laws of flow in rough pipes-nikuradse

8/13/2019 Laws of Flow in Rough Pipes-Nikuradse

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TECHNICAL MEMORANDUM 292

C\S3c\l i

b4 NATIONAL ADVISORY COMMITTEE

FOR AERONAUTICS

LAWS OF FLOW IN ROUGH PIPESBy J. Nikuradse

Translat ion of gStromungsgesetze in rauhen Rohren.VDI-Forschungsheft 361. Beilage zu Forschung auf dem Gebiete desIngenieurwesens Ausgabe Band 4 ~uly/August 1933.

II

i


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NATIONAL ADVISORY COMMITTEE FOR iZERONAUTICS

TECHNICAL I+EMORANDUM 1292

By J Nikuradse

INTRODUCTIONN u mer o u s r ecen t i n v es t i g a t i o n s ( r e f e r en ces 1 2, 3, 4 and 5have gr ea t l y increa sed our knowledge of t u r bu le n t f low i n smooth tubes ,

c ha nn el s, a nd a lo 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 yd 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 l aw s c o n t r o l l i n g r e s i s t a n c e , o nimpact , and on mix ing len g th . The da ta cover th e tu rb u l en t behav ior o fth es e f low problems. The l o g ic a l development would now in di c at e as tu dy of t h e l aw s g o v e rn in g t u r b u l en t f l o w o f f l u id s i n r o ug h t u b e s ,channels , and a long rough p lane sur fac es . A s tudy of th es e p rob lems,b ecau se of t h e i r f r eq u en t o ccu rr en ce i n p r a c t i ce , i s more impo rt amtthan t h e s t u d y o f f l o w a lo n g smooth s u r f ac es an d i s a l s o of g r e a ti n t e r e s t a s an ex te ns io n of our ph ys i ca l knowledge of t u r bu le n t f low.

T u r b ulen t f l o w o f w a te r i n ro ug h tu b e s h as been s tu d i ed d u r in g t h el a s t cen tury by many in ve s t ig a t o r s o f whom the most ou ts tan d ing w i l l beb r i e f l y men tion ed h e r e . H D ar cy ( r e f e r en ce 6 made comprehensive andv e r y c a r e f u l t e s t s on 2 1 p i p e s o f c a s t i r o n , l e a d , w rought i r o n ,a s p h al t - co v e re d c a s t i r o n , a nd g l a s s . W it h t h e e x c e p t i o n of t h e g l a s sa l l p ip e s w ere 1 0 0 me te r s l o n g an d 1 . 2 t o 3 c e n t i m e t e r s i n d i a m et e r .He no ted t h a t th e d i sc harg e was dependent upon the type o f su r fa ce asw el l a s upon th e d i ame te r o f t h e p ip e and t h e s l o p e . I f h i s r e s u l t sa r e e xp re ss ed i n t h e p r e se 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 i scon sid ere d dependent upon th e Reynolds number Re, th en i t i s f ound t h a t


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For a constant Reynolds number, h i n c r e a s e s markedl y f o r an i n c r ea s i n gre l a t i v e ro u g h n es s . H Bazin ( re fe rence 7 , a fol low er of Darcy, c ar -r i ed on the work and der ived f rom h i s own and Darcy ' s t e s t da ta anemp i r i ca l fo rmula i n which t he d i scharge i s dependent upon the slopeand diameter of th e p ipe . Th i s fo rmula was used i n p r ac t i c e u n t i lrecen t t imes .

R v . Mises ( re fe rence 8) i n 1914 did a very va luab le p iece o fwork, t r ea t i n g a l l of t h e t hen-known t e s t r e s u l t s f rom t h e v iewp oi nt o fs i m il ar i t y . He obtained, ch ie f l y from th e obser vat io ns of Darcy andB az in w i th c i r c u l a r p i p e s , t h e f o ll o wi ng f o rm u la 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 th e Reynolds number and the r e la t iv e roughness :

This fo rmula fo r va lues of Reynolds numbers near the c r i t i c a l , t h a t i sfo r sm al l value s , assumes th e fol lowing form:

The t e rm re la t iv e roughness f o r the r a t i o i n which k i s t h erabso l u te roughness was f i r s t used by v . Mises . Proof o f s i mi la r i tyf o r f low through rough pipes was fur nishe d i n 1911 by T. E Stan ton( r e f e r e n c e 9 . He s tu die d pi pe s of two diameters i n t o whose in ner sur -f ac e s two i n t e r s ec t i n g t h r ead s had b een cu t . I n o rd e r t o o b t a i ngeomet r i ca l ly s i m i l a r depths of roughness he varie d the p i t ch and deptho f t h e t h r ead s i n d i r e c t p ro p o r t i o n t o t h e d i ame t e r of t h e p i p e . Hecompared for the same pipe the largest and smallest Reynolds number


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NACA TM 1292

a s t u rbu le nc e se t s i n . In t he c a se o f l e s s s e ve re ly roughened su r fa c e she observed a s low inc re a se o f t h e f r i c t i o n fa c to r wi th t he Reyno ldsnumber. Sc hi l l e r was not abl e t o de termine whether th i s in c reas e goesover i n t o the qua dra t ic law of f r i c t i o n fo r h igh Reynolds numbers , s in cet he ~ 8 t t i n g e n e s t a p p a r at u s a t t h a t t im e was l i m i t e d t o a bo ut Re 103.His r e su l t s a l s o i nd i c a t e t h a t f o r a f i xe d value of Re ynolds number t h ef r i c t i o n f a c t o r i n c r e a se s w i t h a n i n c r e a s in g r ou gh n es s.

L Hopf (reference 11 made some t e s t s a t about th e same t ime a sS ch il le r t o determine th e fu nct ion f Re He performed system-:)a t i c e xpe rimen t s on re c t a ngu la r c hannel s o f va r ious de p ths wi th d i f fe r -en t roughnesses (wire mesh, z in c p la te s having saw-toothed type su rfa ces ,and two types of corruga ted p la te ) . A re c t a ngu la r s e c t i o n was se l e c t e di n o rder t o det ermine t he e f f e c t o f t he hydra u l i c r a d ius (hydra. u l icr a d i u s r = a re a of s e c t i o n d iv ided by wet t ed pe r ime te r ) on t he va r i a -t i on i n de p th of s e c t i o n f o r a c ons t an t t ype of wa l l su r fa c e . A t Hopffssugges t ion these t e s t s were extended by K F r o m ( r e f e r e n c e 1 2 ) . Onthe b a s i s o f h i s own and Fromm s t e s t s a nd o f t he o the r a va i l a b l e t e s tda ta , Hopf concluded th a t t h er e ar e two fundamenta.1 ty pe s of roughnessinvolved in tu rb ul en t f low in rough pip es. These two type s, which hete rms surface roughness and surf ace corruga t io n , fo l low di f fe re nt lawso f s i m i l a r i t y . A surf ace roughness , according t o Hopf, i s char ac te r izedby t h e f a c t t h a t t h e l o s s of h ea d i s independ ent of th e Reynolds numberand dependent on ly upon th e typ e o f wa.11 sur fac e i n accordance with th eq u ad ra t i c law o f f r i c t i o n . He c o n si d e rs s u r f a c e c or r u g at i o n t o e x i s twhen th e f r i c t i o n fa c t or a s we l l a s t h e Reynolds number depends uponthe t ype of wa l l su r fa c e i n suc h a manner t ha t , i f p l o t t e d l oga r i t h mic a l ly ,the curves f or a s a func t ion of th e Reynolds number f o r va r io us wal ls u r f a c e s l i e p a r a l l e l t o a sm ooth c ur ve . I f a i s t h e av er ag e d e pt h ofroughness and b i s the average distance between two project ions fromasur fa c e , t he n su r fa c e c o r ruga t ion e x i s t s fo r sma l l va lue s of


