turbulent prandtl no.

turbulent prandtl no. - The 1992 Max Jakob Memorial Award...

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Unformatted text preview: The 1992 Max Jakob Memorial Award Lecture Turbulent Prandtl Number— l.llilliam M. Kays Ptolesss: Emeritus. Slan'c': bniue’sity. Stanlatd. CA 9430:] Where Are We? Theobjeerit-e rafrnis prepares to examine rrr'rr'cui.[ v the pret'crrtlyuyuilabie experimental data on Tut-burn” Frairdrrr'sitmibprformertr-o-dt'rirertsiona.’nit-indent heraldry layer. artdlforjrrffv rret=efamd firm in a circular tube or after duct. and artcmpt to dran- .tmrrc corichrsmns as m tt-ftera rrmrrers presenrfv stand. Introduction The title is in the form of a question. Before attempting an answer we should perhaps address another question: Why is Turbulent Prandtl Number even relevant today'.J There are many who would argue. and quite convincingly, that recent advances in turbulent fie“ modeling make 11m concept Ur eddi' diffusit ity obsolete, and. without an eddy diffusivity Turbulent Ptandtl Number has no meaning. [See. for esample. Launder [I989] or Nagano and Kim (1988).] if we are talking abottt three-dimensional fiotvs or re-circuiating flan-s. or in other words the completely general turbulent flotv problem. then the concept of an eddy diffusivity certainly loses its usefulness and there is no argument. But if we restrict consideration to the two-dimensional turbulent boundary layer. or flow irt a long pipe or duct. “here in hot h cases the most important turbulent heat diffusion is only in a direction normal to the main flow direction. the use of higher levei turbulence models for J8 percent ofengineering calculations becomes absurtlly complex. A very large number of experimentally determined constants becomes necessary. and even these have not all been satisfac- toriiydeterrnined. 0n the other hand an eddy diffusivity model. in which the eddy diffusivity for momentum is evaluated by either a mixing-length model or the k—e model. and thc eddy diffusivity for heat is evaluated train" the Turbttlent Prandti Number concept. is extremely simple. it iscapuhle ofaecurately predicting behavior in a great variety of situations—in fact in most engineering boundary layer problems that one is likely [D encounter. \\'c:e the Turbulent l’randtl Number a very complex func- tion of a number of variables it would still not be very useful. but all the evidence suggests that it is a number near unity [uhich is essentially the ciassieal Reynolds Analogy) and its departures from unity art: not great except for very low szndtl number fluids. it should be noted that it must be possible to "back out" Turbulent Prandtl Number from any higher order turbulence model. so experimental data on Turbulent Prandtl Number provide a compact database. which must in any case be satisfied by such models. The primary ohjcct' e of this paper is then to examine cri:- ically the presently ava able experimental data on Turbulent Prandll Number [which we will hereafter refer to as l’r-t] for the two-dimensional turbulent boundary layer and for fully developed flu“ in a circular pipc or a flat (Illcl. We will examine C(lnlfll‘fl1rd h_\ the Heat rlnmfer inistor. and pertcnler‘. 2| lltc Nalion al lied Trumlct to (fence. Allatllafrmfi Anfull 5 II. l'fJJ. Alanntcrtpl :3: vctl by lltt: ilca'. Tum'e' lth. nLtt'i IU'J'I: r MO" it‘rl‘i'rcd Dreelnh W93. Key-nun}! Force. any“ on [ uid Melalt. Muchnt: and Sealing. Tu:bn- lrncc. rhaocialr Technical - - 2311 i Vol. “6. MAY 1994 data for favorable and adverse pressure gradients. flows with transpiration. and rough surfaces. but the primary emphasis is ill be on the simple flat plate boundary layer with Ito pressure gradient. and fully developed flow in a tube or duct.{1’-or fttlly developed fva in a tube or duct there is of course a favorable pressure gradient.) We is ill emphasize constant t'iuid properties and velocities sufficiently low that viscous dissipation may be neglected. However. the properties of some of the liquids vary so markedly with temperature that we will he forced in that case to examine data where the properties [primarily viscosity and Prandtl number] do vary considerably. Definition of PM For constant fluid properties and no viscous dissipation. the differential equations that must be satisfied in the boundary layer are [for flow in a circuiar tube the coordinate system need only be changed to cylindrical): Continuity i1] Montennnn (3} The eddy diffusivity concept introduces the following def- initions: . . (in ii L' _. — — :11 «(1‘) { . fl 3 .r r.- a “(5') t5} and Eqs. :2) and [3} become: [in mi 1”] t6] alt Br 3: an fly (1 lfT+L'—' — .._- ._. 'l’ 1 axtbay at [a+tieia_‘l [-1 we fly Jo Notv let us define Turbulent l‘randd Number as: a e l"t-l='fl tit (51 Transactions of the ASME Substituting for (if in Er}. [7}, and after slightly rearranging. (ht: cncrgy equation become I + fuJ"? PM {9] J, .. ._ _ _ Pt PH 31' It is now apparent that ifa solution to the momentum Eq. [6} is available. .2. 1'. and r” are known .1: all points l.‘l the boundary layer and all that is needed so salt-e the energy Eq. [9} for any temperature boundary conurnons is Pr-t. Evaluation of PM if we ntlw‘ substiittil: Eqs. (-1] and [5) into [S] we obtain an equation that can be used to evaluate Pr-I from experimental date or from analytic solutions: at Pr‘ = {10] fly Thus to evaluate Pr-t .1‘ any point in tile boundary layer from experimental data it necessary to measure fuur quan- tities: the turbulent shear stress. the turbulent heat flux. the velocity gradient. and the temperature gradien1.'l'hc dll'lieulty of measuring 2]] four of these quantities accurately at a single point in the bouttdary layer is the reason that direct mcasure- ments of PH are relatively scarce. and the reason that the scatter of experimental data tends to bl: SCI large. The Reynolds Analogy ' Thus: PM: LOCI ' ill} This result is the Reynolds Analogy The remarkable thing is that the Reynolds Analogy is we _ closc to correct for most boundary layer flows. as we shall see. But it is not precisely correct and there are important dcparturcs from l’r-t=1.UU that we will examine. l-It'alua n of PM From the “1.0g” Slope of Velocity and Temperature l‘rofi as The difficulty of accurately determining P:-. from esperi- mental data has bcett mentioned. Hon ever. there is one speeial cast: when: it is not so difficult. As is well known. the yelocity prot‘tic over a substantial part of the simple turbulent boundary layer with no transpiration is logarithmic in form and can be expressed by the “law of the wall": b“ _-2_4-lltt 1" ~:- 5.00 Ill! For moderate and high Prandtl number Fluids for no tran- spiration the temperature profile at er a substantial part of the boundary layer can he expressed by a similar equation: 1‘" =C.ln‘.r"—C: ll-IJ The reason for the limitation of Eq. (l-l] tn Fluids with moderate Prandtl numbers [and here we are referring to mo- lecular Prandtl number. not rurlrutem Prandtl number: can be seen by examining the term in Ilse itrrter brackets in Eq. 19]: The simplest possible model for turbulent heat transfer leads to the result 6n ear Nomenclature 1 up": _+ [Pr PM] The first term represents the contribution of mnlecu r con- “ ll duction. while the second is the turbulent conduction. For \“an Driest constant — _ I arneter (hydraulic diame- “a- Vt‘lOCll‘E at 0Lle “121‘ or r = speetl'tc heat at eonstant Lee: 2xduct spacing] boundary layer Pr_=5_5ur= _ I _ Re-h enthale thiefvtness Reynolds “shear‘r velou.‘ C; = fnfi‘lm‘ CUCfllclcnl=2To" number -— a3ttm2’v mean velocity in a pipe {WT-1» 213301131 Re-m momentum thickness Reyn- z: timc-' crancd veiocity in v D _ pipe diameter olds numbcr=n;n,h direuion I - EDR = eddy diffusivity ratit::e_trr'v t time-averaged temperature instantaneous l‘luctuatina it = heat transfer _ r' instantaneous fluctuating component of \elocity in t- r.'t1cl'l‘tt'.