1!"#$%&$'()*+,( )*-$'(./&01"$&%(!!( Inviscid Flow!!"#$%&$'(2+,($34*$/%(5015(50/(#$%&+-%(/6/&5($%("/7*$7$8*/9(:0$%(+&&-;%($"( 50/(2+,('+31$"(1,1<(=;+3(1(%+*$'(8+-"'1;<(+-5%$'/(50/(8+-"'1;<( *1</;(15(>/"?9( 2! :0/(2+,%(1;/(7+#/;"/'(8<(@-*/;(@A-1B+"%(,0/;/(! C(vC(1"'(p(&1"(8/(=-"&B+"%(+=(;(1"'(5(9(
Inviscid Flow! D"(50/(+50/;(01"'C($=(2+,%(1;/(%5/1'<(8-5(&+34;/%%$8*/C(50/( 7+#/;"$"7(/A-1B+"(8/&+3/%( 3,0/;/(!(&1"(8/(1(=-"&B+"(+=(r! )+;(&+34;/%%$8*/(2+,%C(50/(%515/(/A-1B+"(( $%("//'/'E(50/"C(,/(,$**(;/A-$;/(50/(/A-1B+"(=+;(( 5/34/;15-;/(T(1*%+9( Inviscid Flow 4! F+34;/%%$8*/($"#$%&$'(2+,%(-%-1**<(8/*+"7(5+(50/(%&+4/(+=( 1/;+'<"13$&%(+=(0$70(%4//'(2$705(+=(1$;&;1G9(H/;/(,/(&+"%$'/;(+"*<( $"&+34;/%%$8*/($"#$%&$'(2+,%9(! )+;($"&+34;/%%$8*/(2+,C(50/(7+#/;"$"7(/A-1B+"%(;/'-&/(5+((,0/;/(!(I(&+"%51"59(
Inviscid Flow! )+;(%5/1'<($"&+34;/%%$8*/(2+,C(50/(7+#/;"$"7(/A-1B+"%(;/'-&/(=-;50/;(5+( 5 ( ( ( ( (,0/;/(!(I(&+"%51"59(! :0/(/A-1B+"(+=(3+B+"(&1"(8/(;/,;$J/"($"5+(! :1K/(50/(%&1*1;(4;+'-&5%(,$50(';(1"'($"5/7;15/(=;+3(1(;/=/;/"&/(( 15(?(1*+"7(1"(1;8$5;1;<(%5;/13*$"/((((((((((((((C(*/1'%(5+(( %$"&/(( Inviscid Flow!!=(50/(&+"%51"5(L5+51*(/"/;7<(4/;(-"$5(31%%M($%(50/(%13/(=+;(1**(%5;/13*$"/%C( 50/(4150(+=(50/($"5/7;1*(&1"(8/(1;8$5;1;<C(1"'( ( ($"(50/(2+,( '+31$"(/N&/45($"%$'/(8+-"'1;<(*1</;%9(! )$"1**<C(50/(7+#/;"$"7(/A-1B+"%(=+;($"#$%&$'C($;;+51B+"1*(%5/1'<(2+,(1;/(! O$"&/(((((((((((((((((((((($%(50/(#+;B&$5<(C(2+,%(,$50((((((((((((((((((((1;/(&1**/'($;;+51B+"1*( 2+,%9(( 6
Inviscid Flow 7! P+5/(5015(50/(#/*+&$5<(1"'(4;/%%-;/(Q/*'%(1;/('/&+-4*/'9(H/"&/C(,/(&1"( %+*#/(50/(#/*+&$5<(Q/*'(=;+3(50/(&+"B"-$5<(1"'(#+;B&$5<(/A-1B+"%9(:0/"(50/( 4;/%%-;/(Q/*'($%('/5/;3$"/'(8<(R/;"+-**$(/A-1B+"9(! S(#/*+&$5<(4+5/"B1*(((((/N$%5%(=+;($;;+51B+"1*(2+,C(%-&0(5015C(( 1"'($;;+51B+"1*$5<(((((((((((((((((((((((((((((((((( (((((($%(1-5+31B&1**<(%1B%Q/'9(( T+;B&$5<(U(F$;&-*1B+"( 8 Consider a simple current in a flume (tank) with a shear velocity profile y B B x A A After a time, t, A moves to A and B moves to B. The line AB has rotated and stretched, and all the fluid elements on the line must have rotated and stretched. Viscosity (and turbulent shear) introduce rotation to the flow.