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corrugation he gives the formula

i n which Lo i s the f r i c t i on fa c to r fo r a smooth su r face and i s ap ropor t iona l i ty f a c to r which has a value between 1.5 and 2 f o r woodenpipes and between 1.2 and 1.5 f o r a s p h a lt ed i r o n p ip e s.The var ia t i on o f the ve loc i t y d i s t r i bu t io n wi th the type of wal lsurface i s a l so important , a s wel l a s th e law of r es is tan ce. Observa-

t i o n s on t h i s problem were made by Darcy, Bazin, and Sta nto n re fe re nc e 9 .The neces sary dat a, however, on temperature of th e fl ui d , typ of wallsurfac e, and l o s s of head ar e lacki ng. In more rec ent t imes such obser-va t ions have been made by Fr it sc h refe ren ce 13 a t the sugges tion o fVonkt ,si ng t h e same ty pe of ap pa ra tu s a s th a t of Hopf and Fromm.The channel had a length of 200 centim eters, width of 15 cent imetersand depth varyi ng from 1. 0 t o 3.5 cen tim ete rs. two-dimensional condi-t i on o f f low ex is ted , there fo re , along th e s ho rt a x i s of symmetry. Heinves t iga ted the ve l oc i ty d i s t r ib u t i on fo r the fo l lowing types of w a l lsurf ace

1 smooth2 . corrugated wavy)3 . rough

I f l o o r s , g l a s s p l a t e s w ith l i g h t co r r u g a ti o n s )4. rough

11. r i b b ed g l a s s )


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very small range of Reynolds number. The purpose of t he pr es en t in ves -t i g a t i o n i s t o s tu d y t h e e f f e c t o f c o ar se and f i n e r ou gh ne ss es f o r a l lReynolds numbers and t o determine th e laws which a re ind ica te d. t was,

rthe re fo re , neces s ary t o consider a de f in i t e r e l a t i ve roughness fo ra wide range of Reynolds number and t o de termine whether f o r t h i s con-

rs t a n t t h a t i s , f o r g eo me tri ca l s i m i la r i t y, t h e va lue f ( ~ e ) skthe same curve f o r pipes of d if fe re nt diameter . There was a ls o th erque stio n whether f o r the 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 rk

and var y wit h th e Reynolds number, and whether f o r a vary ing f: t h ekv 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.

wish here t o express my s i n c er e g ra t i t ud e t o my immedia.tesuper ior , P rofessor Dr. L Pr an dt l, who has a t a l l time s aided me byh i s va luab le adv ice .

I EXPERIMENT1 Description of Test Apparatus

The appa ratu s shown i n fi gu re 3 was used i n making th e t e s t s . Thesame appara tus was employed i n the inv es t i ga t ion of ve lo ci t i es f or tu r -bulent f low i n smooth p ipes . The de ta i le d descr ip t ion of the appara tusand measuring dev ices has been presented i n Forschungsheft 56 of theVDI Only a brief review w l l be given here. Water was pumped by meansof a ce nt ri fu ga l pump kp driven by an e l e c t r i c motor em, from t h esupply canal vk, in to the water tank wk, th en through th e t e s t p i p e vrand in to the supply cana l vk. This arrangement was employed i n t h e


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3 . d i sc ha r ge qua n t i ty4 . tempera ture of t he wate rThre e hooked tub e s wi th l a t e r a l a p e r tu r e s were use d t o m ea su re t he

l o s s o f he ad . These t u be s a r e de s cr ib e d i n d e t a i l i n s e c t i o n I 3 . Theve l oc i ty d i s t r ib u t io n was de ter m ine d by means o f a p i t o t t u be wi th0 .2 mi l l imet e r in s i de d iamete r, mounted in th e ve loc i ty-measur ingde v ic e gm a nd 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 fo r Reynolds numbers up t o 3 105 was measured i n a ta nk mbon t h e b as is of depth and t ime. Larger di sch arg es were computed byin t e g r a t i ng th e ve l oc i ty d i s t r i bu t i on c u r ve . Tem pe ra tu re r ea d ings weret a k en a t t h e o u t l e t o f t h e v e l oc i t y- m e as u r in g d e v i ce gm T h e t e s tp ip es were drawn br as s p ip es of c i rc ul a r se c t io n whose d imensions a r eg iv en i n t a b l e 1 The diame ters of th e pipe were determined from th eweight of th e wate r which could be conta in ed i n the p ip e wi th c los ede nds a nd f rom th e l e ng t h o f the p ipe .

2. Fabrication and Determination of RoughnessS i m i l i t u d e r e q u i r e s t h a t i f m e ch an ic al ly s i m i l a r f lo w i s t o t a k e

place i n two p i pes they must have a g eome tr ica l ly s im i l a r form and mustha ve s im i la r w a l l su r f a c e s . The f i r s t re qui r em e nt i s met by the use o fa c i r c u l a r s e c t ion . The se cond r e qui r e me n t i s s a t i s f i e d by m a in ta in inga c o n s ta n t r a t i o of t h e p ip e r a d i u s r t o t h e d e pt h k of p r o je c t ions .t 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 p ro du ci ng t h e r ou gh ne ss

shou ld be s im i l a r . P r o f e s so r D. Thoma's preceden t of usi ng sand f o rt h i s purpose was adopted .

Gr a ins o f uni fo rm s i z e a r e r e qu i r e d t o pr oduc e uni f or m r ou ~ h ne ssthroughout th e p ipe . Ordinary bu i ld ing sand was s i f t ed . Ln o r de r t o


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th e p ipe wa.s f i l l e d wi th sand o f a. ce r t a i n s i ze . The sand was the na l lowed t o flow ou t a t the bo ttom. The p re l iminary t e s t s showed t h a tth e d ry ing which now fo l lows i s o f g rea , t impor tance fo r d ur ab i l i ty . Arying p e r i o d of two t o t hr ee weeks s req uir ed , depending upon th eamount of mois tu re i n the a i r . A u ni fo rm d r a f t i n t h e p i pe , due t o a ne l ec t r i c b u l b pl aced a t t h e l o w er end , h e lp ed t o o b t a i n even d ry i n g .A f t e r t h i s d ry in g , t h e p i p e was r e f i l l ed w i t h l a cq u e r and ag a i n empt ied,i n o rd e r t o o b t a i n a b e t t e r ad h eren ce of t h e g r a i n s . T he re fo l lo w edan o t h e r d ry i n g p e r i o d o f t h r ee t o fo u r w eek s. A t each end of th e pipe ,a l en g th of abou t 10 cen t ime ters was cu t o f f i n o rder t o p reven t anyp o s s i b l e d ec rea s e i n t h e end s ec t i o n s . A f t e r t h e t rea. tment j u s t d e s c r ib edth e p ipe s were rea dy t o be measured.

One of t h e co n d i t i o n s c i t e d abo ve i n d i c a t e s t h a t d i f f e r en t g r a i ns i z e s must b e u se d f o r p i p e s of d i f f e r e n t d i a me te r i f t h e r a t i o rwhich i s t h e g age 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 r e ma in c o n s t a n t .Geometr ical s i m i l a r i t y of t h e w a l l s u r f ace r eq u i r e s t h a t t h e form oft h e i n d i v i d u a l g r a i n s s h a l l b e un chang ed and a l s o t h a t t h e p ro j ec t i o nof th e roughening, which has hydrodynamical ef fe c ts , s h a l l remain con-s t an t . F igure hows th a t vo ids e x i s t between th e g ra . ins . The hydro-dynamica l ly e f fe c t iv e amount of p ro jec t io n k s e q ua l t o t h e g r a i ns i ze . I n o rder t o de te rmine whether the p rev io us ly observed d iametero f g r a i n s s a c t u a l l y e f f e c t i ve , a f l a t p l a t e was c oa te d w i th t h i nJapanese lacq uer th e nec essa ry degree of th in ne ss was determined byp re l i mi n a ry t e s t s ) and rou gh en ed i n accordance w i t h t h e d e s c r i b edprocedure. The pr oj ec t io n of the g ra in s above th e sur fac e was measuredi n th e manner a l re ady descr ibed and t was fou nd t h a t , f o r a d e f i n i t ed egree o f t h i n n es s o f t h e l a cq u e r , t h i s ave rag e p ro j e c t i o n ag reed w i t hth e o r i g i na l measurements of the g r a i n s .


