Lecture-11 - APPH 4200 Physics of Fluids More 2D Potential...

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Unformatted text preview: APPH 4200 Physics of Fluids More 2D Potential Flow Chapter 6 1.! More examples from Chapter 6: 2D, Inviscid, Irrotational flow 2.! Blasius Theorem (Lift and Drag) 3.! (Easy) CFD: potential flow in 2D 1 Cã ~ :J~ x .U . -. (/ Il ~~ J -\ rL ~ T '3 -~ )( ~ \J J 'J ~ l v ), Lrl t7-l t J. ç ~ ~ ~ L~ .. ~ t 0 v X òa 3 x \ x :J .. , \ Complex Velocity Potential \) ~ ~ 1 ~ ~ .. ~ l It i " Ì\ J XL ~ ~ ( ~ * ') .. lA ~ - ... ( G ~ ~ ..~ f.. ~ t lli ~~ i- ~ ~ ~~ J 'oJ ., Vi r¡ \/ u. ,).. ) ~ \J (e i ~ VI :J~ ~ 'v ~\J-S X + ~L:; J \ IJ U-J f- -J 0 tN ~ f ~ ?l I\ ~ i", 1-- U \) ... II l~ J ~ ~ ~ '.. t: \) () ~ \.C, ~ 'J l~ .. ~ f ~ \J lJ ~~ ~ "- I '. ~ 1\ ~X II .. .. ~ .t v" ~ \.\ J. :: I II -. ~ ~ ~ .. .i " \J r.. .0 \. l1 J "- ~ t t II J ~ ~ )'-' l- v i( J lij ) -l -. (cl , ~ a G çt () V\ ~ ? J \: II '" ~ ~ ~ a ~ ~ lU .. ) UJ .) l~ ~ ~ .. ~ \¡ i 0 "- (.! i. \t - L- ~ 't '£ ~ ~ t ~ \J \J ~, ~ t. ,~ \f - -- ~ ,J \ ~ lJ Ç) ~ ~ ~ " i. 't ). ~~ ~~ ~~ l\ () \. \: ~ l j\(' ) lL J 'u li lr 0 't .3 C' (V ~~ ~ ~ ~~ ~~ III Ç) ~ ~ V\ r- 0 ') ~ -! ~ () . l. t)-f .. J~ ~ \ ~Uj ~ ~ "- 0 ~ ci ,v t ~ ~ Ii " l" ~ '.3 () \l fJ" . ~ Ii Q LL (.1 ~ .. l~ ~ X ~ ~ \. '.. J. ~ .l \J ~ IV +- ~ \\ i r' ~ ~ ~ ~ l '0 ;) (': .. '\ Vl 't V) "" t\ \. 3 ) "0 ~. ~I --'- -.. .0 L I-= J t 3 t: v ii ~ ~ lL v t ).. () fà Q. .. ili ~ f- ~ ~ ~ l 2 ~ '" ~ ~ ~ Example Complex Potential J- ~ Ii ~ r). ~ -j )I l( ~ ". .J v: '- ~ ~ ). IV 1- ") "- q: 1 ~ '0 t, II "t Li -: 'X ~ J- ~ "v ~ Lt a: II NI LI () tJ J- j "\ ~ )lNJ ti 'i l- rl '~ :i- ~ J- v' l -\ ~ tJ ~ "" ~ ~ '," ,J f t lV ~ Vi ,J \ \ 3 l- .l ~ u. \ iI Li l LL lt ~i '-'-\- r ( ,) ~ ( V' 5 t t Ii d (- 4 :i .. c: - ) tJ (I _ji."'()~ ~ ~, ,~~ 'co a. .. :J 3 ~ l\ "' J çt () ~ r- L ~ lD X l Another Example \i II Vi .. l~ t- VI \ lQ .. ~ .J ~ ~ VI ~,l l-- ~~ -. . " ~ ~ 'J Q J: l,J V\ 0 ~ .. .. , ~~ ''" le .j ~ f\ ~~ ~ \ (' (i C" çt ~ (I C" q: ~ \. +- , ''' J-- /" ~'X (I ~ ~ 'X ~~ I t¡ )( II "Y.. (J .) X' 8- \. .. "- 't f' lj ~ r" ~ v... '- ~ '0 (.. 'v ~ "t 1 L! tf :ì ~ '~ '- \. J- r- ~ /' II J. .. r" Cl ).. ,. (V "X Q: "' (' I J. /" t' .. ,'-. 