V/Q"$B+"(+=(F$;&-*1B+"( 9 The circulation is where V s is the velocity parallel to the curve, s. dk K ds V s This can also be found by adding all the small circulation elements, dk V/Q"$B+"(+=(T+;B&$5<( 10 To calculate the circulation around the element dxdy dy dx Vorticity is defined as No shear = no vorticity
:0/(O5;/13()-"&B+"( 11 Is there a single function to describe the velocity field in the x,y plane in the absence of vorticity? YES The stream function is defined as Does it satisfy the equation of continuity? YES If the flow is irrotational (no vorticity) then it satisfies Laplace and is therefore additive 0.5m grid u 0 =1ms -1.!=3 3 12!=2 2!=1!=0!!" "!#$% &# ( 1 0 y!=-1-1!=-2-2!=-3-3!=-4-4 0 1 2 3 4 5 6 7 8 9 10 x
W/"/;1*$%/'(*$"/1;(2+,( 13 y V! v 0 u 0 x Inviscid Flow 14! :0/(&+"B"-$5<(/A-1B+"(8/&+3/%(,0$&0($%(1*%+(K"+,"(1%(50/(X14*1&/(/A-1B+"9(! @#/;<(4+5/"B1*(%1B%=<(50$%(/A-1B+"9()*+,%(,$50(50/(/N$%5/"&/(+=( 4+5/"B1*(=-"&B+"%(%1B%=<$"7(50/(X14*1&/(/A-1B+"(1;/(&1**/'( 4+5/"B1*(2+,9(
Inviscid Flow! :0/(*$"/1;$5<(+=(50/(7+#/;"$"7(/A-1B+"(=+;(50/(2+,(Q/*'%($34*$/%(5015( '$6/;/"5(4+5/"B1*(2+,%(&1"(8/(%-4/;4+%/'9( 15!!=(((((1"'((((((1;/(5,+(4+5/"B1*(2+,%C(50/(%-3((((((((((((((((((((((((((1*%+(&+"%B5-5/%(1( 4+5/"B1*(2+,9(Y/(01#/((! H+,/#/;C(50/(4;/%%-;/(&1""+5(8/(%-4/;4+%/'('-/(5+(50/("+"*$"/1;$5<($"(50/( R/;"+-**$(/A-1B+"C($9/9(( 2D Potential Flows!!=(;/%5;$&5/'(5+(%5/1'<(5,+('$3/"%$+"1*(4+5/"B1*(2+,C(50/"(50/(7+#/;"$"7(/A-1B+"%( 8/&+3/( U L y D z x! @979(4+5/"B1*(2+,(41%5(1(&$;&-*1;(&<*$"'/;(,$50(VZX([[\($%(1(]V(4+5/"B1*(2+,("/1;(50/( 3$''*/(+=(50/(&<*$"'/;C(,0/;/(8+50(w(&+34+"/"5(1"'( 16
2D Potential Flows! :0/(]^V(#/*+&$5<(4+5/"B1*(=-"&B+"(7$#/%( 17 1"'(50/"(50/(&+"B"-$5<(/A-1B+"(8/&+3/%(! :0/(4;/%%-;/('$%5;$8-B+"(&1"(8/('/5/;3$"/'(8<(50/(( R/;"+-**$(/A-1B+"C(,0/;/(p($%(50/('<"13$&(4;/%%-;/( 2D Potential Flows 18! )+;(]V(4+5/"B1*(2+,%C(1(%5;/13(=-"&B+"(((((((((((((&1"(1*%+(8/('/Q"/'(5+7/50/;(,$50((((((((((((9(!"(F1;B%$1"(&++;'$"15/%C(((,0/;/(&+"B"-$5<(/A-1B+"($%(1-5+31B&1**<(%1B%Q/'C(1"'($;;+51B+"1*$5<(*/1'%( 5+(50/(X14*1&/(/A-1B+"C(! R+50(X14*1&/(/A-1B+"%(1;/(%1B%Q/'(=+;(1(]V(4+5/"B1*(2+,((
