TURBULENT

FLOW

M.D.

IN

THE

BOUNDARY

LAYER

AND

IN

TUBES

MiI[ionshchikov

UDC 532.542.4

Determination of heat and m a s s exchange in various devices, including nuclear r e a c t o r s , requires a knowledge of the basic laws of turbulent flow. In particular, investigations of flow taws in tubes with rough walls may lead to new methods for the intensification of heat and m a s s t r a n s f e r . In [1] the author considered turbulent flow in the boundary l a y e r and in tubes, using the superpositi0n principle for molecular and turbulent viscosity. In the p r e s e n t work we c o n s i d e r in more detail some of the results obtained in the previous work. We simplify the hypothesis c o n c e r n i n g the variation of mixing length, and calculate the flow in tubes with nonuniformly rough wails. Boussinesq [2] (1877) was the f i r s t to indicate the n e c e s s i t y of introducing a turbulent viscosity eoeffietent in the equations of motion. In 1895 Reynolds [31 considered turbulent motion as the overall r e s u l t of i r r e g u l a r motion of small particles of liquid subject to statistical laws. This approach led him to consider the velocity v e e t o r in the equations of motion of a viscous liquid as a random function in space and time, and to seek the statistical rules for turbulent flow by taking averages of various kinds in the equations of motion. The nonlinear t e r m s in the equations of motion give r i s e to a supplementary t e n s o r T~h = - - p v ~ v ; ,

in the averaged equations, where the b a r indicates the average value. zero, is called the turbulent s t r e s s .

(1)

This tensor, which is generally non-

Reynolds also showed that flow beeomes turbulent when the ratio of inertial f o r c e s to viscous f o r c e s (the Reynolds n u m b e r Re) exceeds a certain value. The Reynolds n u m b e r can be expressed in t e r m s of p a r a m e t e r s of the flow under consideration such as the velocity (for example the mean v e l o c i t y ~ in a tube), the seale of the flow (the internal d i a m e t e r d of a tube), and p a r a m e t e r s c h a r a c t e r i z i n g the liquid (the viscosity ~ and density p : /~e = p~d (2) It has been found that flow in tubes remains laminar if Re < R e e f ~ 2000-2500. the Reynolds number exceeds the critical value,

Turbulent flow begins when

The g r e a t difference between these two flow r e g i m e s is shown in Fig. 1, in which the dimensionless resistance 2d

=-z-"

Ap

pr,-T~,

(3)

(Ap is the p r e s s u r e d drop along a length L of a tube) is plotted against the logarithm of the Reynolds number. The points in the diagram show results obtained by Stanton and Panell [4] and Nikuradze [5] using both wate~ and air. Curve 1 was calculated f r o m the theoretical f o r m u l a = 64 (4) B for l a m i n a r flow. There is good a g r e e m e n t between experimental and theoretical results for a v a r i e t y of media and conditions of flow. When the Reynolds number is between 2000 and 2500 there is a rapid change in the r e g i m e with the development of turbulence. B l a s i u s ' s empirical formula

(5)

~:_: ~ Translated f r o m Atomnaya l~nergiya, Vol. 28, No. 3, pp. 207-220, March, 1970. submitted July 18, 1969.

Original article

01970 Consldt~ults E,ureal~, a diui,sion of D[er~um Publis/Ting Corp<)ration, 227 ~'<,,~t lZth Street, ~'ew Yor/,, N~ Y. lOOll, ill fillets reserved. T/lis ~zrticle cannot be reproduced for any p~rpese w]~at,soec, er without f~<'rmissio,~ of the publis/~cr. I cop) of this article i,s auailahle /?om t/*e publisher for $1,5.o0.

268

i

1-Poiseuille motion 9Water(~,25~cm diameter) .

0,7125 "

.

.

0,361

,,

.

"

2~855

"

"

1,255

,,

,,

,,

Z,~5

,,

.

,,

0,7125.

Air

,,

O,3Sl . .

12,62

.

,~

- Nikuradze's results 2-Computed curve ~2

2

45

4

4~

5;5

~g~

Fig. i. R e s i s t a n c e in a c i r c u l a r tube v e r s u s the logarithm of the Reynolds number. gives good results for turbulent r e g i m e s with Re < 105. N i k u r a d z e ' s f o r m u l a ~

= A lg (R VL) + B,

(6)

with suitably chosen values of A and ]3, gives a s a t i s f a c t o r y description of the r e s i s t a n c e in any turbulent regime. The f o r m u l a s of Bias[us and Nikuradze are of g r e a t [mportcime in c o n t e m p o r a r y hydraulics and are widely used in engineering calculations 9 The P r a n d t l - v o n K a r m a n t h e o r y [6-8], which is the basis of N i k u r a d z e ' s formula, has contributed extensively to our knowledge of turbulence. C o n t e m p o r a r y theories, however, do not give a unified picture of the interaction between the separate p r o c e s s e s of m o l e c u l a r and turbulent exchange which take part in different d e g r e e s in the formation of flow as a function of the surrounding conditions. This fact has led to a m o r e detailed study of turbulent flow in tubes by various experimental methods, and also to many attempts to develop new theories to d e s c r i b e turbulent flow in tubes. The turbulent boundary layer f o r m e d by parallel flow past plates has been investigated in connection with many p r o b l e m s arising in aviation and related technology and in connection with p r o b l e m s concerning heat and m a s s t r a n s f e r arising in m e t e o r o l o g y and oceanography, as well as in the study of flow in tubes. Flow in tubes and flow in the boundary l a y e r have many features in common, although the conditions in the boundary l a y e r are usually s i m p l e r than those in tubes since the tangential s t r e s s in the boundary involves constant p a r a m e t e r s which can v a r y with the tube radius. In the boundary layer, however, there is no general description of the velocity profile valid for the whole turbulent region. Fig 9 2 shows a graph of m e a s u r e d distributions of the dimensionless velocity u / v . v e r s u s the dimensionless distance ~1= y v . / v , f r o m the wall, where v , is the dynamic velocity V ~0/p, To is the friction at the wall, v is the kinematic viscosity, and y is the distance f r o m the wall. The velocity plainly v a r i e s according to a r a t h e r c o m p l i c a ted law. Three different flow r e g i m e s are distinguishable. The first, and n e a r e s t to the wall, is l a m i n a r flow (the l a m i n a r sublayer) in which only m o l e c u l a r viscosity operates. The theoretical c u r v e for this region is known9 In it the velocity is proportional to the distance from the wall (deviation from a straight line in this region is due only to the use of a logarithmic scale). Beyond the l a m i n a r sublayer, for "O~ 8, there is a transitional region which cannot be s a t i s f a c t o r i l y described by theory. F o r values of ~?l a r g e r than 200 a completely turbulent r e g i m e is established which is well described by the known logarithmic law u.