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connecting le g was bent a t an angle of about 60 i n the plane of thef r ee l eg i n o rde r t ha t t h e pos i t i on of t he f re e l eg might a lways beind icat ed. The bent tube was fas ten ed i n th e t e s t pipe by means of as tuff ing box.

Varia t ion of the pressu re readings i n a hooked tube wi th va r ia t ion si n t h e p o s i t i o n o f t h e t u b e r e l a t i v e t o t h e d i r e c t i o n o f f lo w i s showni n f ig ur e 6. Thi s f i gu re i nd i ca t e s t h a t cor rec t r ead ings a re ob t a inedonly i f t he d i rec t i on of t he f re e l eg dev i a t e s not more t han 7.5 fromth e di re ct io n of flow. The intro duct i on of th e hooked tube into thet e s t pip e r e s u l t s i n an increase of pressure drop due to th e res i s tan cet o the tub e. The resi sta nc e of th e two hooked tube s used in measuringmust be deducted from the observed pressure drop pl - p2. The r e s i s t -ance of th e tub e must th er ef o re be known. Thi s va lue was found bymeasuring the pressure drop h i n a smooth pipe i n terms of the d is -charge a t a constant temperature, f i r s t by us ing wa ll pi ezomete r o r i f i ce sand then by measuring th e pres sure drop h a i n te rms of the di schargea t t h e same tem pera ture by means of a hooked tu be . The increme nt a f o requal di scharges i s th e res i s ta nce of th e hooked tubes . The correc t io ncu rv e f o r t h i s r e s i s t a n ce i s g iv en i n f i g u re 7.

t should be noted th a t changes i n di rec t ion of th e tube re su l tb o t h i n an e r r o r i n t he p re ssure read ing and i n an i n c re a s e i n t h ere s i s t an ce due t o t he tube . I f t h e cor rec t ed pre ssure d rop pl - P2 i sdivided by th e observa t ion length 2 (di s ta nce be tween th e holes in th es ide of t he hooked tubes ) , t he re i s ob ta ined the s t a t i c p re s suregradient ,


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a v e r a g e v e l o c i t y u was about 2 meters per second. t was observedt h a t w i th i n a few days the pressure s lope deve loped a pronouncedi n c r e a s e . A m arked washing o f f o f t he l a c que r was in d ic a t e d a t t h esame t ime by de po si ts on t he bottom of th e supply channel . Anotherob je c t iona b le f e a t u r e was th e p a r t i a l washing ou t o f th e sa nd. Thei n c re a s e i n t h e p r es s u re g r a d i e n t i s ac co un te d f o r by t h e i n c r e a s e i np r o je c t io n o f r oughness due t o the washing o f f o f th e l a c qu e r . There-f o r e , t h e method o f f a s t e n in g th e sa nd had t o be c ha nged i n o r de r t oi 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 ri n g t h e t e s t p r o ce d u r e.The p r o je c t ion k of th e r oughness had t o r e ma in c ons ta n t du r ing th et e s t s a nd t h e d i s t r i b u t i o n o f t h e s an d g ra .i ns on t h e w a l l su r f a c e s hadt o remain unchanged.

Adhesion between sand gr ai ns was prevent ed by us in g a v e ry t h i nla c que r . This lacquer formed a d i r e c t c o a t i n g on t h e w a l l a nd a l s o ac o v e ri 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 o f t h e s e g r a i n si n t o t h e l a c q u e r c o a t i n g o f t h e w a l l . The o r i g i na l f or m a nd s i z e o fthe grains remained unchanged. A d e te r mi n in g f a c t o r i n t h i s p ro bl emwas th e degree of th i ckn ess of th e lacq uer which was var i ed by th ea d d i t i o n o f 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 re ma in ed un ch an ge d.Te s ts made wi t h p i pes wi tho ut lacquer recoa t ing showed that the sandwould wash ou t . The recoa . t ing wi th lacq uer was, th e re fo re , adopted .I f o n l y a shor t pe r iod o f d r y ing was use d f o r bo th c oa t s , t h e l a c que rwas washed o f f . I f t he f i r s t d r y ing was sho r t and the se c ond long ,t h e n a l l o f t h e l ac q u e r was a l s o washed o f f . I f t h e f i r s t dr y i ngper i od were lo ng and t h e se cond shor t , t h e r e wou ld a l so be some l o s sof sand. A cons tant cond i t io n of roughness could be obta ined only wheneach lacq uer co a t i ng was dr i ed from th re e t o fo ur weeks. The accuracyof o bse rva tio ns made with t he hooked tube was checked by conn ecti ng th etub e thro ugh a manometer t o a w a l l p i ez o me t er o r i f i c e a t t h e same s e c -t i o n o f t h e p i p e . B o th c o n n e c t i o n s sh o u ld show t h e same p r e s s u r e i n asmooth pip e, t h a t i s , th e manometer read in g must be zero. Hooked tu be s


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11 EVALUATION OF TEST R SULTS1 Law of Resistance

T h e r e s i s t a n c e f a c t o r X f o r f lo w i n t h e p i p es i s expressed bythe fo rmula :

i n w hich i s t h e p r e s s u r e d ro p p e r u n i t o f l e n g t h , d i s the diam-x ii2e t e r , a n d = p F th e dynamic p re ssure o f t he average f lowv e l o c i t y E and i s t h e d e n s i t y . An ex ten s iv e t e s t p r og ram w i th arange of Re = 6 o Re = lo6 f o r t he Reynolds number was c ar r i ed ou t ,m d t h e r e l a t i o n s h i p o f t h e r e s i s t a n c e f a c t o r t o t h e R ey no ld s numberwas s tud ed f o r p ip es o f var i ous roughnesses . Six d i f f e r e n t d e gr e es o f

kr e l a t i v e r o u gh n es s w er e u sed , w i th t h e r e l a t i v e r o u g hn ess - determinedrby t h e r a t i o of t h e a v er ag e p r o j e c t i o n k t o t h e r ad i us r o f t h e p ip e .

I n e v a lu a t in g t h e t e s t d a t a it seemed adv is ab le t o use ins tea d ofk rt h e r e l a t i v e ro ug hn es s - i t s r e c i p r o c a l - Fig u r e 9 shows t o ar kl o g ar i th 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 R ey no ld s

1number f o r t h e r e c ip r o c a l v a lu e s - of t h e s i x r e l a t i v e r ou gh ne ss eskt e s t e d a nd f o r a s mooth p i p e ( s e e t a b l e s 2 t o 7 . The bottom curve i sfo r the smooth p ip e . If t h e c u r v e f o r = f ( ~ e ) s s t u di e d f o r ag iv en r e l a t i v e r o u gh n es s , t h en i t must b e co n s id e r ed i n t h r e e p o r t i o n s


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NACA T 1292

T hi s i s r e pr e se n te d i n t h e f i g u r e b y a s t r a i g h t l i n e o f s l op e 1 : 4 Thec r i t i c a l Reynolds number f o r a l l degrees o f r e l a t i ve rollghne s s occu r s a tabout t he same po s i t i on a s fo r t he smooth p ipe , t h a t i s between 2160and 2500.

Within the second range , which w i l l be t ermed t he t r a ns i t i o n r ange ,th e in f lu ence of t he roughness becomes not ice able i n an i n c r e a s i n gd e gr e e; t h e r e s i s t a n c e f a c t o r X incr eas es wi th an incr eas ing Reynoldsnumber. This t r a ns i t io n range i s p a r t i c u l a r l y c h a r a c te r i z e d by t h e f a c tt h a t t he r e s i s t an ce f a c t o r depends upon t h e R eyno lds number a s we l l a supon the r e l a t i v e roughness .

