Turbulent Prandtl Number Distribution within the Constant Stress/Heat-Flux Region of a Near-Wall Flow

^ 6 m tsoi

Turbulent Prandtl Number Distribution within the Constant Stress/Heat-Flux

Region of a Near-Wall Flow by

G. P. Hammond

LUCHTVAM

HOGESCHOOL DELFT '/AARTTECHNm'.

iUX.'iJ^ ji_i - ^ a 1 .^EK Kluyverweg 1 - DELFT

School of Mechanical Engineering Cranfield Institute of Technology

January 1984

Turbulent Prandtl Number Distribution within the Constant Stress/Heat-Flux

Region of a Near-Wall Flow

by G. P. Hammond

School of Mechanical Engineering Cranfield Institute of Technology

Bedford MK43 OAL, UK

ISBN 0 902937 96 0 £7.50

"The views expressed herein are those of the authors alone and do not necessarily represent those of the Institute."

TURBULENT PRANDTL NUMBER DISTRIBUTION WITHIN THE CONSTANT STRESS / HEAT-FLUX

REGION OF A NEAR-WALL FLOW G.P. Haranond

School of Mechanical Engineering Cranfield I n s t i t u t e of Technology

Bedford MK43 OAL, UK.

ABSTRACT

The turbulent Prandtl number variation within the near-wall region of fully-developed boundary layers is obtained by using p r o f i l e analysis. This is achieved by deriving expressions for the 'eddy' d i f f u s i v i t i e s from Spalding's inner layer formula and i t s thermal analogue. The analysis is performed over two orders of magnitude of the molecular Prandtl number, encompassing most of the common, non-metallic f l u i d s . I t ensures that the l i m i t i n g conditions at high and low turbulence Reynolds numbers are satisfied e x p l i c i t l y . The turbulent Prandtl number d i s t r i b u t i o n is found to be complex, and

dependent on i t s molecular counterpart within the diffusive sub-layer. I t is compared with the distributions implied by

several phenomenological models, and the consequences for the treatment of the near-wall region i n numerical calculation procedures for t h i n shear lao'ers are discussed.

Introduction

The time-averaged momentum and thermal energy equations for turbulent t h i n shear layers contain apparent or 'Reynolds' shear stress and heat-flux terms that need to be modelled i n order to close the equation set. Many and varied turbulence models have been devised for the Reynolds shear stress, whereas by far the most conrion practice f o r modelling the corresponding heat

f l u x is to prescribe a value for the 'turbulent Prandtl number' (Pi^*.)- ^^^ r a t i o of the turbulent or 'edcjy' d i f f u s i v i t i e s for the momentum and heat. The l a t t e r d i f f u s i v i t i e s appear i n the classical Boussinesq formulations for the t o t a l (molecular plus turbulent) shear stress and heat-flux which may be w r i t t e n , using conventional near-wall scaling, i n the form:

T^ = c ^ ^ I ) ^ 0)

+ f^ . *\ <n* h . ^ia\ (ïï*

( ^ • • : )

' • • I f - . l ^ - i T F ' ^ j l 7 l«

s

^e

The simple addition of molecular and Reynolds fluxes, as implied by equations (1) and ( 2 ) , arises from the collecting together of terms appearing in the time-averaged thin shear (or boundary) layer equations. I t needs to be emphasised that this practice amounts to a hypothesis about the nature of the processes of molecular and turbulent interaction in regions where these

fluxes are of comparable magnitude. The v a l i d i t y of the additive formulation must therefore be rather suspect, in view of recent discoveries about the structure of turbulent wall leiyers.

The turbulent Prandtl number was developed by analogy with i t s molecular counterpart ( P r ) , although i t is normally considered to be largely a function of the type of flow rather than f l u i d properties. Despite the widespread use of Pr. in conjunction with numerical calculation methods since the mid-1960's, uncertainty remains concerning i t s dependence as has been emphasised i n the comprehensive reviews by Reynolds [ 1 ] and Launder [ 2 ] . The present con-t r i b u con-t i o n focusses i con-t s acon-tcon-tencon-tion on con-the behaviour of Pr. i n con-the near-wall region of boundary layers where the t o t a l shear stress/heat-flux remain constant normal to the w a l l , and for which the turbulence Reynolds number is low. Reynolds [ 1 ] and Launder [ 2 ] noted that both the available measurements and the various model proposals display contradictory variations i n Pr. as the wall is approached. These observations have recently been reinforced by the

measurements of Snijders et al [ 3 ] i n a fully-developed air flow over a f l a t plate. They found that Pr^. = 0.9 + 0.1 i n the f u l l y turbulent part of the inner laiyer (30 < y < 100, corresponding approximately to the socalled ' l o g -law' region), in agreement with the data of Fulachier et al [ 4 ] . However, the results of Snijders et al display an increase i n Pr. to values greater than unity within the 'diffusive sublaiyer' (y < 30), where the molecular and

