On turbulent flow in circular pipe

ON TURBULENT FLOW IN CIRCULAR PIPE BY S. I. PAIt ABSTRACT The Reynolds equations of motion of turbulent flow of an incompressible fluid have been studied for turbulent flow in circular pipe. The number of these equations is finally reduced to two. One of these consists of mean velocity and correlation between radial and axial turbulent velocity fluctuations u'w' only. The other consists of the mean pressure, the radial and the circumferential turbulent velocity intensity only. Some conclusions about the mean pressure distribution and turbulent fluctuations are drawn. The mean velocity distribution and the correlation u'w r can be expressed in a form of polynomial of the radial distance with the ratio of shearing stress on the wall to that of the corresponding laminar flow of the same maximum velocity as a parameter. These expressions hold true all the way across the tube, that is, both the turbulent region and viscous layer including the laminar sublayer. The mean velocity distribution has been compared with Nikuradse's experimental data to turbulent flow in a circular pipe. It also shows that the logarithmic mean velocity distribution is not a rigorous solution of Reynolds equations. L INTRODUCTION

Actual flows at large Reynolds number are characterized by a peculiar phenomenon known as "turbulence." In such flows, the apparently steady motion of fluids is only steady insofar as the temporal mean values of the velocities and the pressure are concerned ; in reality velocities and pressures are subjected in irregular fluctuations. The essential characteristic of turbulent motion is that the turbulent fluctuations are random in nature. Hence the final and logical solution of turbulence problem requires the application of methods of statistical mechanics. It was Osborne Reynolds (1)~ who first introduced elementary statistical motions into the consideration of the turbulence problem. The motion of a fluid as usually considered in hydrodynamics is obtained by considering the means of the molecular motions, and according to the kinetic theory of gases, for example, the velocity components, pressure, density, temperature, etc. of a gas in motion may be defined and approximately satisfy the Navier-Stokes equations. In turbulent motion, however, even if the fluid is regarded as a continuum, we still have to deal with turbulent velocity fluctuations superimposed on a mean motion. We may write u , -- a , + u,',

(1)

x Associate Research Professor, Institute for Fluid Dynamics and Applied Mathematics, University of Maryland, College Park, Md. 2 The boldface numbers in parentheses refer to the references appended to this paper. 337

338

S . I . Pat

I.I i' 1.

where u~ is the ith c o m p o n e n t of the total fluid velocity, zz~ is the ith mean velocity component, and u{ is the ith c o m p o n e n t of the turbulent fluctuating velocity. If we substitute u~ from Eq. 1 into the NavierStokes equations of motion for an incompressible fluid and use the rules t h a t we say derive for taking means, we obtain a set of equations known as Reynolds equations which differ from Navier-Stokes equations only in the presence of additional terms called the Reynolds stresses or eddy stresses. T h e y form a tensor with components - p u i ' u / , the eddy normal stress components being - o u i '2 and the eddy shearing stress components being - p u { u / (i # j), where p is the density of the fluid. These stresses are the representation of the rate of transfer of mom e n t u m across the corresponding surfaces due to the turbulent velocity fluctuations. The solutions of Reynolds equations will represent properly the t u r b u l e n t flow. However there are only four Reynolds equations for ten u n k n o w n s • the mean pressure, three mean velocity components and six Reynolds stresses. In general, the Reynolds equations are not sufficient to determine these unknowns. T h e Reynolds equations have been extensively studied for the simplest type of turbulence, isotropic turbulence, by Taylor (2), von Kfirmhn (3), Heisenberg (4), Kolmogoroff (5), and others. Excellent reviews for the present status of the statistical theory of isotropic turbulence are given in references 6, 7, and 8. Even in such a simple case, it is not possible to solve this problem completely because the n u m b e r of equations is less than the n u m b e r of unknowns. Our knowledge about the statistical theory of nonhomogeneous turbulence is even more meager (3, 9, 10). There is indeed much to be done before practical engineers can make real use of the statistical theory of turbulence. As far as the mean velocity distribution is concerned, some sentiempirical theories of fully developed turbulent flow have been successfully developed. T h e best known theory among t h e m is Prandtl's mixing length theory (11, 12, 13). The famous von K~rmhn's logarithmic velocity distribution is derived from the mixing length theory (14). Some critical discussions about the logarithmic velocity profiles have been made (15, 16, 17). Even though Prandtl's mixing length theory had successfully predicted the mean velocity distribution in turbulent flow, it is now generally believed to have serious limitations and inconsistencies (18). The mixing length deduced from measurements of the mean velocity is of the same order of magnitude of the dimension of the mean flow, but in theory, it should be very small. The assumptions on the distribution of mixing length are in general not true as revealed from hot-wire a n e m o m e t e r measurements. The boundary conditions at the wall are not satisfied, thus the logarithmic velocity distribution is not a rigorous solution of Reynolds equations.