= _t In 7]~- C. (r

(7)

During the past few decades turbulence t h e o r y has attracted a g r e a t amount of attention in many different countries. P r o g r e s s has been p a r t i c u l a r l y striking in the development of statistical theories, which began with the work of F r i e d m a n and Keller [9], f i r s t reported in 1924. This work, which was influenced by results obtained by Kolmogorov, Taylor, von Karman, Obukhov, and others [10-12], led on the one hand to an understanding of the s t r u c t u r e of turbulence, and on the other had the effect of initiating experimental work to determine the s t r u c t u r e of turbulent flow. Here we only mention m e a s u r e m e n t s of the s t r u c t u r e of turbulent flow usefut in the analysis of flow in boundary l a y e r s and tubes.

269

.... j o~ 8

. . . . . . . V ....

---t

................................ ! 1

F i g . 2.

t,0

0 '

. . . . , . . - - ~ - ~ ? ~ - -,,, .

"tO

Fig. 3.

20

.

.

.

.

.

~t

.

'

I

30

40

! 50

60

1

-.

ZU,

z+=~--.

Fig. 3 Statistical parameters

cg

~,5

1,~ 2,5 3 J,5 '~ V e l o c i t y p r o f i l e f o r t u r b u l e n t flow.

9

I 0

10

20

30

40

50

60

70

80

90

Fig. 4

of boundary-layer

flow.

Fig. 4. Correlation between longitudinal and transverse pulsations in the boundary layer.

components

of velocity

A very detailed study of the structure of turbulence in the boundary layer and [n tubes was carried out by Lau~er [13]. Laufer's results can be used to determine the variation of the turbulent friction --gu'v', referred to v2,, as a function of the distance from a plate. Measurements obtained by Laufer in the boundary layer are shown in Figs. 3 and 4. Figure 4 shows calculated and not experimental points, since here the graph was obtainedby conversion of the experimental data shown in Fig. 3, i.e., points were not obtained by direct measurement of ~ but by the measurement of other statistical parameters related to u'v' . It should be noted that the curves in Fig. 3 were obtained near the wall by extrapolation, since in this region experimental values were measured at distances not smaller than ~7 and amounting to several units. We

~tpt

see from Fig. 4 that, at large distances from the wail, the ratio ~,-~-tends asymptotically

unity. For a wide range of values of ~ this quantity differs only slightly from tends rapidly to zero only close to the wall. It follows from the equation for flow in the boundary du

value, and it

layer

~ - - Ou'v' = To

270

its asymptotic

to

(8)

or dujo,

d~

~'Y

v2, -= .1

(9)

that the fact that -- ~'o_~'is approximately equal to unity implies that the tangential s t r e s s at g r e a t distances f r o m the walt is completely d e t e r m i n e d by the turbulent pulsations, and that the dynamic velocity v , d e t e r mines the scale of the turbulent pulsations. Hence L a u f e r ' s m e a s u r e m e n t s confirm Reynolds' theory concerning the cause of turbulent s t r e s s e s . This is not a trivial result. Analogous m e a s u r e m e n t s should be c a r r i e d out to derive theories for heat and m a s s t r a n s f e r , based on the c o r r e l a t i o n between velocity and t e m p e r a t u r e pulsations o r between velocity and concentration pulsations; such m e a s u r e m e n t s would not n e c e s s a r i l y lead to the same result. It should be r e m e m b e r e d that m e a s u r e m e n t s of velocity pulsations will not lead directly to the velocity profile, since the frictional s t r e s s in the boundary l a y e r is constant and is independent of the m e c h a n i s m which c a u s e s it. We encounter a s i m i l a r situation if we try to introduce m o l e c u l a r - s c a l e liquid pulsations into the equation of motion of a perfect liquid. The equations of motion also contain a s t r e s s t e n s o r analogous to the R e y n o l d s ' t e n s o r , with the difference that pulsations are generated by m o l e c u l a r motion in this case. In deriving t h e N a v i e r - S t o k e s equations we must, however, use the concept of a viscosity coefficient and must postulate a relation between the s t r e s s t e n s o r and the r a t e - o f - d e f o r m a t i o n tensor. The theories of Boussinesq, Prandtl, von Karman, Taylor, and others involved different approaches to the problem of turbulent viscosity. In B o u s s i n e s q ' s theory the t u r b u l e n t - v i s c o s i t y coefficient PT is introduced by analogy with the m o l e c u l a r - v i s c o s i t y coefficient: du

----/~v~-y.

(I0)

It is clear, however, that in c o n t r a s t to the m o l e c u l a r viscosity, which depends only on the state of the liquid or gas, the turbulent v i s c o s i t y cannot remain constant and independent of the distance f r o m the wall since this would imply that only the scale and not the c h a r a c t e r of the velocity profile would vary, while we have seen that it has been experimentally established that there is a change in the c h a r a c t e r of the flow. The theories proposed by Prandtl and yon K a r m a n and by Taylor were attempts to relate turbulent v i s c o s i t y to the local mean-flow s t r u c t u r e . Prandtl, d i r e c t l y considering the Reynolds turbulent-friction stresses,

du set the velocity pulsations u' and v' proportional to the derivative t~v , where l has the dimension

of a length (the mixing length).

