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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? Theobjeerite rafrnis prepares to examine rrr'rr'cui.[ v the pret'crrtlyuyuilabie experimental data on Tutburn” Frairdrrr'sitmibprformertrodt'rirertsiona.’nitindent heraldry layer.
artdlforjrrffv rret=efamd firm in a circular tube or after duct. and artcmpt to dran
.tmrrc corichrsmns as m ttftera 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
threedimensional fiotvs or recircuiating flans. 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
twodimensional 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 mixinglength 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’rt] for
the twodimensional 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.
Keynun}! 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"tl='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 salte the energy Eq.
[9} for any temperature boundary conurnons is Prt. Evaluation of PM if we ntlw‘ substiittil: Eqs. (1] and [5) into [S] we obtain an
equation that can be used to evaluate PrI from experimental
date or from analytic solutions: at Pr‘ = {10] ﬂy Thus to evaluate Prt .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’rt=1.UU
that we will examine. lIt'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_4lltt 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: llIJ The reason for the limitation of Eq. (ll] 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 _ Reh enthale thiefvtness Reynolds “shear‘r velou.‘
C; = fnﬁ‘lm‘ CUCfllclcnl=2To" number — a3ttm2’v mean velocity in a pipe
{WT1» 213301131 Rem 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 timeaveraged 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 Pet = 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 Prt = 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
Red = 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 Italn "log" region prollle slopes
; Prelaure Gradient a DJ!
E ' Experiment lieIn ﬁeh Prl
uncutImam 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] ' =
Rollingworth :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: Prt = C;."l.44 (15] Thus for this unique but rather fundamental ease Prt in the
"log" region eatt be determined simply lrtrm the slope of tlte
temperature profile plotted in sentilog coordinates. Note tha:
this further implics l’rt 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 Prt 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‘rt=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
Prt 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
uhat 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 tensma But1min: :u‘armsuﬂhﬁug'uw "rJetenmmntou—l 3...", =ig. E Analytic solutions . y ccvoid of et . rica.
simple equation gite: 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'tell [‘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 Prt appears as a Function of Pr and I:
however, an 2 PM tun! Eq. 5] is plotted :E a rumLinn
ol'the produetlr_w'v]Pt, u. .tch we Mlleall the turttt '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 curteta
all collapse ‘iu ‘ sa e lo function of Piet :xccdt at very
lcw values cf‘clrr'b. i.e.. values encmtnleted for l" < 100. At
high ralncs ol I‘eL 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 rylowvaluesolFett'
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. . TrPee—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’et 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 Pet. 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
Iheresultscl'thereeentsuecessol’l ect HumericelSimtlnlion
IDNSJ ur‘ beta 2 tbulent ﬂow in an s and the turbulent ex
ternal boundary _ er. DI'M'S involves complete sniutian ol‘ the
tttncriependent 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 ﬂeaJury,
but interesting rte.ttits are hcgtnntng to appear. The results ol Journal at Heat Transler Munch mv—nw(ua: mlunrhcA—u mammo nu: ateg— Mme: t...s=s;x.u:  m v .o. \ ﬁg. 3 PM from 3H5 calculitlans threcs ch eal' Iatittntt are shawr. n Fig. 3. TIc results ct" ..t
and Main (1989i are [or fully developed I'lau . II Flat duct at
a Reynolds number. Red   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. Rerr. The only prehetn mth the 3N5 ca' ul:tlio'
date is that the Re holds numbers are so EDI dire:tt. 11113 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 [latplate
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 thewalte. 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 :enteri..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 Pet:> 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 Prt=ﬂ.85 at high value; or Pet. 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 :lnllrIIM
"net [at annualat mt suntn a runa ..nn.o FltI t auntus;
Hmr—enntaw
“Dimer luquIII
Mrlﬂ ezan
min.
ace—«m ItsW .95.”.0‘... . . .r we we:
FnI 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 Prt data From these
experiments. The following equation is plotted on Fig. 4 as a
reissunahll: approximation to these data: PM = 2.0! Pet + 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 finitedifference procedure
and a wenestablished mixinglength 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 auntcu 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
Prt 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 PrL=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‘rt. 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 Prt. 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 Pet. In fact all of Sleieher's results on
Fig. 4 vary wilh Pet 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" .tsa In. . st?! threw min— may nut
mom19: 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 hate 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 Pet > In are Prl. quite consistent with a scatter of no more that: about '2 10
percent. Virtually all ofthe data for chPet 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 finitedifference procedure and a mi 
inslength model that reproduces the velocity proﬁlcs very iell.
