Chapter 8: Inviscid Incompressible Flow: a Useful Fantasy

Transcription

1 Pofesso Fed Sten Fall 4 Chapte 8: Invscd Incompessble Flow: a Useful Fantasy 8. Intoducton Fo hgh Re etenal flow about steamlned bodes vscous effects ae confned to bounday laye and wake egon. Fo egons whee the BL s thn.e. favoable pessue gadent egons, Vscous/Invscd nteacton s weak and tadtonal BL theoy can be used. Fo egons whee BL s thck and/o the flow s sepaated.e. advese pessue gadent egons moe advanced bounday laye theoy must be used ncludng vscous/invscd nteactons. Fo ntenal flows at hgh Re vscous effects ae always mpotant ecept nea the entance. Recall that votcty s geneated n egons wth lage shea. Theefoe, outsde the B.L and wake and f thee s no upsteam votcty then ω s a good appomaton. Note that fo compessble flow ths s not the case n egons of lage entopy gadent. Also, we ae neglectng nonnetal effects and othe mechansms of votcty geneaton. Potental flow theoy ) Detemne φ fom soluton to Laplace equaton φ B.C: φ at S B : Vn. n at S : V φ Note: F: Suface Functon DF F F + V. F Vn. Dt t F t steady flow Vn. fo

2 Pofesso Fed Sten Fall 4 ) Detemne V fom V φ and p() fom Benoull equaton Theefoe, pmaly fo etenal flow applcaton we now consde nvscd flow theoy ( µ ) and ncompessble flow ( ρ const ) Eule equaton:. V DV ρ p+ ρg Dt V ρ + ρv. V ( p+ γz) t V V. V V ω Whee : ω V votcty flud angula velocty V ρ + ( p+ ρv + γz) ρv ω t If ω e V then V φ: φ ρ + p + ρ φ φ + γz Bt () t Benoull s Equaton fo unsteady ncompessble flow, not f() Contnuty equaton shows that GDE fo φ s the Laplace equaton whch s a nd ode lnea PDE e supeposton pncple s vald. (Lnea combnaton of soluton s also a soluton) V φ φ φ φ + φ φ φ ( φ φ) φ φ φ + + Technques fo solvng Laplace equaton: ) supeposton of elementay soluton (smple geometes) ) suface sngulaty method (ntegal equaton) 3) FD o FE 4) electcal o mechancal analogs 5) Confomal mappng ( fo D flow) 6) Analytcal fo smple geometes (sepaaton of vaable etc)

3 Pofesso Fed Sten Fall Elementay plane-flow solutons: Recall that fo D we can defne a steam functon such that: u ψ y v ψ ω z v u y ( ψ ) ( ψ y ) ψ y.e. ψ Also ecall that φ and ψ ae othogonal. u ψ φ y v ψ φ y dφ φ d + φ dy ud + vdy y dψ ψ d + ψ ydy vd + udy dy u.e. d φ const v dy d ψ const Unfom steam u U ψ y φ const v ψ φ.e. φ U ψ U y y Note: φ ψ s satsfed. V φ U ˆ Say a unfom steam s at an angle α to the -as: ψ φ u U cosα y ψ φ v U snα y

4 Pofesso Fed Sten Fall 4 4 Afte ntegaton, we obtan the followng epessons fo the steam functon and velocty potental: ψ U ycosα snα D Souce o Snk: ( ) ( cos sn ) φ U α + y α Imagne that flud comes out adally at ogn wth unfom ate n all dectons. (sngulaty at ogn whee velocty s nfnte) Consde a ccle of adus enclosng ths souce. Let v be the adal component of velocty assocated wth ths souce (o snk). Then, fom consevaton of mass, fo a cylnde of adus, and unt wdth pependcula to the pape, 3 L Q V da A S Q ( π ) ( b) v O, Q v v m, v θ π b Q Whee: m s the convenent constant wth unt velocty length πb (m> fo souce and m< fo snk). Note that V s sngula at (,) snce v In a pola coodnate system, fo -D flows we wll use: φ φ V φ + θ

5 Pofesso Fed Sten Fall 4 5 And:. V ( v ) + ( vθ ) θ.e.: φ ψ v Radal velocty θ φ ψ vθ Tangental velocty θ Such that V by defnton. Theefoe, v v θ.e. m φ ψ θ φ ψ θ Doublets: φ mln mln ψ mθ m tan y + y The doublet s defned as: ψ m( θ θ θ θ ) snk souce snk souce ψ m tan(θ θ ) tan( ψ tan θ tan θ ) m + tan θ tan θ

6 Pofesso Fed Sten Fall 4 6 snθ snθ tanθ ; tanθ cosθ a cosθ + a snθ snθ ψ cos cos a sn tan( ) a a θ θ θ + m snθ snθ + a cosθ acosθ + a Fo small value of a a snθ amy λy ψ lm m tan ( ) a a + y λcosθ λ am ( Doublet Stength) φ By eaangng: λy λ λ ψ + ( y + ) ( ) + y ψ ψ λ It means that steamlnes ae ccles wth adus R and cente at (, +/-R).e. ψ ccles ae tangent to the ogn wth cente on y as. The flow decton s fom the souce to the snk.