Oct., I953.]

TURBULENTFLOW IN

CIRCULAR P I P E

339

Usually one can only say t h a t the results of the mixing length theory do not hold true for viscous layers and laminar sublayers which are near the wall. Furthermore it fails to predict anything about turbulent fluctuations. In order to understand more thoroughly the mechanism of turbulent flow, it is better to study directly the Reynolds equations without introducing the concept of mixing length. Professor Kampe de Feriet (19) recently studied the Reynolds equations for turbulent flow of an incompressible fluid between two fixed parallel walls, that is, the so-called plane Poiseuille flow in reference (19). The analysis has been extended by the present author (20) such that rigorous solutions for mean velocity distributions between parallel plates, both plane Poiseuille flow and plane Couette flow, have been found which hold true for the whole region between the plates. The plane Poiseuille case has been checked experimentally, with fairly good results, for turbulent flow in a channel. This paper studies the turbulent flow in a circular pipe by an analysis similar to that given by the author in reference 20. A rigorous solution of Reynolds equation for this case is also obtained. The theoretical results are checked with the experimental data. It also shows that the logarithmic .velocity distribution is not a rigorous solution of the Reynolds equations in the present case. II. S T A T E M E N T

OF T H E P R O B L E M

Consider the turbulent flow of an incompressible fluid in a circular pipe of radius a. The cylindrical coordinates r, 0, z are shown in Fig. 1,

Ii,

,~

,z~,~l

i i , l l l l l l , l ~

----

~'trJ

o

iil.~lJ~l/llllllll~lllllll~l

i

FIO. 1.

ii

l,

i

l l l ] ] I f l l ] l l ] l l l l l l l l l l l l l l l l

,Ill,

i i i • Iii

Flow in a circular pipe.

where z = 0 is arbitrarily chosen, and z is extended to both + oo and - ~ . We assume that the mean flow is steady and axially symmetrical, that is, o j = o, Ot

o j = o, O0

..

f = ](r,

(2)

Where ] may represent any quantity of the mean flow such as mean velocity components or mean pressure etc.

34 °

S.I.

PAI

[J. F. [.

T h e i n s t a n t a n e o u s velocity c o m p o n e n t s of the flow are u, v, and w in the direction of r, 0 and z, respectively. We assume t h a t the mean flow has velocity c o m p o n e n t s along the z-axis only, t h a t is, w = ~v(r, z),

a = 0 = O.

(3)

T h e t u r b u l e n t velocity c o m p o n e n t s along r, O, and z directions, respectively, are u'

=

u,

v' =

v,

w'

=

w

-

~v.

(4)

In the Reynolds equations we have the following m e a n values {}f t u r b u l e n t fluctuations u'2(r),

w'2(r),

v'~(r),

u'v'(r),

v'w'(r),

(S)

w'u'(r).

