https://ntrs.nasa.gov/search.jsp?R=19710022769 2020-03-11T20:16:13+00:00Z A TURBULENCE DESCRlPTfON OF COUETTE FLOW* Paul M. Chung Department of Energy Engineering Universi ty of I l l i n o i s a t Chicago *This work i s supported by the National Aeronautics and Space Administration, NGR 14-012-012, The simpiified statistical theory developed earlier by the present authui is made self-containing fo r the wall shear flows, During the course of the formu3.ationP the comparison of t h e present theory, developed from the generalize3 Brownian mcition, with the stochastic theory based on the. Havier-Stokes rc@a%icsn, was made. 'r1ze mutual consistency of the ~ P I W theories was shown, Analysis o f the resulting equations shows that the flow field riatuxally divides itself h t o ttie three layers of the outer, transition, and the inner layers. The contributisn o f the small eddies through viscosity t o a31 observable properti-es is negligible in the outer layer, whereas its contribution t o the mean shear is of first or63er in the transition layer. In the inner layer, a3.3 viScosity effects m e of first order. The comparison crf the Couette flow solutions obtained with the available experkrental results shows that the presept theory clescrj,.bes the rather detailed turbulence structure of the flow field satisfactorily in addition to cofre8tPy predicting t h e mean velocity prafilc and the surface shear. Most of the real turbulent shear flows defy analysis by the classical staskical turbulence theorias becailse of the extreme complexity of t h e theories, Studies o f these flows, therefore, have exclusively depended on the phencllmeno- logical theories such as the Praadtl's mixing-length theory. (See Ear instance, Xeferences 1 - 6.) Many of the important aspects of the flow, howeveus such as dhe turbulence structure of the flow field and the chemical reaction, cannot be described correctly by these phenomeno1ogica.l theories, A simplified statistical theory was developed by the present author (7-9) which contained those stochastic propeqties of the high Reynolds rimer turbu- lence field which are critical in determining the salient turbulence and trans- port characteristics of the shear flow fields. It has been demonstrated ( 8 p 9 1 .. ehat this theoxy can indeed be employed to des-eribe tractably the various pra- perties of a shear flow field, such as the turbuledce and chemical struct~aes of a turbulent diffusion flamePF. which has not been possible before. The previous application^'^‘^' of the simplified statistical theory were limited to the shear fiow fields with no solid boundnries. The .Uzeary was first applied'71 to the Couette flow at one Repolds number as the initial test of the 'theory, However, the experhentally determined dissipation function of a pipe flaw available at that Reynolds number had to be employed,' Alsa, several assumptions had to be made in order to apply the theory all the way to the wall. fn this paper, the structure o f a flow field as affected by a solid boundary will be described by the simplified statistical theory with the same rigor as sat previously applied to the Slow fields with no solid boundarise. For this purpose, the plane Couette flow is chosen because of its simple flow c o n f i ~ - ra t ion. The fluid i s considered to be incompressible and chemically inert, It will be shown that, when the Reynolds number is sufficiently large, the flow field naturally divides itself into three layers: the outex layer, inte-:- mediate (transition) layer, and the inner layer. Solution of the governin? equations leading to such division will be carried out by tihe rnetliad of matched asymptotic expansions, The subdivision of t h e flow field into such three layers, and even the description of the field by the matched asymptotic expansions, are not new in themselves. [See for instance, Reference 10,) Xn the previous works, hswcver, these subdivisions were tho results of the analysis of t h ~ phenomenologically constructed equations with the emperical functions and parameters a priori supplied from the known experimental results. Therefore, the analysis essen- tia2l.y returned what one put in a priori empemrically. Furthemorel the division and the asymptotic matching were all based on the mean velocity profile of i l y , In contrast, the present theory is self-contained. Hence, it is the na tu ra l scaling of the governing equations, that control the behavior of the Reynolds stress, turbulence energy, etc., as well as the mean velocity, which results in the division into such layers. From the present dnalysis, the mean velocity, turbulence energy, and the Reynolds stress profiles as well as the surface shear stress will be de temined as $unctions of the flow Reynolds number. These results will be compared with (4) khe' available experimental. hata of Reichardt and Robertson, .et.al. &st61 A detailed description of the turbulent Couette flow, and that of the general near-surface turbulence structure of shear fl.ows, will be given in t a m s of the present theory, - 3 - %I. GOVERYING