Irrotational Flow• Analysis of inviscide flow can be further simplified if we

assume if the flow is irrotational:1

02

; ;

z

v u

x y

v u w v u w

x y y z z x

• Example: uniform flow in x-direction:

Velocity potential

• equations for irrotational flow will be satisfied automatically if we introduce a scalar function called velocity potential such that:

u v wx y z

V

• As for incompressible flow conservation of mass leads to:20, 0 ∇ V

Laplace equation2 2 2

2 2 20

x y z

Some basic potential flows

• As Laplace equation is a linear one, the solutions can be added to each other producing another solution;

• stream lines (y=const) and equipotential lines (f=const) are mutually perpendicular

along streanline

along const

dy v

dx u

dy ud dx dy udx vdy

x y dx v

Both f and y satisfy Laplace’s equation

u v

y x y y x x

Uniform flow• constant velocity, all stream lines are straight and

parallel0

0

Ux y

Ux

Uy x

Uy

( cos sin )

( cos sin )

U x y

U y x

Source and Sink

• Let’s consider fluid flowing radially outward from a line through the origin perpendicular to x-y planefrom mass conservation:(2 ) rr v m

10

2

m

r r r

10

2

m

r r r

ln2

mr

2

m

Vortex• now we consider situation when ther

stream lines are concentric circles i.e. we interchange potential and stream functions:

ln

K

K r

• circulation

0C C C

ds ds d V

• in case of vortex the circulation is zero along any contour except ones enclosing origin 2

0

( ) 2

ln2 2

Krd K

r

r

Shape of a free vortex

2

2

2

p Vgz const

2 21 2

2 2

V Vz

g g

at the free surface p=0:

2

2 28z

r g

Doublet• let’s consider the equal strength, source-sink pair:

1 2( )2

m

12 2

2 sintan ( )

2

m ar

r a

if the source and sink are close to each other:

sin

cos

K

rK

r

K – strength of a doublet

Summary