M E FLUID MECHANICS II

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Eleanore Burns
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1 Name: Sden No.: M E FLUID MECHANICS II Depamen o Mechanical Enineein Uniesi o Saskachean Final Eam Monda, Apil, 003, 9:00 a.m. :00 p.m. Insco: oesso Daid Smne LEASE READ CAREFULLY: This eam has 7 paes. This eam is closed book. Yo ae pemied o se o on calclao; no ohe aids ae pemied. Thee ae 6 poblems on he eam. Each poblem is o eqal ale. Aemp all 6 poblems. lease ense ha o anses ae clea and leible. Some helpl inomaion is ond on he las pae o he eam. Yo ms en he eam qesion shees ih o eam bookles. oblem A lamina bonda lae on a smooh la plae is appoimaed b a linea eloci poile ien b he olloin eqaion, o eeseam eloci, U, and bonda lae hickness, δ. Thee is no pesse adien, and he lo is o-dimensional, sead, and incompessible. U a. In he absence o a pesse adien, ha ie bonda condiions shold be saisied o an lamina bonda lae eloci poile? Be se o poide a sho descipion o deiniion o each o he ie condiions. b. Fo he linea eloci poile ien aboe, hich o he ie condiions ae saisied? c. Usin he Kámán ineal momenm eqaion, obain an epession o he bonda lae hickness, δ(), as a ncion o onl, o he aboe linea eloci poile. d. Fo a la plae o lenh L and idh W, sho ha he da coeicien can be ond simpl b ealain he momenm hickness a he ailin ede, accodin o he olloin epession. C D δ ( L) L ae o 7

2 oblem The poe,, podced b a ind bine is pimail a ncion o he ai densi,, he ind speed,, and he bine blade adis, R. Seconda aiables inlencin he poe ae he ai iscosi,, he oaional speed, ω, and he mass momen o ineia, I. a. Use dimensional analsis o deelop a ncional elaionship o he poe,, sin all he paamees lised aboe. Based on he inomaion poided in he poblem aboe, jsi o choice o epeain paamees. b. oide a sho phsical inepeaion o descipion o each o he o dimensionless ops om (a). A poope ind bine ill encone a ind speed o 30 km/h in ai a sandad amospheic condiions. A /5-scale, eomeicall simila model o he ind bine is o be esed in a lae ae nnel, hee he ae is a 0 C. The lid popeies ae poided on he las pae o he eam. c. Fo complee dnamic similai beeen he model and poope, ha shold be he lo speed in he ae nnel? Wha shold be he aio o oaional speeds beeen he model and poope? oblem 3 The olloin eloci ield epesens a sead, incompessible, o-dimensional lo o a Neonian lid. The lid has densi,, iscosi,, hemal condcii, k, and speciic hea, c. Gai ma be neleced. a. Sho ha he lo is incompessible. b. Find he local acceleaion o he lid. c. Find he conecie acceleaion o he lid. 0 d. Use he Naie-Sokes eqaions o ind he pesse ield, (,). e. Conside an elemenal conol olme o lid, ih dimensions d and d and ni deph, as shon in Fie belo. Fo he lo ield ien aboe, ealae all he shea sess componens acin on his elemenal aea, and indicae hei diecion (i applicable) on a skech o he elemenal aea.. Based on o analsis o he lo in (a) hoh (e), eplain hehe his lo can be consideed iniscid. d d Fie ae o 7

3 oblem 4 Conside he sead, lamina, incompessible, iscos, aismmeic, ll deeloped Neonian lid lo in he cicla pipe shon in Fie belo. The pipe has a adis R. The lid alls nde he acion o ai. The pipe all moes pads a a consan eloci,, conein he acion o ai. The lo is pel aial, ih () onl, and 0. The lid has a densi,, and a iscosi,, and he pesse in he lid is consan. a. Speci he bonda condiion(s) o he lo eloci, (). b. Use he Naie-Sokes eqaions o ind an epession o he eloci poile, (). c. Compe he olme lo ae, Q. Wha pad pipe-all eloci is needed o eo olme lo ae? R Moin pipe all ( ) Flid Moin pipe all Fie ae 3 o 7

