Potential Flows Stream Functions

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Potential Flows(Stream Functions)

• Fluid Mechanics 4

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Aims

• to provide a method of representing flow in 2D
• to provide insight to mechanisms of lift
• to define vorticity
• inset flow over a rotating cylinder

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Navier Stokes Equations in 2D
Non-viscous, steady-state, ignoring gravity
And continuity
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Vorticity Circulation
Consider a simple current in a flume (tank) with
a shear velocity profile
B
B
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.
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Definition of Circulation
The circulation is
where Vs is the velocity parallel to the curve, s.
This can also be found by adding all the small
circulation elements, dk
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Definition of Vorticity
To calculate the circulation around the element
dxdy
Vorticity is defined as
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The Stream Function
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
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0.5m grid u01ms-1.
?3
?2
?1
Vu01ms-1
?0
?-1
?-2
?-3
?-4
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0.5m grid u00.5ms-1.
?2
?1
Vu00.5ms-1
?0
Streamlines further apart implies lower velocity
?-1
?-2
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0.5m grid v00.5ms-1.
?-1
?-2
?-3
?-4
?-5
?0
Vv00.5ms-1
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Adding two stream functions
?1
?0
?-1
V(u02v02)1/21.41ms-1
?-3
?-2
?-4
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Generalised linear flow
x
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Bernoullis equation
Euler equation in 2D (NS without viscosity)
For an irrotational flow
since
integrating through wrt x
p0 is the static pressure
This applies throughout the flowfield, not just
on a streamline
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Sources (reminder 2D)
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Sources
q is the source strength in m2s-1
The radial velocity decreases inversely with
distance from the source, but the flow rate, q,
remains constant
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Sources not at the origin
For a source in a generalised position (x0,y0)
(In data sheet)
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Source at the origin
?2
?3
?1
?0
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Source and horizontal flow
?5
?2
?3
?1
?4
?7
?3
?6
?0
?6
?6
?5
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Source and horizontal flow (right to left)
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Rankine half-body
Streamlines may not be crossed by the fluid, so
the streamline can represent a body in the flow
(Nb External values have a 102?
discontinuity. This is because arctan is a
multi-valued function.)
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Stagnation point
For a source and horizontal flow
At a stagnation point u0
If the stagnation point is on the axis, yy0
For the example over
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Stagnation streamline
The stagnation streamline defines the shape of
the aerodynamic body. The value of the
streamfunction on this streamline is the same as
the value at the stagnation point.
y0 on the x-axis. There are two solutions, one
for the upper and one for the lower streamlines.
?sq/2
?s-q/2
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Sinks
Basically a source with a negative q
q is the source strength in m2s-1
Flow towards the source, increasing as r-gt0
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Rankine Bodies
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Rankine Bodies
sink source linear flow
for source and sink at general coordinates
(x0,y0) and (x1,y1)
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Rankine Bodies - stagnation points
y
To find the stagnation points, put y00, y10 and
x0-a, x1a
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Rankine Bodies - stagnation points
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Rankine Bodies - example
u0 1ms-1, q20m2s-1, a4m
The body is therefore 12.88m long.
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Doublet
Bringing the sink and source closer together
lessens the effect on the flow. Increasing the
source and sink strength, keeping qaconst,
maintains the effect on the flow, and as a?0, the
Rankine body becomes a circle. This is known as a
doublet. Its most significant use is giving the
streamlines for flow over a circular cylinder
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Source Sink Linear Flow (u01ms-1,q500m2s-1,
a0.1m)
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Doublet - Theory
y
d?
r
r-dr
?d?
?
ds
x
By the sine rule
source
sink
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Doublet - Theory
This represents a series of circles with centre
on the y axis
In cartesian coordinates
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Doublet - examplem 100m3s-1
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Doublet Linear Flowm q2a/(2?)
50020.1/(2?)15.915m3s-1
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Flow past a cylinder
Call the cylinder diameter, d, and the radius, a.
Can find the cylinder radius by locating the
stagnation points.