This approach yields the f o r m u l a "~"~ Pl~ k d~ ]

(11)

for the turbulent friction and the f o r m u l a 92 au

(12)

for the turbulent viscosity. The quantity l r e m a i n s undetermined in this theory, and it is a s s u m e d that it is proportional to the distance f r o m the wail. In this c a s e we have du 1" , ~ py2 ( ~-Y

(13)

and for constant r the logarithmic law can be obtained directly. Von K a r m a n took the f u r t h e r step of a s s u m i n g that l is proportional to the ratio ~du/dy and derived the formula ~ P

/ ~[fi-v~ J

~dy ;

(14)

f o r the turbulent viscosity. Another approach is to apply dimension theory to the problem of the logarithmic velocity profile. We a s s u m e that, f o r large Reynolds numbers, the relation between the velocity and the distance f r o m the wall is independent of the viscosity. Here only %, p, and y d e t e r m i n e the flow, and we have dy

y

(15)

271

The logarithmic law immediately follows. which the logarithmic law holds [14].

This procedure yields a c l e a r e r indication of the conditions under

We note that although dimensional considerations do not yield d i r e c t methods for the construction of the velocity profile in the region where m o l e c u l a r viscosity is important, they do yield a rigorous foundation for the logarithmic law under conditions of strongly developed turbulence when frictional s t r e s s e s are completely due to turbulence. It can be shown that difficulties in taking m o l e c u l a r viscosity into account may be avoided by d i r e c t l y considering the concept of turbulent viscosity as in the original investigations of turbulence. The concept of turbulent viscosity is used in connection with many problems of dynamic meteorology and oceanography. It is also of use in the theory of jets, but we must always r e m e m b e r that it is applicable only when g r e a t a c c u r a c y ls not required or when computed results cannot be verified experimentally. Flow

in the

Boundary

Layer

We use the turbulent-viscosity coefficient and write the equation of motion for the boundary l a y e r in the f o r m du (~ + I~) ~ =-.t0, (!6) where r 0 is the tangential s t r e s s , assumed to be independent of y. It is known f r o m m o l e c u l a r theory that/~ is proportional to the density p, the t h e r m a l m o l e c u l a r velocity Vm, and the mean free path/m: bt = cplmv m.

The coefficient c in this f o r m u l a is calculated by using kinetic theory. Its value depends on the molecular model used in the theory. Different assumptions concerning the bodies modelling molecules and their modes of collision yield different values of c varying approximately f r o m 1/3 to 1/2. The turbulent-viscosity coefficient is d e t e r m i n e d s i m i l a r l y with the length scale taken to be the mixing length t, as in the Prandtl - y o n K a r m a n t h e o r y , and the velocity-pulsation scale taken to be the dynamic velocity v , in a g r e e m e n t with L a u f e r ' s experimental results. In the boundary l a y e r of a liquid this length scale is independent of y, We have ~tT : ply.,

and Eq, (16) becomes (v + v,1) ~d. = v,,, where v is the kinematic viscosity (v = g / P ) ,

(~7)

or d '~ v,.,. tJr, d 'o

t+

i81

L a u f e r ' s experimental results imply that, at sufficiently g r e a t distances f r o m the wall, the turbulent component is dominant in the frictional s t r e s s and so, at g r e a t distances f r o m the wall, the t e r m v , l / v , in the brackets is much l a r g e r than 1 and we have d~]

v,1 ~v

'

where ~ = y v , / v . Hence, if the turbulent viscosity in the boundary l a y e r is to satisfy the experimental logarithmic law, we must have l = a.q, (20) where a is constant. F o r sufficiently large values of y we may assume that y is measured, not f r o m the wall, but f r o m a point at a small distance 50 f r o m the wall.

272

We a s s u m e that this r e l a t i o n , f o r c o n s t a n t a, holds in the whole flow r e g i o n including the p a r t w h e r e m o l e c u l a r v i s c o s i t y cannot be n e g l e c t e d . We will s e e l a t e r that this a s s u m p t i o n is quite g e n e r a l l y , but not u n i v e r s a l l y , v a l i d . I n t h e region~? < 5, w h e r e 5 = v,5___qo t h e r e is no p e n e t r a t i o n of s t a t i s t i c a l o r d e r i n g of t u r b u l e n t exchange e n s u r i n g the d e v e l o p m e n t of t u r b u l e n c e s t r e s s e s and the o n s e t of t u r b u l e n c e . This f a c t d e t e r m i n e s the value of 60 f r o m which we m u s t m e a s u r e the value of y used in (20). Hence

l=a(y-60)

for

y>~0.

(21)

It follows that the equation of m o t i o n in the b o u n d a r y l a y e r is (22)

l y . _-- i ' [ t 4 - a 0 ] - - 6 ) ] d ud~

where a = O for ~ l ~ 6 , a ~=const r 0 for ~l~ 5. I n t e g r a t i n g and using the condition of a d h e r e n c e to the wall, we obtain _~u = ~]

for

q ~ 6,

us

,,~---=lhl[l-I a (B -- 6)] 4- const

for

q>6.

The c o n s t a n t of i n t e g r a t i o n is d e t e r m i n e d f r o m the condition f o r the c o n v e r s i o n of the l a m i n a r into the turbulent profile: ~1==6,

" :=6.

U,

Hence -%:=~}

for

0~! ~6,

(23)

~* --%~--1 lnll-4 a(ll--b)]-~-5 for t~, a

,1~(~.

