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EML 5709 Homework 4 Spring 1997

1. Describe the velocity field corresponding to a source of strength m located at x=0, y=h, and one of equal strength at x=0, y=-h In particular, find and and sketch the streamlines. What direction is the velocity on the x-axis? Use real analysis only. Solution.
2. For the flow of the previous question, find the pressure on the x-axis. Use real analysis only. Solution.
3. Using the results of the previous two questions, can you figure out the velocity field corresponding to a single source located a distance h above a solid ground? And the pressure on the ground? Mirroring an image of the features of a flow into a solid wall is called the mirror method. Solution.
4. Describe the velocity field corresponding to a vortex of strength located at x=0, y=h, above a solid wall along the x-axis. Use the mirror method. Find the pressure on the ground. Explain why the effects of wingtip vortices are much less pronounced when a plane is flying close to the ground. Use real analysis only. Solution.
5. Write down the combined complex potential of a uniform flow in the x-direction of unit velocity and a source of strength at the origin. Draw the streamlines. What is the velocity at large distances? Where does the fluid coming out of the source at the origin end up? Solution.
6. Write down the complex potential of a sink of strength m located at the origin, a source of strength m at z = -a²/l and a source of strength m located at z=-l, with l>a. Show that this represents the flow around a circular cylinder of radius a, i.e. which is induced by a source at z=-l. Solution.
7. For the flow of the previous question use Blasius theorem to find the force on the cylinder. Also use Kutta-Joukowski and D'Alembert. Which result is correct? Why? Solution.
8. Using complex variables, write down the combined potential of a uniform flow in the x-direction of unit velocity, a strong sink of strength at the origin, and a strong source of strength at . Show that for small , we get potential flow past a circular cylinder. Draw the streamlines. Where does the fluid coming out of the source at end up? Solution.
9. Find the potential flow around an ellipse. Assume that at large distances, the fluid velocity is zero. The circulation of any contour circling the ellipse once is . What is the net force on the ellipse? Solution.
10. Find the potential flow around a thin plate airfoil at an angle of attack . Solution.
11. For the previous question, find the velocity on the surface as a function of the position x along the plate. Solution.
12. For the previous question, find the pressure on the surface as a function of the position x along the plate. Solution.

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