Proceedngs of the th WSEAS Internatonal Conference on Flud Mechancs and Aerodynamcs, Elounda, Greece, August -, (pp-) Consderaton of D Unsteady Boundary Layer Over Oscllatng Flat Plate N.M. NOURI, H.R. AHMADI FAKHR, H. Madad, R. Abdollahpour Appled Hydrodynamcs Lab., Mechancal Engneerng Dept. Iran Unversty of Scence and Tech. Narmak, -, Tehran Iran mnour@ust.ac.r Abstract: - The unsteady boundary layer due to small ampltude snusodal oscllaton of a plate n vscous ncompressble flud s nvestgated here usng Random Vortex Method. Whle the plate oscllates n ts own plane. The unsteady boundary layer causes the unsteady velocty profle and shear waves propagaton. The numercal result s compared wth analytcal soluton for the case thahe oscllaton ampltude s small enough to neglect nonlnear convectonal term. The results of RVM for unsteady boundary layer show good smlarty confrmng the ablty of the proposed method. The nonlnear convectonal term can also be taken n to account n RVM, n the cases thahey can not be neglected. Key-Words: - Vscous flud; Oscllatng wall; Unsteady flow; Transent flow Introducton The moton of vscous flud caused by snusodal oscllaton of a flat plate s termed as stocks second problem by schlchtng []. It s not only of fundamental theoretcal nterest bu also occurs n many appled problems; Such as acoustc streamng around oscllatng body []. As early as, M. Emn Erdogan has consdered the flow of an ncompressble vscous flud caused by the small ampltude oscllaton of the plane wall [, ]. Ths moton wll produce, far from the body, acoustc wave of small ampltude. the flow near the body wll, n general, have normal and tangental velocty component relatve to the body. On the body s surface the normal velocty component s fxed by the requremenhahere be no flow through the boundary. And also when the vscosty effects are taken nto accounhe flud n contact wth the body can no longer slp over the body; Instead, t adheres to t. Ths s nohe only effect of vscosty, for n the same way that s precludes slp between flud and sold, t also prevents complete slppage between contguous layers of flud. Therefore, no slp condton at a boundary wll make the whole tangental velocty profle sgnfcantly dfferent from whch would exst f the flud were nvscd. The propagaton of shear waves and unsteady boundary layer are analyzed here va Random Vortex Method, In RVM, the Naver Stokes equatons, n the form of vortcty, s splt nto dffuson and convecton parts, accordng to the fractonal step method. A random Walk method s used to solve the dffuson equaton. So unlke the analytcal method usng RVM, the nonlnear convectonal term s also taken nto account. Problem Formulaton To study the moton of nfnte plate a rectangular system of coordnate s attached to the plate s such a manner thahe plane wall s chosen as x-axs and t oscllates n ts own plane, as sketched n fg..because of vscosty the flud above the plane s also move, bu s clear thahe flud velocty wll have only one component, and ths wll be parallel to the velocty of the plane. Further, ths velocty component can not depend on dstance along the plane so that u = u( y, t),,. Therefore. u =, so the momentum equaton yelds: u u + ( u. ) u = ν (). Analytcal soluton [] When the flud s ntally at rest and oscllaton ampltude s small, the nonlnear term s assumed to be n small order. Neglectng the convecton term, lnear form of equaton s obtaned: u u = ν ()