W it hi n t h e t h i r d r a ng e t h e r e s i s t a n c e f a c t o r i s independent of theReynolds number and th e cur ves f Re ) 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 is i s t h e r an g e w i t hi n which t h e q u a d r a t i c l a w ofr e s i s t a n c e o b t a i n s .

The thr ee ranges of th e curves X f ~ e )may be p h ys i c a l l y i n t e r -p r e t e d a s f o ll o ws . I n t h e f i r s t r an ge t h e t h i c kn e s s 6 o f t h e l am in arboundary layer, which i s known t o dec rease w it h an incr eas ing Reynoldsnumber, i s s t i l l l a r g e r t h a n t h e av er ag e p r o j e c ti o n 6 k ) . Thereforeenergy losses due t o r ou gh ne ss 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 smoothplpe

In t he s econd range t he t h i ckn es s o f t he boundary l aye r i s of thesame magnitude as t h e a v e r a g e p r o j e c t i o n 6 k ) . I n d i v i d u a l p r o j e c t i o n sextend through t he boundary lay er and cause vo r t ic es which produce anad di t i on al l o s s of energy. As th e Reynolds number inc re as es , an


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In orde r t o c he ck t h i s fo rmula e xpe r ime n ta l l y t he va lue was p lo t t e dii n f i g u r e 1 0 a g a i n s t l o g and it was found th a t through the se poi ntskthe re c ou ld be pa sse d a l i ne

The e n t i r e f i e l d of Reynolds numbers inv es t iga ted was covered by p l o t -1 r v kt i n g t h e t er m 2 l o g - a g a i n s t l o g7 his te rm i s p a r t i c u l a r l yk

su i t a b l e d ime ns iona l ly s inc e i t h a s c h a r a c t e r i s t i c v a l u es f o r c o nd i ti o nsalong th e sur fac e . The more convenient value lo g Re l o g mightkbe used instead of lo g k a s may be seen from th e fol l ow ing con sid era-t i o n . From the fo rmula fo r t he r e s i s t a nc e fa c to r

t h e r e l a t i o n s h i p betw een t h e s h e a r i n g s t r e s s T~ and t h e f r i c t i o nfa ct o r may be obtained . I n accordance with th e requirements ofe q u i li b r i un : f o r a f l u i d cy l i n d e r of l e n g t h x nd r a d i u s r

or f rom equa t ion 1)


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log Re fi l o g = 1 0 ~ ~ . 6 6yV*k rlog = const l o g ~ eG l o g i

From equation 5) t h e r e i s obtained:1 rl o g = 1 74f-x k

It i s evident then th a t the magnitude ofcons tant wi-thin th e region of t he quadrat ic law of res is t an ce butwith in t he othe r regio ns i s var iab le depending on the Reynolds number.

rThe prec edi ng ex pl ai ns why th e val ue log Re fl) lo g was used askthe absc i s sa ins tead o f l o g ~ e i) as was done f o r th e smooth pi pe .Equation 58) may now be wr it te n i n th e form1 2 o g r = f logi k v

There occurs here , a s the determining fac to r , th e dimensionless term


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1 rl o g = 0.8 2 l o g 9 )i n which t he va lue of a fhnc t i on f i s de te rmined by equa t i on 8. Thef 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 r a ng e i s due t o t h e i n fl u en c eof v i sc os i t y which i s s t i l l pr es en t f o r t h es 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 l a w e x pr e ss e d i n e qu a t i on 3 i s n ot e x a ct ly f u l -f i l l e d . The t r a n s i t i o n r a ng e , r an ge 11 i s r e p re s en t ed i n f i g u r e 11 bya c ur ve which a t f i r s t r i s e s , t hen has a cons t an t value, and f i n a l l ydrops . The c urves t o be used i n l a t e r c om pu ta ti on s w i l l be approximatedby t h ree s t r a i gh t l i n e s no t shown ( re fe r ences 19 and 2 0) i n f i g u r e 11The range covered by th e quad rat ic law of r es is ta nc e, range 111 i n

v*kt h i s dia gr am l i e s above l o g = 1.83 and c o r re s po n ds t o e q u a t i on ( 5 a ) .These l i n e s may be expre ssed by equa t ion s of t he form

1 r v kJX

2 l o g j = a b l o g vv ki n which t h e cons t an t s a and b v a r y w i t h i n t h e f o l l ow i n g

manner :v k < v k


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vel oc i ty d i s t r i bu t i on s were symmetr ica l , on ly one-hal f th e curve hadt o b e co n sid e red i n th e ev a lu a t io n o f t e s t d a ta . d imens ion lessequation of the form

was se l ec te d t o show th e v a r i a t io n of t h e v e l o c i ty d i s t r ib u t io n wi th th ervalue Ln t h i s e q ua t io n U i s th e maximum ve lo ci ty , and u i s t h e

v e l o c i t y a t a ny p o i nt y d i s t a n t f rom t h e w a l l i n a p i pe o f r a d i u s rTh is r e l a t io n sh ip i s shown in f ig u re 1 2 f o r a smooth pipe and f o r suchv e l o c i t y d i s t r i b u t i o n s a t v a r i ou s de g re e s of r e l a t i v e ro ug hn es s a s l i ewi th in th e r eg io n of th e q u ad ra t i c law of r e s i s t an ce . T h i s f i g u r e i n d i -c a t e s t h a t a s t h e r e l a t i v e r ou gh ne ss i n c r e a s e s , t h e v e l o c i t y d i s t r i b u -t i on assumes a more po in ted form. Our ea r l i e r te s t s wi th the smoothpip e have shown, however, t h a t a s th e Reynolds number inc re as es t heve lo c i ty d i s t r i bu t i on assumes a more b lun t form.

A very s im pl e law f o r t h e v e l o c i t y d i s t r i b u t i o n i n rough p i p e s i suobta ined f rom th e fo l lowing p l o t t i ng . The d imensionless ve lo c i ty *i s shown i n f i g u r e 1 3 p l o t t e d a g a i n s t . The t e rm v* i s t h e f r i c t i o nrv e lo c i ty , v, as p r ev io u s ly in tro d uced . Th i s f ig u re in d ica t e sth a t i n th e r eg io n s away fro m th e wa l l t h e v e lo c i ty d i s t r i b u t i o n s a r es i m i l a r . I f , i n a c co rd an ce w i t h Von ~ a ' m ' n , t h e p l o t t i n g i s f o ru - u f ( f ) , t h e s imi l a r cu rves merge t o form a s i n g l e cu rv e ( f i g . 1 7 .v*The v e l o c i t y d i s t r i b u t i o n s f o r t h e d i f f e r e n t d e gr e es o f r e l a t i v e r o ug h-


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U r 5 5 5.75 l o g l-v*

14)I f t h e r e l a t i o n s h i p cp f 1 o g 7 i s now p l o t t e d f o r r o ug h p i p e s , f i g -u r e s l 5 a ) t o l 5 f ) a r e o b ta in e d, which i n e v er y c a se y i e l d a s t r a i g h tl i n e f o r t h e d i m en s io n le s s v e l o c i t y . Each f i g u r e c o rr es p on d s t o ade f i n i t e r e l a t iv e r oughne ss a nd t o th e se ve r a l Re ynolds num bers r e c o r ded ;

rf i gu r e 15 a ) c o r r e sponds to th e sm a l l e s t r oughness i 507 , f i g u r e 1 5 b )t o the ne x t to sm a l l e s t , e t c . The re i s f u r the r m or e shown on e ve r y f ig -u r e t h e 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 smooth p ip e a s g i ve n b y e q u a t i o n 1 4 .The o b s e r v a t i o n p o i n t s l y i n g on t h i s s t r a i g h t l i n e were o b t ai n e d n o t i na smooth p ipe but i n a rough p ipe a t such a smal l Reynolds number t h a tt h e i n f l u e n c e of t h e ro ug hn es s i s n o t n o t i c e a b l e . These s t r a i g h t l i n e sf o r a g i ve n r e l a t i v e r ou gh n es s s h i f t wi t h an in cr ea si ng Reynolds numbert o a p o s i t i o n p a r a l l e l t o t h a t o f t h e s t r a i g h t l i n e f o r t h e smooth p i pe .A c a r e f u l s t u d y of t h e i n d i v i d u a l t e s t p o i n t s s hows t h a t t h o s e n e a r t h ewa l l sm a l l va lue s o f lo g 7 ) a s w e l l a s t h o se n e ar t h e a x i s l a r g e v a l ue sof log q ) l i e s l i g h t l y above t he l i n e .

