Ameican Jounal of Fluid Dynamics 6, 6(: 7-4 DOI: 93/j.ajfd.66 A New Mahemaical Appoach o he Tubulence Closue Poblem Mohammed A. Azim Depamen of Mechanical Engineeing, Bangladesh Univesiy of Engineeing and Technology, Dhaka, Bangladesh Absac This sudy inoduces a new mahemaical appoach (NMA saing ha he emaining small ems upon dominan balance, if eains he essenial physics of he oiginal equaion, may exis as an addiional equaion. Applicaion of his addiional equaion appeas o educe he unclosed govening equaions fo he wo-dimensional ubulen flow o closed ones. The closed fom equaions due o NMA fo he ubulen bounday laye and ound je ae solved using a Fully Implici Numeical Scheme and he Tidiagonal Maix Algoihm. An oveall ageemen of he obained esuls wih he exising daabase fo boh ypes of flow show he capabiliy of NMA in solving he complee closue poblem of ubulence. Keywods Maximal balance, Minimal se, Addiional equaion, Scaling, Axial slope effec. Inoducion The new mahemaical appoach (NMA seems o have a common oigin wih he mehod of dominan balance which is known o deal wih he pocess of simplificaion of he equaions wih small paamees fo hei soluions. Basically, his simplificaion consides wo o moe ems in he equaion as lage which dominae he soluion, ohe ems being small. The mehod of dominan balance was inoduced by Isaac Newon [] in 67-673 in obaining appoximae soluions of he algebaic equaions wih small paamees and in consideing infiniesimal displacemens fo he developmen of diffeenial calculus, and was subsequenly developed by Kuskal []. Bende and Oszag [3] focused on he simplificaion mehods based on self-consisency and dominan balance fo diffeenial equaions. Fishaleck and Whie [4] used Kuskal-Newon diagam fo he simplificaion of a diffeenial equaion involving a small paamee and fo discoveing all possible combinaions of maximal balance of he equaion. The new mahemaical appoach saes ha he emaining negligible ems upon maximal balance of an equaion may epesen an addiional equaion. This addiional equaion may be useful in solving poblems wih moe unknowns han he numbe of equaions. The sudy of fluid ubulence by decomposing he flow vaiables ino mean and flucuaing componens encounes a siuaion wih moe unknowns han he numbe of equaions, called he closue poblem. This ubulence closue is a * Coesponding auho: azim@me.bue.ac.bd (Mohammed A. Azim Published online a hp://jounal.sapub.og/ajfd Copyigh 6 Scienific & Academic Publishing. All Righs Reseved long-sanding unsolved poblem of classical physics since he ime of Osbone Reynolds [5]. The unclosed diffeenial equaions govening he ubulen mean flow ae called Reynolds Aveaged Navie-Sokes (RANS equaions. As no mahemaical mehod exiss, appoximaions called closue models ae used o obain closed fom RANS equaions. The closue models ae eihe sysemaic appoximaions o inuiion and analogy whee he lae, in geneal, is moe successful. Howeve, engineeing closue models ae sysemaic appoximaions in combinaion wih inuiion and analogy [6]. These engineeing closue models have a quie consideable hisoy saing fom he fis mixing lengh model [7] o he fis k- model [8] and afewads vaious k- ype models. Boussinesq s [9] eddy viscosiy concep is he coe of hese closue models bu he did no aemp o solve he RANS equaions in any kind of sysemaic manne. Simila o engineeing closue models, his pape claims he success of sysemaic appoximaions in combinaion wih NMA in solving he closue poblem. Using he addiional equaion along wih he sysemaic appoximaions due o he ode of magniude easoning, a closed se of equaions fo he ubulen bounday laye flow as well as fo he ound je flow is obained fom hei especive unclosed se. The closed ses of equaions and he equaions fo he emaining unclosed ems ae solved numeically o discen diffeen chaaceisics of he flows and heeby o seek he validiy of NMA.. The New Mahemaical Appoach I is ofen possible o find wo o moe ems in an equaion which dominae he soluion, ohe ems giving only small coecions o he value obained by neglecing hem eniely. In neglecing he small ems, one mus
8 Mohammed A. Azim: A New Mahemaical Appoach o he Tubulence Closue Poblem espec he Kuskal s pinciple of maximal balance which saes ha no em should be negleced wihou a good eason []. In maximal balance, he compaable ems consiue a maximal se and he emaining small ems belong o a negligible se. The NMA descibed heein goes fuhe and saes ha he negligible se, if i eains he essenial physics of he oiginal equaion, may be called a minimal se which acs like an addiional equaion. Thus NMA may povide addiional equaions o ge he closed sysem of equaions fom he unclosed one... Algebaic Basis of NMA The mahemaical jusificaion fo NMA is sough hough an algebaic equaion x, y ax, y x, y bx, y ( whee and ae of compaable magniudes, >>a and >>b. This equaion can be wien o a good appoximaion ino wo equaions as x, y x, y x, y S bx, y ( a (3 whee he scaling paamee S in geneal is S(x,y o a consan fo idenical shapes of a and b. Hence, Eq. ( may epesen he maximal se and Eq. (3 he minimal se. 3. Applicaions of NMA NMA is applied o he unclosed govening equaions fo he wo-dimensional (D ubulen bounday laye and ound je flows shown in Figs. and. Bounday layes ae wall shea flows whee he fluid viscosiy, no mae how small, enfoces he no-slip condiion a he solid wall. This viscous consain gives ise o viscosiy dominaed chaaceisic velociy and lengh nea he wall while ubulence dominaed ones ae found away fom he wall. On he ohe hand, jes ae fee shea flows whee fluid coming ou fom cicula oifice mixes wih he suounding fluid and develops hough hee successive disinc egions, namely iniial, inemediae and developed egions. Iniial egion is chaaceized by he pesence of a poenial coe ha is in lamina sae, inemediae egion by he ansiion sae and developed (self-simila egion by he fully ubulen sae. This je gows non-linealy in he developing (iniial and inemediae egions and linealy in he developed egion. The unclosed govening equaions, addiional equaions due o NMA, pimay closed fom equaions, seconday closed fom equaions, and hei iniial and bounday condiions ae pesened in his secion. 3.. Unclosed Govening Equaions Coninuiy and RANS equaions govening he D axisymmeic ubulen flow v,,u fo a consan popey fluid in cylindical co-odinaes (,, x ae u v (4 x v v p v u v x x w v uv v uv u u u p v u x ρ x x (5 u (6 whee v and v ae he mean and flucuaing velociy componens, simila ae ohe quaniies, v w and u w ae zeo by he cicumfeenial symmey, is he fluid viscosiy. The above coninuiy and RANS equaions epesen he govening equaions fo D plane ubulen flow u, v, in Caesian co-odinaes (x, y, z upon subsiuion of he fee, y, w w / and v / in hem. The sysem of Eqs. (4-(6 is no closed because hee ae moe unknowns han he numbe of equaions. In he nex secion, his unclosed se of equaions is made closed by using he addiional equaion. u u o o Fee seam y Figue. Schemaic of a plane bounday laye flow Iniial egion Figue. Schemaic of a ound fee je 3.. Addiional Equaion due o NMA u Oue flow Bounday laye Shea laye Dimensional analysis povides a gea deal of x x