‘ient=q'u fun—rm]. component of temperature direction - i I?” flfa— {w} turbulent “heat flux." time x distance measured in direc- lr = molecular thermal conduc- averagc of r'. a" product tiun at mean flou- “WY _ I 7" [temperature in "Wall" .I' distance measured in dire:- ‘f P_ranr.l:t mixing. length coordinates = [r—ruJ tion normal to mean flow bu = husselt nLtml:cr=Jtt')K.(' Irrgch' Y‘ distance from “all in “wall P = trme—aycraged pressure r= temperature at outer edge of coordinates" _ rnp’v Pe-t = turbulent l-‘cclct nunt- boundary layer .1. enthalpy thickness of ther- ber=tt_t,fv]Pr I... mixed mean temperature in _ mal boundary layer Pr = molecular Prandtl Itle- a pipe flow momentum thickness of mo- PI c” h::_l_=e:t‘c‘/ku Ht 1“ in temperature at wall surl'aee mentum boundary layer - = we or ota at time- s . y ' ' . - " '-' I - Fund“ numberl m: Eq- dirEESDZFVSf-‘d ‘iIOCIli’ In if Err eddy <1 lfustnty to. heat. see {15} n' instantaneous l'luctu:ttin_t,r n, eddy diffusivity for momen- Pr-t = turbulent Prandtl num- component of velocity in .r I tutti. see Eq. [4! I hCT = LN-‘Irsh‘ I_ direclicm it dynamic viscosity coefficient in. = heat Flux at Wall tr v' turbulent “shear stress." v kincmatie viscosity coeff- Re = lieynolds number for flow time average of u'. u' prod- cient = pm In a pipe = [J V9,“); tier 9 density Re-d = Reynolds number in a flat U‘ mean vcloeity :11 "wall" eo- r shear stress duct, based on hydraulic di- Journal at Heal Transfer ordinates = um, shear stress at wall surface MAY 199:, Vol. 115 .l 235 x ""‘W"'" “" "°" "° "“"‘"' """Nm ""‘ "- Table 1 Turbulent Prandtl numoar Ital-n "log" region prollle slopes ; Prelaure Gradient a DJ! E ' Experiment lie-In fieh Pr-l uncut-Imam uv - saw. ‘ an __ ____________________ ___ _-___a_-- n - - E out ,5]: Er;— 0.1 ‘ Reynolds H.959! various to 6‘00 0.73 5 )tottne (1567: “1.; use 0.85 5 "-"Mt- ‘39 . nuloane H.965] 1572 1531 u.a5 \ 5 Blackwell H.972] 2151 aces mas -- —- -- "/4 - 1- i 2509 June 0.73 did, r...2nrs.a-.r.::s 291.1. 3150 D.” 9 Gibson [.1930 215a 3129 0.51 Simon tlsaol uds 1.756 a.” r a: a 5 arts 0 . r 3 ts 57 ‘26 t. a . a: ., m ramanea H.979} 15.1.42 tstl: a.“ v- tent. 1y :auuh lurtacll Fig.1 Eiamplu ordelerminalion at tuthulenl Prandtl number Irom slope of l“ cunts in the "loo" region. data of Blackwell et at. :1972] Ely] ' = Rolling-worth :199911552 25? tt.as }" )vabout 30. t on- -— n' l" n ltere v = 0.4] for cslcrnal bound- _ I P I ary layers s long as the molccular Prandtl number is near "n 1.00 or higher the molecular conductivity is negligible relative “hunk” “5 a.” 181 o 195 D _ as to the turbulent conduct ity for Y' >30. But at very low Prandtl number the molecular term can be as large or Earger than thc turbulent terra even at very high values of Y' and is not negligible. Note that for the momentum equation. Eu. (6]. the turbulent terttl is always much greater than the molecular term for 1:" >30 and thth the molecular term is always neg- ligible. Tlte logarithmic forms of Eqs. U3} and (H) are a direct result of tlte fact that molecular viscosity and conductivity are negligible in the region 1" > 30. Referring now lo Eq. [ID], in the region from l" =30 to the start of the "u‘ake" region (see below) ttte :utbulcnt shear stress. W. and the turbulent heat flux. t a. . are close to constant. and are equal to tlte wall shear slre . rm divided by p. and the wall heat flux. ou'. divided by pc. for the singular case where there is no asia] pressure gradient. no transpiration. no axial sutt'aee curvature. Noting the definitions of to". l“ . and T'. we can now differentiate Eqs. [ti] and 114] and sub- stitute into Eq. tin} with the result: Pr-t = C;."l.44 (15] Thus for this unique but rather fundamental ease Pr-t in the "log" region eatt be determined simply lrtrm the slope of tlte temperature profile plotted in senti-log coordinates. Note tha: this further implics l’r-t is a constant itt this region, which apparently is the case. At very low Prandt] numbers {the liquid metal range. Pr<D.lt the temperature profile is no longer logatithmic and Pr-t cannot be so easily deduced. nosz it a constant. if there is an axial pressure gradient. Eq. [t3] is still a good representation for the velocity profile and Eq. [[4] can be used to represent the temperature profile. However. o'e' is no longer cottstant through the "log" region though t L' may he so. So again Eo. {IE} is not applicable. The same eonctusion can be reached for flow over surfaces with axial curvature \yltet: it’ll" varies markedly through the "log" tcttion. Figure 1 shows an example of PH evaluated from the log slope of temperature profiles for airl'low along :1 Flat plate. in s case tlte result is E‘r-t=0.tfi although it can be seen that there is somc uncertainty in evaluation of the slope. One of the probleros is that there is a dcarth of good tem- p atltre prot'ilc measurements at high Reynolds ntlmllers and this means that the length of the “log ' region tends to be short attd thus its slope is sometimes difficult to measure ae- curatelt'. Note the three distinct regions on this graph: the zaElVol.116.MAY1994 sublayet from Y" =0 to about 30. the "log" region. and the "wake" region at the outer edge of the boundary layer. As Reynolds number is increased the "log" region extends to higher values of l". while :he “\valte" retains its shape but moves to higher mines of Y". Table 1 shows the results of this method for determining Pr-t For a number of cases for air. water. and an oil. Although some v riation in PM is noted. it seems reasonable to conclude that P . n the "log" region is essentially a constant at about 0.35 for all values of molecular l’randtl number from 0.7 to 64. and there is no reason to believe that it will differ at still higher values of Pr. The range of Reynolds numbers is some- u-hat limited. but note that the one case ofa fully rough surface M Reynolds numbers considerably higher than the others still yields close to Pt-. =U.BS. Turbulent Pruntlll Number From Analytic Solutions Before proceeding to an examination or' the data for ycry low Prandfl number fluids it will be worthwhile to L'xatttitle the results of two different attempts to determine turbulent Ptandtl number analytically. Our purpose will be to attempt to establish the ltey variables upon which Pr—t depends so as to provide a rational scheme for representing the experimcntal data. During, the past 40 } ars dozens of solutions for PM based on a Variety of analytic models have appeared in the literature. Most of these ltave been inspired by the observation in the 1950s that :he experimentally determined heat transfer coef- ficients for flow of liquid metals in pipes seemed to fall well below what would be predicted using models such as the Reyn- olds Analogy. it “as first suggested by Jenkins (tr ‘1] that this was a result ofthe relatively high thermal condue ty ot'vcr, low Prandtl number fluids. and Jenkins proposed a model bascd 0n the idea that a turbulent "eddy" could lose treat by simple conduction during ils Flight normal to the mean flow direction and thus the heat transferred by the turbulent es- Ehanat‘ praccss would be lessened. Most of the other analyses are based on variations of the same idea. but \irtually all comain Free constants. which must be evaluated by comparison with the available experimental data. Mme rmnlly Ynkhot et a]. U93?) present an analytic so- lutiott based on what they call the "renormalization grour‘ Transactions ol the ASME tens-ma But-1min: :u‘armsuflhfiug'uw "rJetenmmntou—l 3...", =ig. E Analytic solutions .- y ccvoid of et . rica. simple equation git-e: by l’altttol e: nl. method." and this result is 31:12:11": Input. The relative :S WUIlII hart—:1. _-' Ir' 1 ’ q; 1:: —.'.| at? J L_l-'t-ell [‘Fr . 't' ’ J r' t *' ‘ = t- i—:.tt§t3| flaws) I “m kPr {a LIP! ’ whch [16) liq alien [ls] prb . _' vet's all Prandt'. numhers. and nl hi; FF and high values of an!» it converges c-:. = 0.83. [t .s not L".ch whelhct this result is exp red ta be alid ever the entire boundarg'layer, nt anty in the “ 3g" region. but w'.