7 Pofesso Fed Sten Fall 4 7 D vote: Suppose that value of the ψ and φ fo the souce ae evesal. v φ ψ K vθ θ Puely cculatoy flow wth v θ lke /. Integaton esults n: φ Kθ ψ Kln Kconstant D vote s otatonal eveywhee ecept at the ogn whee V and V ae nfnty. Cculaton Cculaton s defned by: Γ V d s Fo otatonal flow c C closed contou O by usng Stokes theoem: ( f no sngulaty of the flow n A) Γ V d s V d A ω. nda c A A Theefoe, fo potental flow Γ n geneal. Howeve, ths s not tue fo the pont vote due to the sngula pont at vote coe whee V and V ae nfnty. If sngulaty ests: Fee vote υ θ π π K Γ v eˆ d eˆ ( d ) K and K θ θ θ θ θ π V ds K Note: fo pont vote, flow stll otatonal eveywhee ecept at ogn tself whee V,.e., fo a path not ncludng (,) Γ Γ π

8 Pofesso Fed Sten Fall 4 8 Also, we can use Stokes theoem to show the estence of φ : V ds V ds φc Snce ABC ' AB C ' ABCB A V. ds Theefoe n geneal fo otatonal moton: V. d φ V d dφ d dφ d V φ. ds ds ds d eˆ s ds ( V φ). eˆ Whee: Snce s e unt tangent vecto along cuve s e s not zeo we have shown: V φ.e. velocty vecto s gadent of a scala functon φ f the moton s otatonal. ) ( V ds s space The pont vote sngulaty s mpotant n aeodynamcs, snce, eta souce and snk can be used to epesent afols and wngs as we shall dscuss shotly. To see ths, consde as an eample: an nfnte ow of votces: πy π ψ K ln K ln (cosh cos ) a a Whee s adus fom ogn of th vote. Equally speed and equal stength (Fg 8. of Tet book)

9 Pofesso Fed Sten Fall 4 9 Fo y a the flow appoaches unfom flow wth ψ πk u ± y a +: below as -: above as Note: ths flow s ust due to nfnte ow of votces and thee sn t any pue unfom flow Vote sheet: Fom afa (.e. y a ) looks that thn sheet occus whch s a velocty dscontnuty. πk Defne γ stength of vote sheet a πk dγ uld uud ( ul uu ) d d a dγ.e. γ Cculaton pe unt span d Note: Thee s no flow nomal to the sheet so that vote sheet can be used to smulate a body suface. Ths s the bass of afol theoy whee we let γ γ () to epesent body geomety. Vote theoem of Helmholtz: (mpotant ole n the study of the flow about wngs) ) The cculaton aound a gven vote lne s constant along ts length ) A vote lne cannot end n the flud. It must fom a closed path, end at a bounday o go to nfnty. 3) No flud patcle can have otaton, f t dd not ognally otate

10 Pofesso Fed Sten Fall Potental Flow Solutons fo Smple Geometes Ccula cylnde (wthout otaton): In the pevous we deved the followng equaton fo the doublet: λy λsnθ ψ Doublet + y When ths doublet s supeposed wth a unfom flow paallel to the - as, we get: λsnθ λ ψ Usnθ U snθ U Whee: λ doublet stength whch s detemned fom the knematc body bounday condton that the body suface must be a steam suface. Recall that fo nvscd flow t s no longe possble to satsfy the no slp condton as a esult of the neglect of vscous tems n PDEs. The nvscd flow bounday condton s: F: Suface Functon DF F F + V. F V n Dt t F t (fo steady flow) Theefoe at R, V.n.e. v. R F F eˆ ˆ + e F θ V veˆ ˆ + ve θ θ, n θ eˆ F F + Fθ ψ λ V n v U cosθ θ U λ UR λ If we eplace the constant by a new constant R, the above equaton becomes: U R ψ U snθ Ths adal velocty s zeo on all ponts on the ccle R. That s, thee can be no velocty nomal to the ccle R. Thus ths ccle tself s a steamlne.