Two-Dimensional Potential Flows! )+;(5,+^'$3/"%$+"1*(2+,%C(((((((((((((((((((8/&+3/_(!!"(1(F1;5/%$1"(&++;'$"15/(%<%5/3(( and!!"(1(f<*$"';$&1*(&++;'$"15/(%<%5/3( and 19 Two-Dimensional Potential Flows! :0/(4+5/"B1*(=-"&B+"(((1"'(50/(%5;/13(=-"&B+"(((1;/(&+"`-715/(41$;( +=(1"(1"1*<B&1*(=-"&B+"($"(&+34*/N(#1;$18*/(1"1*<%$%9(:0/(&+"'$B+"%_( 20! :0/%/( 1;/( 50/( F1-&0<^>$/31""( &+"'$B+"%9( :0/( 1"1*<B&1*( 4;+4/;5<( $34*$/%(5015(50/(&+"%51"5(4+5/"B1*(*$"/(1"'(50/(&+"%51"5(%5;/13*$"/( 1;/(+;50+7+"1*C($9/9C((((((((((((((( ((((((((((((((((( ((1"'(((((((((((((((( ( ((5+($34*<(5015(
Two-Dimensional Potential Flows! :0/;/=+;/C(50/;/(/N$%5%(1(%5;/13(=-"&B+"((((((%-&0(5015( (((((((((((((((((((((((((( ( ($"(50/(F1;5/%$1"(&++;'$"15/(%<%5/3(1"'( ((((((((((((((((((((((((((((( ($"(50/(&<*$"';$&1*(&++;'$"15/(%<%5/39(! :0/(5;1"%=+;31B+"(8/5,//"(50/(5,+(&++;'$"15/(%<%5/3%( 21 Simple 2-D Potential Flows 22! a"$=+;3()*+,(! O517"1B+"()*+,(! O+-;&/(LO$"KM(! );//(T+;5/N(
Uniform Flow # For a uniform flow given by, we have and # Therefore, and # Where the arbitrary integration constants are taken to be zero at the origin. 23 Uniform Flow! :0$%($%(1(%$34*/(-"$=+;3(2+,(1*+"7(1(%$"7*/('$;/&B+"9( 24
Stagnation Flow! )+;(1(%517"1B+"(2+,C((((((((((((((( (((((((9(H/"&/C((! :0/;/=+;/C( 25 Stagnation Flow 26! :0/(2+,(1"($"&+3$"7(=1;(Q/*'(2+,(,0$&0($%(4/;4/"'$&-*1;(5+(50/(,1**C( 1"'(50/"(5-;"($5%('$;/&B+"("/1;(50/(,1**(! :0/(+;$7$"($%(50/(%517"1B+"(4+$"5(+=(50/(2+,9(:0/(#/*+&$5<($%(b/;+( 50/;/9( y x