Ameican Jounal of Fluid Dynamics 6, 6(: 7-4 9 undesanding of he physics involved in he govening equaions of a phenomenon ha is indispensable in deemining boh he maximal and minimal ses of he equaions. In view of his, dimensional analysis of he RANS equaions is caied ou by consideing he chaaceisic mean velociy U, velociy flucuaion u, flow widh and flow lengh L as he scales of mean and ubulence velociies, diffusion and convecion lenghs, especively, wih he assumpions [] ha u is compaable o U and is compaable o L. Using hese velociy and lengh scales, he odes of magniude of he ems of Eq. (5 ae esimaed as follows: v OU / L v v, u OU / L x p w x v O u /, v u 3 v Ou / L, O U / L v O U / L. x The ems of Eq. (5 ae wien in he ode of hei magniudes as p w v v x u v v v v u v (7 x whee he fis hee ems ae lage and he las five ems ae small. This equaion epesens a balance among he ineia foce, pessue foce (including nomal foces, ubulen shea and viscous foces. The fis hee ems consiue he adiional maximal se which epesens a balance beween he pessue foce and ubulen nomal foces. The las five ems consiue he minimal se which epesens a balance among he ineia foce, ubulen shea and viscous foces ha is he minimal se eains he mos essenial flow physics of Eq. (7 compaed o he adiional maximal se because he fome possesses he ineia of fluid moion. Hence, accoding o NMA, Eq. (7 may be wien ino wo equaions as x p w v v v v uv S v u v x,, (8. (9 Equaion (9 epesens he minimal se and hence povides an addiional equaion. Compaison of magniudes of uv / x wih he lef side of his equaion povides S C u / U / / L ( whee C is a consan and S=S(x since all he flow scales in i ae funcions of x only. In bounday laye, he flow scales nea he wall due o viscosiy dominance become u Ou and y O / u and ohe scales emain he same. In case of fee je, only he scales u and change hough he flow egions, maked by he diffeence in he flow sae [6]. Tha is in some flow aeas some ems ae lage han ohes and elaions beween he ems vay in diffeen flow aeas. The scaling paamee which compises of an abiay consan and some flow scales, accouns he vaiaion of hose elaions hough is ansvese and axial dispesions. 3.3. Pimay Closed Fom Equaions Equaion (6 by he ode of magniude easoning and unifom pessue assumpion fo hin shea flow (/L<< educes o u u u v u v u. ( x Hence coninuiy equaion (4, addiional equaion (9 and momenum equaion ( fom a closed sysem of equaions. 3.4. Seconday Closed Fom Equaions Compaison of unclosed equaions in Sec. 3. wih closed equaions in Sec. 3.3 shows ha p, u u v v and w w ae lef unknown. Wih a view o obain he equaions fo he unknown ems, Eq. (5 may be wien as being v v p v u v x x uv v ( v v and w w ae equal in axisymmeic flow and v v can be solved fom his equaion fo he known p. Afewads w w can be calculaed fom Eq. (5 because all ohe quaniies ae aleady known. Neglecing he small viscous em, Eq. (6 may be ead as follows whee u u τ p v u u (3 x ρ x x u / uv is he effecive shea sess. In ode o solve he mean saic pessue p, Eq. (3 may be wien by neglecing he small nomal sess em as v
3 Mohammed A. Azim: A New Mahemaical Appoach o he Tubulence Closue Poblem u u τ p v u (4 x ρ x whee by he ode of magniude agumens, lef side is a maximal se and igh side a minimal se of single small elemen. This singleon is ivial, as a esul, he minimal se is obained by including a small facion of he lef side ino he ivial se, as an exension of NMA, which may be called an exended NMA. Equaion (4 is wien in he fom of Eq. ( as u u τ S v u S v u x x S τ p ρ x u u (5 whee S is a small scaling paamee. Now following NMA, he minimal se may be exaced as u u τ p S v u x. (6 ρ x Equaion (4 epesens a balance among he foces due o ineia, effecive shea and pessue. The above minimal se also epesens a balance among he same foces implying ha he minimal se eains all he essenial flow physics of Eq. (4. Hence equaion fo solving p is p u u τ S v u. (7 ρ x x Equaion fo solving he axial nomal sess obained fom Eq. (3 as u x u u may be p u u τ S v u. (8 ρ x x Radial nomal sess in he fom vv can be solved by scaling Eq. ( p v v v S v u x x and he azimuhal nomal sess scaling Eq. (5 as v u v v (9 w w can be solved by w v v p Sv u v x x v u v v. ( Sess w w does no appea in he RANS equaions fo plane bounday laye flow o add o he closue poblem. So i can be obained fom he ubulen sess enso ui u j ui u j u k u k x x j i ij ( 3 which is in analogy o he consiuive equaion fo lamina flow fo ui u j ij / and uk u k uk / xk, and in D flow fo i=j=3 educes o validiy of which is discussed lae. 3.5. Iniial and Bounday Condiions w u v /, ( In bounday laye flow, he iniial condiions ae u, y u o, p, y p o and, y whee u o and p o ae he unifom fee-seam velociy and pessue, and is he geneal flow vaiable. The bounday condiions ae aains fee-seam values a he bounday laye edge, / x a he ouflow and = a he wall. u In fee je flow, he iniial condiions ae o o,, v,, u o, and p o, u o u o, u,.4uo whee o and u o ae he adius and unifom velociy a he je exi. The bounday condiions ae aains ambien condiions a he je oue edge, / x a he ouflow and / a he axis of symmey excep v. 4. Numeical Mehods I is aleady menioned ha eddy viscosiy concep is he coe of all ubulence closue models. As a esul, hadly hee is any sandad numeical scheme fo solving he closed fom Eqs. (4, (9 and (. This necessiaes o wiing he momenum equaion ( in he fom u u u v u νn (3 x using uv u and N / necessiaes o wiing Eq. (9 as, and also x u v v v S v u v dx (4 x whee is he eddy viscosiy and N is he nomalized effecive viscosiy.
