Changes of body mass index in celiac children on a gluten-free diet

TURBULENT BOUNDARY LAYERS IN INCOMPRESSIBLE FLOW J . C . ~ Aemdym~h~ v e ~ ~ Contentn P~ t~u,~ L ] [ q Y W D W d J . 8 0 ~ Tm131~LI~ m J t J t T l,t'rIB 1. X a t r e d t ~ a 2. Basie ~ and mst,hernstie~ fo~nulati~ of the l ~ , b h n 2.1 E q u s t , ~ of motmn 2,.5 ~ om~t.i~. ~,.3 lh~mun, h e t u a t i a m 1. SmzisCic~ themT of t m b u l m ~ 4. ]kxmdm.~,. ~=:'w equmZo.~ in aud,istiml m g ~ ~ & Hy.~w~ mist~m

5 e e S 10 11 IS IV

C s t u ' s ' ~ I L UxnrsmatL ~ T z e n , m, ",-,'- s m ~ , t x m J m' 1"mmezau~ i~TJTDATLT T*JTT~ FLOW

e. ~ , u b l e v ~ l ~ y om-~la~i~n f ~ and speeW~a f 6.1 ~ double ~ l ~ J ~ ewrela~on ~

u

~

Ez'pu'imen~ n=uJu

6,3 Speeq'um funetiem

7. y . n e , ~ d i . p m i m by v i . e e . ~

Ik 9. 10. 11.

M m

2S Jt

SS

~.1 ¥~oeem of turb~eat e m ~ 7 dimipstion ~.~ l , o o ~ y imtmpie turbulaa~ ":.3 ~ - r g y spectrum of locally kotmple t m b a l m ~ Double velocity eq~utti(m= The ~ g y Wan~er amoug d i d e r o t velocity oompementa ~ e eq~'b.-ium ~ o ~ s in ~ t abemr fimr The flew n e ~ the w~U I1.1 f~;,-n-~ity eemldemtkma 11.2 The behaviom- of the ~ n , ;- the immediate vicinity of the wa~ 11.3 F-~q3y ~ 11.4 ~Fm-t.her d e t a ~ of the 8trneture of wall flow 11.S C,eneralJz~tiom of the uaJvenm~ law of the ~

43 47 49 84 54 U 59 60 70

1 l.e Wall ~m, on reegh ~

-n

11.7 Representation of the mean velocity dj~.-ibutiem 12. The free ~ boundary of the turbulesat layer 12.1 8tatktical description of the free stream boundary I

58 ZV

78 I~t

2

J.C. Ro~x 17.2 The prooeva o' s - m ~ . i ~ p r o ~

U

12.3 In/'ere~e of' tutti-.hOe properties from t.ho isltermlttent J i ~ b 12.6 Turbu/ms= d i / l ' ~ in t.he s~.gion of t.~ fi'~ ~ ~ 12.3 Flow outa/dm ~ turbaka~ ~

91 93 94

LAT'/~m

13. Boundm'7 layer on 8 f l ~ p l a ~ with zm'o premm~ ~ U m ~ 13.1 Loe.~ ~ L s ~ ] " 13.2 E z p e r i m m ~ s,em/t~ 13.3 Theon~cal background of'the Ioe,al ,~--.'lar/ty 13.4 TIs~ ~ balan~ 14. F..q-~l;btiv-',* bomsd~-y layeml 14.1 Corn:fit/ram for aqufl/bn-,~ layml 14.2 Experimeat.ed v m ~ t J o a of equ~br/um l~ymm 14.3 Dowmtrmtm s t a i ~ t ~ 15. Boundary l~ymm in srbiLrazT ps.ewure d/~tz~3uiioa 18.1 l~'oa-dJmemionJl r e l x . a e n t ~ i ~ 15.2 Detaib of the motims le, s e p m u ~

96 97 100 105 114 119 120 "124 127 121) 131 158 lt6

xv'. T~c CAZ~qTLL~O~WOJV~ ~UTJ~rm IT- The problem of boundary layer c a l c u l a ~

183

18. I)eacr~ption of the m e a : veloeity profiles 19. Loca/skin fi-iet/oa oo.~cim~ 19.1 Skin friction f o r m ~ 19.2 D~erminat/on of sL.~ fri~'on m ~ m t

boundm7 '-yem

~o~roa]~r L~Z~

186

166 10q far e

x

~

lV0

20. Detm-miuation of ,Ismu-~j mcmm ~ 17,~ 20.1 T h , ~ pm~bh,m 173 20.2 The i ~ im~bi~ 1":9 20.3 The int~.gml of the ener87 dimil:m~.imz 182 21. The turbuleu~ tmundsry l a y ~ on a fiat platm 188 2:.= InZegmtlou of the mmmmtum equ~,;~* 18,q .el.2 Boundary layer ~m ~maoth surf'aoe IS7 21.3 ]~oundsry b.yer on uniformly rough msrfa~m 189 ~I.~L The tot.~ s k ~ ~.:m~e, coefllcle~ 191 22. Tho drape p & m m ~ r equ~.iaa 192 22.1 ~ m p u ~ ! d ~ p e l~u~met~w ~iuat~om 18(I 22.2 Shape 1 ~ equstioas bamzl or, boundary I~yer equ~Jmm 197 22.3 ~,am.;,,-tion of" the shape psrsmeter eq,,-*;on 199 23. Some ~mar.~j ~ Lhe n,~mer~o~L!boundary layer m~=u~aUo~ 204 23.I ~ t ~ v~luc~ 205 23.2 ~ , p p ~ x i z ~ . t ~ ~

20"/

TURBULENT

BOUNDARY

IN INCOMPRESSIBLE

LAYERS FLOW

J . C. P.O~A Commonly nsed symbols C(,,~,I,) C, (=Iv

E(k) F

,I',,

emstaat ~amd by ~ . (Zl.40) c o m e t de~**d by Eq. (11,61) lore1 akin friction e m ~ e i ~ t total akin/~iet/~ ~ e m e ~ , q ~ e m n fuaetkm normalized velocity defect function, ~ normalized bequeaey ~ fun~m

(IS.IS) and (16.2}

N - r/e..ha~ Ho - e-/e, - ~ - p . ~

.f I,

J¢

i=

~ pa,=mm~ ~ by Eq. {lS.ie} •hape p m . = n e ~ defined by ]~q. (18.e} eomtan~ defiaed by ~ (13.12) and (11.~) wsve numbm'

1

P P |

U, F

~ogp u~v/s) ¥ Y £ i

E

mean value of pmsmm pre~uzu ~ u e t ~ t i ~ us +es.F.ms double veloei~ eorretstion I'un~on defined by F,q. (e.l) time mma veloci~ eompmnuW ee~ponenta of ~Ioeity flu~.usti~ frier/on veloe/t7 local veloeit'y of vos'ticity mean veloei~ of vm't~elty pmpqatlon wake, function slmm ~ t m diataa-e of free s~mm boundary from the wail. 1~. 12.~ iatenmittmey factor length ,ode defined by Eq. (!3.17) total botmdary htyer thielmma d~l,~ement thie~w= decried by Eq. (4.1S) thielmam of sublsyer thickness of supedzy~ total rate of enerzy dissip-Ltlon swte of tuzbulent enerl~ dimdpation eddy viaeeaity non.dimensional cQ.ow:linatedefiszed by Eq. (13.17) S

4

,I. 0 _ . . I t ~ ' ~ I le .I

momen~un t h / c k n ~ dermal b~ F,q. (~.1~) ~ ~ bT F.,q. (17.3) .-,~ver~ con~,~S d e ~ e d by Eq. (II.e)

il

+ 1I 0 f

~,/~,, ~ vbMiW n o u ~ ~ d d ~ d by F_,q. (l&l) nnu.dimms/on~ pa'smm.e u s d i e n t penunet~ ~ b 7 F.,q. (16.2) dmJW •,hesr/~ sU~m non.dimens/onsl ~ ehesr/~ ,it.rein d e ~ n ~ by Eqs. (13.18) (1~)

a

tuner/on ~

bT Eq. (l&,J) v'(edS) ~ m p e m ~ of vur~i~e~ ~uruut~m u,,llT. -

OD-

~ U t ~ o n . , ~ outer odse of bounds~ l s ~ . (.y - 8) e ~ d i t i o m M tho wsn (.v " 0)

Cax~=wt I

FUNDAMENTALS OF TURBULENT BOUNDARY LAYER THEORY 1. Introduction In the year 1904 I~ Prandtl ~ showed that in flukis of small frietion the viscoaity s~'ects the eow only in a very t~;- layer in the i m m e d h ~ neighbourhood of tl~e solid body, and thLt the developmentof this l~yer is responldble for the major part of the flow resistance of the body. Very soon it was di~overed that the flew can equally be turbulent w ltTrJ,mr in this kyer, which is now genet~Uy designated ~.he boundary hayer. Since that time a vast number of experimental and theoretieal investigations on this tubjeet have been puhlisheit. Nevertheless, the basic problem of the turbulent boundary layer in an i n e o m ~ fluid i a g far from being mired. Meanwhile the range of inteee~ lure immen~ly inereaaed. Modern developmenta in techuletl seienoe lutve raked many new questions, for inst~ee what l u t ~ to the t u r b u k ~ boundary layer, whe~ at high speeds the friction eatu~t hetvy telnpel~ tare variations, when fluid k removed or injeet~ ~ g b the swrfaoe, and when many other changm are imposed on the boundary iayee. On the other hand the interest in turbulent flow M a phyldea] problem of general importance grew in the same measure -I this problem maisted inemudng effort, to find a Mt~f~ctory Jolution for engineering requlremenq. This article confines itself to a rather amall part of the wide field of turbulent boundary layer ,~roblema. It deals with the turbulent b~.~dsry layer in s steady incompreuible, two4iimensional flow on plan, impermeable walls. (3rest emphasis is put on the representation of the fundamental phenomena, because these form the buis for furthe~ advaneee in more general cues. It is felt that a profound knowledge of the turbulent mee~ankm and the ideas underlying the buie eoncepCiom is of gnutter value to anyone concerned with turbulcut boundary htyem than any of the convenient calculation methods, kithough the existing picture on the motion k still rather incomplete and precarious. A broad chapter is devoted to these problems, which is preceded by an acoount of the fu~amentnl equations and principles pertaining to the theory of turbulent boundar7 layers. Thereafter the beh&viour of particular s

J . C . ROTT.L

6

boundary layers is treated, and in the ~,,~] chapter e n ~ e e r i ~ e~lculstion methods are rev-~ewed. 2. Basle E q u a t i o n s and MathemacI,-~, F o r m u l a t i o n of the P r o b l e m As the first, step we sh,l] discuB the ~ss/o equations for Curbulent boundary layer flow and attempt to formu~to the problem mathematically. We shall consider the two-dimens/onal case c ~ by the flow conditions being independent, on the statistical average, with respect to the z-axis and b e r g 8y~nmetrica] with respect to the ~-plane.

FreeemmmmD tdremmt

Y ~j~ m . .al ~

"am~m'

/ ~-"'1~a-'-~--"r I ~ v(~.y}~=~N(,~) • (

I

~

~

'

d

~'/'f~///'/"

~ o . ~L sket~ o t t ~ boundary ~ w . The z. and z-axes co/no/de with the surface of the solid body and the y - , ~ - / s at right-anKlce to the surface, Fig. 2.1. ThL, cMe corresponds to the flow about & fiat cylindrical body, wh/oh stretche~ to infinity in the direct/on of the z-a~s and moves at right ankles to the z-axis through a fluid at rest. 2.1 E q u a t i o n s of m o t i o n A/though the content of this article k con~ned to steady flow, the equations will at first, be written down for L general motion varisbk in t;me. Th/s ;,, done with regsrd to the subsequent ctiseumon, which is intended to d e m o n s t ~ the di~culties inherent to the theoretical treatment of the problem. ~ e velocity components of the flow are related to a coordinate system fixed in the body. The turbulent boundary layer as weU ss the l~m;nar one is governed by the l~sv/erStokes equ~ions and the continuity equation. It is in the spirit of the bonnda:y layer conception, introduced by L. Prandtl in 1904, that & number of simplLfieat.ions of the complete flow equations are sdm/ssihle. These result from the fact that the r~te of increase in the boundary hayer thicknces 8 in the downstream d/reefion is relatively small, 08[£~ ~ I. In consequence of this, the velocity component F at right angles to the surface is at least one order of maguitude emaUer than the

7 eompo~ent U in d i . ~ o n of the" z - s ~ (r" < U), and the dedvativee .of svemged qmmtitiee with re,spe~, to : remain small in-oomparimn with derivative, with respect to ]7, (e4;. ~Ulaz < ~U[~y). ffmm in addition, the t h i ~ of boundary layer is 8enerally suppmed to be mall eom~ with the d ; m e ~ o l m of the body, the effect of ourv~ ture of the surfaee on which the boundary ltyer developo is neglJtp"ok. The turbulent boundary layer difl'ere from the ~amimtr one in that the velocity vomponent~ fluetmtte randomly with respe¢~ to q ~ o e sad time about a mean valu~ Them random £uctuatinm ~ , for im~neo, respondble for the difusivenets which ;- one of the most striking c h ~ e 8 of the turoulent motion. _q'ne velocity OOml)ommta in the three oo-ord;n~te directions can be introduced in the form in the z ~ o n

U(z, y, t) +,,(z. y, z, t), in t~,e ~ ~ o ~

T'(=. ~, t) ~ v(z, ~, z, t), in the z - d i m ~ o n

o+=,(=. ~, •, O, where the mean values of the Teloeity eomponenta are de~gnated by eapitsl ietten m-.d the fluetuating p a t ia denoted by m a l l l e t t e ~ Team are teveral ways of taking mean values. In the pzmont e ~ e tho-mma values are zimply defined by

: " ~ 2zl J D-.e,

and Lre inat~endent of z by defmitmn of the twt~
/'(:, ~. t)+~,(z, y, :, O. '[~ne mean 'v'alum of ~, e, ~ &nd ,p are zero by definition. Toe deoomlmaition into ~ mean part and s fluctuating l~rt, whiob w ~ first done by O. Reynolds in 18~5, s u g g e ~ it~.l[ since the motion is so oomplex t h ~ it can quantitatively be described only in terms of moan values. Only wit]= the aid of this decomposition does it t~-¢ome possible to appiy the eimpli.rying principle of the boundary lsyer oonoept. According to e ~ m e n t a l observations, the velocity fluvtuatio=s are one order of magnitude smaller th~n the mean velocity U.(=, y, t)

J.C.P.o~.

S

outside the boundary layer (u, e, ~o < U.). The tJ~ree veloci~ fluctu~ ~ion components are of mime order. However, the derivatives of the • elocity fluctu~LiOM with respect to the co-ordin8~ axes ( ~ [ ~ , e t a ) ssaume very h ~ peak values, which are often materially greater than the gnd/ent; of mean veloe/ty 8~71~!~. All ~erential quotients of the veloe/ty fluetustioM are of tim same order of mslpdtude (~tlaz ~t~ ~ a,l~~ l a . . . + e l ~ - ~ , l a z - ~ l a z - aml~ - ~ l a ) . This is an indication of ~ very high degree of randomly distn'buted vortieity, which is, besides the di/~umveness, another i m p o r t a ~ eharseteristie of turbulent flow. For the flow defined in t h k way. the general l~av/er-Stohes equations reduee to

f.~tU+.)+ -

a -

-

~ 1a

~u

/~U

ao ( u + , ,ao ~ + ~av +(v+,~+ ~ 8z . .

Vzu)'

~e+,)+,v,,.

"~ +

-~-+(u+~

+

+(v+o)~+~a= - - ; T + . ~ ,

(2.3) 42.41

where the symbol Vz denotes the Laplace operator, VS - ~ / ~ = + ~ / ~ m + ~/~z=. p is the dendty and ~ - ~,/p is the kinemst/o of the fluid. ~n~logoue to laminar boundary layers the terms / T a F / ~ and VaF/~y are small to the second order when compared with USU/~ and FSU/~y, and are thus negligible. The same is true for the terms ~/7/,~s, ~ F / ~ a, ~F[,~-4, when compared with ~U'/~js; accordingly these terms ace also suppressed. The terms u 817/~ and • 8F/By are at lease one order of ma~n/tude smaller than / 7 8 U / ~ and F~U/d¥; they have not been neglected. Tl;e term u 8V/8= is again small to the second order and thus negligible. By no means negli~ble are terms like ~ [ & , u ~ [ ~ , ~V'~etc., sa will be dizenued in more detail later in ,h~, srti©le. The system of Eqs. (2.2} to 42.4) thus represents a second order approx/marion to the problem. The continuity condition is to be satisfied separately by the mean velocity and the ~luetmtting parts because of Eq. (2.1); h e n m 8/7 aV a=

+ .-

~t 8e ~o --+--+---

o,

(.~.5)

o.

42.6)

Eqs. (s.2) to (2.6) provide the set, of fun,4-,,',ental turbulent~ b o u n d s ~

laye+ 2.2 B o , , - ~ u ' y ¢o,',Wtlo,,The solutions of the system of di~erential equations (2.~) to (2.6) are not uniquely defined until the corresponding boundary eunditio~ are speci~ed. AI for laminar ]ayem, the condition of no aUp at the i s t o be ~ k f i e d : y-0:

U-

U-w-

w-w-0.

At the outer edge of the boundary layer, the 9ow a ~ e s eally the pre~nibed external flow: -~ ~ :

U -

a~uq~d-

U . ( z , 0-

I n addition to thk, the distKoution of the veloeity with rwpeet to :=, y, z is to be given at an initial instant | -- t~ Then the flOW e~ a~y t i m e t ~ to is completely determined. The principal di~culties of • theoretical iolution arke from two (1) the non-linear, three-dimensional ~ of the l q ~ ~ equations, (2) the proper Ipeeification of the boundary oonditionL The L v ~ b | e ~ t b e m a t i e a l methods ans not mdleient to loire in a 8~nend w~.v ~ mt of partial dWerential equations of the .type of the Navier-Stokes equation. Indeed, there are, ]~Nddm turbuia-: flow, many other unsolved problems in fluid dynamics. In order to d e m o ~ strste the di~ieulties involved with the specification of the bmmdsry conditions, consider a semi.infinite ,.hl. smooth tilt plate p l ~ e d in infinite body of fluid at ~ t . l~ow let the fluid begin to move parslkl to the plate and at right angles to the leading edge of the plate~ The velocity may grow gradually from zero up to a fins] eonm~ut value. Then ~ boundary layer forms on both sides of the plate. This earn ~ u be treated by Eqe. (2..~) to (2.6), eventually by means of numerical methods. When U - 0, ~ - ~ - ~ - 0 is aasumed M initial eondifim~ for t -- ~0, the caioulat'on gives th~ leminar solution, ~h~eh eorrespond~ to the Blasius solution of the laminar fiat plate boundary layer after the steady state ~ rea~ed. ~Uhen a eorTe~ponding experiment i~ oondu~:ed in the laboratory, the boundary layer flow changes to the turbulent state of motion at some distance from the ~ edge of the plate. ~ l a t causes this difference? Certainly not the fa¢~ that the fluid and the plate have finite dimensions in the laboratory. After the p-._-e.sge of more than half a century of boundary layer research, the answer is

I0

J.C. Eo~A

e a s y given. The laminar boundary layer be~mea uustahlz ~ some distamee from the leading edge, and the stream of fluid is not fx~e of snail perturbations. The fluid, for eT*,nple the air in s room, is usually in s very unstable state of equifibrinm, even the smallest uon---i¢orm ~h~-ge in temperature due to the motion of an object stint the siz up again, so that i t never comes completelyto rest. These mudl, irregular motions superimposed on the stream in combination with the instability of the laminar boundary layer are the reason why the flow becomes turbulent in a tea/boundary layer. I t / s beyond all dispute the~ neither quantitative nor qualitative affeete of the free stream perturbations are experimentally observed on the mean quantities of the tnrb,.dent layer st s m~c/ently lm~e distanm from the leading edge. This hold& of course, only as Iong u the free strea~n pezfurbstions are a m ~ compL,~[ with the veloe/ty fluetuations in the boundary layer. It is i n f e c ~ from t h k that turbulence obeys ita own law of mot/on. The free streean perturbations are neeemary, however, to initi&ce the turbulene motion. Aeeordingly, i~ sppesrs po~ible in prineiple to e e J ~ thz turbulent boundary layer by numerical integration of the differential equstions (2.2) to (2.6) for the unsteady boundary layer flow-although st s tremendous expense of labour--provided one succeeds in q~eifying properly the state of perturb&tion in the free sate-m= Obviously s rather complex dimtn'bution of the perturbations/s to be ammmed,/n order to obta/n s solution wh/ch resembles s real turbulent~ boundary layer.. The random ehemetan- of the free s t m ~ perturbations_ /m vury probably an indisp~,~hle supposition for the development of turbulent flow. The author is not aware of any attempte to Ikt~l~krthe ~ l f ~ in this w~y, edthongh /t appears not quite hopelees to eonduet such eaJeulations with the use of modem computing maehinm. 2.3 P r e s s u r e fluctuations The pressure is not an independent quantity in the se~ of equations (2.3) to (2.4); it is in f a ~ determined by the velve/ty field. Tn general the pressure can be e ~ d e ~ from the veloc/ty field by ~ the diwn~enco of the complete l~avier-Stokes equations and applying the continuity condition. In this way, the quantity (l/p)VzfP+p) is obtained in terms of velocity derivatives. In the ease of the boundary layer it is, however, more convenient to deduce the mean vsl,e of the pressure P directly from the averaged boundary layer equa~/ons as shown in Section 4. Thus we will give here only the relations for the fluctuating part of the preum~, wh/ch are obte/ned in the following way. First. (l/p)V~(P+p) is determined in the wsy juse mentioned.

e~r..b term. Then f:om 1

1

I

P

P

-v~ - - v ~ +~)- -w-O'+~)

(~.~)

the f o n o w i ~ rdation ,~,,~ts for the fluet~atlng part of the

s~;~

~.'