'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= oll —exp{— rm ' t] where A ‘ =15 pot The dashedline 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 punmt n Fr—l “seamMunro": . .n h on an Fig. 9 Prt 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 Prt somewhere in the reg.on Y' (.30. i.e.,
in the shirlaJer 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 Prt. based on experimental data.
is suggested by Kay; attd Crawford {I093}. lt prosides a rel
atively high value of P't neat' the wall. but approaches 0.35
as uth. attd tltus }“ , Increases: The fullline 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 Prt must be greater that: 0.3.
somewhere near the wall. Note. honever. 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
PTl 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 Prt 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 Prt. 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 Elmtutu. 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 Prt 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 30200 is the range of 1’” where
experimental uncertainty in the least. Figure 10 shows the experimental results of llollingsworth
et al. H989} for water. together with the data of llislrida 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 Prt 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
Prt 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 Prt 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 triplewire. 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! _ .._ .... _.
AIMDH“ mummin . na m. mm
't.
..  m."
M mumm.
severMu”
' —  wowa;
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 Prt 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 Prt in the
sublayet is seen. and it cnuld be inferred that there is a small
effect nftranspiration. llowcver. 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 Prt in the "log" region and. in fact En. Elli} would 'i the
data very well. Tlre EddyDiffusivity 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 Bronuns. an wamunmwm
..  __ _... no.m.w..___.. summm
wannaw: PM ‘ .__. . .nroo‘rnlmen
sisq 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 Prt will have to he inferred by indirect
methods, i.e.. by making complete boundary layer calculations
using an eddydiffusivity 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 law0T
lhEWall. 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 Eddyolnuglvllv ratio so . . .__._ rumorav.”
5: DnuunIﬁuv’ﬁt my”: I DrIWDMSIIPI.23
1. a . . ...__ 'n..2ciar'=..e'.'1i' \rua Vmwl’: anuq ismIss2 IIINV?" annular130m 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‘rt are shown in Fig. ll]. The
Following equation is proposed by Hollingsu‘orth et al. {1989l
as a reasonable in to these data: Prt =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, Prt 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‘rt
just slr rtlJ abate LE1: at the “all. Therelate it appears I':.:tt
Ptt 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 "' ‘ = IIJI 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 ﬁnitediff:rcr.cc pr am far the lial
lingswoth 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 ' ne 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
curre 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
alucs :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
plDlLCd on Fig. 1". The same calculations were carried out as
ngsn'orth dala. blll (his. Itm: Llle i'lollings'i'orlh
equation for Prt appears to be tut) high and the results it n
'.lE Bell data are a lint: rinser. T'..e anly clcsr cunthtmn is
0.35 toastanti is tar: law in the neatstall regian.
We er er. arr: 5c.  :cmii its in the lekuuskas FC
iullS. Close examination at Fi .'.‘ w:tu.d suggest Illﬂl PIL 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‘Jol. 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.iEA.3 Intrausltas ard Slanrsiaushasl The Ptanc'tl number far IlteZultausltasdata "artcd I"rnu!:h
li'tc hauntlut la;.'er t'rcttt 4 53 to 5.13. so there realer v.1 '
EDIEDFODK‘HF :rTrct than For the Holiingstarth data. httt :.c
method at caicul ion does take into consid ation this elitist.
Anatht‘r cztlcrerree that may or may not he of imiatrtance it
that the liollingsn Dtth aspen—amt invnlved a :ather large are
[avid seating length = thus a law enthalpy thicitncss Regn—
olds number. Peeh. 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
irnentwas acornrant 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 later
[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 sarnc "Kidth as h the previous Itsu gums. Again the
results _ gacans: PIt=0.55 artshcwn 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 Prt in
thcrcgion l" c IZI. [ [act the data sligg ll 3r PrI has begun
to rise as earl,» as i'  i or 1. allho “ It is difﬁcult It tell
uhc: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 Rettt=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 ptaﬁletRem= Iﬂ‘.I ‘ yield!
sontcwhal lowr values tn‘ 7” dues the pro , for R9.
u:.h lower v :5 DfPl'd in the nearwall r. Cln halaace the results uJ’ Fig. til provide fair'iy strung surr:crt t'or '.i‘i.‘ suggested F‘rt 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 lteen seen
i' i (he eddy diffusivity [or heat. ts,"
is still very much greater than the male Transactions ol the FSME :
= ﬁl‘rrj.'."3 nnla Q REEF5  sunum“
. 5.”. “tamar... 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 Prt 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 iIJ.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'Prt in the region of 1" .rom O '.D 3. proided tha he Values of eta"; used in this region at accurate. i.c..