Ameican Jounal of Fluid Dynamics 6, 6(: 7-4 3 Fuhe N is wien in a ecuence elaion, using he apezoidal inegaion fomula wih he fis degee polynomial and Eq. (4 in N /, in he fom u Ni, j N i, j i, j whee x u x m f i, j f i, j (5 i, j m x i x i and fi, j v v v S u x v v. (6 i,j The closed se of Eqs. (4, (3 and (5 ae solved numeically using he Fully Implici Numeical Scheme (FINS [] and Tidiagonal Maix Algoihm (TDMA [] fo he soluions of u, v and u v. Thee second-ode upwind inepolaion is used fo convecive coefficiens of Eq. (3. The soluions of p, u u v v and w w ae obained fom Eqs. (7-( by wiing hem in finie diffeence fom excep Eq. (9 in he inegal fom fo he soluion of v v. In bounday laye flow, he soluion of w w is obained by using Eq. (. 4.. Gid Specificaions Schemaic of he compuaional domains fo he bounday laye is L x =.4m and L y =.6m in x- and y-diecions as seen in Fig. and fo he je flow L =d and L x =3d in and x- diecions whee d=4m is he je diamee a he exi as in Fig.. Gid spacing is unifom in x- and vaiable in y- diecions such ha y j+ =Ky j and y =L y (K-/(K nj - whee K =. fo he bounday laye and K = fo he je flow. Hee all he flow popeies ae locaed a he same gid poin as he pessue gadien is absen in he se of pimay closed fom equaions o be solved. The unde-elaxaion facos used fo u, v and N ae equal o.7 fo each. The used numeical scheme is second ode accuae and found o povide conveged soluion in 9 ieaions which is accuae o six decimal places fo he mean velociy u / uo. To avoid he necessiy of special gid specificaions nea he wall o accoun he seep vaiaion of flow popeies, ceain vaiaion of u y is assumed wihin y/<5 in accodance wih Townsend [3] and Klebanoff [4] daa, insead of in Eq. (4, whee =.4 is he von Kaman consan and u is he ficion velociy. 4.. Gid Convegence Tes Gid convegence es is caied ou wih he hee diffeen gid sizes emed as coase, medium and fine fo nixnj equal o 43x6, 46x66 and 5x7 whee ni and nj ae he numbe of gid poins in x and y-diecions fo he bounday laye flow, and x, x34 and x45 in x and -diecions fo he je flow. Figue 3 shows he ansvese pofiles u / uo of he bounday laye a x/l= fo he hee diffeen gid esoluions wih R e =8x 5 whee R e =u o L/ is he Reynolds numbe. Figue 4 exhibis he adial pofiles u / uo of he ound je wih R e =3 4 a he locaion of x/d=3 fo he hee diffeen gid esoluions whee R e =u o d/. Gid efinemen shows successful convegence wih he hee gid sizes. The esuls pesened in his pape ae obained by using he fine mesh. u/u o Figue 3. Mean velociy pofiles a x/l=. Gid poins ninj: 436 (, 4666 (, 57 ( u/u o 3 y/l.5. Figue 4. Mean velociy pofiles a x/d=3. Gid poins ninj: (, 34 (, 45 ( 5. Resuls and Discussion The unclosed se of equaions govening he ubulen flow, coninuiy and RANS, is made closed using he addiional equaions due o NMA. The closed se of Eqs. (4, (3 and (5 is solved boh fo he bounday laye and fee je flows fo he given iniial and bounday condiions using FINS and TDMA. Deails of he scale faco S fo calculaing u v, p, u u, v v and w w fom Eq. (9 and Eqs. /d
3 C f, 5H 3 Mohammed A. Azim: A New Mahemaical Appoach o he Tubulence Closue Poblem (7-(, especively, ae descibed in he nex subsecions. The flow developmen, mean velociy, ubulen shea and nomal sesses, and mean saic pessue fom he pesen simulaion fo boh ypes of flow ae pesened in his secion, and compaed wih he expeimenal daa of Klebanoff [4] fo R =75, Mulis e al. [5] fo R =5, Coles [6] fo R =5 and McQuaid [7] wih lile ai injecion v w / u o 3 fo R =5 fo he bounday laye flow, and compaed wih he expeimenal daa of Fellouah e al. [8] fo R e =3 4, Fellouah and Pollad [9] fo R e =3 4 and Hussein e al. [] in he self-simila egion (x/d 3 fo R e =9.6 4 fo he je flow. 5.. Bounday Laye Flow Simulaion is made fo he ubulen bounday laye due o ai flow ove he fla plae wih zeo pessue gadien a R =7 (R e =8x 5 whee R =u o / is he Reynolds