1- y. "çt ll t" r' çt li o r+ '.. /Î // .--./ .- V' '" l~~ :y 'l " ~\ \1 I~ t\ ) '\ ~ t 1- )~ ~ V\ V' t- ~ o 1 ~ t ~ t ~.~ .j \~ '0 ~ ~ 4 ~ '\ ~ y. I'" i 't ~~ r (c Yet Another Example ". .. t ? , ~ ., ~ '- t .. " .t ~ ( " "- .. .) c; '- ~ ~ ~ -l , v 'V + V) 0 ~ 'N ~ , \, II l; S ~ Q: -. .~ -. II .N ~ ~~ ~ v1 0 '-(. ~ ct II ,,' ". ~ '- ~ .. VI ... IJ ~ '~ '- ': ~ ~ "- \/ "t \ yN II ~ ~ ~ '0 '" ~ ii '\ "''1 c: C) N \~ ~\ ~ ) \ N' ~~ l/ c:\ t" It '- ~ ~ ~ ~ -\~ \( ~ l- ~ ~ J- / Òï i ~ '\ '~ ¡ ~ I ~ i 'Ç\" \\ - ¡ ~..~0 l "" -e ~ ~ x 5 G 0 t/ ~ ~ \J Li: Mass Source .~ -+ L, '" ~ ~ .. (i' r ~\~ l. .J ~ ~ ~ 'i I, ~ tI ~ ~ ~ t ~ \~ \I ~ V' 0 \1 \; c ~\.~ ~ ~~ .3 ( C' \i ~\~ IJ X ~/" I "v )- ~ l/ ~\ i ~ ~ ~ 1.\ ~\~ /" QJ c. ~ ~ \.~ t: li 'b S- '" "- .. ,. r\' ~\~ .. -: '-. ~ ~ 'i ~(~ II NJ ~ ~ J~ Ll r' t+ "- :l '\ ~\~ l, ~ ~" I. ~ I. '- .J ! ii V' ii ~~ ~~ \J \J .~ ~ "- ~~ s" ~~ 0 V" 6 G Y. ~ É 0 ~ "' F ') l 'Ì .. ~ Q ~ Line Vortex Î ~ . .. L .r Li c:l ~ '- II ~ ~ C-(E . 'L li r" r.. ~ \t ~\ ~ II .. 'b " \ ~ C-i~ \ 1I ~ " ~~ ~ 3- ~ ~ ~i ~ l ~ t' Zß c;1 ~~ v i 1I ,, .Y '-! X 'r. c:i ~ . C' .v I ii NI l\ L- \ \~ ~ VI Ç) tJ ~\t ti ~~ .. ~ "\. r-\ £ li ~") \' '\~ h ~~ 0 \l :: " ~ . "- II .0 ~ \ Nl ~~ J \ ~ & L ~~ 1~ '\ v ~ .. lQ \J ~" ~~ Il .. ~J ( .. .. ~ '\ \ 'i 7 1- Q o ~ J ,.. l~ “ Dipole” or Doublet c. t ~ .. ~ Q '- ~ ~ 'Ç G t r ) ~ UJ i 1l l.t "" l( ~ \. ~ ~ t '" 0 '0 VI V ~ V\ I ~ ~ ~l ~ l ~ v'O i- ~ ~ ~J ~ II r r1 '- .3 ~ \/ ~ ( '0 v l '- ~ 1 ll v ~ .r ct tv ~t. ~ " 0 1 ''," (t ~ "" Lc l- i -- -- 1- I~\ (' -- 't- l-\~ i- \. ~ r't + -- k- l ~ V\ ~~ \i li ci ill ~ t ,j .. ~ l ~ r- ¿ (/ lt c- ~(\~ él k 1" ~ JL ~li\ ~ \r'+ ,-' II , 'n .¿ ~ k: ~ . '- I ~ \ l' ~ r~ "- '- ~ Ll ~\ ~ J: i ~ ~ teN t: "~ ~I~ II ./ l'- 3 "( l. t 8 )" -- ~ ).. ~ ii. j (The “easy” complex way) ,J LL () Q. ~ ~ ~ ). ~ t .. " V (L r -' J ~ '1 ~ ( !t ..~ j " Flow Past a Cylinder 'ò 'V I \) +- t-( L o- (ò ~ \' :: Ii Q I l- + ~ :: ii r- r\ '- J 1Uj ,,-1 ~ ol \ I \ ~ ~ CD .. ~ Vì ,v 4CD V' .. .. ~ ~ ~ i \b " 0 ~ ~ ~ r- - ~\ " ic. :: ~ \l ~ \" ~ C) ¿ ii i "i Ô \b ~ .. "\) ~ Ol~\ QI t', :3 \/ ~~ :: ~~ II ~\ ~ CC . C( li c- l' -\ , ~\ t II :: Q) ~' ~ N ~ \\ i Ci .. ~ li /'i. II C' '- ~'\ l- J ~ \! V\ 0 ~ ,. ,. h ~ '- ~ + i- "- ~ ~ :;lLi .) l~ ~ ~ ") ~ .,j ~ J Uj ~ \( ~ _ -tJ ;) "" .. ~l " '.. _0 ! i, ., f i: 3 _ .. I( ~ ~ II 1 ì\1~ ~- k) ~I\ '- Ii \ ~ -: ' "- ~ .. ~ ~ ~ iJ :: ~ ~ -- l -'- '- ~l \ ~ ~ r~~ '- ~ "" 9 ~ -J ~ .. V ~ ~ ~ ~ .. ~ ~ \! 