The Modified Rankine vortex. In your recitation this week, you will develop a ma
ID: 111978 • Letter: T
Question
The Modified Rankine vortex. In your recitation this week, you will develop a mathematical expression for the Rankine vortex which has a constant vorticity from the vortex center to some fixed distance from the center. Beyond that distance, the vorticity is zero. This is an idealized vortex, and we use it for study because its properties are easy to understand and can help us gain insight into how real vortices work. The Rankine vortex has some characteristics of some real vortices (vorticity concentrated at the vortex center), but one shortcoming is that the velocity does not vary smoothly with distance from the center and another is that the vorticity is discontinuous. The modified Rankine vortex (MRV) does not have these shortcomings. Here, you will derive an expression for the MRV and analyze some of its properties. (a) Consider a tangential velocity u_e profile of the form u_e = r/a + r^2 b, where r is the distance from the vortex center. This has the appeal that for very small r (ie, r^2 a) u_e is proportional to 1/r, as in an irrotational vortex. Solve for a and b in terms of r_o and ohm given the conditions: u_e = ohm r, r^2 a. The idea is that the MRV approaches the Rankine vortex as r gets very big or very small. (b) For the MRV, find the value of r at which u_e is a maximum. Also find the maximum value of u_e (c) Make a plot of u_e/r_0 ohm vs. r for the MRV and the Rankine vortex (see equations from recitation). Use graph paper and draw clearly or use a plotting program. Put both curves on the same graph so that you can easily compare them: distinguish them using color or dashed and solid lines. If you draw, guide your sketch by calculating u_e/r_o ohm for each vortex at r = 0, r_o/2, r_o, 2r_o, and 3r_o.Explanation / Answer
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