NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS

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study of the laws governing turbulent flow of fluids in rough tubes, channels, and along ... roughened surfaces the quadratic law of friction is effective as soon ...
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NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS TECHNICAL MEMORANDUM 1292

LAWS OF FLOW IN ROUGH PIPES By J. Nikuradse

Translation of gStromungsgesetze in rauhen Rohren." VDI-Forschungsheft 361. Beilage zu "Forschung auf dem Gebiete des Ingenieurwesens" Ausgabe B Band 4, ~uly/August 1933.

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Washington

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November 1950

NATIONAL ADVISORY COMMITTEE FOR iZERONAUTICS

TECHNICAL I+EMORANDUM 1292

By J. Nikuradse

INTRODUCTION Numerous r e c e n t i n v e s t i g a t i o n s ( r e f e r e n c e s 1, 2, 3, 4, and 5 ) have g r e a t l y i n c r e a s e d our knowledge of t u r b u l e n t flow i n smooth t u b e s , channels, and a l o n g p l a t e s s o t h a t t h e r e a r e now a v a i l a b l e s a t i s f a c t o r y d a t a on v e l o c i t y d i s t r i b u t i o n , on t h e laws c o n t r o l l i n g r e s i s t a n c e , on impact, and on mixing l e n g t h . The d a t a cover t h e t u r b u l e n t b e h a v i o r of t h e s e flow problems. The l o g i c a l development would now i n d i c a t e a study of t h e laws governing t u r b u l e n t flow of f l u i d s i n rough t u b e s , channels, and a l o n g rough p l a n e s u r f a c e s . A s t u d y of t h e s e problems, because of t h e i r f r e q u e n t occurrence i n p r a c t i c e , i s more importamt than t h e s t u d y of flow a l o n g smooth s u r f a c e s and i s a l s o of g r e a t i n t e r e s t a s an e x t e n s i o n of o u r p h y s i c a l knowledge of t u r b u l e n t flow. Turbulent flow of water i n rough t u b e s h a s been s t u d i e d d u r i n g t h e l a s t c e n t u r y by many i n v e s t i g a t o r s of whom t h e most o u t s t a n d i n g w i l l be b r i e f l y mentioned h e r e . H . Darcy ( r e f e r e n c e 6 ) made comprehensive and v e r y c a r e f u l t e s t s on 21 p i p e s of c a s t i r o n , l e a d , wrought i r o n , a s p h a l t - c o v e r e d c a s t i r o n , and g l a s s . With t h e e x c e p t i o n of t h e g l a s s a l l p i p e s were 100 m e t e r s l o n g and 1 . 2 t o 30 c e n t i m e t e r s i n d i a m e t e r . He noted t h a t t h e d i s c h a r g e was dependent upon t h e t y p e of s u r f a c e as w e l l a s upon t h e diameter of t h e p i p e and t h e s l o p e . I f h i s r e s u l t s a r e expressed i n t h e p r e s e n t n o t a t i o n and t h e r e s i s t a n c e f a c t o r X i s c o n s i d e r e d dependent upon t h e Reynolds number Re, t h e n it i s found t h a t a c c o r d i n g t o h i s measurements

k A, f o r a g i v e n r e l a t i v e roughness r'

i s t h e average d e p t h ,d i s t h e r a d i u s of t h e p i p e ; Reynolds number Re = u-v

v a r i e s o n l y s l i g h t l y w i t h t h e Reynolds number ( k of roughness and

r

i n which ti i s t h e average v e l o c i t y , d i s t h e p i p e d i a m e t e r , and v i s t h e k i n e m a t i c v i s c o s i t y ) . The f r i c t i o n f a c t o r d e c r e a s e s w i t h a n i n c r e a s i n g Reynolds number and t h e r a t e of d e c r e a s e becomes slower f o r g r e a t e r r e l a t i v e roughness. For c e r t a i n roughnesses h i s d a t a i n d i c a t e t h a t t h e f r i c t i o n f a c t o r h i s independent o f t h e Reynolds number.

* " ~ t r $ m u n ~ s ~ e s e t izne rauhen Rohren. " VDI-Forschungsheft 361. B e i l a g e zu "Forschung auf dem Gebiete d e s Ingenieurwesens" Ausgabe B Band 4, ~ u l ~ / ~ u @ 1933. s t

For a c o n s t a n t Reynolds number, h i n c r e a s e s markedly f o r an i n c r e a s i n g r e l a t i v e roughness. H. Bazin ( r e f e r e n c e 7 ) , a f o l l o w e r of Darcy, c a r r i e d on t h e work and d e r i v e d from h i s own and D a r c y ' s t e s t d a t a an e m p i r i c a l formula i n which t h e d i s c h a r g e i s dependent upon t h e slope and diameter of t h e p i p e . T h i s formula was used i n p r a c t i c e u n t i l r e c e n t times.