1 -: ~ 1I ~ 0. ~ 0- -iN D ~ ~ ~ J .j .J .. (V Bernoulli’s Principle t:(,. i- cJ ~ \ II -\ ,. ( , '" )~0' ~ ~ ~t~ tV .. .J ¥l 2 0 ~ __ '- .-- ~ Il .:1 Q. ~ ~ ~ 1~ ') ~ -i ~ J\ll i: ~~ l ~~ ~ 12 - - - I~£ (~l.ÚJ~ r~ . lL. 'L c: JJiE()~ I I( V\~,N1l ,~ ~ ) ~ l) ( '" \~ t ll¡ ~( .. t.( ~ tt ~ ~ i V\ ~ ( i:t~ '-~~ "i! ~ .. ~ ~ lt ~ ~ ~ ~ - u C- 9J \ ~ .. - "( .. Vi CD V) 15 ~ :J ~ vi e. N~ :: -\ AI + + r4J (/ O-~ -T '- '1 ~ ~ '" ~.~ ~ Q- (L -. .: F- f' \ l. V f. 1,- V\ .¿ '. 3 \ ;~J~l:~ ~~ -.)~ - \l (l (V .. .: ~ \: (' .. rv ~ ~ a ~ .. \r ~ r- II ~ (" ~~ f' Ii ~ ~ ~ iU ~ .. .. ). l( II Ç) ¡¿J v t lL , J Vi \1 ~ c -i çt 10 Blasius Theorem lel these port, latter I, the point nically Fig. 1. Blasius in 1962, after retiring Fig. 6. Blasius during lecturing, 1925 math Paul Richard Heinrich Blasius (1883 - 1970) was one of the first students of Prandtl who provided a mathematical basis for boundary-layer drag (Chapter 10) but also showed as early as 1911 that the resistance to flow through smooth pipes could be expressed in terms of the Reynolds number for both laminar and turbulent flow. 11 .( (l. ~ L. -J c: \r :J 'v ~ iU. \) \ I \) .. t- -' lL 'X .. ~ '-.. ~ 'V . ~ 0- t, iv v ¡. Ql ~~ \ /"). l .. "- :t / ~t f-~ .J ~- ,. )- ~S~ i " .. ~ ). ~ (¡ a. '- ~,- Q ~ ~ I ( \1 \ II t ~ Q. Q. j -. ~ c '\1 -J ~ .v l o 't "- .. '~i~ ::. \L l "'.. i ~ t/~ / Blasius Theorem ~ rt ~ Q. \ .. ~ D \. ~ ¡ -. -i \l .v L\J o V1 \~ ) ~ ~j. 'v c: x ~~ 'c.). :: ~~ "V + 1rv~x ') ~~ a. ~ Ii ~~ + --I l" C- (' :: -\ f' -t l" tV 2) 0. -tN -l ~ (I r- '- \1 n- "~ J. .v .J .. J ~ ~ ~ (:) Çj ii t1 ~ I,, , \. 1' I ~ '.. ~ '.. t :: x ~ ~ o s: ~ rl J~ .( rJ ", t 3 t~ ~ -t (' 1\ o~ u 'v ~ l1/ ~ \!~ ~~ ~ . .. 1 \ Y. 'I . "- ~~ '-IN ~ ~ "- \ ::J. .Iv '~ ~ )( :3 .'