Lecture 16Final VersionCombinations of Solutions: Solid Bodies in a Potential Flow (Rankine Oval etc.)Cylinde in !nifom FlowCylinde wit" Ciculation in a !nifom FlowPessue #istibution $ound t"e Cylinde%utta&'oukowski (ift )"eoemCiculation and (ift fo $eofoil $**licationsContentsDesign Project InstructionsnowonwwwasPDFfile.(Instructions should also appear as hardcopies via our pigeon hole Deadlineforsu!"issione#tendeduntilFrida$ee% 1& (' Fri. 1( )an. &**+, -u!"issionsheetwillappearonwwwsoon.ish(and also via pigeon holes,CO+B,-$),O-S OF SO(!),O-S: SO(,# BO#,.S ,- $ PO).-),$( F(O/Recall: Can use PR,-C,P(. OF S!P.RPOS,),O- PR,-C,P(. OF S!P.RPOS,),O- fo velocity *otential.Catesian Coodinates:!nifom Flow and Souce: /01 2345I41 67D8/"at "a**ens if we combine...+souce flow unifom + Pola Coodinates:( )

,_

+ xym y U y x0tan 1 ( ) m r U r + sin 1(0)2(3) e*esent com*lete desci*tions of flow field.But w"at does it look like4...(0)(3))o 5a*" lines of constant 1fist look fo S)$6-$),O- PO,-)S.)"ee ...7 V)"us1bot"velocitycom*onentsmustbe8eo9#iffeentiateto5et e:*essions fo velocity com*onents ...,n addition1 "ave s"own t"at fo incom*essible1 iotational flow1 steam function also satisfies (a*lace .;. So can similaly constuct flow solutions by combinin5 S.F. associated wit" unifom flow1 souce2sink flow and line&vote: flow. ,n fact1 we will almost e:clusively use steam function "ee because we ae inteested in *attenofsteamlines 3 0 Calculate e:*essions fo vel. com*onents u, v fomxvyu 1-ote: Bu5e c"oice as fa as selction of *aametes is concenedG Souce sten5t"1 vote: diection of otation1 sten5t" 9 ...#etemine cood.of sta5nation *oint(s) viau=0 , v J7.#eteminevalueofsteamfunction*assin5t"ou5" (sta5nation)*ointbysubstitutin5coodinatesof (sta5nation) *oint(s) into t"e steam function.Setsteamfunctione;ualtot"evalueyou"ave detemined fo *oint in ;uestion.#eteminevaluesof:1y(o1)t"atsatisfyt"is e:*ession and *lot to obtain steamline.C"oose new *oint x,yFom*eviousits"ouldbeobvious"owonecanfindsteamfunctionfoa cylinde (cicle) in a unifom flow ...Cylinde in a !nifom Flow/urn 2an%ine oval into circle by allowin5 ... c Um$c"ieved by movin5 souce and sink close to oi5in ...7 c(imit (c=0)would ultimately Icancel? *aiG7 cwit". const m c ( )

,_

+ 3 3 303tan 1c y xy cm y U y x Recall t"at Rankine oval "ad S.F. ...7 let c-otin5 t"at small fo tan0( )3 31y xyy U y x+ -ote:+e5in5ofsouceandsinkasabove*oducesstuctueknownas #O!B(.).c m 3 flow!nifom oi5in at #O!B(.).nsuet"eiinfluenceemainsbyallowin5mtoinceaseinsi8e. -ecessay limit is...-ow t"en t"e a5ument of tan!" 5oes to 8eo...5ives( )3 3 331c y xy cm y U y x + #efine D7=6L1/ -/214>/0?-trea" Function for Clinder Flow(1,Continued...+oe convenient to wok in *ola coodinatesG S.F. can be witten ...( )

,_

r Ur Urr U r sinsinsin 1Fom .;. (0) can 5et velocity com*onents in usual way...(1,Continued...

