The form of the extended Reynolds analogy for rough surfaces

Pergamon Press Printed in Great Britain

IN HEAT AND MASS TRANSFER

Vol. 5, pp. 99 - 109, 1978

THE FORM OF THE EXTENDED REYNOLDS ANALOGY FOR ROUGH SURFACES

D. C. LESLIE Department

of Nuclear Engineering, Mile End Road,

Oueen Mary College

London E1 4NS

(C~iit~nicated by J.H. Whitelaw) Introduction It seems that the precise nature of the Reynolds analogy is still a matter of debate when the surface is fully rough and various literature

irreconcilable

forms are to be found in the

(eg Dipprey and Saberskey

1963, Whitehead

1976).

This

1963, Owen and Thompson

letter presents what is believed

to be the only result which an eddy diffusivity treatment can give:

this result is found to differ from all the ~orms cited

above. The analysis

is limited to fully developed

(turbulent)

flow in a round pipe.

It is formally valid for any value of + the reduced roughness height e = e u / v , e being the actual roughness

height and u

~w is the wall stress:

the friction velocity

(Tw/p)i/2, where

the results for a fully rough surface

(e+ > 50 say) are of particular interest, because of the current doubt about the form which the analogy should take in this case.

The work is limited to flows with a clearly

99

i00

D.C. Leslie

defined than

lozarithmic

[say] 0.025,

where

2. Preliminary: Lyon

region,

Pr t is constant

requirin Z that e/a should be less

a is the pipe radius.

intezral

[1951)

Vol. 5, No. 2

deduction

showed

surface

that if the eddy Prandtl

everywhere, 1

for a smooth

number

then s3w2(s)ds

(2.1)

1 - aUm fo <+{e(s)/Pr t} St Here

S = r/a, r being

the radial

St

is the Stanton

Um

is the volume

~(s)is K

the eddy

coordinate

number averaze

of the axial

(momentum)

diffusivity

is the m o l e c u l a r

thermal

velocity u(r)

diffusivity

and

w(s) = foS 2S' U(S') u

ds'

(2.2]

m From now on we shall simplify a good value

for internal

by putting Pr t = l, since this is

flows.

The standard

core similarity

u(r)

= u(O)

- u~ hlrl~a)

(2.3)

w(s)

= 1 +

lj:j(s)

(2.4)

j(s)

= 2__ f s ' { h - h(s')}ds' s2 o

implies

equation

S

where

Here f is the Fanning denotes

the spatial

1 2 factor ~w/~PU m and the overbar

friction

averaging

(2.5)

operation

fl° 2sds.

Eqn

(2.1)

can

now be rewritten 1 - ii + i2 + i3 St

(2.6)

EEYNOI/~ ANALOGY FOR ~]](~ SURF~CF~

Vol. 5, No. 2

1 s3ds II = aUm fo v+c(s)

where

i01

2 - f

(2 7)

I I2 = aUm fo sSds x x

E{<

+ E(s)} -I - { v + ~ ( s ) } - i

I s3{w2(s)_l}

z3

=

fo

~+~(s)

12 and 13 are dominated regions

respectively.

(2.8)

(2.9]

ds

by contributions It is quite

Reynolds number Re i s

~

from the wall

and core

easy to show that, when

the

large

12 +

G(Pr)

where G(Pr)

= fo dy+

{

+ e(Y+)}v

-

{i + ~ (y+) }

(2.10)

and (2.11) where p = 2{h 2 _ (~)2}

(2.12]

and 1

Q = fo s 2 j 2 ( s ) h ' ( s ) d s (y=a-r)

In [2.10) y+ = yuT/v then follows

is the usual wall coordinate.

It

that 1 St

This

(2.13)

2 + f

is the canonical

{or a smooth wall. its contribution

However Jayatilleke

+ P +

-Form o{ the extended

(2.14) Reynolds'

The final term is suppressed

is less than

Equation

{low assumption

G(Pr)

(2.14]

since

1%.

is due in essence

{ollowed

analoKy

Martinelli

(that is, he assumed

to J a y a t i l l e k e

(1947]

(1969].

in using the Couette

that the heat {lux varies

102

D.C. Leslie

linearly

across

factor w2(s)

the pipe).

in equation

St so that

there

[2.1)

f +

is now no term

that

the

of the pipe,

the

to smooth large

angles,

methods

are

integral will

walls.

