Ch.8, Voticit (epetition) Velocit potentil Stem function Supeposition Cicultion -dimensionl bodies Kutt-Joukovskis lift theoem Comple potentil Aismmetic potentil flow
Rotting fluid element Chpte 4 Angul velocit: dt d dt d z 1 Fo smll ngles: dt u d ddt v d 1 dt v d ddt u d 1 d d t d v d t d u d t d u t d v d d
Rotting fluid element Chpte 4 Let u dt d v dt d dt 0 d d t d v d t d u d t d u t d v d d u v z 1 Angul velocit z v w 1 w z u 1 Note tht in the cse 0 V V ot 1 1 Voticit: The flow is clled iotionl if 0
Inviscid flow: V Dt D g p V 0 DV Dt g p Eule s eqution Integting long stem line leds to the Benoulli eqution w w u Iottionl flow V,, 0 v z u z v
Inviscid nd iottionl flow If the flow is iottionl the velocitpotentilen V V 0,, z, t u v, w,, cn be defined V u, v w z z Continuit: z 0 Momentum: t p V gz constnt
Stem function: The velocit components cn now be witten s 0 u u v Compe to 0 v Inviscid nd iottionl flow, Potentil flow: Fo D-flow: u v Stem-lines nd potentil-lines e pependicul
: Relit :
: Element flows which cn be combined to cete othe flows: 1. Unifom stem: U U sin U U cos u U v 0 U
:. Line souce/sink: m mln Stength m Q b Q b Volume flow Width noml to the plne v v 1 m 1 0
3. Line vote: K K ln Vote stength K v v 1 0 1
Supepostion: Gficl method Empel: line sink + line vote t the oigin. Connect points whee s v constnt
Supeposition of element flows, empel: Unifom stem + souce = Rnkine hlfbod U sin m U cos mln
Souce in (,)=(-,0) Sink in (,)=(,0) s k mctn mctn m k m s k s m k s tn k tns k k s s m k m s
ctn m m s k s k ctn ctn 1 ctn tn tn 1 tn tn ctn tn tn 1 tn tn tn s k s k s k
Souce + sink t oigin: Doublet d lim 0 mctn m m d sin cos
Kpitel 8 Unifom stem + souce in (,)=(-,0) + sink in (,)=(,0) = Rnkine ovl p k s U sin m The ovl is fomed b the stem line 0 k s U mctn k s k s m k m s L
Rnkine ovl: 4 4 4 m U u X-velocit Stgntion points in (,)=(L,0) 0 4 4 L m U L L L m U U m L U m L 1 U m h h cot 1 1 m h U m U u
Clinde: Let 0 h 1 u L m U i.e. cicul clinde = Unifom stem +doublet d U U sin U sin U cos cos
Kelvin-ovl: Unifom stem + vote pi in = U 1 K ln
Cicul clinde with cicultion Unifom stem + doublet+line vote But let s fist intoduce the cicultion
C Cicultion V cos ds V ds ud vd C C C wdz ds V Iottionl flow: V V ds ds d d dz z d ds V C d 1 1 0 NB! 0 if C is closed cuve is not vlid if C suounds vote cente
C Fo line vote: v v 0 K C V ds 0 K d K ds V Cicul clinde with cicultion Unifom flow + doublet+line vote
Cicul clinde with cicultion sin Unifom flow + doublet+line vote U sin Let the clinde be fomed b the stem line 0 K ln C 1 Cn be chieved b U C1 K ln whee is the clinde dius U sin K ln
Cicul clinde with cicultion Chpte 8 v v 1 U U cos 1 sin 1 K The Mgnus effect
Kutt-Joukovskis lift theoem Velocit t the clinde sufce: Benoulli ields: Rewite fo sufce pessue v 0 v U sin K U p ps U sin p s U K K p 1 4sin 4 sin U U K v D s 0 The dg foce cn be witten s p p cos bd
Kutt-Joukovskis lift tteoem D s 0 Intoduce p p cos bd K U U D b 1 4sin 4 sin cos d 0 n t cos sin d 0 0 D Alembets pdo: The dg foce on ll bodies submeged in n inviscid fluis is zeo v 0
Kutt-Joukovskis lift tteoem Lift foce L L 0 U sin d U U L s 0 b 0 0 b Chpte 8 p p sin bd sin 4sin 0 K b U 0 sin sin 3 U 3 d 4 sin 0 d U K b Ub sin d b 1 v cos sin 0
Kutt-Joukovskis lift tteoem
Comment to empel 8.3 NB! Dt in fig 8.15 is vlid fo Re=3800, in the emple Re=60000. Re=3800; C L =1.44; C D =0.9 Re=60000; C L =0.91; C D =0.7
Flettne-oto Union Rotoplne, 1931. The ship Bucku, built in Hmbug 190, clindes, L=18.5m, D=.8 m. Speed 5-6 knots
Flettne-oto E-Ship 1, built in Kiel 010, 4 clindes + popelles. Speed m 17,5 knots
Comple potentil Comple velocit: Some emples: Unifom stem Line souce Line vote f f Chpte 8 z, i f, df dz i i i i f z Uze z m z ln z 0 z ik z ln z 0 u iv
Comple potentil Confoml mpping f Clinde with cicultion i z Uze mlnz z ik z z 0 ln 0 Joukovski ifoil z 1 z
Cone flow f A n A n e in n detemines the cone ngle n A cos ia n n sin
Cone flow f z Uze i Mpping: z A n with U 1 0 f A n A n e in A n cos ia n sin
Mioing Used to genete wlls
Aismmetic potentil flow 1. Unifom stem: 1 U sin U cos v v U cos U sin. Point souce/sink: mcos m v v Q 4 0 m 3. Point doublet: sin cos Note tht thee is no equivlence to the line vote
Aismmetisk potentil flow, Supeposition is vlid s befoe. Unifom flow+ souce i (,)=(-,0) + sink i (,)=(,0) = Rnkine ovoid Unifom flow+ doublet = sphee 1 U sin sin v v cos 1 3 U 3 3 1 U sin 3
Hdodnmic mss(, dded mss, vitul mss) When bod is cceleted the suounding flid will lso ccelete. The bod will then ppe hevie F m m h du dt Assuming potentil flow KE fluid 1 1 dmv U el mh
Hdodnmic mss Empel, Sphee: KE fluid 1 1 dmv U el mh v U 3 3 U cos v 3 3 V v v el sin dm sin dd Integte: KE fluid 3 3 U m h 3 3 Clinde: m h L
Hdodnmic mss Vibting clinde:
Hdodnmic mss Vibeting clinde: Hdodnmic mss in eltion to clinde mss t ving vbtionl fequenc