The c o n s t a n t s a and 6 in (23) have a s i m p l e p h y s i c a l i n t e r p r e t a t i o n : 5 is the t h i c k n e s s of the l a m i n a r s u b l a y e r , and a is analogous to the c o e f f i c i e n t c in the f o r m u l a f o r m o l e c u l a r v i s c o s i t y . F i g . 2 shows the c u r v e c a l c u l a t e d f r o m (23) f o r a = 0.39 and 6 = 7.8. E x p e r i m e n t a l points obtained f r o m m e a s u r e m e n t s in tubes a r e shown in the s a m e d i a g r a m . The d e t e r m i n a t i o n of t h e s e e x p e r i m e n t a l points will be d e s c r i b e d in d e t a i l in the s e c t i o n d e a l i n g with flow in tubes. When the v a l u e s of a and 6 have been found we can c a l c u l a t e the d e p e n d e n c e of the t u r b u l e n t s t r e s s one?, We e a s i l y obtain the r e l a t i o n u'v'

v~ * - I

d (u/v,)

d~

a (~ - - 6)

= l§

"

(24)

F i g . 4 shows the c u r v e c a l c u l a t e d f r o m (24) f o r a = 0.39 and 6 = 7.8. The points shown c o r r e s p o n d to L a u f e r ' s m e a s u r e m e n t s . It is plain that the c a l c u l a t e d c u r v e is in good a g r e e m e n t with LauferTs r e s u l t s f o r a wide r a n g e of v a r i a t i o n of 7/. N e i t h e r c a l c u l a t e d n o r e x p e r i m e n t a l r e s u l t s a r e a c c u r a t e in the l a m i n a r s u b l a y e r , in fact c a l c u l a t i o n does not y i e l d a point but a t r a n s i t i o n a l r e g i o n which, however, is n a r r o w e r than that obtained by e x t r a p o l a t i o n of L a u f e r ' s r e s u l t s . This f a c t i n d i c a t e s that a does not jump suddenly from_ z e r o to 0.39, but that t h e r e is a r e g i o n in which a v a r i e s s m o o t h l y which is not o b s e r v e d in v e l o c i t y p r o f i l e m e a s u r e m e n t s ( F i g . 2) o r e x h i b i t e d in the v a r i a t i o n of the s t r u c t u r e of the t u r b u l e n c e . We c o n s i d e r this r e g i o n below in o u r study of r e s i s t a n c e in tubes. Flow

in

Tubes

The l o g a r i t h m i c v e l o c i t y - p r o f i l e law has been e x p e r i m e n t a l l y e s t a b l i s h e d f o r flow in the b o u n d a r y l a y e r at the s u r f a c e of a plate, f o r flow in tubes of c i r c u l a r c r o s s section, and f o r flow between two p l a t e s . Flow in tubes (or between plates) is m o r e c o m p l e x than flow p a s t a s i n g l e plate b e c a u s e the tangential s t r e s s is not a c o n s t a n t but d e p e n d s l i n e a r l y on the tube r a d i u s . The l o g a r i t h m i c law d o e s not hold c l o s e to the wall and it cannot s a t i s f y the condition d u / d y = 0 on the axis of a tube. T h e s e f a c t s indicate that a m o r e d e t a i l e d i n v e s t i g a t i o n of flow in tubes is needed. We s t a r t f r o m the e x p e r i m e n t a l l y p r o v e n fact that the l o g a r i t h m i c law holds at s u f f i c i e n t l y g r e a t d i s t a n c e f r o m the wall and f r o m the tube a x i s . F i r s t c o n s i d e r the v a r i a t i o n s in the s t r u c t u r e of t u r b u l e n t flow in a tube. L a u f e r [13] m a d e a c a r e f u l study of t u r b u l e n c e in tubes as well as in s i m p l e b o u n d a r y l a y e r s . F i g . 5 shows the v a r i a t i o n of the quantity

273

.~. along the radius of a tube for a Reynolds number Re = 5.10 ~. Even for this Reynolds number (~/= 55,500), turbulent pulsations a r e completely responsible for tangential s t r e s s , since the experimental points are concentrated along the line corresponding to complete tangential s t r e s s along the radius. Deviations f r o m the line a r e observed for s m a l l e r values of ~?0On the other hand if the quantity --u'v' is related, not to the square of the dynamic velocity v . c o r r e s ponding to the tangential s t r e s s at the wall, but to the square of the dynamic velocity v.~ c o r r e s p o n d i n g to the tangential s t r e s s at the section under consideration, i . e . , to v~ = v~ (t -- #), ~) = y/r, (r is the tube radius), we find that u'u'/v~.~ r e m a i n s constant and p r a c t i c a l l y equal to unity over a wide range including regions close to the wall and close to the axis. Hence, f o r flow in tubes, the scale for velocity pulsations in developed turbulence is the dynamic velocity v.1. We use this r e s u l t in determining turbulent viscosity. We must also r e m e m b e r the need f o r introducing variations in the relation between mixing length and the distance f r o m the wall, which was taken to be linear in the boundary l a y e r . The need for some alteration is clear, since the change in the mixing length near the axis must be taken into consideration. A m o r e accurate e x p r e s s i o n of the relation between I and y is obtained by using a logarithmic velocity profile. The equation of motion in a tube is = ~,

(25)

where ~- is the tangential s t r e s s at the section under consideration, related to the tangential s t r e s s at the wall % by the equation r: : % (i y) (26) -

(~r is the distance f r o m the wall).

-

This equation can be written (v + vT) du :: vl (i -- ~),

(27)

where ~T

P We define the turbulent kinematic viscosity vI to be the product of the c h a r a c t e r i s t i c velocity (the dynamic velocity v,1 for the radius under consideration) and the mixing length l :

Using the relation v. = v ~ ( t - y ) ,

we obtain

and Eq. (27) becomes

(, f ,,,~k ~t---y=) ~ d~, =_. , 4 (1 - F)

(28)

This equation shows that, for Reynolds numbers sufficiently large for the viscosity to be negligible, the logarithmic law c o r r e s p o n d s to a relation between I and the radius given by the equation l a (y-- 5 0 ) V ~ ,

(29)

where a is a constant and 50 is the thickness of the l a m i n a r sublayer; the introduction here of the latter p a r a m e t e r is based on the same reasoning given for its introduction in o r d i n a r y b o u n d a r y - l a y e r theory. Hence the equation for motion in a tube (or between parallel plates) is

[/~ ! ~(y -6)(,1 = ~ ) J ndul'~., ~-~:

i--~,

(30)

where -

!/

_

!