Source (Sink)! F+"%$'/;(1(*$"/(%+-;&/(15(50/(+;$7$"(1*+"7(50/(b^'$;/&B+"9(:0/( 2-$'(2+,%(;1'$1**<(+-5,1;'(=;+3(L+;($",1;'(5+,1;'M(50/(+;$7$"9(!=(3('/"+5/%(50/(2+,;15/(4/;(-"$5(*/"750C(,/(01#/(((((((((((((((( L%+-;&/($=(!($%(4+%$B#/(1"'(%$"K($=("/71B#/M9((! :0/;/=+;/C( 27 Source (Sink)! :0/($"5/7;1B+"(*/1'%(5+(( and! Y0/;/(171$"(50/(1;8$5;1;<($"5/7;1B+"(&+"%51"5%(1;/(51K/"( 5+(8/(b/;+(15(((((((((((((((((((((((((( 28
Source (Sink)! S(4-;/(;1'$1*(2+,(/$50/;(1,1<(=;+3(%+-;&/(+;($"5+(1(%$"K( 29! S(c#/(3($"'$&15/%(1(%+-;&/C(1"'(d#/(3($"'$&15/%(1(%$"K(! :0/(317"$5-'/(+=(50/(2+,('/&;/1%/(1%(\Z;((! b('$;/&b+"(i($"5+(50/(414/;9(l&01"7/(7;140$&%m( Free Vortex! F+"%$'/;(50/(2+,(&$;&-*1B"7(1;+-"'(50/(+;$7$"(,$50(1( &+"%51"5(&$;&-*1B+"((((9(Y/(01#/_(((((((((((((((((( (,0/;/(2-$'( 3+#/%(&+-"5/;(&*+&K,$%/($=(((((($%(4+%$B#/(1"'(&*+&K,$%/($=( "/71B#/9((! :0/;/=+;/C( 30
Free Vortex # The integration leads to and where again the arbitrary integration constants are taken to be zero at 31 Free Vortex! :0/(4+5/"B1*(;/4;/%/"5%(1(2+,(%,$;*$"7(1;+-"'(((((((((((((((((((((((( +;$7$"(,$50(1(&+"%51"5(&$;&-*1B+"(((9(( 32! :0/(317"$5-'/(+=(50/(2+,('/&;/1%/(1%(\Z;9((
Superposition of 2-D Potential Flows! R/&1-%/(50/(4+5/"B1*(1"'(%5;/13(=-"&B+"%(%1B%=<(50/(*$"/1;( X14*1&/(/A-1B+"C(50/(%-4/;4+%$B+"(+=(5,+(4+5/"B1*(2+,($%( 1*%+(1(4+5/"B1*(2+,9( 33! );+3(50$%C($5($%(4+%%$8*/(5+(&+"%5;-&5(4+5/"B1*(2+,%(+=(3+;/( &+34*/N(7/+3/5;<9(! O+-;&/(1"'(O$"K(! V+-8*/5(! O+-;&/($"(a"$=+;3(O5;/13(! ]^V(>1"K$"/(D#1*%(! )*+,%(S;+-"'(1(F$;&-*1;(F<*$"'/;( Source and Sink! F+"%$'/;(1(%+-;&/(+=(3(15(L^1C(eM(1"'(1(%$"K(+=(3(15(L1C(eM((! )+;(1(4+$"5(f(,$50(4+*1;(&++;'$"15/(+=((((((((((((((9(((!=(50/(4+*1;(&++;'$"15/( =;+3(L^1CeM(5+(f($%((((((((((((((1"'(=;+3(L1C(eM(5+(f($%((((((((((! :0/"(50/(%5;/13(=-"&B+"(1"'(4+5/"B1*(=-"&B+"(+851$"/'(8<( %-4/;4+%$B+"(1;/(7$#/"(8<_( 34
Source and Sink 35 Source and Sink! H/"&/C(! O$"&/(! Y/(01#/(( 36
Source and Sink! Y/(01#/(! R<(! :0/;/=+;/C( 37 Source and Sink! :0/(#/*+&$5<(&+34+"/"5(1;/_( 38
