Control of the Laminar Boundary Layer. around a Profile by Suction

Transcription

1 Adv. Theor. Appl. Meh. Vol. 008 no Control o the Lamnar Boundary Layer around a Prole by Suton A. NAHOUI Retorat de l Unversté El Had Lakhdar Batna Avenue hahd boukhlou Mohamed El had Batna Algéra Enantramz@yahoo.r N. KABOUCHE Retorat de l Unversté El Had Lakhdar Batna Avenue hahd boukhlou Mohamed El had Batna Algéra nkabouhe@yahoo.r L. BAHI Retorat de l Unversté El Had Lakhdar Batna Avenue hahd boukhlou Mohamed El had Batna Algéra Bahl@aramal.om Abstrat A lamnar and two-dmensonal nompressble boundary layer around a prole and ts ontrol usng suton s studed numerally. The study s based on the Prandtl boundary layer model. Usng the method o nte derenes and the Crank-Nolson sheme. The study onerns the lat plate ase and the 00 NACA prole. The veloty dstrbuton the boundary layer thkness and the rton oeent dstrbuton are determned and presented wth and wthout ontrol. The applaton o the ontrol tehnque has demonstrated ts postve eet on the detahment pont and the rton oeent. The ontrol proedure s appled or derent porosty lengths speeds and angles o suton Keywords: boundary layer lamnar two-dmensonal nompressble prole ontrol suton and rton oeent

2 88 A. Nahou N. Kabouhe and L. Bah Introduton The present work s to study the lamnar boundary layer developed by twodmensonal nompressble low o ar around proles arplane wng or vane ompressor or turbne blades rom the equatons o the lamnar boundary layer told Prandtl s equatons and ts ontrol by suton. The equatons governng ths knd o problem derental equatons are nonlnear partal derental. Ther soluton an be only numerally. The boundary layer s the thn area n ontat wth the wall o the prole s the seat o the phenomena ausng vsous drag o rton[] wth the several reent studes estmate that about hal o the total drag. Any reduton o the latter would result n an nrease n perormane or a reduton n energy onsumpton knowng that the ossl uels are sare by the day on the one hand; on the other hand the exessve onsumpton o these reserves s onstantly ollowng nreasng nternatonal demand [3]. One way to aheve ths goal s to montor the boundary layer pushng ts pont o detahment towards the tralng edge by applyng suton [5 6] n order to expand ts area lamnar. The study onerns the lat plate and NACA 00 prole. Fg. : Geometral haratersts o a two-dmensonal prole. Nomenlature x y : Cartesan o-ordnate s : Curvlnear o-ordnate. : Dmensonless o-ordnate : Dmensonless o-ordnate

3 Control o the lamnar boundary layer 89 : Outsde the boundary layer υ : knemat vsosty τ : Tangental onstrant. α : Angle o attak. θ : Angles o suton : Dmensonless parameter ( s) = ( s. du ) ( u ( s) ds) p : Stat pressure. e e. Cp : Coeent o pressure Cp = ( p p ) ( ρ U ). u : Speed Component ollowng x v : Speed Component ollowng y V : Dmensonless speed omponent ollowng y δ : Boundary layer thkness : Stream unton : Dmensonless speed omponent ollowng x Re : Reynolds number U : Veloty Outsde the boundary layer M : Mah number vo /U : Veloty o suton Xp : Porosty lengths. Mathematal modellng. Modellng o the boundary layer The problem o a lamnar boundary layer dynam and two-dmensonal nompressble s governed by [789] v x () p u u u v = υ x y x x y ρ v v p v v u v = υ x y y x y ρ g x g y () (3).. Assumptons o the lamnar boundary layer

4 90 A. Nahou N. Kabouhe and L. Bah The thkness o the boundary layer s small enough rope to the prole (δ << ) The omponent o the longtudnal veloty s greater than the transverse omponent ( v <<u ) The transversal pressure gradent s negleted ( ( p ) ) The ores o weght are negleted. The gradent o the longtudnal veloty n the transverse dreton s muh hgher than the gradent speed transverse ( v ) << ( ) By ntrodung these assumptons we arrve at a sale model o Prandtl ollows: v x p u u v = υ x ρ x () At the entrane u ( x. y) = u( y) (5) v ( x. y) = v( y) (6) u ( x0) (7) v ( x0) (8) u ( x ) = Ue( x) (9) The length o the lamnar boundary layer wth ts endpont pont where detahment or the boundary layer hanges n nature and beomes turbulent s dened aordng to the lassal theory by (0). Boundary layer on the lat plate The model s redued to [ 7 8 9]: v x