FOYGR--PTiANCK EQUATION The detailed development of the present simplified statistical theory is given in the previou,~ papers, "-" and it will not be repeated here. It is only pointed out here that the theory resulted in a set of modified Pokker- Planck equations governing the distribution functions of the fluid elemenks and the c'nemi.cal species. These equations have been made self-containing (8,531 for the flow fields with no solid boun,dari@s. In the present section, the Fokker-Planck equation governing the distri- bution function of the fluid elements will be made self-conta.ining for the shear flows with solid boundaries. In the next section, a set of moment equations will be derived from the Fokker-Planck equation in order.to solve this equation by the moment me"c& I due to Liu and Lees (I1' as it was done in the earlier problems. (7-9' The solu- tion of the moment equations for Couette flow and'the discussion of the rrzsults will then fallow. General Form The Fokker-Flanck equation which governs the distribution function of the + 3 fluid elements, f(t,x,u) was formulated (7.8) as, (rfr 3 3 -)1 The distribution function is defined such that f(t,x,u)du denotes the probability -f of finding a fluid element in the phase'cell du about the instantaneous absolute 3 -+ velocity u, at time t and position x . The cartesian tensor notation is employed in Eq. (I), The symbol u represents the average velocity whereas U is the 0 j j instantaneous velocity relative to u Hence, U. = u - u The smol > oj * 3 j 0 3 ' denotes the ensemble average, T~us! for instance, The quantity B represents the characteristic relaxation rate of the non- equilibrium degree of freedom representing the lager energy-containing eddies." K , represents the effects an the fluid elements of the molecular viscosity and 3 the applied mean pressure gradient.* These two quantities, B and K must be j ' properly defined in order to render Eq. (1) self-containing. The characteristic relaxation rate, 6 , is given f7 t 8 ) as r for the two-dimensional flows wherein the predominant mean velocity gradiezt is 8uo/ay The quantities u and y are the mean velocity in the streamwise direction 0 and the ordinate normal to the streamwise direction, respectively. L and ?? are the characteristic lengths respectively of the flow field and the system producing the eddies. In a pure shear-produced turbulence, h = L when L is taken as the shear layer thickness. The subscript denotes the reference condition, *See References 7 and 8. The expression for K will now be discussed for the flows wherein the nrean j pressure gradient is negligible. Wall Effect The present turbulence model (7-9) is built q o n the known phenomenon k12,B3) of the statistical separation which exists between the nonequilibrium, energy- containing, larger eddies and the smaller, equilibrium eddies when the turbulence Reynolds nwnber is sufficiently la.rge. All obsevvable properties of a turbulence field and also the dissipation rate are determined by the'behavior of these larger eddies. The lower wave number region of the spectrum signifying the energy-containing eddies i s represented by one nanequilibrilm degree of freedom whose characteristic relaxation rate is given by ~ q . (3). When there is no solid boundary, this nonequilibrium degree of freedom is sufficient to determine both the observable propeqties and the dissipation rate. Hence, it was formulated(8' for the free turbulence with # where 6 is the characteristic dissipation rate. As a solid boundary is approached in a flow, the characteristic length of the nonequilibrium degree of freedom is expected to decrease, and this decrease is manifested by the usual increasing of the mean velocity gradient near the w a l l - This e f f e c t of t h e decreasing c h a r a c t e r i s t i c length oil the charabteriseic: r.--- l axa t ion xate of ti.,e ncnequ5.libriurn degree of freedom i s already included i n Eqs . ( 3) thror~gh the mean ve loc i ty gra.die11.t. There i s , however, another influence of the s o l i d boundary. A s the ao l ld boundary i s approached, the energy-containing reylon of the spectrim )>roadens; thus , the i n e r t i a l subrange nloves toward the higher w3t-e nu~t~her regior-;, riphis i s c l e a r l y seen, f o r instance, i n t h e f i g u r e s 4 .4 an2 4.5 of IIinze (13' m75Fc!1 show t h e energy 'spectra of the pipe and boundary layer flows obtained by 1,au- f er (I4) and KleSanoff, et . a l . (15) As it was mentioned e a . r l i e r , the d i s s i p a t i o n r a t e i s contro l led by .t2ae energy-containing eddies w h e n thn turbu.lence Reyno3.d~ nrunber i s s u f f i c i e n t l y la rge . More s t r i c t l y speaking, however, it i s contro l led by t he region adjacent t o the low wave number end of t h e i n e r t i a l subrange (I2 'I3' which cor~csp;lldr t o the high wave-nu~vher end of the energy colxtaining region. This upx2e.c en3 o f t h e energy-contaxning region moves 5rito the higher wave ~mmber murn f a s t e r tha? does the rroncqui.1ibriuii-i degree of freedom represent isg the e n e r g y - c o ~ ~ t a ~ n - ing as a whole, a s t h e wall i s appcoached.. That i s , the eddies conC::aL- l i n g t h e d i s s i p a t i o n r a t e a r e more d i r e c t l y a f fec ted by tke s o l i d boundzry 25 compared t o ehc energy-containing eddies a s a whole, E; t he re fo re , m t x r r e f l e e t j " t h i s d i r e c t eEfect of t h e s o l i d boundary on the d i s s ipa t ion with t h e app?r~?r-!