11 Pofesso Fed Sten Fall 4 We can also compute the tangental component of velocty fo flow ove the ccula cylnde. Fom equaton, ψ R vθ U + snθ On the suface of the cylnde R, we get the followng epesson fo the tangental and adal components of velocty: v θ U snθ v How about pessue p? look at the Benoull's equaton: p p + ( v + vθ ) + U ρ ρ Afte some eaangement we get the followng non-dmensonal fom: p p v + vθ Cp (, θ ) ρu U Fo the ccula cylnde beng studed hee, at the suface, the only velocty component that s non-zeo s the tangental component of velocty. Usng vθ U snθ, we get at the cylnde suface the followng epesson fo the pessue coeffcent: C p 4 sn θ Whee θ s the angle measued fom the ea stagnaton pont (at the ntesecton of the back end of the cylnde wth the - as). Cp ove a Ccula Cylnde Cp Theta, Degees

12 Pofesso Fed Sten Fall 4 Fom pessue coeffcent we can calculate the flud foce on the cylnde: F ( p p) nds ρu C p ( R, θ) nds A A ds ( Rdθ ) b bspan length π F ρu br ( 4sn θ)(cosθˆ+ sn θ ˆ ) dθ π Lft F ˆ CL ( 4sn θ )snθ d θ (due to symmety of flow ρu br ρu br aound as) π Dag F ˆ CF ( 4sn θ )cosθ d θ (dálembet paado) ρu br ρu br We see that C L C D s due to symmety of Cp. But how ealstc s ths potental flow soluton. We stated eplan that vscous effects wee confned to thn bounday laye and t has neglgble effect on the nvscd flow fo hgh Rn flow about steamlned bodes. A ccula cylnde s not a steamlned body (lke as afol) but athe a bluff body. Thus, as we shall detemne now the potental flow soluton s not ealstc fo the back poton of the ccula cylnde flows ( θ > 9). Fo geneal: C C + C D C D FORM DRAG C D SKIN FRICTION DFORM DRAG DSKIN FRICTION : dag due to vscous modfcaton of pessue dstbuton : Dag due to vscous shea stess Steamlned Body : C Bluff Body : CD FO > C DSK DSK > C D FO Ccula cylnde wth cculaton: The steam functon assocated wth the flow ove a ccula cylnde, wth a pont vote of stength Γ placed at the cylnde cente s: λsnθ Γ ψ Usnθ ln π Fom V.n at R: λ U R U R snθ Γ Theefoe, ψ Usnθ ln π The adal and tangental velocty s gven by: ψ R v U cosθ v θ θ ψ R U + snθ + Γ π

13 Pofesso Fed Sten Fall 4 3 On the suface of the cylnde (R): ψ R v U cosθ θ R π Γ V d v dθ,.e., vote stength s cculaton θ v θ ψ Γ U snθ + π R Net, consde the flow patten as a functon of Γ. To stat lets calculate the stagnaton ponts on the cylnde.e.: Γ vθ U snθ + π R Γ K sn θ β / 4πU R U R Γ K Note: K β π U R So, the locaton of stagnaton pont s functon of Γ. K Γ θ s (stagnaton pont) β UR πur ( sn θ ),8 ( sn θ. 5 ) 3,5 ( sn θ ) 9 >( sn θ > ) Is not on the ccle but whee v v θ Fo flow pattens lke above (ecept a), we should epect to have lft foce n y decton.

14 Pofesso Fed Sten Fall 4 4 Summay of steam and potental functon of elementay -D flows: In Catesan coodnates: u ψ φ y v ψ φ In pola coodnates: φ ψ v θ φ ψ vθ θ y Flow φ ψ Unfom Flow U U y Souce (m>) Snk (m<) mln mθ Doublet λ cosθ λ snθ Vote Kθ - Kln 9 Cone flow / A( y ) Ay Sold-Body otaton Doesn t est ω These elementay solutons can be combned n such a way that the esultng soluton can be ntepeted to have physcal sgnfcance; that s, epesent the potental flow soluton fo vaous geometes. Also, methods fo abtay geometes combne unfom steam wth dstbuton of the elementay soluton on the body suface. Some combnaton of elementay solutons to poduce body geometes of pactcal mpotance Body name Elemental combnaton Flow Pattens Rankne Half Body Unfom steam+souce Rankne Oal Unfom steam+souce+snk

15 Pofesso Fed Sten Fall 4 5 Kelvn Oal Unfom steam+vote pont Ccula Cylnde wthout cculaton Unfom steam+doublet Ccula Cylnde wth cculaton Unfom steam+doublet+vote Keep n mnd that ths s the potental flow soluton and may not well epesent the eal flow especally n egon of advese p. The Kutta Joukowsk lft theoem: Snce we know the tangental component of velocty at any pont on the cylnde (and the adal component of velocty s zeo), we can fnd the pessue feld ove the suface of the cylnde fom Benoull s equaton: p v vθ p U ρ ρ Theefoe: ρ ρu p p ρ Γ U U snθ Γ + p + ρu ρu sn θ snθ Γ + πr 8π R πr A+ Bsnθ + Csn θ whee ργ A p + ρu 8π R ρu B Γ π R C ρu

