STEADY, INVISCID ( potential flow, irrotational) INCOMPRESSIBLE + V Φ + i x. Ψ y = Φ. and. Ψ x

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1 STEADY, INVISCID ( potential flow, iotational) INCOMPRESSIBLE constant Benolli's eqation along a steamline, EQATION MOMENTM constant is a steamline the Steam Fnction is sbsititing into the continit eqation, v and is a soltion to the continit eqation whee, v EQATION CONTINITY V k i V V

2 d d γ d γ 1 dv ω tan γ ω, tan α α ( γ γ ) fo iotational flow, ω, dv d 1 fo small d d d d otation and 1 v sbsititing to pove this, γ 1, angles 1 tan γ 1 dv d dv d d d is a soltion to this eqation whee, γ is thevelocit Potential

3 φ and ae pependicla d d d vd whee is constant d d d v d d dψ ψ d dψ vd d whee is constant d d d v d d d d ψ d v v 1

4 LAPLACE s EQATION i i soltion combined i i soltion combined i i i soltion combined soltion combined F F F F soltions, F, can be added Laplace's eqation is linae fnction and soltion whee F is a z F F F F the fom, Laplace's eqations is of

5 It can be shown that and satisf Laplace's eqation 1) sbstitte the epessions fo ψ into the iotational condition ) sbstitte the epessions fo into the continit eqation Fo Continit. v fom the definition of v d d sbstitting,, Fo the condition fo iotational flow, v d fom the definition of sbsititing, d d, v d d d, V ( ) ( )

6 EXAMPLE V ) ( p p ) v ( p p V p p Eqation Benolli's veif soltions, d v d v d d O O O ρ ρ ρ

7 NIFORM FLOW SOLTION, v, integating, constant choose,, at, in clindical coodinates whee. cos, sin cos in clindical coodinates, sin -

8 SORCE FLOW SOLTION π tan π q whee tan In Catesian Coodinates, π q ψ 1 q R R R ln π q whee, In Catesian Coodinmates ln π q d π q q d d v, q volme flow fl q π π

9 POTENTIAL VORTEX SOLTION Potential Vote cons tan t cons tan t Solid Bod Rotation π D ω π ω C solid bod otation viscois, otatioanl flow potential vote inviscid, iotational

10 Ciclation Γ cons tan t ciclation, Γ Γ Γ Γ π cons tan t cons tan t cons tan t π Γ π d cons tan t Γ π S Potential V d s d Vote and d Γ d π Γ d π Γ π d Γ d π Γ d π Γ ln π

11 CIRCLATION intensit of otation in a contol volme d Γ Γ dγ S S v V ds V tangent to sface d ds v v dd dd vd v V V tan gential v v d S v dd S ω z ds

12 DOBLET and ( 18-1) q sin k, 1 1, soce, π DOBLET sink pls soce X 1 X a sin q 1 π X tan( 1 ) as fo small q a sin π K sin sepaated b a distance a ( ) a ( ), tan( ) q π a X X a 1 Since K sin 1 K cos K cos

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14 DOBLET PLS NIFORM FLOW ψ K ψ sin nifom flow K sin sin - is a closed sface π, K R ψ ψ ψ K sin a cicle of doblet doblet of to get a cicle of adis stength K eqied R, cos cos R sin sin - at the font stagnation point at the ea stagnation point the sface of cos - sin R cos cos R cos R cos the clinde, sin R cos R cos

15 STAGNATION POINTS stagnation points at at, the tailing point and π, the leading point at R sin cos R, v cos, R R π cos cos cos at all and R R cos R cos R cos

16 ' sin sin ( ' 18) Benolli's Eqation p p p ρ ρ p ρ sface p p sface sface ρ ρ (4 sface sin ( ' ( 1 4 sin ( -18)) ' -18)) ' d distance to ea stagnation point

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18 DOBLET IN NIFORM FLOW doblet stength K

19 Jakowski Tansfomation (of z 1 b a z z the potential flow, field)

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21 MAGNS EFFECT lift geneated b a otating clinde in nifom flow nifom Flow Doblet Potential Vote K Γ ψ sin ln ln π π K cos Γ R cos π K - Doblet stength Γ Potential Vote ciclation d d stagnation points 1 d d,, R

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24 Jokowski Tansfomation z1 az a 1,b 1 b z nifom Flow 4 m/sec 5 degee angle of attack Cicle with K4, cente at (-.1,.8)

25 Ktta Condition Viscos flid can not make the shap tailing edge tn of the indal flow soltion. Rea stagnation point mst be at the taining edge. Add vote stength to achieve this condition.

26 Jokowski Tansfomation z1 az a 1,b 1 b z nifom Flow 4 m/sec 5 degee angle of attack Cicle with K1.4, cente at (-.1,.8) Foce

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