- ~ :: ... )- ~ .r ~ -- l\ ~ /" - ~ ~~ '.. + ~ ,\ -\,\! ~ "~ :i 'V :: '- '-.) ~~ ~ :: \ -\ (' to Q, "1", + c:~ L- \1 + Ce t) ~~ ... I 'v \\ .. i \J \ o 12 ~ 0 ,J L!Y W ~ i .. c: v ). ~ Example: Flow Past a Cylinder rcv~ '-.) I'" ~~ 'a- -t f\ ~ ."' li -... L D ~ ~'. L ~ 'J \\ J L (' ~~ \l ~ 0 (( '- \/ ~ .. ~ ~ t "\I ~ ~~ \ lo t) '- ~ l\ ~\: ":~ \i x. :)~ :: J l- ç: N ri ~~ + 2) \ f' i ~ldJ ) II ~ ~ (' ,~ (i :J + :s ~ I 'i ~~ \ CV .: (J r-~ 3\li ~~ ~ "t j\i\ "- (i :: l\ ,3 ~ II ~ II 0 d /' ) .. ll ~ - "" l~ ~ 0 J Q \\ C' ~ ": \:\~ :i r\ ~ ~ 13 ~ ~ ,l ~ .'\ (Problem 6.9) ~ ~ r- 3 LL -. 0 ~ r ~ ~ (~ 0( "" ). \J .. '- ~. ~ -11 ~ \. .. ~ . \) ~ Flow Past a Rotating Cylinder ~ ~\l ~ qj u -i .\l \Y ~ ~ li ~ '.v ~ \\ ~ - In (. l~ -. ~ ~ '-\~ II ~ ~ \ k: \ t ~ \L ('~ + ~ c. \' ~ ~ "' 4:\~ r-- ~ f' VI 0 CJ \'\ c ~ -- + c. "j 'II l- ct '- .5 + L l/ -l ' -. '- (. ~ ~ ¿ ~ ./ =\~ "+ ~\~ + 'i :: L¡ -l" "" J 'i -~ t' ~ ~ t: Ji .:\ ~ \, rv r ~ .: rv ~ ~ j ~ ~ , Q "" ~ 0 - ).. 't ~ \i .V' t \j -' ~ ': V' l :J \) J \ LL .. :5 li ". 0 II ~ r' ~" t.1 i ~ l' i .L ::''-t: ') It ~ ~ -r~ t. ~ ~ .. "" + ~ ~ l- l - +, ii \l C' "' \I ci .. '-\. . I. ''" li ~ k 'V -i ri l\ ~ ~ 3" N~ ~ ~ C" c: .. '= V\ 0 - '" ~ ~ ê ~ ~ l ~'. ~~ .. ~ , ~\ ~ ~ rt \ ~ ~ .. -- 0 lì cl \ \i -- 0 ~ l f" :: ~ '-~ t: ~~ ',- ' '-v- t:~ ~ J\~ ~~ \I 0 ~ :: ! ~ rV "r N 'v L-i ~ 3~~ 0 \5~ ~ ~ 14 ~ .. A~ J. "I ) ,0 . ) ~~ "" ,~ '" ~~ ~~ t~ ~J '\ ~ "" :: ~ -i v i') l- C) l.j l t tr 'tJ t- C1 t '\ \. v 1 iL " ,./""" f0 t t ~ r: ': ~ )- \J t \) .. ..~ ~ l: cv l\ ~~ (' ~. ~ U '0 II C' ).. \\ 0l ~l" l. N )( ~ r4 ~\" II ~ ~ )) " ~~ .. II "' \t ~ \. C.i !U ~ F- \l l~ ,lLQ .. " ~ t '0 ~ ~ ~ - -£ c .:0 qJ t J~ r" s- s' '0 t , , ~ ~ J t.. tJ l J :l t ( /j r- Q ~ ~ I: r ~ ~ ( c v '- " -- )" :? -ct -. V ':r t -j 3- ~ ~ ti ri ~ ~ \ ci \ l\ J~ ::J- ) "- "l 't 't u. cD .3 0 1 ~~ w . '- J\ v . " .. I. L. .. "- L- .. ~ J ~ r- C; F: lL ~ 1- ~ ~ .t Q. (: C" F 2 ~ J ~- -. ~ ~ v -- c: lu ( ': 2 -. .. '0 Numerical Solution to Potential Flow in 2D 15 ~ ) ,~' - ~ tl ~ "S ~~ ~~ "a tC ~ )( .c rl ~~ .l l¡ ~ eV Y- ~ ~ "- l ~ .( + 'X "- ~ Cô l- -T \ 'X ~( +- O: ~ 'j (l l ~ '/ ~ ~ l.. \J t..u ~ ltJ iL .. U- Q ~ .. -~ lL ~ ~ ~ ~ .. I:. i .. Finite Difference Approximation .. i ~\N 'cÕ ~ .j-\ C1 I ~l(\ -+ ~ T\~ rb rb ~r T~ -. ~ ¡v\õ)~ )( ~, ". " I ~ ~ 'X L /" '- .).i \. L r'j ~ I ''''i ~ + 'i ~ 'X '- .. ~ l N i '~ ,, .- rJ y. .~ ~~ ~ .; ~ l ~ ~ L )( .. 