Proceedngs of the th WSEAS Internatonal Conference on Flud Mechancs and Aerodynamcs, Elounda, Greece, August -, (pp-) Ths s a dffuson equaton. Therefore f the flud starts a= and mparts some momentum to the flud n contact wth t, would expechs momentum to be dffused slowly nto the flud. The motons of flud after all transent effects have dsappeared; snce the plane s oscllatng as: ( t U U e % = ) P () The flud velocty s also expected to depend harmoncally on tme. Therefore: t u( y, t) = U% ( y) e () So t yelds: U% ( y) + K U% ( y) = () K = ( + ) % ν () The soluton s: % t ( + ) y δν ( + ) y δν u( y, t) = e ( Ae + Be ) () where: ν δ = ν % (9) Snce for y the velocty must be small, we must set B=. Also at y= the flud velocty s equal to that of the plane, so that A= U and: ( ) (, ) y δ u y t = U e cos ν t y % δ () ν The flud therefore, also oscllates harmoncally n tme, buhe oscllaton lag those of the plane, and has very small ampltude far from the plane. fg. depcts relatve-velocty profles at varous tmes durng one oscllaton. In the fgure tme s measured from the pont durng a cycle when u= U at y=. π It s seen that for % t the maxmum flud velocty ampltude s ahe plane y=. However, for y> n facs locaton n the flud s gven by: ( ) π y t = t δ max ν % () Thus one f the feature of the oscllaton, namely, the pont of maxmum flud velocty s seen to be movng nto the flud wth velocty % δ = % ν.. The Numercal method Equaton () can be wrtten n the form of vortcty: + ( u. ) = ν () Ths equaton called vortcty transport equaton and may be splt nto lnear dffuson and nonlnear convecton equatons accordng to the fractonal step method of Chorn [9,], gvng. ν = ν () = ( u. ) () where s vortcty vector. The dea of the fractonal step method s to solve these equatons sequentally rather than smultaneously. The sequental soluton means that at each tme step the dffuson equaton s solved usng the state of the flow ahe end of the prevous tme step as the new ntal condton. Then the convecton part s solved usng, as the ntal condtons, the soluton of the dffuson equaton for the currenme step. By takng the convectve term nto accounhe nonlnear problems wth large ampltude oscllaton can also be solved. The transport of vortcty due to dffuson n random vortex method s mplemented by dsperson of a fnte number of vortex elements wth fnte and constant vortcty accordng to a -dmensonal Gaussan statstcs. Ths based on the fachahe green functons of -dmentonal form of equaton () s: [] R G( y, t) = exp y πt () t In dentcal to the probablty densty functon of Gaussan random varable η wth a zero mean and a standard devatonσ : P( η, t) = exp η πσ () σ t If σ = The green functon of dffuson equaton n -dmenson s: R G( x, y, t) = exp ( x y ) πt + () t whch s equvaleno: G( x, y, t) = G( x, t) G( y, t) () where G( x, t ) and G( y, t ) have the same form as n equaton ().then the correspondng probablty densty functon s the product of two -dmensonal probablty densty functons: P( η, η, t) = P ( η, t) P ( η, t) (9) So the soluton of equaton () s smulated stochastcally by a -dmensonal dsplacement of vortex elements n two perpendcular drectons usng two sets of ndependent Gaussan random numbers, each have a zero mean and standard Dt devaton of σ =. To construct an algorthm the vortcty n the flow s represented by a number of dscrete vortces, whch are gven a random Gaussan moton, or random
Proceedngs of the th WSEAS Internatonal Conference on Flud Mechancs and Aerodynamcs, Elounda, Greece, August -, (pp-) Dt walk wth zero mean and varance of where Dt s the tme step. These vortces are generated on the surface to satsfy the no slp boundary condton. Such thahe surface of the body s represented by m panels. Each of whch s allocated a vortex dstrbuton of Γ per unt length, Ths vortex dstrbuton s then dscretzed nto a number of pont vortces, such thahe crculaton of each vortex beng less than some prescrbed maxmumγ max and the dstrbuton Is such that made lnear velocty profle on the panels wth respeco y and zero resultanangental Velocty ahe central collocaton pont. The panels heght s chosen