Reynolds fluxes are comparable. This is contrary to the earlier measure-ments of Blom [ 5 ] i n t h e i r own laboratory, and to those of Fulachier et a l .

The aim of the present contribution is to resolve the uncertainty, out-lined above, over the behaviour of the turbulent Prandtl number within the inner layer. In order to achieve t h i s , distributions for the eddy d i f f u s -i v -i t -i e s are developed from Spald-ing's analyt-ic funct-ion for the -inner layer velocity p r o f i l e [ 6 ] , and i t s thermal analogue derived independently by Hammond [ 7 ] and Snijders et al [ 3 ] . The analysis is restricted to a molecular Prandtl number range of 0.7 < Pr < 70.0, which covers most of the comnon, non-metallic fluids including a i r , water and technical o i l [ 8 ] . The resulting Pr. - d i s t r i b u t i o n is compared with the limited data available for a i r [3-5,9,10], and with previous algebraic models [11-14]. The exotic inner layer variation and molecular Prandtl number dependence of Pr. obtained from p r o f i l e analysis is f i n a l l y j u s t i f i e d on the basis of the severe con-straints imposed by i t s defining equations (1) - ( 3 ) , coupled with i t s l i m i t i n g conditions at high and low turbulence Reynolds numbers.

Profile Analysis

The variation of the mean (time-averaged) velocity p r o f i l e across the constant stress inner layer i s constrained by the following l i m i t i n g

conditions:

u+ = 0 and - ^ = 1 at y"^ = 0 (4) u"*" = 1 Jln y\ B, for y"*" > 30-50 (5)

K

The former restrictions (4) are imposed by the so-called ' n o - s l i p ' condition at the w a l l , while the log-law (5) results from simple near-wall scaling arguments at high turbulence Reynolds numbers. Attempts to develop a single inner layer expression for u"*" in terms of y"*" were unsuccessful i n meeting these constraints, until Spalding [ 6 ] observed that by inverting the problem a corresponding function for y^ = y"*^ (u*) could be readily obtained.

Inspection of the l i m i t i n g conditions (4) and ( 5 ) , together with the need to ensure agreement with experimental data i n the intervening region, led him to propose a continuous analytic function of the form:

n=4

^ n .

(6) + + -A

y = u + e Ï - 1 - ) (icu"^) / n ! tcu r~« + n ,

viscosity becomes, via equation (1) + _ dy*

m _du^ - 1 (7)

Thus, d i f f e r e n t i a t i o n of the velocity p r o f i l e expression (6) and substitution into equation (7) yields a continuous inner Isyer turbulent viscosity d i s t r i b -ution: + m K e n=3 n=l e - 1 - > (KU*) /n ' (8)

which is shown i n Figure 1. The £« ~ y p r o f i l e was determined numerically for the present purposes, using the Newton-Raphson i t e r a t i v e method to solve the coupled equations (6) and (8). In calculating this d i s t r i b u t i o n ,

Brederode and Bradshaw's recommended values for the log-law constants [15] were adopted : K = 0.41 and B = 5.2.

Recently Hammond [ 7 ] and Snijders et al [3] derived independently an inner layer mean temperature p r o f i l e expression analogous to Spalding's velocity function ( 6 ) . Hammond's expression was o r i g i n a l l y developed i n a rather more complex form applicable to the complete temperature p r o f i l e within the plane w a l l - j e t . However, i n a constant heat-flux, near-wall layer his

formula reduces to that of Snijders et a l :

n=4

1 - y (<eT^)"/n

n=l

= T Pr"^ + e

/e"^

where A = KQ B^. This expression satisfies l i m i t i n g conditions analogous 6 0 8

to these imposed on the velocity p r o f i l e :

r +

0 and

dL

dy^

= Pr at y 0

T"^ = — ün y"^ + B.(Pr), for y"^ Pr > 30-50

(10)

. (11) Kader and Yaglom's recommendations for the thermal log-law constants [ 8 ] were adopted i n the present stucty. These imply a value for the turbulent