‘ will use it: in the "log" region only. in this : ua 'on Pr-t appears as a Function of Pr and I: however, an 2 PM tun! Eq. 5] is plotted :E a rum-Linn ol'the produetlr_w'v]Pt, u. .tch we Mlle-all the turt-tt 'rn Peder nurtt'aet, Pe—I. it is plotted for thrcet. cs :tfPramJ '_'mbl:.'. Ill. 0.0!, and 0.001. The important point is that tlte curt-eta all collapse ‘iu ‘ sa e lo function of Pie-t :xccdt at very lcw values cf‘clrr'b. i.e.. values encmtnleted for l" < 100. At high ralncs ol I‘e-L the cunt aapronehes 0.35. and {or Pr nenr |.lJlJ and higher there is lirt' m _.ton eoutse consistent w-‘h what was obs:r mentaldatn |:-:e~iaasl3-.Aet rylowvaluesolFe-tt' Prantltl number becomes re . consistent serrations :‘or liquid metals in the log re it- A simple empirical equation that fits the asymptotic :urres very well is: Pr-. . TrPe-e—D,33 {lit Thts. equation 5; similar in [arm to an cql!:ll'.on s"gges'°c' by Reynolds tl9‘l5). but in :haL case PEA: is an ell' ve average ' entire boundary Inge.- anc' l’e-t is tlte pmducl of the lnun'thet and the Re nolds number. This .‘c .he:'. suggests that Pr is close 0 aunique fun:- tion of Pe-t. But whether this is o It; ' .. "lag" region. or own the cntire'l: I d'I ' _ mains a questi .-.. The other analytic sol Inns that we need to consider are Iheresultsc-l'thereeentsuecessol’l ect Hume-ricelSimt-lnlion IDNSJ ur‘ beta 2 tbulent flow in an s and the turbulent ex- ternal boundary _ er. DI'M'S involves complete sniutian ol‘ the tttnc-riepende-nt Navies—Stakes and thermal erctg.» equ' are. For a turbulmt :‘low. Supercamptttets. an ettormous fi ite- dtrlfiliflce grid. and many he" ‘s u"-:crn :t'JIElEim'l are flea-Jury, but interesting rte-.ttits are hcgtnntng to appear. The results ol Journal at Heat Transler Munch mv—nw-(ua: mlunrhcA—u mammo nu: ate-g— Mme: t...s=s;x.-u:- - m v .o. \ fig. 3 PM from 3H5 calculitlans threcs- ch eal' Iatittntt are shawr. n Fig. 3. TI-c results ct" ..t and Main (1989i are [or fully developed I'lau . II Flat duct at a Reynolds number. Re-d - | LOCK). wher ' e Reyitc' '3 num- ber .; based on the hydrauitc diameter. ltich Is 2 limes the duet width. Titre: u: .. s of Prandtl number are considered. 0.1. E'Jl. 1.0. Very strnt'ar results are rep al.1j19‘3 . Kasagt et al. {1091] ca Llala a 'glttly lower Reynolds number. _ :tt' . al. [19. latte l'or thesimple turbulent hannda ayerona Flat plate with no pressure gradient. at Pr = 0.1. 0. . and 3 III Here the momentum thickness Reynolds number. Re-rr. The only prehetn mth the 3N5 ca' -ul:tlio-' date is that the Re holds numbers are so EDI dire:t|t. 1111-3 the ' ' ' .' Be ataett magna numberolpuints t'rctt'. what appears to be the "lag" region have been plotted 0-. Fig. 1. Two Ltmclusiorts are appaJe I: [1] PM doesappe ta reasonably :crrelate these .TSIIIIH and (2} they coincide very closelv with the Y1 . rt ,..d with Eq. (1?). .' re the DI‘US calculation again when we examine the cnpctintexal data. J 3 n a El In G Experimental Measurements of PM in the "Logarith— mic" Region In Fig. 4 there is plotted a Large amount of data covering the :rtl'..'t': Prantltl number range of awailaale c.it,'Jc!lm:l'll ;. Some of the :l a an: Elam experiment tit Fullydmeioped :‘low in ' nd others are :'r tr. cape m :5 on external [lat-plate y layers with no pressure gradient. With twp exe- - tions [liallingswot'th et al. {1959] and Z ciauskas “9871] lites: data are for "r: . e'unlunted using Eq. ['0). and with two 0 he." exception data taken from what. s be cc! no he the "lug" rcgio is. excluding :ltesu'dlayer regtnn and thewal-te. The latte execp 'ons are I:.e data of Brettthont J Krebs {I991}. are at Sherift‘ and Kane Ijl‘iSl]. boll: ol wh :‘I‘. wo'e Oblainccl at ur "at a. pipe -:enter|i.-.c using a technique I t involved in- jectto: ol‘ a sntall amount of hot [aid Isodiurn in this case) and ttteasuting the diliusion ol the temperature pulse. The data tn the central portion a“ e graph arc pet 131'in the results of exp. irnenls with a he data at low values or' Pc—I arc Largely for .tquid metal. data for Pe-t:> 13') are for water and higher Pranc'tl not her liquids. Note that there apnea s In he a continuum nFslates lrorn the Ltquid metals to the high Prandtl number fluids. and that all or I'.-.c cal: tend toward about Pr-t=fl.85 at high value; or Pe-t. But a difficulty is apparent i the liquid metal region. The Btelnhcrst and Krebs {1992] and the Sheriff and Kane [1981] da:a are includec‘ on this graph even though they were not abtained in Lhe "lug" reginrt because they seem to -:crrcs]:c-'.d MA'K1994, Vcl. ‘16i23F run. new»: use not nine :lnllr-IIM "net [at annual-at mt sun-tn a run-a ..nn.o Flt-I t aunt-us; Hmr—enntaw “Dimer luquI-II Mrlfl ezan min.- ace—«m Its-W .95.”.0‘... . . .r we we: Fn-I Fig. a turbulanl Prnndtl number In the "Iogarlrnmie" region. 011053.: Prcfifi rm rattan as Fig. 5 Skuplnslti et al.. experiments very closer to Eq. (1?). which, it will he recalled. closer approximates lhc ‘r‘akhot rcsttlts and the various DNS results. A consistent picture would emerge were it not for the data of Fuchs (191‘3). “'hieh also correspond closer to Eq. (17}. Un- fortunately the Fuchs data are not in the open literature. but are reported by Bremhorst. and also by Lawn [l9??}. with the implication that they are data from the "log" region. All of the other liquid meta] data [Butrr et al. [1968}. Sleicher et al. (1933}, HDchrciter (19m. and Brown et al. “9570] yield values of Pr—t that are considerably higher titan Eq. ([1]. al- though the scatter of data is considerable. These results differ from the previously cited data in that they were all obtained from experiments in which Nusseit numbers as well as tem- perature profiles were measured. 1: will now be instructive to compare measured Nusseltpum- bers with what would be predicted using Pr-t data From these experiments. The following equation is plotted on Fig. 4 as a reissunahll: approximation to these data: PM = 2.0! Pe-t + 0.35 [is] Figures 5. t5. and '.‘ show predicted Nussell numbers for three sets of experiments, using Eq. (18). together with the relevant es perimenlal data. .-‘\It are for fully developed flow in a circular pipe: the first two are for fully developed mrrsravrr treat rare. and the third is for fully developed constant surface (emper- orare. The momentum and energy differential equations are ordinary differential equations for these cases and calculations were tirade using a relatively simple finite-difference procedure and a wen-established mixing-length and eddy diffusivity model of turbulence. Equation IR} was used across the enlire pipe. but the sublayer region [for very low Pr fluids] and the cen- terline region are rather insensitive to the value assigtted to PH. 288 J' Vol. 116. MAY 1994 aunt-cu fig. 1' Sleleher at al.. erporlmonts Figure 5 shows the extensive data of Skupinski ct al. (1965} for NaK, Pr=0.0153. Calculations were also made using a constantvalueforPrAt=lJ.SSsothat theinl'luenceoftltehigher Pr-t can be clearly seen. it is apparent that calculations using Er]. (13} fit Lite data as well as could be desired. and over the entire extensive Reynolds number range. The resttlts of using Err. (1?) are not shown. but the). would lie on a Curve ap- proximately halfway between the results using Eq. [[8] and the results for Pr-L=0.BS. Note that the effect of the higher PM is to decrease Nusselt number. and this is the effect that has been consistently observed. Skupinslti et al. did not meas- ure l-‘r-t. nor do they provide profiles front which PM can be deduced. so it is only by this indircci procedure that PM for their data can he determined. Figure 6 shows the data of Pluhr t:t al. [l‘JfiS] and include data for both NaK and Hg. Since the Prandtt number of these various experiments varies from ilnl9 to 0.029. two predicted curves using Eq. [lit] are ineluded. one for Pr = 0.019 and the other for Pr ..: 0.029. These data are not totally consistent with one another. but it appears that Eq. “8] provides a fair ap- proximation [or Pr-t. it can be seen on Fig.