Near

no longer

formula

the t e m p e r a t u r e there easily

extended

to rough

The wall

profile (see

section

the flow

fails

extended

to

§4].

and

to eqn

[2.1)

eddy

through

diffusivity there

is no

for a rough

wall.

[2.14)

is a statement

in the turbulent logarithmic

is limited

turns

Therefore,

the result

is a w e l l - d e v e l o p e d

He e v a l u a t e d

(f/2) I/2.

P = 15.6

wall

analogy

profiles

to

[2.15)

velocity

applicable.

that

[2.15]

to rough walls

a rough

analogous

now be shown

and to modify

in the previous

Reynolds'

the

+ ~P

of order

and found

deduction

is to replace

by w(s),

logarithmic

3 Extension The

effect

G (Pr)

P assuming

centre

The

Vol. 5, No. 2

core

region,

and this

It

about

that,

provided

result

is

walls.

form of the

logarithmic

velocity

profile

may

be written u

+ = A' (e +)

[Schlichtin g 1968). rough wall

and

fully

This

+ Bin

~ e

£ormulatien

rough walls:

(3.1)

covers

in p a r t i c u l a r

smooth,

partially

the standard

smooth

formula + u

is o b t a i n e d

+ Bln

y

(3.2]

+ Blne +

(3.3]

by puttin Z A'

Similarly

+ = A

= A

the wall

form of the

T + = A ~,( e +)

+Bln

logarithmic y+

temperature

profile

is

(3.4)

Vol. 5, No. 2

REYNOI/3S ANALOGY FOR ~

where

SURFACES

103

T+ = pCpu qw T[Tw - T(y) }

qw and T w being the wall

(3.5)

temperature

and the wall heat flux.

For a smooth wall A ' ( e +) = A + G(Pr) G(Pr)

being defined by eqn

(3.6)

(2.10).

The core similarity (see eqn

+ Blne +

form of the velocity

profile

(2.3)) may be rewritten u = u

where U

m

(3.7)

+ u {h - h(s)}

(or u) is the

(volume-averaged)

Near

mean velocity.

m

the wall,

this has the asymptotic u = Um

The standard

+ u

{h

logarithmic

form

- Bln

a

Y

friction

u

=

_ j

(3.6)

+ O(a~ ) }

factor formula

{A' (e +)

- h

+ J}

+Bln

(3.9)

ae

T

is obtained (3.8).

by matching

the logarithmic Taking

profile

terms

account

(3.8)

average,

The cupmixed mean temperature the quantities

Q

{{)+

are given

(3.10)

is believed

in the Appendix.

as

m

of

is a~ + J - h +

(3.10)

O{[a~]in[y]}

is defined by eqn

P and Q are specified

details

of the heat flux

that the equivalent

profile

qw EBIn pCpU T

+

{3.1) and

the mean temperature T

it may be shown

= Tm +

in eqns

agree.

of the curvature

for the temperature T(y)

terms

already

and of the need to define

a cup-mixed eqn

the constant

by eqns

[A.6) while

[2.12)

and

(2.13):

In this form the result

to be new, but the work

is essentially

due

104

D.C. Leslie

to Squire [3.10]

(1953].

and

then

Matching

using

(3.9]

Vol. 5, No. 2

the constant

terms

in

[3.4]

and

to eliminate

the term Bin(a/e),

we

derive 1 _ pCp(Tw-T m) St Um = ~ + It follows [2.14]

from

[3.3]

is a special

As before,

we have

being

small.

rough

wall.

{ A ~ ( e +) and

case

[3.6]

that

out

using

the form

recommended

[private

of h(s)

of order

communication]

[1976),

au Y

(f/2) I/2 as for a really

has evaluated

by the eddy

viscosity

P and Q distributior

namely

for s < I - ~i

(s) = --6--~i

[3.11].

of P and Q

generated

by Townsend

equation

relation

not be negligible

4. The evaluation Hassid

general

the term

it will

[3.11]

the s m o o t h - w a l l

of the more

struck

However,

- A' (e +) } + P +

} [4.1]

au'r(1-s) B

where

s = r / a as before,

distribution

falls

for viscous

effects

(4.1] and

is s u f f i c i e n t

for

s > 1 -

~1

while ~l is an adjustable

to zero at the wall

s = 1.

in the n e i g h b o u r h o o d to generate

constant. It must

of the wall,

the a s y m p t o t i c

This

be m o d i f i e d but the form

equations

[3.8)

(3.10). The

parameters

B and

pressure

drops

pipes).