small enough to place under the lamnar sublayer so ther lnear velocty profle role as a boundary condton and force the velocty profle to be lnear near the body. The convecton term s then taken nto account wth movng the vortces wth ther nvscd veloctes n the Lagrangan scheme. So the new poston of the vortces due to the convecton and dffuson s gven by: t+ t x = x + u Dt+ η x t+ t () y = y + v Dt+ η y where ( x, y ) and ( u, v ) represented the poston and velocty vector of ' th vortex ame t, and ( η, η ) s Gaussan random translaton vector. The x y vortces velocty ( u, v ) s calculated usng potental velocty around desred geometry and velocty nduced by other vortces. Concluson The problem under consderaton s a snusodal n plane oscllaton of nfnte plate wth velocty ampltude U =. and oscllaton frequency = π, the consdered flud s water wth µ =.9 and ρ =. The plate oscllaton s consdered n two cases: The RVM and analytcal results are compared for small ampltude wall oscllaton n statonary flud, so as assumed n analytcal soluton the convecton term s small enough to be neglected; Comparsons between analytcal and numercal result for ths case n fg., fg. and fg. show the capablty of RVM for unsteady boundary layer consderaton,. ferences: [] H. Schlchtng, Boundary Layer Theory, McGraw-Hll, New York, 9 [] N. Tokuda, on the mpulsve moton of a flat plate n a vscous flud, J. Flud Mech., Vol., 9, pp. -. [] M. Emn Erdogan, A note on an unsteady flow of a vscous flud due to an oscllatng plane wall, Internatonal Journal of Non-Lnear Mechancs, Vol.,, pp. -. [] Denns SCR., The moton of a vscous flud past an mpulsvely started sem-nfnte flat plate., IMA Journal of Appled Mathematcs, Vol., 9, pp.. [] K. Stewartson, On the mpulsve moton of a flat plate n a vscous flud (Part I). Quarterly Jnl. of Mechancs & App. Maths., Vol., 9, pp. 9. [] K. Stewartson On the mpulsve moton of a flat plate n a vscous flud(part II). Quarterly Jnl. of Mechancs & App. Maths., Vol., 9, pp.. [] MG Hall., The boundary layer over an mpulsvely started flat plate, Proceedngs of the Royal Socety of London, 99, pp.. [] S. Temkn, Elements of Acoustcs, John Wley & Sons, 9 [9] A.J. Chorn, Numercal Study Slghtly Vscous Flow, J. Flud Mach., Vol., 9, pp.-9 [] G. H. Cottet, P. Koumoutsakos, M. L. Ould.Salh, Vortex Methods wth Spatally Varyng Cores, Journal Of Computatonal Physcs, Vol.,, pp. [] N. R. Clarke, O. R. Tutty, Constructon and valdaton of a dscrete vortex method for the twodmensonal ncompressble Naver-Stokes equatons, Journal of Computatonal Fluds, Vol. (), 99, pp. -. [] J. S. Marshall, J. R. Grant, Penetraton of a blade nto a vortex core: vortcty response and unsteady blade forces, Journal of Flud Mechancs,Vol., 99, pp. -9. [] N. M. Nour, S. Eslam, Modelng of freestream turbulence wth random vortex method (RVM) n ncompressble flow and soluton for crcular cylnder, 9 th Conference of Flud Mechancs, Shraz, Iran. [] A.F Ghonem and Y.Gagnon, vortex Smulaton of Lamnar calculatng Flow, J. Comp. Phys., Vol., 9, pp.-
Proceedngs of the th WSEAS Internatonal Conference on Flud Mechancs and Aerodynamcs, Elounda, Greece, August -, (pp-) fg. schematc of n plane oscllaton of half plane wall (stocks second problem) fg...... *t= *t= π / *t= π / *t= π / *t= π /...... u /U fg. nondmensonal-velocty profles at varous tmes durng one oscllaton Numercal sult Analytcal sult.... u /U fg. Analytcal and Numercal results of velocty profle nt= π
Proceedngs of the th WSEAS Internatonal Conference on Flud Mechancs and Aerodynamcs, Elounda, Greece, August -, (pp-) Numercal sults Analytcal sult.... u /U fg. Analytcal and Numercal results of velocty profle n π t= fg. Numercal sult Analytcal sult.... u /U fg. Analytcal and Numercal results of velocty profle n π t=