numbe and is he momenum hickness. The shapes of. u / U and / L ae idenical [] o x ha endes Eq. ( o S=C. The numeical scheme is found o convege o a sable soluion fo C =3. which appeas in Eq. (5 hough f i, j. 5... Flow Developmen Developmen of he bounday laye is shown schemaically in Fig. whee is hickness is locaed a u /u o =.99. The gowhs of and epesened by he Reynolds numbes R and R ae shown in Fig. 5 agains x/l whee R =u o /. Thee he saicase bounday laye edge obained fom he pesen simulaion is shown by he bes fi. I seems fom he figue ha bounday laye hickness is one ode highe han momenum hickness which is consisen wih he heoeical esuls []. -4 R, -3 R 3..4.6.8 Figue 5. Seamwise gowh of he bounday laye. R (, R ( Figue 6(a shows he seamwise vaiaion of he ficion C f w / u o x/l coefficien agains R whee w u is he wall shea sess. Mualis e al. [5] daa of ficion coefficien added in he figue fo compaison ae seen in accepable ageemen wih ha of he pesen simulaion. Figue 6(b shows he seamwise vaiaion of he shape faco H agains R whee H is he aio of displacemen hickness o momenum hickness wih hei usual definiions. The value of H in he figue indicaes a ubulen flow alhough he Reynolds numbe is low. Compaison shows ha Coles [6] daa ae in good ageemen wih ha of he pesen simulaion. Figue 6. Seamwise vaiaion of (a ficion coefficien: pesen (, Mualis e al. daa [5] (, (b shape faco: pesen (, Coles daa [6] (o u/u o 5 5 5 5 3 4 Figue 7. Mean seamwise velociy pofiles a x/l:.3 (, (, ( Mean seamwise velociy u / uo fo he bounday laye is ploed in Fig. 7 agains he ansvese disance y/l a he seamwise locaions x/l=.3,.6 and. As he seamwise disance inceases, he bounday laye gows and velociy wihin i deceases due o inceasing loss of momenum a he wall. Dimensionless mean velociy u u / u * is shown agains y uy / in Fig. 8 a hee diffeen locaions x/l=.3,.6 and. Mean velociy pofiles follow he log-law u ny / A, wih consans =.4 and A=5 which ae close o hei widely known values. The velociy pofiles have he logaihmic egion beween y 3 and y 6 whee he uppe limi depends on he Reynolds numbe of he flow. y/l R
-u'v'/u * u + (v/u o /(/L Ameican Jounal of Fluid Dynamics 6, 6(: 7-4 33 3. 5 y +.5. y/ Figue 8. Mean velociy pofiles in semi-log axes. Lines as in Fig. 7. Log-law ( Figue. Mean ansvese velociy pofiles. Lines as Fig. 7 5... Flow Popeies Mean seamwise velociy u is ploed in similaiy co-odinaes in Fig. 9 a x/l=.3,.6 and using he velociy scale u o and lengh scale. The scalings ae found o collapse he velociy pofiles vey well. The u velociy pofile of he pesen esuls is compaed wih Klebanoff [4] daa and found o be in good ageemen. Mean ansvese velociy / / / L v is ploed in he similaiy co-odinaes v u o and y/ in Fig. which shows good collapse of he ansvese velociy pofiles igh fom he wall ouwad. In he figue, hese velociy pofiles show an ouwad flow a all x- locaions of he bounday laye. I is noewohy ha he plo of v -velociy pofiles does no collapse in similaiy co-odinaes v / uo and y/ (no shown which is consisen wih he exising lieaue []. u/u o..4.6.8. Figue 9. Mean seamwise velociy pofiles. Lines as Fig. 7. Klebanoff daa [4] (o y/.8.6.4...4.6.8. Figue. Reynolds shea sess pofiles a x/l:.3 (,.6 (, (. Klebanoff daa [4] (o Figue displays he collapse of u v pofiles in similaiy co-odinaes uv / u and y/ igh fom he wall a he posiions x/l=.3,.6 and. This shea sess is calculaed using he eddy viscosiy fomula N u y uv. (7 Compaison shows ha Klebanoff [4] daa ae in good ageemen wih he pesen simulaion. Again, he shea sess is calculaed by inegaing Eq. (9 along wih he coninuiy equaion as x y/ uv u v / y v vdx (8 whee S=C no shown in he equaion. This shea sess is depiced in Fig. agains y/. I is appaen ha calculaed shea sess wih S=3. is in excellen ageemen wih Klebanoff [4] daa and hus validaes Eq. (9 as an addiional equaion o achieve ubulence closue.