-~rng of the boundary. Also, t h e i13creasing r o l e of the smal l , equil ibrium (near-equilinri7,Ini eddies on the observable p roper t i e s m d s t be manifested i n K . Since ehese i eddies %re small i n s i z e and in near-equj-lihrium, t h e i r contribu.tiorn:s t o ehe obser~rable p roper t i e s sl-ioulcl bs i n 30cal in character . This means t h z t ~ ~ C L L cont:cibutions shouid hz Su~1?ctior~s o f t h e local proper t i e s , and not fun:.t~ c 11. ~ 7 f the two-point cor^;rela.tions, et .c , , 13rhich a r e the p:ropertles of izhe l a r g ' x eddies expressed rn ehe presen t .chcary hy t he c h a r a c t e r i s t i c r e l axaz ion r , ~ t r - Irr t 7 i . e ~ of t h e above d i s c u s s i o n p the K. i s expressed fo r t h e yt?aL1,.--.'r,i.irii~1-1 1 Jence a s , wi th , for the Couette flow, In eq. (61, the f i r s t t e r m r e p r e s e n t s t h e e f f e c t of nmlecalar viscus%"; the 'Q d i s s i p a t i o n whereas t h e second term, I < . , t h a t on t h e observable p r o p e r t i e s , 3 In Eq. ( 7 ) de f in ing t h e c h a r a c t e r i s t i c d i s s i p a t i o n r a t e , t h e c h a r a c t e r i s - t i c s i z e of t h e eddies r e spons ib l e f o r t h e energy inpu t t o the i n e r t l ~ l sul)rarqz i s consj-derecl t o be of t h e o r d e r of t h e dis-cance from t h e w a l - 1 . However, 2s Y -?- o r the s i z e of t h e s e eddies shouEd nu t beccgie zero on physical g r o ~ ~ n d s . 2: t h e flow reg ion r~ecir a \ g a l l , tlie m i n i m c a n s i z e , y*, of the eddies which can LC? considered as bekrlg a t t h e low wave nun&er end of an i n e r t i a l subrangc ~ r i ; t he of t h e o rde r oT t h e t f r ickness af the layer with in which the v i s c o s i t y cfl^c t- on C he observable p r o p e r t i e s i s of f i r s t o rde r . As it w i l l be seen i n a subsequent s e c t i o n , t h e viscosity e f f e c t :irst nlanifests i t s e l f on t h e inran shear as t h e wa l l i s approached. This l a y e r will he c a l l e d t h e " t r a n s i t i o n layer" . T h e minimum s i z e of t h e e d d i ~ s , y" , i s , t h e r e f o r e , of t h e order of the t r a n s i t i o n l aye r . his q u a n t i t y will he defined more p r e c i s e l y i n Sec t ion I V , It is c l e a r t h a t t h e t r a n s i t i o n l a y e r t h i ckness and, t h e r e f o r e , y* w i 2 ~ be func t ions of t h e f l o w Reynolds nunber.. Furthermore, )7*/L<) The effect of these small eddies on f, that is, the expressxon f 3 ~ K_ wxl; J be obtained from the ~avier-Stolces equation folSowincj the stochastic analyses of thak equation given by I,undgreii (I6' and Fox. (I7) By comparing the p r e s i n t Fokker-Planck equation to the equations derived from the Navier-Stokes eqpat ion , (2.6) the con~istcncy of the present theory with that developed by Lundgren and Fox (17 1 can also be seen, In the works of Eundgren and Fox, it is first considered that t l ~ e > c ;--stn ( n ) --> --t -7 -P A" an n-point distribution function of fluid elements, f (t,x1,x2,----- x .v ,v n 1 Z P -I. ----- v n ) , wllere n may be as large as it is necessary to describe a turkia!encm -I. -> -5- -3- field. Then the detailed ensemble average of any function G[V (r t) ,v, ( x 2 , t) . --- I Z L i + ---v (x ,t) 1 is given by, n n The Gar on a funmion is employed to represent the particular enserrdole aTiez;lc:ing defined by Eq. (8) , The lower hierarchy distribution function is then defined in terms af" f ( i l l and the Oirac delta function as, i 3 The one-point distrib~~.tion f(t,:<., ,la ) defined above is the same 2s 5 s * h -?- --k present distributiarr func t i on f !t,x ,u) , Fron Eq ., (-9) , -P 'dv 1 TJow, by obtaining the expression for --- in terms of space derivativzs I'. 3-r; at the ? a 2 f a f - 3 F T C f b - u 2 + K K -- 3 - 2 1 - - at j ax i oj' 3 au . au j j 3 j A comparison of the Lundgren's equation, Eq. (11) , with the presellt eclua- tion, Eq. (121, shows that the evolution toward the isotropy* is governed l ~ y the third term of each equation. The infiuence of the molecular viscosity on f is given by the Last t z r n of Eq. (11) which nmst be equal to the last two terms of Eg. (12). In the i-:ilc;a-L development of the present theory given in reference 7, the entire viscosity effect, K was expressed as, j ' IhTI-ien Eq.. (13) i.s employed, the last two terms of Ey, (12) simply be~afics, ---I ---- "itelaxation of the nonequilibsim degree of freedom toward equilibrium, ,* - Six? l a s t term of Eq. (11) can be also wri t t en , with ."Lhe irse of E q s , (5, a!::' (9) , as, ThercFore, t h e e f f e c t of vsscos i ty on f expressed by Lundgren ' 16) i s I .;enl-; c a ~ ( 7 ) t o t h a t Z ~ : ~ t i a L l y developed in t he present -&eory, 9 2 The expression, f ( a u ,/axwax ) , however, depends on t h e two-point l i s t v , - J r; ('" and s o u ( t7 , bu t ion firriction f '2' as well as on f, ds w a s showin by Lundqron is it was discussed e a r l i e r , t h e dissl:patiora r a t e 2.n the high Reyricl.cts--.ix~rb(-ir turbuLex~ce i s c o r ~ t r o l l e d by t h e l a r g e r energy-containing cdd ie i . ?