,_

330 cos0r#Urur

,_

+ 330 sinr#Uru /"eewe used...U#CB.C% t"at t"is flow eally does e*esent a cylinde in unifom flow.-tagnation points?(&,(:,1@. (:, ?7 7 u7 uSubstitute t"ese an5les into .;.(3) ... 5et to 7 set and ru# rr#r#U

,_

33330 0 7 cos 7# rr#r#U

,_

33330 0 cos 7 Bence1... Sta5nation *oints:( ) ( ) 1 and 7 1 # #-urface -.L. V3L=1 by substitutin5 one sta5. *oint into .;. (0)...( ) 7 7 sin 7 1

,_

# U# U #-ow 5et e;uation foSuface S.(.by e;uatin5 .;. (0)to 8eo ...Ur3# r o$s e;uied suface S.(. is cicle wit" adius U # P(O): 7 Fo : Fo UrContinued...Velocit Co"ponents on cylinde suface ae obtained1bysettin5 r=#, fom...

,_

330 cosr#U ur

,_

+ 330 sinr#U u 1@. (&,1@. (:,7 rusin 3 U u+i5"t"avee:*ectedtofindt"atadialflowcom*onentis8eoon suface & flow cannot *ass t"ou5" (solid) cylinde wallG-ote also t"at ma:imum flow s*eed occus atw"ee it is + U U 3 and 33>and3 es*ectively.+eans in bot" cases (to* and bottom "alf of cyl.) flow is fom left to i5"tGOn to* ne5ative value as velocity *oints in clockwise (ne5ative an5le) diection. On bottom in anti&clockwise (*ositive an5le) diection.Finally1 note symmety of flow about bot" t"e x! and y! a:es. /"at does t"istellyouaboutt"e*essuedistibutionont"ecylindesuface9 emembe t"e Benoulli .;uationG=nifor" FlowA Dou!let 'Flow over a ClinderContinued...Since we can now mat"ematically descibe 9...wecan1in*inci*le1alsodescibeflowt"ou5"anabitayaayof cylindes as1 fo instance1 t"e flow s"own in t"e *"oto below. /e sim*ly need to *ut seveal doublets in ou unifom flow.Cylinde wit" Ciculation in a !nifom Flow /it"out*efomin5calculation1canseein*ecedin5flownonetliftoda5on cylinde since *essue distibution on suface symmetic about x! and y!a:is..2K ' -ote t"at t"is does not violate t"e flow aound cylinde: line vote: *oduces aucom*onentofvelocityonly.Bence1weaestillad"ein5toconditiont"atflow cannot *ass t"ou5" cylinde bounday. /okin5 fom S.F. fo cylinde in unifom flow additional inclusion of line vote: 5ives:( ) C r $rr U r + lnsinsin 1 oi5in at #oublet flow!nifom oi5in at vote: (ineconstant$bitay!se esult t"at adius of esultin5 cylinde is :$nd set :U## $ C ln (1,(1,( ) # $ r $r Ur U r ln ln sin 1 +

,_

( ) # r $r Ur U ln ln0sin

,_

#r$r#r U ln sin3

,_

Velocit Co"ponents

,_

330 cos0r#Urur22R Ku Usin 1r r r-_ ___ _ _ _ ___ ___ ' ,nodeto5eneateliftneedtobeaksymmety.$c"ievedbyintoducin5line vote: of sten5t"1 $1 at oi5in w"ic" intoduces ciculation .Continued... So1 on suface %r=#&1 velocity com*onents ae:7 ru#$U u + sin 3 Suface Sta5nation *oints also need:7 uU #$3sin-ote: By settin5 vote: sten5t" 8eo (%J7)1 ecove flow ove cylinde in unifom flow wit" sta5nation *oints at 1 7 Plottin519 C"oose value fo $19 -ow fist 5et value of S.F. for=#1... t"en setS.F.e;ualtot"atvalue19t"encom*iletablervs'angle9)"is5ives *aticula steamline t"ou5" sta5nation *oints. )"enc"ooseanyot"e*ointinflowfieldnotonsta5nationsteamline19 deteminevalueofS.F.fot"is*oint19setS.F.e;ualtot"atvalue19t"en com*iletablervs'angle9)"is5ivessteamlinet"ou5"t"ec"osen *aticula*oints9)"enc"ooseanot"e*ointinflowfield9etc(com*ae flow c"atfom be5innin5 oflectue).Fovaious valuesof$ t"e followin51 flow fields eme5e...7 $ 0 $3 $ > $Continued...Can now also descibe flow t"ou5" an abitay aay of cylindes w"en eac"oft"emisotatin5G(-ote:,n*"otobelowcylindesaenot otatin5)Pessue #istibution $ound t"e Cylinde)oevaluate*ess.oncyl.sufaceuseBenoulli.;.alon5S.(.t"at oi5inates fa u*steam w"ee flow is undistubed. ,5noin5 5av. foces:3 33030SSU p U p + + flowd undistube!*steam sufacecylinde On Re&aan5in5...