R . v. Mises ( r e f e r e n c e 8) i n 1914 d i d a v e r y v a l u a b l e p i e c e of work, t r e a t i n g a l l of t h e then-known t e s t r e s u l t s from t h e viewpoint of s i m i l a r i t y . He obtained, c h i e f l y from t h e o b s e r v a t i o n s of Darcy and Bazin w i t h c i r c u l a r p i p e s , t h e following formula f o r t h e f r i c t i o n f a c t o r h i n terms of t h e Reynolds number and t h e r e l a t i v e roughness:

This formula f o r v a l u e s of Reynolds numbers near t h e c r i t i c a l , t h a t i s , f o r small v a l u e s , assumes t h e following form:

The term " r e l a t i v e roughness" f o r t h e r a t i o

-kr

i n which

k

is the

a b s o l u t e roughness was f i r s t used by v . Mises. Proof of s i m i l a r i t y f o r flow through rough p i p e s was f u r n i s h e d i n 1911 by T. E . S t a n t o n ( r e f e r e n c e 9). He s t u d i e d p i p e s of two diameters i n t o whose i n n e r s u r f a c e s two i n t e r s e c t i n g t h r e a d s had been c u t . I n o r d e r t o o b t a i n g e o m e t r i c a l l y s i m i l a r depths of roughness he v a r i e d t h e p i t c h and depth of t h e t h r e a d s i n d i r e c t p r o p o r t i o n t o t h e diameter of t h e p i p e . He compared f o r t h e same p i p e t h e l a r g e s t and s m a l l e s t Reynolds number o b t a i n a b l e w i t h h i s a p p a r a t u s and t h e n t h e v e l o c i t y d i s t r i b u t i o n s f o r v a r i o u s p i p e d i a m e t e r s . P e r f e c t agreement i n t h e dimensionless v e l o c i t y p r o f i l e s was found f o r t h e f i r s t case, but a small discrepancy appeared i n t h e immediate v i c i n i t y of t h e w a l l s f o r t h e second case. S t a n t o n t h e r e b y proved t h e s i m i l a r i t y of flow through rough t u b e s . More r e c e n t l y L . S c h i l l e r ( r e f e r e n c e 1 0 ) made f u r t h e r o b s e r v a t i o n s r e g a r d i n g t h e v a r i a t i o n of t h e f r i c t i o n f a c t o r X w i t h t h e Reynolds number and w i t h t h e type of s u r f a c e . H i s t e s t s were made w i t h drawn b r a s s p i p e s . He o b t a i n e d rough s u r f a c e s i n t h e same manner a s S t a n t o n by u s i n g t h r e a d s of v a r i o u s depths and i n c l i n a t i o n s on t h e i n s i d e of t h e t e s t p i p e s . The pipe diameters ranged from 8 t o 21 m i l l i m e t e r s . H i s o b s e r v a t i o n s i n d i c a t e t h a t t h e c r i t i c a l Reynolds number i s independent of t h e type of w a l l s u r f a c e . He f u r t h e r determined t h a t f o r g r e a t l y roughened s u r f a c e s t h e q u a d r a t i c law of f r i c t i o n i s e f f e c t i v e a s soon

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NACA TM 1292

a s t u r b u l e n c e s e t s i n . I n t h e c a s e of l e s s s e v e r e l y roughened s u r f a c e s he observed a slow i n c r e a s e of t h e f r i c t i o n f a c t o r w i t h t h e Reynolds number. S c h i l l e r was n o t a b l e t o determine whether t h i s i n c r e a s e goes over i n t o t h e q u a d r a t i c law of f r i c t i o n f o r high Reynolds numbers, s i n c e t h e ~ 8 t t i n g e nt e s t a p p a r a t u s a t t h a t time was l i m i t e d t o about Re = 103. His r e s u l t s a l s o i n d i c a t e t h a t f o r a f i x e d value of Reynolds number t h e f r i c t i o n f a c t o r k i n c r e a s e s w i t h a n i n c r e a s i n g roughness. L . Hopf ( r e f e r e n c e 11) made some t e s t s a t about t h e same time a s

S c h i l l e r t o determine t h e f u n c t i o n

X

( -:) .

= f Re

He performed system-

a t i c experiments on r e c t a n g u l a r channels of v a r i o u s d e p t h s w i t h d i f f e r e n t roughnesses ( w i r e mesh, z i n c p l a t e s having saw-toothed type s u r f a c e s , and two t y p e s of corrugated p l a t e ) . A r e c t a n g u l a r s e c t i o n was s e l e c t e d i n o r d e r t o determine t h e e f f e c t of t h e h y d r a u l i c r a d i u s (hydra.ulic r a d i u s r ' = a r e a of s e c t i o n d i v i d e d by wetted p e r i m e t e r ) on t h e v a r i a t i o n i n depth of s e c t i o n f o r a c o n s t a n t type of w a l l s u r f a c e . A t H o p f f s suggestion t h e s e t e s t s were extended by K. F r o m ( r e f e r e n c e 1 2 ) . On t h e b a s i s of h i s own and Fromm's t e s t s and of t h e o t h e r a v a i l a b l e t e s t d a t a , Hopf concluded t h a t t h e r e a r e two fundamenta.1 t y p e s of roughness involved i n t u r b u l e n t flow i n rough p i p e s . These two t y p e s , which he terms s u r f a c e roughness and s u r f a c e c o r r u g a t i o n , f o l l o w d i f f e r e n t laws of s i m i l a r i t y . A s u r f a c e roughness, according t o Hopf, i s c h a r a c t e r i z e d by t h e f a c t t h a t t h e l o s s of head i s independent of t h e Reynolds number and dependent only upon t h e type of wa.11 s u r f a c e i n accordance w i t h t h e q u a d r a t i c law of f r i c t i o n . He c o n s i d e r s s u r f a c e c o r r u g a t i o n t o e x i s t when t h e f r i c t i o n f a c t o r a s w e l l a s t h e Reynolds number depends upon t h e type of w a l l s u r f a c e i n such a manner t h a t , i f p l o t t e d l o g a r i t h m i c a l l y , t h e curves f o r X a s a f u n c t i o n of t h e Reynolds number f o r v a r i o u s w a l l s u r f a c e s l i e p a r a l l e l t o a smooth curve. I f a i s t h e average depth of roughness and b i s t h e average d i s t a n c e between two p r o j e c t i o n s from a the s u r f a c e , t h e n s u r f a c e c o r r u g a t i o n e x i s t s f o r small v a l u e s of b a and s u r f a c e roughness e x i s t s f o r l a r g e v a l u e s of b' A summary of t h e t e s t s of Hopf, F r o m , Darcy, Bazin and o t h e r s i s given i n f i g u r e s 1 and 2, t h e f i r s t i l l u s t r a t i n g s u r f a c e roughness and t h e second s u r f a c e c o r r u g a t i o n . Hopf d e r i v e d f o r t h e f r i c t i o n f a c t o r k w i t h i n t h e range of s u r f a c e roughness t h e f o l l o w i n g e m p i r i c a l formula:

i n which

r'

i s t h e h y d r a u l i c r a d i u s of t h e channel \

(r' =

2F F;

F

=

area

of c r o s s - s e c t i o n ; U = wetted p e r i m e t e r ) . This formula a p p l i e s t o i r o n p i p e s , cement, checkered p l a t e s and wire mesh. I n t h e case of s u r f a c e

corrugation he g i v e s t h e formula

i n which Lo i s t h e f r i c t i o n f a c t o r f o r a smooth surface and 6 i s a p r o p o r t i o n a l i t y f a c t o r which has a value between 1 . 5 and 2 f o r wooden p i p e s and between 1.2 and 1.5 f o r asphalted i r o n p i p e s . The v a r i a t i o n of t h e v e l o c i t y d i s t r i b u t i o n with t h e type of wall surface i s a l s o important, a s w e l l a s t h e law of r e s i s t a n c e . Observat i o n s on t h i s problem were made by Darcy, Bazin, and Stanton ( r e f e r e n c e 9 ) . The necessary d a t a , however, on temperature of t h e f l u i d , type of wall surface, and l o s s of head a r e lacking. I n more r e c e n t times such observ a t i o n s have been made by F r i t s c h ( r e f e r e n c e 13) a t t h e suggestion of Von k t & using , t h e same type of apparatus a s t h a t of Hopf and Fromm. The channel had a length of 200 centimeters, width of 15 centimeters and depth varying from 1 . 0 t o 3.5 centimeters. A two-dimensional condit i o n of flow e x i s t e d , t h e r e f o r e , along t h e s h o r t a x i s of symmetry. He i n v e s t i g a t e d t h e v e l o c i t y d i s t r i b u t i o n f o r t h e following types of w a l l s u r f ace : 1. smooth 2 . corrugated (wavy)

3. rough I. ( f l o o r s , g l a s s p l a t e s with l i g h t c o r r u g a t i o n s )

4. rough 11. ( r i b b e d g l a s s )

5 . toothed (termed saw-toothed by From) F r i t s c h found t h a t f o r t h e same depth of channel t h e v e l o c i t y d i s t r i bution (except f o r a. l a y e r adjacent t o t h e w a l l s ) i s congruent f o r a l l of t h e s e types of s u r f a c e s i f t h e l o s s of head i s t h e same. T e s t s i n a channel with extremely coarse roughness were made by Treer, ( r e f e r e n c e s 1 4 and 15) i n which he observed t h e r e s i s t a n c e as well a s t h e v e l o c i t y d i s t r i b u t i o n . mom t h e s e t e s t s and from those of o t h e r i n v e s t i g a t o r s , he found t h a t t h e v e l o c i t y d i s t r i b u t i o n depends only upon t h e shearing s t r e s s , whether t h i s i s due t o v a r i a t i o n i n roughness o r i n t h e Reynolds number. The numerous and i n p a r t very painstaking t e s t s which a r e a v a i l a b l e at t h e p r e s e n t time cover many types of roughness, but a l l l i e w i t h i n a

very small range of Reynolds number. The purpose of t h e p r e s e n t invest i g a t i o n i s t o study t h e e f f e c t of coarse and f i n e roughnesses f o r a l l Reynolds numbers and t o determine t h e laws which a r e i n d i c a t e d . It was, t h e r e f o r e , necessary t o consider a d e f i n i t e r e l a t i v e roughness

r k

for

a wide range of Reynolds number and t o determine whether f o r t h i s conr s t a n t - t h a t i s , f o r geometrical s i m i l a r i t y , t h e value h = f ( ~ e ) i s k ' t h e same curve f o r pipes of d i f f e r e n t diameter. There was a l s o t h e r question whether f o r t h e same - t h e v e l o c i t y d i s t r i b u t i o n s a r e s i m i l a r k and vary with t h e Reynolds number, and whether f o r a varying f: t h e k v e l o c i t y d i s t r i b u t i o n s a r e s i m i l a r a s s t a t e d by V. K&~I&II. I wish here t o express my s i n c e r e g r a t i t u d e t o my immedia.te s u p e r i o r , Professor Dr. L. P r a n d t l , who has a t a l l times aided me by h i s valuable advice.