-'e:. - , L*' : .- " .v. (.. .' L iiu, ,?.on of f (8 tt .,ax ax i per ta in ing -to di.ssipati.or; rn~xst depend on the .t.~,co-- 3 K K ' point d i s t r i b u t i o n Euncti.on. On the o the r hand, as i t wa.s explainecl :;.'I -2i";~ pxl-aqz:apks sirecedi-r~g Eq, (6) , the contri.butio~z of the small, near--e~]tiiiibri.al:! c?ddies on t he observable p roper t i e s , such as f, n~s.k be in chal-aci-er, Tklus, it-. must depend only on t h e one-point. d i s t r i b u t i o n function f, 2 I n the m r k of reference 7 , t h e d i s s i p a t i o n por t ion of f ( 2 u 'ax ax .in.ir~~h j' K. depends on t h e two-poir~t d i s t r s b u t i o n function was supplied from the a v = . ~ i a b l e experimental dara, In the suhseq~ent analyses of references 8 and 9 , i-Ee d : s s . ~ - p:,ti'crr+ ~ v . 2 ~ expressed i n terms o f zhe evolut ion r a t e of the larger ecidics a s Cqs, (4) and (5 ) . ';he con t r ibu- ioc or' t h e sr;mc*ll eddies oil f through v l scc;:t :, is ncgl igrb l e when 'trk-re is na s0li.d boundary. Therefore, the Poklcer- "7 src I.: cquackan was made se l f - con ta in l rg with U q s , (3 ) through (5 ) f o r f r e e i ~ r n u i - ~ ? c e - That p o r t i o r ~ of E (i -u , / a x ax ) perrair-rng t o t h e cont r ibut ion of 1-be spa l i 3 I< K ed3ies t o E i s now dererml.nc4 by equatj-ny the Last two terms of Eq. (12) zo c\c last t e r r n of Eq, (11) Before these terms a r e equated, however, the la?>*c i =ii n 3f Eq, (11) i s manipulated i n t o a more convenient form a s follows. By expressing the Di-rac delta function as the limit of a Gauss ian tu r i ( ' t l .~ :~ , Pox (2.71 obtained, , k d V . -f 3 <2 a 2 av av 3 2v - - m m , -3- - 6(v - u) = lim - -- ax ax a~ 312 5 2 ax ax ax ax C K j 5-"0 n r; K K K K 2 f ~ - u 1 ( T : ~ - un) av av + m m m n -- 2 2 2 exp a x ax - (v,< - uK) / r r, k' K T h e above expression is manipulated i n t o a form somev?!-lat d i f f e r e n t f r c m t%iL obtained by Fox a s , 3 a - V . 4- -L v - 1 2 - 6 ( v - u) = - V 2 % - -- aUom aUon a 2 f 3x ax a~ ax ax ax ax au au K K j 1s' K K K TR n Equation (27) b r i n g s oue c l e a r l y t h e t w o aspects of the viscous e f f ec t , The f i r s t . quantity on t h e right-hand s i d e of Eq, (171 depends on the ciale-g*uint d i s t r i b u t i o n function only, and i t is the c o n t r i b ~ l t i o n of the small edciios t o ,. t he abservahle propert ies . . The second quar-itity, on the o the r hand, depe;zS;'s or1 the two-point distribution funetxon, and it i s the d i s s i p a t i o n t e r m which dcpen(2a Qn t h e 1 2 rqer eddies, *3-. , ecquat i~g the l a s h two terns of Eq. ( 1 2 ) t o t h e right-hand s i d e of FJq, (LY), it i s s r n n that t he f i r s t of the l a s t two terms of Eq. 112' 1s c q ~ a i to the last t e r m of Eq, ( 1 7 1 , and, Equation (12) governing f i s now self-contained with t h e use of Zq. (181 a This equation i s transformed i n t o the fol-lowing by consj-dering f t o .be a func- i -?- t i o n of the r e l a t i v e v e l o c i t y u ins tead of the absolute ve loc i ty u- The q u a n t i t i e s B and 8' are given by E ~ S . (3) and (7 ) , r e spec t ive ly . Solut ion of Eq- ( 1 9 ) f o r Couette f l o w and discussion of the resulGs c c n s t l - t u t e the remainder of t h i s paper. .2 The solutj.on of Eq. (19) w i l l be obtained by t h e moment method due t o Liu and Lees as i n the previous papers. (7-9) For t h i s purpose, w e first obta in the general ized moment equation f a r s teady s t a t e by mult iplying Eq, -f 619) by a function Q(U1, and i n t e g r a t i n g it with respect t o the v e l o c i t y space, The p a r t i c u l a r moment equations governing the Couette flow i s now derived from Eq. (20) by f i r s t expressing where 2 2 ( u - u 1 3-v 4 - W 2 - 1 - 0 l - 3 / 2 2E1/3 fox V?J 8 (2/3.rrE19 2 2 2 1 ( u - u 1 + v 3 - w - - 0 2 f 2 - f o r V I O , 3/2 2E2/3 - (22) f 2/3nE2) and f and f are zero for VO, respectively, l 2 The coordinate system for kine Couette flow is given in Fig. 1, !vote thdr u, v, and. w without the tensor indices are the x , y, and z components of the instantaneous absolute velocity, respectively. Also, U , V, and w witlzout the tensor indices are the x, y, and z components of the relative velocity, respec- tively . There are four unknown functions of y in the expression far f, whicl? are Elf EZP uol, and u Four particular moment equations are, therefore, generated 02 - from Eq. ( 2 0 ) by substituting four different, suitable functions for 9 nn t h a t equation, In the previous analyses, (7-9) the values chosen for Q were 1, U, iW, and U U . In the following analysis, the four values of Q = 1, U. u', and I T 2 will K ic be employed, because the resulting moment equations with these valves of Q are a little simpler than those resulting from the previous values of Q, By compar- ing the limiting homologous solution of the present moment equations with :he hom~logous solution obtained earlier,"' it will be shown that these two sei:E of Q give about the same results. First, the substitutions of Eq. (21) and Q = 1 into Eq. (20) simply gives, . 