16 Pofesso Fed Sten Fall 4 6 Calculaton of Lft: Let us fst consde lft. Lft pe unt span, L (.e. pe unt dstance nomal to the plane of the pape) s gven by: L pd pd Lowe On the suface of the cylnde, Rcosθ. Thus, d -Rsnθdθ, and the above ntegals may be thought of as ntegals wth espect to θ. Fo the lowe suface, θ vaes between π and π. Fo the uppe suface, θ vaes between π and. Thus, L R uppe π ( A + B snθ + C sn θ ) snθdθ + R ( A + Bsnθ + C sn θ ) π Revesng the uppe and lowe lmts of the second ntegal, we get: L R π snθdθ π π 3 ( A + Bsnθ + C sn θ ) snθdθ R ( Asnθ + Bsn θ + C sn θ ) dθ BRπ Substtutng fo B,we get: L ρ uγ Ths s an mpotant esult. It says that clockwse votces (negatve numecal values of Γ) wll poduce postve lft that s popotonal to Γ and the fee steam speed wth decton 9 degees fom the steam decton otatng opposte to the cculaton. Kutta and Joukowsk genealzed ths esult to lftng flow ove afols. Equaton L ρ uγ s known as the Kutta-Joukowsk theoem. Dag: We can lkewse ntegate dag foces. The dag pe unt span, D, s gven by: D pdy pdy Font Snce yrsnθ on the cylnde, dyrcosθdθ. Thus, as n the case of lft, we can convet these two ntegals ove y nto ntegals ove θ. On the font sde, θ vaes fom 3π/ to π/. On the ea sde, θ vaes between 3π/ and π/. Pefomng the ntegaton, we can show that D Ths esult s n contast to ealty, whee dag s hgh due to vscous sepaaton. Ths contast between potental flow theoy and dag s the dálembet Paado. The eplanaton of ths paado ae povded by Pandtl (94) wth hs bounday laye theoy.e. vscous effects ae always mpotant vey close to the body whee the no slp bounday condton must be satsfed and lage shea stess ests whch contbutes the dag. ea

17 Pofesso Fed Sten Fall 4 7 Lft fo otatng Cylnde: We know that LρU Γ theefoe: L Γ CL ρu ( R) UR π Γ Defne: vθ Aveage vθ dθ π π R π CL v U θ aveagfe Note: Γ V d s V d A V Rd θ θ θ c c c Velocty ato: aω U Theoetcal and epemental lft and dag of a otatng cylnde Epements have been pefomed that smulate the pevous flow by otatng a ccula cylnde n a unfom steam. In ths case v θ Rω whch s due to no slp bounday condton. - Lft s qute hgh but not as lage as theoy (due to vscous effect e flow sepaaton) - Note dag foce s also fay hgh Flettne (94) used otatng cylnde to poduce fowad moton.

18 Pofesso Fed Sten Fall 4 8

19 Pofesso Fed Sten Fall 4 9

20 Pofesso Fed Sten Fall Method of Images The method of mage s used to model slp wall effects by constuctng appopate mage sngulaty dstbutons. Plane Boundaes: -D: ln ( ) m [ + y ] + ln[ ( + ) + y ] ln[ ( ) + y ][( + ) y ] Q m + 3-D: φ M ( ) + y + z + ( + ) + y + z Smla esults can be obtaned fo dpoles and votces:

21 Pofesso Fed Sten Fall 4 Sphecal and Cuvlnea Boundaes: The esults fo plane boundaes ae obtaned fom consdeaton of symmety. Fo sphecal and ccula boundaes, mage systems can be detemned fom the Sphee & Ccle Theoems, espectvely. Fo eample: Flow feld Souce of stength M at c outsde sphee of adus a, c>a Dpole of stength µ at l outsde sphee of adus a, l>a Souce of stength m at b outsde ccle of adus a, b>a Image System Souces of stength ma at a and lne c c snk of stength m etendng fom cente a of sphee to a c dpole of stength a a 3 µ l at a equal souce at and snk of same b stength at the cente of the ccle l

22 Pofesso Fed Sten Fall 4 Multple Boundaes: The method can be etended fo multple boundaes by usng successve mages. () Fo eample, the soluton fo a souce equally spaced between two paallel planes w( z) m m m, ±, ±, [ ln[ z ( 4n + a) ] + ln[ z ( 4n + a) ] [ ln( z a) + ln( z + a) + ln( z 4 + a) + ln( z 6 + a) + ln( z + 4 a) + ln( z + + a) + ]