~

~

-

,~. . -

~I'

~+ (~.8)

where the boundary layer -~,~plifieations have again been taken into oonsidemtion. The pressure fluctuations ~ i ' y Poimoa's d~We:eatial equation. By applleatJon of Green's theorem, the value of the Fmmm~ at any point can be found as an integral over all sl~a¢~ Far points not too cloee to the

. ~ . / a u h &.(x') I,'1

+'_

+ _

~'T; + a~e; - a:~/J-'~T"

(2J)

There p is measured at position x, the velocities are m e s s w ~ a t x', and r -- x - x ' . It may be noticed that the total pressu~ fluetuatio~ are the sum of one part, ~hich k proportional to the mean velocity gradient ~U]~, and another part, which is established by i n ~ o a of different velocity fluctuation eomponentL 3. Statlstlca] T h e o r y of T u r b u l e n c e T~e task of turbu;enee theory is not the deter~;.ation of the ind/~dual motior, from an exactly given initial Te|oeity distribution the determination of the Mtstistica] properties of the typical motion. This fact and ~he observation that the statistical properties of the typic~l motion are unaffected by the properties of the small free stream perturbation suggests treating the turbulent flow as a problem of ststb. tica] mechanics. This conception considers the turbu]enoe as a ~ v e n

1:~

J . C . ROZTA

phenomenon, and om/ts the question as to how this motion h ~ d~wlo t ~ i from the initially calm flow. The bsa/s of the theory is sgs~n Eqs. (2.2) to (2.S),where ~Ul~t and 8 P'/~ v~nlmh for a steady mean flow, together with principles of prol~biItty theory. Several po.~btlifies have been tried bu~ none has yet led to sneceu. Indeed, turbulent motion is more complex than for example the motion of the molecules of a g ~ , 8in~ the motion of a continuous flu/d at any point inJiuenoes the mot/on at other points of the 8ow space. It seems not possible t.~ de~ribe turbulemt fluid motion by a shnple model I~ is not the purpo~ of this article to diseuas in more detail the various ektempts which have been made along this line. The reader is re/erred to the current literature on this subjectt, ao, ~. However, the descr/pt/on of turbulent motion in terms of statistical quant;ties has revealed many important relations, and has often formed the b u i s for further experimental inv~mtisatiorua In fs~t, most of oar ~ ut knowledge on the mechanism of turbulent flow is the result of combined e~'orts of mO, ist3ee3theory and experimental meeazeh. Chapter II of t h k article prments more details on this subject as far u they am useful in me,kinK the belmviour of turbulent boundary layers better 4. B o u n d a r y L a y e r Equations In Statistical M e a n Values Since strictly theoretical methods hr,ve not yet been developed to an applicable stage, the cun~nt calculation methods fcr turbulent boundary l~yers are based on statistical averages of the fundamental boundary layers Eqs. (2.2) to (2.6). The average of the flow propertiea is conveniently taken with respeet to z-direction, in secordan0o with Section 2. In other words the

opemt/on

'I

~..,,m 2el

at-&

is applied to e~eh term of the equatioue. The subsequent disc-radon will be confined to steady mean flow. Consequently, all d/~erentis3 quotients with respect to time vanish, when the average is taken. Then, with the use of Eq. (2.6) the Eqs. (2.2) and (2.3} become

U_~ + Fao" ~

~

tap

a~p

m

~v

m a:

+

8~ @

-

I ~P

. . . . , pay

(4.2}

g ' ~

J ~ a d a ~ " Zayerm in l ~ m p ~ z ~ e

Flmo

IS

while both sides of Eq. (2.4) s~e identically equal to zero, ~noe the product mea~ values G~ and ~ v.~;.h by symmetry. When expezimonte are eondueted, it is move oonveniont and has _beoo__meeommon pract,iee to take the average with respect to tame. The svex'agee taken with respect to time and z-dL-ec~on are equive3ont., of eourse, when the mean flow is steady and two-dimonsimad. The term ~z'~/az of ~q. (42) i~ at l ~ one order ofml4~aitude amLIIer than the term ~ / ~ end will oonmqueutly be auppremed. Thon Eq. (4.2) can bo integrated with rmpect to y t o siws -

P

~s.

_ _ +eonL P

(u)

The oonstaut of integration is deters;ned by the eonditlons that out.. side the boundary layer (.,V;~ 8 ) ~ - 0 and the mean prmsure equsk the external pressure P... Accordingly from Eq. (4.$) follows P _ .

P. n

P

_1,1.

(4.4)

P

W h ~ this equation is diff'erontiated with s,mpect to z and introduced in Eq. (4.1), the b o ~ d z r y byes' equation for the mean flow is obtained., which is conveniently written as

This equation difl'en from that for the laminar boundary htyer_by the oeonn~.uee of the differont~d quotients of terms - u e , ~2 and vs. The shearing and normal etnmes-respoetively and are known u Reynolds staeesm. The mean value ef the total shearing stwes is then given by

Sinee all ve]cclty component, vanish at the surface, the mean value af the wall shearing stress is • -

-

(4.7)

where t.he subscript w denotes values at y - 0. Equation (4.5) is also called the momentum equation for the mean

1~,

J. Q Ro'r'r.~

flow, for its terms descr/bo the Ices of momentum of the mean flow by the action of pressure, Eeynolds and visoous streem8. Some years, ago, there was some controversy in the literature ooncemin~ which of the fluctuation terms should be included in the momentum equation. The controversy was ~nmsod by certain which came tc Hght with the analysis of experimental reenlte. Later, it was f o ~ d that the mean flow k not exactly two-dimensional in laboratory e ~ h n e n t ~ , and t h e n d e v i ~ o n s are nesponsible for the observed disorel~aeie~a. The form of Eq. (4.5) k now p a e r ~ l y ancepted u l~.ing the correct one. The term a(,,;I-oS)l~ is usually of seeondar7 importance, hence its uqleet is j ~ in mint ~ By no mean~ however, may it be coneluded from thk, thst the termx u~/Sz, e~lSy eto. in Eqs. (2.2) to (2.4) are negligible. ActlmUy, them terms play an importamt role in the mecbankm of turbuknk flow, u he out.,,~t in Chey~" rr The momentum equation for the mean flow, in the form 8ivm by Eq. (4.5), represents a second order ap__pro~mstion to turbulent flow near the wall; neglect of the term ~(us_es)/~, leads to only a firB order appro~mn~ion. The term ~s-eS)lB= becomes more important when the region of separation is spproaehed. Some oheek on the order of msgn/tude of the individual terms of Eqa. (4.1)and (4.2) is available from an expcdmeut~'inve~gxtion of 8 turbulent boundary layer in rising pressure, made by Randbom and Slogarao~. Fig. 4.1 gives a sam.pie of these memmrements, where the terms am plotted u non-d;m,ew/onal qmmfiti~, using the momentum thickness 0 and the ex-ternal velocity U. as reference lengthand velooity. The term ~,~U]8-~, already neglected in Eq. (4.1), w ~ found to he of the order of 1/1000 of v~/T/~_,t, and therefore was not inelnded in Fig. 4.1. Furthermore, the term &gs[~ was found to he too smsJl oompared with the other term~ of Eq. (4.1) to be plotted. The cdryo of the experimental ~ between left- and right.hand s/dee of Eq. (4.1) is slso included inFi I. 4.1. The e~ktence of this di~erenee, which should be zero, is attributed largely to uncerte~ntyin the determination of U~UlOz. From the ev~,uation of the terms of F.~1. (4.2) it w ~ found that the terms ~ s [ ~ and VOF/O~, which were n~glectod in Eq. (4.2) becsum of the boundar~ layer simpli~eatlonx, were considersbly pin,tiler, then the other terms. Moreover, the change in statio preesure aero~ the boundary layer produced by the term ~ z is below 1% of the tion p r~sure p~e2/.• everywhere in the flow investigated, although the term ~vS/~y reache~ locally very high value~. At the outer edge of the boundary layer ~ 7 / ~ and the turbulent velocity fluctuations drop to zero. Therefore, the following relation

15 ,

I I

I ~

llii

4,L u~~

I ,I

I~.

liE lie

f

m

J i

i

¢

l~a. 4.1. Distn'b~ms d t~ms a p p m r i ~ in the e q u s t k ~ cd' mcMan fm • tm4~m,t ~ in ~ pm.sm~ ~ 8s~Ibmm , ~ t SloSm~t (Statk~ 4). The tz~vidusl term. of ~ (4.1) and (l.2)smmult/plid

e/u.,.

(a) ~ . (4,.1) m

Q ~/7 eU

fY 8U X

f

0 b,~

o..,~

dP,.

pU..~ dz (~) Eq. (4~)

# o D

~

vJ

~

0F

~Y

8U ~Y

~J

18

J.C.

Rosa

holds for the mean vslue or. the premum g m d i a t d/7.. I dP,.

zT. d=

"

p

d='

(4.s)

wkich is obts/ned from Bemoulli's theorem. If" this is introduced in Eq. (4.5), the following form for the momentum equation for tho flow is obts/ned d/7., a _ 8/7 a-:

17-•+F-•

The ~Tu~onding boundsry oonditioM am y-0: y -~ cO:

/7.1, /7 -. Urn.

F-0~ (4.10)

Equs~/on (4.9) is to be used in combiner/on with the contintdty Eq. (2.5), but it rosy be noticed thst tkis set of equations is/ndetermins~. In other words, there are more u n ~ o w n s than equations. Therefore, it mint be complemented by additional relstious or hypotheses eoneeming the Reynolds .-~esses. It is really this problem to which them has not yet been found s astidsetory sx~,w~r. All the effcrt8 ofexperimentsl turbulence r ~ e s z ~ dre.le sround this central problem, to find out the w~y in which the Reynolds iCzmm~ a~m related to the mmm flow properties, or to find s model of flow, which ensbles us to csJculste this relstion. In coaeluding this section, we ~ give the momentum integral equsCkm ~ u t used by yon K t r m / n , which is obtained by i n t q m t i n g Eq. (4.9) with respeat to y. The mean veloeity component F is eliminsted by the use of the oontinuity equstion (2.5). The yon X / r m / n momentum integrsl equation is most often used in the form de d/7.,Idz +e(s+a) - cd~ (4.zz) d= /7. where the following coneeptd are intr~ucod: Momentum thieJmee8

O

Displacement thlc~neM B* -

! - -~--~) dr,

(4.1a)

5"t,rb~mt ~ m , ~ 7 Y,ayer~ in I , u ~ m ~

~

IT

Shape~ - e./e,

(4.;1)

Local skin friction eceffi~ent ©t-

~B

The term f)(u"/-~)/~= of Eq. (4.9) is neglected in Eq. (4.11). The derivation of the momentum integral equation taking all f l u ~ teru~ earefully into conside~tiou is given by re,, Lees. The momentum integral equation without neglecting the Reynolds normal stress term of Eq. (4.9) i8 found in Table ~. In .most ea~e8 the form ~q. (4.11) is su~eient for practical 5. Hypothetical Relations The IU-indp4d d i ~ d t i m involvod in the problem ~ stated in the preeeding ee~o-~, were ~ in the early days of flow r m e a r ~ Sinee then many investigators h~ve endeavoured to obtain at ~eut an alrp~ximste so)ution by introducing some emptriea] hypothesis, baled on a simp|e model of the turim]e~t motion. Again ~md again new proposals of th~ kind have been put forward, and 'probLhly.furt.her st~ tempts wfl] follow. The c]om~ examination of the tcrbu]enoe meehanbun, which beeame poaibie by the continuous imprvvement in expetimental methods, revealed always very soon the entire inm~cienvy of the aannned flow model Refinements of e~dstmg relations, 8ugwestod by new e ~ e n t a ] observations, cannot be suocessful, sin~ on the one hand new unkno~w maguitudel appear, for which additional assumptions are required, and on the other hand the user is discourag~ by i n ~ mathem~ti¢~ d!~vuRies, particularly sinue the genemlis~ tion of such relations remain~ highly questionable. Despite their shortcomings, many of the older relations are still being used with varying suceen for the lack of snyth~g better. A generally applicable ~ t i ~ is not yet awfilabk. J. B o ~ e s q , who was the first to attach the problem of finding a suitable relation between the alypa~mt shearing stress --ptw im~ the mean velocity, introduved the concept of a turbulent exchange ooefficient "r, such that the Reynolds shear stress is expre~ed by eU which is' aualogous to ~'et~on'# law of fluid friction. The ~ t u d e Iike the k~nematie viscosity v, can be interpreted as being a product of

s veloe/t7 and • length, and is now mostly called "eddy viscos/~y ~'. The uncterlying idea is that in turbulent flow ~ flu/d a ~ o m e r s tions of maeroeeopie ~ are moving as & whole and thua work in s way as the free molecules of 6 gas. The fluid au]omem~ons , or odd/on, Ic~e. of course, their indiv/d--1;ty ~ a -hor~ time. wInle new ones are continuously created. The distance they trawd in the transverse direction is called the miTing length |, and the velocity with which they move in relation to the surrounding fluid is of the order of the fluctuation velocity ¥ - ~ / e s . Hence the eddy viaeoeity can be written aa

• ,

- Yt

(s.2)

The motion of the moleoulu of • gas is determined by the enthslpy of the gas ,rod is not a~'ected by the ~,dd-motion, m t ~ t molecular viscosity appears to be ~- property of the fluid. In contrast to th/s turbulent motion is maintained by the mean fluid motion and varies wi/~ it. When the mesh motion stops the turbulence will also come to mat. For ~ r e s i n ILtt~ h . . been gained ~om the Bou~inmq

formula Eq. (&l) Ltone. In an sttempc to ob*-,,;~ s, genera!ly appUe~ble formula, Pnm~tlTS f o ~ ! his we!l.know~ ~ length theorem, s c c o n t i ~ to-which the Reynolds -heating m ~ m is to be ealoulated fi~m (s.3)

Here, the tra~.mveme component of velocity fluetuatiom,

- !I--~ .

(s.4)

~= estimated by mesne of the plausible assumption that the mean of the eddies and the m;~ng length are of the same order. The roiling length k ~fll an unlmown function in Eq. (5.3) but it k not influeneed by the velocity, and seems to be s function only of flow geometry. l~andtl attempted to proceed with spee/nl assumptions concem/ng | appropriate to the individual type of flow. Some early inv-~tipt i o n ~ , us were concerned with the eaIeuI&tion of the ~ length fi~m meuured mean vel¢city distributions in turbulent boundary layers The inverse problem of calculating the boundary layer with speei~ed easumptiona for the mixing lengttt was treated by Szablewakim and very recently by Philip;S. However. the a~umptiona introduced can hardly give confidence in the results, when al~plied to other cMeg for wh/ch no ex'perimentsd .wsults are available. An attempt to obtain s generslly v~idrelat/on for the mixing length is ava/lsbleuvon]~L-n~Ln'a

•~,~n--ity h y p o t h m i ~ , which k ~ on the ~ p a d t i c n that the ztru¢~ ture o f turbulent motion is ~ - ~ n , , everywhere and that the length ~ l e s t any point is determined by the distribution of mean velocity in the ~--~ediste neighbourhood of the point. With these u s u m p t i o ~ the s ~ t i o n . z -

(5.5}

is derived for the , , , ~ S length in L two-dimenlional mM~ shear flow, where K denotes an empiric~d eoustant. I n fact, each of the relations Eqs. ($.1) to (5.5) ~equh~ iome emldrio~ information, if they are to he u ~ d for qugntitsUve ca.lculatlona. Aimzt from this, the actual tmhAviour of turbulent flow diffem appree~b]y from that m~,ceted. This is a~o true for a number of ~m~L, r e l a t i u which need not be reviewed here. Therefore the relatious are very olden not reoonc~ble with experimental observationL The moat Nuious objection is that the Reynolds shear is related to local flow quantities only, whereas e ~ o r d i n g to the ellipticld c ~ o f the N~vie~ Stokm equations, the Reynolds shear is actually infinenoed by the ennditions ,~t every point of the flow field. This is best illustrated by the relztious for the pressure finctuatious obtained in Section 2. T h k stztement may he modUied in the e a ~ of turlmlent boundary l~yeM in that the P,e~molds shear is nud/~y influenced ~oy ~ eon~itio~, while downstream ennditions have only ~ r y little effeet. ~ is • oon~.quence ef the boundary k y e r ~.,,plifio~tion~ which reduce t . ~ Navier-Stohes equationJ to s di~erentisl equation of pa~bulie type in the c u e of lsminsr IsyenL Another contrast between turbulent motion and moleeuisr motion of gases may help to illuminate the situation. Whereas the velocity Of the molecules is very high and the length of free path betw¢~l two oolli~mm k extremely small, the reverie is true for turbulent ~qow: The fluctuating velocity is -.; lea~t one order of magnitude mnaller than the mean velocity, and the mi]~n~ length is comparable in aise wi~h the thickness ofthe lay?r. Thus when an eddy mo~es a distance I in the transverse dL--ection, it ~;ll travel rbther a long di~ance in the downstream direction at the u n : e time, as is Been from experimental rema]ts represented in Section 6. At the end of its travel the eddy will find itself st a locztion w h ~ e the conditions are ve~f different from those ~'here tbe eddy origin&ted. So far very little progress has been made in the attempt to find relation between Reynolds she~ring ~tren and mean flow on the bask of simple hypotheses. :Nevertheless, the ~oussinesq formul~ and ~ o m e extensions of it have found very. u~eful applieatiorm, although some investigator~L ~ mentioned the possibility that a va~ue for - u v may

20

.1". C. Ro'rr.a

event-~ny be observed st lo~.fions where the mean velocity ~'sd/ant a/Z/By is zero. But such c u e s are certa/nly not important in b o u n , : k q layers. In most cases, a. v'h'tusl eddy viscosity can be defined by re..

writi 2q. (&l) . . . MIP

- aO'lay" (5.6) The interpretation of the Reynolds ~hem-Jng mr,rein as a turbulent t r a n ~ port process suggests an srufiogy with the turbulent ~ r t phenomens of other qnntitiee, for i m v ~ c e heat. This lesdm for axLmple to the z~.lstion '*""

I

F

for the rate of turbulent best transfer in the y-dL-vetion, w h e ~ ~' il the mean value o f t e m ~ r a t u m and T denotes the temperature c~ k the speeifla heat st constant p n m u m and ~ is the exchange coefficient tbr heat. The analysis of e ~ t ~ with the use of Eqs. (5.6) and (5.7) shorn that the two eddy d i ~ i v i t y ooemeienta ~ and ,q are rouglfly of the same magnitude, cq being mmally smnewh~t lauqfer. The ratio of the two coefficients, which k called the turu-~ulent Pmndti number, k found to be of the order 0.; ~ , J q ~ 0.9. This analogy between various turbulent t r a n ~ o r t ~ i~ one of the few principles which can be applied with some degree of reliability in s more general way. It eeems even permimdble with some t;on~ to extend the Bou~inesq concept to the remaining eomponentm of the Reynold~ stremes, which form s second ordm" sym,~m-io temmr. In turbulent flow the prineipsl axe~ of P,e),noldm ~ are not nemmarily coincident with the principal axes of rote of attain, but for mo~t turbulent shear flows the two sets of principal axes are nearly coincident. Towel48

And T . i ~ t k i l l ~ h~ve con~dez~t the e~'ee(; of a ~

~-

ditiomd rote of s t r ~ on a pla.e she~ring ~ot~on, where it w u supthat th~ di:eetion~ of principsl ~ and tho~e of prlnvipal retrain ewe identical. If tim hdtial state of strain ia of the ~ r m

•

a las

aplas

o

0

0

0

0

o /

the extended Boumine~1 r~Istion becomm

\

°I

•-

--AKf

OF

/ 8F

~Wx

(5.S) whero ~ denotes the sumll ehsnge in stzem or at~z~, sad c, is S i c by Eq. (5.6). This rrlstion rosy be we/ul in the omm of" sli&ht3¥ threeAi,-ension~l turbulent bounclsr 7 ls~n~

Cz~.rzs II UNIVERSAL ASPECTS OF THE MECHANISM OF TURBULENT BOUNDARY LAYER FLOW 6. D o u b l e Velocity C o r r e l a t i o n ]Functions a n d S p e c t r u m Functions In order to become a~q-A|nted with the complex nature of turbulenk flow it is not sufficient to investigate only those ~low quantities which are of practical interest and appear in the boundaz7 layer equations. ~ - y attempts have been n'u~e to obtain greeter ins/ght into the motion. V'mual observations are onlyseldom successful, since the motion is too complicated and, in most cues, too ~'sst to be followed by eye. Much information has been gained by m e u u r e m ~ n ~ of various s ~ d s tical quantities which include .~mnltaneous mesauremen~e of m o r e then one velocity component. Two concepts have found a wide~resd appUcation in this respect, namely, the double velocity correlation functions and the spectrum function. It is recognized, however, that the i n f o r m ~ tioa obtained from these functions is restricted and its interpretstion sometimes of a speculative nature. 6.1 T h e d o u b l e velocity c o r r e l a t i o n f.u~'tion The double-correlation is generally defined M the covariance between velocity components at two separ~ed pointa taken at diferent times, ~ . 6.h

R , ( x . r. 0 - s~(x)sj(x+r. 0-

(e.1)

Here we secept temporarily the standard practice of denotin~ the various directions by sumces ~, j, etc, so that ul(x) is the instantaneous value of the ~ component of the velodty fluctuation at the point of position vector x. The separation in space of the points is denoted by vector r, and t is the time delay of the observations, i.e. when u,(x) is observed at time h, then ~j(x + r, ~) is observed at time tz - h + t. The overbar indicates that a mean value is ~o be taken in the sense outlined in Section 4. It is generally s ~ u m e d that the motion is statistically independent at points sufficiently far separated in space or (and) time, so 22

t h a t ~ tends to zero as r or t a p p r ~ . h m ;-~-;~'y. The form o f ~ to mstisfy the oondition of interchangeahility o f ~ and ~ , Z~(x., r, t) - 2Zlj('~+r, - r , - ~,

hu (6.2)

- n d the oondition o f oontinui~V

~-~s~j(=, r,.t) - 0.

(e.s)

where the repested su~3z j i~ to be regffirded u being ~ m m ~ po~ible w l u ~ of the ~ The mmditicm F~t. (e J ) ~

mteg~ ~

o v ~ an to t h e

for a eonstant value of rz. TK~I equation zmmlta from t h e requiremeat t h a t t h e immmta=eoue flux of fluid across any closed gadaee il ze~o,-and ]~a is zero f i r s ±morrs± ao. F,q. (eA) impoml • kinenm~ condition on the form of the space time c o n ~ t i o u f u ~ i m ~ to which P,~ m u ~ h s v e n e ~ t i v e values over s oertcin r m ~ o f r .