Here we assume t h a t these m e a n values of t u r b u l e n t fluctuations are functions of r only. This is due to the fact t h a t we consider a problem which has the same statistical properties at all planes z = constant. m . FUNDAMENTAL S O U A T m N S

We assume t h a t the instantaneous velocity c o m p o n e n t s of the flow satisfy the Navier-Stokes equations which in cylindrical coordinates (11) are as follows: Ou

10ru 2

10vu

Owu

v2

o-7 + ~ 0-7- + 7 o-T + o~lo,

(

pot Ov

10ruv

10v 2

Ovw

+ ~

,,20,,) v~u

r~

r ~ 00

'

(6)

v~v -

r~ + r~ ~o

'

(7)

uv

o-t +-~ o~- +-~-g + ~-~ + r p rOO + ~

Ow

10rwu

10vw

Ow"-

o-? + -~ o-T + ~ TO- +

1 op . . . . . + ~V°-w, -Oz p Oz

(8)

where 72 = 02 Or s +

1 0

1 02

02

r Or -4- r2 002 + Oz 2

The equation of c o n t i n u i t y is 10ru

10v

Ow

7 o~ + r o0 + U~ = 0.

(9)

Substituting p =p+p',

u =a+u',

v = O+v',

w =w+w',

(10)

Oct., z953.]

TURBULENT

FLOW

341

IN CIRCULAR P I P E

into Eqs. 6 to 9 and averaging, we have O~ lOr~ ~ 10~ 0-7 + r 0-7 + r O--ff 4-

Ow~ Oz

~2 = _ _~IOP r

1 0 r u '2 r Or

O-tt 4- r

O~

O---r-+ r - ~ + Ozz 4- r

10r~

o--[ + r

1 0~o

(

p Or + v

V~a

10v'u' r 00

p r00 4- v

~

2 0~)

r2

r~

Ou'w' Oz

4- v'-~2, (11) r

V2~ 4- r2 00

r2

10rUlV p

1 0 V ;2

OWlV I

UlV I

r

r 00

Oz

r

Or

O¢ 2

--,

(12)

10~

o--;- + r o--o + oz

, oz + ~ v ~ 10rwru r

10WlV l

r

r

Or

1 ara 1 0~ aw r 0-~ 4- r ~ 4- ~-z = O.

00

OW r2

Oz '

(13)

(14)

Equations 11 to 14 are known as Reynolds equations in cylindrical coordinates, which are the fundamental equations for turbulent flow of incompressible fluids. There are ten unknowns/3(1), ~, 0, ~v(3), u ' v i, etc. (6). The four Reynolds equations are not sufficient in general to find the ten unknowns. However in our present problem, because of the simplicity of the configuration of the problem, it is possible to get some information from these fundamental equations. IV. TURBULENT FLOW IN A CIRCULAR PIPE

If we substitute Eq. 3 into Eq. 14, we have 0~ --

Oz

=

0,

or

w -- zv(r).

(15)

We see that the mean velocity zv is a function of r only. Substituting the values of Eqs. 2, 3, and 5 into Eq. 12, we have 1 dY/~ Pvt

---4---=0. r

dr

U/V t

r

(~6)

342

S.I.

PAI

[J. F. I.

I n t e g r a t i n g Eq. 16 we have constant u'v' = - y2

(17)

But on the wall of the t u b e r = a, u'v' = O, hence the c o n s t a n t in Eq. 17 must be zero. T h u s tttV p = O,

for the whole velocity field. In this t y p e of flow, the correlation of two transverse t u r b u l e n t fluctuating velocity c o m p o n e n t s is zero for the entire flow .field. E q u a t i o n s 11 and 13 become

10p p Or

---

10/5 -O0--zz - v

1 0 r u '2 r Or

+

#2 r

(02w

10w) -~-(r2 q - r - ~ r

-

o,

lOrw'U'=o" + r Or

(118)

(19)

Now we have five u n k n o w n s w(r), p(r, z), w'u'(r), u'2(r) and v'2(r), but only two Eqs. 18 and 19. We c a n n o t solve this problem completely, but it is possible to find some relations a m o n g these five u n k n o w n quantities. V. N O N - D I M E N S I O N A L

FORMS

OF FUNDAMENTAL EQUATIONS

We define the non-dimensional coordinates as follows: = -,

~ .....