3 (p-p w /u o -u'v'/u * u' i /u* 34 Mohammed A. Azim: A New Mahemaical Appoach o he Tubulence Closue Poblem 5.8.6.4...4.6.8. y/ 4 3..4.6.8. y/ Figue. Reynolds shea sess by Eq. (9. Lines and symbol as in Fig. - - -3 Figue 3. Mean saic pessue pofiles. Lines as in Fig.. McQuaid daa [7] (o Soluion fo mean saic pessue wih efeence o he wall saic pessue p w is obained fom Eq. (7. The p / is ploed in Fig. 3 agains nomalized pessue u o y/ a he seamwise posiions x/l=.3,.6 and. The S 4 / L close o he wall obained soluion equies and S.7 / L away fom he wall. The pofile of p p w / u o a x/l= (R =7 is compaed wih he expeimenal daa [7] a R =6 and found in qualiaive ageemen whee he disageemen is due o lowe p w in he expeimen induced by he fluid injecion. The soluions of u u and v v ae obained fom Eqs. (8 and (9, especively, fo S =5 and S =.3. The soluion of..4.6.8. y/ w w is obained using Eq. (. Figue 4 displays he sesses u u, v v and w w as funcions of y/ a x/l=. Klebanoff [4] daa fo he nomal sesses added fo compaison exhibi good ageemen wih ha of he pesen simulaion. Figue 4. v v / u Tubulen sess pofiles a x/l=. (, w w / u (. Klebanoff daa [4] (o 5..3. Esimaion of Scale Faco Scale faco S is esimaed fo calculaing / u (, u u uv fom Eq. (9, p, u u and v v fom Eqs. (7-(9 by compaing he odes of magniude of he shea sess em wih he ineia as S=C, compaing he pessue wih he shea sess as S=C (/L, compaing he axial nomal sess wih he pessue as S=C 3 and compaing he ansvese nomal sess wih he pessue as S=C 4, especively, whee C, C, C 3 and C 4 ae consans. Equaion (7 fo p possesses paameic scale faco while Eqs. (8 and (9 fo u u and v v possess consan scale faco. As a esul, he soluion of p equies a small value of C nea he wall and a lage value away fom he wall due o he pesence of viscosiy dominaed flow scales nea he wall and ubulence dominaed one away fom he wall. 5.. Round Je Flow Simulaion is made fo he ound fee ai je wih 4 mm diamee and op-ha velociy a he exi wih R e =3 4. The numeical scheme is found o convege o poduce a sable soluion fo. 5 S x / d a x/d 5 and fo S= a x/d >5 whee S appeas in Eq. (5 hough f i, j. Such expessions fo S come fom he educion of Eq. ( a x/d 5 due o u /U ~x and /L~x -, and a x/d >5 due o he consan values of boh u /U and /L. 5... Flow Developmen In he pesen simulaion, iniial egion eminaes a x/d=5. a which u c =.98u o and inemediae egion eminaes a x/d5 beyond which he je gows linealy. The obained locaion fo he lae egion is close o he daa of Fedman e al. [3] fo R e =.4 4 and Xu and Anonia
/ /d Ameican Jounal of Fluid Dynamics 6, 6(: 7-4 35 [4] fo R e =8.6 4. Diec numeical simulaion of a ound je pefomed by Boesma e al. [5] fo R e =.4 3 shows ha self-simila sae of he Reynolds sess appeas a x/d35, alhough geneal consensus [6] is ha fully developed egion occus appoximaely a x/d3. The condiions fo self-similaiy obained fom axial RANS equaion ( by neglecing he viscous effec dicae U=U(x o consan [7] bu pesence of he pessue o nomal sess gadien in he equaion esics U o be consan. Axial decay of he ceneline mean velociy u c is shown in Fig. 5 ha follows he invese elaion wih downseam disance given by uo / u c Ad x x (9 o whee A is he mean velociy decay consan. Expeimenal daa [8] added fo compaison indicae somewha highe decay of he mean velociy. Cuen simulaion of he je is found o yield A= 6.5 and x o = -.69d which ae close o he exising daa [3, 8]. Gowh of he je half-widh / is ploed in Fig. 6 whee / is he adial disance fom he je axis o which mean velociy is half of he ceneline velociy. Pesen simulaion illusaes linea gowh of he half-widh as / x / d. 8 / d B (3 fo B=8 a x/d>5. Fellouah and Pollad daa [9] povided fo compaison exhibi close ageemen wih ha of he pesen simulaion. Mean axial velociy u / uo is pesened in Fig. 7 agains he adial disance /d a he axial locaions x/d=3, and 5. As he axial disance inceases, he je gows due o enainmen of he ambien fluid and is maximum velociy deceases due o he loss of momenum by he ineacion wih he ambien fluid. 6 4 u/u o u o /u c 5 5 5 3 x/d 3 /d Figue 7. Mean axial velociy pofiles a x/d: 3 (, (, 5 ( Figue 5. Ceneline mean velociy. Pesen (, Fellouah e al. daa [8] (o 3 5 5 5 3 x/d Figue 6. Gowh of je half-widh. Pesen (, Fellouah and Pollad daa [9] (o 5... Flow Popeies Radial pofiles of mean axial velociy u / uo ae ploed in Fig. 8 a axial posiions x/d=3, and 5. Fellouah e al. [8] daa added fo compaison show easonable ageemen wih hose of he pesen simulaion. Again, adial pofiles of his velociy ae displayed in Fig. 9 in similaiy co-odinaes u / uc and /(x-x o a x/d=, and 3, and found in good collapse. Hussein e al. [] daa of u -velociy ae found in accepable ageemen wih he pesen esuls. Mean adial velociy v is ploed in Fig. in similaiy axes v / uc and /(x-x o a x/d=, and 3. The v -velociy in Pope [] fom Hussein e al. lase dopple anemomee daa included fo compaison ae found in easonable ageemen wih ha of he pesen simulaion bu un off a /(x- x o.5 and he velociy becomes negaive. This is because of moe decay of he je in Hussein e al. (A - =.7 causes moe enainmen of he ambien fluid owads he je ceneline compaed o he pesen simulaion (A - =.6.