-11hat Eq. (I31 does not actually fit the data of Ruhr cl al. preeisely. Figure T shows the constant surface temperature data of Sleicher et al. ll9i‘3] together with the predicled Nusselt num- bers using Eq. {18]. it will be seen here that at the higher Reynolds numbers still higher PH than glvs‘l'l hi‘ EEI- U5} is apparently irnnlied. However. the same thing is seen on Fig. -1 where Sieieher's results are considerably higher than Eq. [t8] at the higher values of Pe-t. In fact all of Sleieher's results on Fig. 4 vary wilh Pe-t in a way that seems inconsistent with the other data. lfthis is a result ofthcconstant surface temperature rather than constant heat rate boundary condition. [hen this is unfortunate because it adds another eompti ation to the problem. In fact, ‘r’oussef et al. [1992} suggest that there is such an influence. Transactions of the ASME a" .ts-a In. . st?! threw min— may nut mom-19: r. . it m: .3. Im v. Fig. 5 Ellen oi Pr—t on lamusratura prolilns So this is where the matter stands in the liquid metal region. Tltere is a major discrepancy between the analytic solutions [and this is of particular concern uith the DNS ealeulationsl arid the experiments where Nussclt numhers hat-e aciuaily been measured. Equation {J S) is suggested as a reasonable basis for calculating actual heat transfer rafts. hut there remains an uneasy feeling that there could be some consistent error in liquid metal heat transl'er experiments. One can only hope that this is not the case and that there is sontc other explanation for the discrepancy. Referring again to Fig. .1, all of the data for Pe-t > In are Pr-l-. quite consistent with a scatter of no more that: about '2 10 percent. Virtually all ofthe data for chPe-t s: 100 have been obtained from experiments wilh air: above 100 thc data are for mater, glycol.- and an oil. The results or tlte DNS calcit- lutions in Fig. 2 are vcry consistent nith the experiments. So far we have considered only data in the "logarithmic" region ot‘ the velocity profiles. The data for air and the higher l-‘randtl number fluids show nothing unusual in this region. but in the Sltbr'rlyer it is quite a different story. Let us now examine the available data for air. including the entire bound- ary layer. Turbulent Prnntltl Number for Air Figure B shows a plot of nondirnensional temperature. T". as a t'une:ion of nondirnensional distance from the wall Y', both in u'trlieoordiaerex. The line labeled Thermal Wall Law is a plot or the following equation. which is simply a ‘ocst l'it to a very large amount of data from numerous experiments for air in the "log" regiott [see also Fig. l] for the simple flat- plate turbulent boundary layer with no pressure gradient: 1" = 2.0?Sln Y' +3.9 (19} [Slight variations or this equation will be found in the literature. but the differences are small] Equation {19] then represents very closely the bulk of the experimental data for air in the "log" region. The results or tttn sets of calculations are shown on Fig 3. lloth were made using a finite-difference procedure and a mi - ins-length model that reproduces the velocity profilcs very i-ell. |'h_e Van Driest equation was used for tlte sublayer and "log" regtons. and .1 constant mixing length was assumed in the wake region. The Van Driest equation is: l= o-ll —exp{— rm ' t] where A ‘ =15 pot The dashed-line curve shows the result of using l-‘r‘t =0.ES constant throughout the boundary‘ layer. As can be seen. in Journal of Heal Transier u - men was mm..- m m. we... mm aura pun-mt n- Fr—l “seam-Munro": . .n h -on an Fig. 9 Pr-t data for air thce'log“ region the curve i Icl to Eq.tl91 hut lies about It] percent below it. The only a profile can be calculated that coincides with Eq. [l9] in the ' ug" r on is to int:oduce a higher value for Pr-t somewhere in the reg.on Y' (.30. i.e., in the shirlaJ-er region. Later ue will examine temperature profiles for water and for an oil that will illustrate this et'fect even more strongly. The following equation for Pr-t. based on experimental data. is suggested by Kay; attd Crawford {I093}. lt pros-ides a rel- atively high value of P'-t neat' the wall. but approaches 0.35 as uth. attd tltus }“ , Increases: The full-line curve on Fig. 8. calculated usit; be seen to fit Eq. [19] perfectly in the "log" region. This then. is the basis for feeling the Pr-t must be greater that: 0.3. somewhere near the wall. Note. hon-ever. that in the liquid mcfal region it ntaltes little difference because the heat transfer by eddy conduction in the sublaycr is virtually aluays very much smaller tltan that by molecular conduction |scc liq. [9}]. Figure 9 showsgxperintentally determined point values for PT-l l'rom Eo. {ll}; for air from eight different experiments. all plottetl as a func:ion of Y'. All of these d a with the exception of those of Hishida el al. {[986}. are It‘Jr external flat plate boundary layers with no pressure gradient. The Ilish- ida data are for flow in a pipe. in addition the DNS results of Kim and Main {1989). and of Bell c: al. (1992:. are plotted as solid lines. Finally Eu. {it} is plotted as a thin line. Between Y in and about “<10 tlte experime't al data are in t' y close agreement. The dry )Ct.‘ nt‘the n at higher \-: ues of Y' is partially caused by the difficulty of accurately measuring the four components ol' Eq. {ID} [which is probablyr the eattse for the high values reported by Fulachicr tlU‘lZJI. and partially the fact that in the “wake” region Pr-t seems to tend toward a value of about 0.5— . Note tha= oolh ot' the IJNS calettlatiotts have this characteristic - e probably more accurate in this r on than the experimental data. The presumption then is that experitttents at higher Reynolds nurn- bt'rs would 1101 show this drop front 0.35 until higher values of Y' are reached. As has been mentioned earlierI the limited Reynolds number range of the available experimental data is one d!— the difficu this in getting a clear picture of Iltt: behavior of Pr-t. The more inter- g region on Fig. 9 is at talues of l" helnw about 30. 'I hls ts ol' course the viscous sublayer region. With the exception oi one point try l-‘uln chier [and this proba bls‘ is a result of experimental uttcerlaitaty], the experittrcnts that extend deeply into this region sltow a marked increase in Pr- 1. The results of Hishida. and of Blackwell, are almost iden- MAY 1991:. Vol. 116l239 _ ldcqugpcfllu a. Dr . o r Elm-tutu. v. .0; I 'r h -ro .ro Fig.1D Comparison oi data tor an and water tical; Snijders et al. [l983} show a substantial rise although at a higher value of Y'. But the disturbing fact is that neilher of the DNS calculations show this behavior, although they do show a modest upward bulge in PM in the range of l" =40— .ifl. Furthermore. mine of the experiments short an upward inrlge in Pr-t in the 40—50 range despite the fact that this is a range in which there is substantial agreement hetween the ex- periments. and the range 30-200 is the range of 1’” where experimental uncertainty in the least. Figure 10 shows the experimental results of l-lollingsworth et al. H989} for water. together with the data of l-lislrida et al. {1986], and Blackwell et al. (I912). for air. The data For water are almost identical 10 those for air! it is worlh noting further that we see here Ihe results of three experiments. in- vnlving two different fluids. ver)l different measuring tech- niques. two different laboratories. one in Japan and one in the US, and over a time span of about IT years! Bell er al. {1992) attribute these differences to errors in the experiments, and further suggest that there are compensating errors in com‘ putations based on the experimental data. However. the er:- perirncntal data on Pr-t involve nothing more than direct measurements of mean velDCil)‘ and temperature. So what can one conclude? The only real difference between the experiments and the DNS calculations is that the latter were carried out at very mttch lower Reynolds numbers. For example, the Bell