For s m o o t h - w a l l e d

formula

[3.9]

[for fully

reduces

to

~l can be determined

developed pipes,

flows

the

from m e a s u r e d

in s m o o t h - w a l l e d

logarithmic

friction

round factor

Vol. 5, No. 2

REYNOI/)S ANAIfX~Y FOR ~

=

and

the

values

use

this

set.

recommends friction from

C

+Bln

B = 2.5,

C =

Schlichting

B = 2.46,

factors

those

C - value

of 1.75

if c(s)

has

1.75

(1968),

by this

A

-

h

data

quoting

over set

+

J

(4.2)

very

we

With

shall

(1935),

range,

differ

values.

well:

Prandtl

the w o r k i n g

do not

105

the

significantly A = 5.5,

a

gives J

the

h-

=

fit the

by our p r e f e r r e d

-

Now

; C

C = 2.00:

given

given

a

SURFACES

=

3.7s

form

J = B

(4.1)

then

1 3 1 + in [i + 1 + 2 ~ i 4~ 1

(

1 2 1 3 - 2 ( i + i-2~i )

and

~1 = 0 . 1 6 makes h - J = 3 . 7 5 , With

this

(4.3)

in

agreement with

value

P = 18.2, He has

also

(1967)

fit

the

P appears

to

than

our

of

the

determined

of heat

this

as

and

geometry,

Comte-Ballot

channel

using

(1965).

Barnett's

He f i n d s

that

geometry.

of Petukhov

P = 27.

experiments

to l i q u i d

to the

found

inferior

with

but

transfer

to

correlation

(3.11),

~l by f i t t i n g

f = O.O46Re-i/s regard

the

re c o m m e n d e d v a l u e

*In a s t u d y

channel

Q = 32.8

be i n s e n s i t i v e

form

finds (4.4)

measurements of

Encouragingly, precisely

Hassid

P and Q i n

P = 17.4, so t h a t

(1'

(1976)~

Q = 15.5

calculated to

of

Townsend

This

is

(1970)

somewhat h i g h e r

on s m o o t h w a l l s ,

metals

(Leslie

1977),

standard

friction

factor

it s h o u l d

be equal

to O.21.

to a P r a n d t l - t y p e

fit,

is

leading

for

I

formula I now

to ~l = 0.16.

106

D.C. Leslie

which

(2/f) I/2 is rather large,

very accurately.

Finally,

Vol. 5, No. 2

do not determine this quantity

continuing the logarithmic profile

right to the centre of the pipe

(Martinelli assumption)

gives

P = 5B2/2 = 15.6.

5. Conclusion and recommendation Equation

(3.11) seems to be the only possible outcome

of an eddy viscosity analysis

(that is, of the assumption that

the eddy Prandtl number is constant)

and it should be used in

preference to the formula of Oipprey and Sabersky to that of Owen and Thompson Q term

(1963).

(which has been dropped)

The eddy diffusivity treatment

With an allowance for the

a P-value of 20 is appropriate. is crude,

would no doubt give different results. almost inconceivable

and better methods However,

[3.11).

In general the functions A~(e +) and

wall

the

from function

it does seem

that they would support a form of Reynolds

analogy different from eqn

determined

(1963) and

experiment. G(Pr)

In

the

special

can be c o m p u t e d

from

A,(e +)

case eqn

of

must

be

a smooth

[2.10),

References Barnett, P. G. (1967) "Eigenvalues of the Orr-Sommerfeld equation for laminar flow and some turbulent mean profiles." AEEW-R523. Comte-Bellot, GeneviEve (1965). "Ecoulement turbulent entre deux patois parall~les." Publications Seientifiques Techniques du Ministere de l'Air, no.419. Oipprey,

et

O. F. and S a b e r s k y , R. H. ( 1 9 6 3 ) " H e a t and momentum transfer i n s m o o t h and r o u g h t u b e s a t v a r i o u s P r a n d t l numbers". Int.J.Heat and Mass T r a n s f e r 6,329.