(20)

Let r0 be the shearing stress on the wall of the tube, t h a t is, at = 1. r0 is a c o n s t a n t or r a t h e r a p a r a m e t e r in the present problem. We define a reference velocity

v * = x/7'

(21)

and the corresponding Reynolds n u m b e r R* R*=

U*a

(122)

P

T h e non-dimensional m e a n velocity is defined as V = U* = V(n, R*).

(23)

Oct., 1953.]

TURBULENT FLOW IN CIRCULAR PIPE

343

T h e average of the mean velocity o.ver a n y section ( --- c o n s t a n t is Uo, t h a t is,

2f0°

Uo = -~

~rdr;

U* -

2

f0

VTdn.

(24)

T h e corresponding R e y n o l d s n u m b e r is

Uoa Ro = - -

(25)

Y

T h e coefficient of friction of the pipe is defined as G

~

TO

pUd/2"

(26)

Hence

•JC]

U* R* - 2 - = Uoo = Ro"

(27)

T h e non-dimensional pressure is defined as _

~ -

p o =

p U *~

~(~,

(28)

,7, R * ) ,

where t50 is a reference pressure which m a y be t a k e n as the mean pressure at the point ~ = 0, 7 --- 1. T h e non-dimensional forms of t u r b u l e n t fluctuations are Ut,W t

y(7, R*) -

U. ~ ,

ai(7, R*) -

U . ~,

(29)

ut2

(30)

•7.)t2

(31)

a2(•, R*) = U , 2. In non-dimensional form, Eqs. 18 and 19 become

O~ 1 dTal nu an n a7

l(d2v a~

ldv

R*~.d,~ + ~ - ~ , !

0"2 = 0, n

(32)

ld,y +rid,

-0.

(33)

344

S.I. P,i

[J. F. I.

T h e b o u n d a r y conditions are V(1, R*) = 0, dV

d7

-

0,

at

(34)

7/ =

0,

(35)

y(1, R*) = 0,

(36)

¢1(1, R*) = 0,

(37)

~=(1, R*) = 0,

(38)

~(0, 1, R*) = 0.

(39)

and Equation 39 is obtained by the proper choice of :0. Integration of Eqs. 32 and 33 gives ~5(~/, 7, R*) q- ,r1(,1, R*) 4-

f

n 0" 1 - -

0"2

d7 = A~/4- C,

(40)

.-,1

1

dV

R* ~ ~

A~ ~

-- ny = T

4- B,

(41)

where A, B, and C are constants. From Eqs. 39 and 40 we have C = 0. Hence the mean pressure & becomes ~(~, 7, R*) = A~ 4- P(7, R*),

(42)

where P(7, R*) is a function of 7 and R* only, but independent of ~. The shearing stress on the wall of the pipe in non-dimensional form is R* =

dV

d7

,

at

7 = 1.

At 7 = 0, both y and d K/d7 are finite, hence 23 = 0. then, (43) in (41), we have A=-2,

B=0.

(43) Substituting, (44)

Equations 40 and 41 become 6+¢,+

~1

W 0"1

~

0- 2

&

=

-

1 IdV R* d~ 4- y = 7.

(45) (46)

Oct., I953. ]

TURBULENT

F L O W I N CIRCULAR P I P E

345

It is interesting to note t h a t the five u n k n o w n quantities are divided into two groups V and y in one group and ~5, el and ~ in the other group. T h e correlation u ' w ' influences only the mean velocity profile, while the transverse t u r b u l e n t velocity intensities u '2 and v'2 affect only the mean pressure distribution. VI. MEAN PRESSURE DISTRIBUTION

