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Fluid Dynamics Problem Set

This document contains 4 problems related to potential flow analysis: 1) Given velocity components, show the flow satisfies continuity and obtain the stream function. The flow is also irrotational, so the potential function is obtained. 2) Given a stream function, show the flow is irrotational and find the corresponding potential function. 3) Sketch streamlines for a given stream function and derive the velocity expression. Calculate the vorticity. 4) Write the potential function for a source-sink combination. In the limit of a small distance between them, derive the velocity potential, which is a doublet singularity.

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97 views1 page

Fluid Dynamics Problem Set

This document contains 4 problems related to potential flow analysis: 1) Given velocity components, show the flow satisfies continuity and obtain the stream function. The flow is also irrotational, so the potential function is obtained. 2) Given a stream function, show the flow is irrotational and find the corresponding potential function. 3) Sketch streamlines for a given stream function and derive the velocity expression. Calculate the vorticity. 4) Write the potential function for a source-sink combination. In the limit of a small distance between them, derive the velocity potential, which is a doublet singularity.

Uploaded by

Likhith
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
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Potential Flow

Problem 1
In a two-dimensional, incompressible flow the fluid velocity components are given by: u = x-4y
and v = -4x-y. Show that the flow satisfies the continuity equation and obtain the expression for
the stream function. If the flow is potential (irrotational) obtain also the expression for the
velocity potential.

Problem 2
A stream function in a two-dimensional flow is ψ = 2xy. Show that the flow is irrotational
(potential) and determine the corresponding velocity potential function φ.

Problem 3
A flow field is described by the equation ψ = y-x2 . Sketch the streamlines ψ = 0, ψ=1, and ψ = 2.
Derive an expression for the velocity V at any point in the flow field. Calculate the vorticity.

Problem 4
Consider the superposition of a source of strength 𝑚 at (−𝑎, 0) and a sink of strength 𝑚 at
(𝑎, 0). Write the potential function for this combination. In the limit of 𝑎 being much smaller
than any other length scale in the system, derive the equation for the velocity potential of this
combination. This singularity is called the doublet. Hint: use cylindrical coordinates, use the
approximation ln(1 ± 𝜖) = ±𝜖. You might want to consider an arbitrary point (𝑟, 𝜃) and do the
simplification. The origin can be the midpoint between the source and the sink.

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