~J ; /~=: ~]0 ; ?"

qo

,-,. v

;

~

6o

=--, r

Integrating this equation and using the condition for smooth matching at the boundary with the linear range ( i . e . , for y = 5 and u v . = 5), we have u =rloy=q

274

for q ~ 8 ;

for~--5

_ L/iV

O~

5550

I

F o r m u l a (31) gives the velocity in the whole range of variation of ~ f r o m 0 to 1. On the tube axis (for ~ = 1) it satisfies the condition d u / d y = 0. It can be shown, however, that for small values of ~ = 1/~ 0 and S (31) differs f r o m (23) by only a small c o r r e c t i o n f (~) which is nonzero only n e a r the tube axis, i . e . , (31) can be written

o,2 o,1 ~z o,~ 0,4 o,5 0,6 o,7

o,s o,~-~

Fig. 5. C o r r e l a t i o n between longitudinal and t r a n s v e r s e components of v e l o c ity pulsations in a tube.

We

or ~_>6, where

used this formula

-,t US

--

| (/

lrt [1 + a (t]-- 6)1 -F 6 +- / (q),

(32)

where f0?) is a small c o r r e c t i o n (for small fl and 6) defined to be the difference between (31) and (23).

to obtain the variation of the velocity in a tube.

The difference (u/v,)axperimental~ according to the law

- f0?) obtained from

measurements

must

vary as a function of

= ~ l r , if =-a (~-- 6)1 +6, which holds in the boundary layer. In p r a c t i c e this means that experimental points obtained close to the tube axis which are not in a g r e e m e n t with the logarithmic law are c K s e to this line. Results are shown in Fig. 2. E x p e r i m e n t s should be c a r r i e d out to d e t e r m i n e whether deviatiovs of experimental points f r o m the logarithmic law a r e due to i n a c c u r a c i e s in m e a s u r e m e n t , i m p r o p e r l y conducted e x p e r i m e n t s (possibly with s u r f a c e s insufficiently smooth), or whether they indicate an actual unknown phenomenon. Resistance

to

Flow

in

Tubes

For flow in tubes the resistance

is given by the formula

Ap - - 8 ( v~)2, ~ = 2-L-" 9u2 where ~ is the mean velocity over the c r o s s section of the tube given by the integral I

0

We divide the range of integration into two p a r t s : f r o m 0 to 5 where a = 0, and f r o m 5 to 1 where a = eonst 0. The f o r m u l a s for ~/v, and k are u ... O,

~2 ~:

where

/lncz_3,

2a -

2

8

[

(33)

t I , rv, =,,~,rl,~ -~-; ~=i--K(~-6); 5=,~5 0

..... 5I~;

and e is the small c o r r e c t i o n related to f(~) which can be neglected in practice. Eqs. (33) give a p a r a m e t r i c relation between Re and k with p a r a m e t e r p. The constants a and 5 have already been determined f r o m the velocity profile. In the limiting case of v e r y large Reynolds n u m b e r s and c o r r e s p o n d i n g small N i k u r a d z e ' s relation i = A lg (R~ V~) + B

fi,

(33) b e c o m e s

Vz

with A = 2.08 and t3 = - 1 . 0 4 . These values differ only slightly from N i k u r a d z e ' s values, the difference being due to the g r e a t e r a c c u r a c y of our constants determining the velocity profile.

275

To obtain the laminar profile we must set a -- 0 everywhere. In this case we wi[l have laminar flow in the whole tube and the thickness of the laminar (parabolic) layer will be equal to the radius and 5 = 1. In this special case we have the Poiseuille law 64

Be Curve 2 in Fig. 1 shows the relation between and also many points obtained experimentally.

k and Re, calculated from

(33) with a = 0.39 and 5 = 7.8,

It is important to observe that, under different experimental conditions, the breakdown of laminar flow occurs for different critical Reynolds numbers Recr in the range 2000-2500. If the laminar regime breaks down for large critical Reynolds numbers, the resistance first increases slowly and then suddenly increases to a value corresponding to a turbulent regime in which the turbulence resistance law holds. If the regime breaks down for small l~eer , the resistance increases sharply to a value smaller than that for a turbulent regime and subsequently to the value for a turbulent regime corresponding to a final value of a for large Reynolds numbers. In this case the breakdown of stability of the laminar regime is, so to speak, premature, and does not Occur under conditions suitable for the development of turbulent flow. At the present there are no reliable data concerning this transitional regime, and we therefore start with the simplest assumptions possible. Since laminar flow occurs for a = 0 and turbulent flow occurs for a = const ~ 0 we assume that, in the breakdown region between the initial departure from the Poiseuille law and the arrival at a turbulent region (in the case of large values of Recr), the coefficient a is a linear function of I/fl or, equivalently, of Re. To each value of/3 there corresponds a value of a, and substitution of the value of' fl in (33) yields the resistance coefficient k. Calculated results are shown by curve 3 of Fig. I. For the case of premature breakdown we must make a further assumption concerning the value of a attained after the sharp increase, although there is no reliable data available. We assume that a varies linearly from zero to 0.39 when I/fi varies from 60 to 103. Calculated results are shown [n Fig. 1 (curve 4). The Introduction of a variable a in the calculation of turbulent stress would ensure a smooth transition to Laufer's curve. However, turbulent stress has not been measured for small ~, and there are no results for the range in which calculations must be carried out. Further experimental work is needed. In all our reasoning, we started from an analysis of experimental results. Certain basic facts have been established, but there remain areas requiring detailed investigation. It is possible that a further improvement in the accuracy of experimental results will lead to improved representations of results. There is a certain amount of freedom left within the framework of the above theory, since changes in the values of a and 5 might yield descriptions of phenomena which, if not more accurate in whole flow, might at least be an improvement in the range of Reynolds numbers in which we are interested. Turbulent

Flow

and

Wall

Roughness

Turbulent flow in the boundary sufficiently marked irregularities.

layer and in tubes is considerably

influenced by wall roughness

with

Resistance in rough-walled tubes was studied by Nikuradze [5]. His results, contain*~d in all texts and reference books, are shown graphically in Fig. 6 for several values of k = k/r, where k is the mean height of roughness protuberances and r is the tube radius. Nikuradze described four different turbulence regimes. I. Laminar for low Reynolds

flow, following Poiseuilie's law for any amount numbers.

2. Turbulent flow in which the resistance law is the same

of roughness.

as for smooth

tubes.