U! m

Fro. 6.1. The doable velocity Oonmistion rune.tirosRu - ~ . I n most c u e s the double oorrelstion functions are um,d in the normalized form .~j(x, r, t) - /~j(x, r, t) = /~.(x, r, t).

~,s(x, o. o)

(6.6)

~uj(x)

]~'or r - 0 snd t - 0 we obtain by definition

~ j ( x , o, o) - z.

(e.e)

From this norm-~;-~ function, ~ r t a i n inte~-al length and thne sesles can be derived,

9

(rl •

'JT{x) - | R ~ ( z , O, t) c~,

(e.s)

which give an ides of the ~ ~ze of eddy. and the a v e r s p time the e d d y takes ~ pass a fixed point at position x. It is also noticed that fi~m the double velocity function the cova~a-ce of two velocity derive.tires can be obtained by aifferent,latlon. The general form is

~-~= x ) ~

x + r' ') " ~R''(x" r" ') - Oalh'(x" ' Or.~rtr' --.

(6.9)

In p a r t i c - l v , when r and t tend to zero, the mean~products of velocity derivatives, wh/ch appear in the equations for the viscous energy ~l;~pation, Section 7, can be ~ from ~.

-

~,R.(x, 0, 0)

~ R . ( x , 0, 0)

(8.t0)

The first term on ther/ght-hemd side of this equation is usually negligible. I n general the space time correlation function R,S is &rather complex function of vector r and time delay t. Experimental inve~Lige~ions are, therefore, mostly restricted to special eases. One spec~"-*tiun is the covariance of velocity components at two separated points at the same time (i.e. t =- 0), the so-called sps~e correlation function 'Ro(x, r). Another specialization is the covsrbmce of two velocity components At the same point at ~ifferent times, the ~o-called autocorrelation function fRO(X, g). O. I. Taylor Ia7 suggested that, if the mean velocity of flow wh/ch carries ~,he eddies, is very much greater than the twrbulent velocity fluctuation, one may a~sume the sequence of changes in velocity At a/L~ed point to be simply due to the p a a s a p of an unchanging pattern o,P turbulent mot/on over the poi~-t, Le. ~(=, y, z, 0 = , , , ( z - U t , ~-, z, o).

(8.1t)

This approximation wM ori~nally introduced for turbulence produced by a urdform stream through a grid, but it also applies, with some reservations, to turbu:ent boundary layer flow. This leads to R,j(x, r., r~,, r~. t) - R . ( x , r z - Ut. r~, r:, 0), (e.l~)

-~d. e ~ ' - l l y , . t

loz~tudimd ~

~.ktion between the sutocon..]stion f u n ~ o n e~d the

corrclstion is obCsined, n a a ~ y ,~,~x~ ~) - oP~j{x, - Ut. o, o).

(e.]s)

Another important eonseqoenoe from Tsy)or's hypothesk is the rektion between d e r i v a t i ~ with rmpect to z and ~, e t

These rektions are generally used to deter-~,- experimentally the ~ e s of the spatial var~tions of the velocity component4 in the mean flow ;dh~Uon by ana]yxis of the velocity fluctu~Uons st • fixed point. TurbuleBoe theories nowsdsyx seoepted rely most3y on the idez that turbulent mot/on oonxiets of superimposed m o ~ o ~ herin S • wide range or length scales. It is • oo~,~on ~ to ~ in term~ of eddi,whe~ the turbulence is de~ribed qu~tativcly. An eddy ~ u hardly be imagined u • clearly defined strnctm.e. Cloeer to the ~ bebsviowr is the definition of an-eddy ~ s region over which the ~hnul~J-..~ua ve]o~ty fluotuatiom are oorrelated. It may be assumed that the oovarlanoe (with zero delay, t - O) between the veloeities st ~ pointl is unaffected by motions wh/ch hsve ]en~.h eoales much a ~ l l e r than the distance r b e t ~ e a the Poimm. At the largest values of r, for whieh the ecrrektion is different fzom zero, the ocrrektion w~l be d e m e n t e d by the effects of tbe ]zrgest eddies prment, and st any smdler vsloe of r the correlation will z~oe.ive oontn'butions from all mot~o~ with ) e n ~ scales greater than r. Thus from the -~-pe of the oort'elst/on functio~ some information •l~out the eddy structure ~.u be gained. In particular, large ~ a ~ t i o n s of the oorrelation ~or renal] r indiez*e the presence of very small e d d i ~ S p e c ~ mention ahould be m ~ e of the sttempts of Townmnd 14x and his co-worke~, to desaKoe the motion in terms of • few types of eddi~, and to provide the neoema-y evidenee from the tion functions. The caleulation of the oamdstion funotions for vortex arnmgements, u presented in Itef. l~S, ~ very instractive remdts. The reverse, that is, the inferenes of the ~ - a c ~ m of • system ~" eddies from the correlation function is, howev~, an ambiguous prooem, and depend:. =ooording to T o ~ d ' a own zt,,.temontx4a-.on the prejudioes of the o ~ . 6~ E x p e r i m e n t a l rein•Its The dou'Dlevelocity correlation functions can resc~y be m e a m n ~ t~o hot wire probes placed in the boundary bayer, and the ap/n~pr~te 3

-°n

J'. C. ~

eIectrical equipment. Such meeEaremeats have been made by several authom, and the nmults have revealed some ext:eme~y vsJ-ffible deta/k of the flow pattern. Mort o f the experiments are confin~ to particular space oorrelations (w~o delay time, t - O) along the main co-ordiaa~e ~ Schubauer and K]ebe.uoflq-~ have m e u u r e d the spaoo oorrektion functions of the longi~,~t,,-I velodty f i u c t ~ t i o n u along the longitudinal and transverse axes in s boundary layer in an adverse pressure gmd/ent.

0

4"

~

£$

28

fJ

S°

&

turbu]emt boundsq k ~ " in an edve~e p~mum uudiem Se.hubw~ m d K h ~ u o G ~ . V~mdo is

i

lib So

Noekoatli mile ILopreeentativo , p ~ o correlation ourvee fi.om t h e n measurements are shown in Figs. 0.2 m d 0.3, in order to give an ides of the distances over which the lougitudinsl '~elooity Hu(~t'ustions are ,.,on'elated ~ m p e a ~ with the boundary' layer thieknem. I t is seen from Fig. 0.2 that s t r s n ~ e ~ e eonmlstion eads~ over much oF the boundary 16yet thiekneel. With the region near flow separation excluded, f l u c t u s ~ m at the centre of the layer ~re related to thoeo everywhere else in the same croa see° tion. A small negative correlation is found between point8 in the ~ y e r and those outside. This can be explained by the continuity requiremeu~, given by Eq. (6.4). Another noticeable feature is the a]most aharp peak of the correlation functiom. The locally high curvature near rz - r r - 0 indicates the presence of s wide range of eddy sizes. From

f'vrb,,k~ aou,u~anj f a ~ - , i , I r . e ~ p m m ~ ~qo~o F.,q. (6.10) it f o l ] o ~ a

~

th~

(e.~) The high values of eurv~*m'e at the p e ~ of the m r r e l a t i ~ functions thu~ ;,,,p~y ~ v, d m of (~,/e=)~ ,,,-d ( ~ ) ~ .

)

s.

Fro. e.a. l z , ~ t . d i . ~

~

f!

~m,kt.i~

f e a e t t m , J ~ . . ~ ~. !.,) fee •

turb.Ieat bouudm-y ] a ~ in ... s d m proton, er~Umt ~ Sehutmar u d ~ o k u e ~ m

Very e x t e ~ v e meum~menta of ~ time oormlst~on f u ~ , ~ o ~ of the longitudinal velocity 8uctuLtloni in s boundary ILyer on • ~ t plate have been made by Fsvre. G&viglo, and D u m ~ u . u . Some of their resulta are prme=ted in the ~ Fh,et, ;,, order to e:nu:dm Tsylor's hypotheak ~, stated in Eq. (S.U). sutoeorre~tion and lu-n~itudinal ~ a e e cot:elation functions • t A;~rent distances from the are e o m ~ ~ith eseh othe~ (Fig. 6.4). The longitudinal q ~ e eerrelation curves (the ~ e ~ ..~ s ~ o e t Pand]el to the l ~ t e ) m to lie ~ to the autoeorrelation e u r v u for y/~ ~: 0.03, if the tramformation t - - r s / U is used. ~'he di~erences noti__r~e~_~ negii~ble in the oentraI part of the layer ~18 - 0.24), and have oplxNdte signs on the two sides of this zone. I t seems that to a first appro~mation Tsylor'l hypothe~_'s may be applied to the turbulent boundary layer for pmi~ions not too close to the w=11. Tn this omme~tio~ the reader is referred to Ib theoreLica] e:camination of Tay!or'e hypothesis by l,in e4. The longitudinal space time eorrelatio~ functions shown ~n Fig. 6.5 give some information on the life his*.ory of the eddies. ~ c h correlation curve has ~ m a g n u m at a delay time t~. ~'hich i n ~ a t a r~te proportioua] to the dJstanoe rs. ~ e ratio rs/t=,'7~ ,- 0.8 indicates th&t the

I-¢

y/~ ~0-P4

,-k

r

CkE

0-4

o

O

|

4

•

•

~t,

I~

•

~

mm

Fro. 0.4. C h e ~

-.'~t,

b

ms4m

oF TATi~'8 h y p o t h e s i s b y e o m l ~ r i J m l of" auto-oorrele,

t~c~ ~cl ~ = s i , ~ a ~

~e

oam~r~a~ ~ s ~rbuha~ bounchu~

on A fls~ plsto ~

F&~e. G~vi~io. ~

J

0 A~-to-eorrelation J ~ . , ( 0 , e , - r , - ,'. - 0 • Lonsitudinal ~

eoemr~i-,,

,~..( -r,/~,

J - 0, , , - ~., - 0.

HI

I-I

O.II

J

0-2 0 -4

]?Za. ¢ &

-it

•

Lo~tucUmd

ozz,*h~ plate ~

2

4 6 Time.

R~-time

• m Me

con~latio--

Fs~.rs, G s v ~ t l o . D u n ~ , - . ml~.

J -

&

r=-0

•

rs -- 2 6 - ~ , m m

0

r:-

~8mm

1(I.8 r a m . r r -

rs ~

r~,U,,

J0

I~

ia t~

- ~,

oouncl~-jr

.•

lATer

y/8 - 0.24, [7. - 1~-0,

0. P. -

0.

-- 0.81

-0-79

eddim ,travel in the flow d i r ~ t i o n with • ve|ocity of about O-SDm. I t is seneral]y taken for stunted t h ~ any individual eddy hM • limited duration of life. This is ahown by • decrease of the ,,,,,~,.,um value of the oorrelstion u rz grows. The envelope of the eurvm Sivee the oon~lstion for o p t ~ u = delay time la~ When rz is given, ~he spaee-time correlation a~..(rz, r , , 0o tm) with optimum delay t~ reackea • sns~m u m m - ~ o r u m for o u e value of r r. Corresponrl;-~ values of rz and r U form • line o f m s z i m u m o o r ~ a t i o n with 01~|'num delay, F i p . 6-8 to

6.10.

line goes through the fixed point A. The linm of ms=imum

)

.-4

-.q

-;

-i

0

I

|

i

4

Fso. 6.6. 8ps~-time om-ratatio=J in the boundmT ~,ym" on • s . (r, ,, O) shoe Fawe, OsvilJUo, D u m ~ "

O Along the line of maximum eorr~tlon with Optimum ¢k~y ~ foe • Akmg • mean ~ I ; , .

with sero ¢le~.

eorrdaticn outside the boundary, la~er eolncide with the mean stream lines, b u t they £iffer markedl 7 within the boundary layer. The oorrmponding correlation functions are presented in l~ig. 6.8. As seen from these figures, the oorrelation o o e t e i e n t with optimum delay time retainl high values for distaneea rz/8 along the mean flow upstream and downstream, which are aplor~able at ¥/8 - 0-03 and incren~ wRh y/8. On the average the psasage of a large eddy can be identified over a dictanoe about one ord~.r of magnitude larger than the boundary ]myer thic~esa. Actually, the m A ~ m u m correlation curve correepondJ •pproximately to the a u t o e o ~ ] a t i o n function in a system moving with the mean velocity along a stream]inc going through the fixed point.

30

J . C . Boz-~

Another i n ~ , ~ t i n g feature of the turbulent boundary iaye~ i~ seen f~om the tra,,~ver~e space-time correlation~. 1 ~ . 8.T. One finds that the correlsAion re~ches & m , ~ , , , m , , for a value of d e ~ y time, whieh k ftmct/on of the distance r~ of the two obeerv~tion point~ The delay i~

neg~iva for poa/tivs r, (swsy from the ws],l) ~

,po~ti~ for ne~p~vs

r~ (towards the wall), 'l"heee meamzr~memts sngse~ an eddy strueturo I

I

m,)m mS M

!1 )mmm~._,,.L

4 St~

"~

I b~mDq

o

a

m.

"

O-d

0-2

#

' -

"t

J

I

f __

o,;

-"',,.,

II -7 M~

"10 -II

"t~ - 4

"Z

0 Tlm~,

~

4

6

II

iO

I~

[4

16

mle~

Fro. &7. Tmn~wmB epaoe-thne eor~lstlon ha the bounds~ ls]m." ~a s eat pl~t~ ~ Fsv~. Gs,~io. Dum~L: y -- 8 ~

| -

34 r a m . y / I -- 0.24. r : - v, - 0.

~7, -

12-00 m / t e e .

l~vin ! an obJique pattern in which ~he outer ~ are in adv~uje of those pszt~ situated closer to the wall This orient~l~on of the eddies must be s~triouted to tho continuous distortion of the turbulence field caused b:F the nonuniform mean velocity dietfibutdon. The measurements of Favro ct a L u include also longitucT;--I, tr~,~-verse and lateral space- "thne correlations, Le. apace-tlme correlations, in which all three components of separation rz. rv, and rs are d ~ e r e n t

from zero. ~ 6.6 to 6.10 a~ow'the so ~dled spao~time k o - e o r r e l ~ o n with optimum delay, that it, s~f~oes ~enemted by the positiom of oonstant ~alue of t h e eorrelation function with optimum delLy, for Po~ition~ of the fixed point. The sp~oe-time iso-eowelation f~om are wider along the mean flow upstream and downstream than

lip

7

,~|~ ' i 9,4-

ol ~'. IF.-O.e

J " AminO'| 04

.---.@,ll o-all iI

f°'s •

)-Q

e-,

Jl

H

FJJ J !

1~. 6 ~ Spe~-tlme ~ o a ~a~ with ~ deisy 8, the bound~.kLy~ on • ~ l~W ~ F a ~ G s ~ l ~ , DO=~Lm U..SI, .. ~ 9 0 0 ,

~t/J .. 0.03, • . ,

li~

of ,,,~,,,,,

optimum d ~ y for ~

emmlatima with

v~

per~ndic~lar to it even for A mndl dist~noe of the fixed point from the ~31 (.V/8 - 0-03). They are ~ , ~ e t z i ~ l with respect to r , and their a;~enaions are of the same order, transversely ~nd l&terally. The sbeolute ai,~enaions of tho sp~ce-time i ~ e o r r e | a t i o n ~ deere~e u the fixed point &pproa~hes the wall. These mcasv~u~ents are extremely ins~.ructive, slthovgh they give only an ineon:plete picture of the motion, ~uce only the fluctustion eomponent in the dh~ction of z - ~ i s is o o n s i d e r ~ Neverthc]esa they rosy in,tire the reader'a hnagination with regard to the pre~lli~g type cf motion.

J . C . ~

The concept of ~ s e e - ~ m e correlstion functions can be applied s b o to fluctu_q*~ quantLtiee othe~ than velocity components. One example is the pre~ure fluctua~om, t r s n z ~ t t e d fi~nn the flow to the w~[]. ~easurementa of ~ t i m e ~orrelatiom of such wall premure fiuctu~ tions, have been mad~ by W'~marthZ=Z - ~ - g two barium t~tansbe -S-~J - ] ~

-t~

-4t~ .44

-4.0 -0-S

¢

O-9

~O

I-!b

4.0

[ •

~

•

•

O-i dP..O.l~

l~a. 0.9. Spe~o-timo bo.correl,C.kmsurf'som with optimum delsy in tl~ boundary Isyer ~a a ~-t pbstm alan" Fsvm, Gav~ito and ~ t e (7,Sl, - ~900. y18 - 0.16, B - line of" mmdmum ~ with optimum delsy tar g ~ m e,.

pressure transducers with s ~;~meter of 4.1 ram. The measuremm~8 provide very -raluAble infoz'm~on additional to that obtained from the velocity correlation m ~ e n t 8 already siren. The boundary layer W~ produced on the inside wBll of a pipe. The r e s u l t , M shown in ~'qg. 6.11 in normalized form, reeemble v e t7 much the lonffitudinal spsce-time velocity correlstions of Fig. 6.6, For each distanee r: of the transduee~ there is ms optim u m delsy ~a at which the correlstion is a rn,L~m,,m The ratio rz/f~s~m was found to be appro~m~te!y 0.82, which equals the correapondin8 value determined frnm the velocity correLstion measurements, l~q~. 0.& "l'no wall pre~ure fluctuations are swept along the plate with the same velocity as the large eddiee. It must, however, be ~n~itted that the results will spply only to preesure fluctuations, whose length s ~ l e is

P~o. e, Z0. 8 ~ ~ o u ~ ,riCh o p t ~ 6uisy ~m the ~ hby~ on • ~U~ p i ~ s ~ r Y a m , G~vislio, l ) u m ~ H Uol~l, - 27900. yll m 0.77, m ,, ~ c~ m u ~ u m eomhLtkm

I.C

.

.

.,? ~!

I,

mute

}~'a. ~.ll. 8p~,-t,~nr~ eorTe~t,i~ of ~

p r N s u ~ fluct~t,loa in the bou.~sry btyer v i t h zero premure ~rsciJen~ after 'iS"dlms~h.tet | -- l~Smm, ~7,0 ,,. llSm~bec. •

O A

r# r.5 m 3 1 - ? r a m ,

76ram, 152 r a m ,

- -

t,.U®

-. 0-81

0-81 O-~J2

34

,L C.. Ro'x'r.~

larger than the diameter of the trA,~duosr, Le. ~ 0.38. Sinai]or scale pressure fluctuations will partially cancel e~eh other on the ta'ansduvor fa~e. An i n ~ theoretical oont.n'bution to tim ~mdy of the oovariance of the wall pmssuru fluctostiom h u very recently bee,, give=

by r.m~ und llod~aXm. 6.3 Spectrum functlon~ From the msthemsti~J point of view the three-
different sizeL ~

Four~ trandorm is defined by

r) = f ~ , ~ k} exp(v~- l]kr)d,~}

(s.le)

where k ia the wave number veetor sad d ~ k ) - c ~ x d ~ r ~ k the element of volume about k. The quan~ty ~t~(x, k)d~(k) denotes the contribution to ~uj(x) of the volume element d~(k). For many purpoees,

the inte~Fsted spectrum function is used, Le. .~'t~x, k) - r ~,,~(x, k)d,S'(k). ((I.17) J The inteiFstion externals over the spherical surfsos of radius k, where k is the wave number magnitude and 0.8(k) is the element of spherical sur/sce. Of special interest is the energy speetmum, defl~)d by'-

~(~ k) - i f w.(x, k)da(£)~

(6.x8)

This fun~,tion 2r(x, k) represents the averu6e densit7 of turbulent energy in the wave number space at wave number k. From the viewpoint of the experimenter, the one-dimensional spectrum function in the direction of mesa flow, whie.h k defined u

• ,~x. a,) - ; f ~(x, k)dIudk, (6.19) I

- x f. 'R'( r~, o, o) oo~kls) drs l c!~ml the most interest. If T6ylor's pstlllJtg ofEq. (S.I1) is adopted, the one-d;mensionsl spectrum function can be found from the cortesponding frequency spectrum which is r e ~ i l y measured with electrical

spectrum an~yser8. P~tween the frequency n of the flu~ust~on snd the wsws number component ks exists the J:" "

F~m -W"

(6.=0)

~ben the f~equeney apeetrum function is related to the one-dimen~omd wsve number spectrum by

~ . . / ~ , ~ ) - ~--.[~,(- t.) + , ~ t . ) ] ; ~'t~,O b the n

~

(e.~)

fzequency spe~t~m function, thst is I

f

T~(~)d~ - 1.

Another impor~,t t~Ist~on fo]lowi~ f ~ m Eq. (e.lej is

w,~x. k)~.,,h d,O0. 4

w~oh ~ o / ~ ~ d to ~ ~

ten~ l~e ( ~ / ~ . ) ( ~ / ~ )

~

th.

The form of spectrum fim,~om ~ to them7 end ecqmziment wm be d k o u s ~ in the subsequent m ~ o u ~ 7. 1~ex'~.y Dispersion by Ylteosit~ The whole work doze by the friction dra~ of s body, wrapped in • turbu}e~'~ l~Nm.ndlury]~yer. is eonTerted into he~t by viJeou8 ~ pstion. T'ne mean rote of tots] energy d/ssil~tion per unit volume of ~ is exanm~ rz ~v xt ¢ ~ ~

(7.1) This equsfion is derived from the expre~on for the viscous energy dissipation ~iven in st~ndanl text boo'~ (e.g. SchlicbtingU0) by spldicstion of the boundary l~yer ,~mplific~tior,s and taking t ~ mean value.