9- ~ Z( ,J Y- .. ~ (( 7 "; l:' t J j Ul ~ ~ ~ -. ~ ~ ~ -- .. 'J ,/, \ " ~ ~ ~ ,or ~ ~ I" -. .) ~ \ c- .a .. 4- ~ 'j- \ r- U 0 á (\ l- ~ ). .. ~ -: + -- ~ ). '- ~ .q N 'i ~ -- -y. ). C" \ r')( ~ 'Q )l .. ~ .. -). )i J- ~ 16 ~ lN .J ~ ~ 't '/ LL l~ t (j vi Three Examples \, , \i \. \. \. '\ \. ~ "- t; 4; ~ '-t ''- ~ \i l;t "- ~ '\ "" "' " ~ 1 ~ '- \. \. "" "- "- '" "- " "- \. " " \. " ~\ 1 s~ Mathematica is available at all Columbia computer labs. See http:/ /www.columbia.edu/acis/facilities/software.html 17 Problems in Ch 6 • • • “Doublet” (or dipole) potential Drag on a “half-body” Potential flow around an ellipsoid 18 Problem 6.1 19 ~ 0- ;: ~ -J ~ lJ ~ ~ Problem 6.1 rVv ,.. +- ~ ~ t. \ ,t u 4- ~ , -. VC i N. +- , (~l N ~\; 'V \ rt '- +~ '- "t ~\l; ~ + r- L- \~ ~ '- ~ l- 'V I' x 'V .N ~ \¡; . '- (( ~ t VQ t.\k: .~ r-\ t J \'1 \. '- "- L ~ l J ~ ~ ~~ C-\ ; .. Ii rf' ~ ~ .)- "0 c. ~ ~ \rl (f ~ ~ t Q: ~ V\ /.. t ti\ o ~Q ~~ l2 0. ~ ~~ ~~ :t I ~ \j \~ \b J iù fZ \I )-. li l~ _ C- ~~ 'i lt .. 20 Problem 6.2 21 e ~ l. Q: .. ~ ~ N , '- Problem 6.2 N I (l ~ t\~ ''- ~ ;. ~ ~~ ~ ~ 'f ~ ~ -~ "- ~ ~ ( ii 'D ~.I~ ~ + 'b N cJ ll~ '+~ .. -it ~ ~ ~ ,:J ~\~ t- ¿ ". II -L\ ~~ J~~ - (' ~~ ~') (' +- ii r. :: ~ -i 11' -+ ~ ~ "" ~~ Li I ~~ ~ \' ~ q~ c- L ~,¡ ~l\ ~~ i- ~ l~ N :: ii 3 lI~ ~i~ ~ ~ :) \\ '~ ~ tA ). ~ .. V) t" 0 (j :: ~i~ ~ /" l" \' (J + t' ! ~ r + '.J\) ~ 'V - 1- "' ~\rJ rI o ~ ~( ~ .. 'i ~ 1\ -F ~ ) i\~ l( 'J: :: :: \: 'Ii -' Ii It:: ) 'J D " 0 ~ 0 Vi l\ \. ". ri~ II C1 ..~ ,+ \~ 1 r ~Vl CL ~ ~ .) G) ~ \.- \J ~ ~C) il~ . t\ ( :: ~ \t (i vi ct ~ fJ \I ~ e: 0 3l rt ~~ 22 G) ~ ~ V ! ~ .. ~ (Ü . '- (" ( r- \ Problem 6.2 (cont) ~IN lo "" ii '" ~~ lJ .. ~ , QJ i \~ r Q ~ \i ~ ~ t ~ ~ -Vi Ci \:: '- ~ ~ l -~ r\ ~ I" l \r r- ~ - V1 )" a ~ ~ ll t\ \. \, 't I-L I: ~ ~ t- 0 ~ ~ ~ ~ ~ N l ~ \: r- ~ 0; ~ l 1 \ \b c\ + ~t 't \ t;' i~l ~I ( ~ ~~ 'V ~ ~\~ () t ~l~ ~~ l\l ~t J; ~ .~ '- .l l ~\ ? \t -- r- ~\~ l~ ~~ ~ 'l ~ I \1 ,\ ~,~ (~ '1~ ~ ~~~ L L- t:(' ~ u~ 1I ( r:\ ~ ~~ -" V" t t' _. ~~ o ~ Q II .~ ~ ~.~ t: '. C5 a V" Q fS 23 Problem 6.4 (Rankin Ovoid, ca. 1871) 24 ~ _..