3. Turbulent flow with a resistance ceofficient depending on the Reynolds roughness. The second and third regimes

occur for intermediate

Reynolds

This type of flow occurs

number

and the relative

numbers.

4. A self-simulating regime in which the resistance is independent of the Reynolds numbers. Here the resistance depends on the relative roughness. This regime develops for high Reynolds numbers.

276

~gloo~.

Expt.

.

Calc.

-

...... - y

3,0

3,5

/

0,0

~,5

5.0

5,5

go

g5

7,0

z#n

F i g . 6. D e p e n d e n c e of the r e s i s t a n c e c o e f f i c i e n t in c i r c u l a r t u b e s on the R e y n o l d s n u m b e r and on the r o u g h n e s s .

In v e r y r o u g h t u b e s (in N i k u r a d z e ' s e x p e r i m e n t s with k: = 1/15) the t u r b u l e n t r e g i m e is d i f f e r e n t f r o m t h a t in s m o o t h t u b e s , and c h a n g e s d i r e c t l y into a l a m i n a r r e g i m e . H e n c e in t h i s c a s e t h e r e is no t y p e 2 regime. T h e a s y m p t o t i c b e h a v i o r of the r e s i s t a n c e a s a function of the r e l a t i v e r o u g h n e s s i s d e s c r i b e d b y N i k u r a d z e ' s f o r m u l a [15] )~.....

t

(34)

F o r t h e t r a n s i t i o n a l r e g i m e we h a v e a univ e r s a l f o r m u l a which N i k u r a d z e o b t a i n e d b y p r o c e s s i n g h i s e x p e r i m e n t a l r e s u l t s . N i k u r a d z e g a v e a q u a l i t a t i v e e x p l a n a t i o n of the b e h a v i o r of the r e s i s t a n c e in the s e c o n d regime. T h i s e x p l a n a t i o n is b a s e d on the f a c t that, if the R e y n o l d s n u m b e r is r e l a t i v e l y low, the t h i c k n e s s of the l a m i n a r s u b l a y e r is g r e a t e r than the h e i g h t of the p r o t u b e r a n c e s f r o m the w a l l and the r o u g h n e s s h a s no i n f l u e n c e on the flow, j u s t a s in a p u r e l y l a m i n a r r e g i m e . A s the R e y n o l d s n u m b e r i n c r e a s e s and the t h i c k n e s s of the l a m i n a r s u b l a y e r d e c r e a s e s the r o u g h n e s s b e g i n s to p l a y a m o r e i m p o r t a n t p a r t in the f o r m a t i o n of the flow. T h e c o n c e p t of t u r b u l e n t v i s c o s i t y c a n be u s e d in the i n v e s t i g a t i o n of flow p a s t a r o u g h w a l l . c o n s i d e r flow in the l a y e r c l o s e s t to a p l a n e w a l l . A s b e f o r e we use the e q u a t i o n of m o t i o n du.

-~

"~ @ = 7 =: 4 .

First

(3S)

The q u a n t i t y ~ is the s u m of the m o l e c u l a r v i s c o s i t y ~ and the t u r b u l e n t v i s c o s i t y , w h o s e f o r m we will immediately establish. I t w a s shown a b o v e that, f o r a s m o o t h w a l l , the k i n e m a t i c t u r b u l e n t v i s c o s i t y f o r the l a y e r c l o s e s t to the w a l l can be e x p r e s s e d b y the f o r m u l a vT, :: a ( y - - 6o) v,.

(36)

In the p r e s e n c e of r o u g h n e s s and f o r the l i m i t i n g c a s e of high R e y n o l d s n u m b e r s , we h a v e a s e l f s i m u l a t i n g r e g i m e d e p e n d i n g o n l y on the r o u g h n e s s . T h e r e g i m e c a n be o b t a i n e d by u s i n g the h e i g h t k of the r o u g h n e s s p r o t u b e r a n c e s a s the c h a r a c t e r i s t i c d i m e n s i o n and the q u a n t i t y v, i n t r o d u c e d a b o v e f o r the v e l o c t t y - p u ! s a t t o n s c a l e . H e n c e f o r the s e l f - s i m u l a t i n g r e g i m e we h a v e v, 2 = air,

(/~ - - {3o),

(37)

w h e r e a 1 is a c o n s t a n t with the s a m e p h y s i c a l i n t e r p r e t a t i o n a s the c o n s t a n t a. L e t a t = a. We a l s o a s s u m e t h a t a l l t h r e e v i s c o s i t i e s ( m o l e c u l a r v i s c o s i t y , t u r b u l e n t v i s c o s i t y d e t e r m i n e d by the d i s t a n c e f r o m the wall,

277

and t u r b u l e n t v i s c o s i t y d e t e r m i n e d b y the h e i g h t of the p r o t u b e r a n c e s ) a c t s i m u l t a n e o u s l y }.n a l l flow r e g i m e s , and t h a t the t r a n s i t i o n f r o m one r e g i m e to a n o t h e r d e p e n d s on the r e l a t i v e c o n t r i b u t i o n of the d i f f e r e n t v i s c o s i t i e s f o r c o n s t a n t a (which c a n in g e n e r a l d e p e n d on k ) . Hence the t h e o r y is b a s e d on the a s s u m p t i o n t h a t the s u p e r p o s i t i o n p r i n e i p i e h o l d s f o r the v i s c o s i t y in the l a y e r c l o s e s t to a r o u g h w a l l : ~ :: v -! a (~ - ~,) ~ , -.: ~ (~: -

G ) * ~,-

(38)

The s i g n i f i c a n c e of the a s t e r i s k in (38) will be e x p l a i n e d b e l o w . We w r i t e the e q u a t i o n of m o t i o n in the l a y e r c l o s e to the w a l l in the f o r m (39)

(v + av. (g--4o)~ av, (k -~G)*I W o r in the d i m e n s i o n l e s s f o r m It : a ( q - - 5 )

', a01a--6)*l d(~,.~.) dq

t,

where gv,

kv.

I n t e g r a t i n g and using the c o n d i t i o n u = 6 f o r ~ = 5, we o b t a i n .-~-=:i--ln.. . I ! ~..{.a~01--6)

-!5.