2 2 The successive substitution of Q = U, U , and V along with the subs%i2?sntian a£ Eq, (21) for f into Eq. (20) then give a set of three differential equations after some manipulation. These equations are suitably nondimensionalized and mitten as, d$e - = ( 6 ~ r l " ~ d24 dY 2 E - dy2 l/2 + Cfi2 $8 d Y + $(f) a 4 *Z 2e2 + 3 1 2 ~ ~ + - (6?-i1l/~ Eai = 0 Y + Y* e 2 + 9 - dY - b2 Y 4- Y* 4 dY* * 2 d2e2 - E a i - - - - = O d?12 where, and u* is the friction velocity defined as (7 w /p)'/2 where .r w is the surface shear stress. v In deriving E q s . (23) through (25) , B and B given by Eqs. 13) and (7) , 1 1 2 respectively. have been simplified a little by replacing 43 IC U K >'I2 by ;U K LT K The only undefined parameter in Eqs. (23) through (25) is Y* = y * / ~ where y* is the minimum eddy scale explained in the sentences following Eq. (7). As it was explained t h e r e , y* is of the orde r of t h e t r a n s i t i o n l a y e r k h ~ c k n e t - s , It will., t h e r e f o r e , be determined i n t h e next s e c t i o n du r ing t h e discussiorr o f t h e n a t u r a l s c a l i n g of t h e va r ious flow reg ions , Finally, t h e boundary cond i t i ons a r e given a t t h e two solid boundaries as fol lows : In the subsequent s o l u t i o n , hnwever, Eqs. ( 2 8 ) will be rep laced by a more con- venien t s e t of boundary cond i t i ons a t Y = I. IV. BEHAVIOR OF THE GOVERNING MOL%NT EQUATIONS A study of Eqs, ( 2 3 ) through ( 2 5 ) shows t h a t t h e r e a r e two small. parameters, E and a , which govern the general behavior of these equations when R e is large, I t w i l l be shown i n t h i s s e c t i o n t h a t the sca l ing based on these two para- meters na tu ra l ly d iv ides the flow f i e l d i n t o t h e th ree regions of outer, transi- t i o n , and inner l aye r s . The solrltion of Eqs. (23) through ( 2 5 ) w i l l tl-ien be ab- ta ined i n t h e next sec t ion by the method of matched asymptotic expansions for the l a rge Reynolds numbers. We consider t h a t the Reynolds number i s s u f f i c i e n t l y Large such that there e x i s t s t h e following order relati-onslzip , Each of t h e th ree l aye r s w i l l be discussed i n t h e following for 0 2 - Y 5 1. Outer Layer - (I This layer is d.efined a s t h a t region wherein U = 011), We expand t h e dependent v a r i a b l e s a s , Sn view of t h e or3er rcLationships set for th by Eqs, 1291, the tcrris cf Ofa) arc consisten~ly r e t z~ incd along wit11 t h e terms of O (1) 4 * A l s o , a r we shall. see preserktly, Y* .; O f r ) and, hence, ir does not. i.?ippa.r in Eqs, (30: through ( 3 2 ) . Eqr.iatians (30) through (321 s l m w that , to the Tirst order, the cffr:::is. o f the smedler eddies through ~riscosity on t h e observa.bEe properties is neql3ci:ihlc in a"- .cn is negl jq ibfe at Y = I, This means tha t tkc- j,-ri%scl-it *Such is the case with the aeher two layers a l s o - equations should descr ibe a t Y = I a homoLogous flow analyzed e a r l i e r . (8) Ti le f a c t t h a t the flow i s honologous a t the cen te r of a Couette f l o w when Re i s Large kaas been experinentaI2y es tabl i shed by Robertson. C5l Equation ( 3 0 ) through (32) can be r e a d i l y solved, and t h e r e r e s u l t , st Y = I, where ( 3 3 ) The Reynolds s t r e s s and t h e turbulence energy can be constructed from qoo and Ooo as (see R e f , 8 1 , A s it was explained following Eqs. (221, t he present governing equations, Eqs. (23) through ( 2 5 ) , a r e obtained by taking the d i f f e r e n t moments o f the Fokker-Planck equation r a t h e r than t h e moments taken i n reference 8, A carnpari- son of Eqs. (33) and (34) with the corresponding so lu t ions of reference 8 si?sws t h a t the two s e t s of so lu t ions agree wi th in 10%. Figure 2 shows t h e Reynolds s t r e s s and turbulence energy computed by Eqs, (35) and (361, and those computed i n t h e e a r l i e r work. (') It a l s o conta ins the experimental r e s u l t s of Reichardt'" and ~ o b e r t s o n ' ~ ) obtained a t the center of Couette flow. TransiCion Layer This layer is defined a s t h a t region of Y = O ( E ) from the wall . We f i r s t def ine t h e s t re tched independent va r iab le as , The dependent va r iab les are expanded i n t h e manner same as Eqs, (29), A s u b s t i t u t i o n of the above s e r i e s i n t o Eqs. (23) through (25) gives, t o the f i r s t order , F i r s t we no t i ce i n the above equations t h a t t h e second order d e r ~ v a t i v e term appears i n t h e momentum equation, Eq. ( 3 9 ) . Thj-s means t h a t the contri- bution of the small eddies through v i s c o s i t y t o the momentum t ranspor t i s o f 2 first order i n t h i s l aye r . Isowever, t h e cont r ibut ions t o t h e energies , <:U 7. 