Jayatilleke C. L. V. (1969) Article in "Progress in heat and mass transfer, volume I" edited by U.Grigull and W.Hahne. Pergamon P r e s s , Oxford.

Vol. 5, No. 2

Leslie,

REYNOI/3S ANALOGY FOR ~

SURF~

107

D. C. (1977) "A recalculation of turbulent heat transfer to liquid metals". Letters in Heat and Mass Transfer, 4, 25.

Lyon, R. N. (1951) "Liquid metal heat transfer Chem. Eng.Progress 47, 75.

coefficients".

Martinelli, R. C. (1947) "Heat transfer to molten metals". Trans. Amer. Soc.Mech. Engrs. 69, 947. Owen, P.

R. and T h o m s o n , W. R. ( 1 9 6 3 ) " H e a t t r a n s f e r rough surfaces". J.Fluid Mech. 1 5 , 3 2 1 .

Petukhov,

Prandtl,

B.S. (1970) "Heat transfer and friction in turbulent pipe flow with variable physical properties". Adv. Heat Transfer 6,504. L. [1935) Durand's

"The mechanics of viscous fluids" in W. F. Aerodynamic Theory, Vol. III, p.142.

Schliehting, H. (1966) "Boundary-layer McGraw-Hill, New York. Squire,

across

theory",

sixth edition.

H. B. (1953). Article "Heat Transfer" in "Modern Developments in Fluid Dynamics, High Speed Flow" ed: L. Howarth. Clarendon Press, Oxford.

Townsend,

A. A. [1976) "The structure of turbulent shear flow". second edition. Cambridge University Press, Cambridge.

Whitehead,

A. W. (1976) "The effects of surface roughening on fluid flow and heat transfer". Ph.D. thesis, University of London.

Appendix When the flow and heat transfer are fully developed [and the wall heat flux is constant) enthalpy

balance

equation

dT _ qw

dr where

£(s)

sw(s)

(A.I)

~(s)

The molecular

and w(s) thermal

since it will be neKliKible

the molecular Prandtl

of the

in the core region is

is the eddy viscosity

[2.4) and [2.5). suppressed

pCp

the first integral

is defined by eqns

diffusivity compared

number is very small,

covered by this analysis).

Now

has been

to ~(s)

a case which

(unless is not

108

D.C. Leslie

e(S)

where h(s) so that

au

h , s(s)

T

[A.2]

is the core similarity

[A.1]

[2.4],

function

defined

by eqn

[2.3],

may be rewritten dT ds

From

=

Vol. 5, No. 2

qw pCpU

-

the integral T(s)

[A.3]

h' (s)w(s)

= T(O)

of this equation

+

[

qw pCpU T

h(s)

is

+

(A .4]

m(s)

S

where

m(s)

= ~o h' (s") j (s")ds"

The next step

is to form the cup-mixed

1 u(S) Tm = ~o 2S - m =

T

[A.4]

pCpU

and

mean

temperature

T(s)ds

[A .6]

E~(.~

fo 2sds

E~ +

x from eqns

qw

+ O

[A 5]

~

[3.7].

+

h(.~]

[A.7]

This may be simplified

Tm = To + _ _ pCpU T

(s~] x

to

(s)_{h'~ _ (~)2 _ ~

[A.8) where,

as before, 1 operation fo2Sds.

the overbar Combining

qw T (S) = T m + -p C-p U + {m(s)

-~

denotes

the spatial

[A. 4] and ~

h(s)

averaging

[A.e] we have

- h} +

+ h 2 - (~[)2} [A.9]

As s - ~ l h(s)

+ Bln[~s)

+ J + O(l-s)

[A.IO]

Vol. 5, No. 2

(of eqns

REYNOLDS ~

(3.7)

m(s) Eqn

(3.10J

and

(3.8))

+ m(1)

from this

+ O{(l-s)In

of the main P = m(1)

and

FOR ~

text

then

- m + h2 -

SURFACBS

it follows

109

that

[ll--~sl} follows

(A.II) from

(A.9],

with

(h) 2

[A.12)

Q = hm - hm These and

can be rsduoed

(2.13)

by parts.

(A.13) to the s i m p l e r

by ohanginK The w o r k

the ordeP

is tedious

foFms

given

of integPation

but

in sqns and

not difficult.

(2.12)

integrating