The term A a s i n d i c a t e d by e q u a t i o n 1 3 ) h s a c ons ta n t value inthe r e g ion o f the qua dr a t i c l aw o f r e s i s t a nc e . n t h e t r a n s i t i o nu2rregions and 11 however, A depend s upon t h e Rey nol ds number Re ,I

ka nd on th e r e l a t i ve r oughne ss i n suc h a manner th a t A e s s e n t i a l l ykdepends only on the produc t Re fi - i n a cc or da nc e w i t h e q u a t i o n 7 a ) .r

From equ ati on 6b )


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NACA T 1292 17

ve l oc i t y c u rve unde r c ons id e ra t i on . Pa r t i c u l a r c a re must be use d i nt h i s d e t e r m i n a t i o n a t medium di st an ce s from th e wall , s in ce, on th e onehand, th e value of y c m o t be a c c u r a te l y ob ta in ed f o r p o i n ts n e ar t h ewa l l , a nd fu rthe rmore t he v i sc os i t y ha s a not iceab le inf luence here , andon the other hand, a re gu l a r de v i a t i o n a lways oc c urs f o r po in t s ne a r t heax is . The value of A a s found i n t h i s manner f o r a l l v e l o c i t y c u r v e s

v k ( se e f i g . 16 ) . The formas t h e n p l o t t e d a s a func t i on o f l o aof curve A a s a f u n c t i o n o f l o g s Yery s i m i l a r t o t h e c ur ve f o rv rt h e r e s i s t a n c e la w o b t a in e d by p l o t t i n g v*k2 l o g a g a i n s t l o gkf rom equa t io n 8 ) .

Ana lyt ic a l proof of t h i s re l a t io ns hi p may be obta ined by the samemethod a s th a t used f o r th e smooth p ipe re f e re nce s and 21 ) . I naccordance wi th equa t ion 13 )

o r , i f t h i s e q ua ti on i s w ri t te n f o r t h e p ip e axi s , t h a t i s u = U y = r :

u - u f th e r e may be obta ined by in te gr a t io nrom the equa t ion --- - rt h e t e r mu - T i - 1 7 4v*


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S ub s t i tu t ing e qua t ion ( 1 8 ) in to e qua t ion ( 17b) a nd d iv id in g by v,

and then f rom equa t ion (16b)r2.83 A B l o g D

o r w i t h B = 5.75

The d e s i r e d r e l a t i o n s h i p b et we en t h e v e l o c i t y d i s t r i b u t i o n and t h e l awof r e s i s t a n c e i s g iv e n i n e qu a t io n s ( 1 5 ) and ( l o b ) . I t may be expressedi n th e f o l lowing fo rm

F i gu r e 1 6 c o n ta i n s i n a d d i t i o n t o t h e v a l ue s o f A computed from thev e l o c i t y d i s t r i b u t i o n s by e q ua t io n I ? ) , he computed values obtainedf rom th e l aw o f r e s i s t a nc e by e qua t ion ( 19b) . The agreement between thevalues of A determined by these two methods i s s a t i s f a c t o r y .

By t h e same method as i n f i gu r e 11 t he c u r ve A may be r e p r e se n te dv*ka s a f u n c t i o n sf I c y Within th e range of th e law of r es is tan ce

where t h e efi ec.1 3f vi:;cos: t y i s n o t y e t p r e s e n t t h e l aw f o r smoothp i p e s a p p l i e s , : .lit :


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and w i t h i n t h e zon e of t h e q u ad ra t i c law o f r e s i s t an c e :v*kl o g > 1.83

T hese ex p re s s i o n s d e s c r i b e w i t h s u f f i c i e n t a ccu racy t h e l aws o f v e l o c i t yd i s t r i b u t i o n and o f r e s i s t a n c e f o r p i p e s w i th w a l l s ro ug he ne d i n t h emanner he re con sidere d.

F i n a l l y , i t w i l l be shown b r i e f l y t h a t th e Von K ( r e f e r e n c e 2 )e qu a ti on f o r t h e v e l o c it y d i s t r i b u t i o n

d e r i v e d a n a l y t i c a l l y on t h e b a s i s of h i s hy p o th e s is of s i m i l a r i t y ,ag re es wi th th e exper imen ta l da ta . The t e rm K i s a u n i v e r s a l c o n s t an to b ta i n ed fr om t h e v e l o c i t y d i s t r i b u t i o n . Ln f i g u r e 17, the curve drawnt h rou g h t h e ex p e r i men ta l p o i n t s ag ree s a l mos t ex ac t l y w i t h t h e cu rv e fo rt h i s equa t io n . With very l a rg e Reyno lds numbers where th e in f luen ce o fv i s c o s i t y i s v e r y s l i g h t 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 c c o rd in g t o VonKarman s t re at me nt do no t depend upon th e ty-pe of w a l l su r face nor upont h e Reynolds number. Good agreeme nt wi th K 0 .3 6 i s o b t a i n ed b et weene x p er i m en t a l a nd t h e o r e t i c a l c u r v e s f o r s u ch v e l o c i t y d i s t r i b u t i o n upt o t h e v i c i n i t y of t h e w a l l . I t may be concluded from t h i s th a t a t ad e f i n i t e i n t e r v a l y , fro m t h e w a l l , th e ty pe of flo w and th e momentumchange a re independen t of t he type o f wa l l su r fa ce .

I n o r d e r t o i nc lu de t h o s e o b se r v at i on p o i n t s f o r v e l o c i t y d i s t r i b u -u - ut i o n s wh ich a r e n ea r t h e w a l l t h e t e rm as eva lu a ted f rom thev*


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in te re s t t o consider fo r comparison the equat ion which Darcy refe renc e 6)o b ta ined in 1855, on th e bas is of c ar ef ul measurements. H i s eq u a t io n fo rv e l oc i t y d i s t ri b u ti o n , i n t he no t at i on of t h i s a r t i c l e , i su - u 5 . 0 8 j l z 12v*

In f i g u r e 17, equat ion 23) i s r ep re se nt ed b y a f u l l l i n e and equa-t i o n 2 4 ) by a do tt ed l i n e . The Darcy curve shows good agreement excep tf o r p o i n t s n ea r t h e w a l l where L < 0.35. This imperfection of therDarcy fo rmu la i s due t o th e f ac t t h a t h i s o b se rv a t io n s n ea re s t t h e w a l lwere for = 0.33. Up t o th i s l i m i t th e agreement of equa tion 2 4 ) withrth e observed d a ta i s very good.