Prandtl number of 0.85 at high turbulence Reynolds numbers, where Pr. = K/KQ, giving KQ = 0.48. The molecular Prandtl number dependence of the corres-ponding 'additive constant' is given by [ 7 , 8 ] :

2 /

E *

^ m

e :

FIG. 1

FIG. 2

Ed^y viscosity distribution

Earlier attempts to develop a thermal analogue to Spalding's analytic function, f o r example by Smith and Shah at Cranfield [16] and by Gardner and Kestin [ 1 7 ] , involved a r b i t r a r i l y specifying a value for Pr^ of unity across the whole of the inner Icorer. Such formulations also f a i l e d to reflect the correct Pr-dependence of Bg {equation (12) },although they nominally s a t i s f i e d the wall boundary conditions (10).

The 'edc(y conductivity' within the constant heat-flux region, where q* = 1 , can be rewritten in the form:

+ dy* 1 dT^

^l ' ^. - Tr (13)

via equation (2). Its distribution maiy therefore be obtained by different-iating the temperature p r o f i l e expression ( 9 ) , and substituting into equation

(13) to y i e l d :

r _+

-'Y (KeT*)"/n

n=l

The Eg - y p r o f i l e , computed from the numerical solution of equations (9) and (14) i n a similar manner to the eddy viscosity d i s t r i b u t i o n , is plotted i n Figure 2 for values of Pr = 0.7, 7.0 and 70.0. The corresponding Pr.

-profiles were then readily obtained from equation ( 3 ) , and these distributions are plotted as the solid lines i n Figure 3. They display a rather exotic molecular Prandtl number dependence i n the diffusive sublayer. However, this behaviour is simply the consequence of the restrictions placed upon Pr. by i t s defining equations (1) - ( 3 ) , coupled with the l i m i t i n g conditions represented i n equations ( 4 ) , ( 5 ) , (10) and (11). Launder [ 2 ] expressed some scepticism about the presence of a modest peak i n Pr. close to the wall as exhibited by several a i r flow measurements (Pr = 0.7), but this occurrence is seen to be supported by the present analysis. The dramatic rise in Pr. to extremely high wall values when Pr > 1 has no practical significance. An examination of equation (2) indicates t h a t , although the turbulent Prandtl number rises steeply as y ^ 0 in such f l u i d s , e l -^ 0 and so the turbulent heat-flux

q^ ->• 0. The variation i n q. is i l l u s t r a t e d i n Figure 4.

The reviews by Reynolds [ 1 ] and Launder [2] emphasise the sparseness and contradictory nature of the Pr. measurements i n the diffusive sublayer. This is mainly due to the well known d i f f i c u l t i e s i n making measurements of the

25 20

p^

l

k:

'~ rWJrlLt AIM ALT blo 1

WASSEL AND CATTON MODEL: r€W COhHHUhNTS E L

" ^ x . - ^ ° ^

T^'^^^^'^"^^"--»-f

10

10^

FIG. 3

Near-wall turbulent Prandtl number distribution

Five experimental data sets for air are plotted in Figure 5, where they are

compared with the Pr^ - distribution obtained by profile analysis for Pr = 0.7.

The measurements were made in essentially fully-developed turbulent boundary

layers on smooth surfaces, in which the inner layer extends in practice to

about y"*" = 200-300. Outside this limit the influence of the free-stream

conditions is felt, and Pr. might be expected to differ from its log-law value

(=

The present inner layer profile lends support to the data of

Simpson et al [9] (who evaluated Pr^ from the mean properties alone) and

Snijders et al [3] . The trend in the measurements of Antonia et al [10] is

also correct, although the values are a l i t t l e high. Only the early

flux-based measurements of Blom [5] displa^y an opposite trend as the wall is

approached to that obtained by profile analysis.