En. rit
300:! it: 111 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 thoun. Journal of Heat Transler e.tpe:i;tily at the highcr values ol Fr. So tt stern: thal Prt 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 Prt 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 Prt 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 calcelarnd curve is also includcti For the cast.
it; ere the Van Drtest equation is used allt ay tut e vrall.
This plottoes 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. Prtappc tobcprimarily
a function via turhttlent Pe:ict number. Po tinfvtPr. At
large values of Pel. Prt approaches a ettnstant '..a]u¢ of about
0.85. At small values of Pel. Plt ases indeﬁnitﬂy. For
gases Pet is sufficiently large so that r__ '°.'td; to be close to
car rttirtthc"log"ltgicnt ' rhigherPrandtlnumberliquids
PrI 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‘et [and thus luu 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 Pet reg: is an aptno ...at:
and less good lit to lite TCilIIIS cf several ..quid metal exper
imcnls in w";ch temperature proﬁlesand Nut5e" ' '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 neightcrhood ctr3.541? Formoderate and high Pra
b:r l‘ltlitis. Tittse regions hate 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 Prt
LII] t‘or air) at about l" ..:4n whereas the capcritncns show
no such peel: at all but rath a sharp rise  l—JZh 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’rt becomes lower Nth 1991. Vol. 116il 293 and appmchn LC!) as [he wall is clnsciy approached. The
expBrim nial Call .Dr very high Prandll number ﬂuids conﬁrm
lhis 'a:1 and indiraic that Prt 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 lDOC' produces results rcry alas:
tc Ihc expnimcnlai dala (Il'El' Lliis cnlin: Prundtl numbL'. range.
in the liquid metal Prnndtl number rangz there an: virtually
llﬂ “minimal data in line subia  re ' in. but mnIrL"_a.' conducwn is so dominant in this mgran Ihatit is ..0 "Eccﬁsgrg'
rc knclw Pr—i ancmaiely. The DNS caieulaLimis suggest that
Pri 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 Prr Io decrease. and a famrabie (i g
ati‘.'r'i pressure E1. cnr :ausing Prc in crease. How: :r.
lites! results are [ar from conclusive. '.‘ Trampiraziun [Hmring eir suciian) appareﬂy has links
effect upon PM. 8 Surface roughncss has lilzlem no nfi‘ccl. ?:Ir a r‘uliy mush
surface rim1e 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 cJn
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 Simlane Heal EI'J
' . "BI. 30 DP.1‘5 I18
A. Brim] .'\ Ii! AnlnriaJt.A..In:Emwne.1 \h. '(‘olirmrioulmdcrx Kin1:.
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]: Dinelem TIlr'UJ 'Du:( Flow,
pfrrm and Mm marry. vii 31. in x. p . mix—mils. BaﬂHI' \' ‘I' :1 EL. :99! _ :awwnrr It! Turbulclll Boundary '_A):.'
Pranﬂ] thumb“: and SpaccTim: 1c mper am Curd aims." AMA _'.1nrrrai’,
Val. 30, ND. 1. FE. 354: nzil D. M.. Frraij‘ﬂ J H,.lnd
SikaIiDr of ii“; 31'. Hr Bumj:
In Iii: 113mm" DJ'F Id .‘WKI'IJJKL. Elanaell. ‘i 7.. K45. W. H.. a1”
Emnda: Lager an a Poroui Plane: A1
Beharor Wiih Ariease I'Irisue Gn m:
ecimze. Di.1ai5n.Dq1.sinlechrdcu . '
Cu. Au; Blzn. I I‘D, Dircn Numerical
$Ilbminli .o Hr.lTi6. Ti 'nm'
r. m Ulm'.. SunfauL Ewaim: 'Ilnl Br:rnniminn on :Turbulcni Fran :ill Niln.
'Mr i: . Devious; Tri am 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.llId ﬁlm .3. I95
"e'mizi Dislri'huuon and Tra'isl'e: :f Hra' in a I and Mimi].
'.'n. ‘9. p. ‘— BL H 
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054. C..' A H.. and Falrhikr. R. F '36}. "Temp(mitt ’.‘:fi:< mi EdUII' mumm in mm MenH  J. rim. 1; h'a'l. ".1. .535‘.
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Iiiill. H.. and Erica. II. 13
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 Spring '10
 Dr.JamesKlausner
 Heat Transfer

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