HW.III.1998 - θ to find radial and tangential components...

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AE 3003 Homework Set #3 Due on November 16, 1998 1. In 2-D flows, show that streamlines (along which ψ is constant), and lines along which the velocity potential Φ is constant are orthogonal to each other. Stream function C1 Stream function C2 Velocity potential Lines Hint: the solution is in the text book, in Chapter II. 2. In the figure above, show that the difference between the stream function values C1-C2 equals the volume rate of flow (per unit width in a direction perpendicular to the plane of the paper) of the fluid that flows between these two streamlines. Hint: the solution is in the text, in Chapter II. 3. Given the stream function for a doublet: r θ π κ ψ sin 2 - = find the velocity potential φ associated with this doublet. Hint: Take derivatives of the stream function with respect to r and
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Unformatted text preview: θ to find radial and tangential components of velocity ( ) v and r v . Equate these to corresponding derivatives of the velocity potential . Integrate ∂ ∂ ∂ ∂ and r to get . The answer will be r cos 2 = . 4. Using the MATLAB script given in a recent handout as a starting point, construct contours of the following stream function, which corresponds to the superposition of a uniform flow and a doublet: -= ∞ 2 2 1 r R y u You may use any numerical value for the velocity u ∞ and R (Being unimaginative, I would use unity). Visually verify that this represents flow over a circular cylinder of radius R....
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