calculations are at a momentum thickness Reynolds number of 669 while the Blackwell results are {or about mm. and the Ht‘rllittgswortlt results are for I552. The Reynolds number difference may be the reason for the dis- crepancy. but this discrepancy must remain one of the unre- solved problems with turlrulcnt Prandtl number. Further difficulties in the sublayer region will appear when we examine heat transfer measurements at very much higher Praiidl] num- bers where most of the temperature profile is in the viscous sublayet. A Air. Effect of Pressure Gradient Figure l] shows the data of Blackwell et al. il9?2l for two cases ofan adverse pressure gradient in the flow direction. and lhcdata ni‘ Rt'rgannv et al. [1984} for a Serrig favorable pressure gradient. The Hlackwell data for no pressure gradient are shown in dark. Once more the Blackwell data show a steep rise in Pr-t in the sublay'er. From these results one could infer that there is indeed an effect of pressure gradient with an adverse pressure gradient causing a decrease in Pr-t and a favorable pressure gradient causing an increase. The data of Orlando el al. [I911]. Fig. 12. are not so conclusive. The Orlando experiment involved a technique using a triple-wire. which allowed simultaneous measurement of all of the components of Eq. [l0]. However. the size of the pro he made it impossible to make measure merits in the sublayer. although the measurements in the "log" and 2!DIVoI.t16.MAY1994 2! _ .._ .... _. AIM-DH“ mum-min . na- m. mm 't. .. - m." M mum-m. sever-Mu” ' —- - wow-a; or _ _.._ . . r . .. Y. I, m Fig. tt Effects of pro ssure gradienl is... .. . .._.._..... A Anallvm a _ _ _ P'-I .. ' ' _ 3' or . . . . . . .' :2. .= ..z w W fig. 12 Ellecl of adverse pressure gradient "wake" regions may be among the mUsl accurate ever made. Note that these results show a decrease or Pr-t in the it ake as discussed earlier. The only conclusion that can be reached is that there may be an effect of pressure gradient as indicated, bin that there are insufficient data to quantify this effect. However. the dis- crepancy between the two DNS calculations on Fig. 9 may be due to the favorable pressure gradient in a duet flow. Air. Effect of Transpinllion Blackwefl also made measurements for :1 boundary layer with both "blowing" and "suction." all with a moderate ad‘ verse pressure gradient. These are shown on Fig. l] together with Blac ku'ell’s results for the same pressure gradient bttt with no transpiration. Again the tendency toward high Pr-t in the sublayet is seen. and it cnuld be inferred that there is a small effect nftranspiration. l-lowcver. any such effect is pretty well masked by the probable experimental uncertainty and the only conclusion that can be drawn is that if there is such an effect it is small. Air, Effect of Surface Roughness On Fig. 14 are shown the data of Pimenta et al. (1919} for flow 0 e an aerodynamically lull}I rough surface. Pimcnta again used the technique developed by Orlando and was unable to make measurements close to the wall. However. for a fully rough surface one would not expect to find a \ iscous stthlatycr. These mung suggest that roughness has virtually no effect on Pr-t in the "log" region and. in fact En. Elli} would |'i| the data very well. Tlre Eddy-Diffusivity Ratio Very Close to Ille Wall Harare aetempting to analyze the available data at high Prandtl number [which “ill include the data for n'ttrer as well as those for much higher Prandtl numbers] it would be well to examine the behavior of the momentum eddy diffusivity at Transacllons ol the ASME Bro-nuns. an wamun-mwm .. - __ _... no.m.w..___.. sum-mm wanna-w: PM ‘- .__. . .nroo‘rnlmen sis-q men - Imam .9 al. u . .._.. a. ._. ._.__...__ n; _... _... . . .._ .__ n mrmrwmmmmmmm r. Fig. 1-! Air flow over a Iully rauin surlace points very close to the wall. i.e.. at values or Y' c 5. At high Prandtl number it is precisely in this region when: much of the temperature profile resides. This can he seen by again examining the inner bracketed term in EL]. [9}: L. Pr Even though the eddy dilrusivity ratio. gun. is very small in this region. the molecular conduction term. lr’l’r. can become the same order of magnitude or even smaller. Thus turhulent heat transport. and PH. can be important even though tur- bulent momentum transoort is negligible. link» will also he referred to as BDR in this pttpcr.| The reason for this concern with Fur-'9 near the wall is that it is Virtually impossible to ntake experimental measurements in this region. so Pr-t will have to he inferred by indirect methods, i.e.. by making complete boundary layer calculations using an eddy-diffusivity model. and this cannot be done with- out accurate data on tofu. The Van Dricst Eq. {20) provides a convenient way to eval- uate {trr'in and permits fairly accurate solution of the mo- mentum equalion, i.e.. the velocity profile. [It later the suggested value forA ' =25 is dctcrmlncd so that it leads to the law-0T- lhE-Wall. Eq. (I31 at values or Y' outside of the sublayer. However. solution to the momentum equation is totally in- scnsllm: to values 01': or’v for l" < 5. quite unlike the situation For the energy equation at high Prandtl number. it is thus necessary to look more carefully at to!» in this resinn hula": “sing to use the Van Driest equation for the energy equation 3‘ high Ptandtl numher. IThe most likely source for accurate information on 21,1":- in this region very close to the wall is the [3N3 calculations. On F‘s. IS there is plotted e.“st as a function or Y‘ from the DNS Calculations of Beth et al. “992). Also plotted is enr’v its cal- Journal ol Heal Transler Eddy-olnuglvllv ratio so . . .__._ rumor-av.” 5: DnuunI-fiu-v’fit my”: I Dr-I-WDMSIIPI.23 1. a . . ...__- 'n..2ciar'=..e'.'1i' \rua Vmwl’: anuq ism-Iss2 III-NV?" annular-130m warm-“murmur r Io roe - Y‘ on Fin. ts Waler, Pr: S.63—B.19[Hollllngswuflh| culated using the Van Driest equation. Although these results are tier)l close for l" )5, there is a progressively larger dis- crepancy as Y‘ Approaches 0.0. Further csami. Ition or the Van Driest results indicates that an!» varies as Y‘ ' as the wall is approached. lt has been well established on theoretical grounds that and». should vary as Y‘ ‘ near the wall. Assuming that this dependence would be valid to Y‘ =.. the following equation is proposed: c_u.-"v=D.DDll"] :22) Equation [23] fits the DNS results reasonably well down to l" = 0.] . as can he scen. so it is proposed that in all suh= quen: calculation Eq. [22] be used For l“ < 5 and that the Van Driest equation he used for Y‘ >5. PM for Water Figure 16 shows a temperature profile from the water data of Hollingsworth et al. [1939} plotted as T'. a function of Y' . Note first that a greater part or the temperature- \‘erlulitltl takes piaee in the suhlayet as compared to air [see Fig. 1!. Hullingsworth's results for l-‘r-t are shown in Fig. ll]. The Following equation is proposed by Hollingsu‘orth et al. {1989l as a reasonable in to these data: Pr-t =l +0.555— tanh [D.2(Y‘ —'i.."~l] {l3l It was previously noted that PM = 0.55 can be deduced from the slope of Hollingsworth's temperature profile in the "log" region. and Eu. {2!} does indeed approach 0.85 as l" is in- creased. As will be seen later when still higher Prundtl number data are examined, Pr-t cannot possibly reach the high Values in— dicated by Eq. (23) in the region Y‘ nil—5. it appears that Prv I must be near 1.00 at and close to the wall. and the DNS MAY1994.V01.116J291 kflafifl hi ‘ / maAnK1~»')wY»<Lvmmnlxlaiv:5 an mannnaara- . - . 5.. . a -no Fi; t? Wallet. =r = LES—5.35 IEukaushu anti Slenclaualtast :ak‘u‘ations ol Kim and M and Bell et a1. indiratc l-‘r-t just slr rtl-J- abate LE1: at the “all. The-relate it appears I':.:tt Pt-t mu . increase to high mines in the rcightcrhmd :rt' 'r" =1Il._ and then decrease tr: Cit:ch tn 1.00 at the wall. A telllElIh'E and rather crude suggestion is that I3 "' ‘ = I-IJI mt 41-: l" (j. and then Eq. (23) ._-:tr fig. (211] applies thereafter. Thr: value 1.0‘.