3s

3. C,. Roc't~.

The ~ part of Eq. (7.1), ~(aUl~y) z, repreeenta t h ~ fi~ction of the t]~lll~tio~.I'~'h/Ch ~J due to the f~el~ ll'l~.,lol~it~_~ ] ~ J ~ t and iz called direct ,];,~pstion. The re,,,,~;-;,,~ L-action is the mean turbulent energy ,.];u;iNtk~n

+ (_~ + Ov~z" ~k~

~

s

('t.2)

Except for region~ very elom to the wall the turbulent energy dksipstion is by far the more important 7.1 P r o c e ~ of turbulent e~erSy dlssilmtloa The akin friction drag of turbulent boundary l~yers k &ppmxim&tely proportional to the second power of the velocity, and thus the work done by the £rietion dra 8 u well as the drag itself are nearl.y independent of the viacosity. This seems to be at vad_'ance with Eq. (7.1) where the viscosity appears u & multiplying f ~ t o r in the e q u ~ o n for the viseouJ d~sil~tioa. Itmuot thum be concluded that a e h a n p in the free stream veloeity has the effect of vary~g the spatial differentia] quotienta of the velocity componemta st a faster race than the veloeitim f.hem~lveL The ge~rclly sc~-.pted erldsnstion of this str;1,1,,$ faet, as flu" , turbu/en~ dkdp~tion is concerned, is b a u d on the notion that eddies of widely' diferent sizes hsve no direct in~uenoe on one another, and thst an eddy exchanges energy st an appr~cisble rote only with eddim of similar size. As a consequence the turbulent energy dimip~tion appea~ to be eompamb]e with some type of o u e a ~ pro~eu, in which the l s r p s t eddies lose ener~, to eddies one order of m s ~ t u d e smaller, which lo~e enorgy to mnaller eddies in thmr turn, and so on until the eddies eonoerned are so small that they lose so muoh energy by set/on of v ~ o u J stresses that no ~ l e r eddies can be formed. The rate of dissipation is determined by the first few stages of t i n n i e r from the eddiee containing the most energy. The magnitude of the viaec~'ty s~ects only the size of the smalleat eddLu. From dime~onad r e a ~ g it fol]ows that the turbulent energy dlBipation can be expressed • -

c

t0

where (g-i)In is 8 chamcterkt/o velocity ((~)~n =, (~i ~el+trz)uz} and i0 is an sppropr/aZe length scale for the Jarge edd/~, comparable

with Prandt1's mixing ]enir~h.~CiJ s ~;--enslon3ess constant depending on the structure of the ~ eddy motion. The eorrmpondinS terms of the boundary layer Eqs. (S~) to (~.4), which keep the shore-mentioned ~ueade ~ in motion, are the pr~duct~ ~ ! ~, eaul~,y etc. It thin becomm clear that throe tenm m y not be .e~lecte~ The.role of the direct em.l~V d i u i p ~ o n and the mec3anism by which it is governed will be discussed in Sections 11 and 15. 7.2 Locally isotropic t u r b u l ~ c e The ides tlmt, for high Reynolds numbers, the emall eddies are not eorrelstod with the large ones implies t ~ t the motion of the small eddies is isotropie, which memm that the 8truoture is invariant under ro~ation of the axes of refemnee. ~ waa first sugHmted by KolmogorovS~,u who i n ~ d u e e d the eoneept of locally isotropie turbulonee. Prandtl sz, v. W ~ e and Onsa~er~ came independmt]y to the same eonc]us~n. Y~hen the motion of the part of the tur~a]euoe which i~ respomible for the ~imom d i u i ~ t i o n is isotropie, the ezprm~ou for the turbulent energy dia~l~tion, ~.q. (7.2), e m be s;mplified~U ~ the continuity equstion and the deKuition e f "~otropy, the followin~ rektions m obtained.

-

-

'~+

-

~

-

+

-

"

"4"

-

2

(~.4)

"3

Heaoe Eq. (v.2) reducm m •

-

1~

,

(7.5)

which w u f~mtgiven by TaylorU~. 7.~ Energy spectrum of lo~VSy Isotroplc turbulence The validity of the iuotropie re]ations with respec~ to turbulent boundary l~yers has been examined experimentally by KlebanoffU and Sandborn and Br~un~ae. Some of the terms of Eq. (7,2), meu,~,d in a fiat p]~te boundary ~ayer, are p|otted against the d/mension]eas distanoe

38

J'. C. ~ , x - , . ~

T~om the wall in l~.g. 7.1. The agreement with the isotropic relstions is exoe!les/t for large distances and fair for positions eloser to the walL Since the concept of l~,~]ly iaotropio turbulenoe may be considered to be & limiting case for infinitely large Rey'nolck. numbers, it ia beUzvvd that the d~J~rep&~~_ ill Fig. T.1 ~ & collleqllm~ of tho ~ y e l y low P.eynoldm number of the experiment. The oonditiona un,ie~ which local isotmpy can be ~ in turbulent shear flow, will be d k e u u e d 0"8 O-Y

7

"ill

0"4~ O'S 0.4

4

0"$ 0-~ "

o~ m

Ool

I

0

04

O-t

0"$

0.4

0-5

0"t;

i

0.7

0-8

0.tl

I.O-

v 1~o. T.L ][Y.mibutiou of clLmi~t~on d e ~ U m

;,, the turbulem b o ~ .

din7 lsym-~a a fist plsto vmsus the dktsnos bom tl~ wall sttme Kkbmso~U 17,~IIF - T.lOI 0 2Ut%a= I

_ ,___t

• ~U=t% a=/

& 2P~-~j! x

0 2O'=t%a=l

zTs~a=l

in Seotion 10. The obaerv&tion ~hAt the rate of turbulent energy diuip~tion is determ;ned b 7 the rste of energy t r s r ~ e r from the energy cont~i.ing eddies (of size 10) to the smaller edKtes, and the idea that energy transfer occurs in stages n g s e e t a t h s t eddiea snell in comperkon to Zo will be in & state of ~t~tisticsl equih'brium. The strdeture of thia equilibrium range is determined by the energy dimipatton , and the viseoeity v. I f t h o structure is exprem~d in term= of the energy epeotrum function, defined by Eq. (6.IS), simJ.larity considerations lead to &

Turbu~nt ~ o u . ~

Laee~ in l r ~ n ~

~

SS

spectrum funetien of the form where A - (~i/t) 1/4 ;. the |ength ~ade of the ]oe~]]y ~ o p i e tm4mkm~ and f is • univenud function of ~ . At nmeisntly hish Reynolda humbern, part of the range of univenud equilibrium may not e o n ~ ap/n~chtbly to the viscous dis~pstion, so that in a certain Imbranp of

wave numberl (Ipo < k < I/~), the motion is independent of the vi,eo.Aty. For this ,ubr,~ge. Eq. if.e) takes the form

E(t) ~ ~ L - ~ ,

(~n)

where z is 8 univenud mnstant. The theoreties] deter~;-~tton of the function f in Eq. (7.6) is one of the eentral problems of turbulenee research. An exaet solut/on has, however, not ~ been found. A ~nl4nnp~eal Lttsek on the energy tnnsfer proem is b,u e d on the k l ~ that, , ; , , n t , to Bcwminesq'au umnnptlm~, an eddy viscosity of the smaller eddies ae~ C~ the gradients of the larger eddieL According to Heisenber~ the equstion ~or the m ~ of energy lou 8{/:) of the part of the speetaum of w~ve numbers s m a I ~ than k ean be written u

e

This }ms of energy in lartly eo~verted into hest direetly by the motion of wsve number eompenentl smaller thtn k, " " * 7

8,(~) - . f ~J(~)k~dl ~, e

and partly tranafen~ to the part of the a l ~ t m m with w~ve number larger than k, namely

0

The eddy viscosity ~ k ) ;I p.-~xlueedby the part of the speetrum with wave humbert larger than ~. Heisenberg proposed the form

/(Ect")

.Tlk) -. f 4t---~-~-- ] dk", k

(V.O)

40

3. C. ]~n-rA

where ~ k a numerical uon~tant. This expression together with Eq, (7.8) possesses the a~umed p r o . ~ t y thst the energy transf~ is mostly ee*~bl;,h~! by the wave number compouenta in the neighbourhood of k The parts of the ~pectrum having wave numbers much larger and much amaller than k oontr;bute only Little ~o the energy transfer. For the r a n ~ of un/veruJ equilibrium i t / s requi,red t hzt the energy loss ,~(k) in Eq. (7.$) ehou]~ be independent of the w~ve number k and equal to the rate of turbulent energy dimipation ,. With tide condition, Eq. (7.8) in combi--tion with Eq. (7.9) represents an equation/'or the energy spectrum, which has the eolution

.<.,-

8~ s~

8~s_ 1.-4~a

(7.1o)

When ~ < (3~,/S.,S):/,. the q~etmm redum~ to / 1 8 e a~J

2(t) - f .--.--|

£.-'~,

(7.xl)

which is in SSTeement with Eq. (7.7) for the non-viscous submnge. For extremely high w~ve numbers, k ~, (3~s,/8~3) v4 - (3~/8)Stql/A) the spectrum has the ssymptotio form ~. 2,1 /

"

This spectral law is. of course, n , t expected to go down to molecular

sealm. An experimental e x t m i n ~ o n of the theoretical relations k rendered

dimcult by the fMt ths~ only fi-~Iuenoy q:ectmm lunge,,- can be measured, which are related to the oorresponding one-~,n--.~cnsl w~ve number spectrum funet/on. The one-dimensional spectrum functions, defined by Eq. (6.19), can bederived from ~he energy spectrum. With the condition of kotropy and Eq. (6.19) there restdts for the frequency speotrmn functiona

_

_

,~ f

ts+(2'm/U)s~(k)dt.

(7.14)

~.Jmlff When the solution Eq. (7.10)/s introduced, the frequency spectrum fmictions can be calculated. The result cannot, however, be given in s

9 " u r b u ~ ' ,B~Ta~V Z,e~Ver.i~ I ~ e o m p r e e . ~ F/me

41

eloeed form. For the non-vleeoua subnmge, when 2='~]P~ ( 3 ~ ] 8 ~ ) u ~ , ~ e L-equeney speetru~ r u n . o - - have t h e form 2 f 1 8 / 8 e%t ' l ,

~'¥,,.c-)

-

~.r..(.)

" ~U {~% kk' )~,=(9"/2m~,w=.

p.le)

i

h

a-O

uo

?.Ct U

*

sOj

@lU'I

1~0. 7.t. Fa,quene7 .peetram fuaettom of the iongitedi~ £~tion

ia the eute~ ~ of the turbulent bomulary Isyer on • tS~t plate ~ ~ * t Uoll, -- 7"10' • y/S-O'~

,~t,/a-e-~.

Olda-l-O

The uymptot3© forms, when 2w~JU ~, t 3 ~ , r ] 8 ~ 'l. m'~ --

2f2

t

~

--

2e

8

(v/,,.,), mr ~z

The so]utio~ for the spectrum ftmct,ions are in good agreement with 4

4~

~r. C. Eorr.a

e~perimental results for t~rbulence in s ,,,~irorm stream behind when the constant is choaen u ~ -- 0-4. There is. of course, some d o u b t as to whether the power - 7 is correct in the h/Sheet wave number r s n ~ Some attempts haws also been made b y H e i s e n b e r ~ and P~oudman aud Re/deS to determine the p r o ~ of energy t r s n ~ - theoretically. T h e n calculations are b ~ on the s ~ m p t i o n that the statistical mean values of fourth and second.order velocity' products .am related in the ~ n n c r ~ppropriate to ~ normal prob&b/~ity distribution. A c c o ~ - ~ to Heisenberg's csloulstions ~ is o f the order 1-0. ,o"

g ff

Io0

to'"

~

I.O

L~tJ_ e

tO

,o'

g~l

u

Fro. 7.3. ~ p m b o n of ~s spect=~ with ~ ~louL~ted fl~om the me~ured u 8 spectrum ,,.~-- tJ~ isot~.opio rehs~oa Eq. (7.19), ~1oL~nofr.~ y/a - 0 . ~

U , I / , - 7.1~.

The frequency spectrum functions of u I from m e m m ~ m l m t 8 by KlebanolrJs ,~ vm,-lousposition= in s fistplate boundary IAyer m m giv~ in Fig. 7.2. In the outer region &t y/8 - 0-58 and 0-8 there Js an extensive region where the spectrum varies aa n -6n corresponding to the non-viscous subrange. All or the spectra indicate the e~dstenee of s region varying as n -7 at the h/jqh-~,quency end. In Fig. 7.3 the measured frequency spectrum o f v z s t y/~ =. 0.58 is compared w/th that calculated from the measured uz spectrum ,,.~ng the isotropio relation

-

~r

d

(7.1e)

which can be derived from Eqm (7.13) and (?.14). It is ~een that the measured spectrum agrees with the ealcu~ted one-at the higher quenciw, for which loual isotropy can be expected. It might be ~meluded thst in the outer region of the boundary l~yer, where the Reyno)ds number of turbulenoe is ara.~ele~t,ly high, the theo~ of locally kotropi~ ~bule~oe eaa be applied. As the ~ is a ~ however, the non-isotropy becomes si~n~eant. ~ will be dkeumed in more detail in Seeti~ I L 8. Double Yelocit'y Equxtiows

The general mechanism which maintains the ~t~ti~ieal equih'bfium of a turbulent shear flow involves much more t.h~ the proeeu of dksil~tion. The essential fea~uru, as they appear to us now, will be Eiv~m in the m~bsequeut sections. For the purpose of these ~ , it appears to be usef~l to derive a ~.t of double velocity equ~ona from the boundary layer equa~iom. The ~ s t of these e q u ~ o m is obtained by multiplyt~ Eq. (2.2) by the fluetustion eomponent ~ and then teeing the average. ~ the uae of the eou~uuitT Eq. (2.8), i n ~ of bound=y lsyer ~ , - p l ~ i o . ~ and ~ . ' - . m ~ m e n t , t h k equation ras~u

This is an equation for the mean square of the longitudinal velodty fluctup,tion t, in the steady boundary layer. 81-,:1.. equ~ianJ e_w derived for ~ e mean square of the nv~.-al (v), and lsteral (w) eomponemt, if Eq. (2.3) and (2.4) are multiplied by ~ and ,e respecUvely.

'

+

-

0.

(s.:)

The s-,---ation of these three equations gives a rektion for the kinetic energy ~z--/2of the veloelty ~fluctuations, when

ga. (,~z+~+~).

(8.4)

According to Cornels the vis.-otity term of the energy equation ~ -

4~

,T. C. P,os-nk

c o , ~ of only ~ possa~le efi'e~, nzm.ly, the r~te of d~ssip~ion of turbulent energy to h e ~ (.), and the rote of transport of turbulent e n e r ~ by v~oou~ forees (div T).

~(uV~.+ ~V:~+wV~)

- divT-,.

(S.~)

where T is ~ vector flux of turbulent energy. Since the rote of turbulent energy ~mp&tion is given by Eq. (7.2), d/v T is determ/ned by Eq. (8.3). ~ the boundm-y layer only the component .'n the y-diroetion of the v~:oua transport n__ee~_be considered. Henco div T reduces to

- ,~-(~,+~

(8.8)

It may be remarked *hAt, in s ,,~,,~I,L,wsy, the mean term of each of the Ecp. (S.l) to (8.3) oe.u be int,erpt'¢,~t u the ~ eueru cUip~of tim component in question, & mmsport by viacou forcm and an e x d m ~ of turbulent enersy ~ the veloeity componentL However, such ~ deeompoaition of the action of the visoous terms sppears to. be krelevant with regard to the subNquent 4:Uaeumon, and nor,, therefore, be ~arried out hem in If the w term is oub~i~ted by 2q. (8.4) and (S.6) in the ~ of F,q~ (8.1) to (8.3) and the continuity Eqs. (2.5) sad (2.6) are mmc], tJae foUowing equation is o ~ for the c o n m r v a ~ n of turbulent eneq;y

+,,~+,-~(t~z+~ oy,

~,,~(,,z + ~ + ~ ) +pip] + (8.7)

t) - u.

.A~ ,,.~di~onal ~ t i o n results, w h ~ Eq. (2.2) multiplied by ~, is added to F,q. (?..3) multiplied by s. The &write of this equs~'on is ÷

+ -~u~+up/,o)-~u~.m+,,Vtf].

O.

(s.s)

This is ~ relation for the Reynolds shear stresL The reaa-ran6ements in the set of Eqs. (S.X) to (8.3), (8.7), ~ d (8.8) are made in order to obtain relstions in which the aignificanco of all ternm csn most eaaily be e=plained. mghly imr-~ortant are those terms which contain ~adients of mean velocity components, ~ . a u s e they represent the predomln~lt ~'ect

of the distortion o f , turba]ence fiekL This i~ essiiy ~ by m n . sidering an infinitesimal volume element of fluid, which ha~ veloeity components u and v relative to the mean velocity'. In • mz~dl time raterval At. this volume element travels • distance ~y - vat in the direction of y-sxk. At the end of the time i n t e r ~ l , in which any e.fleet of e=termd forces on the veloei~ of the volume element is negligible, the longitudinal component of the velocity with zmTmct to the mean velocity is u - ( ~ U / ~ ) t ~ . The ~i-etie energy of the u~omponent is new

By taking the average, it is soen that the term - ~ U I ~ y denotes the rate at which the longitudinal component gains energy by the a~tion of ~U]~y when all other eifeets are assumed to be absent. If the velocity at the end of the time interval is multiplied by - v , we obtain

-v( ~,--~t~-tI - -~+~t+-. On taking the average again, this d~owl that the term ~i~Ulbj is the rate of production of - u v in the s~sence of any other effc~t, w1~m tim turbulence field k distorted by • velocity gradient ~U]~. The term - uv~U/~y ~ also be interpreted u the work done by the meffiu flow against the Reynolds shear t~.rem.-Therefone, it iJ also e~lled the produo• ion of turbulent energy. The production terms ~ , ~ U I ~ and e ~ F l i ~ •re in molt. ~ one order of magnitude smaller than I,~Ulby and they van be negleet~ when the term ~(uz-v~)/~z i~ considered to be negligible in the boundary layer Eq. (4.6). The viscosity termB have already been mentioned and relmment mainly the viscous dissipation. The terms consisting of mean produets of pressure fluctuation and velocity gradients (p~/~z etc.) will be treated in ".he next section. The rem~inlng terms represent transport effects. The first two terms in each equation can easily be identified as the eonTee•ire transport of the quantity in question (e.g. ]/2ul) by the mean flow. The terms preceding the viscotm terms describe the tliffusive transport of the quantity in question in the direction of ~/-axis, which is achieved by turbulent movements. The mean triple products ~f turbulent velocity fluctuation components are usually called turbulent diffusion, ~ ' h e r e u .the product terms of the pressure fluetu~ion and • velocity component are called pressure ditfusion. •

ie~

The volume element need not be identiea~ wtt~ ~ tlBdy in | b e sense of ~;~tn~

theol.

aS

3. C . ~

Spec.king of energy conservation, the conversion of the kinetic energy of the free stream in the turbulent boundazy layer occurs in The fir~ stage is the extract-ion of k~etie energy from the mean flow due to retardation of the fluid that entrains the layer. The l u t stage is the viscotm dissipation of kinetio enerSy i n ~ h e ~ . The entire proesm is deecrihed by two equations. The first one is obtained by multiplying the boundary layer momentum equation for mean flow, Eq. (4.9), by 17. Thia equation is convm~ientiy written u

½117.~...z~

a~ a

+ ,a17 - _ a17 --

--

o~

--

~,~I~

I~17X l

} - o.

(8.9)

The second equation is Eq. (8.7). The first term in brzekets of Eq. (8.9) represents the loss of kinetio enersy of the mean flow. The nz~ct two terms are production terms, which sppesr here with signs opposite to those in Eq. (8.7). The following three terms indicate the t r - , , J e r of energy and the last term gives the direct disslp&tion.. The two equations ( 8 j ) and (8.9) ~ be combined to

[,.,

+ v ,,y-+

a

I

a+ + +-'.>+

-,.~[i(~+q2)+~*]+E

- o,

(8.1o)

in which the production terms &ppear no longer. The letter E denotes the total energy dissipstion according to Eq. (7.1). This equation (8.10) expre~m the fact that the loss of total kinetio energy (mea~ flow and fluctua~inn) is trmmferred to such poqitiorm in the layer where it is converted into hest by viscous dJeaip~don. One might expect that the given double velocity equationa m~y serve to construct together with Eq. (4.9), a more complete set of equations for calculating the turbulent boundary layer. However, the set of equbt~ons is indeterminate, even to & higher degree tIum Eq. (4.9) is =]one. Some attempts h&ve b~en made to complement the equations by semiempirical relations, based on ideas aimilar to the hypothetical relations reviewed in Section 5 81, 9e. But the suece~ of these attempts is poor as far as ready calculation methods are concerned. Nevertheless, very

Tu~o.~

B,,un~a~ La~,., in l , w , , ~

Flow

4?

va]--b|e contributions to the understanding of t.he m e e h u ~ m of Lm-bulent b o u n d a ~ layers result from the double vele~it,y equations in combination with experimental re~ult~ and further theore~ier.] c o ~ i d m tiOnL 9. T h e E n e r g y T r a n s f e r a m o n g Different Velocit7 C o m p o n e n t s The ".Jlroe~|merulioll~ e . ~ Of tm~o~eDt flOWis to • gT~tt extez~ attributable ~ the pressure fluctuations. In the double ~ " ~ M ~ tions the action of the pre~um fluetu~tioM is sub~.,~-T]y c~utsined in the terms p ~ / a z etc. The sum of these terms of EqL (8.I) to (S~) is zero u L consequenee of the continuity relation. Henoe these terms do not contribute to the balance of the total turbulent energy, Eq. (8.T). Their Ju~tion m u ~ be m t ~ p ~ M ~ ~ energy transfer among the different velocity components. Since from theoretical arguments isotmpk tur~ukmce is statistically the more probable state, it is expected that, in Lnkotropie turbulence these terms transfer energy from the larger fluetuat/on components to the smaller ones, thus giving the flow the tendency to isotropy. Experimental investigations of homolpmcou~ Lnkotmpie turbuhmoe by Townmmdzu and M nu m ~ that there is in fact an energy transfer in the Lbove-mentioDed statue, though t]~ rote of this t r ~ , ~ r is not LS ~ as might be ex-peef~ st ~ I t w u found by Uberoi thLt in Lxisymmetrie turbulence the larger (radial) o o m p o n ~ t , Io~ more energy due to viscosity t h ~ by trLnder to the mnaller {axial) component, l~evertheless, the smaller component ~z receives enoush energy by tr~,~e~ to compen~te completely for its decay, when the ratio Levz/wz - 2. Furthermore, it was ahown by these experiments that the -,~,*" s~t.le motion is becoming botmp/e at • fasterrate tha~ the largeeddies.In 1~oeroi',e z ~ t e the tarbulenco waL not locally isotropie at the be~n-~-g of decay, but it bcoame,soby the ti,-e the velocity compouents w e ~ in the m~io ~z/uz _ I.~. An e ~ * t e of ~h~. energy redistribution made by Rott~H, is based on the a~umpfiou that the ra~e of transfer is proportional to the degree of ~-;,otropy. If the turbulenee is u s m n e d to be locally i s o t r o ~ so thLt the ~ energy cUsaipation of any component is ¢~ - ~, - ,~ -- ,/3, then the ratio of the e ~ e r ~ tran~er to the partial energy d ~ i l ~ t i o n can be expressed as

;,(e,,la:). •vhere k is an empirical constant, , , the partial energy ~;a~pation of the ~-component, end ~ = ( u ~ + ~ + w~). This relation is c o m l ~ e d

48

J . C . RoT~a

with the experimental reeulte of Uberoi in Fig. 9.1. The mnstant k is chosen equal to 1/0-7 in accordance with the odgimd l~tperre. It is seen that the vaJ_uce from the me~uremente ~ larger ro,~aly by • factor two. I'l

u O

1"2

(aJLg"0 I N

•

CJ

•

OO@

(- .0.41

o

-O-Q

['.