-u'v'/u c 36 Mohammed A. Azim: A New Mahemaical Appoach o he Tubulence Closue Poblem u/u c 3 /d Figue 8. Mean axial velociy pofiles. Lines as in Fig. 7. Fellouah e al. daa [8] (x/d: 3 (o, (o, 5 (o seen in saisfacoy ageemen wih Fellouah e al. [8] daa, alhough he sess level is somewha diffeen in he simulaion. Lowe decay of he je in he pesen simulaion (A - =.6 han he expeimen (A - =.8 causes lowe level of u o as appeas in he figue fo u v / u c a x/d=3 u v / bu nomalizaion by u c masks his fac a x/d= and 5. This shea sess is ploed again in Fig. agains /(x-x o a x/d=-3. Daa of Fellouah e al. [8] and Hussein e al. [] added fo compaison show close ageemen wih ha of he pesen simulaion. u/u c 3 /d 5..5 /(x-x o Figue. Reynolds shea sess pofiles a x/d: 3 (, (, 5 (. Fellouah e al. [8] (x/d: 3 (o, (o, 5 (o Figue 9. Mean axial velociy pofiles a x/d: (, (, 3 (. Hussein e al. daa [] (o -u'v'/u c v/u c 5..5 /(x-x o - 5..5 /(x-x o Figue. Mean adial velociy pofiles. Lines as in Fig. 9. Daa fom Pope [] (o Reynolds shea sess u v / u c is depiced in Fig. agains /d a x/d=3, and 5. This shea sess is calculaed using eddy fomula in Eq. (7. The pofiles of u c ae u v / Figue. Reynolds shea sess pofiles a x/d: (, (, 3 (. Fellouah e al. [8] (o, Hussein e al. [] (o In ode o seek validiy fo pimay closue, u v is calculaed fom Eq. (9 fo S=43 in he iniial egion, x / d. 5 S 75 in he inemediae egion and x / d S in he developed egion whee S fo diffeen egions ae obained by simple compaison of magniudes of he appopiae ems as shown in he nex secion. This calculaed shea sess is seen o incease apidly in he downseam fom x/d=-3 as shown in Fig.
-u'v'/u c ( p-p o /u c Ameican Jounal of Fluid Dynamics 6, 6(: 7-4 37 3 indicaing ha S canno accommodae he axial evoluion of ubulen shea sess unlike he bounday laye flow. This is because ubulen sesses gadually incease and hen decease along he je flow, i.e. hei slopes change fom posiive o negaive conay o he bounday laye flows as obseved in expeimens and simulaions (e.g. [9, 3]. While S.6 x / d obained by compaing he magniudes of he ems as above bu wih S insead of +S a x/d>5 (exemplified lae o accoun such negaive slope effec (NSE poves o be able o pedic he evoluion of he shea sess as demonsaed in Fig. 4. Fellouah e al. [8] and Hussein e al. [] daa added fo compaison ae found in good ageemen wih he pesen simulaion.. 5 in he iniial egion, S 35 x / d in he inemediae egion and S. x / d in he developed egion. Hee S fo he developed egion of he je is obained by aking NSE because pessue also changes slope a some disance downseam along he flow. Figue 6 displays p / u c as a funcion of /(x-x o a x/d=-3 whee he ceneline pessue is seen o decease in he downseam as may be obseved fom Quinn [9] daa. 6 4-3 /d 5..5 /(x-x o Figue 5. Mean saic pessue pofiles a x/d: 3 (, (, 5 ( Figue 3. Reynolds shea sess by Eq. (9 wihou NSE. Lines as in Fig. 3 ( p-p o /u c -5 -u'v'/u c -. 5..5 /(x-x o Figue 6. Mean saic pessue pofiles a x/d: (, (, 3 ( 5..5 /(x-x o Figue 4. Reynolds shea sess pofiles by Eq. (9 wih NSE. Lines as in Fig.. Daa of Fellouah e al. [8] (x/d: (o and Hussein e al. [] (o Soluion fo he mean saic pessue is obained fom Eq. (7 fo posiive pessue a he je exi wih efeence o he ambien pessue p o. The nomalized pessue p / u c is shown in Fig. 5 agains /d a he axial posiions x/d=3,. 5 and 5. The soluions ae obained fo S 5 x / d Soluion of he nomal sess u u is obained fom Eq. (8 along wih he coninuiy equaion fo he. 5 scale facos S 8 S 3 x / d. 5 x / d fo he iniial egion, fo he inemediae egion and S=-6 fo he developed egion consideing NSE by equaing he odes of magniude of he pope ems. The nomalized sess u c u u / is ploed in Fig. 7 agains /d a x/d=3, and 5. Fellouah e al. [8] daa of he nomal sess povided fo compaison ae found in excellen ageemen wih hose of he pesen simulaion. Obained soluion wih NSE a x/d>5 poves o be able o pedic he
u' /u c u' /u c 38 Mohammed A. Azim: A New Mahemaical Appoach o he Tubulence Closue Poblem axial evoluion of he nomal sess as obseved in Fig. 8. While calculaed nomal sess wihou NSE, like he shea sess, ae seen o incease apidly in he downseam fom x/d=-3 (no shown. Fellouah e al. [8] and Hussein e al. [] daa included hee fo compaison ae found in easonable ageemen wih he pesen simulaion. displays v v / u c agains /(x-x o a x/d=-3. Fellouah e al. [8] and Hussein e al. [] daa included fo compaison show fai ageemen wih ha of he pesen simulaion..5 8. 6 5 v' /u c 4 3 /d 3 /d Figue 7. Axial nomal sess pofiles. Lines as in Fig. 5. Fellouah e al. daa [8] (x/d: 3 (o, (o, 5 (o Figue 9. Radial nomal sess pofiles a x/d: 3 (, (, 5 (. Fellouah e al. [8] (x/d: 3 (o, (o, 5 (o.5 8. 