‘ is based on the D245 resu'. . .-'-\.c-:uall;.- this b ' err 'ttzts IiLIlc client in the Prandtl number range :3. water, gher T‘tandtl numltcrs. Using this model. including liq. [21ft liar rJrr't in II neat- wall regian and the V57. Dries: equation for ir" > 5. calcula- tions were made inga finite-diff:rcr.cc pr am far the lial- lingswo-th as The resulting :tandimt‘ns .. tcmpereture pull—tic ts n an Fig. lb Whch II passes I rough :he data almost as well as could be desired. [it shcuit‘l be added that usi _E,t|. 13) all the wa', to the wall. instead ol Pr-.= ' 0". makes little :lift‘rrena: in this result. but it makes .1 large :1 {create '- n-e examine higher l’randtl numkcr data. nlsa . eluded an ti: _ure and shown as a dotted the lclllpfiulure profile calculn't‘c' usi'r =0_S.5 :uns...nt. Not IhE'. in th: "10;" region ' s cttrtc is parallel t:l thedata but IS "-3 percentbelow. Aswanhc case :‘n. this is further evidence that a ll'_lrer value For Pl'-'. is required in the rice:- wall region. Further it rlalfnns were made using an cmpirttal t‘it m the cur-re tnr . _ " from the DNS rcsultscrl liell el al. (1992] from Fig. 3. As car. he scan l".c Drv'S r|'_|.'_|lE begin to depart from 1'" exprrimcmal data at about Ir" El and flu: dit‘fescnrr is aluc-s :2: PH indlcatcd b_v th: T’rartc'tl liquids .ature Although ihc temperature '3“ unit .I'I a- _ :tzlt: using the at. ual prapeuiesaf state! as a functior. aft :rnpetatusc. hul ll'c varia'hte :trnirert y prablem is still a vexing one when analye etqaetinteatal data. :dwtng zny conclusions we Eltould examine sttme anal data for wa :. The result; of lekauskas and Slan- ciaus'sas {1987] are available in \'e.1' complete form and :t:e pl-DlLCd on Fig. 1". The same calculations were carried out as ngsn'orth dala. blll (his. Itm: Llle i'lollings'i'orlh equation for Pr-t appears to be tut) high and the results it -n '.|lE Bell data are a lint: rinse-r. T'.-.e anly clcsr cunthtmn is 0.35 toastanti is tar: law in the neat-stall regian. We er er. arr: 5c. - :cmii its in the lekuuskas FC- iullS. Close examination at Fi .'.‘ w:tu|.d suggest Illfll PI-L is lndrcdrct'f high mute range or ‘ frmn 5 to 9.:tndth rdrrs to below 0.85 a: Y' fl berm-.- becoming about 0.85 the LIUI‘JI' rcgicrz. This 3.- ;J of behavior does nnl seer. likely. 292 f‘J-ol. 116, MAY ‘99-! .2 Jump,“ mwmya ._ In a, adamam—a. _.__ a” Buwhr‘wflu' m» .v. et urinate tom—:5 Fig. la Trensfasner all. P: =dB.i-EA.3 Intrausltas ard Slanrsiaushasl The Ptanc'tl number far IlteZultausltasdata "artcd I"rnu!:h li'tc haunt-lut- la;.'er t'rcttt 4 53 to 5.13. so there realer v.1 '- EDIE-DFODK‘HF :rTrct than For the Holiings-t-arth data. httt |:.c method at caicul ion does take into consid ation this elitist. Anatht‘r cztlcrerre-e that may or may not he of imiat-rtance it that the l-iollingsn Dtth aspen—amt invnlved a :ather large are [avid seating length = thus a law enthalpy thicitncss Reg-n— olds number. Pee-h. whereas :r; the Zuvauskas experiment arr: virlttal origins hi the momentum and energy tct. . ary layers wet: apparently the same. The Z'Jkiiuikas experanent w s a ranstam‘ hear raw emerr'mtenr while the l-[ollingsuo‘tll r- ir-nentwas acorn-rant 52.91100 .' nerasurr Exile! new. [I :5 )Im not clear uhetltet 0: not well ,tt:rature \a aticn has a significant cl'lcst. Pr—t I'ur an Oil Zukauska; and Sian:iaukas ( | 98?) present some faith: enm- plctc :iata f a flat plate boundary layer "sing a transformer mi. The Pie." 'tl number vattcd through huundary lat-er [rum 48.6 'u 54.1, so there is a substantial variable property offset. T'r virlual orig s at the tuo baundarv layers were appull'nli; 'he same. Again tltis was a (DIE-'9! new: rate ex- periment . hr: results for two temperature prc le. are shown {In Fig. :8. Tn: solid curse show; the alt“ results using the s-arnc "Kidth as h the previous Its-u gums. Again the results _ gacans: PI-t=0.55 artshc-wn as a :lashrd line for reference. Once more (hL‘ ncacssh i; For a It -lr value tar Pr- I in she sttblavcr is _ enr. Thfic rt‘s IS stranglv s'.‘7.|rt:trl the .-.eerl :‘or a high Pr-t in thcrcgion l" c I-ZI. [ [act the data sligg ll 3r Pr-I has begun to rise as earl,» as i' - i or 1. allho “ It is difficult It- tell u-hc:hcr thc5trep rise in T' is not dues It: IOC‘i‘JSTiITlCnlflI uncertainty Note that far if" :10. l‘t- — BS is again quite leasnnahl: .5. httlterscmc result is ' tnn: rrr‘thc two profilgg 's for Re-ttt=1I2'.'" which is low and aparaaching the 669:15 Bcll [3N5 res \ and 3 . a. much higher value For PM is implied in the near- all region than the UNIS calcarln ,on; mid, 0n the atlte: hand I'rtt' particular ptafiletRe-m= Ifl‘.I ‘ yield! sontcwhal low-r values tn‘ 7” dues the pro , for R9. u:.h lower v :5 DfPl'd in the near-wall r. Cln halaace the results uJ’ Fig. til provide fair'iy strung surr-:crt t'or '.i‘|i.‘ suggested F‘r-t model. PM at Very High Pranall Numb" At wry high Prantlll number the tempereatrc profile tnuves (105:! and. Closer 1-: l' ‘ mall to the point where it is almost mtirnly inside of Y" . But it is sti rcryrnuclr gOrerneu br 3 rudiment ex:ltartge pr :css be:aus has already lt-een seen i' i (he eddy diffusivity [or heat. ts," is still very much greater than the male Transactions ol the F-SME : = fil‘rrj.|'."3 nnla Q REE-F5 - sun-um“ . 5.”. “ta-mar...- se—Lu; .m Flg.19 Full, flaeeloped new in a eirenartuue itcat. l-‘Pr. Althccgl: it would be very :lif‘cult to make my curate expert. . measucn'tertts in this reg r and DNS ale; tians at hlgh ?r are e dently no: yet practt It. it is still possible to infer the 'aeha m of Pr-t by ind 1 means. Thete is a considerable body or' eapelintental date cr. heat I:anst'er to high Prandtl :tumbe.‘ fluids for fully de'uclnpcd Flaw in a circular pipc. antl there at: a number of empit‘cal car- 1:] ions cf these data. Two ccrrelations for Nuss '1LIITIbCI' that yield. Let}: close to the same results. aztd the are nrivt' rather generally accepted a; defin'l' are those u. Slciehc.‘ and Ron 193‘s}. and cffiniclins "fit. The Slei he.- and Ftnuse :qcatian i5. Nttzi i-IJ.DJSR:“Pr" [14) where 910.334.24.11“ Pr}, s=0.333+o.5e'°-"" The finiclinski equation is'. _ -. n Nu: the icon lFlt..._l [25> [1.0+ [2.71:.’3}"’(PH"—I.DI}] .I" Calculations :an be carried out to solve the morent um and enflrgy equations :rt varicus Reynolds and Fratttitl numbers n the eddy diliusivity and turbultnl Prandll n miner models mi above. This s a rather simple finite- .ilcrcnct: c=|- Cllic..iDl'| sin-:9 the anpltcenle equations are ordinary :liffere’ al equatiuns. Figure 19 SIICWS the :esults of such :alculatiuns fcr two dill’erent Reynoid; numbers and Prandtl numbers from 1 m um. I-‘o ese calculations Eq. t2 .. it re 5. the VanD 'est Eq. [4 was used for 'r" >5. and a constant eddy diffusivity was used in the :23: ccntcrlinc region. The calculated hictio: caclt‘tt‘ients ue:e within a fractio: or per- cent ' the wcll- newts Kariiian—Nikuradse equation. PM = Li]? wa. _se:l [or if" from D [U 5. and the Kay; and Crawford Es]. [1]) was used fcr Y‘>$. [Equation {Ill and the Ho worth Eq. {23] give close to the same values For iJET.] I9. It is account that the result; .118 very Jcse tc bath th Sleicher and House equation and th: Gnielinski equal ion over the entire ?randl5 number rangl. At P.’ = I'JCIC [Iris becomes a particularly 'e lcsr ct'Pr-t in the region of 1" .rom O '.D 3. pro-ided tha he Values of eta";- used in this region at accurate. i.c.. En. rit 300:! it: 1-11 the empi cal equations for Nusselt number s ub— Iained alter a very large rang; or‘ Prandtl number. which is probably more than coincid r.‘.a|. Had ei r Eq. [3.1). the Kuy5 equat' n or St] Hullingswcrth equation, been used I'm P _ the calculated cunts would [ic considerably tclou those thou-n. Journal of Heat Transler e.tpe:i;tily at the highcr values ol Fr. So tt stern: thal Pr-t must be near 1 no very close to the waJl “.07 was usedfn this arse], consist with (it': DES caicttl Ito ;. However. the data for air and water [and ever: the tra sic—mar oi|| stern to indicate much higher values it'. the .1510" n:' Y" frut S to 30. P surna'tly Pr-t then approaches 0.35 a: _ r values oi l". DILhCILIgh this would have little influence [or Prattdl numbers srcal han IOO. This proposed model wi:lt a L‘tJl tau: P.