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by: Norval Douglas


Norval Douglas
GPA 3.84


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This 3 page Class Notes was uploaded by Norval Douglas on Thursday October 22, 2015. The Class Notes belongs to FR 3 at University of California Santa Barbara taught by Staff in Fall. Since its upload, it has received 15 views. For similar materials see /class/226935/fr-3-university-of-california-santa-barbara in French at University of California Santa Barbara.




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Date Created: 10/22/15
VOLUME 84 NUMBER l9 PHYSICAL REVIEW LETTERS 8 MAY 2000 Heat Transport in Turbulent RayleighB nard Convection Xiaochao Xu Kapil M S Bajaj and Guenter Ahlers Department ofPhysics and Quantum Institute University of California Santa Barbara California 93106 Received 19 January 2000 We present measurements of the Nusselt number N as a function of the Rayleigh number R in cylindrical cells with aspect ratios 05 S T E Dd S 128 D is the diameter and d is the height We used acetone with a Prandtl number 039 40 for 105 S R S 4 X 1010 A t of a power law N MR7 over limited ranges of R yielded values of ya from 0275 near R 107 to 0300 near R 1010 The data are inconsistent with a single power law for N For R gt 107 they are consistent with N aa39 112R14 17039 17R37 as proposed by Grossmann and Lohse for 039 Z 2 PACS numbers 47277i 4425f 4727Te Since the pioneering measurements by Libchaber and coworkers 12 of heat transport by turbulent gaseous helium heated from below there has been a revival of in terest in the nature of turbulent convection 3 In addi tion to the local properties of the ow one of the central issues has been the global heat transport of the system as expressed by the Nusselt number N AERA Here As qdAT is the effective thermal conductivity of the convecting uid q is the heatcurrent density d is the height of the sample and AT is the imposed temperature difference and is the conductivity of the quiescent uid Usually a simple power law W E MR7 1 was an adequate representation of the experimental data within their resolution of a percent or so 4 Here R agdsATKV is the Rayleigh number a is the thermal ex pansion coefficient g is the gravitational acceleration K is the thermal diffusivity and V is the kinematic viscosity Various data sets yielded exponent values 7 from 028 to 031 56 Most recently measurements over the unprece dented range 106 S R S 1017 were made by Niemela et al and a t to them of Eq 1 gave 7 0309 5 but even over this extremely wide range these data did not have the resolution to reveal deviations from the functional form of Eq 1 Competing theoretical models also made pre dictions of powerlaw behavior with 39y in the same narrow range 2678 For example a boundarylayer scaling the ory 28 which yielded 7 27 1 02857 was an early favorite at least for the experimentally accessible range R S 1012 It was generally consistent with most of the available experimental results However very recently a competing model based on the decomposition of the ki netic and the thermal dissipation into boundarylayer and bulk contributions was presented by Grossmann and Lohse GL 7 and predicted nonpowerlaw behavior Accord ing to GL the data measure an average exponent 39y as sociated with a crossover from 39y 14 at small R to a slightly larger 7 at much larger R In the experimental range the effective exponent ysff E dlnWdlnR which should be compared with the experimentally deter mined 39y is close to 27 and depends only weakly upon R 00319007008419435741500 Within the typical experimental resolution of a percent or so it has not been possible before to distinguish between these competing theories Here we present new measurements of WR over the range 105 S R S 4 X 1010 for a Prandtl number 039 E VK 40 Our data are of exceptionally high precision and accuracy They are incompatible with the single power law equation 1 and yield values of 73 which vary from 0277 near R 107 to 0300 near R 1010 In particular the results rule out the prediction 28 7 27 For R 2 107 a much better t to our results can be obtained with the crossover function W airrerz burr7R37 2 proposed by GL 7 for 039 2 2 We used two apparatus One was described previously 9 It could accommodate cells with