From Eq. 45, two practically interesting conclusions m a y be drawn for the mean pressure: (a) Since ~1(~) and a2(~) are independent of ~, the mean pressure decreases linearly with increase of ( in the same m a n n e r as in laminar flow. (b) From experimental d a t a of Fage, 3 we have t h a t near t h e wall ~ 1 , ~2 > ~i and near the center ~ = 0, ~1 = ~ . T h e n the m e a n pressure in the fluid given by Eq. 45 will be greater than or equali'to !the mean pressure on the wall if we substitute these experimental ~, and ~ in Eq. 45. This point differs entirely from the turbulent flow in a two-dimensional channel where the mean pressure in the fluid is less t h a n or equal to t h a t on the wall (20). On the center of the pipe, from Eq. 45, we expect t h a t gl = ~2, otherwise the mean pressure would be infinite. In Fage's experimental data, one sees t h a t gl is always equal to g2 for various Reynolds numbers, at the center of the pipe. VII. MEAN VELOCITY DISTRIBUTION

Because the mean flow is axially symmetrical, we have the boundary condition (35). Substituting Eq. 35 into Eq. 46, we have y(0, R*) = 0.

(47)

T h e correlation u ' w ' is zero on the axis of the pipe. Integration of Eq. 46 gives V(n, R*) = ~ - (1 R*

- - ~2) _ R *

f~

ydn.

(48)

If y(~, R*) - 0 , Eq. 48 reduces to the velocity profile of laminar Poiseuille flow. The m a x i m u m mean velocity which occurs on the axis of the pipe is given by the following formula U~ -U*

=

V(0, R*) = R* 2

R* f01 y d s .

(49)

It is advisable here to find an expression of velocity distribution which satisfies the differential equation (46), the boundary conditions a Figure 131 (p. 394) of Reference 11.

346

S.I.

PAI

[J. F. I.

(34) and (36) and also the experimental results. T h u s the velocity distribution will be an exact solution of the Reynolds equations. It should be noticed t h a t because in this problem the n u m b e r of u n k n o w n s is more t h a n the n u m b e r of the differential equations the solution will not be unique. But if a suitable velocity distribution m a y be found, it will give us some information about the general characteristics of the velocity profile in the central portion of the pipe as well as t h a t near the wall. In t h e l a t t e r case, it will give us some more rigorous results to c o m p a r e the o r d i n a r y semi-empirical concept of laminar sublayer. It was found t h a t the solution of V(7, R*) m a y be written in the following form

V=FiU.. [1 + ao7 ~ . + ,z~7~,,],

(so)

where n is an integer larger t h a n or equal to 2. In order to satisfy the b o u n d a r y conditions (34) and (36) and Eq. 46, we have S--/~

a~ = - - - n-l'

a~,, -

1 --S

n-1

,

(51)

where s

.

R* rru~ . . 2U~

TO

. 2U,, /z--

TO

r,

,

(52)

t h a t is, s is the ratio of the shearing stress r0 on the wall of the pipe for the t u r b u l e n t flow to the corresponding shearing stress r, on the wall for the laminar flow with the same m a x i n m m velocity U,,. Hence the m e a n velocity profile is

v(7, R*) = F ~

s--n72+ l --s

1 + -n --

1

]

n--Z-11 7~" '

(53)

and the correlation y is n(s-

1)

y(7, R*) = s(n - 1) 711 - 7 ~ - - " ] .

(54)

If s = 1, Eq. 53 reduces a u t o m a t i c a l l y to the case of laminar flow and Eq. 54 is identical to zero. In t u r b u l e n t flow s is larger t h a n 1. T h e value of n has to be d e t e r m i n e d from experimental data. F r o m Nikuradse's m e a s u r e m e n t s (21) of t u r b u l e n t flow in circular pipe, for the case where Reynolds n u m b e r 2 R 0 - - 2 U o a / v = 3240 X 108, the m e a n velocity distribution is found as follows: V,~(7, R*) = 1 -- 0.20472 -- 0.7967aL

(5.5)

Oct., 1 9 5 3 . ]