2 and , a r e s t i l l negl ig ib le . The above po in t s were f i r s t mentioned i n t h e paragraphs following Eq, 171, It was a l s o explained t h e r e t h a t y* should be of the order of the t r a n s i t i o n - l aye r thickness, t h a t i s Y* = O ( E ) . The present so lu t ion is not too sensitive t o t h e p a r t i c u l a r choice of the value f o r Y* provided thag it is O I E ) , It i s l i k e l y t h a t Y* < E because t h e s i z e of t h e eddies generated by a shear layer i s always smaller than t h e l aye r thickness. We l e t , i n t h e present s tudy, Lnner Layer This l aye r is defined a s t h a t region adjacent t o the wall whose thickness is of t h e order Ea. Note t h a t ca is the rec iprooal of u*L/v which is of t h e order of the turbulence Reynolds numbex. The s t r e t ched independent v a r i a l e is defined a s , Note t h a t the inner va r i ab le , 2 , is t h e usual nondimensionalized ordinate wi th which the various experimental r e s u l t s near t h e w a l l a r e co r re la t ed . The dependent va r i ab les a r e expanded as , Subs t i tu t ing t h e above s e r i e s i n t o E q s . ( 2 3 ) through- ( 2 6 ) , w e oh tain, t n f i r s t order , We see i n the above equations t h a t all t h e second order derivatives are of f i r s t order i n t h i s l aye r . The con t r ibu t ions , the re fo re , of the small eddies through viscosity t o a l l observable p roper t i e s a r e important i n t h i s Layer, In f a c t , it is t h e viscosity which enables t h e flow field t o f i n a l l y sat i-sfy the wall boundary condi t ions given by E q s . (271 and ( 2 8 ) . V. SOLUTION FOR EXIGB EEYNOI;IIS MUM13ERS The f i r s t order equations f o r the t h r e e l a y e r s developed i n t h e preceding sec t ion can be separa te ly solved and asymptotical ly matched. However, E q s , ( 2 3 ) t h o u g h (25) show t h a t the ou te r and transiti.011 l a y e r s can be analyzed cogether w i t h no added mathematical d i f f i c u l t y . By handling these Payers toge the r , one can e l iminate one matching thus improving t h e accuracy of the f i n a l results, In t h i s sec t ion , t h e f i r s t order equations f o r t h e ou te r - t r ans i t ion layer* and t h e corresponding equations f o r t h e inner l aye r w i l l . be formulated, The so lu t ions of these equations w i l l then be obtained. F ina l ly , these so lu t ions w i l l be asymptotical ly matched thus constructing a uniformly v a l i d so lu t ion f o r the e n t i r e flow f i e l d . Before we begin the ana lys i s , however, the boundary condi t ions , E q s , ( 2 8 1 , w i l l be replaced by a s e t of more convenient ones. Boundary Conditions The boundary condit ions a t Y = 0 a r e those given. by Eqs. (27) . Ewations (28) w i l l be replaced by a s e t of t h e c o n d i t i o n s e a t Y = 1 t o be determi-ned in t h e following manner. A s it was mentioned e a r l i e r , it has been experimentally e s t a b l i ~ h e d ' ~ ) that a homologous condit ion p r e v a i l s a t t h e cen te r of a Couette flow when R.e is suf - f i c i e n t l y l a rge . The homologous so lu t ions have been obtained previously'81 and a l s o i n Section I V a s Eqs. (33) and (34 ) . Tlie boundary condit ions a23 Y = 1% are, therefore , replaced by the homologous condit ions a t Y = 1. This means t h a t we seek t h e so lu t ion of Eqs, (23) through (25) which s a t i s - f i e s t h e boundary condi t ions a t Y = 0 given by Eqs. (271, and t h e p a r t i c u l a r *Combined l aye r cons i s t ing of the ou te r and t r a n s i t i o n layers . 2 " c ~ ; & i n a t i a n a of ii,. Q:, (d+/dY) which sat.isf y t he homolognus ~ O ~ U ~ Z G ; , ?.'38. (331 st Y =: 1 . Xn addition, t k i c so'f,zrtion m?~s"isa t i s fy the condi.t;:lon of ; = Equations (27) an2 t h e conzi t ions a r ' r = 1 specified above ro:ist~klri-,c se:rcn bouu~dary contitLons whereas ~ q s , ( 2 3 ) tkurougil ( 2 5 ) ccrmprxse a s i x t h o rde r sys test, Mowevn,~, these equat4.ofis cantsir, a n unspecified parsmeter c . Tkrercfaxe, fcr 8 p a r t i c t ~ l , ~ ~ set of tile boundary conditi.ans given at Y = It, there exists a par- t icu lar value of E for which E q s , (2 3) through f 25) can be solved to sacx s f y thc seven boundary conditionsA Since r is a ur-,ic~ue func t ion of Re, each soXutrc3ri corresponds to d p a r t i c ~ i a ~ Xe, Ln order to combine the transition layer with the outer layer , t h e teems cr 0 ( e ) must be re ta ined , T ~ I Y J ~ , w e expand, 2 2 2 4 5 (a, Y ) = $:(I?) + a R, ( Y ) i O(a ) .A As before, klle terms of C(i:r,f will. be c o n s i s k e n t l y retained for the first oicdelc in view of the above expansions* A s u b s t i t u t i o n u f E g s . (48) in to Eqs. ( 2 3 ) tkxxough ( 2 5 ) gives , tn first order, w1.1ere Eq- (425 has been employed, As 5.t was explained i n the precedirig subsection, Eqs. (49) th~a~,!g?i (51.) 