-t near 1.00 close to [i1L‘ wall. and thcn an abrupt discontinuous rise to a much higher value at l!" = S. is obviously not Very elegant. but Ihc dearll‘. ol reliable data in l':ti; region makes anything better rather :iiffizuit. A further bothersome [act is that the d: hat .tuprmTt a high value for Pr-t in lhc stthlaycr are all . on: up ilTlC‘nlE on external constant pressure boundary layers, wliile I..e dale the: support a low value very near the wall are from mperi‘ menus with ['Jlly derelapcc l'lorw tn :np:s whcrc lhcrc is a fafira'tic atrial pressure gradient. the that a calc-elarnd curve is also includcti For the cast. it; ere the Van Drtest equation is used allt ay tut e vrall. This plot-toes a. good measure at' the sens ol' the alcu— lotions to :hcaltcuracy ct‘eur't. i: th: region C.L§\’-‘ to I. c wall. Summary and Conclusions 1 In the “loga ' region. Pr-tappc tobcprimarily a function via turhttlent Pe:ict number. Po tinfvtPr. At large values of Pe-l. Pr-t approaches a ettnstant '..a]u¢ of about 0.85. At small values of Pe-l. Pl-t ases indefinitfly. For gases Pe-t is sufficiently large so that r__ '°.'td; to be close to car rttirtthc"log"ltgicnt ' rhigherPrandtlnumberliquids Pr-I tttost defini: v a consl The r" ictt oi Pe—t where P-—I ts. igh is almost Eitiufiihc 111 he Pr :dlt nttmtc: long: of Lite quuill rnclals. 2 Two different equations, Eqs [11'] and [I3] ' oilered For th: dependence of PM on P: . The prime. difference belweett lites: Equations is i.'. the low I‘e-t [and thus lu-u Pr] region. Equation (1?: is a good iii to the DNS celctl " and the ‘t'akhct anal} and to :itpcrittenu'. data chat in the ct trline rcgioncf a pipe. Ec__a'iutt tilt]. which vtcltls higher values ol‘ Pt-' 'n the low Pe-t reg: is an apt-no .-..at: and less good lit to lite TCilIIIS cf several ..quid metal exper- imcnls in w";ch temperature profile-sand Nut-5e" -' 'mhers wcl: ttteasured LiJTC l3'.'l‘h ason for this dis:repa _ isnctcle but Eq. (15] is reso- ded it‘thc :xpc'rtmcnlel data on K 5:1: number al low Pr are to he believed and it ' desirtfd It: malt: calculations of Husselt nurntcr consistent with these ..s- :Jalfi. 3 l.-. th wake“ m5.on afanekternal turbulent buntciuy layer. .1... . e _nt:rline rtgio' t'or ['.:||5'dc\'{‘|0p(‘d turbulent Flaw in pipe. I- :Ipparenlly tends tc decrmse tc values in In: | num- neightc-rhood ctr-3.541? For-moderate and high Pra b:r l‘ltlitis. Tittse regions hat-e not been extent.» I but fcrLtmately the helm in! of i impurlanot.‘ in calculating heat Iran“. . or the “1th” region PM is usual ' suffictenlip t’- Thcrc is compelling euideiwc that for ait [ in t'.-.e suhl vcr Itgicn ' <Jfllt than in the "logarithm '= region. But here there is a rr _ r dtscrcpenw between ..e DNS c _u|.11t-:tns and l'ne expert- rncnls. The DNS calculations show a. pca's value oi Pr-t LII] t‘or air) at about l" .-.:4n whereas the capcritncns show no such peel: at all but rath a sharp rise - l—J-Zh highcr values it: the region nl‘ 'r“ from E '.D about 15 This behavior is desrrikod by either ut‘Ecfi. {2|} and |:23]. A difference be- Iwec: the DNS t‘fl ' lions and th‘ captrimcnts is tltat the DHS calc-.:!ations have been carried riul at very low Reynold; numb rs, hut whether this is the reason for the difference is I'JCl E.C:ll'. 5 The DNS caltu'.ati:lns indicate that T’r-t becomes lower Nth 199-1. Vol. 116il 293 and appmchn LC!) as [he wall is clnsciy approached. The exp-Brim nial Call .Dr very high Prandll number fluids confirm lhis |'a:1 and indiraic that Pr-t mus: be close :0 1.0)1': 111v: region or i" {ram 0 m S. A model using Pr—i—_ LO? for ‘F' from II In S and lhcn either Eq. [21] or (2.3] for Ir" >5 [or calcularicns from Pr=7.l110 l-DOC' produces results rcry alas: tc- Ihc expnimcnlai dala (Il'El' Lliis cnlin: Prundtl numbL'. range. in the liquid metal Prnndtl number rang-z there an: virtually llfl “minimal data in line subia - re ' in. but mnIrL"_|a.' conducwn is so dominant in this mgr-an Ihatit is .-.0| "Eccfisgrg' rc- knclw Pr—i ancmaiely. The DNS caieulaLimis suggest that Pr-i approaches LOO at the wall jusl as at highsr Frandll IIIHTItUS. 6 E. n:rimenlal data [0.- air gges! [hat Lhere be a pmsurc gradient effect an Pr—i an adverse (pa ill'rC] pics, sure gradiruL cans ig Pr-r Io decrease. and a famrabie (i g- ati‘.'r'i pressure E1. cnr :ausing Pr-c in crease. How: :r. lites! results are [ar from conclusive. '.‘ Trampiraziun [Hm-ring eir suciian) apparefly has links effect upon PM. 8 Surface roughncss has lilzlem no nfi‘ccl. ?-:Ir a r‘uliy mush surface rim-1e is iirrually n3 :ubiayer region so me :iailatle data are all in the ' 'logarirh Refere m‘ea , |c;'l"i1i5rct‘cri:. icnian. Judcspubliralianscantzining cxul data an Iurnulmt Frandtl number. pubiicarions c-Jn- mining daia I'xom which lLIIb'JIEJ'II 'Prnndil number may be deduced. and. publicalions :claicd indirrcl .: Lo ihc turbulent Prandtl lumber pmblem. References DEIIII. H,{".. lrd'fi'a'liu. EL. 1971‘, "RESPE- _ rma 51:: C an n Sim-lane Heal EI'J ' . "BI. 30 DP.|1‘5 I18 A. Brim] .'\ Ii! AnlnriaJt.A..In:Emwne.1 \h. '(‘olirmrioulmdcrx Kin-1:. Fund Numb“ m a 1mm“: 'Plzlle Mali M'emzlrcl'ln‘i Journal .13' Hm: :MJ' .lfu: Trauufiv‘ ‘ynl. 50. 1:0 “'1. "I. .‘D ED. Anlnr. 11.5.. and El J.. '9912."Turbu|.:lll =ririlillll‘liilll . in (he .‘inr- ' I1] .‘fm .rmufr' . n. 'i‘. E‘P- IW5 19]“. Antonia. R. -._. 3:: k III. I W‘ilb. RtFnDIéS 5m: Calcu'aumu ir. : I'uI]: Din-elem TIlr'UJ 'Du:( Flow, pfrrm and Mm marry. vii 3-1. in x. p . mix—mils. BaflHI' \' ‘I' :1 EL. :99! _ :awwnrr It! Turbulclll Boundary '_A)-:.' Pranfl] thumb“: and Spacc-Tim: 1c mper am Curd aims." AMA _'.1nrrrai’, Val. 30, ND. 1. FE. 35-4: nzil D. M.. Frraij‘fl J H,.lnd SikaIiDr of ii“; 31'. Hr Bum-j: In Iii: 113mm" DJ'F Id .‘WKI'IJJKL. Elana-ell. ‘i 7.. K45. W. H.. a1” Emnda: Lager an a Poroui Plane: A1 Behar-or Wiih Ari-ease I'Irisue Gn m: ecimze. Di.1ai5n.Dq1.sin-lechrdcu . ' Cu. Au; Blzn. I I‘D, Dircn Numerical $Ilbminli .o Hr.lT-i6. Ti- 'nm' r. m Ulm'.. SunfauL Ewaim: 'Ilnl Br:rnnim|inn on :Turbulcni Fran :ill Nil-n. 'Mr i: . Devious; Tri a-m Bounan Layz." am 1m=naimal He: Tramll.‘ :Caifrltf Pam. Lersaille. ul. 1 FL" 2 2, Alan. _ Brnznnrfi, K in: Krebs. L.. |99‘_' Esp: IIrKrIIII)‘ Dflt ilnrd Tlrbulrll'. rnrrrtwrl'mrul J“! .Efel.\vtlt.15_ M 2. pr). 1 ' . ad.B.H.l|lId film .3. I95 "e'mizi- Dislri'huuon and Tra'isl'e: :f Hra' in a I and Mimi]. '.'n|. ‘9. p. ‘— BL H - in Lia: u Halal. 4:! {NINE-HE! and ' PM: IISLIE. .und BEJIMKY. R. E = ElTes. ui fiupmilnfiovcd YCI: Corvm'liul' III 1| ..rr. l'|:\ .wla! Junrrvui‘ a! Mr)! um chzr Tmmfcr, Vol. I'. pp. full - 054. C..' A H.. and Falrhikr. R. F '36}. "Temp-(mitt |’.‘:fi|:< mi EdUII' mumm- in mm Men-H - J. rim. 1; h'a'l. ".1. .53-5‘. Cnrrbcu.A.l..rL'I:cuia. R. .\ and rum ,L .198 Tummy: Fundtl Lurflbcr am Snnra.CI1:.-a:'criui: urn mum: Mimi; Lm ' Immo- n'm ' qurnifufflul and His. Clua. L. r ' C'ErularJll. hi.'rrna.m.la. 2. p. ill. mm“! (I! Hui and Men Tmufli‘r. \"ui. 1‘3. hi). 294.‘Vo|.1|6, MAY 199-1 Fuflii. H.. '93. “Heir Transfer no F min: Scdium " l'l'fiii. Imlilm Inr i-‘alioricm uzignru: ' f‘ISH' rlnd. EIH. Her iii. ran. 2-“. IaAJIIir L.. I977— “Conribulion ) 'fludt d1 : lCI.ci.d5 ("IanIDi dr‘. nu'm u: c |i'('l.1lq'.1r :zm en: much: iin'ir ihuin'u. ' Thu: Dunc." c: ' . L {Eli dl: Pun-(3.1. Frlnx. ' "II E.. 1rd I:th z: I. 19' . "Erpc menu] huh; I u.'.i.I Swim: and Html'J; hirrwnuni . L'EI. ill. 02. )0.“qu o! Hui air: AMI; i—ularli . L m: Ania. |955 “5m oeraiure an: -rl 1 i-Iucmauo || :r.cual lubuml Fir-us] inner-an! )0.»qu o_r Hur a uns; Trees! Nu. ' . 9!"-‘.|9'. Huck. D I... Satan“. 1.1.. Lung. F. T 'J-(cnilna m of 'Iu'bulcm P.’alic.|| Nun . Slrt‘am I-irhilr ' I1 \‘clrflD‘Jul m rm. up .:rr.in:n|:l rm Boundary Lari-r; Ufldc.