347

TURBULENT FLOW IN CIRCULAR P I P E

where V~ = # / U m , n = 16 and s = 12.94. T h e theoretical curve of (55) and N i k u r a d s e ' s d a t a are plotted in Fig. 2. In the m a j o r portion of the pipe, the theoretical curve checks fairly well with the experimental d a t a b u t near the wall of the pipe, the experimental d a t a are much higher than the theoretical curve. T h e present a u t h o r believes t h a t N i k u r a d s e ' s d a t a near the wall are too high to be true, for instance on the wall, instead of zero velocity, N i k u r a d s e found t h a t V~(1) = 0.546, at R0 = 1620 X 103 which is e v i d e n t l y not quite correct. In order to see the c u r v a t u r e of the mean velocity distribution near the wall, if we t a k e N i k u r a d s e ' s experimental d a t a on the wall 7 = 1 10

e., : Vm 0~

y,,, : i - a 204 7"- o. 796 ~*" --

- --

V,.: 1-0.204

~

-0250~ ~

0 c 4 - -

o NJkuradse

,I

of pi'pe

.2

Oat~

Re.3Z40,/O J =~

.3

.4

.~

'~" %

, V,=24aO cj~(e¢

.6

.t

Radial Dlsta.ce From the Center o f the Pipe

.9

t.O

~,ll ~ ~ pi~

FIG. 2. Velocity distribution in a circular pipe.

as one of the b o u n d a r y conditions, we can a d j u s t the coefficient of Eq. 55. T h e suitable mean velocity distribution is then, at R0 = 1620 X 103, V,(7) = 1 - 0.20472 - 0.250732.

(56)

E q u a t i o n 56 is also plotted in Fig. 2. It gives a much b e t t e r a g r e e m e n t with experimental d a t a and it shows t h a t near the wall the mean velocity distribution b e h a v e s like 1 - a272 - a~,732. In order to check the t h e o r e t i c a l curve (55) with experimental data, the a u t h o r believes t h a t refined m e a s u r e m e n t s , such as those b y h o t wire a n e m o m e t e r , should be m e a s u r e d near the wall, in order to determine a c c u r a t e l y the mean velocity distribution near the wall to c o m p a r e with the theoretical curve. T h e correlation y corresponding to Eq. 55 is, at R0 = 1620 X 103, Y(7) = 0.98357(1 -- 73°).

(57)

348

S.I.

PAI

[J. F. I.

Equation 57 is plotted in Fig. 3. Since there are no experimental data available for circular pipe, it is not checked with experimental data. The coefficient of friction Cs is given by the following formula from Eqs. 24, 27, and 48 ~ ~f~I = R* R* ff

(58)

nydn.

Substituting the value of y of (54) into (58) we have

1) G

]

(59)

s(n G 1),"

4 L1 -

LO

/ q

0.8

/

0.~

04

/ 0t

0.2

Center of the P,pe FI6. 3.

0.4

06

7=z. ~tT/jP

Correlation y = ~ -

08

J.O

Woll of ~thePipe

distribution in a circular pipe.

VIH. LOGARITHMIC VELOCITY DISTRIBUTION

Similar to the case of turbulent flow in a channel (19), the necessary and sufficient conditions for the existence of logarithmic velocity distribution of yon Kfirmfin in a circular pipe are as follows :4

6U*/v,

(a) There exists a universal profile with abscissa where 6 is the distance from the wall, and ordinate as obtained experimentally by Nikuradse (21). In the present notation, we have

w/U*

V(n, R*) = $[(1 - n)R*],

0 _< n _< 1.

4 This derivation is also similar to those given in References 15 and 16.

(60)

Oct., 1953.]

TURBULENTFLOW IN CIRCULAR PIPE

349

(b) The average mean velocity over a section is given by the following formula 2

f01 Vndn

= a + [3 log R*,

(61)

where a and fl are constant. This formula is sometimes known as Stanton's Law. It is interesting to note that in derivation of the logarithmic velocity distribution for turbulent flow in channel, a different form of this Stanton's Law (61) should be used (19). Substituting Eq. 60 into Eq. 61, we have 2

f01 ~,[(1

(62)

- n)R*-]ndn = a + # log R*.

Changing the variable ~ to t = (1 - n)R*, Eq. 62 becomes fo '~* (t - R*)4~(t)dt = R*2 -~- + a + / ~ log R*.