-•,ii -o.@ -O-S -0,4 -O,~i -0-| -0.I •" . ~/~!

•

O.t

i

0-|

0-$

P -.c,'w/ID

Fro. 9.1. The rst.io of ~ t.mn~er due to pr~sen ~etusainns to ~ di~ip~tioa ~ the d ~ of ~ W ~ ' In a ~ W metrio ~ ~ bqm me~ursmm~ by Ubeeo/.1,. Since in any stress held s system of principal sxm can be found, in which the shear stre~ components are zero, the relation Eq. (9.1) can adso be spphed to the term p ( ~ / ~ + ~ [ ~ ) of Eq. (8.8). In this wsy we obtain for locally isotmpie turbulence in a uniform me,.n flow

1 ~+ As is seen from Eq. (2.9), the pressure fluc~u&tions are produced partly by the mean pre~ure ~ a d / e n t ~U[~ ~ d the fluctuating qu~-ltity ~ [ ~ , and partly by the interaction of two fluctuatin8 veloe/ty component~; so thst a substantisl fraction of the e n e r ~ redistribution • mong the velocity components is achieved by the mean velocity dient. Consequently, when the me~n velocity is not uniform the redistribution of energy is an integral peat of the distorting process. In t h k connection, & theoretical treatment by Batehe|or and l~roudm~r.~ of initially ~otropia turbulence under the action of & rapid uniform cKstortion is of interest. In tl-~s investigation only the part of the pressure

T - ~

B~un~a,y I~e~m i. l.~omprz~%k ..FZow

49

fluctuation ~chieved by the m~-- velocity gradientl and the distortion terms are taken into acenunt, while the mutual intersction of turbulent veloci'ty fluctuatious and the ~.~ous terms are neglected. Towns; end 14e applied the results of this paper to the zpe~'*! e a ~ of--;fonm plane strain. I t could be shown that the effect of the gistortion term is reduced to • remLr-ksble degree by the pressure fluctuation term, Results simZlLr in magnitude were obtained for shear flow near ~ solid wall through • leasex~,t e s t ~ t ~ by 11ott~ee. The part of the e n e z ~ rod~r/bution ,zhieved by the ~ velocity gn~lient thus r e v e ~ itself as • resistance of the turbulence ~ ; n a ~ becoming by distortion, and is benoe • counterl~rt to the tendency to i~otn)p~ achieved by the interaction of the fluctuating ve]oe.-;t7 oomponent~ It would be very desirable to h~ve directly measured values ofp,~/~z etc. in order to confirm the above conclusions, but the experimerd~ determination of pressure fluctuations is • ~;mcult task. P.emmtly Kohsshi~ constructed a device for pressure fluctue.tion measurement& In combination with • hot wire anemometer, it was pose/hle to determine thecorrelation between prmsure and velocity fluc~uatiom in the wake of • cylinder. A'further develOlnnent of this method in order to measure the correlation between veloc/ty derivative and fluctuations in boundary layers would oerCa/nly be very valu-.bk.

10. The E q u i l i b r i u m Conditions in T u r b u l e n t S h e a r Flow T h e following dbcussion will be co-~-ed to p l ~ e shear flow, wbem only the e~e~ti~ effecte are in ~tion. To th~ p=pose a,~ transport terms in F,qs. (8.1) to (8.3), and (8.5) are n e g l ~ Th&t the convection terms (U~uz'-/~÷ r ~ / ~ ) etc., are not indislx~4~le il eui~y s e ~ by the fact that there sre types of shesr flow in which these terms vanish, Such a type, ~vhich k ,im;lar to a boundary ]~v~ in some imlz~-tc ~t reqx~ta, i~ for example the fully dcvelo,-,~d turbulent flow t~.,mq~h • two~hnensional channel of constant width. The diffusion terms of e ~ b equation disappear upon integzstion through the thickness of the boundar T layer. These terms do not eont=ibute to the overall belanoe of each velocity component. From this it can be concluded that these terms a~e not absolutely ~ecc~ary for the equilibrium of shear aow either, although no turbulent she~r flow can actually be constructod in which the diffuEion terms are completely absent. YTith these simpllfying mnml~ tiorm, the distorting terms, the pressure velocity gradient product, and the viscosity terms are in competition. This system of redstiona is sho~m di.~gr&mmafically in Fig. I0.I The effect of the ta-ansport terms ~'Lll be discussed i&ter.

J. C. R o T ~ When s Reynolds shear stre~ and a velocity g2"a~Uent a/7[~j are pre~ent, the whole rate of enersy production, which is equal to the ener~7 ai,~pation, is supplied to ~luctu~t/ons of the u-component. The energy dissipated by the terms ~ V ~ ~ , , and ~wV~ ,- ,,, is hs~ucecl by the enersy gained from the ~-~omponant by the term I/pp,~[~ The ~ntensifies of the ~- and ~ e o m p o n e n t s are therefore e~l~ected to 8olo~e ~

sM4r ~rees

-~-;F Balanc'e of

4,nq~j~

(nerqll r N hilt U~14m,

OIwu'Na

Vlw,mdly

g

P.olo~:l d

dlstorfi4a

l~o. I0.I. Y~lu~la'[um of turbulent eheedr ~)w st hu~s Iteynolda numb~L Valuu in squsro brsekete apply to |oea/ly k t r o p ~ tw.bu/mwL be smeller than that of the v-component. An equ/l/br/um value of the Reynolds stress is g u a ~ t ~ e d by the l u t equation. The distortion term -u~U/ay is balanced by the action of 1/pp(~/~,/÷~e/~) and the v/seosity term (~(vV~+ tIVze). The proceas may briefly be summed up as f611ows. Lon~tudina] velocity fluctuations e a s t because of ~he simultaneous ex~d~ce of the Reynolds shear stress and mean veloe/ty gradient, transverse and lateral velocity fluctuations exizt because of the pressure fluctuations, and the Reynolds shea~ stress exists because of the A~multaneous exktence of transverse Teloeity fluctuations and mean velocity gred~ent.

Tu~u~st .5m,nSa,'~ .f,o+~er.~i,t I,~,omp,'e~m~ jr2om,

Sl

When the Reynolds number is sufficiently high to m~nn~ load .isotropy for the dissipating eddies, the viscosity terms in the emeslqybalanee are equs.l in mat~tude. (to.t) and the viscosity term of the equation for ~e is zero

~(vV~z+~Vs~) - O.

(~o.2)

There then results simply

p-J~ " - ~ m

1~

1 ~

1

(so.,q

L. this limiting eue two-thirds of the energy production is tmnders~l from the longitudinal component to the normal and is.end component

'l"be dktortion term -;".elPl.~ is completely i.d.,cod .by tim, ImmUre,

velocity gradient product .t l ept~,/av + a,,ia:). The c o n e m n ~ is ",~,em independent of the ~%en/tude of the Reynolds s t r e ~ The fluctuation eomponents ~ in constant mtim + a / ~ and ~ l u S , , ~ d the eoemcient - zvl(us~)ln, has a well defined value. The relation between

the mean velocity gradient and the Reynoh:k shee.r ~ deponda, d oourse, on the size of the eddies c o n ~ g energy which in turn on s c l ~ ~ e length of the flow (i.e. channel width, boundary layer thickness), and is •

-

~

(lo.~)

--'IKIP

relation is already suggested by the mixing length theory. AJthough these inferenoes are e~a~ighLforward, ~ ]]tt~ is seen of whst actually takes p]soe. FL,~t it might be ~ m&tterof dispute, whether local isotrupy can be expected for'the di~ip~ting part of the spectrum in eo.uilibriam shear flow. In fs~t, experimental investigations on the distortion of homogeneous turbulence (TownsendX4', UberuPmz) show clearly that the distortion affects the ~rhole range of eddy ~dzes including the ~Luipating structure. Howe~er, t~e degree of anisotcopy produeed is determined by th~ ratio of the mean ve]ecity gradient ~ET[~y ' 2 to s representative turbulent v e l o c i t y ~ i e n t , say [(~/&)]~z. Townsend came to the conclusion the~,l~c)zJx~z must be st least

52

J . C . ROTTA

two orders of m~,nLtude greater than aU/Oy if loc~ isotropy of the dissipating eddies is possible. B y means of the relation for the turbulent; energy d;ulpation the ratio (SU/&j)/[(O~/Oz)z]l/s may be replaced by

Furthermore , can be replaced by -uvaU/Oy. Then with the a/d of Eq. (10.5), it can bz shown thLt

which gives the more convenient criterion •

lo4

(1o.7)

IP

as a condition for local isotropy. It is seen that ~oea/isotropy for the dissipating structure is just & question of Reynolds number. W'th laboratory experiments, Jl; apI~m'a to be generally imgomu'ble to est6bIJeh Reynolds numbers high enough to me,=t the condition of lees//sotropy. ~ is probably the r e u o n , why ~ m d b o r n and B r a u n ~ found no evidence for local kotrooy in the boundarj layer. Some calcul~tious ~u~ed on simple easumption= with respect to the energy transfer and dissipation terms have been carried out by R o t t a H in order to support the picture of turbulent shear flow= as sketched in Fig. 10.1. I t can be seen from the details oi"these e a l c u ~ t i o ~ t l ~ both paxts of the energy redistribution terma--viz~ the part depending on t h e mean velocity grmiieut and the part ach/eved by the turbulenl; .quctuations a l o n e - a r e ~uall.v impcrtant. Recently the behaviour of weak homogeneous turbulence under a uniform distortion h u been studied theoretically by Peamon v4 by means of linearized equations. It follows from t h ~ paper too that a steady turbulent ahem- flow cannot be achieved by the interaction between mean flow and the turbulent flucf,u~tion and by visoous tanma alone. Horeover, since the pre~ure fluctuations are determined by an integral over the flow field, the redistribution of energy is affected by the ~patial distribuzion of OU/Oy. Thus in the general case the ntato of equ/librium is quantitatively influenced rather by the distribution of U over y than by the local value of 8UI~. ='~aere am some more effects ~n the eddy structure, which cannot be seen from the equations of Section 8. The change of spectrum functions during a rapid d/stortion of isotropic turbulence hM been inveeftigated theoretically and experimentally by Towz.sendl4S. There is no doubt that such effects are also present in equilibrium shear flow. In addition,

~'urbu/e~ ~ o ~

J ~ y e ~ in l ~ o m p v e a ~ k Jv/o~

~t

the energy t r m ~ e r among the eddies of dif.-Tent sizes is dimetly afl'eoted by the distn'bution of mezn velocity, if the.mean motion has • eomplexIy var$ing l ~ r s L This is seen by writing down the equAtiom o f motion for the double velocity correlation functionN. Such ooudit/ons rosy occur, when the iMude of the mean motion, u depon~mg on I0=~

i

i

\ j.l-~ o-, .~/

04).'

~I

O"OI

,

,

Oq

4-0

IO-O

I/

1,m. 10.~. l~equmey .pectnm f m ~ o m o~ ~ t o e # t u d ~ fluetutlems at dillenmt di~aaeew from the wall of the mz4mlmt boond ~ l a ~ " ea ~ a-t plate h~an mmauremema by ~ e b m ~ r a ~ d ~ a U.,~I, - 7.10% £ . - UFaz(0)/6 - longitudinal male of u ~uett~tioa. 0 ~/t - o ~ £,/t - o-zs w ~/a -

o-57

£./a

- o-5

(a lee)l(e i@% (e: 71+t)l(e: les ) em. is ,,m,,]l become,, com. parable to the mean eddy size. Tchen~ l has diseussed this in more detail and, by the use of dimensional argument., he came to the conclusion that there will be s subrange in which the energy spectrum vzrim with £-2, when the scale of the mean motion is amalI in ee.mparison with that of the turbulence. ~eamarementa of the frequency spectrum made near the wLll and in the outer region of a turbulent boundary layer #how in fact L remarkable difference in shape, Fig. 10.2. At y/~ - 0.033 there is a frequency range with the shape n-l, which eorresponde to the prediction by Tchen. Similar reg~t8 have been found also tbr turbulent flow in a pipe by LaufereO.

54

~ C. I ~

The equilibrium conditions in a turbulent boundary layer are rendered- still more complex by the action of the transport terms. The~ contribution to the enerKy balance will be discussed in Section 13. It is thus seen thst the ~ow a~ & certain point, is affected by the surrounding field for several reasons. The constant defined by Eq. (I0.~), although independent of vi~wsity, u long u the Reynolds number is high enough is subject to & number of fs~.ors. This emphasizes again thst the ideas underlying the Bouminesq relation ~nd the re;r/rig length theory am rather incomplete with respect to the e~tual behsviour of turbulent shear finw. 11. T h e F l o w n e a r the Wall

11.1 S i m i l a r i t y c o n s i d e r a t l o u The fiow in the pro~m~ty of the wall deserves specis/cons/demtion. Here, the motion of the ~uid/s" restricted by the preeence of the ws/l so that the ~kcous disaipstion increue8 and eon~quently the v ~ c i t y fluctu~tious deeren~ when the wall is approached. As s eonm~enco, the fraction of shear stress achieved by direct action of the viscosity glows more and more while the l~ynolds shearing stre~ - u v decreases. At the wall itself, the velocity .fiuctu&t/onp v~nish~ and so does the Reynolds shc.~r; thus ;he laminar law, ~U -

-/p,

¢u.I)

is valid here. Very important is the experimental fact that the d s m p h ~ action of the viscosity forces d e ~ very rspklly with d/stance fi~m the ws]], so t h ~ the trans/tion from thJ~ laminar law to the fully developed ~.ow, in which ,~/7l~ is h e l l i s h l y small compared with - u e , is a~omplkhed withiu s very t.h;n layer. Since the thiekness 8, of this aubl~yer k small in comparison with the boundszy layer thickness, further simp]~e~tions of the governing equstions of motion are admi~ tible. ~t psrticulsr, sll deriwLt/ves of mean values with r e s p ~ to z are n e ~ ' b l e for sumciently small vslues e l y . With this supposition it follows from the continuity Eq. (~.S) that I v . 0, and the boundary lsyer Eq. (4.5) Sires upon integration

This means that the &vere~ shesring atre~ • is consklered to be constant b~th respect to all three direet~ous, of space throughout thk part

o

-

[

of the boundary layer and equak the wail shearing stre~ v~ I t ia generally assumed that the mean ~elooity diztribution in this region k completely determined by the magnitude of the shear stre~ at the the density p, the kinematic viscosity ~, a~d the d i ~ m c e y from the wall. and e~n thereforebe expressed by the ,~.'~l.~dt'ylaw

V - ,..,f

(zt.)

whm

v+(v,/p)

(11.4)

i. the ~ e t i o n velocity, and +fk z u n i v e n ~ function of ~ J v . The univernal law of the wall ,,~;~en in +.h;, form, was first given by Prandtlm in 1932. The g r i t ~miflcance of this law ia that it seems to be valid everywhere in turbulent flow along a eoUd walL T h u it applim for flow in ,,'l.,it,.,,.,eliland pipm as well as boundary l ~ y m with and without extermd pressure gr~__ient~ Outside the n b l a y e r but still within the region of constant abesring strces, the tu~bu}ent motion approaches a ,~tate of equlUbrium to that discussed in t h e pr___-,~Ting eectio~ The form for the vslodty digmbution in this fully developed turbulent .-~gime k derived from Eq. (11.3) by ~ifferenti~tion

,_..,,

(11J>

Since the right-hand aide mm;t be independent of the viscosity, it follows that -

-

m

.

Or

.

~

m

--,+ for

where ~ is a u n i v e r ~ | constant, which has been found from exporimente to be nftbe order 0.4. The thickness of the sublayer is about ~u~[, -. 60 Integration of (11.6) gives for large vffiluce of yu,[,

. ~,l_in~___~,+c[fl_1 for

~ ~. "(11.Y) J whom C is a constant of in*~gratLon. This ~ailogaritl~ai¢ law first d~'h-ed by n,~ng the --~i~ing length theory, where the = b ~ length was assumed to be proportional to the distance from the walL However, the notion that the flow is Rim;|ar ill a~] planes ~ / - oonstaut, and differs only in the scale, which is proportional to y, leads in a more general manner to the universal law, Eqs. (11.6) and (11.7). The mean velocity d~,ribution over the w~ole range of wall distanves is shown in the generally used semi-logarithmic representation on Fig. 11.1. U

L;f

•

30

|

|,

:o

J

/ I

I

o

O

j

TM

!

IO

*06

iOa yms*

"7

Fla. I L l , ~ ~ h , , , ~ ~ of ma~z Teloci~y dist~'budom ~ the v i c i n ~ ot ,~ m z o o ~ will ~ q ~ vsu Drim~m ~ poiata fm • anoo~ ~ pipe ~ Lms~.m OR.Tnoldsnamb~ •

..

.

S0.00#

., 500,000

3.0

Z.I

2-G 1-5

0

,,J-v- mm~.

,i

f f t

O

f

IG

ZO

~

3Q

4Q lur B

SO

60

70

80

l:~a. 1L2. D~e~Jo~e~ plot of LurbuJeneei-I~wiUee ~ r ~ Jumaot~ walL Experimental poinW for • srnoo~h pipe 6f~er Laafer.W -qymbola u im Flg. 11.1.

Fxo~ ~zictly theoretical srgument~, the r~ppomition of • ~,~n*rity of the wall flow iz only justLfied, if it ineludce any ~ a t i ~ e a l quantity. I t has become standard practice to plot experimental raw.Its in • dim,~,~onleu ~ a y in aoeozdmce with thcee similarity zel~tinn~ ~ . llJ~

,-o

zo.~i,.,

~..

~

O,~ O-$

!

i~.4/.~'~ ' •O , Z

•

,

*O

20

~O

40

6O

tO

?0

no

~0

m

l ~ z l l ~ l~eyzml,k ab~z ~ mar • z~z.nh wall ~ ~

and ~ ~

m~z~m~ m ~ m ~ in, ~ p[l~ by L ~ . ~

ahowm the mere nqmLw :eot vLlum of the turbulence i n ~ The P.eynoldz mh~ring ~ ~, computed from F_,q. (11.2) with the m e of (zz.5) =~z (zz..e), ~ r

ff-[

It is seen from this rr~lstio= thgt the spparent ahear strew - ~ proa~ee rapidly the value ~rwJp, u the diztance fxom the wall incaws~a, Fig. 11,~. The correlgtion oo~.flieient -uv/~/uz~/z4 is also given in Fig. 11.3. The non.visoom mnilaxity in the fnl]y turbulent p u t of oonstant e h e a r ~ ~zem implies s c o n m n t oor~'~tion o c e t B d ~

-~/~/~Zv,~Z. Since aooon~g to F~t. (ZZ.8)the Itey~old. ~pproaches z c o ~ t value, it follow1 tl:~t the turbulent intemitim within this region mz~t bc oaz~ant multiples of the friction velocity, and in continuation of such ideas, trip}e product8 of the velocity oomponenta must be vonstant multiples of the 3/2-power of the fxiotiozz velocity and so on. An attempt to correlate the e~veml turbulent intenaif,y components has been made by Hehnbo]d 40. The only part which im excluded from the general ~milarJty is the di~dpating range of eddies, which is of course ruled by the a~-~,ionof visvomty.

The scale of the larger eddies is determ;-ed by the~ distance from the wall. Hence the application of the ~ m i l . ~ t y relation to double velocity enrrelstion functions and spectrum functiens leads to form~ 1/~

(t~.9) (l~.Zo) In the fully turbulent region, the functions are independenb of 9~.lv. Some doubts amto the general validity of this simi~-ly have been raised by TownsendlU. ~ore recent experiments by Grant ~ indicete ths~ the longitudinal seale of the la,r~,,, eddies is posibly governed by "the . outside flow conditions rather than by wall ,~m;1A,ity. Apar~ from ~h;, possibility, it may he recalled that there is observed a subrange near the wall, wi~.a;n wh/eh the spectrum function varies ~s k -i in aeoordsn~e with a theoretical prediction by Tchen Townsend pointed out the fact that the sim,.t~neous val/dity of the adm;1Arity, Eq. (11.10), mesas that a substantial part of the spectrum is i n ~ t with respect to variation of dktance from the n i l 11.2 T h e behaviour ot the flow in the i m m e d i a t e vic/n/.'y of the wall The bel~viour of the velocity ,quctu~tions sod Reynolds shear stress in hnmediate vicinity of the wall can theoretically be studied by means~ of the continuity equation and the condition of no slip st the wall Sneh considerations have been used by Reichardt~, Townsend.14a, Elrod=, and Sz~blewakil=. The f l u c t u ~ o n velocity aomponents are zero at any i ~ t a n t u-.

• .. w--

0,

t'or ~/--

0.

(11.11)

Con~quentJy ~z

az

~z

az

~z

•= 0

for

y=

0.

