LectureSummary(3February)

ME2135FluidMechanicsII LS71

Vorticity] (Greekalphabetzeta)isakinematic quantitydefinedas:

Vorticity istwice themeanangularvelocityofafluidelement:

Vorticity ] =0 Irrotational orpotential flow fluidelementsarenotrotating(Z =0)

Vorticity ]z 0 flowisrotational fluidelementsarerotating (Z z 0)

SummaryofLectureon3February

2] Z

v ux y

] w w w w (2.2.1)

(2.2.2)

LectureSummary(5February)

ME2135FluidMechanicsII LS81

Circulation* isthecounterclockwise lineintegralofthevelocityaroundaclosedloopC:

Circulation=lineintegralofvelocity=areaintegralofvorticity

DifferentialFormofMassConservationEquation orContinuityEquationfor2Dincompressibleflow

SummaryofLectureon5 February

C C A A

v uV ds udx vdy dxdy dxdyx y

] w w* w w G G> > (2.2.9)

0u vx y

w w w w (2.3.1)

LectureSummary(5February)

ME2135FluidMechanicsII LS82

Definestreamfunction suchthat

automaticallysatisfiestheContinuityEquation:

(i) Linesofconstant\ correspondtostreamlines oftheflow(ii) Changein\ between2streamlines=volumeflowrate between2

streamlines:

(iii) Aflowisirrotational ifandonlyif\ satisfiestheLaplacesequation:

Laplacesequationislinear cansuperposeelementaryirrotationalflowstogeneratemorecomplicatedirrotational flows

SummaryofLectureon5 February

u vy x\ \w w w w (2.4.1)

,x y\ \

,x y\ \ 0u v

x yw w w w

(2.3.1)

2 1q \ \ (2.4.8)

(2.4.10)2 2

2 2 0x y\ \w w

w w

LectureSummary(10February)

ME2135FluidMechanicsII LS91

Elementary2Dirrotational flows:SummaryofLectureon10February

(a)UniformFlow

Uy\

(b)LineSource

2q\ TS

(q >0)

(c)LineSink

2q\ TS

(q

LectureSummary(12February)

ME2135FluidMechanicsII LS101

Uniformflow +Source FlowpastHalfRankine BodySummaryofLectureon12February

1tan2q yUy

x\ S

(2.7.1)

0\ 0y or 2cot Uyx yqS

0, 4qU

LectureSummary(12February)

ME2135FluidMechanicsII LS102

Source +Sink:SummaryofLectureon12February

Doublet:

2 22y

x yP\ S

2sqP

12 2 2

2tan2q sy

x y s\ S

LectureSummary(12February)

ME2135FluidMechanicsII LS103

Uniformflow +SourceSink Pair FlowpastFullRankine BodySummaryofLectureon12February

0\ 0y or

12 2 2

20 tan2q syUy

x y s\ S

2 2

2 2 1 2 cot 2x y y Us y

s q ss sS

LectureSummary(17February)

ME2135FluidMechanicsII LS111

Uniformflow +Doublet FlowpastNonRotatingCircularCylinder

Onsurfaceofcylinder: ; ; (dAlemberts paradox)

SummaryofLectureon17February

0\ 0y or

2 2qs y Uy

x y\ S

2 2 2qsx y aUS

2

2sin 1aUrr

\ T

(2.7.16)or

' 2 sinv U T ' 0u 21 4sinpC T 0L 0D

SS

LectureSummary(3March)

ME2135FluidMechanicsII LS121

' 2 sin2

v Ua

T S*

FlowpastNonRotating CircularCylinder+Irrotational (free)vortexFlowpastRotating CircularCylinder

Onsurfaceofcylinder: ;

(KuttaJoukowski Theorem;MagnusEffect) (dAlemberts paradox)

SummaryofLectureon3 March

' 0u

0D

2

2sin 1 ln2a rUr

ar\ T S

*

4 UaS* 4 UaS* 4 UaS* !

0sin 4 UaT S

*

L UU *

LectureSummary(3March)

ME2135FluidMechanicsII LS122

VelocityPotentialfunctionI existsifandonlyiftheflowisirrotational ( ):

In2Dirrotational flow,streamlines intersectequipotentiallinesatrightangles linesofconstant\ A linesofconstantI

VelocityPotentialfunctionI satisfiesLaplacesequationduetocontinuity (massconservation):

LaplacesEquationforI islinear complexsolutionscanbeobtainedfromsuperpositionofsimplesolutionsforI

SummaryofLectureon3March

(2.8.3)

(2.8.6)

0V] u G G

V gradI I G

uy x\ Iw w w w v x y

\ Iw w w w

(2.8.5) (2.8.6)

2 2

2 2 0x yI Iw w

w w

LectureSummary(3March)

ME2135FluidMechanicsII LS123

MethodofImages Whenasource,sink orvortex isplacednexttoawall,animage is

addedtocanceloutthenormalvelocitycomponenttothewall

SummaryofLectureon3March