6 5 v' /u c 4 Figue 8. Axial nomal sess pofiles wih NSE. Lines as in Fig. 6. Daa of Fellouah e al. [8] (x/d: (o and Hussein e al. [] (o Soluion of he nomal sess v v is obained by inegaing Eq. (9 along wih he coninuiy equaion fo S.4 5..5 x / d. 5 in he iniial egion, S=75 in he inemediae egion and S=56 in he developed egion. The scale facos ae deemined by compaison of he magniudes of he suiable ems of he equaion whee v v does no appea as axial deivaive and does no equie o conside NSE fo he developed egion. The sess v v / u c is pesened in Fig. 9 agains /d a x/d=3, and 5. Fellouah e al. [8] daa of /(x-x o v v / u c (aken hee abou wice of hei v /u being he values ae somewha less han he c expeced added fo compaison ae found in excellen ageemen wih hose of he pesen simulaion. Figue 3 5..5 Figue 3. Radial nomal sess pofiles a x/d: (, (, 3 (. Daa of Fellouah e al. [8] (x/d: (o and Hussein e al. [] (o Soluion of he nomal sess w w / u c is obained fom Eq. ( along wih he coninuiy equaion fo S= a x/d 3. The sess w w / u c is depiced in Fig. 3 agains /d a x/d=3, and 5. The pofiles of w w / u c ae displayed in Fig. 3 agains /(x-x o a x/d=-3 whee Hussein e al. [] daa added fo compaison ae found in accepable ageemen. 5..3. Esimaion of Scale Faco In ode o esimae he scale faco, nonlinea gowh of he je in he developing egion (x/d 5 is assumed as /(x-x o ~ x and linea gowh in he self-simila egion
Ameican Jounal of Fluid Dynamics 6, 6(: 7-4 39 (x/d>5 as ~ x x o in accodance wih he lieaue [4, 3]. The assumpion of linea gowh of he je a x/d>5 is in excellen ageemen wih ha of he pesen simulaion. Expeimenal and compuaional daa (e.g. [9, 3] show ha u u inceases apidly as linea in x a x/d 5 and hen deceases as x - a x/d>5 indicaing he axial vaiaion of he flucuaing velociy scale u fo he same egions. In deemining S fo Eq. (9, equaing of he odes of magniude of u v / x and v fo he viscosiy dominaed iniial egion (x/d 5 yields S=C, and equaing of u / x v and uv / x fo 5 x/d 5 yields S=C x.5. Then compaison of he magniudes of u / x uv / x along wih NSE fo x/d>5 povides C / xl S u / L / L v and (3 o which on subsiuion of he scale funcions becomes S=-C /x. Scale faco is esimaed fo calculaing p, u u, v v and w w fom Eqs. (7-(, especively, by he compaison of magniudes of he appopiae ems. In esimaing S fo Eq. (7, compaison of he pessue and shea sess fo x/d 5 povides S=C /x wih wo diffeen values of C fo he iniial and inemediae egions and equaing of he pessue and ineia along wih NSE povides S=-C /x fo x/d>5. In evaluaing S fo Eq. (8, compaison of he nomal sess and shea sess fo x/d 5 povides S=C 3 /x wih wo diffeen values of C 3 fo he iniial and inemediae egions and equaing of he nomal sess and pessue fo x/d>5 along wih NSE yields S=-C 3. Evaluaion of S fo Eq. (9 is made by simply equaing he adial nomal sess and shea sess as S=C 4 x fo he iniial egion and equaing he nomal sess and pessue yields S=C 4 wih wo diffeen values of C 4 fo he inemediae and developed egions. S is evaluaed fo Eq. ( by equaing he azimuhal and adial nomal sesses as S=C 5 wih a single value because axisymmey equies ha v v ww ove he enie je flow. w' /u c w' /u c 8 6 4 3 Figue 3. Azimuhal nomal sess pofiles. Lines as Fig. 9 8 6 4 /d 5..5 /(x-x o Figue 3. Azimuhal nomal sess pofiles. Lines as Fig. 3. Hussein e al. daa [] (o 6. Fuhe Discussion Compuaional esuls show ha C = 3. fo he addiional Eq. (9 in he bounday laye flow implying ha u v / x and uv / x ae of equal odes of magniude bu u / x v > uv / x while C = a x/d>5 fo he je flow implying ha u / x v and uv / x ae of equal magniudes. Moeove, v is seen lage han uv / x in he iniial egion of he je a he ime of calculaing he shea sess fom Eq. (9. In mos ubulen shea flows, Reynolds nomal sesses bea he elaion u u vv ww. (3 Using he ubulen sucue paamee uv / u u vv ww. 5 afe Badshaw e al. [33] and he shea v afe coelaion coefficien u / u u v v. 5 Klebanoff [4] along wih Eqs. (8 and (3, he condiions uv / uu.3, uv / vv. 8, uv / p can be deived fo he bounday laye flow. In he fee shea flow, he values of.35 and.4 fo he ubulen sucue paamee and he shea coelaion coefficien afe Pope [] give he condiions uv / uu. 7, uv / vv. 6, uv / p.4. Resuls fom he pesen simulaions a he las axial posiion of hei calculaion domains ae seen o closely saisfy he above condiions acoss some facion of he shea laye fo boh ypes of flow. I is o be noed fo he bounday laye flow ha in addiion o he ageemen wih Klebanoff [4] daa, he esimae of w w using Eq. ( ae