I. EXPERIMENT 1. Description of Test Apparatus

The apparatus shown i n f i g u r e 3 was used i n making t h e t e s t s . The same apparatus was employed i n t h e i n v e s t i g a t i o n of v e l o c i t i e s f o r t u r bulent flow i n smooth pipes. The d e t a i l e d d e s c r i p t i o n of t h e apparatus and measuring devices has been presented i n Forschungsheft 356 of t h e VDI. Only a b r i e f review w i l l be given here. Water was pumped by means of a c e n t r i f u g a l pump kp, driven by an e l e c t r i c motor em, from t h e supply canal vk, i n t o t h e water tank wk, t h e n through t h e t e s t pipe vr and i n t o t h e supply canal vk. This arrangement was employed i n t h e i n v e s t i g a t i o n with medium and l a r g e values of Reynolds number. An overflow was used i n obtaining observations f o r small values of Reynolds number. The water flowed through t h e supply l i n e 2 2 , i n t o t h e open water tank wk, and a v e r t i c a l pipe s t r , connected with t h e tank, conducted t h e overflowing water over t h e t r a p and down through t h e overflow pipe f r . The flow i n t h e t e s t pipe could be t h r o t t l e d t o any d e s i r e d degree. A constant high p r e s s u r e i n t h e water t a n k wk was r e q u i r e d i n order t o a t t a i n t h e highest values of Reynolds number. Observations were made on: 1. l o s s of head 2 . v e l o c i t y d i s t r i b u t i o n i n t h e stream immediately a f t e r l e a v i n g

t h e t e s t pipe

3. d i s c h a r g e q u a n t i t y

4. temperature of t h e water Three hooked t u b e s with l a t e r a l a p e r t u r e s were used t o measure t h e l o s s of head. These t u b e s a r e d e s c r i b e d i n d e t a i l i n s e c t i o n I , 3 . The v e l o c i t y d i s t r i b u t i o n was determined by means of a p i t o t t u b e w i t h 0 . 2 m i l l i m e t e r i n s i d e diameter, mounted i n t h e velocity-measuring d e v i c e gm, and a d j u s t a b l e b o t h h o r i z o n t a l l y and v e r t i c a l l y . The d i s charge f o r Reynolds numbers up t o 3 X 105 was measured i n a t a n k mb on t h e b a s i s of depth and t i m e . Larger d i s c h a r g e s were computed by i n t e g r a t i n g t h e v e l o c i t y d i s t r i b u t i o n curve. Temperature r e a d i n g s were t a k e n a t t h e o u t l e t of t h e v e l o c i t y - m e a s u r i n g d e v i c e gm. The t e s t p i p e s were drawn b r a s s p i p e s of c i r c u l a r s e c t i o n whose dimensions a r e g i v e n i n t a b l e 1. The d i a m e t e r s of t h e p i p e were determined from t h e weight of t h e water which could be c o n t a i n e d i n t h e p i p e w i t h c l o s e d ends and from t h e l e n g t h of t h e p i p e .

2 . F a b r i c a t i o n and Determination of Roughness S i m i l i t u d e r e q u i r e s t h a t i f mechanically s i m i l a r flow i s t o t a k e p l a c e i n two p i p e s t h e y must have a g e o m e t r i c a l l y s i m i l a r form and must have s i m i l a r w a l l s u r f a c e s . The f i r s t requirement i s met by t h e u s e of a c i r c u l a r s e c t i o n . The second requirement i s s a t i s f i e d by m a i n t a i n i n g a c o n s t a n t r a t i o of t h e p i p e r a d i u s r t o t h e d e p t h k of p r o j e c t i o n s . It was e s s e n t i a l , t h e r e f o r e , t h a t t h e m a t e r i a l s producing t h e roughness should be s i m i l a r . P r o f e s s o r D. Thoma's precedent of u s i n g sand f o r t h i s purpose was adopted. G r a i n s of uniform s i z e a r e r e q u i r e d t o produce uniform r o u ~ h n e s s throughout t h e p i p e . Ordinary b u i l d i n g sand was s i f t e d . Ln o r d e r t o o b t a i n a n average g r a i n s i z e of 0.8 m i l l i m e t e r diameter, f o r example, s i e v e s were employed having openings of 0.82- and 0 . 7 8 - m i l l i m e t e r diameter. A Z e i s s t h i c k n e s s gage was used t o o b t a i n t h e a c t u a l average g r a i n s i z e by t a k i n g a c t u a l measurements of t h e diameter of s e v e r a l hundred g r a i n s . These sand g r a i n s were spread on a. f l a t p l a t e . The d i a m e t e r s of t h e i n d i v i d u a l g r a i n s were t h e n measured w i t h t h e Z e i s s t h i c k n e s s gage (having a n accuracy of 0.001 mm) by s l i d i n g t h e p l a t e . For t h e c a s e c i t e d t h e a . r i t h m e t i c a 1 average was found t o be 0 . 8 m i l l i m e t e r . A micro-photograph of uniform s i z e (0.8-mm d i a m e t e r ) g r a i n s a s reproduced i n f i g u r e 4 f u r n i s h e s some i n f o r m a t i o n r e g a r d i n g g r a i n form. P r e l i m i n a r y t e s t s had i n d i c a t e d t h e manner i n which t h e p i p e s could be roughened w i t h sand. The p i p e p l a c e d i n a v e r t i c a l p o s i t i o n and with t h e lower end c l o s e d was f i l l e d w i t h a very t h i n Japanese l a c q u e r and t h e n emptied. A f t e r about 30 minutes, which i s a p e r i o d s u f f i c i e n t f o r t h e d r y i n g of t h e l a c q u e r on t h e p i p e s u r f a c e t o t h e "tacky" s t a t e ,