(63)

Differentiating (63) twice with respect to R*, we have ~(t)=

-In

+~__flB+ f l l o g t ] .

(64)

Hence from Eqs. 60 and 64 we have •

(65)

Equation 65 is the logarithmic velocity distribution which does not however hold for ~ = 1. Usually one says that Eq. 65 holds only in the region 0 < n < 1 - ,, where ~ is the thickness of a laminar sublayer. Since the logarithmic velocity distribution does not satisfy the boundary condition at the wall, it is not a rigorous solution of Reynolds equations.

IX.CONCLUSIONS The Reynolds equations have been studied for the turbulent flows in circular pipe and the following conclusions may b e drawn : 1. The correlation of the radial and the circumferential turbulent fluctuating velocity components u'v' is zero for the entire flow field. 2. The mean pressure distribution may be divided into two parts. One varies in a similar manner to that in laminar flow, that is, it decreases linearly with the axial distance along the pipe. The other part is influenced by the radial and the circumferential turbulent intensity

350 _

_

S.I.

PAl

m

[J. F. I. B

#2 and v'~. From the experimental data on u p2 and v '2, we conclude that the mean pressure in the fluid is greater than or equal to that on the wall of the pipe. From the equation for mean pressure, we see that near the center of the pipe, u '~ must be equal to v'2. It is verified by experimental results of Fage. 3. The mean velocity distribution and the correlation u'w' may be expressed by a polynomial of the radial distance, with the ratio of the shearing stress on the wall to that of the corresponding laminar shear stress of same maximum mean velocity as a parameter and an empirical constant. These expressions hold true all the way across the pipe, that is, both the turbulent region and viscous layer including laminar sublayer. The mean velocity distribution has been checked with Nikuradse's experimental data. The general shape of the theoretical curve agrees fairly well with the experimental data. But Nikuradse's experimental data near the wall are too high to be true, and the wellestablished condition of zero slip on the wall was not observed in Nikuradse's data, so that in order to check quantitatively the mean velocity distribution near the wall some refined measurements are needed. 4. The logarithmic mean velocity distribution is not a rigorous solution of Reynolds equations. LIST OF SYMBOLS radius of the circular pipe coefficient in mean velocity distribution (50) constant constant constant coefficient of friction defined by (26) any quantity of mean flow (2) exponential factor pressure a reference pressure a c o m p o n e n t of pressure defined by (42) radial distance Ro = Uoa/v Reynolds number R* = U*a/v Reynolds number a a2 A B C Cl ] n p po P r

S

~

TO/TI

t time u radial component of velocity u~ ith component of velocity uj jth component of velocity Um maximum velocity in the circular pipe U0 average mean velocity defined b y (24) U* = ~ / r - ~ reference velocity v circumferential component of velocity V = ~/U* non-dlmenslonal axial component of velocity V,~ = ~/U~, non-dimensional axial component of velocity w axial component of velocity

Oct., I953.]

TURBULENT FLOW IN CIRCULAR P I P E

35I

y z

non-dimensional correlation coefficient defined by (29) axial distance constant /3 constant radial distance from the wall of the pipe = r/a non-dimensional radial distance O angular displacement coefficient of kinematic viscosity tl = z/a non-dimensional axial distance p density of the fluid ¢1 non-dimensional radial turbulent intensity defined by (30) ¢2 non-dimensional circumferential turbulent intensity defined by (31) r~ laminar shearing stress on the wall r0 shearing stress on the wall certain function non-dimensional pressure defined by (28) - - (bar) means average values ' (prime) means fluctuating value without bar or prime: instantaneous values. REFERENCES