,_

333030UUU p pSS Substitutin5 fo flow s*eed 5ives...) sin 3#$U u + 1 7 ( #u( )

,_

,_

+ 33333 33030030UuUUu uU p p#S ( ) ( )3 3 32 sin 3 030 + U # $ U U )'

,_

+ 33 3sin=sin = 030U #$U # $U p pS 9 diffeence in *essues between suface and undistubed fee steam(1,In particular for non.rotating clinder where 5'*?{ } 3 3sin = 030 U p pS(&,33sin = 030 Up pCSp#ef.:Pessue CoefficientOnly to* "alf of cyl. s"own.Continued...)'

,_

+ 333sin=sin = 030U #$U # $Up pCSp Bualitative !ehaviour offor various values of .

#U $ Bestwayofinte*etin5above5a*"sistot"inkofflowvelocityandadiusbein5constant w"ile vote: sten5t" is inceasin5 fom one *lot to ne:t. #U $ /"en*lottin55a*"s,didnote:*licitlys*ecifyvelocityoadiusG,sim*lyuseddiffeent numeic values foin ode to illustate be"aviou of 5a*". , "ave not consideedif any of t"ese cases may not be eali8able in eality o notG. 1 cyl. of )o* : FA . 0 3 1 cyl. of Rea : 7 ( ) cyl. of Bott. : A0 . = 3 > 1 cyl. of Font : 0= . > Continued....;uation (0) 9 9 can be used to calculatenet lift and da5 actin5 on cylindeG )'

,_

+ 33 3sin=sin = 030U #$U # $U p pS Sketc" ($)Sketc" (B) ,n Sketc" (b) ...( ) ( ) sin sin p p p LS( ) ( ) cos cos p p p (SBence1 inte5ation aound cylinde suface yields total ( and # ...( ) 37sin d # ) p p LS( ) 37cos d # ) p p (Sw"ee)iswidt"(into*a*e)ofcylinde.Substitutin5fo*essueusin5 .;. (0)1 and inte5atin5 (most tems do* out)1 leads to followin5 esults:) $ U# )U #$U L

,_

3=3037 ( O1 lift *e unit widt":( ) U $ U)L 3/husC drag DeroE a re"ar%a!le resultF/heore" Lift)ou%ows%i 5utta Parado#s 3le"!ertG dGContinued... -et lift is indicated in sketc" below. ... -ote t"at if a line vote: is used w"ic" otatesinmat"ematically*ositivesense(anti&clockwise)t"enesultin5liftis ne5ative1 i.e. downwads. U)LULFinalnotes?Bowislift5eneated4...Fomsketc"aboveandfom *essue*ofiles*lottedealieitisevident"owt"isis*"ysically ac"ieved9 Beakin5 of t"e flow symmety in x!a:is means t"at flow ound lowe *at of cylinde is faste t"an ound to* & t"is means t"at *essue is lowe oundbottom and so a net downwad foce esults. -oticet"atsymmetyiny!a:isisetained9symmetyof*essue onleft&"andandi5"t&"andfacesisetainedandsot"eeisnonet da5 foce. %ee* in mind t"at ou analysis was fo an ideal fluid (i.e. t"eeisnoviscosity).,naealflowwouldfoe&aftsymmetybe etained4 (astly1sinceliftis*o*otionaltociculation1wewis"tomake ciculationla5eto5eneateala5eliftin5foce.,na**licationsof aboveflowt"isisac"ievedbys*innin5cylindeto*oducela5e voticity9butist"eealimitto"owmuc"ciculationwes"ould *oduce4