t h e pipe wa.s f i l l e d with sand of a. c e r t a i n s i z e . The sand was t h e n allowed t o flow out a t t h e bottom. The p r e l i m i n a r y t e s t s showed t h a t t h e d r y i n g which now f o l l o w s i s of grea,t importance f o r d u r a b i l i t y . A drying p e r i o d of two t o t h r e e weeks i s r e q u i r e d , depending upon t h e amount of moisture i n t h e a i r . A uniform d r a f t i n t h e p i p e , due t o an e l e c t r i c bulb placed a t t h e lower end, helped t o o b t a i n even d r y i n g . A f t e r t h i s drying, t h e pipe was r e f i l l e d w i t h l a c q u e r and a g a i n emptied, i n o r d e r t o o b t a i n a b e t t e r adherence of t h e g r a i n s . There followed a n o t h e r d r y i n g p e r i o d of t h r e e t o f o u r weeks. A t each end of t h e p i p e , a l e n g t h of about 10 c e n t i m e t e r s was c u t o f f i n o r d e r t o prevent any p o s s i b l e decrease i n t h e end s e c t i o n s . A f t e r t h e trea.tment j u s t described t h e p i p e s were ready t o be measured. One of t h e c o n d i t i o n s c i t e d above i n d i c a t e s t h a t d i f f e r e n t g r a i n r s i z e s must be used f o r p i p e s of d i f f e r e n t diameter i f t h e r a t i o

E'

which i s t h e gage f o r s i m i l a r i t y of w a l l s u r f a c e , i s t o remain c o n s t a n t . Geometrical s i m i l a r i t y of t h e w a l l s u r f a c e r e q u i r e s t h a t t h e form of t h e i n d i v i d u a l g r a i n s s h a l l be unchanged and a l s o t h a t t h e p r o j e c t i o n of t h e roughening, which has hydrodynamical e f f e c t s , s h a l l remain cons t a n t . Figure 4 shows t h a t voids e x i s t between t h e gra.ins. The hydrodynamically e f f e c t i v e amount of p r o j e c t i o n k i s e q u a l t o t h e g r a i n s i z e . I n o r d e r t o determine whether t h e p r e v i o u s l y observed diameter of g r a i n s i s a c t u a l l y e f f e c t i v e , a f l a t p l a t e was coated w i t h t h i n Japanese l a c q u e r ( t h e n e c e s s a r y degree of t h i n n e s s was determined by p r e l i m i n a r y t e s t s ) and roughened i n accordance w i t h t h e d e s c r i b e d procedure. The p r o j e c t i o n of t h e g r a i n s above t h e s u r f a c e was measured i n t h e manner a l r e a d y d e s c r i b e d and it was found t h a t , f o r a d e f i n i t e degree of t h i n n e s s of t h e l a c q u e r , t h i s average p r o j e c t i o n agreed w i t h t h e o r i g i n a l measurements of the g r a i n s .

3. Measurement of S t a t i c P r e s s u r e Gradient Measurement of s t a t i c p r e s s u r e g r a d i e n t d u r i n g flow i n smooth p i p e s i s u s u a l l y made by piezometer h o l e s i n t h e w a l l s of t h e p i p e . Marked e r r o r s r e s u l t , however, i f l o s s of head i n rough p i p e s i s determined i n t h i s sane manner. These a r e due t o t h e f a c t t h a t t h e v o r t i c e s which r e s u l t from flow around t h e p r o j e c t i o n s produce p r e s s u r e o r s u c t i o n , depending on t h e p o s i t i o n of t h e a p e r t u r e . For t h i s reason t h e hooked tube was adopted f o r observing t h e s t a t l c p r e s s u r e g r a d i e n t . T h i s t u b e had a r e c t a n g u l a r bend a s shown i n f i g u r e 5 and was mounted i n t h e t e s t p i p e s o t h a t t h e f r e e l e g was p a r a l l e l t o t h e d i r e c t i o n of flow. L a t e r a l openings o n l y were bored i n t h i s f r e e l e g . The o u t s i d e diamet e r d of t h e t u b e was 2 m i l l i m e t e r s . Other f e a t u r e s of t h e tube a r e i n agreement w i t h t h e s p e c i f i c a t i o n s ( r e f e r e n c e 16) s e t up f o r t h e P r a n d t l p i t o t s t a t i c t u b e ( ~ t a u r o h r ) . The f r e e l e g was placed a.t a d i s t a n c e from t h e w a l l e q u a l t o 1 / 2 t h e r a d i u s of t h e t e s t p i p e . The

connecting l e g was bent a t an angle of about 60' i n t h e plane of t h e f r e e l e g i n order t h a t t h e p o s i t i o n of t h e f r e e l e g might always be indicated. The bent tube was fastened i n t h e t e s t pipe by means of a s t u f f i n g box. Variation of t h e pressure readings i n a hooked tube with v a r i a t i o n s i n t h e p o s i t i o n of t h e tube r e l a t i v e t o t h e d i r e c t i o n of flow i s shown i n f i g u r e 6'. This f i g u r e i n d i c a t e s t h a t c o r r e c t readings a r e obtained only i f t h e d i r e c t i o n of t h e f r e e l e g d e v i a t e s not more than 7.5' from t h e d i r e c t i o n of flow. The introduction of t h e hooked tube i n t o t h e t e s t pipe r e s u l t s i n an increase of pressure drop due t o t h e r e s i s t a n c e t o t h e tube. The r e s i s t a n c e of t h e two hooked tubes used i n measuring must be deducted from t h e observed pressure drop pl p2. The r e s i s t ance of t h e tube must t h e r e f o r e be known. This value was found by measuring t h e pressure drop h i n a smooth pipe i n terms of t h e d i s charge a t a constant temperature, f i r s t by using wall piezometer o r i f i c e s and t h e n by measuring t h e pressure drop h + a i n terms of t h e discharge a t t h e same temperature by means of a hooked tube. The increment a f o r equal discharges i s t h e r e s i s t a n c e of t h e hooked tubes. The c o r r e c t i o n curve f o r t h i s r e s i s t a n c e i s given i n f i g u r e 7.