(1) O. REYNOLDS, "An Experimental Investigation of the Circumstances which Determine Whether the Motion of Water shall be Direct or Sinuous, and of the Law of Resistance in Parallel Channels," Phil. Trans., Vol. 174, Part III, p. 935 (1883) ; "On the Dynamical Theory of Incompressible Viscous Fluids and the Determination of the Criterion," Phil. Trans. Vol. 186, Part I, p. 123 (1895). (2) G. I. TAYLOR, "The Statistical Theory of Turbulence," Part I-IV, Proc. Royal Soc., Vol. A151, p. 421 (1935). (3) Tri. VON K.~RM.~N, "The Fundamentals of the Statistical Theory of Turbulence," J. Aeronautical Sci., Vol. 4, pp. 134-138 (1937). (4) W. HEISENBERG, "Zur statistischen Theorie der Turbulenz," Z. Phys., Vol. 124, p. 628 (1948). (5) A. N. KOLMOGOROFF, "Dissipation of Energy in Locally Isotropic Turbulence," C. R. Acad. Sci. U.R.S.S., Vol. 31, p. 538 (1941), Vol. 32, p. 16 (1941). (6) H. L. DRYDEN, "Review of Statistical Theory of Turbulence," Quart. Appl. Math., Vol. 1, pp. 7-42 (1943). (7) TH. VON K.~RM~,NAND C. C. L:N, "On the Statistical Theory of Isotropic Turbulence," Advances in Applied Mechanics, Vol. II, New York, Academic Press, 1951, p. 1. (8) H. W. LIEPMANN, "Aspects of the Turbulence Problem," Zeitschrift far angewandte Mathematik und Physik, Vol. III, pp. 321-342,407-426 (1952). (9) P. Y. CHOU, "Pressure Flow of a Turbulent Fluid Between Two Infinite Parallel Plates," Quart. Appl. Math., Vol. 3, pp. 198-209 (1945). (10) J. ROTTA, "Statistische Theorie nichthomogener Turbulenz," Z. Phys., Vol. 129, p. 547 (1951), Vol. 131, p. 51 (1951). (11) S. GOLDSTEIN, "Modern Developments in Fluid Dynamics," Vols. I and II, Oxford, Clarendon Press, 1938. (12) C. T. WANG, "On the Velocity Distribution of Turbulent Flow in Pipes and Channels of Constant Cross Section," Trans. A. S. M. E., Vol. 68, pp. A85-A90 (1946). (13) W. SZABLEWSKI, "Berechnung der turbulenten Str6mung im Rohr auf der Grundlage der Mischungsweghypothese," Z. A. M. M., Band 31, Heft 415, pp. 131-142 (1951). (14) TH. VON K~RM-~tN,"Mechanische Aehnlichkeit und Turbulenz," Nach Ges. Wiss. Goettingen, p. 68 (1930). (15) C. B. MILLIKAN, "A Critical Discussion of Turbulent Flows in Channels and Circular Tubes," Fifth Int. Cong. for App. Mech., Cambridge, Mass., 1938, pp. 386-392.

352

S.I.

PAI

[J. F. I.

(16) R. voN MIsEs, "Some Remarks on the Laws of Turbulent Motion in Tubes," Theodore yon K~rm~n Anniv. Volume Contribution to Appl. Mech. Calif. Inst. of Tech., Pasadena, Calif., p. 317 (1941). (17) S. GOLDSTEIN,"The Similarity Theory of Turbulence and Flow Between Parallel Planes and Through Pipes," Proc. Roy. Soc. London, Vol. A159, pp. 473-496 (1937). (18) G. K. BATCItELOR,"Note on Free Turbulent Flows, with Special Reference to the Two Dimensional Wave," J. Aeronautical Sci., Vol. 17, pp. 441-445 (1950). (19) J. KAMPE DE FERIET, "Sur l'Ecoulement d'un fluide visqueux incompressible entre deuxplaques paralleles indefinies," La Houille Blanche, Vol. 3, pp. 1-9 (1948). (20) S. I. PAl, "On Turbulent Flow Between Parallel Plates," J. Appl. Mechanics, Vol. 20, No. 1, pp. 109-114 (1953). (21) J. NIKURADSE,"Gesetzmassigkeiten der turbulenten Stromung in glatten Rohren," Berlin, V. D. I-Verlag, 1932.