Source and Sink 39 Doublet! :0/('+-8*/5(+&&-;%(,0/"(1(%+-;&/(1"'(1(%$"K(+=(50/(%13/( %5;/"750(1;/(&+**+&15/'(50/(%13/(*+&1B+"C(%1<(15(50/( +;$7$"9(! :0$%(&1"(8/(+851$"/'(8<(4*1&$"7(1(%+-;&/(15(L^1CeM(1"'(1( %$"K(+=(/A-1*(%5;/"750(15(L1CeM(1"'(50/"(*/g"7(1("(eC(1"'( 3"((((C(,$50(!"(K//4$"7(&+"%51"5C(%1<(]"!#$% 40
Doublet! )+;(%+-;&/(+=(3(15(L^1CeM(1"'(%$"K(+=(3(15(L1CeM(! a"'/;(50/%/(*$3$b"7(&+"'$b+"%(+=(1"ec(3"((((c(,/(01#/( 41 Doublet! :0/;/=+;/C(1%(1"e(1"'(3"(((((((,$50(]"!#$((! :0/(&+;;/%4+"'$"7(#/*+&$5<(&+34+"/"5%(1;/( 42
Doublet 43 Source in Uniform Stream! S%%-3$"7(50/(-"$=+;3(2+,(a($%($"(N^'$;/&B+"(1"'(50/( %+-;&/(+=(3(15LeCeMC(50/(#/*+&$5<(4+5/"B1*(1"'(%5;/13( =-"&B+"(+=(50/(%-4/;4+%/'(4+5/"B1*(2+,(8/&+3/_(( 44
Source in Uniform Stream 45 Source in Uniform Stream! :0/(#/*+&$5<(&+34+"/"5%(1;/_(! S(%517"1B+"(4+$"5(+&&-;%(15((((((((((((( ( (:0/;/=+;/C(50/(%5;/13*$"/(41%%$"7(50;+-70(50/(%517"1B+"( 4+$"5(,0/"(((((((((((((((((((! :0/(31N$3-3(0/$705(+=(50/((((((((((((((((( ((((((((((&-;#/($%((( 46
Source in Uniform Stream! )+;(-"'/;7;+-"'(2+,%($"(1"(1A-$=/;(+=(&+"%51"5(50$&K"/%%C(50/( 2+,(50;+-70(4+;+-%(3/'$1(1;/(4+5/"B1*(2+,%9( 47! S"($"`/&B+"(,/**(15(50/(+;$7$"(501"(1&5(1%(1(4+$"5(%+-;&/(1"'(50/( -"'/;7;+-"'(2+,(&1"(8/(;/71;'/'(1%(1(-"$=+;3(2+,9( 2-D Rankine Ovals! :0/(]V(>1"K$"/(+#1*%(1;/(50/(;/%-*5%(+=(50/(%-4/;4+%$B+"( +=(/A-1*(%5;/"750(%$"K(1"'(%+-;&/(15(NI1(1"'(d1(,$50(1( -"$=+;3(2+,($"(N^'$;/&B+"9(! H/"&/C(( 48
2-D Rankine Ovals! @A-$#1*/"5*<C( 49 2-D Rankine Ovals! :0/(%517"1B+"(4+$"5%(+&&-;(15( (,0/;/(((((((((((((,$50(&+;;/%4+"'$"7((((((((((((((( 50
2-D Rankine Ovals! :0/(31N$3-3(0/$705(+=(50/(>1"K$"/(+#1*($%( (*+&15/'(15((((((((((((((((,0/"((((((((((((((((((((((((((C$9/9C( 51 2-D Rankine Ovals 52
2-D Rankine Ovals 53 Flows Around a Circular Cylinder 54! O5/1'<(F<*$"'/;(! >+51B"7(F<*$"'/;((! X$G()+;&/((