-

It should be noted t h a t changes i n d i r e c t i o n of t h e tube r e s u l t both i n an e r r o r i n t h e pressure reading and i n an increase i n t h e r e s i s t a n c e due t o t h e tube. I f t h e corrected pressure drop pl - P2

is

divided by t h e observation l e n g t h 2 , ( d i s t a n c e between t h e holes i n t h e s i d e of t h e hooked t u b e s ) , t h e r e i s obtained t h e s t a t i c pressure gradient,

4. Preliminary Tests A mixture of sieved sand and white lacquer i n a d e f i n i t e proportion was used t o f i l l a pipe closed a t t h e bottom, i n t h e manner of Professor D. Thoma ( r e f e r e n c e 17). The mixture was then allowed t o flow out a t t h e bottom. After a drying period of about two t o t h r e e weeks, prelimi n a r y t e s t s answered t h e question whether t h e hydrodynamically e f f e c t i v e p r o j e c t i o n of t h e roughening remained constant. The pressure drop was measured a t hourly i n t e r v a l s f o r a given Reynolds number f o r which t h e

I

This f i g u r e i s taken from t h e work of H. Kumbruch, c i t e d herein a s reference 16.

average v e l o c i t y u was about 20 m e t e r s p e r second. It was observed t h a t w i t h i n a few days t h e p r e s s u r e s l o p e developed a pronounced i n c r e a s e . A marked washing o f f of t h e l a c q u e r was i n d i c a t e d a t t h e same t i m e by d e p o s i t s on t h e bottom of t h e supply channel. Another o b j e c t i o n a b l e f e a t u r e was t h e p a r t i a l washing o u t of t h e sand. The i n c r e a s e i n t h e p r e s s u r e g r a d i e n t i s accounted f o r by t h e i n c r e a s e i n p r o j e c t i o n of roughness due t o t h e washing o f f of t h e l a c q u e r . Theref o r e , t h e method of f a s t e n i n g t h e sand had t o be changed i n o r d e r t o i n s u r e t h e r e q u i r e d c o n d i t i o n of t h e s u r f a c e d u r i n g t h e t e s t p r o c e d u r e . The p r o j e c t i o n k of t h e roughness had t o remain c o n s t a n t d u r i n g t h e t e s t s and t h e d i s t r i b u t i o n o f t h e sand gra.ins on t h e w a l l s u r f a c e s had t o remain unchanged. Adhesion between sand g r a i n s was prevented by u s i n g a v e r y t h i n l a c q u e r . T h i s l a c q u e r formed a d i r e c t c o a t i n g on t h e w a l l and a l s o a c o v e r i n g on t h e g r a i n s no t h i c k e r t h a n t h e p e n e t r a t i o n of t h e s e g r a i n s i n t o t h e l a c q u e r c o a t i n g of t h e w a l l . The o r i g i n a l form and s i z e of t h e g r a i n s remained unchanged. A d e t e r m i n i n g f a c t o r i n t h i s problem was t h e degree of t h i c k n e s s of t h e l a c q u e r which was v a r i e d by t h e a d d i t i o n of t u r p e n t i n e u n t i l t h e o r i g i n a l g r a i n s i z e remained unchanged. T e s t s made w i t h p i p e s w i t h o u t l a c q u e r r e c o a t i n g showed that t h e sand would wash o u t . The r e c o a . t i n g w i t h l a c q u e r was, t h e r e f o r e , adopted. I f o n l y a s h o r t p e r i o d of d r y i n g was used f o r b o t h c o a t s , t h e l a c q u e r was washed o f f . I f t h e f i r s t d r y i n g was s h o r t and t h e second long, t h e n a l l of t h e l a c q u e r was a l s o washed o f f . I f t h e f i r s t d r y i n g p e r i o d were l o n g and t h e second s h o r t , t h e r e would a l s o be some l o s s of sand. A c o n s t a n t c o n d i t i o n of roughness could be o b t a i n e d o n l y when each l a c q u e r c o a t i n g was d r i e d from t h r e e t o f o u r weeks. The a c c u r a c y of o b s e r v a t i o n s made w i t h t h e hooked t u b e was checked by c o n n e c t i n g t h e t u b e t h r o u g h a manometer t o a w a l l piezometer o r i f i c e a t t h e same sect i o n of t h e pipe. Both c o n n e c t i o n s should show t h e same p r e s s u r e i n a smooth p i p e , t h a t i s , t h e manometer r e a d i n g must be z e r o . Hooked t u b e s checked i n t h i s manner were used f o r t a k i n g p r i n c i p a l o b s e r v a t i o n s . F i n a l l y , a d e t e r m i n a t i o n of t h e approach l e n g t h

4

was made. d V e l o c i t y d i s t r i b u t i o n s were observed f o r t h e l a r g e s t r e l a t i v e roughness k -- 1 ratio The v e l o c i t y a t v a r i o u s d i s t a n c e s y from t h e s u r f a c e r 15 was determined f o r Reynolds numbers of Re = 20 x lo3, 70 x lo3, and

.

150 x 1 03 a t v a r i o u s d i s t a n c e s from t h e e n t r a n c e

-.d X

T h i s was e f f e c t e d

by c u t t i n g o f f p o r t i o n s of t h e t e s t p i p e . T e s t s show t h a t changes i n t h e approach l e n g t h have small e f f e c t on t h e Reynolds number. The approach l e n g t h i s somewhat s h o r t e r t h a n t h a t f o r smooth p i p e s , (fig. 8).

The approach l e n g t h

X

d

= 50

%

40

was used a s f o r smooth p i p e s .