(tl.19)

I t followl fi-om the continuity equation that 8B

~

~Z

8z

--0

for y - 0 ,

(11.13)

However, the derivatives ~/~y, ~w/~9, and higher order derivatives with respect to 9 have finite values. Again from the continuity equation

- P,.P~,,Z+'~ ,,,Bo,.'m~ Layer, i,~ I n c o , n ~

F/~

59

~-hich has thus also s S-;t¢ value. ~Vith th~se conditions, the first three deriwtives of the Reb'nol~ shear stress are -m

ms--m,

m

Ox.zs) m

(~ue)

Introducing Eq. {II.14) for ~ ] ~ r ~ gives •

-

-

-

0.

c-.-)

l~ow, since eli derivatives of mean values with respect t0 z are auumed to be negligilAe, the first term on the r i g h t - h , , d -;de of ~_x/. ( I 1.18) v~i,hes. Thi, gives ' ~q,w

~ , age

-

,3w , ~ / ~ x

.

Now the question is whethe~ or not. t b e ~ exists • oorrel~ou b e t R m

eg/e~ ,~d r.r/~z and between eu/~ ,ad (ew/ez)e/~. Th~ que~oa cannot be definitely de~ided either by theoreti~l arguments or by experimental results st-the present t;-,e. The experiments, l~g. II.~, show ~ d e e d &~ear]y ~onstant eorrel~t/on eoemcient down to p o s i i / ~ lying deep within the sublsyer, but the extrapolation to the wall is uncert~n, l~resumably ~v--/~.I is different Lwnn zero for y - O. ~ t h the e ~ n p t i v n t]~t the third order derivative of ~vvis d~eTent from at the wall, the mean velocity distribution extremely el(me to the w~] i8 obt4dned by integration of Eq. (I 1.2), and il give= by

11.3 E n e r t y balance The investigation of the energy balance in the wall layer is important in the ~ttempt to improve understanding of the motion in th;. ~gion, and to realize its significance in the theory of boundary k y e r flow. Several authors ss, 6o. sz. x4s have been :oneerned with this problenL

S0

,T. C, Bm, r~

From the outer part of the boundary layer energy is t m m ~ r t s d st s zzte (~,~/p)Z7to the region of the wail layer. T / ~ is partly changed into turbulent energy and partly converted into heat by direct dismluttion To see this multiply Eq. (11.2) by dU/dv, sad the energy balance is t h u wzitten - , I d(r wU)

__dU

/ dU \s

(zz.~-z)

EquAtion (&7) for turbulent energy conaer~tion in the boundary htyer reduee~ for the th;~ ltyer eloee to the wall to --dUd

+,,,(~.vz.u,+ , ~ f + w~.w) - o,

(11.2~)

i f E q . (8.5) is used. Eq. (11.22) contains Idn~e terms, viz. the turbulenb emergy production, s ~ term including turbulent energy difueion and pressure d~A~teion, and the viscosity term. The difaeulty in making up sueh an energy baJanee lies in the ~wurate determination of the pertinent terms of Eq. (11.22) from available measurements, xince t number of terms o e e u t ~ g cannot be mes~ured by known experimental methods, Furthermore, hot wire mewuements in the eublayer e~e affeetsd by the preeeuee of the will and the ~ u l t a am thul not adw~ya eompletely reli~ie. The energy production - ~'e(dU/dy) aa well as the direct vkeoua dkeip&tion v(dU/dy)S can be ealculated fi~nn the mean velocity distribution without further mesmuemmts of" any other turbulent quantity. The mean velocity di~n~oution is known with a relatively high d q m e of aceumoy. It is easily seen by di~etwst~tion of Eq. (11.21) with respee* to the quactity dU/d¥ that - ~ d U / d y ) h a 8 meadmum for ~lZT/d v -, u,z/2, that is, when the P.eynolda s h e ~ and the viaooul shear ~a~ea are equal in mqnitude. The m ~ mum turbulent energy production is _dU) v~ -~:--. - --. U y maz 4~

(U.2S)

The direct d;~ipation has the same value sa the turbulent energy production at this station. Fig. 11.4 shows the t,~bulent energy production and the direct di~ipation in non-dhnemdonal form. The energy production reaches its m a ~ m u m at ~u,/~ ~ 11, as given by the Lurersection of the two curves. It may be noticed that the bulk of the direct ~tion takel p~ce in a very narrow region, 0 4: ys,/r ~ 15.

61 In onler to cteter~;~ste the vise~uJ term offal. 1 1 ~ from me~umme~ts, the relstion of ~q. (S.5) ;,, to be used, wh/eh booomes on

]~.q. (s.e)

,~,,V~+ ~vs~+~Ww] = , - - ~ + ~ ) - z .

(llJ4)

The f i ~ term on the rlght-hand rode is the rate of energy diBudoa by viscous form, and can be ealculat~ f~ozn tbe measured t u r b ~ e e intc~uitim shown on F ~ . 11,2. The turbulent e n e r ~ ~ b u ~ t i o n given by Eq. (7.~), ~ be w~itSe= ,,,

mature,

The rust three terms can be obtained experimentally by tbe upe ~ the dit~ere~tistioa method introdueed by T ~ d , adopting T~ylor~ hypotbeak, Eq. (6.14). The fourth to ninth terms ean be determined~mn I'O 0.t)

~"

O-ll

0"? 0"6 O'& 0"4

D'3 O-r

0-! 0

I0

20

3C

40

SO

GO-

1'0

IlO

DO

ymr

Fxo. 11.4, ~ t e of turbulent energy produotion and the rst4 of viseouJ dimip~t~on near the ~ after Lsufer.*e

aZ

J. C~ R o r r ,

the space correlation function, using the relations Eq. (6.15). Although it appears possible-on principle, to determine all mean square derivatives of Eq. (U.25), oaly the fourth and seventh terms can actually be obtained in this way. The hot wire arrangements necessary fox measurement of transverse velocity fluctuation makes the application of the method described impracticable. Me,q.surements of the disaip,~tion derivatives in close proximity to the wall have been made by Laufere° in & pipe and by K'lebanot~2 in a boundary, layer on a fiat plate. The results are in fair agreement with each other, rn addition to this a few remarks about the behaviour at the wall can be made with the aid of the continuity equation and condition of no al]p as described in Section 11.2. From +..

u -. a ~ ÷

(tl..~e) ~ - b~V+

+...

it follows thM

~a .. a t ~ s + a x a ~ + _ . (liSzT) u~ = b t z U z + ~ + . . . ~trgl

(-:I'

(I].2s) - bt~ + 2 b t b ~ + ....

-

-

Thus (Su/~)z and (Sw/Sy)s at the wall can be calculated from the tribution of intensities. All other dluipation derivatives are zero at the wall and start with yS or higher powers. The experimental results of I~ebanoff are given in Fig. 11.5 together with extrapolations up to the wall. It will be noted that the condition of local isotropy, Eq. (7.4), according to which all terms of Fig. 11.5 must be equal, is not ~t~sfled in tbis region of the boundary layer. The curves become more widely separated as the wall is approached. We note that the derivatives with respect to y and z are much larger than the derivatives with respect to z, and this illustrates the very small scale nature of the turbulent motion in the trausverse and lateral directions as compared with the longitudinal motion in the region close to the wall. In order to obtain an accurate measure of the dissipation all nine derivatives of Eq. (11.25) must be knowv. Since only five terms can be measured the remaining

four must be esthnated. Laufer and Klebeno~" proposed the use o f ' h e isotropic relations

(#--

o

(11.29) %~zt

for t h ~ purpose. Actually, (~u/~/)# and ( ~ / ~ ) s have finite but v e ~ ~ifl'erent value8 at the wall, T h e r e ~ ( ~ l ~ / ) s is zero for ¥ -- 0. Therefore, we assume ( a w / ~ ) s to be a~ shown by the d u h point curve in O'OZ4

O" 020

..,.~ (~-) .,~,,, ,., ,.o

\,

O-Or6 g

U 0-012

i

.. , - . ~ . ~ . ( ~ ) . , . ,

,..,,.,

O" 00@ ,

T

Imlllle)

--

0-004

:.~. 20

40

60

O0

lOG

120

i40

160

m

Flu. 1 I.S. Dist~.butlon of dimip~tion deri~st~vm near the wilt. bicmm~mentJ i~, KJebanof~J. Dotted ~ am eztrupohLtionL

2~rm \ S z /

"

C~'l'

o 2-"~",,~,~/

•

2~'X~zl

v

2,,,',~e/

" (--~'7

,~-,,--,,,.,rm

;'6.

o,"

64

J.C. 11or~

l~g. 11.~, and ( ~ l ~ ) Z h put equal to ½(~/~'.)z.TheremaininS terms are estimated by Eq. (11.29), although (~/~.)z is cert-~inly smaller than (a~/~) z at positions near the wall. The term ~ l ~ y z can be esthnsted flora the dL~tribution of intens/ty. The entire viscosity term of Eq. (11.2.~), calculated in this way, is shown together with the rate of turbulent energy production in ]~g. 11.6. The viscous diffusion and the rate of ~;m~pation have oppos/te signs and are equal in magnitude at the

Jill .

-O-] 0

4

•

12

. te

ZO

Z4

Z|

Lt

34J

40

44

48

3~p

Fz~ 11.6. Turbulent onar~ balance ~

the ~

Eq. (11.2S).

wall itself, as can be shown by F_,qs. (11,27) and (11.28). In eonz:quence of this, the entire viscosity term tends to zero for ~ .. 0. When the d/stance from the wall is increased, the ~mlpation terms of ]~lg. 11.5 show & tendency to sstiafy the condition of local isotropy. If we adopt the relation Eq. (10.7) and put Io - 0.4y, we find from Eq. (il.6) that local isotropy in the wall layer is to be e ~ when • 2.5. 104.

Since, at the same time, y must be roughly on= order of ~agnitude smal/er than the boundary layer thickness, local isotropy in the tango of the wall s/mi]arity can be observed only in boundary Iayers of exo tremely h/gh Reynolds numbers. According to Eq. (11.22), the neo~ative sum of the turbulent e n e r ~ production and the viscosity term is equal to the turbulent ~ o n ° In the fully turbulent region v(½(~z + vz + ~cz)+p/p) is constant because of the s i m / l ~ t y , as already mentioned. There/ore the rate of energy

T~bu~nt Bou~Aa~ Loyola ~ I n o o m ~ ¢

~

G~

production must be ent~e]y eompeusate¢~ for by the rate of dissipa~ic~' From t b k it follows that energy production aud energy ~ p s t i o n esch ~plno~-b

--~.

(11.30)

= . -- - - @

,-, )' is incrommd. On the other hand, quite close to the wall the energy. production term probebty v,m;~es faster than the other terms. The v/soosity term is then b*~-~__~e~by the pressure difusion term. From Eqs. (II~7). (IL28), and (II.22) it oan be ~how= t h a t ( m ~ble~w~au~

• Ol.Sl)

~TJ --qW

"

~or

0.

"

l r -

Since no definite slop of v~(½q2+ vz)/~z osn be identified in Fig. 11.6 the pressure ,4Hr-usion term e~(~lp)l~ wm be very m ~ n for S / " O. Over the most of the viscous part of the wail laye~

~ley(~[~ /2(,,, + := + ,~)

+;o/p]}

-

is positive, thus indicating an energy ~ ' from the r e ~ o ~ n e ~ the wall to the outer part of the lsyer. The wZi~bility of d~;, ~ n e n t ~. ",..," of eoune, on the ~ 'with w h i ~ the y'booei~ term ~ be deter~;~ed, lrlebanofl~ found, ]~om an energy balanoe of the emt~re boundary layer, that the integral of the energy ~;.~pation was too small Consequently, he sugges4~i that there might be s =mall region near the wall (3m,/v < '2~), where the rate of d i J p s t i o n exoeeck the ~ of energy production. This m e ~ . an energy ¢ti~mon towards the On the other hand m e ~ . , ements of triple ~velocity c~dat~ona, made by Laufer in s pipe, give s turbulent energy ~;ffusion ~w~y from the wall. This oontradict/on can be overcome by ua-m;ng a ~ -;on ~]p of opposite sign, which k ;,-deed quite pl~,-,;ble. The ]mowledge of the ra~e of eneqO, ,~;,~;pation in the ~ k y e r beoomes important in the oaloulat~on of turbulent boundary layer~ by m~mu~ of the energy i n t e ~ equstion. The i n ~ of the rate of energy, dissipated ( d i r ~ plus turbulent ~.;~;pat/on) wjt~;- a ] s ~ be~wee~ the wall and a point ~, l)-Lug in the fully turbulent region of the universal wall flow. i~

f

•J

Edy

J,o

0

o -

d~/

d y - t~(~z+ ~+ wZ)+p/p]

~ , , : U ( , / ) - ~,[½(u~ + ~ + ~.~)-,.

(11.32)

~,i.,,].

However, &definite conclusion with r e g s ~ to the value of +

+

cannot be reached from the measurements available. In any case, the high rate of wo~k done emphas/r~ the importance of the region near the w a l l This will become still clearer from the enerzy balance of the entire boundary layer, ~ in-Chapter 11.4 F u r t h ~ details of the s t r u c t u r e of wall flow l~eesntly some-attempts have been made by several authors to explore the mechanism of thewall flow in more deta/I. 3feasm~mante of va.dous components of space double velocity correlation func'./ons, carried out by Grant ~ show that the scales ~n ",,he direction oi" the z - u i s are large compared with the scales in the y- and z-directions, which are comparable with the distance of the wires from the wall. To reconcile Ay

II

/ //////

I /

I//

/ /

[

.

\'Itl , AIX /

FXa. II-?. Struotum of the "two-d/mens/on~" jets ha the we,U 16yet after Towusead'4'. (The f,,n ~ repn~en~ the edp of the jet. The duhed limm s m i ~ ~t-tive to the mean v , ~ i t y s~ the ~ of the

these rmults with the well-estsbliahed almqarity law of the wall, Townmmdz41 distinguishes between the total turbulent motion ~ud the part respormible for t[,e Reynolds shear ~ This latter motfon, called ";mivered motion", behaves in the ~,ay r e q u £ ~ by th~ wall ~mi;~rity while the remainder of the motion, the "irrelevant motion", is char~teristio of the particular flow and does not interact with the u n i v e r ~ motion nor contribute to the Reynolds shear s t r e a The ahape of the correlation function suggests that the motion may be 8 two-dimeuo clonal jet, origingting at the edge of the viscons layer, with its surrounding induced flow, Fig. 1l.T. The jet is nstural;y pauing through a velocity gradien~ tntnsverse to ira direction of motion, and turbulent exchange will kwp the longitudinal velocity w/thin the jet consistently less than the mecn velocity and the velocity outside cons/~tently greater. picture is supplemented, and to some extend supported by visual observations, which are descril~i by l~line and l~unst~fler~s ~s follows:

~Yhen dye is ~troduced s~ery near the wall into a turbuleut boundary Ia)~r, it is found that the patteru of flow appears to e o ~ t of an array o.~ "islands of hesitation" aud longitudb~1 vortices which ,impart • wispy appearance to the flow; these are interspersed with areas of faster moving fluid. The islands of hesitation appear as long stretched illsments in the d/rection of flow which move downstream more alowly than the surrounding fluid. The vortices apparently originate as a breakdown or rollup along the edges of the isls~ids of he~t~t/oL" The prhnsr 7 orientation of the vortex elements is longitudinal, t h a t is, in the ~low direction, but each vortex stands s t a slight angle to the wall so its d i s t a n ~ from the wall i n = e u e , as it moves downstream. After the

arook -MI) , .oa.~***~

i~o kollil

.

]m,lof~ql w, @f hqHB|lfl lira

l~a. 11.8. Ibi*4Fs.~mstic sket~ o/"the flow model in the wsU is~m d t]w /'u]}y d ~ D ] o ~ tm'Iouiqmt Ix)u]K~u~" ] & ~ &J't~' ~ a~][" Run~.z~kn~. vortex element reaches s certain critie~l distance f r o m t h e wall, it breaks Ul~ into a typical turbulent ' ~ a s h " by a process which k too rapid for the eye to follow. A sketch of thcee details is given in Fig. ILS. A recent L~ve~igstion by SandbornlO4 on t h e slmtt,y nature of the tur-' bulent flow seems to oonfl~n these obeervatinnL l~o definite information oonoer~,g ~:hy the observed ps/~em exists is available. Explanations have been suggested by Townsend and by I~ine and R ~ e r . Tbe i ~ t a b ~ i t y of the flow near the outer edge of the sublayer and the action of the outer flow are certainly decisive elements. The outer flow operates mainly by pressure fluctuations transmitted to the sublayer. A very remar "ksble model for the sublayer flow h a l been proposed by EL~stein and Li~, ~;, ~rhich has some resemblance to the observations of Kl]ne and Runst~ller at least with respect to the unstable

68

J'. C. Ro'~.x

,'.~ of the flow. The flow is considered aa an inhereJ,tiy unsteady proce~, in which the sublayer alternately builds up and decay~ ag~;n It is assumed that, at a given i n a t ~ t a rather high flow velocity continues almost down to the walJ. The high veloe~-ty gradient at the wall calla for an extremely high viscous shear there, which is not trausmitted into the free fluid due to the ltck of velocity gradient farther away f i ~ n the wall, but which decelerates the adjacent fluid, creating 8 viseoeity controlled eublayer, which gmwe in thickness with time. We have thug an unsteady 1Amlnear(or nearly ]Amlnar) flow bounded on one side by the rigid wall, on the other rode in & rather ttuder-deter~ined ~ * n - e ~ by turbulent flow. One may imagine that after a oertaln g ~ w t h time, fluid of high velocity is again transported from the turbulent regime to the wall; this re.establl-hes the initial state. Since .*urbulent mi~n~ is much more effective t ~ m viscosity, it must he expected that the cublayer fluid is accelerated by the turbulent outaide fluid to about ira own velocity in a time much shorter than that which wee required for the visooug build-up oi' the subl~yer. After this the cycle ends and a new cycle beghm. In order to obtsin e'x'presaions which can be handled mathematically further simpli~eattiousa m necemary. It is s m u m e d tlmt the ~ over which simultaneous growth of the mblayer occurs is in all directions large compared with the sublayer thickness. The bounchtry layer Eq. (2.2) thus reducel to au. ~s •

-.

(u.:3s)

where u is the in~mtaneoug velocity. A solution of t h e e q u ~ o n for the period of growth is

Q

where ~ - y/2~/(vt), and U0 is the velocity at time t - 0. Furthermore it is eamumed that the procesa of growlng continues over a time interval | =, T, and the duration of dec~y is negligible small With these eaenmptions the mean velocity distribution, the shear etrea at the wall, a~d the root mean square of the longitudinal velocity fluctuations can be calculated by integrating the inatantaneou~ values resulting from Eq. (11.34) over the time. The results, which depend on a eingle par~meter TUo~/v are shown in Fig. 11.9 and FiE. ll.10 compared with T.atufer's experimental resulte. This model is certainly not in full accord w~.'ththe actual behaviour of the sublayer flow. The mean velocity clistribution even violates the

:

.,

oont~uity oondition at the wall (the seoond term in the power oerim for U is proportio-A! to ~z i n s ~ o f ~ in a~oorda~oe with Eq. (11.~0)). Perhaps the most serious defe~t of the model k the entire ~;-,~g~rd of the three-~;mension~l c h a n ~ e r of the flow., Despite th~se -horteOm;n~; the proposed model attract8 interest _not only beesu~e of the

/

tO

I&

I.I

e

~

II

o,...rw i ~ ~-

I'

,/

''

&O~,O00 " " I

I $,SO0

/

!

•0

I0

20

4D

30

SO

iO

TO

Yet

l~a. i1.9. M e ~ v e l o u r , di~n'ouUo~ in t h e s u h k : ~ - ~ ~ d the ~ 0 O M ~ t J b0, L s u f ~ , , , ; " l q 8. 11.1.

to t ~ lnDo'/p.

~ e o r 7 o." ~;-,~J;- a n d / , i n for ~

$-0

1"," .

2"S 2"0

1.0

0-5

0

-% l0

}"1"

20

3(=

40

m~O

60

?0

Y~

Fzo. 11.10. L o n ~ t u ~ , ' ~ velocit-y flu~tuationa ;,, the mublaT~" a ~ o ~ ; n ~ to the theory of Einst#.~ and I~zs for v L r i o ~ v~lues of ~,e psnm~t~.

TU, I/,. O • ~tea~.ment~ by I~m'ez~ u ;,, l~q;. 11,2.

3. Q Rox':A

?o

surprisingly good agreement between calculated ~ud measured turbu. lent intensity ;cl~tributicn, but esl~ciaUy because it is felt t h ~ the underlying unsteady character is a highly important charaeterkt/e of the ~ow. 11..5- G e n . ~ , , ' - t l o n of t h e u n i v e r s a l l a w of t h e w e f t A very' valuable cont~bu~on to our "knowledge on turbulent boundary layer flow has been made by Ludwieg and T~flm...~7, by whom the e~dstence of the universal law of wall ~Iow has been experimentally veri~ed in boundary layers with nonzero pressure gradients.The experimental data of various boundary layers come/de near the wall, when /7/,,, is plotted versus log ~ , / , as ahown in Fig. U . l l , thus c o n R . - ~ . g

o o o O"

,~.+

4.4.

¥ o x;l~

..+:o

N ire qD ~

I0 IO

mj

/

a~

i~o. U.U. Universal veleeity d~stribution for turbulent boundary layers ~ith no~zet~ l~a~sure gradients on smooth w s ~ egret Ludwi~ sad T~nann.T. • ~ L:.'em.n, il,lndlea~ dP,,/d,I

l".a,

, -, 2"84.

lO-:

cf - ' 0 - 0 0 1 5 8

adverse p ~ m = e gradieu~

x

--

2 . 5 9 . 10 - s

0.00206

0

-

5.15.10

d

- -0-2.10-:

O.O011g 0.00364 l'avoursbb pmmm~ grsdimst

- -

-8

/7/., - 5-7~ log

+ &2.

,o'

~ur~,,/e~ J ~ d a ~

f~yer~ in Incomj~eam2de~

;;!

the general validity of the xinn~ar~ty r e l ~ o n s Eq~ (II.3) and (If.?). result has led ~o a number of new approaches to the c~]cu~tion o f turbulent boundary layers and is now generally accepted as a con. stituemt part of the theory. Theoretically an ir~uenee of the pressure gra~ier.t is anticipated sinee an ~Iditional term ycIPm/dz ~ appear in Eq. ( I I ~ ) and ,.h~invalidates the underlying suppoait~on of a mmstant shear stress. In this connection an estimate of the s h e ~ stress di~ ~bution near the wall '-- in order here. With the assumption that the universal velocity dktribution is valid, Co)eelshen calculated the shear stress distribution by introducing the velocity distr/bution accor~:'~g to Eq. (U.3)/nto Eq. (4.~), which givm upon integration dP,,,

du, ~ /r e I U-~~ |

e •Iyw~1

(].z.ss)

0

The integral can r e ~ i ] y 14 evaluated using the univenud velocity distTibut~on. ~umerics] values have been given by C~olesxs. ZForthe ~ g i ~ ou~cle the sublayer the L~t~.gml e~n be ~ l u s t e d analy'.i~lly to Ilrll~qP

w

"

for

~"

• 60.