5 Control o the lamnar boundary layer 9 u v x u = υ () Among the results o ths study we quote: The thkness o the boundary layer s gven by x δ ( x) =. 9 () R ex Or the Reynolds number s gven by U x R ex = (3) υ The loal rton oeent s gven by 0.66 C ( x) = () R ex.3 On the boundary layer proles NACA.3. Equatons o the boundary layer.3.. Change landmark We propose to study the boundary layer n urvlnear oordnates (s y); wth the oordnated who represents the surae ontour prole and perpendular to the oordnated s. Frst we must make the mathematal model no dmensonal by makng the ollowng hanges [7] s = (5) L = y U ( s) e υ s (6) u ( ) = = Ue (7) v s U e ( s) V ( ) = U ( s) υ (8)

6 9 A. Nahou N. Kabouhe and L. Bah The mathematal model takes the ollowng orm [] ( ) V (9) V = [ ] (0) The parameter depends on the speed dstrbuton potental beyond the boundary layer wth: V = V ( ) () ( 0) () V ( 0) (3) ( ) = () At the entrane ( ) s arbtrarly hosen away rom the edge so the mathematal model s redued to the equaton Falkner-Skan [7] 3 3 (5) Wth the boundary ondtons: (0) = (0) ( ) = (Pont o detahment).3.. Numeral resoluton o the equatons o the lamnar boundary layer Typally solvng non-lnear derental equatons governng physal problems appealed to the approprate numeral methods. In ths ase prevous studes show that nte derene s more sutable [] wth the use o the sheme CRANCK - NICOLSON. Thus the transormaton o derental equatons wth algebra equatons s obtaned ommonly reerred to the modellng.

7 Control o the lamnar boundary layer 93. Modellng o derental equatons.. Modellng o the equaton o momentum The modellng o the equaton o momentum s dong term by term t leads to the nal equaton ater reorganzed terms o the equaton o momentum whh s D C B A = (7) Wth the oeents whh are dened by: ( ) = V A (8) ( ) = B (9) ( ) = C (30)... Modellng o the equaton o ontnuty Smlarly the equaton o ontnuty s ( ) = D C B A V V V V (3) The oeents are determned by = A (3) = B (33) = C (3) = D (35)

8 9 A. Nahou N. Kabouhe and L. Bah.5 Algorthm solvng equatons o the lamnar boundary layer The sortng system dagonal o the equaton modelled o momentum s determned by the algorthm Thomas [3]..6 Modellng o the orgnal prole The ntal prole taxed at ( ) s determned rom the soluton o the equaton o Falkner-Skan [0 ].7. Method Prnple Shootng Ths method allows the passage o a derental problem wth the boundary ondtons wth a problem to ntal ondtons. Indeed t s to adust the ntal ondtons so that the boundary ondtons are met.e. orretons to aheve target [7.0].8 Algorthm resoluton o the mathematal model o the lamnar boundary layer The trouble shootng steps are [7].8. The hoe o the barrer sets through ts leadng edge parameter o the ntal orm. Ths allows the resoluton o the equaton Falkner-Skan.e. obtanng all values aordng to the ntal vertal..8. The determnaton o the speed outsde..8.3 The alulaton o the shape parameter or all statons along the prole..8. The alulaton o the values or all o the orgnal staton..8.5 The resoluton o the equaton o momentum by the algorthm Thomas..8.6 The alulaton o values and speeds or the new staton..8.7 The test the detahment ours we stop the alulatons you go bak to the stage (.8.5). 3 Dsusson o Results The tests o the numer ode developed were onduted on a lat plate as well as a prole NACA00. Dstrbutons o pressure on these proles are determned by searhng the nluene o the angle o attak and Mah number o low undsturbed on the latter the pont o detahment s determned rom the speed proles on the surae prole. A tehnque or ontrollng the boundary layer by suton s appled to test the nluene o several parameters suh as the sope o

9 Control o the lamnar boundary layer 95 aspraton the Mah number the dreton o the angle o eeton and the amount pumped. To valdate the alulaton ode we onduted a omparson o the results o the prole wth sold surae and those obtaned by the method o BLASUIS plane to the plate [ ]. 3. Controllng detahment dependng on the sope o suton By applyng the tehnque to ontrol the boundary layer by suton we an see rom the gure () that the delne rom the pont o detahment towards the tralng edge nreases. By nreasng the sope o aspraton ths development s slower beyond a range o 30% suton o the rope [ ] Suton sope (xp) Poston o detahment (x/) Fg. : Eet o the sope o suton on the verge o detahment on NACA 00. M.3 α and vo / U = Montorng detahment dependng on the angle o suton For angles ntake below 87 taken on a prole NACA00 ther values have no postve eet on the verge o loosenng on the ontrary they enourage detahment but rom ths angle the pont o detahment toward the rearward edge o a remarkable manner the angle s hgh over the urrent detahment s delayed towards the tralng edge up to a value o 80 but 80 angle s not pratal we so ust to the angle o 70.