11. EVALUATION OF TEST RESULTS

1. Law of R e s i s t a n c e

The r e s i s t a n c e f a c t o r t h e formula:

i n which e t e r , and

* dx

X f o r f l o w i n t h e p i p e s i s e x p r e s s e d by

i s t h e p r e s s u r e drop p e r u n i t of l e n g t h , d i s t h e diamii2 = p F , t h e dynamic p r e s s u r e of t h e average flow

v e l o c i t y E and p i s t h e d e n s i t y . An e x t e n s i v e t e s t program w i t h a range of Re = 600 t o Re = lo6 f o r t h e Reynolds number was c a r r i e d o u t , m d t h e r e l a t i o n s h i p of t h e r e s i s t a n c e f a c t o r t o t h e Reynolds number was s t u d ' e d f o r p i p e s of v a r i o u s roughnesses. Six d i f f e r e n t d e g r e e s of

k determined

r e l a t i v e roughness were used, w i t h t h e r e l a t i v e roughness

by t h e r a t i o of t h e average p r o j e c t i o n

k

t o the radius

r

r

of t h e p i p e .

I n e v a l u a t i n g t h e t e s t d a t a it seemed a d v i s a b l e t o use i n s t e a d of k r t h e r e l a t i v e roughness - i t s r e c i p r o c a l F i g u r e 9 shows t o a r' k' l o g a r i t h m i c s c a l e t h e r e l a t i o n of t h e r e s i s t a n c e f a c t o r t o t h e Reynolds number f o r t h e r e c i p r o c a l v a l u e s

1-

- o f t h e s i x r e l a t i v e roughnesses k

t e s t e d and f o r a smooth p i p e ( s e e t a b l e s 2 t o 7 ) . The bottom curve i s f o r t h e smooth p i p e . If t h e curve f o r X = f ( ~ e ) i s s t u d i e d f o r a given r e l a t i v e roughness, t h e n it must be c o n s i d e r e d i n t h r e e p o r t i o n s o r ranges. Within t h e f i r s t range, t h a t of low Reynolds numbers, t h e roughr n e s s had no e f f e c t on t h e r e s i s t a n c e , and f o r all v a l u e s of - t h e

k

curve X = f ( ~ e ) c o i n c i d e s w i t h t h e curve f o r t h e smooth p i p e . T h i s range i n c l u d e s a l l laminar flow and some t u r b u l e n t flow. The p o r t i o n of t u r b u l e n t flow included i n c r e a s e s a s t h e r e l a t i v e roughness d e c r e a s e s . A s l o n g a s laminar flow e x i s t s , t h e r e s i s t a n c e f a c t o r may be e x p r e s s e d as :

T h i s i s r e p r e s e n t e d i n f i g u r e 9 by a s t r a i g h t l i n e of s l o p e 1:l. Within t h e f i r s t p o r t i o n of t u r b u l e n t flow i n smooth p i p e s f o r a Reynolds numb e r up t o about

Re =

lo5

t h e B l a s i u s R e s i s t a n c e Law ( r e f e r e n c e

18)

hol'

TM 1292

NACA

T h i s i s r e p r e s e n t e d i n t h e f i g u r e by a s t r a i g h t l i n e of s l o p e 1:4. The c r i t i c a l Reynolds number f o r a l l d e g r e e s of r e l a t i v e rollghne s s o c c u r s a t about t h e same p o s i t i o n a s f o r t h e smooth p i p e , t h a t i s , between 2160 and 2500. Within t h e second range, which w i l l be termed t h e t r a n s i t i o n range, t h e i n f l u e n c e of t h e roughness becomes n o t i c e a b l e i n an i n c r e a s i n g degree; t h e r e s i s t a n c e f a c t o r X i n c r e a s e s w i t h an i n c r e a s i n g Reynolds number. T h i s t r a n s i t i o n range i s p a r t i c u l a r l y c h a r a c t e r i z e d by t h e f a c t t h a t t h e r e s i s t a n c e f a c t o r depends upon t h e Reynolds number a s w e l l a s upon t h e r e l a t i v e roughness. Within t h e t h i r d range t h e r e s i s t a n c e f a c t o r i s independent of t h e Reynolds number and t h e c u r v e s X = f ( R e ) become p a r a l l e l t o t h e h o r i z o n t a l axis. T h i s i s t h e range w i t h i n which t h e q u a d r a t i c l a w of resistance obtains. The t h r e e r a n g e s of t h e c u r v e s X = f ( ~ e )may be p h y s i c a l l y i n t e r p r e t e d a s f o l l o w s . I n t h e f i r s t range t h e t h i c k n e s s 6 of t h e l a m i n a r boundary l a y e r , which i s known t o d e c r e a s e w i t h an i n c r e a s i n g Reynolds number, i s s t i l l l a r g e r t h a n t h e average p r o j e c t i o n ( 6 > k ) . T h e r e f o r e energy l o s s e s due t o roughness a r e no g r e a t e r t h a n t h o s e f o r t h e smooth plpe

.