23 Pofesso Fed Sten Fall 4 3 () As a second eample of the method of successve mages fo multple boundaes consde two sphees A and B movng along a lne though the centes at veloctes U and U, espectvely: Consde the knematc BC fo A: F(, t) ( yt) + y + z a DF V eˆ ˆ ˆ R Uk er o φr Ucosν Dt 3 whee ϕ cos Ua ν, φ R 3 cos ν fo sngle sphee R R Smlaly fo B φ U cos ' ν R Ths suggests the potental n the fom ϕ Uϕ + U ϕ whee φ and φ both satsfy the Laplace equaton and the bounday condton: φ φ cos ν, R R' R a R' b (*) φ φ, cos ν ' R R' R a R' b (**) φ potental when sphee A moves wth unt velocty towads B, wth B at est φ potental when sphee B moves wth unt velocty towads A, wth A at est

24 Pofesso Fed Sten Fall 4 4 If B wee absent. 3 φ a cos cos R ν R ν, 3 a but ths does not satsfy the second condton n (*). To satsfy ths, we ntoduce the mage of n B, whch s a doublet dected along BA at A, the nvese pont of A wth espect to B. Ths mage eques an mage at A, the nvese of A wth espect to A, and so on. Thus we have an nfnte sees of mages A, A, of stengths,, 3 etc. whee the odd suffes efe to ponts wthn B and the even to ponts wthn A. Let fn AAn & AB c b a b f c, f, f 3 c, c f c f b, a 3 c 3, b 3 f 3 ( ), c f 3 adus whee mage dpole stength, dpole stength 3 sstance cosν cosν cosν φ wth a smla development pocedue fo φ. R R R Although eact, ths soluton s of unweldy fom. Let s nvestgate the possblty of an appomate soluton whch s vald fo lage c (.e. lage sepaaton dstance) c c R ' R + c c cosν R cosν + R R c c R R cosν cosν + + R' R R R c c c c Consdeng the fome epesentaton fst defnng µ and u cosυ R [ uµ + µ ] R' R

25 Pofesso Fed Sten Fall 4 5 By the bnomal theoem vald fo < ( ) + α + α + α 3 + Hence f u µ µ < α, 3 α and 3 ( n ) α n 4 n [ ] α + α ( u µ µ ) + α ( ) + P ( u) + P ( u) µ + P ( u) µ Afte collectng tems n powes of µ, whee the P n ae Legende functons of the fst knd (.e. Legende polynomals whch ae Legende functons of the fst knd of ode zeo). Thus, R R R< C: + P ( cosν) + P 3 ( cosν) + R' c c c c c R> C: + P ( cosν) + P 3 ( cosν) + R' R R R Net, consde a doublet of stength at A cosα ( Rcosν c) φ 3 R' c ( R + c cr cosν) ( R + c cr cosν) Thus, cosα RP( cosν) 3R P( cosν) R< c: φ R' c c cν c 3c R> c: φ P ( cosν) + P 3 ( cosν) + P 4 3( cosν) + R R R Gong back to the two sphee poblem. If B wee absent 3 a φ cos ν R R ' usng the above epesson fo the ogn at B and nea B < c, R R, ν ν ' a a arp ' ( cosν ) φ cosν R c c 3 a cosν φr + 3 c

26 Pofesso Fed Sten Fall 4 6 whch can be cancelled by addng a tem to the fst appomaton,.e a cosν ab cos ν ' φ 3 R c R' to confm ths a a3 R'cos ν ' ab cosν φ 3 3 c c c R ' φ ' a cos ν ' ab cosν ( R b) + ' hot ' R c c R Smlaly, the soluton fo f s 3 ' 3 3 b cosν ab cos ' 3 φ R These appomate solutons ae conveted to Ο ( c 3 ). c ν R To fnd the knetc enegy of the flud, we have K ρ φφnds φφnds + SA S B ρ K n A U + A U U + A U φφ ds SA+ SB A ρ φ φ ds A, A ρ φ φ φ φ dsb n, A ρ ds ds n φ ρ φ A B n n A B A B whee ds πr snυdυ 3 3 π ab A π 3 a ρ, A ρ, 3 A π 3 b ρ, 3 c πabρ K M' U + UU 3 + M' U : usng the appomate fom of the potentals 4 c 4 whee M ' U, M ' U : masses of lqud dsplaced by sphee Comple vaable and confomal mappng Ths method povdes a vey poweful method fo solvng -D flow poblems. Although the method can be etended fo abtay geometes, othe technques ae equally useful. Thus, the geatest applcaton s fo gettng smple flow geometes fo whch t povdes closed fom analytc soluton whch povdes basc solutons and can be used to valdate numecal methods.