(11.3k)

As is seen from l~.q. (I 1.35) the variation of the she~r s ~ s s is by the der/v~t.ive of the friction velocity and the pressure i ~ d / e n t . Both terms are of the same order. While the variation of shear stress sm~ll in the fiat plate boundary layer, considerable changes o e e ~ if there are external pressure gradients. A rough estimate may 1~ made by means of the usumpt~on of a constant local ~ fri~/on ooeme]ent e4 = 2(u,/Um)z. We them obtain from Eq. (4.8) d.~ m

~

p~, d z "

(II.36)

Substitution in Eq. (ll.SS) gives for a const4nt value of cr

"r

~

1

-G

d

.

(11.s~)

~

J.C. Ror~

Some ree~lts computed by means of t h k equar,ion are given in s dimenzionlem form in Fig. 11.12 and show that the varia£ion of the shearing stre~ dependa on the vnlue of ~ and is u s u r y much mnaUer than ts expeef~d.-from the wall constraint ~ r [ ~ - dP,~dz alone. T h ~ may help to explain why an in~uence of the p r e ~ u ~ gradient on the velocity distribution near the wall is not observed in most of the / f

.o

.~,

/ /

no

?~,o

-~

F~o.. 11.12. Variation of the ~earing strem ~ the weJl duo to an external peemum gmdimt a~eov41-*to Eq. (ll~ro). L o ~ aldn frietion eeelMeisat et -. nouteat. experiment& However, t h k will very probably not hold up to s r b i t r a ~ y high vnlues of v/(vwu,)dP./dz. I n p a r t i o u ~ , the u n i v e r ~ velocity distribution will certainly not apply when the region of separation k spprosehed, where the wall shear 8trem tenck to zero. A theoreticol attempt to estimate the efl'eet of d/'~/dz on the veloeRy distribution, has been made by Szablewaki~ on the basis of the m;~,~g length concept; the variation of the wall shear stress in mean flow direction w u , however, neglected. Consequently these results e~e useful only if

(ll.sv) 0

r

S z a b ~ ' o invmtig~ion~ include also the u-~ting e,~e of r~=o abearing ~treu, for ~hich the velocity distribution n~y be e ~

b2

as ~ been shown by Stmtfordm. In the fully turbulent part Of the flow, the relative motion is again independent of the visooaity and, in.

of F.,q. (IL6),

obtain (zt.so)

This givel upon inteSnmon s t/leP. The oonstaut K0 is exlz~ted to be of the u=ne order of m a ~ t u d e u in Eq. (11.8), Townsendt H suggested s slightly higher value, v k . ~ 0 - 0-5. The constant of integration Co may be of the order 1 or L After sulz~ituting t h e distribution of ~ . ( l l ~ g a ) into the relation (11.37) and putting (70 m 0, we have d /1 dP,= ~ 8

u the condition that the terms on the ]eft-hand aide of F.~,. (4.6) Lm negl~ible. Some ~ e n t a l vedfio~tion of Eq. (I1~9~) is given by Stmtfordm. ~Ieasurements of turbulenoe q u a n t i t ~ made by Robertson and CalehutpO in a diffuser boundary layer near separation indicted t h ~ the turbulence intensity, its rite of production, and ira rote of tion are greatly in excess of .~.~;1~. quantifies for eonstsnt pressure layers. The effect of s pressure gradient on the flow in the eublayer ham been t r a m experimentally by Shib,emitm#=0. 11.6 Wall flow on r o u g h surfaces In the foregoing dis~mmaionit h u been assumed that the surfaoe of the wall is perfec~|y smooth. If the surface is not ideally smooth there is the possibility that ordy a part of the tangential forces exerted on the m~face is due to the viscous shearing ~treu pBU/by. The rems~de= oon. aist8 of a form drag, which is produced by the pressure distribution on the roughness e]emcnta. Sometimes this form drag may a u u m e very high v~ues, if flow eeg:xation occurs on the rougtme~es. The sum of the 6

74

J . C . Rcsr~

tangential components of pressure and visoos/ty forces gives & mean resultant friction force, whom amount per unit area is equivalent to a mean wall shearing stress o f value rw. The disturbances produced by the flow around the roughneases will affect the turbulent motion very near the waiL ~.nis means that the roughnemea act as artificial vortex genemtom and counteract the clamping effect o f the

viscous forces producod by the pro~mlty of the wall d

•

:

led

Fro. 11.13. lfee,a ~m' ~eld a~ the ~ •vith s p ~ rouzhaem ~

~,A';Ollh~lo d g o

e d p or'a, ~a,~ pl6te oovm~ Schlichttnt~.

- Curvm of oomt,m~ v~oci~ (1) U CD 0

-

4.TS m~eo

e-O m . ~ e

G U

s.Ts

CD

s.6o

•

,.is

5.U 5-0

0

4.O

-

4.so

- - ~ L o n g i t u d i n a l ,.-or~cee aftra.

Sehultr,-Granowue.

I f the boundary layer is considered on a fiat plate with the surface uniformly covered by roughness elements of size small compared with the thickness of the layer, it can easily be shown in a similar m a n n e r as at the beginning of this section that there e . ~ t s a laver of constant mean shear stress adjacent to the wall, where the motion is entirely determined by the maguitudes of shear stc~ss, density, kinematio viscosity, and the scale and geometry of the r e u g h n e M e ~ . I f consideration

0-

is w - ~ - e d to 8~,o.~etrieally similar roughnemes, the effee,, of the rough; n e a is deter,-~,~ed by a repr~ent~tive length seals, say k, {size of grain). Th~ implies that the ]~w-of simi]ari~3, Eq. (ll.3)is expanded to

l~urthermore, it is generally a ~ - p t e d that the direct influence of the surface roughness is felt only near the wall, and that farther away from the wall, but still within the region of constant stress, the flow pattern is independent of the roughness and visooe/ty. Wi~in this rel~on (II.6) a~] (II.7)are exae',]y valid witb the exeeption that the constant of integration is a ftmetion of l ~ , / r , that is /7-~,

I n ~ ' " +C

.

"

11-1.4oi

V

It is seen ths~ outside the sublsyer the effect of roui0mem ,~-~¢eets itself merely ~, a shift of the velocity profile in the semi-loLq~J~.h,~e plot. The known experiment~ invesl~g&tio~ are confined to the mean velocity distribution in the fully turbulent region. The author k not awareof any invest~atiuna on what actually happens quite close to the r o u g h n m elements except for a few experiments on o i ~ k and widely spaoed roughness elements. These measurements are h i l l y i n f ~ SchliehtingX0s measured the flow field near a fiat plate oovered w i t h spherem. The flow pa',tero as illustrated in Fig. 11.13 ~ v - ~ that the. 1 ~ velocities were measured l ~ i ~ d the r o ~ of epherm, thus imiic&ting a negative wake effect. SimgLr results have been found by Jseobs 4s behind a single roughness eleme~t~ According t o aa interpret&tion by Schultz-~runowU' a secondary flow in the fomz of lengitud~nal vort.iees is produced u shown by the dashed eurvm. This pieture _re~_rubles very much the one later proposed by Townoend in connection with the flow on smooth surfaces, Fig. 11.7. In Townmmd's picture the jets emerging from the sublayer origin~ts in +.he free ga1~ in which no spheres are present over the whole length of the plate. These results may giye an ides on the manner the ro~ghnemes affeet the ttrrbu]ent flow near the wall. From lq'~umdse'~~ measurements on flow through pipes, whose surfaces were uniformly covered w-"th san,] grains, it is seen that three different degrees of roughness can be distinguished: (I) H~/drauZica;Q/a,nooth {0 ;~ ~,~,/v ~ S): The roughness grains are entirely imbeclde~ in the soblayer. No e~ect of the roughnem is observed. (2) ~'ranm'en~ ro~3~ncae (S ;~ k,u,/v ;~ ~0): The constant of integrat ion 0 begins to decrease with ~,1~,

3. C. R o r r z

76

(3) C o m p ~

ror~h (£~d~ ;' 70): The size of tl~e roughness elemente is such that the Reynolds number of the flow in their vicinity is large, so thst the viscosity of the flnid has comparatively little ~-fluence on the whole motion. Then, for distances not too small, the mean velocity dist~bution is completely independent of viscosity ac~nfing to the relaffon /I

y

_~

/7 - u , | - l n ~ - - +@r|

(11.41)

where the constant Cr depends on the type of roughness. Com-

p . ~ n of Eq. (Zl.41) with (11.40) y/elds .C(--~)

=. C4v_ llr,g /v~,1, '"

(11.42.)

~ u s e of the great variety of geometrical forms of the sur[ace roughnessea, the number of parameters necessary to describe the snrfa~ properties k very" high. Consider for example & surgsco having merely one ~.'-g~ type of rou~ane~ element. Obviously, the flow on such s wsU will depend not only on the f o r e and height of the roughness elements, but also cn the density', that is the number of elements per unit ares. Furthermore, the manner in Which the roughness elements are distributed on the surfaco will also be of ~-euence. This variety complicstes the problem to s high degree. There is no alternative other than deter•rd.~,~ experimentally the function C(/~u,/v) for each type of roughness. The date for vazious types of r o u g h n ~ as collected by C1Ausert0 are replottsd ;.n Fig. 11.14. When ]c~,lv is im~e.iently high, the l~elAt/on Eq. (11.42) is generally valid. This is seen from the straight line asymptotes cf Fig. U.14. I t is therefore poaible to compare the scale kr of an erbitrary type of roughness with thst of s standard roughneee, giving the same value of (~ according to Eq. (U.40), when ~, and v sam unchanged. I t is customary to use Nlink-ad.~'s sand grain roughnels as standard roughneas. In the completely rough regime any type of roughnose can thus be reduced to sn eqxt[vs[ent sand grain roughnems of scale "~, say. The experimental determLnstion of the equivalent sand grain roughness for s number of different rougin~ess types has been e~rried out by Scblichtingloo in a special test channel 11.7 R e p r e s e n : a t l o n of the m e a n velocity distribution For s number of problems, for example the turbulent transfer of h~.at, it is deaimbIe to "know the mean velocity distribution not only in the ful]y turbulent regime sccording to Eq. (I1.40) but also in

~'urbulen~ B o u , ~ W LoWer* i,~ lw.o, n j ~ e

Jv/¢~

~/

the sublayer and the transient regime. Itia, of eou.W6, posst'ble to determine the velocity profiles very near the wnll from available meamaremente in eombinati0n with the known laminar law Eq. (II.l). The d~ta can be given numerically i n , table, u w u done by C.~lee ~ . A numbe~ I

!

_

g~e) '

f

\ -zo

,-o

nO

W~

o~

1~o. ll.)~ The e~eet of T~-iow type, ~ ~ ~ ~ veloc~:ycliJtt~sti~ ~ to F~q. (11.4~0),:~plc~k~od~ C~amlm~o • W. L. Moo~

4L Rand (leku~)

~'. ~. ~ --

~ S,u,,pks:y. (Fin---)

- .~-Lkursd~" sand ~

Colebmok/Whiten: 49% amootb,

47% ~

gr-~-

45% i m p

V uulform -,~L of attempts h~ve been re&de to compute the mean velocity distribution by means of hypothetical relations. This problem has ~ t t r s c t ~ the interest of many investi~tors in the hopo that such" caleu]~tions might contribute to the elari~cation of l~henomena in the tranment regime s~d to the e ~ m a t i o n of the effect of surface roughnesses. With the use of B o v ~ n e s q relation, Eq. (11.2) can be e x ~ s s e d by dU

"rw = p[v+¢r~--., aN

(11.43)

7S

J.C.

P,o~A

where the eddy -viscosity ~. ha# formally to satisfy the following conditions. Approaching the wall, ¢, tends to zero. For very large distm~cca. the eddy viscosity ~, becomes ,, ~ v and is required to asenme a form which is consistent with the relation, Eq. (I1.6). For the n ~ i o n in between, the flmetion +~ has to sumumo a distribution, which e n s m ~ that Eq. (11.43) yields upon tntegmtion the relation Eq. {II.40) with the accurate value of the constant of integration 0 ( / : ~ , / , ) . It;is quite instructive to col~eet the known relations (Table 1), espeeia/ly since in more than one case the authors had obviou-ly been unaware of proposals slzeady published. The simplest amumption, which ia b a s ~ on no other hypotb.~_', but the validity of Eq. (11.43) end the existence of a ~ml|A~,~y law Of the form Eq. (II.3), leads im.m,dittcly to the rektion '

(II.44)

I r -- ,r,(uv. v, y).

This equation forms the basis of the methods proposed by Squ/.-~u end P,eichardt sT. A number of investigu~m used Prsndtl'e mLldng length hypothesis together with various a~umptions ccncer-~-S the r n i ~ , ~ length L Deksler Is suggested a more general relation for the eddy v i s ~ t y of the form

dU O U •, -

,,(,,y. e,

OU

~

i,

in which the , , i ~ , g length formula Eq (5.3) and voa Kda-m~'e relation Eq. (5.5) are included. But sc~,-~ny the d/ffe:entlal quotients of third ~,,,t ~ o~er rarealmmmto-~l. In Table 1 the various methods ~ u subdivided into three groups, namely those based on Eq. (11.44), those based on the mixing length formula, and those b a n d on the relation Eq. (11.45), which eamlot be classified in the two other srou m. Most authors h~ve div/ded the whole rs~ge of we21 ~ c e a into two subranges, for which ,lln'erent relations are given. 0nly two authors give c~ntinuons relations for the entire wall layer. Einstein and Li'e attempt is not Kqted here, since this method is ber~d on entirely d/fferent ideas. Van Driest'am equation, which was suggested by an tmdogy with the oscillating plate in lamivsr flow, and RelchA~dt'ss7 relation give perhaps the most satisfactory representation of the turbulent velocity profile near the wall. These relations provide a continuous distribution of mean velocity and Reynolds shear s~rees. According to van Dx~e~.'s relation the Reynolds .hear stress vanishes like ~3~ at the wall, whereas with Reichardt's assumption the apparent shear at.was venial-ca as the cube of distance from the wall, which is in a ~ . e m e n t with the conclusion drawn from the ccntinuky equation. The

."

~-

2,

V A

O

h

I

%

II

v

.-

V

A

D

b

+ - -,...,.. .,,

+

.,

.

l

'

~

o.

~

"

V

~ 1

V

.,.

+"~+ '

%

V"'

0

OAllb

,+ 'P

.

,

i : "i J

-

'-i i :

t--

v. ~

-.,'.

.....

I=

, ,,

.~

~

=

~I

~.|Jj

_

,i:+

31k

I

| I

P,

iP

.=.

_.=

I

"+

I":

"

80

d'. C. ]3,m'zA

velocity profile aa calculated from van Driest's relation is plotted in 11.1.

The relations given by Rotta 0~ and van Driest=o are extended also to the velocity distribution on rough surfaces. The action of the roughnee8 can be interpreted as being equivalent to a reduction of the viscoua sublayer. ~ suggests the following shnple model for the representstion of the mean velocity profile. I t i~ assumed that the universal law of the wall Eq. (11.3) applies, when the plane of reference is shifted

__.

Y~. ILLS. Turbulent mean volooity distribution ne~- &rough surfaeo. (I) P~ne of reference foc the universal velocity distn'but,ima ae¢o~;-~

to F,q.(IL~.

beneath the surface by an ~mount of ~yr, as shown in Fig. 11.1~. '~-lae phme of reference moves with the velocity U(~t/r) opposed to main flow direction. Then the velocity dktributien is given by

For large values of y, for which the semilogsz~thnfio 1.aw according to Eq. (11.7) holds, Eq. (11.48) sasumes the form U f, u , [ l h (y+ Ayr}u. +_.

C(0) - J ( - ~ ) ]

.

(I 1.47)

Shoo, for sufficiently high values of y, In(y + Ayr) ~, In y, the comparison of Eq. (11.47) ~'ith Eq. (I].40) yields

C(-~)-

C(0)-f ( - ~ ) ,

(11.48)

which eatabl~hes a relation between U(~:rU,/v) and ~.~rU,[v. The varia. tion of ~9ru,/~ as ~ function 1~u,/v for ~ ~ e ' s sand gr~in rough-

mOO0

/

I00

J t.©

0"!

m.o

mo

t~

"

m~

m~

It, mr

~Om ll.Ze ~

A y e , / , of the plane of refurm~ for mad gss/a ~ml~and earmspmsd~ to Eq. 01.49). O mad Sndn m,,.Shz~ O(k,t~./,) ~ to Eq. (II.4~). (7(0) -- 6-~., ¢, -- I;4

.-..--

....

c(t,4=,l,) = e e o r d ~ to F-,q. ( n . ~ ) ,

G, - S-t

F,q. (It.~l), O, - S.t.

lmmmummm IC

~

1.0

--

t

,

*

I0

1~o. 11.17. S e m g o g a r i t h ~ c

•

•

p]ot of m ~

I0"

I

t

iOJ

velocity d i s t r i b e t i ~

mrtooth and rough walk secor~i-~ to ~,q. (11.46) with £y, uT/, for sand g r i n roug~,',Ms from Fig. l l . l e .

s2

c.

ne~ and for the rel~ion

C(-..~)-C(O)-~ I n [ l + ~'r="ex"p(~c[C(O)-Cr])],(11.49)

wh/ch applies to commercially ruugh inner surfaces of plpes, is shown in Fig. 11.16, where/(~z,/v) w ~ calculated from van Driest's relation. For very large roughnessm, when both the relation (11.5o}

and Eq. (11.42) apply, we obtain M m Eq. (11.48) A u, -

( 1.51)

which gives Ayo. ~ 0"035~ for sand grain r o u g h n e a ~fetn veloeiey, profiles calculated M m Eq. (11.48] ~ r sand ~ roushnesa axe plotted in

11.17.

12. The Free S t r e a m Bo,mda.T of the T u r b u l e n t L a y e r If turbulenc~ measurements are ramie at a fixed point in the outer part of the boundary layer, it is generally ol~served that, as the free stream is approached, the flow becomes intermittently turbulent and non-turbulent. This owand-o~ ohm-sorer of the turbulence, which k also observed in wake and jet flow, has been definitely est~blkhed eJ being a manife6t~fio., of the irregular but distinct outline of the bonndsr 7 layer as it moves down~mmm. The sharp boundary between turbulent and non-turbulent flow ie clearly seen from Fig. 12.1, which represents a spark shadowgraph of a turbulent boundary 18yet on a hollow circular cTlinder in supersonic flow. However, this . h , . p boundary must not be confused with the lim~t M usu~ly defined by a negiigible mean velocity defmt. The intermitt~ncy is e a r l y observed by oscilloscope records of the u-fluctuation, and the record= can be used to gi~e a quantit~tiws estimate of the intermittenoy factor y, that is the fraction of the time the flow is turbulent at a particular position. With the appropriate equipment the intermittency ~ c t o r can also dLre~ly be measured (Townsend 144. Corrsin and K.Mler17) from the hot wire signal of u or &t[~, ~nantity ~/~t is more suitable than ~ to decide whether the flow is momentarily t-.trbu]ent or not. Another method introduced by Townsend 14a involves the measurement of values of the squares and fourth powers of fluctuat"-ng quantities (e.g. velocity derivatives) and deter° mines the intermRtency factor f ~ m the "flattening factor" of the probability density of the intermittent sla~.~. The results of the di~'ereut method~ are in good agreement with eae.h other.

12.1 Statisdcal description of the free s t r e a m b o u n d a r y Under o r d i n a ~ cire-,--tanoee the Lee stream bouniiary e~n alwsyl be eoutidex~i u , eontinuoea m n - f ~ ; there wifi be no "ieYmdJ" of turbulenee out in the free t~ream d i ~ n u e e t e d f~om the ~ body of turbulent fluicL The in~umtaneoua ~ e e of the dmrp free boundary Iv(z, :, t) from the wall is t random varisble stationary in z and time and non.et~tionary in : , Fig. 12.~. I t k of eourm poem'ble for to be a mulfiple~-v~ued function, but the ~eurrenee of multiple 6 o u n d o r y betweez tur~utom

~_.o_.,_,,.,,o , _ _ ~ ~ _ _

•

~

t

WOII

Fro. 12.2. Sketch of the ttwbeYmt betmdK7 Isy~ sR4r K i ~ , values appears to be suefieiently ra~e for the .Mmuaptie, of s ainglevalued Y to be applicable with good a ~ u r a ~ . With throe aMmmptimm the i n t e r m i ~ n e y factor ~ , =) ia simply the I.~bLbmty of finding the free m ~ a m bcmndary at t ~ e e larger t h ~ 7 ,

~.V, z) - prob [.V • Y'(:; z, t.) < a~].

(12.1)

Sinee 7fY, :) is diferentiable, -8~,/c~y is the probabili W density o f 1v. The intermietency faeWr y is zero for large dietanem.a.nd tendJ to unity ae the region eloler to the wall is approached. One i m ~ t ~ t ~ t . i c a ] measure of the free t~re~m boundeyy i, its average |oeafie~,

(xu) 0

O

The standard dcwk,tion

,,C=) -

~¢~_ ~,~-~ _~.

[(Y-' ~ ) , ] x ~ .

_

l~,s dyJ

0

[2f(.v- Y)~ ay] ~n 0

(x2.s)

84

J.C.

Row~

is a suitable m ~ of the w i d t h of the intermittent zone, t h a t is, of the amplitude of *.he waves in the flee stream boundary. I n t e r m i t t e n c y factors obt~Qed from experiments are given in Fig. 12.3. The experimental results can be a p p r o ~ m ~ t e d v e r y closely by means o f the Gaua-;~,, prob6bility distribution, , _

~'f

I

eX'p(-- (Y2 -Y)S).