3 . Exponential LawEven th ou gh t h e v e lo c i ty d i s t r ib u t io n i s adequately descr ibed byeq u a tio n 1 3 ) o r eq u a t io n 2 3 ) , t i s sometimes convenient t o have an

exponential expression which may be used as an approximation. Prandtlfrom a dimensional approach concluded from the Blasius l a w o f r e s i s t an cet h a t t h e v e l o c i t y u n ea r th e w a l l d u rin g tu rb u len t f lo w v a r i e s wi tht h e 117 power of th e di st an ce from t h e w a l l , references 22, 23, and 24 ,t h a t i s

u = ay117 2 5 )i n which a i s a co n s tan t f o r each v e lo c i t y cu rv e . t i s t o be empha-s i z e d t h a t t h e exponent 117 holds only fo r smooth pipe s i n the range of


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I f l o g i s p l o t t e d a s a f u nc t id n of l o g t h e r e r e s u l t s a s t r a i g h tl i ne wi th s lope n . Th i s r e l a t ion sh ip i s shown in f igu re 18 fo r va r iousdeg rees of r e l a t iv e roughness and a l s o fo r a ve loc i ty d i s t r i bu t io n i n asmooth pipe. Al l of th e vel oci ty d i s t r ibu t io ns fo r rough p ipes showni n t h i s f i g u r e l i e w i t hi n t h e r an ge o f t h e q u a dr a ti c l aw of r e s i s t a n c e .t i s ev iden t from the f igu re th a t w i th in the r ange o f r e l a t i ve rough-ness inv est iga ted here the exponent n inc rea ses from 0.133 t o 0.238.

From the recorded curve f o r th e smooth pip e n 0.116. I n or d er t odetermine t he va r ia t io n i n the exponent n wit h th e Reynolds number f o ra f ixe d re la t i ve roughness , the value of log g a s a func t ion of log 7has been determined fo r vario us Reynolds numbers and f or a re la t i v e

rroughness - 126. The change of s lop e of th e li n e was found t o bekvery sl i g h t with va ri at io ns of Reynolds number: The sm al le st recordedvalues of Reynolds number l i e wi th in th e region defined as range ofthe r e s i s t a nce l aw where the coe f f i c i en t o f r e s i s t ance I i s t h e samea s fo r a smooth pipe; th e next la rg er Reynolds numbers l i e i n range I( t r a n s i t i o n re g i o n ) , and t h e l a r g e s t i n ra ng e I (quad ra t i c l a w ofr e s i s t a n c e ) . F i g u r e 18 shows that points on the pipe axis deviate fromthe loc at io ns ob ta ined by the e xponent ia l law.

4 Prand t l s Mixing LengthThe well-known expre ssion of Pr an dtl ( re fe ren ce s 1 26, 27, and 28 )f o r t h e t u r b u le n t s h e ar i ng s t r e s s i s :

The determinat ion of the mixing length f rom the vel oci ty p ro f i le smay be e a s i l y ca rr ie d out by means of eq uation ( 27 a) . B y rearrangement:


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f ro m t h e v e l o c i t y d i s t r i b u t i o n s . T h i s s somewhat d i f f i c u l t i n t h eduv i c i n i t y o f t h e p i pe a x i s s i n c e t h e r e t h e v a l u e s of b o t h Z and dy

ar e very smal l . The p rocedure necessary to ob t a in th e va lue o f asa c c u r a t el y a s p o s si b l e h a s b ee n d e sc r i be d i n d e t a i l i n a p r e v i ou sa r t i c l e ( re f er e n ce 5 .

he d i me n si o nl e ss m i xi ng l e n g t h d i s t r i b u t i o n a r r i v e d a t i n t h i smanner f o r l a rg e Reynolds numbers ly in g wi th in the r ange o f th e q uad ra t i cl aw o f r e s i s t a n c e h a s be en p l o t t e d i n f i g u r e 19. The curve shown sth a t ob ta ined from observat io ns on smooth p ipes , expressed accord ing t oF rand t l i n t he form:

There e x i s t s , t h e r e f o r e , t h e same m ix in g l e n g t h d i s t r i b u t i o n i n rougha s i n s mooth p i p e s . T h i s f a c t l e a d s t o t h e c o n c l u si o n t h a t t h e m e ch an ic sof t u rb u le n ce , e x ce p t f o r a t h i n l a y e r a t t h e w a l l are independent ofth e t y p e o f w a l l s u r f ace .

I n o r d e r t o p r e s en t i n a compact f orm th e v a r i a t i o n o f t h e mix in gl en g th d i s t r i b u t i o n w i th t h e Rey no ld s number and w i th t h e r e l a t i v ero ug hn es s, t h e r e i s p l o t t e d i n f i g u r e 20 t h e t e rm l o g a g a i n s t t h e

verm l o g l o g w ach of the cu r v es drawn fro m th e t o p t o t h ebottom of t he f ig ur e corresponds t o a given Reynolds number which i s2i n d i c a t e d a s a p a ra m et e r. S i n ce h a s i t s l a r g e s t v a l ue s n e a r t h eY

w al l s , t h e p o i n t s f o r t h a t r e g i o n a r e i n t h e u pp er p a r t of t h e f i g u r ea nd p o i n t s n e a r t h e p i p e a x i s a r e i n t h e l o we r p a r t . The c u r ve s dr wnf ro m l e f t t o r i g h t c on ne ct p o i n t s o f e q u a l z -v al ue . These curves ar e


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There remained t o be determined whether i n thes e two pipes f o r agiven Reynolds number th e re si st an ce fa ct or would be the same andwhether the funct io n f Re) would yield a smooth curve.

There was fu r th er t o be determined whether the velo ci t y d i s t r i bu -kt i on s fo r p ipes wi th equal re la t i ve roughness are s imi la r and howrth ey vary wit h th e Reynolds number. The measurements show t h a t th e re i s

a c t u a l l y a f u n c t i o n X f ~ e ) .The ve loc i t y d i s t r ibu t ion s fo r a givenre la t ive roughness show a ver y s l i g h t dependence on t h e Reynolds number,but on the o ther hand, the form of the v elo ci ty d i s t r i bu t i on i s morepronouncedly dependent on th e re la t iv e roughness. As th e re l a ti v e rough-ness incre ases , th e vel oc it y di s t r i bu ti on assumes a more pointed form.study of t he q uestio n whether the exponential law of P ran dtl a ls o applie dt o rough pipes showed th a t v el oc ity di s t r i bu ti on s may be expressed by nexponential l a w of th e form u uyn i n which th e value of n inc rea sesfrom 0.133 t o 0 .238 as the r e l a t iv e roughness inc reas es .

Experimental da ta were obtained f o r six d i f f e r e n t d e g re s s of r e l a -4t i v e roughness wit h Reynolds numbers ranging from Re 10 t o 10 I ff low condi t ions are cons idered d iv ided in to thr ee ranges , the observa-t i o n s i n d i c a t e d t h e f o ll ow i ng c h a r a c t e r i s t i c s f o r t h e l a w o f r e s i s t a n c ei n each range.

In range I f o r s m a l l Reynolds numbers the res i s tan ce fa c t or i s thesame f o r rough a s fo r smooth pip es. The proj ec tio ns of t he rougheningl i e e n t i r e l y w i th i n t h e l a min ar l a y e r f o r t h i s r an ge .

In range I1 t r a n s i t i o n r an ge ) n i n c r e as e i n t h e r e s i s t a n c e f a c t o rwas observed f o r an incr ea si ng Reynolds number. The th ic kn es s of th elaminar layer i s here of t he same order of magnitude as th a t of t hep r o j e c t i o n s .


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express ion i s

v*ka b l o g 2 l o g jri n which th e valu es of a and b are d i f f e r e n t f o r t h e d i f f e r e n tranges

The v e lo c i t y d i s t r i b u t i o n i s given by the general express ion;

i n which = 5.75 and = 8.48 w i th in t h e r eg io n of t h e q u ad r a t i cv klaw of res is tan ce, and i n th e other regio ns depends al so upon .The re la t ion sh i p between the ve loc i ty d i s t r i bu t io n l a w and the law ofr e s i s t an ce i s found t o be:

/in which = 3 75 a s determined from th e Von K IEUI v e l o c i t y d i s t r i b u -t i o n law


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was used t o obt ain the va ria t io n of th e mixing leng th with th ed i s t ance y from the w a l l The fol lowing empirical equat ion resul ted:

This empir ica l equat ion i s appl icab le 0 4 y t o l a rg e Reynolds numbersand t o t h e e n t i r e range of the quadr at ic l a w of re si st an ce wherevis co s i t y has no inf luence

Translated byA A BrielmaierWashington UniversityS t . Louis MissouriApril 1937


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REFERENCES

1. Prandt;, L . : Turbulenz und ih re h ts te h u n g . Tokyo-Vortrag 1929,J Aeronaut. Res. I n st ., Tokyo Imp erial Un ive rsi ty, N r 65, 1930.

