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Material Type: Notes; Professor: Sucosky; Class: Fluid Mechanics; Subject: Aerospace and Mechanical Engr.; University: Notre Dame; Term: Fall 2009;
Typology: Study notes
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Source in uniform stream
Combined velocity potential and streamfunction
Uniform flow Source
Therefore, the combination of a uniform flow and a doublet is expressed in cylindrical coordinates as:
Streamfunction:
Velocity potential:
Velocity field
vr
v
The velocity magnitude is:
V^2
Stagnation point
At b along the x-axis, the velocity of the flow due to the source cancels that due to the uniform flow.
at r b and : vr
v
vr ( r b , ) 0
Equation of the streamline passing through the stagnation point
At the stagnation point:
Therefore, the streamline equation through the stagnation point is:
Doublet
A doublet is the combination of a source and sink
Combined velocity potential and streamfunction
Sink Source
Streamfunction
Therefore, the combination of a uniform flow and a doublet is expressed in cylindrical coordinates as:
Streamfunction:
Velocity potential:
From the combined streamfunction:
Using the identity 1 2 1 2 1 2
, the expression can be rewritten:
where:
Therefore:
The streamfunction is obtained by taking the inverse tangent:
The expression above is the streamfunction for the combination of a source and a sink.
Therefore, the streamfunction and velocity potential for a doublet can be expressed as:
Flow around a circular cylinder
The flow around a circular cylinder can be represented by combining a doublet with a uniform flow.
Combined velocity potential and streamfunction
Uniform flow Doublet
Therefore, the combination of a uniform flow and a doublet is expressed in cylindrical coordinates as:
Streamfunction:
Velocity potential:
Boundary condition
the flow. Therefore, the streamfunction must be constant along the perimeter of the cylinder.
Streamfunction:
Velocity potential:
Velocity field The velocity field can be derived using the definition of the streamfunction in cylindrical coordinates:
vr
v
Points where the maximum velocity is attained on the surface of the cylinder:
at r a : vr
v
v max
Circulatory flow around a cylinder in a uniform stream
Circulatory flow around a cylinder in a uniform stream can be obtained by combining a vortex, a uniform flow and a doublet (i.e., a vortex and a flow around a cylinder)
Combined velocity potential and streamfunction
Vortex Flow around cylinder
2
2
Therefore, the streamfunction and velocity potential for the circulatory flow around a cylinder in a uniform stream can be written:
Streamfunction:
Velocity potential:
Velocity field
vr
v
Stagnation points on the cylinder
Stagnation points are points where the velocity is zero.
at r a : vr
v
v (^) ( r a ) 0
