2-d Ideal Flow II
x-direction, i.e.
ux = U
uy = 0
ux = @
x-direction, i.e.
ux = U
uy = 0
ux = @
2-d Ideal Flow II
df
Stream function
= Uy + C
df
Stream function
= Uy + C
df
df
Stream function
= Uy + C
df
Stream function
= Uy + C
Sink/source ow
Potential vortex
Combining solutions
Laplace's equation
Rankine body
Drag forces
d'Alembert's paradox
We could do the same thing for the potential function, to nd
= Ux + C1
Sink/source ow
Potential vortex
Combining solutions
Laplace's equation
Rankine body
Drag forces
d'Alembert's paradox
We could do the same thing for the potential function, to nd
= Ux + C1
Sink/source ow
Potential vortex
Combining solutions
Laplace's equation
Rankine body
Drag forces
d'Alembert's paradox
We could do the same thing for the potential function, to nd
= Ux + C1
polar coordinates, so
This is a constant for r = const, i.e. a circle.
Similarly, evaluate the stream
x
y
Sink/source ow
Potential vortex
Combining solutions
Laplace's equation
Rankine body
Drag forces
d'Alembert's paradox
Potential vortex
If we swap the potential and stream functions around, we get
another type of ow :
= K log r ; = K
x
everywhere except the origin.
vortex. Examples include :
hurricanes, tornadoes, bathtub
2-d Ideal Flow II
Continuity conservation of mass. So far we have applied this
in an integral formulation.
V
However can also write this in dierential form { dealing with
ow properties at a given point :
@ux
@
@x2 + @2
@y2 = 0
is called Laplace's equation. It is linear, so if 1 and 2 are
solutions, then
This means that we can combine simple potential ow
solutions to solve more complex problems.
Also, also satises Laplace's equation { can combine stream
functions.
R = Uy = Ur sin
S = m
2-d Ideal Flow II
Sink/source ow
Potential vortex
Combining solutions
Laplace's equation
Rankine body
Drag forces
d'Alembert's paradox
Bold line could be the boundary of a body { Rankine fairing.
Velocity components
ur = 1
Sink/source ow
Potential vortex
Combining solutions
Laplace's equation
Rankine body
Drag forces
d'Alembert's paradox
Bold line could be the boundary of a body { Rankine fairing.
Velocity components
ur = 1
Sink/source ow
Potential vortex
Combining solutions
Laplace's equation
Rankine body
Drag forces
d'Alembert's paradox
Bold line could be the boundary of a body { Rankine fairing.
Velocity components
ur = 1
Include a sink as well to give a Rankine body
Source
Sink
Stream function gives lines in space { streamlines { along
which the uid ows. It also gives the uid velocity at points
along the line.
Bernoulli's theorem states
2 u2 = constant
along a streamline. If we combine these we can calculate the
pressure at any point in the ow.
However this sometimes gives odd results. For example, the
ow around a circle of radius a has the potential function
= u0
r +
a2
r
cos
u = 2u0 sin
p = p0 2u20 sin 2
2-d Ideal Flow II
2 a cos d
2-d Ideal Flow II
Fx =
Uniform stream flow
Uniform stream flow