2. Von K , Th : Mechanische jihnlichkeit und Turbulenz. Ggtt ingerNachr., Math.-Phys. Klasse 1930 und Verh. d . 3. in te r n . Kongr. f .techn . Mech., Stockholm 1930 (~ to k h o lm 931), Bd 1. (Avai lablea s NACA TM 611.3. Nlkuradse, J Widerstandsgesetz und Geschwindigkeitsverteilung vontur bu len ten Wasserstrijmungen i n g la tt e n und rauhen Rohren.Verh. d. 3. in te r n . Kongr. f . tec hn . Mech., Stockholm 1930(stockho lm 1931) Bd. 1 p. 239.4. Prandt l , L . Zur tur bu len ten str;mung i n Rohren und l&g s P la tt en .Ergeb. d . Aerodyn. Versuc hsan st. zu ~ Z t t i n g e n ,4. Lief. , 1932,

p 18.3 N i h r a d s e G e s e t d s s i g k e i t e n d er t u r bu l en t e n ~tr0 m un g n g l a t t e nRohren. VDI-Forsch.-Heft 356, 1932 .6 . Darcy, H. Recherche3 er g ri m en ta le s re la t i v es au mouvement de 1 eaudan$le s t uyaux . Memoires a llAcademie d . Sc i ences de 1 ' I n s t i t u t eimperial de France, Bd. 15, 1858, p. 141.7. Bazin, H . : m l r i e n S e s n o u w l l e s s u r l a d i s t r i b u t a t i o n d es v it e s s esdans l e s tuyaux, Memoires 1Academic d. Sciences de I n s t i t u t ede France, I3d 32, N r 6, 1902.8. v. Mises, R . : Elemente de r tec hn isc hen Hydrodynamik. Le ipz ig,


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F r i t s c h , W : Der Einfluss der Wandrauhigkeit auf d i e tu rb u len teGeschwindigkeitsverteilung i n Rinnen . Z . angew. Math. Mech. Bd 8,1928, P 199

Treer , M F. : Der Widerstandsbeiwert bei turbulenten str6'mungendurch rauhe angle. Phys. Z . Bd 9, 1929, p. 539.Treer, M F .: Die Geschwindigkeit sve r te lungen be i gradlinigentur bul ent en str mungen. Phys. Z. Bd. 9 , 1929, p. 542.Kumbruch, H. : Messung s trz me nd er L uf t m i t t e l s ~ t a u ~ e r g t e n .

Forsch.-Arb. 1ng.-Wes. Heft 240, B e r li n 1921. ( ~ v a i l a b l e sNACA TM 302.)

Hoffmann, A . : Der Verlust i n 90~-~ohrkri jmmern i t gleichbleibendemKreisquerschni t t . M i t t hydrau l . Ins t . T . H . M;inchen, publishedby D Thoma, Heft 3, 1929, p. 55 .

Blasius , 8 : Das j iha l ichkeitsgese tz b e i ~ e i b u n ~ s v o r ~ k e nn~ 1 G s s i ~ k e i t e n . orsch.-Arb. 1ng.-Wes. Hef t 131, Be rl in 1913.Prandt l , L . : Rei bun gsw ider stan d. Hydrodynamische Probleme de sSch i f f san t r i eb s . Published by G. Kempf and E . Fsrs te r 1932 , p . 87.Prandt l , L . : Neuere Ergebnisse der Turbulenzforschung. Z . V D I Bd. 77,

N r 5 , -1933 , p 105. ( ~ v a i l a b l e s NACA TM 720.)Prandt l , L . Z u r tur bul ent en StrGrnung i n Rohren und lan gs P la tt en .Ergebn Aerodyn. Versuchs anstal t . ~ S t t i n g e n ,4 . Lief, 1932, p. 18.Frandt l , L : Ergebnisse der Aerodyn. Versuchsanstal t G tt in ge n ,3. Lief, 1327, p 1.


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NACA TM 1292 9

27. Prandt l L Berich t i h e r neuere Turbulenzforschung. Hydraul ischeProbleme Be rl in VDI-Verlag 1926 p. 1

28. Prandt l L h e r a u s ge b il d et e T ur bu len z Verh. 2. intern. Kongrf . t e c hn . Mech. ~ G r i c h 927 .


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T BLE

l M E N S I O N S O F T E S T PIP S

d in s id e d i ame te r 2 outl t l e n g t h2 approach leng th x t o t a l l e ng thl measur ing leng th f r e l a t i v e t o t a l l e ng thd

L measur ing leng th I


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ii cm/s v cm2/sr /k = 507

dyn/cm3 log Re

k = O . O l c m0.000351.000574.00084O.ooog75.000966.001525.00167.001g5.00230.ooegl.00285.00347.00372.00410.00496.oogg7.00718.00878,01087

.01085.01255.01378,015150202.0245.0314.0372.0435.0458.0501.0565

.0760-0975.I310.1585

15.4520.225.027.327.334.436.840.444.046.450.055.958.561.869.076.084.494.0103.5106.0114.0119.8126147162184201217223234248287325375412

4.1144.2304.3224.3624.3624.4624.4914.5324.5684.5914.6234.6724.6904.7164.7634.8064.8514.8984.9404.9735.0095.0255.0495.1005.1435.1995.2365.2705.2815.3035.3265.3775.4305.4935.534

0.0118.0118.0118.0118.0118.0118.0118.0118.0118.0118.0118. o n 8.0118.0118.0118.0118.0118.0118.ox=.011201120112.0116.0116.0116.0116.0116.0116.0116. o n 6

.0120.ox20.01200120

l o g ( l 0 0 A

l o g r / k = 2 . 7 0 50.51.64-79.86.881.051.061.161 . 171.241.311 . 411 . 481.521.621 . 651.741.791.861.891.901.921.932 .OO2.022.052.072.062.072 102.082.062.072.032 01

d = 9 . 9 4 c m0.456.438.417.407.403.381,380.366.365.356.347.333.324.320.307.303.292.286.278

.274.274.272,270.262.260.255.253.255.253.250.252

.255.253.258.260

0.000.OOO.083.117.114.2122 3 6.267PO.322.348.391,407.428.470.508.549.593.638.661.694.713.733.781.829.878.g19.944.959.9711.004

1.0531.1071.1721.214

2 log r /kn 2 83stv&log

4.955.355.755.956.026.486.556 806.877.057.257.507.727.858.058.088.458.588.788.858.898.958.979.179.259.299.369.359.369.459.429.359.369.259.19

u

0.815.819.824.825.824.825.830.829.832.832.834.836.835,838.839,842,841.844.843,845848.845.847.846.847.849.847.849.849.846.851.847.84g.849.846


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k = 0.02 cm d = 9.94 cm log r k = 2.401

ii /s v cm2 s dyn/cm3

2.022.102.12

log e

0.769.884,966

log(100 A

9.259 .9.530.836.840.839

2 log r/k v*klog ;. 2.83 Cfi

IU


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TABLE 4


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r/k 60k 0.02 m d = 2.434 cm log r/k = 1.78

cm/s

I k 0.08 cm 9 8 cm log r/k 1.78 I

cm2/s dyi/cm3 log Re log 100 X v klog -2 log r/kQ \li


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cm s v c a / s ayn/a3 lo g Re log(1 00 L 2 log r /k log C

k 0.16 cm d 9.64 cm log r/k 1.486


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B cm/s v cm2/sr/k = 15

3 dyn/cm3

30.834.537.442.046.651.056.060.661.266.469.477.080.095.099.5105.01115ll8.0124.0131.0133.4149.0169.0196.5214266325364375647484