4 Mohammed A. Azim: A New Mahemaical Appoach o he Tubulence Closue Poblem found in excellen ageemen by DeGaaff and Eaon [3] wih hei expeimenal daa. 7. Conclusions Two-dimensional coninuiy and RANS equaions fo he mean moion occu wih six unknowns fo he bounday laye flow and wih seven unknowns fo he je flow. In his sudy, one addiional equaion fo he bounday laye flow while wo addiional equaions fo he je flow ae obained fom he ansvese RANS equaion using NMA and wo ohe equaions ae obained fom he seamwise RANS equaion using exended NMA o achieve he ubulence closue. The closed fom equaions ae solved numeically ha povide soluions fo he mean seamwise velociy, mean ansvese velociy, mean saic pessue, and Reynolds shea sess and nomal sesses. Resuls exaced fom he simulaion of boh ypes of flow ae found in oveall ageemen wih he exising expeimenal daa ha poves he effeciveness of NMA in achieving he complee ubulence closue. REFERENCES [] I. Newon, Mehods of seies and fluxions, The Mahemaical Pape of Isaac Newon III (67-673, Cambidge Univ. Pess, Cambidge, D.T. Whieside ed. 5-7 (969. [] M. Kuskal, Asympoology in mahemaical models in physical sciences, Poc. Conf. a Univ. of Noe Dame, S. Dobo ed., Penice-Hall, New Jesey, 963. [3] C.M. Bende and S.A. Oszag, Advanced Mahemaical Mehods fo Scieniss and Enginees (McGaw- Hill, New Yok, 978. [4] T. Fishaleck and R.B. Whie, The use of Kuskal-Newon diagams fo diffeenial equaions, Pinceon Plasma Phys Lab-489, New Jesey (8. [5] O. Reynolds, On he dynamical heoy of incompessible viscous fluids and deeminaion of he cieion, Phil. Tans. Roy. Soc. A 86, 3 (895. [6] P.A. Dubin and B.A.P. Reif, Saisical Theoy and Modeling fo Tubulen Flows (John Wiley and Sons,. [7] L. Pandl, Übe die ausgebildee ubulenz, ZAMM 5, 36 (95. [8] W.P. Jones and B.E. Launde, The pedicion of laminaizaion wih a wo equaion model, In. J. Hea Mass Tansfe 5, 3 (97. [9] J. Boussinesq, Theoie de l'ecoulemen oubillan, Mem. Pes. Acad. Sci. Ins. F. 3, 46 (877. [] S.B. Pope, Tubulen Flows (Cambidge Univesiy Pess, Cambidge,. [] D.A. Andeson, J.C. Tannehill and R.H. Pleche, Compuaional Fluid Mechanics and Hea Tansfe (McGaw-Hill, NewYok, 984. [] L.H. Thomas, Ellipic poblems in linea diffeence equaions ove a newok, Wason Sci. Compu. Lab. Repo, Columbia Univesiy, New Yok, 949. [3] A.A. Townsend, The sucue of he ubulen bounday laye, Poc. Cambidge Phil. Soc. 47, 375 (95. [4] P.S. Klebanoff, Chaaceisics of ubulence in bounday laye wih zeo pessue gadien, NACA TN-378 (954. [5] J. Mualis, H.M. Tsai and P. Badshaw, The sucue of bounday layes a low Reynolds numbes, J. Fluid Mech., 3 (98. [6] D. Coles, Measuemens in he bounday laye on a smooh fla plae in supesonic flow, I. The poblem of he ubulen bounday laye, Cal. Ins. Tech. JPL Repo no. -69 (953. [7] J. McQuaid, Expeimens on incompessible ubulen bounday layes wih disibued injecion, ARC Repo no. 3549 (967. [8] H. Fellouah, C.G. Ball and A. Pollad, Reynolds numbe effecs wihin he developmen egion of a ubulen ound fee je, In. J. Hea and Mass Tansfe 5, 3943 (9. [9] H. Fellouah and A. Pollad, The velociy speca and ubulence lengh scale disibuions in he nea o inemediae egions of a ound fee ubulen je, Phys. Fluids, 5 (9. [] H.J. Hussein, S.P. Capp and W.K. Geoge, Velociy measuemens in a high Reynolds numbe, momenum conseving axisymmeic ubulen je, J. Fluid Mech. 58, 3 (994. [] H. Schliching and K. Gesen, Bounday Laye Theoy (Spinge, Heidelbeg,. [] M.A. Azim, On he sucue of a plane ubulen wall je, ASME J. Fluids Eng. 35(8, 845 (3. [3] E. Fedman, M.V. Ougen and S. Kim, Effec of iniial velociy pofile on he developmen of he ound je, J. Popul. Powe 6, 676 (. [4] G. Xu and R.A. Anonia, Effec of diffeen iniial condiions on a ubulen ound fee je, Exp. Fluids 33, 677 (. [5] B.J. Boesma, G. Behouwe and F.T.M. Nieuwsad, A numeical invesigaion on he effec of inflow condiions on he self-simila egion of a ound je, Phys. Fluids, 899 (998. [6] H.E. Fielde, Conol of fee ubulen shea flows, Flow Conol: Fundamenals and Pacices (Spinge-Velag, Gemany, 998. [7] H. Tennekes and J.L. Lumley, A Fis Couse in Tubulence (MIT Pess, Massachuses, 97. [8] N.R. Panchapakesan and J.L. Lumley, Tubulence measuemens in axisymmeic jes of ai and helium. Pa : ai je, J. Fluid Mech. 46, 97 (993.
Ameican Jounal of Fluid Dynamics 6, 6(: 7-4 4 [9] W.R. Quinn, Upseam nozzle shaping effecs on nea field flow in ound ubulen fee jes, Euo J. Mechanics B/Fluids 5, 79 (6. [3] D.B. DeGaaff and J.K. Eaon, Reynolds numbe scaling of he fla plae ubulen bounday laye, J. Fluid Mech. 4, 39 (. [3] P.C. Babu and K. Mahesh, Upseam enainmen in numeical simulaions of spaially evolving ound jes, Phys. Fluids 6(, 3699 (4. [33] P. Badshaw, D.H. Feiss and N.P. Awell, Calculaion of bounday laye developmen using he ubulen enegy equaion, J. Fluid Mech. 8, 593 (967. [3] L. Boguslawski and C.O. Popiel, Flow sucue of he fee ound ubulen je in he iniial egion, J. Fluid Mech. 9, 53 (979.