‘ Hip]: 'anfcm'r: Per-Ci. L|1ix oi Baylor. OH. in Muir-t. L'cll. v -Sfl '1 Equation far ||-_1I .EId Liz“ Tnufilf. i1 Tur- bultl'l I’m! an (in: Eng!!! Vol IO. 11!! 9-.lE5. Cumin.“ .n1-.l Emmi 'TI-( I: ran anmmi INJmhe- :n hum-ram Prol‘rnforl-km .anireni. . -L-u::n.]’i;ic.=lun'. um. “4:51;. .l‘r: 'TGI—ITII 3. A . aud Smith. J. 'L. I as. "TurI1L|::Il i-i:u-. Trumlc.‘ From Scrum In: P. :rfarc 'i'ol l' " r.— svm' Jon-m: I ql‘Hflu and .i {:5 Tmmfi'r. . . . . "'ln'hu'cnl .Ixandll burrich \l'il in a hur-‘A'all AM Jonah. rol ‘ 43.1668. ' riI'DII. 1.. Juli :iiruli. F.. IWl "A. [\(apraJ A. ' . unhnr Heat Tnmlr :irlmh'orm'Ji-umci :‘ Furl: Jhlinr lDr Ifiih . L'J'h'm.‘ am .irm rmch .Dl. lama. u. nguz. m. 113mm. n r-anzicn Prue: or Hm and Mo muu aim lllr WaE Rvgiu‘l :-I’£Tu :ulm; Pin I-‘low " Mmmr'rp SII'H‘ m Frit. u1u.'."{ar:.' Tm'ml’rr L'UQQTMZV. Hrmil:ll:l: rimming cor... in. IN. Hoiiimfimonla D N . Km an: Fred: than :f lhl: TLr::I CCIICfl-VC Sir-run." RrpO'l hirer. _J|§| Steinem. Ul. Sui-lard. Cv‘i. Srpi. Innhins. 2 . I95 inn al lh: Ed inocIdumiily “'ilh E’muflfl MCduluS and [K UK! in plldLllol'. ol' Turbu'llllr Hui T'ansfu Cmfficil 1! m.- HM’ “3m and fluid mining; murmur. Em :ifarxl Lirm min: Fuss. lurid. Iii-ill. H.. and Erica. II. 13 Pull IIIJ Schmidt Numb-.- _ Vol pp. IE-U- [SF Kadrr. _ A 1:] Yig'urn. A. M Turbuienl wall Fl.er i. fr'.'(rr'm‘r':nci' :5. 3:. A‘ml‘fl. Kadrr. A. .JEI , Tum-mun- micinnzie: Div I93. "Arum ||'( Predimon of Turbnlm {mama-Hm}J'Iu.'nefu)'Heciurfl.‘lfm m»? 'H II 2nd Mm Tlaml':r lEWi "-:I Hi I! mai' r:f.'i’na.' air-n drag; thfi'r. ‘u‘ol. mi Cancrmzii'fln T-‘rclfil-e' in Full! 'i‘ur- hLirII'. “Guitar;- szx Inlwvuu'nm.‘ Jamie; qr Hm arfl ms.- Marina ‘ :. N:.9. pp. I541 4.1. Mini. E.. Kuroda. F... and Hurt: \I.. IQEQ. N'uurinl III‘TSIiSaIiEII c-I' Mar-Wall Tmhuni: Hsz ::I rr :kjng Inln Marni Iii: Unmads Heal . 3. F. 99$. andan’Ii'éJZ, IWI. “Elinul Numei 1 Eim'zléiinr .alar Field in s Tm-Dimemiuaai TLr'JuImI (Runnt' lefllk LM‘JIOSILI'I an Tam." I'lrw ff; '.. Tfi'ii. L'niu. of Munich. , 5cm. 9-: l. . ' K2_-L“' hi. an: Crawford. hi. E.. I993. can I-zclin: "Ill and “us Trinircl. 'irird Ecluon. Infill»- IiiE. 5'“- Yuri. Him J.. Ind Izmprm nf Passive Scalar: i:i :Tilrhilear E'han 1zl - - Pawn-dug; :fS nimmizcclfiyllwnmm w rwiawwMa-im Talk-nee. rum. 5.11 9. I59. Sprinan-L'crlsg. Mrl n.i{:id:iL-c:n. P1. E Gin. 0.111 50. a 5.1.5.. Iw‘ Flam ' Mun-[Mar J'numu' «Hem I-‘l‘i‘ l' W)" l-In J us or hibu enl Hui has: Fifi-Ear. Vol. 3!. Na. L4undr. E... Indiamlrlwffi‘s. E.. 1979. "Amiitsliun ul'a Eecnni Marital Inhale-1:! i..osu.'= ID H SI and him Tmmporl in Tllln Shcal FIrvwi—I. ‘hi'fl- 5c:und-Ho mm Closure Pram—and Future?" .Im. 10.513. :4. :p. 2mm. .I”TII_K|1L|=3IT:maerxu F'ui. ualbnain LiqL-id Helit “ -0.r' HNI-vnd.‘vl'u: rmm- .oz. 20. No. In :Ip. m5- LEDIILS. L.. Ihrflllly. [J .alld.'-I:L£ Simulaiim o.’ r‘mir: Hui 15m ii: 3 mm: Joanna of Him am} Mui- rrwrxl' M "1|ir.J.B :9‘3l.”3i Nu'hclltai thanan Flow. (urn—r Sa. .‘Go. as I'D. .149-llél. hi F.. lure. ‘Musurm'enlol‘ :Iirm. 'J. 34.. Picknl. P E.. End Tail Transaciions orlhe ASME a; .-.nu- n UQII finial Tnl‘d chnl Pin-ill Numbcrs Ill Elll III TIIIJH.” nan-fiaiiuwulhzvm'i cier m! Mm Tmufw. \ol. i9. 1): iw-SDJ. Maehawn . Kan-eds. 1‘” Kabansiii “y. and Yxlusnflii. IL. [9; 'An bpfliflkmfl Sud) 011 III: Span-am: Edd! DIITUiI II. of “1 II in A F'al-F‘Iuli: Tirk‘uIrlK BCMI'JSIH' La} " (m'rrnuihE‘m-JI Jami“)? -1_l Han cm1'.Ll'L:.r I'm-myth i.--:i “10.3. ml. mi- i995. m horn. .t . and 13:12.5. 5 p." 3- i'urrmguolmi’ Jnumm' a; Hard mi Arm ringer. Vol. n HM. MOI-I'm. R I .and K"'S. W. \I Dir. my. men. a”: '1; nganci. Y.,and in. EC. IDSE .n wall TuibulcniSIi Ear Flo-a I.‘ ASHE IoL-uu-ui Lsnr I In 955 a. V0 :p. Sin-5:59. 0 .mdmA. F” \Ion. .1 .'.. :11 3113's. Ti". 31.. 197- TiuDLIrfll [{JIIFSCII all an and. \iwrcriuiT : Boundary Lana Su'ajn-i : l'mxlnaiim. Similar 2nd ‘Harilblc Wall Tenacrliiiie." pori HMT- 7. Thainm-imm. Dhisior.. Depl. al Mun. Enni.. Siaaforu U. "A. Fainl. v. E, anfi Had. M. P... I969 5am: C-bxr 1m :n 5km . .ii'or. Ind lencily l’mfils Il‘. ELI: Dn‘el-mcd Pipe and Char.m1?'lav-:."J=I'MN 9f Fi'm'dmrhuam. Vol is. m Isi .mi. Fury. A. IL. and H-zflilian. P H 3T6. "An Eqmincmai film}! of Tn:- cnlmi Cummin- Hat Tunil’u nrr I Phi Plan.“ harm-I ofFinri'dJlr- rim-in. vol. 1?. pp. 155-366 _ P.lli=i..1..‘IrI. KI Murral. ll. J.. and I: I . 39“? "Tile SliuclLitc-l : Boundary Law. a Rival. Wall Will: R «an; and “ET. Tlaml'n." ASKE forum. :J Han Tnssrufibi IC'. :p. I9l-I9!_ - QLafirhy. 3.. ail-J Quark l'}‘-" Men. “menu of the Radial and Tan- gzinjal Edd: Uiffum' «s DI rIea: :nd Mm Trxurni n .1 Han: Tuk. Lire/- irci‘i':li::l JOWIT‘N Earh'fli. an"? JFII‘IJ Tlaufi‘l. ‘v'n'I. l F11. :lW-IHT. ‘TI'4 Predicrim ol' TLrnulz-i: Prandil and humid: I9!-l. “TEID'JICIII Prlii:i| Numb-r in Cir:qu pp. :isx. 196.— Hcpc-n 'ci. P \lT-' TI|£Iinmci=iclzr Elanlar'l Elan-:1 . CA. Aug. '.\ Tara-Equation Mods "-2: Hm Irnnwcn IIU. “359. Reynolds, w_ c.. Kn 58w. Waaiimzion. DC“. Rnguinv. E‘. 5., Zrhclisih. V. P. SIIiihm'. E. ‘1. In: mexh. A. . I9!4. "Some Ami ur‘l- [bulz‘fl .eai 1 arisi'tr iii an: ind Flaw. Dn :irizab'r Sullaaea. 1.1M. now." Jm'fllfl‘r a} HM! :Ma' Ma: Tm.u;'(', \'D|. _v. No. 3. Eli. IE I-IES'J. Sheri" NZ. and Kane, 5}." Ifil. "Soflini: Eddy D-" .Y :I Hf] lem- ursrxnismacimilai Duct,” rm'rrnari'mci'Jm-maInJ' (Emmi: Him i'm-nyrr. Vo|.Ll.|1rI.1:5-Jll. S‘lii :n.T. W Mnl'fal. R. J.. Johnston. J. P.. and K3. Nu. HMT—J 2.1.‘irrmia'ciimfliv nus—at Mach. Engine Sun rm]. CA. New. w .. and Klin: 5. L. I958. NASA Hem: i. .- I980.Re'pu:l .h anrcmunin. SimcannJi ' of III: 'i‘u'tulcu F'IILII Nmit: mn'mrvi' Jounim' affirm mid Mani Transfer. \‘ol . I ; l-U. Stupiuli. E . Tire! ' .Jnd v: m. L.. 100$. "Dc '"linaliDlI d» WI "menu" -. un may “Ammonium 4 r.\ '1 use ci a- .—..Hm.m.'»,mmnmr Mm rm- r». Vol. 3.;; $17.9" -iurr Fioi'iifi Ior "An F1p¢rimnial Ema? ii IL: Simian." biw- . ‘ I‘D SIeicI‘.eI'. C. \ . Ill: :IlseI .kl. W” I97 '1‘ L‘Ommunl Cuntlalifli lDr Hm Tnnsirrm Ennmii: Mil Lunatic I’rapcr Fluiibin'l'urh J..-“K. 9' _ I 02'. i'J—‘EHIt‘uoflcl' Jo'flmci' If”! “ME: in! Tun-r “er, Vol. IE. pp. 67: 6E]. SIN-(I‘LI \A'._Gc.wan. N. 8., In: ‘W iliu'ii‘l. 3.196 Eddr DiIT and Tempe'amre Piafil'el I'or Tur:u|-.i' Hus Tianil’.. io Waiu m PICK). cam " Frag. 51m11057L-m Srr. El. :9. F )2 Sni' mi. A.'I . Knpp'iL . l. . \I . Heir-van. . Mid ilrl'Lt). 0.5.. I975. "Ail Erperim: 1la- Del=m.n:Iion of ihi Tliriaiileix I-‘rnlidi 51mm in me Ini'xr EDEMJer Lag-n- For Air Flan- Orcr 1 Flu " c.” Sixth immunian Hm! Tmng'cr ConIr-TME. Toioam, Us] 9. 5 -33. Enii1(:i..A.L Rom». A. .‘.I.. am] N'inu mu. D Dru-Rninalicn iJ.r . (a: Air FIW Own : F.al Pkic. wemmmnvf Jun'rrm" v," Him! c.n:{_\fi:s: me'et. \‘ul. '6. No. i, u:. I .. 4“. Eubrhi'.ani:i!. C. S. and AK uni]. R. A. "Elfc: of Rental 3 \‘Link: DII .‘J SIIIRIIEIy HI Mini 'lJIhuk :il Bu'znflan‘ Lflfll 'lirrrnrnna'icfluuwi 2f :jl'kul um! .‘t’ais: Dir-Eff". “J.. 1‘. Nu II p:. ISSJ- IE-lE. Ihielbah'. W. H .544». Vi ar-J .\ mi, R. J.. [969. Inuit :ilu HHT- 5. Ttr'inoscielicrs D V. Devi.ol}|och.Eiizi.1n-riii:.SLanIu-d Ui Sianru'_. CA. Mr. “Easel. r‘\. T.. :1d Czuun. I97), "Crixiila'Jon of 'IJI'Iluxr! Tannin LGels Owe! Flll Plals “'ilh 3|.rlllfll I‘Iisicnimaluginl TIieui—ie: of Tui:|i I'eli:= and Van': :Ie anbuflelit Pralidil Nair-bet" i'nnel.u.'.'ocai'im:l.1a.' of Half “Mam Tmnzfi‘r. \'o.. I6. |1D iii-1.1553. Yalhw. V .CISIEI. S ..:iil:lYl|1IIo:. * . I951. cal Tralnf. in._U.':=I:i'. 5— I. P'v: Flaw i'nrevmii'unamym'qr Hevrand .U'cs: «rm. val . I. pp, I‘ _1_ Ybuasef 51.5 rhyme. Y .li'd Taglwl. Transfer “odd [:1 Pkdk’ljnu ‘I'uibulnnl Tiisimi Cum-dam." Mimi 35.ND.1I.JP.W IDL Zii k: iiskas. A.. and Stein's Jskza. 1!... I957, Hem I'rwufeu'r Tun‘ud'cni Fmid Hon, HCIIIISZ'iEk Publ::hingCu:p.. Washington. DC. I9!J. "All E ‘PCIEI'CI'III l.. 1992. "A IIE-Equalicn Hnl ._r-n=| i Uiidci Aiaiirar- 'al! :i .‘m'imi’ :rHeu: Mde li-uww. ' ...
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turbulent prandtl no. - The 1992 Max Jakob Memorial Award...

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