a height up to d 3 cm The other was similar except that its three con centric sections were lengthened by 20 cm to allow mea surements with cells as long as 23 cm In both the cell top was a sapphire disk of diameter 10 cm A highdensity polyethylene sidewall of circular cross section and with diameter D close to 88 cm was sealed to the top and bottom by ethylenepropylene 0 rings Four walls with heights ranging from 070 to 174 cm were used and yielded aspect ratios F E Dd 128 20 10 and 05 The bottom plate had a mirror finish and contained two thermistors The uid was acetone 10 From Eq 8 of Ref 11 we estimated x E 1 00012AT for the ratio of the temperature drops across the top and bottom bound ary layers Our temperature stability and resolution was 0001 C or better We measured and corrected for the con ductances of the empty cells and applied corrections for resistance in series with the uid The bath and bottom plate temperatures usually uctuated by no more than a few mK Usually AT was stepped in equal increments on a logarithmic scale holding the mean temperature close to 3200 C Results of our measurements in four cells of different P are shown in Fig 1 For each cell they cover about two decades of R and collectively they span the range of R from 105 to 4 X 1010 There is a dependence of WR 2000 The American Physical Society 4357 VOLUME 84 NUMBER 19 PHYSICAL REVIEW LETTERS 8 MAY 2000 101 f 105 106 107 108 109 1010 1011 R FIG 1 The Nusselt number as a ll lCthl l of the Rayleigh number Solid squares F 128 open circles F 30 solid circles F 10 open squares F 050 plusses Ref 13 open triangles Ref 14 solid line Ref 15 upon F as already noted by others 12 For compari son we also show in Fig 1 the recent results of Chavanne el 0 13 for 0 2 08 and F 05 plusses and those of Ashkenazi and Steinberg 14 for 039 1 and a cell of square cross section and F 072 open triangles There is good agreement with the former considering the differ ence in 0 The latter are about a factor of 16 larger than our results this difference seems too large to be attributed to the difference in 039 or the geometry and remains unex plained The solid line just above the open circles in the figure more easily seen in Fig 2 corresponds to the fit of Eq 1 to the data ofLiu and Ecke 15 for 0 4 F 2 1 and a cell with a square cross section The agreement with our data is excellent considering the difference in geometry Figure 1 does not have enough resolution to reveal de tails about the data Thus we use the early prediction y 27 as a reference and show log107fR27 as a function of log10R in Fig 2 If the theory were correct 07 I 39r 1 F F 212 A 7 7 I 074 4 01 v n Z 076 i V O 5 078 39 2 3908 W 7 082 J I I 1 1 I 1 5 6 7 8 9 1O 11 10910 R FIG 2 Highresolution plot of the Nusselt number as a Jnc tion of the Rayleigh number The symbols are as in Fig 1 4358 WR27 should be equal to M ie independent of R If Eq 1 is the right functional form but 7 differs from 27 then the data should fall on straight lines with slopes equal to 39y 27 Our F 128 data solid squares are at relatively small R and one might not expect Eq 1 to become applicable until R is larger The F 30 data ac tually show slight curvature but in any case would yield y lt 2 7 The smallerF data are clearly curved show ing that Eq 1 is not applicable with any value of 39y In order to make this conclusion more quantitative we show in Fig 3 effective local exponents 39yeff derived by fitting Eq 1 to the data over various restricted ranges each cov ering about half a decade of R The ts yield values of 39yeffR which have a minimum near R 107 Within our resolution 39yeff is independent of F To the extent that 39yeff does not depend upon F it is possible to write W R F as JWRI fFFR 3 in terms of a scale factor f F 91 and a scaling function F R which is independent of F In Fig 4 we show log10R27FR as a function of log10R Here we chose arbitrarily the F 1 data as a reference and as signed them the value f F 1 1 One sees that the data for all F collapse onto a universal curve Next we compare the predictions of GL 7 with our data These authors defined various scaling regimes in the R 039 plane For 0 2 2 they expect that crossover between their regions I and 111 should be observed For that case Eq 2 is predicted to apply One way to test this 16 is to