An Analytic Potential Solution for Incompressible 2D Channel Inviscid Flow with Wall Injection

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1 A Alytic Potetil Solutio for Icompressible D Chel Iviscid Flo ith Wll Ijectio Coreliu BERBENTE, Steri DĂNĂILĂ Correspodig uthor Deprtmet of Aerospce Scieces, POLITEHNICA Uiversity Buchrest Spliul Idepedeţei, 0600, Buchrest, Romi berbete@yhoo.com, steri.dil@upb.ro DOI: 0./ Abstrct: The preset pper itroduces lyticl potetil solutio for the icompressible flo i D chel ith orml ll ijectio. This solutio is cpble to support resoble ijectio flo rtes s compred to the geerl flo rte i the chel. A strtegy to fid solutios for loger chels i cse of icresig ijectio velocity ith the distce from the etrce, usig smll umber of the solutio terms is poited out. Exmples of clcultio re give. Key Words: D chel flo, lytic solutio, ll ijectio.. INTRODUCTION The possibility to obti potetil solutios to icompressible flo ith ijectio s, fter our koledge, ot thoroughly studied. Severl ppers [], [] tke ito ccout vortex solutio geerted by the ijectio itself, lthough the superpositio of eigefuctios do ot stisfy the olier vortex equtio of Helmholtz []. Of course much ttetio is pid to ll ijectio i coectio ith boudry lyer cotrol [;5].. EQUATIONS AND BOUDARY CONDITIONS Oe cosiders the icompressible flo i D chel (Fig.), symmetricl ith respect to the chel xis. The flo is ssumed icompressible, sttiory d potetil. By itroducig the strem fuctio ( z, y), []: v z ; vy y, () z oe hs to solve the prtil differetil equtio for the strem fuctio: 0. () x y I order to solve the problem the boudry coditio hve to be specified. At the etrce, velocity is costt d prllel to chel lls: vz(0, y) vz0 v x () The ijectio velocity is orml to the ll d vrible log the ll: v ( z, ) u ( z). () y O the xis, the symmetry of flo imposes vy (,0) z 0, z 0 (5) INCAS BULLETIN, Volume, Number / 00, pp. 0 5

2 A Alytic Potetil Solutio for Icompressible D Chel Iviscid Flo ith Wll Ijectio Fig.- Geometric cofigurtio The boudry coditios for the ellipticl PDE () should iclude the hole frotier of the chel domi; to prts of the frotier re required by coditios () d (). As coditio t the chel exit, z L, oe ill cosider the exit velocity prllel to the chel xis, ssumptio correspodig to high vlues of L. Therefore: vy ( L, y) 0 (6) Further by the chge of vribles: z z / L, y y /, vx, y (7) here the ubrred e vribles re dimesioless, oe obtis the equtio: (8) 0 L z y the velocities beig give by: (9) v z, v y y z The boudry coditios for the e uko re: (0) 0 for z 0 d 0 y ; y (0b) 0 for z, 0 y ; z (0c) 0 for y 0, 0 z ; z (0d) u( z) for y, 0 z ; z here u ( z ) u ( z)/ v. x. THE ANALYTIC SOLUTION Oe looks for the eq.(7) solutios of the form: (, zy ) ZzYy () () () Z( z) d Y( y) beig obtied by seprtio of vribles z, y [6]. By itroducig (9) i eq. (7) oe obtis to ordiry differetil equtios hich re esily solved uder the form: INCAS BULLETIN, Volume, Number / 00

3 Coreliu BERBENTE, Steri DĂNĂILĂ INCAS BULLETIN, Volume, Number / 00 L L Z( z) Acos( z) Asi( z) ; Y( y) B exp( y) B exp( y) Ai, Bi, i ; d beig rel rbitrry costts. The strem fuctio d the correspodig velocity compoets re: (, ) cos( L L zya z) Asi( z) Bexp( y) Bexp( y) () L L Acos( z) Asi( z) Bexp( y) Bexp( ) y y ; (b) L L L L A si( z) A cos( z) Bexp( y) Bexp( ) z y. (c) Imposig the boudry coditio (0c): L L L L A si( z) A cos( z) ( B B ) 0, () yields: B B. (5) At the etrce, z 0, from (0): A B exp( y) Bexp( y) 0, (6) oe obtis A 0. (7) Replcig (5) d (7) i () results: (, zy ) A si( z )sih( y ) ; (8) A si( z)cosh( y ) ; (8b) y A cos( z)sih( y ). (8c) z here L/. At the exit, z, the trsversl velocity should vish. So, from (8c) results the equtio: cos( ) 0, (9) hvig the solutio: ( ),,,... (0) Becuse the eqs. () d (8) re lier, the geerl solutio is: y Asih( y)si( z) () v A cosh( y)si ( z); () (b) z vy Asih( y)cos( z ) (c)