Steady Cylinder! )*+,(1;+-"'(1(%5/1'<(&$;&-*1;(&<*$"'/;($%(50/(*$3$B"7(&1%/( +=(1(>1"K$"/(+#1*(,0/"(1"e9((! :0$%(8/&+3/%(50/(%-4/;4+%$B+"(+=(1(-"$=+;3(41;1**/*(2+,(,$50(1('+-8*/5($"(N^'$;/&B+"9(! a"'/;(50$%(*$3$5(1"'(,$50($#]"9(!i&+"%51"5c((((((( ((((((((((((((((((((((( ( ($%(50/(;1'$-%(+=(50/(&<*$"'/;9( 55 Steady Cylinder! :0/(%5;/13(=-"&B+"(1"'(#/*+&$5<(4+5/"B1*(8/&+3/_(! :0/(;1'$1*(1"'(&$;&-3=/;/"B1*(#/*+&$B/%(1;/_( 56
Steady Cylinder 57 Rotating Cylinder! :0/(4+5/"B1*(2+,(+=(1(-"$=+;3(41;1**/*(2+,(41%5(1(;+51B"7(&<*$"'/;( 15(0$70(>/<"+*'%("-38/;($%(50/(%-4/;4+%$B+"(+=(1(-"$=+;3(41;1**/*( 2+,C(1('+-8*/5(1"'(=;//(#+;5/N9(! H/"&/C(50/(%5;/13(=-"&B+"(1"'(50/(#/*+&$5<(4+5/"B1*(1;/(7$#/"(8<((( 58
Rotating Cylinder! :0/(;1'$1*(1"'(&$;&-3=/;/"B1*(#/*+&$B/%(1;/(7$#/"(8<(( 59 Rotating Cylinder! :0/(%517"1B+"(4+$"5%(+&&-;(15(! );+3( 60
Rotating Cylinder 61 Rotating Cylinder 62
Rotating Cylinder! :0/(%517"1B+"(4+$"5%(+&&-;(15(! F1%/(\_(! F1%/(]_(! F1%/(h_( 63 Rotating Cylinder! F1%/(\_(( 64
Rotating Cylinder 65! F1%/(h_( Lift Force! :0/(=+;&/(4/;(-"$5(*/"750(+=(&<*$"'/;('-/(5+(4;/%%-;/(+"(50/( &<*$"'/;(%-;=1&/(&1"(8/(+851$"/'(8<($"5/7;1B"7(50/(%-;=1&/( 4;/%%-;/(1;+-"'(50/(&<*$"'/;9( 66! :0/(51"7/"B1*(#/*+&$5<(1*+"7(50/(&<*$"'/;(%-;=1&/($%(+851$"/'(8<( */g"7(;i; + C(
Lift Force! :0/(%-;=1&/(4;/%%-;/((((((((1%(+851$"/'(=;+3(R/;"+-**$(/A-1B+"($%( (,0/;/(((((($%(50/(4;/%%-;/(15(=1;(1,1<(=;+3(50/(&<*$"'/;9( 67! H/"&/C( Lift Force! :0/(=+;&/('-/(5+(4;/%%-;/($"(&(1"'('('$;/&B+"%(1;/(50/"( +851$"/'(8<(( 68
Steady Cylinder 69 Rotating Cylinder 70
Lift Force 71! :0/('/#/*+43/"5(+=(50/(*$G(+"(;+51B"7(8+'$/%($%(&1**/'(50/(.17"-%(/6/&59(!5( $%(&*/1;(5015(50/(*$G(=+;&/($%('-/(5+(50/('/#/*+43/"5(+=(&$;&-*1B+"(1;+-"'(50/( 8+'<9(! S"(1$;=+$*(,$50+-5(;+51B+"(&1"('/#/*+4(1(&$;&-*1B+"(1;+-"'(50/(1$;=+$*(,0/"( i-j1(&+"'$b+"($%(%1b%q/'(15(50/(;/1;(b4(+=(50/(1$;(=+$*9(! :0/;/=+;/C(:0/(51"7/"B1*(#/*+&$5<(1*+"7(50/(&<*$"'/;(%-;=1&/($%( +851$"/'(8<(*/g"7((#( ) _(! :0$%(=+;3%(50/(81%/(+=(1/;+'<"13$&(50/+;<(+=(1$;4*1"/9( 72
73 74
75