I n t h e second range t h e t h i c k n e s s of t h e boundary l a y e r i s of t h e same magnitude as t h e average p r o j e c t i o n ( 6 Z k ) . I n d i v i d u a l p r o j e c t i o n s extend through t h e boundary l a y e r and cause v o r t i c e s which produce an a d d i t i o n a l l o s s of energy. As t h e Reynolds number i n c r e a s e s , an i n c r e a s i n g number of p r o j e c t i o n s p a s s through t h e l a m i n a r boundary l a y e r because of t h e r e d u c t i o n i n i t s t h i c k n e s s . The a d d i t i o n a l e n e r g y l o s s t h a n becomes g r e a t e r a s t h e Reynolds number i n c r e a s e s . T h i s i s e x p r e s s e d by t h e r i s e of t h e c u r v e s h = f ( ~ e )w i t h i n t h i s r a n g e . F i n a l l y , i n t h e t h i r d range t h e t h i c k n e s s of t h e boundary l a y e r h a s become s o s m a l l t h a t a l l p r o j e c t i o n s e x t e n d t h r o u g h i t . The e n e r g y l o s s due t o t h e v o r t i c e s h a s now a t t a i n e d a c o n s t a n t v a l u e and a n i n c r e a s e i n t h e Reynolds number no l o n g e r i n c r e a s e s t h e r e s i s t a n c e . The r e l a t i o n s h i p s w i t h i n t h e t h i r d range a r e v e r y s i m p l e . Here t h e r e s i s t a n c e f a c t o r i s independent of t h e Reynolds number and depends only upon t h e r e l a t i v e roughness. T h i s dependency may be e x p r e s s e d by t h e formula

In o r d e r t o check t h i s formula e x p e r i m e n t a l l y t h e value

fi

was p l o t t e d

i n f i g u r e 10 against l o g

and it was found t h a t through t h e s e p o i n t s k t h e r e could be passed a l i n e

The e n t i r e f i e l d of Reynolds numbers i n v e s t i g a t e d was covered by p l o t -

r v*k a g a i n s t l o g 7. This term i s p a r t i c u l a r l y 6 k s u i t a b l e dimensionally s i n c e it h a s c h a r a c t e r i s t i c v a l u e s f o r c o n d i t i o n s

t i n g t h e term

1 -

2 log -

along t h e s u r f a c e .

The more convenient value

be used i n s t e a d of

v*k log v '

tion.

l o g Re

6-

log

might k a s may be seen from t h e f o l l o w i n g c o n s i d e r a -

From t h e formula f o r t h e r e s i s t a n c e f a c t o r

and t h e f r i c t i o n t h e r e l a t i o n s h i p between t h e s h e a r i n g s t r e s s T~ f a c t o r X may be obtained. I n accordance w i t h t h e requirements of e q u i l i b r i u n : f o r a f l u i d c y l i n d e r of l e n g t h dx and r a d i u s r,

o r from e q u a t i o n (1)

i n which

and

v,

r o =E is

the f r i c t i o n velocity.

There r e s u l t s

log(Re fi) - l o g Tk

log From equation

V*k

= const

=

y)

10~(~.66

l o g ( ~ G) e

+

-

r l o g i;

(5) t h e r e i s obtained: 1 f-x

r k

2 log - =

1.74

It i s evident then t h a t t h e magnitude of constant wi-thin t h e region of t h e quadratic law of r e s i s t a n c e but within t h e other regions i s v a r i a b l e depending on t h e Reynolds number.

r k

log(Re fl) - l o g - was used a s

The preceding explains why t h e value

t h e a b s c i s s a i n s t e a d of l o g ( ~ efi) as was done f o r t h e smooth pipe. Equation (58) may now be w r i t t e n i n t h e form 1 fi

2 log

r

=

k

f log )k:v

(

There occurs here, a s t h e determining f a c t o r , t h e dimensionless term

which i s t o be expected from t h e viewpoint of dimensional a n a l y s i s . The r e l a t i o n s h i p

-6

2 log

rk

=

( "

f log

a s determined experimentally i s shown i n f i g u r e 11 ( s e e t a b l e s 2 t o 7 ) f o r f i v e degrees of r e l a t i v e roughness. The s i x t h degree of r e l a t i v e roughness was not included because i n t h a t t h e assumption of geometrical s i m i l a r i t y probably did not e x i s t . It i s evident t h a t a smooth curve may be passed through all t h e p l o t t e d p o i n t s . The range I i n which t h e r e s i s t a n c e i s unaffected by t h e roughness and i n which a l l pipes have a behavior s i m i l a r t o t h a t of a smooth pipe i s expressed i n t h i s diagram ( f i g . 11) by t h e equation

1 -

6

r

0.8 + 2 l o g

2 log k =

(9)

i n which t h e value of a f h n c t i o n f i s determined by e q u a t i o n 8. The f a c t t h a t t h e t e s t p o i n t s l i e below t h i s range i s due t o t h e i n f l u e n c e of v i s c o s i t y which i s s t i l l p r e s e n t f o r t h e s e s m a l l Reynolds numbers. This i n d i c a t e s t h a t t h e law expressed i n e q u a t i o n 3 i s not e x a c t l y f u l f i l l e d . The t r a n s i t i o n range, range 11, i s r e p r e s e n t e d i n f i g u r e 11 by a curve which a t f i r s t r i s e s , t h e n has a c o n s t a n t value, and f i n a l l y drops. The curves t o be used i n l a t e r computations w i l l be approximated by t h r e e s t r a i g h t l i n e s not shown ( r e f e r e n c e s 19 and 20) i n f i g u r e 11. The range covered by t h e q u a d r a t i c law of r e s i s t a n c e , range 111, i n t h i s diagram l i e s above l o g v*k = 1.83 and corresponds t o e q u a t i o n ( 5 a ) . These l i n e s may be expressed by e q u a t i o n s of t h e form 1 -

-

JX

i n which t h e c o n s t a n t s

r 2 l o g j;

a

and

b

+

a

=

b log

vary with

v k

v v k %

i n t h e following

manner :

4'x -

k 1118 + 1.13 l o gv L

2 log:=

v

k

=

2.81

-

0.588 l o g v*k