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EML 5709 Homework 4a Spring 1995

1. Describe the velocity field corresponding to a source of strength m located at , and one of equal strength at In particular, find and and sketch the streamlines. What direction is the velocity on the x-axis (y=0)? What can you say about the flow corresponding to a single source located a distance h above a solid wall? Mirroring an image of the features of a flow into a solid region is called the mirror method. Give the pressure on the wall. Use real analysis only.
2. Describe the velocity field corresponding to a vortex of strength located at , above a solid wall along the x-axis. Use mirroring. Explain why the effects of wingtip vortices are much less pronounced when the plane is flying close to the ground. Use real analysis only.
3. 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 . Use Cartesian coordinates. 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? Use real analysis only.
4. Write down the combined 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 end up? Use real analysis only.
5. Repeat questions 1 through 4 using a complex potential. In questions 1, 2, and 3, verify that the imaginary part of the potential is constant on the wall.
6. Find the velocity and pressure on the surface of a circular cylinder in a uniform stream of magnitude U. The circulation around the cylinder is . Integrate the pressure forces on the cylinder and verify that the Kutta-Joukowski law is satisfied.
7. Repeat question 6, but use the Blasius theorem to find the force.
8. Make question 4.4 in the book.
9. Make question 4.5 in the book.
10. Make question 4.6 in the book.
11. Writing , express the pressure on the surface of a flat plate airfoil in terms of . Approximate the expression for small and rewrite in terms of the physical coordinate x. Now integrate this pressure distribution to find the lift and drag force on the airfoil. Draw these pressure distributions. Comment on the results. Find the circulation and verify that the Kutta-Joukowski law is satisfied.
12. Find the shape of the streamlines for flow around an ellipse. Assume that at large distances, the fluid velocity is zero. The circulation of any contour circling the ellipse once is . Is this potential flow? What is the net force on the ellipse?

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