10 1Ó^ 10 • 10' ^ \i r ^ > ^ Pr ^ ^ 07

~~^

7 0 1

Phenomenological Near-wall Models

A number of modern computational s o l u t i o n procedures f o r t h i n shear l a y e r s , such as t h a t employed by Cebeci [ 1 2 ] , i d e a l l y require an algebraic r e l a t i o n f o r the t u r b u l e n t Prandtl number t h a t i s e x p l i c i t i n y . The present Pr. - d i s t r i b u t i o n obtained from p r o f i l e analysis i s rather complex, and i s only i m p l i c i t i n y . I t i s therefore desirable t o examine a l t e r -native algebraic expressions f o r Pr. t h a t might y i e l d reasonable agreement w i t h the values i m p l i e d by p r o f i l e a n a l y s i s . Several attempts have been made t o develop such expressions based on models f o r the t u r b u l e n t t r a n s p o r t mech-anisms, or phenomena, occurring i n the d i f f u s i v e sublayer. Three popular phenomenological models [11-14] w i l l be examined below, from among those

BLOM (19701

ANTONIA ET AL (1977) FULACHERET AL (1977) SNIJDERS ET AL (1983

FIG. 5

Turbulent Prandtl number : comparison of p r o f i l e analysis and experimental data

e f f e c t s on Prx^ was t h a t due t o Jenkins [ 1 1 ] .

One o f the f i r s t models t h a t took account of molecular Prandtl number He extended P r a n d t l ' s con-ventional mixing length analysis to allow f o r i n t e r n a l , molecular-type

d i f f u s i o n processes w i t h i n 'eddies' during t h e i r t r a n s p o r t , or l i f e t i m e . This enabled him t o derive an expression f o r EQ and, by analogy, f o r e „ . The

o m

20

JENKINS (1951)

CEBECI/NAAND HABIB(1973) WASSEL AND CATTON (19731

10

FIG. 6

Turbulent Prandtl number d i s t r i b u t i o n according to phenomenological models

Pr. 2 1 n=oo TT^ "^ / . n 2 -T- < 1 - exp n=o n=a»

^ • < ^ ^ I ^

Pr. - profiles obtained from this complicated expression are plotted in Figure 6 again f o r values of Pr = 0.7, 7.0 and 70.0. Equation (15) was solved numerically f o r the present purpose, using the e - d i s t r i b u t i o n from p r o f i l e analysis. Reynolds [1] argued that Jenkins had employed an incorrect form of the diffusion equation i n the derivation of equation (15), but that the corrected version does not materially alter the results reported here.

The Jenkins formulation display's a similar trend in Pr. across the diffusive sublayer to that from p r o f i l e analysis, except very close to the w a l l . I t is interesting to note that Jenkins cast doubt i n his paper [ l l ] on the

v a l i d i t y of his relation within the sublayer, although subsequent researchers have employed i t there [ 9 , 1 2 ] .

Cebeci [12] derived expressions for the edcjy d i f f u s i v i t i e s of momentum and heat by using van Driest's damped mixing length d i s t r i b u t i o n in the sub-lawyer, obtained from a Stokes-type flow model. Combining these d i f f u s i v i t i e s led to an expression for the turbulent Prandtl number of the form:

p = K [1 - exp (- y ^ D ^ ) ] (^gj

<eD - exp (- yVOg)]

where D and D. are the velocity and temperature 'damping lengths'

respectively. Cebeci i n i t i a l l y employed this model to compute the properties of several air-flows using values for the empirical coefficients in equation (16) of K = 0.40, Kg = 0.44, D = 26 and D^ = 35 for momentum-thickness Reynolds numbers greater than 5000. Na and Habib [13] subsequently extended Cebeci's model to a wider range of f l u i d s by correlating the Pr - dependence of Dg from turbulent pipe-flow data over the range 0.02 < Pr < 15, to y i e l d :

n=5

where Cj = 34.96, C^ = 28.79, C3 = 33.95, C^ = 6.33 and C^ = -1.186. The Pr. - d i s t r i b u t i o n obtained from the Cebeci/Na-Habib model, using t h e i r

original coefficients, is plotted in Figure 6. I t can be seen that the trend within the diffusive sublayer is the opposite to that implied by p r o f i l e analysis when Pr > 1. This behaviour results from a fundamental weakness i n the van Driest-type formulation, which does not meet the require-ments imposed on the edciy d i f f u s i v i t i e s by the wall boundary conditions (4) and (10).

The f i n a l model to be examined is that due to Wassel and Catton [ 1 4 ] , who applied 'damping terms' d i r e c t l y to the eddy d i f f u s i v i t i e s . They were anxious to develop a model for Pr. that would display a molecular Prandtl number dependence within the diffusive sublayer, yet asymptote to a constant value in the f u l l y turbulent part of the inner lawyer. Their expression:

k [l - exp (- kjel )]

^^t = I T T ? F ; \ (17)

^ S ^^ [ 1 - exp (- k^/el Pr)J

has four adjustable coefficients, k , that enabled them to meet these require-ments. The original values adopted by Wassel and Catton for these coeffic-ients are given in Table 1. The resulting Pr. - d i s t r i b u t i o n , computed using the e - d i s t r i b u t i o n from the present p r o f i l e analysis, is shown in Figure 6. I t displays a similar variation to that implied by both p r o f i l e analysis and Jenkins' model [11] . In order to achieve an improved

quantitative agreement with the former, the values of k were adjusted. These new values are also given in Table 1 , and the corresponding Pr.