10 96 A. Nahou N. Kabouhe and L. Bah Angle o suton (θ) Axal dstane (x/) Fg. 3:The eet o the angle o suton on the detahment o a prole NACA00. M = 0.3 and α vo / U = Controllng detahment dependng on the amount pumped For small amounts o lud suked on a range o xp and an angle o 90 eeton the pont o detahment remans unmoved. Then the more we aspre as ther pont o detahment moves to the edge but when you reah a value o 0 pont detahment beomes nsensble to any quantty and more. So we should not aspre too beause the exess low s not useul Quantty suked (vo/u) Axal dstane (x/) Fg Eet o the amount pumped on the verge o separaton. M.3 α and θ = 90

11 Control o the lamnar boundary layer Eet o ontrol by suton on the thkness o the boundary layer The moleular partles adhere better wth the wall by applyng ontrol by suton ths s due to the absorpton o these partles loated ust to the wall allowng the aeleraton o partles above where the thkness o the layer lmt beomes thnner. Thkness o the boundary layer(δ/) NACA 00/ Suton lat plate Axal dstane (x/) Fg. 5: Thkness o the lamnar boundary layer on a porous prole NACA 00 and on alat plate sold. Xp=0.6 M=0.3 α=0 θ=0 and vo/u= Eet o ontrol by suton on the oeent o rton Followng the mplementaton o the suton ontrol over the length o the prole the partles n dret ontat wth the wall aeleratng and the rton oeent s lower than that obtaned wth the same prole sold. 000 Coeent o rton (C) Sutn on the entre wall Sold surae Axal dstane (x/) Fg. 6: Dstrbuton o the oeent o rton on a NACA 00 and wth ontrol plane on a lat plate. Xp.6 M.3 θ and α.

12 98 A. Nahou N. Kabouhe and L. Bah Conluson A numeral study s proposed to analyse the behavour o a lamnar boundary layer and two-dmensonal nompressble around wth proles and porous sold surae. The proles onsdered n ths study are proles NACA00 and lat plate. It has ormulated the mathematal model based on the equatons o Prandtl hosen or the study o the boundary layer. A hange o varables s ntrodued to transorm the system o derental equatons wth two varables nto a sngle non-lnear derental equaton regular n one varable. The resoluton may not be possble as numeral methods thereore t has appealed to the method o Shootng and the algorthm Thomas. The results were used to study the nluene o the sope o aspraton the angle o suton and the amount pumped. The results showed that the suton gves better results whh are a key obetve sought by several researh organzatons avaton. These postve results are onsstent wth the theoretal predtons and valdated by the expermental results obtaned by several agenes n the avaton ndustry. More sope suton s mportant the angle and the amount sphoned are optmal and the results are better. Indeed resultng n a oeent o rton and a lower thkness o the boundary layer thnner. Reerenes [] A. PANTOKRATORAS The Falkner-Skan low wth onstant wall temperature and varable vsosty. Internatonal Journal o thermal Senes 5 (006). [] C.A. FLETCHER Computatonal tehnque or lud dynams Volume. Sprng Verglas Berln Hedelberg GERMANY 99. [3] C. MICHAUT Indésrable traînée. (007) [] C. MICHAUT La lute des aérodynamens pour rédure la onsommaton des avons. (007) [5] D. DESTRAC ET J. REAUX Réduton de la traînée des avons de transport subsonques [6] H. SCHLICHTING Boundary layer theory. Ed M GRAW-HILL 967 [7] J. COUSTEIX Couhe lmte lamnare Aérodynamque. Ed. epadues. Toulouse.988 [8] J. P. NOUGIER Méthodes de alul numérques Volume. Ed MASSON Pars 00 [9] P. BREGON La lamnarsaton des volures permettrat de rédure de 0% la onsommaton de arburant des avons. (8/06/000) [0] R. PANTON Inompressble low. Ed. John Wlly and Sons In New York.983. [] S. CANDEL Méanque des ludes. DUNOD Pars 995. [] T. Neal A Boundary Condton or Smulaton o Flow Over Porous Suraes. 9th Appled Aerodynams Conerene June - 00/Anahem Calorna. Reeved: Marh 7 008