~ rt (Rankin Ovoid, ca. 1871) ~ ~ I .L ~ ~ ~ l c- ~( I~ ~ t t\ ~ ~ (~~ l/ -rt '- .3 :r ~ "" l,~( rv ~ I I ~ ,. ~ '- ~R Il c\ C" -l ~~ x. :r I ~ \ r-l ~ /" rJ tj t~ ~ 1/ Figure 6.8.3. Streamlines in an axial plane for a combination of a point source, a point sink of equal strength, and a uniform stream. ~ Il '( \f\:J -+ ~ +1 l' It 'IJ V\ i- , o. ~ \. ,)( ~ I 'f l a ~ .. \/ /- .. ~ ~ () ~~ .( x' i F- ~ C/ J~ t '- u ) II~ ~~ ~ II \ty \) ~~q V\ t - \j Q ~ ') ~ I~; ~ r ~ '" L; -; LL l- ri ~ f~ ~ ~~ i ~ "" (6.8.33) iJ &. erated by. a body moving in the direction 8 = 0 when "- rN orm stream of speed U in the direction 8 = 71 (so as to give a flow axes fixed in the at .infinity are used), we have \1 (\ ~ L gth m at T= d,8 = 0, a sink of strength. -m at r = d,8 = 71, .and .a (' ce of strength m at the origin is - (m/471) cos 8. Hence, for a source of 't ~ o \L ~~ j lr\ "- ') ~ "- ( '- 1 It ~ \n .mline is closed. The stream function describing the flow due to a 1/ = - m cos 81 + ~cos 82- l Ur2 sin2 8. Oc ~ ~ ~isof symmetry C) .. () J CJ \ ~ ( ,~ ~ ~ :r , '- :t 4. .. ~ '" ~ Q- ~ Problem 6.4 xis, it is evident that, provided the source lies upstream of the sink, none e streamlines coming from infinity flows into the sink and the dividing 471 471 ~ niform stream. :just one source of strength m and one sink of strength - m are placed on 1 - _"..n...:".. :,. ..,.~"" _10""'' 'h.." .&"",,,...0 t. Q '" "l'Iri,:''k ,.1"'r\ ("hl'"'11~ +hO-t:+fIoltt"l;"n.o,- ,~ ';r J- " \4r1 +l (( \l " Ii Q X l- ~ (. II ., "" \D "v t ~ 't , '- ~ V) ~ '- ~ \; l'~. ~ V\ ',.. ~ "- tV C" ~ 'v i+ ~ ~~ \ £\ '- ~ '- ~t i~ ~J~ ~ CJ- ~ \t' "-.~ I ~ ~- ': 'S \1 t\K ¡~ \1 .. ct "- ~ \-i~ ,~ 4. Ç) co ") u. to :i \! .. ~ I 25 ~ " ~ cL Q .. .ç 1.11 ~ Jl ~ ~ V ~ ~ Problem 6.4 (cont) l\r -i -- ( l: :t l.c lÇI ~ '- lL ~ /\ N QJ L .. ~ "-C ~ \1 ~ \~ )~ ZlN l ..\ ~ '- .. I ~ I ~ 'f ,-C :: (I i\ ~ 't '- ( ~ ~ ~ .cl~ c~ L l( ~ ~ ~\ ~ ~ ~ V \- c. () ~ L( ~ l\ ~\~ H B~ \ 'i - -!IJ "- ~I ,0 iJ ~ ~ ~ \j ., '- ~ ,. i - ( 't f' 'c /I t: -\i '- (l l:) Ç: Y. () \. 26 Summary • The complex potential is defined on the complex z-plane (z = x + i y) and contains both the velocity potential, ϕ(x,y), and the streamfunction, ψ(x,y), or vector potential. • In 2D, potential flow is very quickly calculated using today’s computers. 27 ...
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