plot y WfFR14039112 as a function of x R5280584 If the prediction is correct the data should fall on a straight line y a bx with a and 19 equal to the coefficients in Eq 2 Our data are shown in this parametrization in Fig 5 The solid line is a leastsquares fit to the F 1 data The fit is extremely good The coefficients are a 0326 and b 236 X 10 3 in good agreement with the coef cients 032 I I 03 7 a E39 7 t I gt03 028 39 39 026 5 10 11 10910 R FIG 3 The effective exponent as a ll lCthl l of R Solid squares F 128 Open circles F 30 Solid circles F 10 Open squares F 05 Solid line logarithmic derivative of Eq 2 with a 0326 and b 236 X 10 3 Dotted line 7 27 VOLUME 84 NUMBER 19 PHYSICAL REVIEW LETTERS 8 MAY 2000 0735 f O74 0745 7 39 logioi R FR 075 39 0755 39 10910 R FIG 4 Highresolution plot of the scaling ll lCthl l F R de ned by Eq 3 The symbols are as in Fig 1 We used the scale factors fF 0933 1000 1131 and 1186 for F 05 10 30 and 12 respectively The solid line corresponds to a t of the prediction of GL to the F 1 data estimated by GL on the basis of other experiments 17 At small R the T 3 data deviate slightly from the GL prediction This may be because the values of R are too small for the GL function to apply GL estimate that their region I has a lower boundary below which the Reynolds number of a largescale ow which they predict to be given by Re 2 0039R12056 is less than about 50 This occurs when R 2 16 X 107 corresponding to R5280584 178 This value is indicated in Fig 5 by the small vertical bar At large R the T 05 data also deviate slightly from the GL t to the T 1 data However we feel that these deviations are so small that one cannot assert that they exceed possible systematic errors In the range of R where they occur the temperature differences are already quite large and small effects due to deviations from the Boussinesq approximation cannot at present be ruled out 05 048 39 A 046 39 N 11 b 044 R14 042 i V 04 FR 038 39 036 39 034 39 39 39 20 40 60 R5286584 FIG 5 Plot of FRR140 1m as a function of R5280 584 The symbols are as in Fig 1 The straight line is a leastsquares t to the F 1 data The small vertical bar indicates an estimate of the lower limit of applicability of the GL prediction It would be desirable to make measurements in a much larger cell where these Rayleigh numbers could be reached with more modest temperature differences Finally we note that small deviations from the GL pre diction Eq 2 might occur because 039 is not suf ciently much larger than the crossover value 039 2 2 above which Eq 2 is applicable It would be desirable to determine F R using other uids which have larger Prandtl numbers However for these uids larger cells will be required in or der to reach the same maximum R The logarithmic derivative 39yeff of Eq 2 based on the parameters a and 9 determined from the t to the T 1 data is shown as a solid line in Fig 3 For R 2 107 the line agrees quite well with the values determined by local ts of Eq 1 to the data The small seemingly systematic deviations which do exist at large and small R correspond to the deviations from the straight line in Fig 5 and were discussed above for that parametrization Finally we ask whether the agreement between the the ory and the data is sensitive to the details of the GL pre diction For large 039 the theory predicts crossover from I to 111 and Eq 2 7 For somewhat smaller 039 2 1 however the crossover should be from I to IV and the prediction then reads 7 W 0039112R14 bR13 4 As a function of 039 GL estimate that the transition from Eq 2 to Eq 4 occurs near 039 2 but there is some uncertainty in this value Thus in Fig 6 we compared Eq 4 with our results for T 1 by plotting y WR140112 as a function of x R1120112 If Eq 4 is applicable this should yield a straight line As can be seen the data deviate systematically from the t Thus a transition from I to IV at 039 4 is inconsistent with our data In Fig 7 we show the deviations 5WW from ts of Eqs 1 2 and 4 to the T 10 data For Eq 1 they I I I I I 55 6 65 7 75 R1126112 FIG 6 Plot of WR140 1m as a function of Rllzallz If Eq 4 is correct the points should fall on the straight line a leastsquares t The data deviate systematically showing that Eq 4 is not the right functional form 4359


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