get correlation between thedata for water and air .If we replot the data in the form of 0 against j we see from Fig. 6that we have a reasonable correlation over the whole region . Now it mightbe thought that Fig. 6 proves that George and Capp [2] were correct in theirclaim that the wall shear stress is not a relevant parameter in any correlation .However we must note that we only have data for a very limited range o fGrashof number. Just how limited this range is can be seen if we rememberthat the Grashof number is analogous to the square of the Reynolds number .We would not be able to judge the trends in the profiles for a forced flow i f* The heat transfer data from the present experiments correlate as :Nux = 0.11 Grx 0 .3 3

8 2

then

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we only had data for Reynolds numbers up to 3 times the transition Reynold snumber .Cheesewright [8] has argued that the neglect of the wall shear stress byGeorge and Capp (2 ] is only acceptable if the wall shear stress v Grashofnumber relationship is of the form:1P g Q (T w TW)X = GrX/

3 f ( Pr ) (6 )The data in Fig. 3 do not support Eqn. (6) . There is some scatter but ifa relationship of the form

TwA Grnp g s(Tw -T )Xis fitted, n is found to be -0.26 which is consistent with the results of veryrecent numerical calculations by Henkes [9] who suggests that :

Twp g R(T w -T )X

-1/ 4Gr xThe present data are neither sufficiently accurate nor sufficiently extensive t ocheck the suggestion by Cheesewright [8] that in the fully turbulent part of th enear wall region the temperature data should correlate as :

Qo0 = Ao + Al (Tw TO UT In S + AZ S-1/3 (7 )However, both the present data and work of Henkes [9] indicate tha tthe effect of the wall shear stress should become increasingly important as on egoes to higher and higher Grashof number. This implies that the correlation sshown by the data in Figs. 4 and 6 should not be expected to hold at ver y

high Grashof number. There is thus an urgent need for data extending up t oGrashof numbers of 1013 and 101 ' which are typical of conditions in anumber of nuclear reactor situations .Fig. 5 shows that the approach of George and Capp [2] does notcorrelate the velocity data to any significant extent. The data show a clea rdependence on streamwise distance even when the temperature difference acros sthe layer is constant. The implications of this can be seen from the work ofCheesewright [8] who has shown that the general form of the velocity profil ein the whole of the near wall region is :[ g 1 3 (TwT.)]1/ 3

erimentally we know that is not X dependant, and nT and Pr are bydefinition X independent so it follows that the observed X dependence of thevelocity data must be interpreted in the general form, as a dependence on Y +and hence on the wall shear stress . This is consistent with the known form

U =F((,Y+,Y/nT,Pr)

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of the velocity profile in the viscous/conductive sublayer as used in Eqn . 1 .The derivation of a form of the velocity profile for the fully turbulen tpart of the wall region, corresponding to that suggested in Eqn . 7 for thetemperature profile, has proved to be difficult but the work of Cheesewrigh t[8] suggests that one of the terms should be B O UT Ln Y+ as found in anordinary turbulent boundary layer where B O is an unknown constant. Fig 7shows a selection of the present data plotted as :U- 3 .5UTLnY+ Yagainst [ g R (Tw -To ) ] 1/3 nT

The correlation is not perfect but it is much better than that in Fig. 5. Thecomments made above, about the increasing importance of the wall shear stres sas one goes to higher and higher Grashof numbers, are equally applicable tothe velocity correlations and the lack of a solid theoretical basis for th eparticular form of correlation used in Fig. 7 means that we should be carefu lin extrapolating it to very high Grashof numbers .The correlations in Figs. 6 and 7 could be used as a basis fo r'artificial' boundary conditions in finite difference computations of turbulen tnatural convection flows, but for such a procedure to be used in any mediu mother than air one would need to know the influence of the Prandtl numbe ron the Nusselt-Grashof relationship and on the dimensionless shear stres sGrashof relationship. While there is some data on the former there is nodata at all on the latter. It is clear from the data presented here that theuse of the George and Capp profiles as a basis for 'artificial' boundaryconditions, as has been done in the work of Thompson et al [10], is likely t olead to errors both as one goes to media with Prandtl numbers different t othat for which the profile constants were derived and perhaps mor eimportantly, as one goes to higher Grashof numbers . The use of thcorrelations obtained in this work would clearly be better than the use of th eGeorge and Capp [2] profiles but it is very clear that more work, particularl yat high Grashof numbers, is needed before such procedures can be accordedthe same status as the use of 'log law' boundary conditions in the computatio nof ordinary turbulent boundary layers .5 CONCLUSIONS

1. For a turbulent natural convection boundary layer on an isothermal ver-tical plate in air the wall shear stress correlates as:Tw= 1.0 Grx

o .z sPg B(Tw -LOX

2. Over the limited range of Grashof number for which data is availabl ethe temperature profiles correlate as 0 = F(c) .3. The wall shear stress is expected to be a parameter in the correlationof the temperature profiles at Grashof numbers above approximately 10 13 .4. The velocity profiles can only be correlated when the wall shear stress i sincluded as a parameter .6 REFERENCES

1 . Fujii, T., Dennetsu-Kogaku no shinten, vol. 3, p.1, Yookendo, 1974.