2 2 must satisfy at Y = 2, the part icular combinations of O m , and (d$/'dY] n: ~kii.<:h s a t i s f y Eqs, ( 3 3 ) , and the ~ o n d i - t i o n of 4) = 2 , Note t11a.t the terms of 0 (E) appear in E q s , (49) thro?.xgkt (51) bei:r;i...ilse of the "zraxisi.t..ior~ layer, Oute r lyycr docs riot contain them as seer1 ir~. E q s , ( 3 0 ) T h ~ r i ? u ~ ? i (321. Therefore, to bc consisren"i with %he f i r s t csdex s o l u t i o l ~ , it is only required that E q s , (49) thrcuqh ( 5 1 ) satisfy the boundary conditions at Y - 1 2 3 The quantitres tbr0 and 0 ; appearing in Eqs . t49) through (51) are fir::? re- placed in terms of (d$/dT)w by thc use of Eqs. ( 5 3 1 . ~ q * ~ a i i o n s (49) rhrn~",rjn. 152) are t h e n manipulated into the !%oil.uwinc; set of c q ~ a t i o n s . With the boundary condit ion that, Above se"cis a l w a y s sa t i s f ied t o O ( E ) for all. (d4/dY)m, Hence, w i t h Fc, (56) satisfied by Eq. 1521, a l l boundary conditions a t Y = 1 are now saLisf icC by E q s . (5.2) through (56) . The parameter E is determined Lhrough t h e subsscpent matching w i th the inner scl lut ion. The standard sciution, o f course, e x i s t s fo r the cubic equation, Ey. (5.31, Equation (52) i s i n t e g r a t e d numerically t o s a t i s f y Eq. ( 5 6 ) . Inner Layer The s t r e t c h e d inner variable i s t ha t def ined by Eq. ( 43 ) . Ta be c o n s i s t e n t w i t t i t h e c u t e r - t r a n s i t i o n layer, t h e terms of O ( E ) will be r e t a ined , Hence, the dependent varj-ables a re expanGeC! as , 2 Z 2 0 (a, T} = O ( $1 + O ( a ) 0 Again, t h e terms of O(a) w i l l be cons i s t en t ly r e t a ined f o r the first ordee i n view of Eqs. (57) . k subs t i tu t io r l of t h e above s e r i e s i n t o Eqs, 6231 tlarough (251, and t h e subsequent manipulation r e s u l t , t o f i r s t order , i n t h e following equations. 2 I n der iv ing t h e above equations. $_ and 632 have been replaced i n terms of ( d $ / d ~ ) ~ by the use of Eqs. (33) . Also, Eq. (42) has been employed, Equations ( 5 8 ) through (60) must s a t i s f y t h e w a l l boundary condi t ions of Eqs. ( 2 7 ) a t 5 = 0. The o t h e r set of t h e boundary condi t ions a r e determined through t h e matching t o be described below. , Matching The ou te r - t r ans i t ion Xayer and t h e inner l aye r w i l l now be matched t o first order. From t h e matching, t h e parameter E and t h e remaining boundary conditions for Eqs. (58) t h o u g h (60) w i l l be determined f o r a chosen (d$/~3Y)~. Beginning w i t h Eqs. ( 4 8 ) ' t h e one term inner expansions of the one term ou te r expansions '18' become, a s Y + 0, where it has been an t i c ipa ted t h a t 4 ( 0 ) w i l l vanish. 0 Next, using Eqs. (571, t he one term ou te r expansions of t h e one term i n n e r expansions become, a s 5 -+ m , Matching the terms of Eqs. ( 6 2 ) t o those of Eqs. ( 6 3 ) ' t hese r e s u l t , * The subsc r ip t w denotes IT = 0. . The f i r s t of Eqs. ( 6 3 ) is t h e add i t iona l boundary condit ion which t h e outer- t r a n s i t i o n l aye r equations must s a t i s f y i n order t o accomodate.the matching. Therefore, f o r a given homologous boundary condit ion a t Y = l represented by td+/dYIm, the re e x i s t s a p a r t i c u l a r value of E for which @ (Of = O can be sa t i s - 0 f i ed . Once t h e f i x s t order ou te r so lu t ion i s obtained, the rnatchi~~g f i r s t order inner so lu t ion is obtained by numerically in teg ra t ing Eqs. 658) thraugh (603 to s a t i s f y the wall boundary condi t ions and t h e remainder of E q s . ( 6 3 ) . The s t ruc ture of the Couette flow has been described and discussed i n Section I V i n terms of ce r ta in singular behavior of the governing equations, The solutions of the equations obtained i n the preceding sect ion w i l l be discussed herein, Figures 3 and 4 show the typ ica l matchi-ng between the inner and the outer - t rans i t ion layers fo r the mean veloci ty u turbulence energy r Reyna9.d~ 0 K K 2 s t r e s s , and the stream-wise component of the turbulence energy m e plot ted on Fig. 7 since the experimental measure- 2 m n t of was taken by Robert- ~ u n . (61 A s it is expected from the singular behavior of the governing equations discussed i n Section PV, remains prac t ica l ly constant u n t i l the viscosity e f f ec t becomes manifest i n the t r ans i t i on layex. The influence of the viscosity on becomes manifest only with the = r iva l of the inner layer as i t w a s K K evident i n t he discussion of Section IV. The present theory i s seen t o cor rec t ly p red ic t the order of magnitude 2 and the general behavior, with respect of Y , of measured by Robertson- (61 2 increases near the wal-1, This is evidently caused by the eday s t re tching i n the x di rec t ion near the wall. e The coef f ic ien t of f r i c t i o n , C f f computed i n t he present theory and those experimentally obtained by ~ e i c h a r d t ' ~ ' and Fbbertson 15r6) a re shorn i n Fig, 8 