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Chapter 6: Problem 124

\(\mathrm{A}\) flow field is formed by combining a uniform flow in the positive \(x\) direction, with \(U=10 \mathrm{m} / \mathrm{s},\) and a counter clockwise vortex, with strength \(K=16 \pi \mathrm{m}^{2} / \mathrm{s},\) located at the origin. Obtain the stream function, velocity potential, and velocity field for the combined flow. Locate the stagnation point(s) for the flow. Plot the streamlines and potential lines.

Short Answer

Expert verified
The stream function for the combined flow field is \(\psi = Uy + K ln r / 2蟺\). The velocity potential for the combined flow field is \(\phi = Ux - K胃 / 2蟺\). The velocity field \(u\) and \(v\) are \(U - K/2蟺r sin(胃) \) and \(K/2蟺r cos(胃)\) respectively. The stagnation points can be determined by setting these equations equal to zero and solving.

Step by step solution

01

Determine the stream function

Begin by determining the stream function for a uniform plane flow. According to the provided flow field, the stream function for a uniform flow \(U\) in the positive x-direction is given by \(\psi_{u}= Uy\). The stream function for a counter-clockwise vortex located at the origin with strength \(K\) is given by \(\psi_{v} = K ln r / 2蟺\). So, the combined flow field stream function is \(\psi = \psi_{u} + \psi_{v} = Uy + K ln r / 2蟺\).
02

Determine the velocity potential

Next, compute the velocity potential for the combined flow. The velocity potential for a uniform flow \(U\) in the positive x-direction is given by \(\phi_{u}= Ux\). And for a free vortex flow, the velocity potential is given by \(\phi_{v} = -K胃 / 2蟺\). So, the combined flow field velocity potential is \(\phi = \phi_{u} + \phi_{v} = Ux - K胃 / 2蟺\).
03

Determine the velocity field

Now, find the velocity field for the combined flow. This involves differentiating the stream function with respect to \(y\) to get the \(x\)-component of the velocity (\(u\)), and with respect to \(x\) (changing the sign) to get the \(y\)-component of the velocity (\(v\)). The velocity field \(u\) and \(v\) will be \(U - K/2蟺r sin(胃) \) and \(K/2蟺r cos(胃)\) respectively.
04

Find the stagnation points

First, set the equations derived in the previous step to zero which will yield the solutions for \(r\) and \(胃\). With the substitution of \(r = y / cos胃\), one can simplify the equations to the form \(f(y, U, K, 胃)=0\). The solutions of these equations yield the stagnation points.
05

Plot the streamlines and potential lines

Finally, using the stream function \(\psi = Uy + K ln r / 2蟺\) and the velocity potential \(\phi = Ux - K胃 / 2蟺\), plot the streamlines and potential lines. The streamlines are the curves \(蠄 = constant\) and potential lines are the curves \(蠁 = constant\). In the plot, the stagnation points are the intersection points of the streamlines and potential lines.

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