4 A Alytic Potetil Solutio for Icompressible D Chel Iviscid Flo ith Wll Ijectio here: L vy, ( ),,,... () L z The costts A,,,... re give by the ijectio coditio: Asih( )cos( z) u( z ), () hich ill be ritte s follos: Bcos( z) ( z) () ( z) u( z)/ um ed, BAsih( ) / ume d I the bove reltios, the me ijectio velocity umed u ( z ) dz 0 u med s itroduced, defied by: (b) I order to determie the coefficiets B from (), oe uses the orthogolity properties of cosie fuctios uder the form: 0, for m ; cos( mz)cos( z) dz (6) /,for m. 0 (5). RESULTS I the folloig severl pplictios for differet chel prmeters L/ re preseted. Tkig ito ccout tht the ijected flo rte divided by the geerl flo,, rte c be defied s: r u L L Q umed, vx L r u oe c obti expressio for the me ijectio velocity: umed rq L. (8) I Tble re give the expressios of the coefficiets B A for severl distributios of the ijectio velocity, s ell s the limit forms of the strem fuctio for L/. Q med Tble Coefficiets of series developmets ( z ) B A / u lim med / ( z) ( ) ( ) L ( ) ( ) ( ) L ( ) y rq si z sih ( ) L si( z) y rq sih r Q (7) INCAS BULLETIN, Volume, Number / 00

5 Coreliu BERBENTE, Steri DĂNĂILĂ ( z ) ( ) 6( ) 6( ) ( ) sih L ( ) ( ) y rq si( ) z The distributio ( z ), lthough very simple, hs udefied vlue for vz t z =.This is due to the disgreemet betee the boudry coditios t this oly poit. The ucertity disppers i the limit L/ ( log chel). The other to ijectio ls, hvig ull ijectio velocity t z =, re cocordt d besides led to more rpid covergece rte for the series. Fig..-Ijectio velocity profiles cosiderig terms i series developmets. A solutio usig three terms. Let us cosider the ijectio velocity of the form: (, zk, m) C(, km)cos( z), (9) INCAS BULLETIN, Volume, Number / 00 here the coefficiets re: 5 k m m k C ( ), CC, C k C C. (0) The prmeter k represets the vlue of ( zk,, m ) t chel etrce heres the product m / represets the curve slope t z =. I Fig. the curves ( z,0.5,), ( z,0.,5) d ( z,0,8) re represeted, suggestig grdully icresig of the ijectio up to mximum s oe moves to the chel ed. This cofigurtio c be pproprite to some combustio problems [;]. 5. CONCLUSIONS A potetil solutio (stisfyig ideticlly the Helmholtz vortex equtio ulike some the existig vorticl solutios) is cpble to tke over ijectio (or suctio) orml to ll i D icompressible flo. A lyticl solutio s obtied for sttiory iviscid flo tht c be used further s frotier coditio for clcultio of the boudry lyer. Limit behviours for very log chels re obtied cosiderig fiite rtio of the ijectio flo rte to the geerl oe.

6 5 A Alytic Potetil Solutio for Icompressible D Chel Iviscid Flo ith Wll Ijectio Iterestig ijectio velocity profiles ith grdully icresig up to mximum i the lst prt of the chel c be obtied usig smll umber of series developmet. 6. ACKNOWLEDGEMENTS The preset ork hs bee supported by the CNCSIS UEFISCSU, project umber PNII IDEI 00/007 (cotrct umber 09/007). REFERENCES [] C. Zhou, J. Mdjli, Improved Me Flo Solutios for Slb Rocket Motors ith Regressig Wlls, Jourl of Propulsio d Poer, vol.8, o., pp.70-7, 00. [] J. Mdjli, A.B., Vys, A.B. d G.A. Fldro, Higher Me-Flo Approximtios for Solid Rocket Motor ith Rdilly Regressig Wlls, AIAA Jourl, vol.0, o. 9, pp , 00. [] E. Crfoli, V.N. Costtiescu, Dimic fluidelor icompresibile, Editur Acdemiei, Bucureşti, 98. [] V.N. Costtiescu, Dimic fluidelor vâscose î regim lmir, Editur Acdemiei, Bucureşti, 987. [5] H. Schlichtig, K. Gerste, Grezschicht-Theorie, Spriger, 997. [6] I.Gh. Şbc, ŞABAC, Mtemtici specile, (vol.), Editur Didctică şi Pedgogică, Bucureşti [7] C. Berbete, S. Mitr, S. Zcu, Metode umerice, Editur Tehică, Bucureşti, 997. INCAS BULLETIN, Volume, Number / 00