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2. George, W.K. and Capp, S .P ., 'A theory for natural convecton boundarylayers next to heated vertical surfaces', Int. J. Heat Mass transfer, Vol . 22, pp .813-826, 1979 .3. Hoogendoom, C .J . and Euser, H., 'Velocity profiles in turbulent freeconvection boundary layers', Int. Heat Transfer Conference, Toronto, Vol. 2 ,pp . 193-198, 1978.4. Cheesewright, R. and Ierokipiotis, E.G ., 'Velocity measurements in anatural convection boundarylayer', Paper NC31, 7th Int . Heat Transfer Conf. ,Munich, 19825. Cheesewright, R. and Ziai, S., 'Distributions of temperature and localheat transfer rate in turbulent natural convection in a large rectangular cavity' ,Proc. 8th Int. Heat Transfer Conference, San Francisco, USA, Vol . 4, p. 14651988 .6. Chokouhmand, H., 'Convection naturelle dans 1'eau le long d'une plaqueverticale chauffee a densite de flux constante', Division D'Etude et de Develop-pement des Reacteurs, C.E.N . SACLAY B.P . No. 2, 91 190. GIF. su r .YVE'FI'E ,France, 1978.7. Ierokipiotis, E .G ., 'The study of the development of a turbulent natura lconvection boundary layer using laser doppler anemometry', Ph .D Thesis, Uni-versity of London, 1983 .8. Cheesewright, R., Faculty of Engineering, Queen Mary College, Universityof London, Research Report (EP5037), 1987 .9. Henkes, R.AW .M ., Personal communication, Department of Applie dPhysics, University of Technology, Delft, Netherlands, December 1987.10. Thompson, C.P ., Wilkes, NS . and Jones, I .P ., 'Numeical studies o fboundary driven turbulent f low in a rectangular cavity', International Conferenceon Numerical Methods in Thermal Problems, Swansea, July 1985 .NOMENCLATUREGrx local Grashof numberg specific gravitational forceNux local Nusselt numberP r Prandtl numberQ. Q/PQw wail het fl u xTw wall temperatureTo , ambient temperatureU local mean velqcity in the X-direction1 3U

TT shea( T w - T .)velocity ] _ ( Tw/p) 1 / 2X distance up the plate from the leading edgeY distance normal to the plat eY + Y UT/v

cc thermal diffusivity1 3 coefficient of thermal expansionS T YTX)T.0 T - Tw) / fI' - T.nT George andCapp [2] inner length-scale,

for constant wall temperature = 2/ 3[ g R( TwTo)]1 / 3

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v kinematic viscosityp density

a Gr. 7 .2 0 x 10 1 0 Gr.= 3 .94 x 10 1 0e Gr.= 2 .05 x 10 1 0

8 0

60 -

-1 0 1 2 3 4 5 6Y (m m )

Fig. 1 Velocity profiles close to the wall

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0

n o

8? 3 -so

p

4

2 -

1 -

o Gr=2.05x10 1 0Y

A GrY=5.40x101 0

o GrY = 3 .9 4 x 101 0

00

pI A0

o oAi i p 0 o1 O OppO O O Oq 00 0 0 o

A o c 0p o O6 e e

O

4O

0 0

Y (mm)

Fig. 2 Linearised velocity profiles close to the wall

2 4 6 8 1 0

3 .0

2 .5o AT564 K AT=50 Kn AT532 Ko AT519 K

N 3

0 . 5

Fig 3

0.0

on n o o o 0to.. . . . . .......2 3 4 5 6 7 8 9 1 0Gr . 1 0 1 0Dimensionless wall shear stress v Grashof number

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a. . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . .

CI'p

P r e s e n t O r = 2 .05 x 1 0 1 0 Present Gr = 3 . 9 4 x 10 1 2o C h o k o u h m a n d G r = 3 . 1 9 x 101 2A C h o k o u h m a n d O r = 1 . 8 6 x 1 0

. . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . .. . . .1 0 1 0 0 1 0 0 0Y l 1 1 TFig . 4 Temperature profiles in terms of the George and Capp correlation. . . . o . :0 oe G r .= 2 .05x10 n n

. . . ... . . . . .. voten

10 =

5 =

25

20 -I 0o Grx 7 .20x1 0

Gr=3 . 94x10 1 0x

0. ..a.. : ..

4 r o0 1 1 .

D il R a se

o

0 . 1 1 0 100 1 0 0 0YIrlT

Fig. 5 Velocity profiles in terms of the George and Capp correlation

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Fig. 6 Temperature profiles correlated in terms of the conduction lengthscale

%A. 4 1 oe P 0 0.. . . . . . . . . .II

.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .. . .. . . . . . . .. . . . . . . .. . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .'2 %o

. . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .;.4Y NuX x0 . 60 .8 i1 .0 1 S IG r = 5.40 x 1 0Gr = 3 . 94 x 1 0x 1 0a Gr = 2 .05 x 1 0x 1 01 010 100. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .t

o.*0 1 0Gr x = 7.20 x 1 0O r x 3 .94 x 10 1 0 Or = 2 .05 x 10 1 0x-60 '0 100 1000Ylrtr

Fig. 7 Velocity profiles correlated by separating the 'forced' and 'free' part s

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Cheese Wright Et Al 1988 Correlation of Experimental Velocity and Temprerature Etc