27 Pofesso Fed Sten Fall 4 7 Functon of a comple vaable Confomal mappng eles entely on comple mathematcs. Theefoe, a bef evew s undetaken at ths pont. A comple numbe z s a sum of a eal and magnay pat; z eal + magnay The tem, efes to the comple numbe so that;,,, 3 4 Comple numbes can be pesented n a gaphcal fomat. If the eal poton of a comple numbe s taken as the abscssa, and the magnay poton as the odnate, a twodmensonal plane s fomed. z eal + magnay + y y, magnay, eal -A comple numbe can be wtten n pola fom usng Eule's equaton; Whee: + y z + y e θ (cosθ + snθ) - Comple multplcaton: z z ( +y )( +y ) ( - y y ) + ( y + y ) θ ( θ e e e - Conugate: z + y z y z. z + y -Comple functon: w(z) f(z) φ (,y) + ψ (,y) θ + θ )

28 Pofesso Fed Sten Fall 4 8 If functon w(z) s dffeentable fo all values of z n a egon of z plane s sad to be egula and analytc n that egon. Snce a comple functon elates two planes, a pont can be appoached along an nfnte numbe of paths, and thus, n ode to defne a unque devatve f(z) must be ndependent of path. δ w w w φ+ ψ ( φ+ ψ) PP( δ y ) : δ z z z dw φ + ψ dz Note: w φ( + δy, ) + ψ( + δy, ) δ w w w φ + ψ ( φ+ ψ) PP( δ ) : δ z zz y ( y) dw φy + ψ y dz dw Fo to be unque and ndependent of path: dz φ ψ and φ ψ Cauchy Remann Eq. y y Recall that the velocty potental and steam functon wee shown to satsfy ths elatonshp as a esult of the othogonalty. Thus, comple functon w φ + ψ epesents -D flows. φ ψ φ ψ.e. φ φ and smlaly fo ψ. Theefoe f analytc and y yy y + yy egula also hamonc,.e., satsfy Laplace equaton.

29 Pofesso Fed Sten Fall 4 9 Applcaton to potental flow wz ( ) φ+ ψ Comple potental whee φ : velocty potental, ψ : steam functon dw θ φ + ψ u v ( u uθ ) e Comple velocty dz ' θ ' ( θ + α ) ' ' e ρe whee ρ (magnfcaton) and θ θ + α (otaton) Tangle about z s tansfomed nto a smla tangle n the ζ-plane whch s magnfed and otated. Implcaton: -Angles ae peseved between the ntesectons of any two lnes n the physcal doman and n the mapped doman. -The mappng s one-to-one, so that to each pont n the physcal doman, thee s one and only one coespondng pont n the mapped doman. Fo these easons, such tansfomatons ae called confomal.

30 Pofesso Fed Sten Fall 4 3 Usually the flow-feld soluton n the ζ-plane s known: Then W ( ς ) Φ( ξ, η) + Ψ( ξ, η) ( f ( z) ) φ (, y) + ψ (, ) w( z) W y o φ Φ & ψ Ψ Confomal mappng The eal powe of the use of comple vaables fo flow analyss s though the applcaton of confomal mappng: technques wheeby a complcated geomety n the physcal z-doman s mapped onto a smple geomety n the ζ-plane (ccula cylnde) fo whch the flow-feld soluton s known. The flow-feld soluton n the z-plane s obtaned by elatng the ζ-plane soluton to the z-plane though the confomal tansfomaton ζf(z) (o nvese mappng zg(ζ)). Befoe consdeng the applcaton of the technque, we shall evew some of the moe mpotant popetes and theoems assocated wth t. Consde the tansfomaton, ζf(z) whee f(z) s analytc at a egula pont Z whee f (z ) δζ f (z ) δz θ ' δς ' e, δ z e θ ' α, f ( z ) ρe The steamlnes and equpotental lnes of the ζ-plane (Φ, Ψ) become the steamlnes of equpotental lnes of the z-plane (φ, ψ). zϕ ςφ ψ ψ z ς s the ζ-plane..e. Laplace equaton n the z-plane tansfoms nto Laplace equaton

31 Pofesso Fed Sten Fall 4 3 The comple veloctes n each plane ae also smply elated dw dw d dw ς f ' ( z ) dz dς dz dς dw dw ( z) u v ( U V ) f '( z) ( ζ f ( z)) dz dζ.e. veloctes n two planes ae popotonal. Two ndependent theoems concenng confomal tansfomatons ae: () Closed cuves map to closed cuves () Reman mappng theoem: an abtay closed pofle can be mapped onto the unt ccle. Moe theoems ae gven and dscussed n AMF Secton 43. Note that these ae fo the nteo poblems, but ae equally vald fo the eteo poblems though the nveson mappng. Many tansfomatons have been nvestgated and ae compled n handbooks. The AMF contans many eamples: ) Elementay tansfomatons: az + b a) lnea: w, ad bc cz + d n b) cone flow: w Az c) Jowkowsky: w ς + c ς n d) eponental: w e s e) w z, s atonal ) Flow feld fo specfc geometes a) ccle theoem b) flat plate c) ccula ac d) ellpse e) Jowkowsk fols f) ogve (two ccula aeas) g) Thn fol theoy [solutons by mappng flat plate wth thn fol BC onto unt ccle] h) multple bodes 3) Schwaz-Cstoffel mappng 4) Fee-steamlne theoy