a s the functions of the Reynolds number, 3Jso shown i s the empirical cuve which Robertson found t o be the bes t f i t t o h i s experimental points and t o t h e experimental points abtained i n the c i rcu la r Couette flow. "O' Since a secondary flow can eas i ly e x i s t i n a c i r cu l a r Couette fJcw, the appl icab i l i ty of the cir- cular Couette f l o w r e s u l t s to the plane Couette flow s tud ies is somewhat in doubt* Figure 8 shows t h a t the present theory gives tlhe values of Cf wlsich are .be- tween those experhenta l ly obtained by Reichardt") and Robertson ( 5 p 6 f for the plane Couette flow, The s impl i f ied s t a t i s t i c a l theory developed by t h e present author ea r l i e r has been made self-containing f o r t h e shear flows along solid b o w ~ d a r i c s - This has been accomplished i n part by formulating the expression f o r the contribution of t h e small eddies t o t h e observable p roper t i e s from t h e ava i l ab le stochastic analyses (16f17' of t h e Navier-Stokes euqati-on. During the course of t h i s formu- l a t i o n , t h e comparison of t h e present theory, developed from t h e generalized Brownian motion, /7t8,95 with the s t o c h a s t i c theory based on the Navisr-Stokes equation, 'I6 ' l i ' w a s mde. The mutual consistency of the two t h e o r i e s was slnum1. The present theory n a t u r a l l y d iv ides the flow region along a wall i n to t h e three regions of the oixter, t r a n s i t i o n , and the inner l aye r s , The contribution of t h e smaLL eddies t o t h e observable properties throilgh v i s c o s i t y is negligible i n the ou te r layer whereas i ts contr ibut ion t o the mean shear is of f i r s t c~rdex i n t h e t r a n s i t i o n layer. I n the inner l a y e r , a l l v i s c o s i t y e f f e c t s axe oF first order. T?- ~ r i n comparison o f t h e Couette f L o - i so lu t ion< obtained with t h e ava i l.able experimental r e s u l t s shu~eed t h a t the present theory descr ibes t h e rather derailed 2 s t r u c t u r e of t h e flow f ie ld ' , such a s , s a t i s f a c t o r i l y i n actdition to ca3:- r e c t l y predicti.nq Llie mean v e l o c i t y p r o f i l e and the surface shear stress, I t has been shown t h a t the present s impl i f ied s t a t i s t i c a l theory can be employe4 "co analyze the real wall, shear flaws t r a c t a b l y . This woslr was supported by the National Aeronautics and Space A&inistiratic~xe WGR 14-012-012. The encouragement given by Mr. C. Goodwin and Dr. P. Nachtsheim of Ames Research Center is greatly appreciated. The author is also grateful to Mr. T. J. Marcisz, a graduate student in the Energy Engineering Department, for his help in obtaining the numerical results of this paper. 1. P. A. Libby, ARS J. 32, 388 (1962). 2. D, E. Rasner, Chem. Eng. Sci., 19, l (1964). 3. W e Ha Webb and L, A, Exomas, ALAA J,, 3, 826 (1965). 4. V. Ha Reichardt, 2. Angew, Math. Mech., 36, 26 (1956). 5. J. M. Robertson, in Proceedings of the 6th Midwest Fluid Mechanics Conference (University of Texas Press, Austin, Texas, 1959) p. 169. 6. J. M. Robertson, H. F. Johnson, Journal of the Engineering Mechanics Division, Proceedings of the American Society of Civil Engineers, December, 19'70, p. 1171, 7. P, M. Chung, AIAA J., 7, 1982 (19691, 8. I?. M. Chung, Phys. Fluids, 13, 2153 (1970). 9. P. M. Chung, A I M Prepxint No. 70-722 (19701; also to he published in A i k A J. (1971). 10. T. C. Chawla, with P Negligible WaLI Stress, Ph. D = Thesis, Department of Nuclear Engineering, Penn, State University, State College, Pa., June (1969). 11, C. Y. Liu and 2. Lees, in Rarefied Gas Dynamics, L. Talbot, Ed. (Academic Press Inc., New York, 19611, p. 391. 12. G. M. Batchelor, The Theory of Homogeneous Turbulence, (Cambridge University Press, New York, 19531, Chap. 6. 13. J. 0 . Hinze, Turbulence, (McGraw-Hill Book Company, New York, 19591, Clnfiaps, 3 and 4. 14. J. Laufer, NACA Tech. Repts., 1174, (19543, 15. P. 5. Klebanoff and Z , W . Diehl , J. Research NBS, 50 , (1953). 16 , T. S, Wndgren, Phys, F l u i d s 12 , 969 (19671, 17. R. Fox, Solu t ion f a r t h e Cor re l a t ion Functions i n a Homogeneous S s c t s i c Ix- compressible Turbulent F i e l d , Sandia Laboratory Report SC-RR-70-912, Albuqucrquc, N . M,, February, 1971. Also t o be publ ished i n Phys. F l u i d s (1975), 18. M. D. Van Dyke, Pe r tu rba t ion Methods i n F lu id Mechanics, Academic Press, N e w York, (1965) . 19. T. S. Lundgren, Phys. F l u i d s , 1 4 , 225 (1971). 20. M, Couette, Ann. De Chcmie et de Physique, Ser . 6 , 21, 433 (18901, i . 2 : FtEYNOZDS STRESS AND TUREULENCE ENERGY AT THE CEI\dTR OF COUETTE FLtOU FIGURE 3 : TYPICAL ASY!,PTTJTIC MPiTGHING 3ETkWXN TIE OTS'P'ER-TWJ\ISZTTTJN ZiJYEP, AfJD THE XPWER LAYER .IIRBI- OUTER- TRANSITION SOLUTION INNER SOLUTION EXPERBM ENTAL DATA h REPCHAROT ( R e = @ ROBERTSON f Re = 10,0010) PRESENT THEORY 0 0,2 0,4 0e6 Oa$ %,O Y f ?:GUY33 5 : 34EA1.Q ilELOC ETY P WUFI LES

A turbulence description of couette flow

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