(40)

f o r ~?___6 F o r a r o u g h w a l l we o b t a i n a l o g a r i t h m i c law, b u t with an i m p o r t a n t c o r r e c t i o n due to the p r e s e n c e of the f a c t o r

1 in the s e c o n d t e r m of the e x p r e s s i o n in s q u a r e b r a c k e t s . In o r d e r to be a b l e to m a k e i + a (~1~--~)* a c o m p a r i s o n with e x p e r i m e n t a l r e s u l t s (the m o s t r e l i a b l e of w h i c h h a v e b e e n o b t a i n e d f o r r o u g h tubes) we a s s u m e , a s in the c a s e of s m o o t h t u b e s , t h a t the v e l o c i t y p r o f i l e c a n be a p p r o x i m a t e d b y the p r o f i l e f o r the w a l l l a y e r f o r a p l a t e with a c o r r e c t i o n f o r the r e g i o n c l o s e to the tube a x i s . We have l--In

i+

g--5

+5~f(*t),

(41)

where = y . ~ = %. -r, -7, ~, = ~-i- a (~- - 8)* ;

i ~ =-

i = .~,j--~ 9

A s b e f o r e , the r e s i s t a n c e f o r r o u g h t u b e s is t a k e n to b e ~..

In Eq. (33) f o r ~ / v . ,

s---A--

(42)

b e s i d e s the c o e f f i c i e n t fl, we i n t r o d u c e a new c o e f f i c i e n t fll r e l a t e d to ft. We now

have

(43) \ 2 !

j

where ~ , - ] ~ - ~ a ( k- - - 6-) % ~

~ n0

~ ,',,.I'~'

~zt=i r' ~~ ( 1 - - 8 ) ;

8---- 6 = 5 8 ; -7-

and e is the n e g l i g i b l y s m a l l c o r r e c t i o n r e l a t e d to the function f0?) [ s e e (32)]. Eq. (43) can b e u s e d to c o m p u t e a l l flow r e g i m e s c h a r a c t e r i z e d b y d i f f e r e n t v a l u e s of k. F o r the l i m i t i n g t r a n s i t i o n to a l a m i n a r r e g i m e we m u s t r e m e m b e r that the t e r m e in (43) d o e s not a p p e a r if we s e t a = 0 f o r the c a l c u l a t i o n of fi in t h i s c a s e .

278

F o r high Reynolds n u m b e r s (fl = 0) and small [: we have for the self-simulating r e g i m e the relation t

k--

which coincides with N i k u r a d z e ' s f o r m u l a a l m o s t to the values of the coefficients obtained f r o m values of a and 8 In N i k u r a d z e ' s original f o r m u l a the coefficient B 1 was determined experimentally and was an extra constant.

It remains to explain the significance of the asterisk in (38) and subsequent relations. Nikuradze's experiments show that, for high Reynolds numbers, lhe level of the self-simulating regime is well characterized by the n u m b e r k (the m e a n height of roughness protuberances). However with decreasing Reynolds numbers, the curves in Fig. 6 behave differently at the transition to the smoothtube regime. In some e a s e s the transition to the smooth-tube r e g i m e is r a t h e r sudden, while in others th0 transition is delayed (this is especially evident for so-called technically smooth tubes). This difference oa~ be explained by taking into account not only the mean height of roughness p r o t u b e r a n c e s but also their dispersion, [. e., the m e a n - s q u a r e deviation of their heights from the mean value. If the dispersion is small (the half-width of the p r o t u b e r a n c e - h e i g h t distribution curve is small), transition f r o m the roughness r e g i m e to the smooth-tube r e g i m e will be rapid. We obtain this regime if we set ( k - 6~)* = k - 60. Fig. 6 shows r e s u l t s of c o m p u t e r calculations under these conditions for a = 0,39 and 6 = 7.8 (the continuous curves). If the p r o t u b e r a n c e - h e i g h t d i s p e r s i o n is large the transition wi!~ take place gradually since m o r e p r o tuberances will be completely submerged in the laminar sublayer as the thickness of the l a y e r close to the wall increases. For slight roughness the coefficients a, al, and 6 are constant. Their dependence on the roughness becomes apparent for greater roughness in the calculation of the velocity profile. It is an interesting experimental fact, however, that in the formula for the resistance, these coefficients occur in a combination which is independent of the roughness even when it is very marked. In order tobe able to analyze the dependence of a, al, and 6 on roughness, we need more detailed experimental results concerning velocity profiles than are available in the literature. The asterisk in the expression ( k - 60)* indicates the necessity of taking the relation between the protuberance-height dispersion and the thickness of the laminar sublayer into consideration. We must thus take account of the distribution of protuberances with respect to their height. No precise data is available concerning height distributions. We can only assume that in Nikuradze's experiments, in which roughness was created artificially, the roughness was more uniform and the protuberance-height dispersion smaller than in experiments with technically smooth tubes. We can calculate the effective-roughness dispersion from the behavior of the resistance curves when the smooth-tube regime is approached. We assume that heights kp of protuberances,

referred to the radius, are normally distributed: (/~p--k)2

p:=

~

I e

2~2

(44)

where ~ is the standard deviation, Then the mean height ~ of p r o t u b e r a n c e s extending beyond the limiting thickness 6 = 60/r of the lamin a r l a y e r is given by the f o r m u l a ov

(~_~. Hence

279

2~2

e

(45)