32 Pofesso Fed Sten Fall 4 3 The technques of confomal mappng ae best leaned though the applcatons. Hee we shall consde cone flow. A smple eample: Cone flow. In ζ-plane, let W ( ς ) ς.e. unfom steam π θ. Say ς f ( z) z π θ 3. w ( z) W ( f ( z)) z.e. cone flow Note that -3 ae unt unfom steam. n n n w( z) Uz UR cosθ + UR sn nθ, whee.e. φ UR n cos nθ, ψ UR n sn nθ ψ UR n sn nθ const.steamlnes φ UR n cos nθ const.equpotentals z Re θ dw dw dς nuz nur e ( nur cos nθ + nur sn nθ) e dz dς dz ( u u ) e θ θ n n ( n) θ n n θ u u θ nur n nur θ < ( π ) n ( π ) < θ < ( π ) cos nθ n sn nθ < u >, u < n u < n, u θ < n.e. w ( z) Uz θ epesents cone flow: n unfom steam, n 9 cone

33 Pofesso Fed Sten Fall Intoducton to Suface Sngulaty methods (also known as Bounday Integal and Panels Methods) Net, we consde the soluton of the potental flow poblem fo an abtay geomety. Consde the BVP fo a body of abtay geomety fed n a unfom steam of an nvscd, ncompessble, and otatonal flud. The suface sngulaty method s founded on the symmetc fom of Geens theoem and what s known as Geens functon. ( G ΦΦ G) dv G Φ ds () V S S + S B+ S S Φ n G n whee Φ and G ae any two scala feld n V (contol volume bounded by s nfnty S body and S nseted to ende the doman smply connected) and fo ou applcaton. Φ velocty potental G Geen s functon Say, G δ ( ) n V+V (.e. ente doman) whee δ s the Dac delta functon. G on S Soluton fo G (obtaned Foue Tansfoms) s: -D souce at of unt stength, and () becomes Φ S S B Φ G G Φ ds n n G ln,,.e. elementay

34 Pofesso Fed Sten Fall 4 34 Fst tem n ntegand epesents souce dstbuton and second tem dpole dstbuton, whch can be tansfomed to vote dstbuton usng ntegaton by pats. By etendng the defnton of Φ nto V t can be shown that Φ can be epesented by dstbutons of souces, dpoles o votces,.e. o Φ Φ S σ GdS : souce dstbuton, σ : souce stength S S B S B G λ ds : dpole dstbuton, λ : dpole stength n Also, t can be shown that a souce dstbuton epesentaton can only be used to epesent the flow fo a non-lftng body; that s, fo lftng flow dpole o vote dstbutons must be used. As ths stage, let s consde the soluton of the flow about a non-lftng body of abtay geomety fed n a unfom steam. Note that snce G on S Φ aleady satsfy the condton S. The emanng condton,.e. the condton s a steam suface s used to detemne the souce dstbuton stength. Consde a souce dstbuton method fo epesentng non-lftng flow aound a body of abtay geomety. V U + φ : total velocty U : unfom steam, φ : petubaton potental due to pesence of body U U (cosα ˆ + snαˆ) : note that fo non-lftng flow Γ must be zeo (.e. fo a symmetc fol α o fo cambeed fled α αolft ) K φ ln ds : souce dstbuton on body suface π S B

35 Pofesso Fed Sten Fall 4 35 Now, K s detemned fom the body bounday condton. φ V n.e. U n+ φ n o U n n.e. nomal velocty nduced by souces must cancel unfom K steam ds U n n ln π S B Ths sngula ntegal equaton fo K s solved by descetzng the suface nto a numbe of panels ove whch K s assumed constant,.e. we wte M no. of panels K ln ds U n n π S,,M,,M + z z dstance fom th panel contol pont to whee ( ) ( ) vecto along th panel. poston Note that the ntegal equaton s sngula snce ln n n at fo ths ntegal blows up; that s, when and we tyng to detemne the contbuton of the panel to ts own souce stength. Specal cae must be taken. It can be shown that the lmt does est at the ntegal equaton can be wtten K K ln ds π n S S M K K + ds U π n S whee n n + n [( ) n + ( z z ) n ] n whee ( ) + ( z z ) z z n z