(12.,)

Devistiona from this sy,,,,=et':ical distribution must of course o c c u r f o r small and very large y, since the boundary conditions on the two aidel ,-2

I.O

-.T

O.Ik

-~"

o.a

o.4

o-a

o

O'Z

0"4

0-1

O'G

I-O

I.|

1.4

Y

7

Fro. 12.3. Dktdbutien of intermRtenoy t u t o r ~ rne~u~m~mm m • turbulent b o ~ " layer oa • fl~ plat4 ad~u. ~ .

V.l,/,, • from flAt*,,-;n~ factor

-

? . 104.

x vorticity-~1,,,

0 u-m,,,,

'f

Solid line: y -- ~/(2~.)-----~ e x p ( - ~) dy with C - 5[(Y]~)-0"78], v'2¢ " 0-2& • r~ ~astly different. According to Klebcmoff's meMur~ments the mean position of the free stream boundary is found to be at the pmition =- 0.788, and the standard deviation is ~ - 0.148. For a boundary layer on a rough plate the values are F - 0-828 and ~ - 0-158 according to measurements of Corrsin and lr.~tler 17. Additional statistical

°

]no~-~ties hsve been measured by Consi" and F~istler11, of which we will only mention the duration of turbulent or non-turbulent flow between two su _cee~___'vealternations at any par~icnlar position. The aver-. age time interval T between the entry of the hot wire probe into one turbulent reLmne and the entry into the next one, ~ at i~atio~ 0.72 t; y]8 t; 0-98 is in non-dimensional form approY1mlte~ T/7.

-

:~.5.

(12.@

12.2 T h e p r o c e s s of v o r d d t y p r o p a g a t i o n The distance fro-, the wail of the mean position of the free stream boundary increases in the downstream direction, so that entruinmont of non-turbulent fluid takes p|~oe. The fluid permeating through t h e free stresm boundary must he put into s state of turbulent motion. Since s distinctive w . - k of turbulent fluid k its high level of v o r t i d t y fluctuation, the essential setion of the free ~ boundary ;- to traaamit vorticity to tiuid of zero vorticity, which oan be achieved only through the tangentisl forces due to visoosity. The instantaneous border zone lying between turbulent fluid and non-turbulent fluid ;therefore a region in which visooul for~es play s predom;-,~ According to C o r r ~ and l~istler1~ this narrow zone in which the v o r ~ city fluctuation level drops from values ~ o of ful]y t , , r b u l m t flow to practically zero, is termed the "viscous n p e d s y e r " . :In eomm-_ quenee of this concept the turbulent ~uid of the bound*ry }syer k entirely ~'rapped in s visoous layer, which ooc~nl as the n b l s y e r s t the rigid wall and as the superlayer at the free stream boundary. We shall study the phenomer~ in the superlsyer by meJml of t h e vorticity equstion for three-dimeruqional v~cous flow, which k obtained by taking the curl of the Navier-Stokes ec~atiom. I t k

where tempvrar~y again the various d i r e ~ o u s are denoted by md~Soes i, j etc. Terms ~ t h repeated suttees are to be r e ~ d e d u summed over all three possible ~alues o f t h e m.uTiz,In Eq. (I~-6) ~ - ~ 8u~8=I is the vorticity component in the direction of the =t-a~ia, where , ~ is the cyclic coefficient, which is zero unless i. j, k are all different, is equsl to 1 f f t h e y are in the order 123123, and k. equal to - 1 if t h e y are in the order 132132. We consider s plane turbulence front in ~ fluid of zero mean veloe/ty and in which the vorticity has ~lso s z~-ro mean value, ~ - 0. Any •gradient of the mean flow which may be present in s real turbulent

boundary layer may be neglected here. This ~ n t is in the = . p l a n e and moves in the d/rection of the positive ~-sxis with the v e l o c ~ F*, ~ . 12.4(&). In order to discuss the equ/h~orium ofvortic/ty propagation, it is morn eonverdent to choose 8, co...ordina~ system wh/ch moves with the M n t , that is the turbu/ence M n t is at ~ and non-turbulent ~u/d

y

KI)li/vrbulnnt (luid (o! roetl

NOntvrbvlent fluid /~ n ~

y

t_.

t_.

; "///////////..'/////////////~/

~)~//.F//.,'////~'///f////.,'."o'////f

t

Yurtulenf fluid

Ysrt~dtlnt tlvld

I Yurbvlen(:o front (b)

(o)

[at

roof}

l~a. 1~.~. Pleura turbulen~ (~) Fluid witho~ mma m o * ~ (6) ~ =ymtom~ to-the rnmt. is entrained at the rste F~,i~ Fig. 12.4(b}. The vortioity equation (12.6~ for cq multiplied by ~ , and summed over all tl~-ee oomponents gives

after Cdd

the sven

} The orders of magnitude o£ the terms in ~

F

• .

' ~"

(-)'~

equstion Lre respectively: •

(')'~.

The fins~ term o n the le/~hnnd side is the convect/on, the ~ ¢ o n d one the turbulent dJ~usion of vorticity fluctuations. The first term on the r~ght-h~n& aide repre~nts the ra~e of production of vorticity. The seeend and third terms give the v~ecous difl=uaion and the viscous ~ ; p s tion of vo~tie/ty. I t w~s Taylor L~ewho firs~ pointed out t h s t the term ,o~,u ~ I ~ i is ~ m e u u r e of the rote of vorticity intensity produced by diffusive stretch/rig of vortex ~laments. This term is positive. In homogeneous turbulence the first term on the right-hand side bLlances (appro~,-ately) the third one (because a~lz/~t ~ 0). The main r e u o n $or the rel&tively sharp formztion of the turbu/ence front is that the rste of vortJcity production increases with the msgnitude of vorticity already present. The term = ~ . t / a = 1 is thus responsible for the tendency of the vorticity grsdisnt to steepen, Fig. 12.5. In the region of h/gh ~ s surplus amount of vorticity is produced, which has to be trsus~erred

by diffusio-to positiuns where there is a deficitof vor~icit~yproduction. The diffusion term can compete with the other terms only if the tl~iekn e ~ ~ of the euperlayer is of the o ~ e r of the e.haraet~,~-~ length scale A - (,]~)-x~t of the micro structure of the turbnlenoe, which governs the conditions in the s~perlAyer. This is ,een from an intimate of t h e orders of magnitude of t h e terms of ~ . (IS.V). I t k ~ -- ,]~, (e~/~r)z

~ ~i~

-

(e~/~F~,

e ~ ~

~ (,~)v,.

The

pa~tleu~

terms have their orders of magnitude written underuea~ the equation. From this esthnate we find tha~ the velocity of p r ~ t i o n is of the ~rder F,,p ,," (o,)u*, (IS.e) and the thie~rne~ of the superl~yer k ,a~4

,s~

=

~.~.

. (ts.~)

Under such eaummptions all the terms of Eq. (12.7) are of the same order of magnitude.

Turbulent

;

.,.

:

I

s.p,~o~,,

lli~nturl)vle*!

I f~'

1~o. 12.5. Sketch of the dktn'bufion of w ) r ) ~

interior7 ~, the mBioo

of the turbul~-~ee front, pTe~ented in a eo-ordin~te , ~ e n f~ont, after ~ 4rod Y.~mtlerl'.

fixed to the

As another effect which contributes substsnthdly to the whole prooess of vorticity prupssstion is the fact that the superlAyer cannot rema~n plane and must be eontinuous|y contorted by the variotm scales of the turbulent motion. The enlargement of the luperhtyer mxrfa~ so produced results in a great increa~ of the mesu propagation ~ve]oeity of the free stream boundary. The transmission of vorticity to nonturbulent fluid through the superlayer is only the first stage of a more complex process. The est.imatad velocity F*~ is to be considered u a local propagation velocity, di~cted always at right.ang!es to the lomd

88

J.C. RoT~

tangent on the free stream boundar3r. The contor~on is. generuted by the velocity fluctuation and must o~cy the continuitT condition; however, the role of the Ioeal propagation ve]locity is to preven~ an indefinite growing of the amplitude of contortion even in the c a ~ when the generating velocity a~ca Cor an arbitrary length of z;-w. The leluflng feat'u~ may be demonstrated by s ~mple model

, -/

t_

~a.

v~

12.8. Contortion of tho turbulenoo fzont b y a 8implo aino wavo

(1) Plane f'z~nt at time f - 0 (2) front after s short tlrno of' deformation; ~ u a of em-vatml at the valley k ~ thaa st the (3) state of deformation at a later time; a 8harp peak is formed s t the vs/ley. (4) Asymptotic shape of the contor'~d Front (| - a~); front morea with

Vpo - ,/(~:/2)+ V*,p - 3Vo, p.

'We will assume that a turbulence front movin E relative to the fluid , with the velocity Y ' . ~ ~. subject to a deformation by a ~ g l e ~ne w s ~ velocity distribution i

as sketchedin Fig: 12.~, where ~ i s the mean square of v. At the crest of the wave the front moves with the velocity ( ~ / 2 ) ~ + F%p; at the valley it moves with velocity U*~-(~£/2) ~r~ which k in the opposite direction at the beginning if V'(~Ig) ~' ~ . ~ - S/nes F'.~ is alwsys directed at right-sngles to the tangent on the e o n t o r t ~ front, the radius of curvature decreases more rapidly at the valley ~.h~, at the canto and tends f i n ~ y to zero, that is at the valley a sharp peak is formed. R is easy to show that the front ~ be deformed until finally every moves with the same velocity

in the direction of V-L~a. The =h~pe of the front w!]] obey then t h e

+

-

V,'.

1 ~ . 12.6 shows the stste of deformation at different times. Although the actual beh~viour of a free stream boundary will oerta/nly be di~erent from this .~,-ple model, two results should be nofiesd: The mean pro~gation velocity gl~ as given by l~.q. {12.10) k ~ t l y affected by the velocity fluctuLtioni, and the surf~oe of the ~sup,-l~yer is en]aa~ed to a sufficient extent to secure the vortie3ty supply of the entrained fluid. These statements ~ v ~ y probably apply to a real free stream boundary. If the Reynolds number of the turbuIenoe is sumeiently high, the 1o~1 prol~q~ation velocity V',p in Eq. (1~.10) is negligib1.e. This is seen. from Eqs. (7.3) and (12.8):

where 10 is the l e n ~ h scale of the energy c o n ~ - ~ g eddies. In this ease the rate of entrainment of non-turbulent fluid w~l not dei>end on the magnitude of vis~oaity, as was first pointed out by Townsend 14s. This phenomenon is a counterpart to the process of e ~ e r ~ dissipation ~t high Reynolds numbers, in which the rate of energy 7

~'. C. P.orr.a

90

diedpstion is independent of viseoaity, although kinetic energy can be converted into heat only by molec,d*- friction. VerificAtion of this result requires the determination of the mean propagation velocity from the growth of the distance Y..~|nce F ~ is the propage£ion velocity relative to the fluJcl, one can write

dY d=

--

,

(I~.II)

7(F)

where 17( I~ sad V( I~) are the componen~ o~ the mean velocity at the podt/on y - ~. This equation is only sn appro~,-,ation, sines the wrinkle a,mplK'ude of the free boundary is Msumed hem to be ,m,]~ sad st ~ - F the mean velocity ~ of the turb~ent fluid is ~ t u a l l y somewh~ ie~ than that of the non-turbulent fluid. At the edge of the flgt plate boundary layer (.q m ~) the mean velocity component P' k given by the d - ~ ! t i o n of the d/spkcement thictme~s 8*, Eq. (4,13)

Sinmexp~iments show

that / 7 ( ~ ) ~ U ( 8 ) - U . . , one can inter F(~) ~. F(8), so tlmt the menn propagation velocity e ~ be oaleulat~ sppro~dm,~tely from FT* -

/7.(d ~

The meuurements by Klebsao~ give V~/'O.,--0-00~ ~ d V'e'-70 , ( Y) - 0.02, from which VT* = 0.415V' follow.. The oorresponding values for the rough ~ layer measured b y Corrsin sad ~.tlerX7 are FT* -. 0.0114, ~ ( ~ ) / U . , ( ~ ) - 0.09-, sad FT* .. 0"STV'~. The value F f - 0.71~/~ for our simplified model, when compared with *.,heseresults, is ~ ~'easonable one, although no qusatito~,ive aiFesmeat is expected. The quantitative ~;~'erencesmay exkt for esveral reasons. It is perhaps important that say particular eddy has s limited duration of life, so that the asymptotic contortion, as shown for | - co in Fig. 12.6, will not actually be e~tabLished, and the mean propagation velocity is consequently reduced. V'mual observations and also the simple model suggest that the dominant type of large eddy motion is a motion whic~h may be described a4 jets of turbulent fluid emerging fix~m the interior of the layer into the relatively undisturbed flow outaide. According to Townsend14~ the mean velocity' gradient causes the jet to follow a curved path and the flow pattern may resemble that Iketched in Fig. 12.7. This picture is supported by measurements of the spa~e

~'urb~h~ J~our,aartJ ~3/era in I ~ o m p r u m ~ ~

91

eorrelstion fun'etion 'P.~ over the =y-plane by Grantm, where the fixed ~vire was at a distance y -- 0-7~ and the other ~ w u moved along lines between the fixed wire and t]=e wall. The correlation function so obtained ahows a considerable asymmetry agreeing with the suggested jet deflection. By the,way t h k ssymmetry, which can also be seen from tho shadowgraph of Fig. 12.i, is in the same sense aa observed with t h e tmnsveres space.time oorrelations for the longitudinal velocity £uctustion of Fig. e.7, where the fixed wire w u ~t y - 024&

-

~%

Turbullm! flow

l~a. 12.7. Structure of • ,ulxing je~ in the ou*.er part of a boundary i a y ~

after Towmend~". The full line reprment~ the boundm7 of the turbu. lent fluid. The daahed iinm are stemmtinm relative to the mesa at the h u e of the jet.

12.3 Inference of n:rbulence properties from the tntermittent

,tta I t is sometimes desimb|e to eompu-te the ~ ;th~iea] p r o e m of theturbu]enes inside the bulges of the free strea.~ boundary ~ s ]mowledge of the .corresponding statistical properties of the intermittent plus the intermitteney factor ~,. The conditions h~ve been clarified by Corrsin and I~istler17. Consider ~ny l~rticu]Lr property of the flow a(t), Fig. 12.8, which m~y be for instanee t velocit7 eompone~t, s temperature, s chemical e o n t ~ ; n ~ t i o n etc. The mean value of quantity is A -- ~ and the fluctuation of a(t) is

a(O -

a

(12.It)

m

with the definition a(t) .- O. In ~ ,~.-nar wsy the qusntity a{O esn ako be averaged ever the fraction of time the flow is turbulent at the point of observation and over the time the flow is nonturbulent. To this end an intermittent signa| ~(t) must be introduced ~rhieh indieates whether the flow is momentarily turbulent [~(t) .- 1] or Don-turbulent ~8(t)- 0]. Obvious]y ~ - j 3 : . . . - y the intermitteney fsctor. Hence the meau vaJue A~ &rd the corresponding fluctuation aT wit.l~i-

9~

& C. lt~Tl-..l

the turbulent fluid are de~.ned by

a.:(t) - ~ ( l ) - , t ~

for

~(I) - 1,

(1-°.le)

ancLthe mean value A~i and fluctuation ~N(I) withi, the non-turbulenll enid ire given by

(1-7)~a~ - a ( l ) - A x

_.( • il

""I

[1-~(t)]~.(l)> for

(12.17)

J(l) - o.

(l~.ls)

__l...r I G

I

0•

J! il!ll, J ltl! IIII ' i i i' II

t! II1

Fie. I~L Random varied/on e l i phys/csl q 1 ~ i t y a(1) in/ntermittently turbulent flow. ~(l} is an intermittent ~ ind/eat~nK whether tho flow is momea~r/~;" mrbule~ (~ - I) or non-turbulent (~I - 0}. From Eqs. (I~.15) and (12.17) reenlt4 a relation between A, AT, A ~

and y, nsmel~ ~.:.

(1 - ~ ) ~

- ~.

(I~.I0)

In Eqs. (12.18) and (12.18) a( 0 can be replaced by a(t) o f E q . (12.14)

to give aT(t) . . a(t) - (1-1,)(AT-Ax).

(12.20)

Further relations between statistical quantitiee can be derived from these equations, in which the fluctuations in the turbulent fluid and the non-turbulent fluid are expressed by t h . ordinary fluctuation a(t), the difference in mean values A T - A ~ "and the intermittency f~ctor. An important relation between the mean squares reads 7<.,~>T+(1-y).~

= ~-i-~(I-~)(A~-A~)~

(12.22)

where the brackets < >T and < >~ express averages taken within the turbulent fluid and the non-turbulent fluid resp~tively. This equation

Tu,~u~.~ .Bou.~,"9 Layer,. ~,. l , ~ m p , ~ n ~ k Flow

is obtLined by the folloving Inoo u:e. The

98.

of Eq.

is

multiplied by ~(t) and sTeraged, the square of Eq, (lYe21) is multiplied by (l-~(J)) and averaged; the sum gives Eq. (12.22). Usually it is r.asumed that the fluetust/ons in the non-turbulent fluid axe negl~p'ble ( (a.~s >~ . 0) and the mean values are equal (AT -- Ax). Under sue.h circumstanoes from Eq. (IE.29-) the resu~ -

--

(12

)

T is obtah~l. This relation is very often used to oompute mesa squsre~ within the turbulent fluid from the ordinary mean squares. However, it must not be overlooked that the applicLb~ity of this relation is eont,ingent upon the restrictive conditions Ax -. AT and < a ~ . ~ .. 0, which redues Eq~ (~.~0) ~nd ( ~ . ~ l ) to These conditions are certa~.]y met by vorticity fluc~ations for in,canoe. The results bocome however lees aecu~te if Eq. (12.~) is applied to velocity fluctuations, since the mean velocity in the turbulent p s ~ is observed to be somewhat ,-,,~ler them wi~.~,~-the non-tufoulont fluid. If desired, more'reliable information on the s ~ u ~ u r e of flow w i ~ h , the turbulent fluid ea~ be obtained from o~illogmph tmoes by a l a b o ~ u s procedure. The ~bove-mentioned method of determl-;-~ the i n t e r n ~ tency ~ from the fl~ttening ~ is ~ on the relation ]~l.

12.4 T u r b u l e n t diffusion In the region of the h'ee s t r u m boundary As s eonsequenes of the d i i ' ~ e nature of turbulent motion, amy property of the turbulent fluid is s l n ~ d into the newly erested turbulent zones behind the superlsyer. This proeeso is su integr~ part of the whole process of turbulenee propagation. For many purposes it is important to know the mean rate of ctiffu~on in the direction of p r o p s p tion of the free ~ m boundary. Although this phenomenon is in general v e ~ complicated, so that quantitative eomputsfions are not posaibl~, ~n estimate c~n e~a~v be made for the outer l ~ r t of the intermittent zone. The turbulent transport l~ooess is here essentially s oonvection by the emerging jet~ The simpli~ed model introd _ue~__in the discussion above uu~ests that the aTerage convective velocity in the outer tips of the jets equnls to a first approximation the propagation velocity FT*. if the local propa~tion velocity U*sp is small. In boundar T }ayer flow a second part is produced by the fact that the amplitude

J.

94

c. Rolrrr

.

of the wavea of the turbulence tint incmwa M the bulges or jeta am conwcted by the mcau ffow. This part can be calc&tmzi &om the growth rate of the etaadard deviation du/&, if it is swumed that the probability distribution of the position of the turbulence knt M @en by Eq. (12.2) mnah adaaged and that Taylor’s hypothka Eq. (6.11) cau te applied. This additional convection velocity 7s at the poition y can then be titteu (12.25)

Introduction

of 8y/@ from Eq. (124) givea (12.20)

The total convection ia thus

v&city

component in the dim&on .

v, -

VP+

Pa

of the y-axh (12.27)

and the di5ixion of a quantie IX(~)t&m aa mean value over the torhq-. Iemb&id is given by &.28) <~+4%&r~ If the traaapoti

quantity is zero in the non-turbulent MA, ordinary mean value of the dBuaive flux of the quantity 6 is a

- Y
-

GYAT.

the

(12.29)

Gf special interest with respect to the entrgg balance in the boundary When the rsletion Eq. (12.23)

leyer is~the turbulent energy dSuaiou

for the mean squam of velocity fluctuations is adopted, turbdent energy di8uaion can be edcdatai hm ua+d++

-=

2

VS. 2

the rate of

(12.30)

This r&&ion is of courw applicable only at dietancea y beyond the mean position P of the &e stmam boundary. Calculated results kom Eq. (12.30) will be dikcut~ed together with the energy balance in the outer part of the boundary layer in Chapter III. 12.5 Flow outstde

the turbulent

re@nm

Outside the free &ream boundary of the turbulent flow, there are alao small random velocity fluctnstiona prweht. but sheae fluctuations

k very nearly imztational ‘Ikxetioal mtudiss made bi Phillip~‘~ and Stewar@ me baeal on the notion that the izt&ational velocitp fluetatiopr M uniquely dekmnined by the distribution of the ~elo&y oomponent normal to the boanding surfem between turbulent and nonturbulentfluid. The detemination of the irmtational flow on one tide of an Unite plane, whemthe nomal velocity at this plane ia l stationq random function of po&ion, ladn to the result that the mean qnare velocity normal to the plane im equal ti the mm of the m68n qu8ru of the velocitiw in the other tara arthogonal dixmctiom. &aw tbi~ plme coinciduJ l&h the mean po8ition of the fmle otmbln bounduyand b thum ndy parallel to the wall, it follow9 thd L

P-z+a.

m(12d1)

The kinetio energy of the fbtuationn is found to be invemly pmportionrl t48the fourth power of the dktanoe &om the plan% whi& -ma

to be ‘m agreement with azperimcstal o-tiow . and K&W’ have shown that the Pwold~ &besr *rhdlr terma in the momentum equation of maa anotion cam be aspumed fntamr of the Rapolda nond stmsmae, when the oondition of jr+tiondity of the Yelocig fbhlationq Coda

h-h *h

.

.-, ._

. *:

-(*.SZj

is applied with thea dAioruJ ad tLfter.introducing -Eq. (12.31),.gke ._ . momultum eqmh (4.9) of mean motion mboa ta (12.22)