-distribution is shown in Figure 3, where i t is directly compared with the p r o f i l e analysis solution. Agreement is seen to be reasonable using these new coefficients except, of course, very close to the w a l l . In view of t h i s , and the algebraic simplicity of Wassel and Catton's formulation, their model appears to be the most suitable f o r use in modem numerical calcul-ation procedures for thin shear layers of the three phenomenological models examined here.

TABLE 1

Coefficients in the Wassel-Catton Pr. Model

k j kg k j k^

Original values 0.210 5.250 0.200 5.000 New values 0.255 1.850 0.300 1.350

The fact that large discrepancies exist between the measured variations in Pr. across the inner layer, led Launder [ 2 ] to suggest that model pro-posals should be assessed by comparison of the resulting mean temperature profiles and wall heat-fluxes. In the present context i t was possible to

compare the temperature profiles implied by various Pr^ - models with those obtained from p r o f i l e analysis. The local temperature gradient given by each model, when inserted into equation (2) together with the e^ d i s t r i b -ution given by p r o f i l e analysis, was integrated numerically away from the w a l l . A discretization scheme based on 50 ' g r i d points' expanded

geo-m e t r i c a l l y frogeo-m the wall to the point where y = 1000 was adopted. The temperature gradient obtained from equation (9) was also numerically

i n t e g r a t e d and used as the base temperature against which to compute the e r r o r , e(T ) , i n the predicted temperature. This base temperature was used

i n preference t o the actual temperature given by p r o f i l e a n a l y s i s , as i t had the e f f e c t of f i l t e r i n g out e r r o r s due t o the common d i s c r e t i z a t i o n scheme. The v a r i a t i o n o f e(T ) w i t h molecular Prandtl number given by the Cebeci/Na -Habib and Wassel-Catton models at y = 825 i s i l l u s t r a t e d i n Figure 7.

• 60 •40 • 20 »— at - 2 0 - 4 0 -60, Pr FIG. 7

Mean temperature e r r o r analysis f o r various t u r b u l e n t Prandtl number models

The e r r o r associated w i t h adopting a constant value o f Pr^ = 0.85 was s i m i l a r l y computed. The Wassel-Catton model w i t h the new c o e f f i c i e n t s y i e l d e d the smallest e r r o r , although i t can be seen t h a t a l l the models

display s u b s t a n t i a l e r r o r s f o r Pr > 20 (15 55 per c e n t ) . These c a l c u l -ations may exaggerate the e r r o r t h a t would r e s u l t from i n c o r p o r a t i n g the Pr. - models i n t o modem computational procedures f o r t h i n shear l a y e r s . Such methods are also influenced by the s p e c i f i c a t i o n of the s t a r t i n g p r o f i l e s , the eddy v i s c o s i t y model, and the wall and free-stream boundary

y* = 825

TEBECI/NA & HABIB Pr^ MODEL (1973) ^Prt =0 85

\«ASSEL & CATTON Pr^ MODEL (1973):

< ORIGINAL COEFFICIENTS -NEW COEFFICIENTS

conditions. In practice, i t is l i k e l y that there i s l i t t l e to choose between adopting the Wassel-Catton model with i t s new coefficients and a constant value of Pr. = 0.85 (they both y i e l d similar errors, although with opposite sign). The former might be preferred on the grounds of physical realism, while the l a t t e r for simplicity.

Concluding Remarks

The turbulent Prandtl number variation within the near-wall region of boundary lawyer flows, where the t o t a l stress/heat-flux remains essentially constant normal to the w a l l , has been obtained using p r o f i l e analysis.