532560640675788

lo g Re

0.772-772.767775.769.765.765-767.763.760.765.764.760.756.761.758.759.757.756.758.755.755.758.755.756.756,753.755.756.756.754.754.752.756.756.755

l o g r / k = 1 . 1 7 62.142.132.092.062.041.971.941.941.901.871.921.921.851.821.81~1.781.801.741.771.771.721.741.731.721.751.731.721.741.751.731.721.721.711.741.721.73

l o g r / k = 1 . 1 7 675.586.595.0108.0128.5150.0184.021293.5

218246248254

log(100 X

0.0126.0126.ou6.0126.0=6.0123. O E ~.0123.0123.0123,0123.0123.0123.0123.0123.012 3.0123.ole3.0123.0123.0121.0123.0123.0122.0121.0121.0120.0120.0118. o n 7n 7.On7.0108.0108.0108.00g8

0.00995.01260.01505.01920.02392.02g50.o3600.04220.0439.0526.0559.0695.0767.lo97.1192,1370.1526,1765.1930.2147.2280.282.364.493.580.goo1.3501.6801.7762.5402.9823.6114.0195.1005.8097.900

k = 0 . 0 8 c m3.7703.8203.8553.9053.9554.0004.0414.0764.0794.1144.1334.1794.1964.2704.2904.3144.3404.3664.3864.4104.4254.4664.5204.5904.6304.7254.8114.8654.8854.9655.0005.0425.0985.1555.1795.285

k = 0 . 1 6 c m

1.1881.2391.2761.3171.3771.4351.4771.5111.5201.5721.6191.6411.7181.7371.7671.7911.8221.8411.8651.8841.9241.9792.0492.0872.1842.2782.3222.3422.4222.4582.5002.5662.6082.6362.746

2 log r/k

d=2 .412cm0.696.699.707.712.717.730.734.736.744.751.740.744.754.760

.756.769.763.778.772.772.782.785.780.781.777.780.781.7TT.776.779,781.780.781,778.781.779d = 4 . 8 2 c m

9.699.669.579.469.409.239.149.139.039.069.068.898.808.858.698.748.548.648.648.498.548.538.528.568.548.528.558.588.548.528.518.468.548.528.54

0.0132.0132.0132.0132.0128.0128.0127.0126.0120,0118.0118.0118,0098

2.913.824.605.968.4211.5017.3019.1023.0024.230.931.433.0

lo

4.4404.5004.5404.5964.6854.7224.8454.8694.9294.9495.0025.0055.097

1.751.751.731.721.721.751.751.731.721.731.741.751.73

0.775.777.778.780.781.777,775.778.780.779,777.775.778

2 83 cB i

0.756.755.756.758.755.757.757.755,755.754.756.754.755

1.8991'9571.9982.0552.1442.2102.3002.3272.3912.4092.4602.4642.555

8.598.548.538.528.558.598.548.518.548.598.54


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u = v e l o c i t y a t ny p o i n ty = d i s t a n c e f ro m w a l liidR e = = Reynolds number

= a v e r a g e v e l o c i t yd = d iame te r o f p ip ev k i n e m a t i c v i s c o s i t y

-r0 = s h e a ri n g s t r e s s a t w llp = d e n s i t yk = a v e r a g e p r o j e c t i o n o f r o u g h n e s s

0 .936 3039 .5 2 9 .2 8

= r e l a t i v e r o u g h n e s sk

1 Tov* = jy = f r i c t i o n v e l o c i t y

1u C S Y cm/sdy

98.0 265.0150 225 3646 5 11 6 402180 69 432190 50 458201 36 487208 29.3 505220 20. 8 5312 30 1 5 . 9 5 5 2237 13. 1 5682 4 2.5 1 0 . 8 5 8 12 4 7 . 5 8 . 7 5 5 9 12 5 1 . 0 7 . 0 1 6 0 02 . 4 .7 4 6082 5 4 . 5 2 . 9 3 6 1 1 . 2254.8 2 .08 611.6255.0 612.0

ddy5853061 7 8112990725 1

3 1 .82 6 .22 1 .41 7 . 01 1 . 67 . 35.2

1 . 5 39 O

Iu cm/s

1 . 6 8 68 .6 8

25205736256526897197583 8 . 7 7 8 8812832848861871875

u cm/s52360867027618038328749129409609789921003

1008

dy8434442561821 2 8l o 17255453730241 6 . 5

1 0 .2

dy10225 2 63092221 5 31 1 984645 2 .543.635.52 8 .41 9 . 2

1 2 . 08 . 776877 7.31 10101 11


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u v e l o c it y a t ny p o i n ty d i s t ance f r o m w l lRe nd Reynolds number

ave rage ve loc i tydiameter of pipe

v k inemat ic v i scos i ty

v+ f r i c t i o n v e l o c i t yT~ s h ea r in g s t r e s s a t w l lp d e n s i t yk average pro jec t io n of roughness

re la t ive roughnessk


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TABLE 10

u veloci ty a t any pointy distance from wall

iiRe Rey nol ds number-u average velo cityd diameter of pipe

: kinematic viscosity

v f r i c t i on ve l oc i t yP~ s he ar in g s t r e s s a t w l l

de ns i t yk average pro jec tio n of roughnessre l a t iv e roughnessk


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T BLE r

d cmu cm/s1 0 2 ~m2/s10-3 R ev*

v*kl o g

u v e l o c it y a t ny po in ty = distance from w a l ludRe Rey nold s numberaverage veloci ty

d diame ter of pipev kine mat:^ v iscos i ty

Tv, \lp f r i c t i o n v e l o c i t yT, s h ea r in g s t r e s s a t w a l lp d e n s i t yk average project ion of roughness

rela t ive roughnessk IUuro


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T BLE 12r /k 30 6

d cmu cm/s102 v cm2/s10-3 Rev* v&l og

u ve l oc i t y a t any pointy distance from w a l lR e = - =y Reynolds number

average velocityd diameter of pipekinematic viscosity

t-1u1

v, f r ic t ion veloc i tyr O shea r ing s t r e s s a t wa l l

dens i tyk average projection of roughnessre la t ive roughness


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Figure 1. Relation between the resi stance factor ~ and the Reynolds number fo r su rface roughness.The numbers on the cur ves indicate the test resu lts of vario us investigators.)


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Figure 2. lation between the resistance factor 1 . and the Reynolds number for surface corrugation2(The numbers on the curves indicate the test results of various investigators.)


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N C T 292

Figure 5. Hooked tube fo r m easu ring s ta t ic between wanand o bserv at ion point is


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NACA T 292

F i g u r e 7. Co r r e c t i o n c u r v e f o r d e t e r m inin g s t a t i c p r e s s u r e .a i s r e s i s t a n c e o f h o o k e d t u b eh s res is ta nc e of smoo th p ipe


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2.6 2 8 3 0 3 2 3 4 3 6 3 8 4.0 4 2 4 4 4 6 4 8 50 5 5 5 6 8 6 0Figure 9.- Relation e tween log 100h) nd log Re.


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l 0 / 2 / 4 1 6 / 8 2 4 2 2 2 4 2 6 2 8

and l o g zigure 10. Relation between l k

N C TM 292


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F i g u r e 12. Rela t ion between and wi thin the region o t he quadra t i cU rlaw of resistance.


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8 / 2 /.6 2.0igure 14. Relation between ?? nd log .v


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N C T


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292


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2 4 6 8 O 1 2 / 4 1 6 18 2 0 2 2 2 4 2 6 2 8 3 0 32

F i g u r e 16. Rela t ion be tween 5 .75 log 83 - 5.75 log r + 3.75

NACA T 292


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Figure 17.- Relation between E nd :).v


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igure20. e l a t ionbe tween log 10 and log q;)


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v kigure 21 Relat ion between and log

laws of flow in rough pipes-nikuradse

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