36 Pofesso Fed Sten Fall 4 36 Consde the th panel S cosδ ˆ + snδ ˆ, n S ˆ snδ ˆ + cosδ ˆ n n snδ, nz cosδ S B K ln ds U π n + SS, : ogn of th panel coodnate system, S S : dstance along th panel

37 Pofesso Fed Sten Fall 4 37 V U + φ K φ ln ds π S B K V n ds U n n ln π K Let + M K π S S B n ds S n K K + I π I l l I ds U n U sn and RHS U ( α δ ) ( α δ ) sn, then M ( ) n + ( z z ) nz RHS, I ds S ( ) + ( z z ) l ( S nz ) n + ( z z S n ) nz ds ( S nz ) + ( z z S n ) ( ) n + ( z z ) nz + S ( nnz nzn ) ( ) + ( z z ) + S [ n ( ) + n ( z z )], n snδ ˆ + cosδ ˆ z z S CS + D + AS whee A cosδ B C D I I + ds B ( ) snδ ( z z ) ( ) + ( z z ) sn( δ δ ) ( ) snδ + ( z z ) cosδ S D Χ Χ depends on f q< o > whee + S l l ds + C ds DI + CI whee Χ S + AS + B I ln Χ M K K AI q 4B 4A ( ) n + ( z z ) nz ds U n RHS ( ) + ( z z ) + S π RHS [ cos α snδ + snα cosδ ] U sn( α δ ) U K K + I π M RHS : Mat equaton fo K and can be solved usng Standad methods such as Gauss-Sedel Iteaton. ds

38 Pofesso Fed Sten Fall 4 38 In ode to evaluate I, we make the substtuton S S z z S S + + S S + whee S dstance along the th panel l S z z n S n S δ δ sn cos Afte substtuton, I becomes ( ) ( ) ( ) ( ) ( )( ) [ ] ( ) ( ) ( ) ( ) sn cos sn cos sn cos z z z z z z z z A B δ δ δ δ δ δ ( ) ( ) ( ) [ ] cos sn 4 4 z z A B q δ δ.e. q> and as a esult, E A S E q A S q I + + tan tan whee E A B q ( ) ( ) Χ + Χ + B B Al z l C E A E A l E CA D C I CA D AI C DI I l ln tan tan ln ln whee B AS S + + Χ Theefoe, we can wte the ntegal equaton n the fom ( ) M U I K K δ α π + sn

39 Pofesso Fed Sten Fall 4 39 k I π k I π K RHS whch can be solved by standad technques fo lnea systems of equatons wth Gauss- Sedel Iteaton. Once K s known, V U + φ And p s obtaned fom Benoull equaton,.e. V V p U Mentoned potental flow soluton only depend on s ndependent of flow condton,.e. U,.e. only s scaled On the suface of the body Vn so that VS C p whee V S V S tangental suface velocty U V S Φ S U S + ϕ S ϕ VS U S + U cos ( α δ ) + QS S whee U S U ( ˆ ˆ ) ( ˆ ˆ cosα + snα cosδ + snδ ) φ K φs ln ds S S π SB M K φs ( ln ) ds π : tem s zeo snce souce panel nduces no tangental S l ln ) S flow on tself. ( ( ) ds J S ( ln ) whee S l S S + S z z cosδ, S snδ z S {( ) S + ( z z ) S } z

40 Pofesso Fed Sten Fall 4 4 C J V S U V U K J π cos α δ + p, ( ) l ( ) n + ( z z ) nz ds ds l S S ( ) + ( z z ) ( S S ) S + ( z z S S z ) S z ds ( S S ) + ( z z S S ) z whee S cosδ, S z snδ S S + z z S + +, ( ) ( S ) S AS B ( ) cosδ + ( z z ) snδ + S ( S S S S ) λ D CS + D ds S AS C ( ) cosδ + ( z z ) C cos J ( δ δ ) ( D AC) cosδ cosδ snδ snδ snδ DI + CI DI + C ln Χ AI C D AC l A A C l + Al I + E + + ln Χ tan tan ln E E B cosδ + z z snδ D AC ( ) ( ) [ ( ) cosδ ( z z ) snδ ]( cos( δ δ ) [( ) cosδ + ( z z ) snδ ]( snδ snδ cosδ cosδ ) ( )[ cosδ snδ snδ cosδ cosδ cos δ ] ( z z )[ snδ snδ sn δ cosδ cosδ snδ ] ( )[ cosδ ( cos δ ) snδ snδ cosδ ] ( z z )[ snδ ( sn δ ) cosδ cosδ snδ ] ( ) snδ [ cosδ snδ snδ cosδ ] ( z z ) cosδ [ snδ cosδ + cosδ snδ ] D AC whee sn ( δ δ ) E z z z + B

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