This was achieved by deriving expressions for the corresponding distributions for the edcly d i f f u s i v i t i e s of momentum and heat from Spalding's inner layer velocity p r o f i l e formula [ 6 ] together with i t s thermal analogue. This approach ensures that the l i m i t i n g conditions at high and low turbulence Reynolds numbers are s a t i s f i e d e x p l i c i t l y . The Pr. variation was found to be complex, and molecular Prandtl number dependent within the diffusive

sublayer, where the molecular and Reynolds fluxes are of comparable magnitude. These rather exotic distributions are a consequence of the inherent l i m i t -ations imposed by the framework within which the concept of the turbulent Prandtl number was o r i g i n a l l y developed. I t i s therefore unlikely that any further insight into the transport mechanisms within the near-wall region can be obtained using this conceptual framework. Nevertheless, improved computations are possible via the development of Pr. - models which better reflect the inner lawyer behaviour indicated by the present analysis.

I t must be emphasised that p r o f i l e analysis is not exact as, for instance, the loglaw constants adopted are empirical and subject to i n e v i t -able uncertainty. The derivation of the eddy d i f f u s i v i t y variation between the l i m i t i n g values at high and low turbulence Reynolds numbers also

involved an element of empiricism. However, these l i m i t i n g conditions place severe constraints on the behaviour of Pr. within the diffusive sublayer. Indeed, they are so r e s t r i c t i v e , and the size of the sublayer so small, that i t is ^ery unlikely that the inner layer Pr^ - d i s t r i b u t i o n could d i f f e r s i g n i f i c a n t l y from that obtained by the present analysis.

The Pr. - model of Wassel and Catton [ 1 4 ] , using improved coefficients, was found to display the best agreement, of the models examined here, with the d i s t r i b u t i o n obtained by p r o f i l e analysis. Adopting a value of Pr^ = 0.85 was found to give similar errors in the computed temperature p r o f i l e s , although both y i e l d large errors for Pr > 20. I t would seem t h a t , u n t i l

better Pr. - models are developed, a more satisfactory procedure for handling the steep temperature gradients near the wall with high molecular Prandtl number fluids is the 'wall function' approach used in the Patankar-Spalding method, and incorporated into the GENMIX computer code [18]. These wall functions are based d i r e c t l y on the log-law expressions (5) and (11), and avoid the need to perform computations within the diffusive sublayer as is done, for example, by Cebeci [ 1 2 ] . Provided the wall functions reflect the correct Pr - dependence of the thermal log-law additive constant, they are l i k e l y to give reasonable inner layer temperature p r o f i l e and wall heat-flux predictions in moderate pressure-gradient boundary layers when used in

conjunction with a value of Pr^ ^ 0.85.

Acknowledgements

The work reported here forms part of a research programme which has been p a r t i a l l y supported by the UK Science and Engineering Research Council

under research grants GR/A/4431.2, GR/B/5010.2 and GR/C/2419.0. The author is grateful for t h i s support.

The author is also grateful for the care with which Mrs. D. Hoffman prepared the camera-ready typescript, and Miss D. Roberts and Ms. C. Keen traced the figures.

Nomenclature A = K B

B velocity log-law additive constant

Bg temperature log-law a d d i t i v e constant

C empirical constants i n the Na-Habib Pr^ - model

Cp f l u i d specific heat at constant pressure, J kg"^K"^ D"*" van Driest velocity 'damping length'

Dg van D r i e s t t e m p e r a t u r e 'damping l e n g t h ' e(T ) error in predicted temperature, %

k^ empirical coefficients i n the Wassel-Catton Pr^ - model n integer number

Pr molecular Prandtl number (= v/a) Pr^ turbulent Prandtl number (= E^/EQ)

q total (molecular plus turbulent) heat-flux, W m~^ qx^ turbulent heat-flux, W m"

=

q/q„

+ q

T mean (time-averaged) temperature, K T^ wall temperature, K

T^ ' f r i c t i o n temperature' {= (Up c u ) , K T* -= (T„ - T) /T^

u mean (time-averaged) velocity, m s"^ u^ ' f r i c t i o n velocity' { = ( T ^ / P ) },m s"^

T W

u+ = u/u

y distance from and normal to the wall, m

y* = u^/v

Greek symbols

a f l u i d thermal d i f f u s i v i t y , m^ s'^

e 'edcjy' or turbulent viscosity, m^ s"^

'm

et, =• e„/v m m

Eg 'eddy' or turbulent conductivity, m^ s"^

K von Karman's constant

Kg thermal analogue of von Karman's constant

V f l u i d kinematic viscosity, m^ s"^ p f l u i d density, kg m~^

T total (molecular plus turbulent) shear stress, N m"^ T^ wall shear stress, N m'^

W

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