The continuous curves in Fig. 6 show the results of calculations for N l k u r a d z e ' s experiments and for experiments with technically smooth tubes. It appears that the artificial roughness in N i k u r a d z e ' s experiments is c h a r a c t e r i z e d by a relatively low standard deviation ( a / k ~ 0.23-0.3) . Technically smooth tubes yield a l a r g e r standard deviation ((r/E ~ 1.5) o These results are in a g r e e m e n t with the qualitative c h a r a c t e r i s t i c s of roughness in N i k u r a d z e ' s experiments and in experiments with technically smooth tubes as shown by photographs of the protuberances [15, 161. It is interesting to note that, for low protuberance-height dispersions, the asymptotic behavior of the r e s i s t a n c e coefficient is determined only by ~: and is independent of ~. F o r large ~ (including or/} = 1.5) the r e s i s t a n c e depends asymptotically not only on k but also o n e , so that the asymptotic value is determined by a l a r g e r value of the effective roughness than that corresponding to uniform roughness. This explains the recent experimental observation of higher r e s i s t a n c e in technically produced tubes than in uniformly roughened tubes. The introduction of roughness dispersion yields a m o r e detailed description of r o u g h n e s s statistics, but does not exhaust the possibilities in this direction. When more experimental results are available it may be n e c e s s a r y to use distribution taws depending on m o r e statistical p a r a m e t e r s , whose experimenta~ determination must be provided for. When more detailed information concerning roughness structure is available, we might be able to improve our theory for all cases. The r e s i s t a n c e in rough tubes was calculated by analogy with smooth tubes from the profile obtained for the flow c l o s e s t to the wall. We now show that this is p e r m i s s i b l e . Consider the equation of motion in a tube du

r

where j== ~]Ir,

Assume, as for the l a y e r c l o s e s t to the wall, that the overall viscosity is made up of: 1) the molecular viscosity u; 2) the turbulent v i s c o s i t y VT1 given by the f o r m u l a %L :=: v . a ( g - - 50) ( I --~), used above for a smooth tube, and 3) the turbulent viscosity %2 := v*~12 depending on the roughness. As above we use the dynamic velocity

for the velocity scale of the turbulent-velocity pulsations, and we use the c h a r a c t e r i s t i c length

We now have and the equation of motion ay

or, in dimensionless form,

If~ ,-~ (~-~) (~ -J)-= ~ ( ~ - ~)* (1.-.,7)1 d~/j~ .... ~ _.~-. dg

In the limiting case of small fl and e we have du/u,

and so we obtain the logarithmic law

280

{

'~

u

__t ln[(g--6} i (]c--5)*] ,r

Calculations taking m o l e c u l a r v i s c o s i t y into account are analogous to those f o r smooth tubes and yield a formula which converts into the corresponding f o r m u l a for the wall layer, just as in the case of smooth tubes. Our analysis of s e v e r a l c a s e s of turbulent flow has yielded evidence that the idea of turbulent viscosity can be used to explain v e r y complex phenomena and is applicable in calculations not hitherto possible. The e s s e n c e of our approach is as follows: the behavior of the turbulent viscosity, which is related to certain other p a r a m e t e r s such as the distance y f r o m the wall or the height of the roughness p r o t u b e r ances, is derived f r o m the asymptotic p r o p e r t i e s of the solution. To obtain the total viscosity we use the superposition principle. This method has been shown to hold for mixed m o l e c u l a r and turbulent exchange as well as for the more complicated e a s e of a rough wall. A basis exists for believing that this approach can be used in investigating s e v e r a l new problems [17]. A s i m i l a r method might be applied to other problems concerning turbulence. The purely hydrodynamic results derived above can be generalized and applied in the improvement of f o r m u l a s used for calculating heat and mass exchange in many different devices, also in the study of c u r r e n t s and exchange p r o c e s s e s in the atmosphere. F o r m u l a (40) gives a b e t t e r description of the ground l a y e r of the a t m o s p h e r e than the so-called l o g a r i t h m i c - p l u s - l i n e a r law, obtained by the superposition of limiting r e g i m e s . The law derived above was derived in a m o r e natural way by applying superposition not to the final r e s u l t s but to the exchange coefficients. The author wishes to thank I. S. Kudryatsev and L.N. Sazykin for p e r f o r m i n g the calculations. LITERATURE

1.

3. 4e 5~ 6. 7. 8~ 9.

10. 11. 12. 13. 14. 15. 16. 17.

CITED

M.D. Millionshchikov, Turbulent Flow in the Boundary L a y e r and in Tubes [in Russian], Nauka, Yioscow (1969), J. Boussinesq, Essai sur l a T h ~ o r i e d e s E a u x C o u r a n t e s , MSm. Pr~s. P a r Div. Say., P a r i s (1877) O. Reynolds, "On the dynamical theory of incompressible viscous fluids," Phil. T r a n s . Roy. S o t . , London (1895). T. Start,on, "The mechanical viscosity of fluids," P r o c . Roy. Soc., A, 85, 366 (19II). J. Nikuradse, "Gesetzm~ssigkeiten d e r turbulenten Strhmung in glatten Rohren," VDJ Forschungsheft, No. 356 (1932). L. Prandfl, "Neuere E r g e b n i s s e der Turbulentzforschung," VDJ, 77, No. 5 (1933). T, yon Karman, Some Aspects of the T h e o r y of Turbulent Motion, P r o e . Internat. Congress AppL Mech., Cambridge (1934). G.I. Taylor, Philos. T r a n s . Roy. Soc. London, A, 215(1915). A.A. F r i e d m a n and L.V. Keller, Differentialgleiehm~gen fur die Turbulente Bewegung e i n e r K o m p r e s siblen Flussigkeit, P r o c . F i r s t Internat. Congr. Appl. Mech., Delft (1924). A.N. Kolmogorov, Dokl. Akad. Nauk SSSR, 30, No. 4 (1939). T. von K a r m a n and L. Howarth, P r o c . Roy. Soc., A, 16..__44(1938). A.M. ObL~khov, Dokl. Akad. Nauk SSSR, 3._22, No. 1 (1941). J. Laufer, NACA Report No. 1174 (1954). L.D. Landau and E.M. Lifshits, The Mechanics of Continuous Media [in Russian], Gostekhteorizdat, Moscow (1953). J. Nikuradse, "Strhmungsgesetze in rauhen Rohren, VDJ Forsehungsheft, No. 361 (1933). B. Bauer and F. Galavics, Mitteiiang aus Betrieb und Forschung des Fernheizkraftwerks des Eid. Teehnische Hochschule in Ztirich, II (1936). M.D. Millionshchikov, "Isotopic turbulence in a field of turbulent viscosity, " Letters, Zh. ]~ks. i Tekhn. Fiz., i_~0, 406 (1969).

281