4 oblem 5 A hin shee o smooh la plood lies la on a ooop, as shon in Fie 3 belo. The plood has dimensions 3 m m. The eih o he plood is 90 N. The coeicien o saic icion beeen he oo and he plood is S 0.. Wind a sandad amospheic condiions (se he ai popeies ien on he las pae o he eam) blos paallel o he oo and plood in he diecion indicaed in Fie 3. A bonda lae oms on he ooop ih is oiin ( 0) a he ooop ede m ahead o he plood. a. Reein o Fie 3, ha ind eloci U ill eneae enoh icion o dislode he plood om he oo? The hickness o he plood can be neleced. In aiin a o anse, make se o one o he coelaions poided on he las pae o he eam. Be se o jsi o choice o coelaion. m 3 m.5 m.5 m U lood m Rooop 0 Fie 3 The same piece o plood is no eeced as a sin acin nomal o he ind, as shon in Fie 4 belo. The ind speed is 0 m/s. The plood is moned on a smooh clindical pole ha is 0.5 m in diamee and 3 m hih. b. Calclae he bendin momen a he base o he pole cased b he ind oce. Cleal indicae an assmpions. Some sel inomaion is ond in Fie 5 and Table. c. I he sace o he clindical pole as ohened, ih an aeae ohness heih o ε mm, deemine he edcion in he bendin momen. Cleal indicae an assmpions. Fom an enineein sandpoin, is i oh ohenin he sace? Wih he aid o a cleal labeled skech, poide a phsical eplanaion o he edcion in bendin momen. d. Ae seeal eas o epose o modeae ind speeds, he bols holdin he sin epeience aile cased b aie. Wih he aid o a cleal labeled skech, eplain he oiins o he aie aile om a lid mechanics pespecie. 3 m lood sin m Fie 4 Clindical pole d 0.5 m 3 m Uniom ind U 0 m/s ae 4 o 7

5 Fie 5 Table (daa ae independen o Re) Reeence: F. M. Whie, 999, Flid Mechanics, 4h Ediion, Boson: McGa-Hill. ae 5 o 7

6 oblem 6 a. Wha is mean b he em sead lo? b. Wha is he inciple o Dimensional Homoenei? c. Wiho iin an eqaions, poide a phsical descipion o he bonda lae displacemen hickness, δ*. d. Wih he aid o a cleal labeled skech, eplain ho a lamina bonda lae ndeoes ansiion o a blen bonda lae. e. Wha is he niesall acceped ale o he ciical Renolds nmbe o he ansiion o an lamina lo o blen lo?. Wha is mean b he sepaaion o a bonda lae? Eplain ho pesse adien inlences sepaaion. Gie o eamples o enineein los ha illsae he conseqences o bonda lae sepaaion.. Wh does a blen bonda lae hae a hihe skin-icion da han a lamina bonda lae? ae 6 o 7

7 ae 7 o 7 SOME USEFUL EQUATIONS AND INFORMATION ( ) 0 ( ) ( ) Φ T k T c T c hee Φ Clindical Coodinaes, Incompessible Neonian Flid Flo Gadien: e e e ˆ ˆ ˆ Conini: ( ) ( ) ( ) 0 Conecie ime deiaie: Laplacian opeao: -momenm: ( ) -momenm: ( ) -momenm: ( ) ( ) oici componens: ( ) ω ω ω Fla-lae Bonda Lae Coelaions Blasis Solion: ( ) / Re C /7h oe La: ( ) 7 / Re 0.07 C opeies o Ai a Sandad Condiions (pam 0 ka, T 0 C). k/m 3, Ns/m opeies o Wae a 0 C 998 k/m 3, Ns/m

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