Fundamentals of Fluid Mechanics Assignment 10 — Potential Flow II  
Instructions
Write your solutions in single column format, with one statement following another vertically. Write your solutions neatly so that they are easy to read and verify. Don't write one line with two equal signs. Highlight your answers using a box. Failure to do this will result in a lower score and fewer comments on my part.
11.13.19
Question #1
Consider flow past a circular body of radius $R$ with a blunted base as shown below.
A10Q1.png  ./download/file.php?id=5805&sid=a882b88f8912e9c31fcc3c1d3983a8e1  ./download/file.php?id=5805&t=1&sid=a882b88f8912e9c31fcc3c1d3983a8e1
As a first approximation, assume the base pressure $P_b$ is constant across the base and that the flow on the circular part of the body follows potential-flow theory. Do the following:
(a)  Sketch the pressure coefficient, $C_p=(P-P_\infty)/(\frac{1}{2}\rho_\infty q_\infty^2)$, as a function of $\theta$ for the assumed flow, with $0 \le \theta \le 2\pi$. The angle $\theta$ is zero on the positive $x$ axis. Explain the basis of your sketch, e.g., maxima and minima, etc.
(b)  Compute the drag on the body. (It is not zero because this is not an exact solution).
Question #2
An interesting way to generate an exact solution for flow past a two-dimensional bump is to use a streamline for flow past a cylinder as follows:
A10Q2.png  ./download/file.php?id=5802&sid=a882b88f8912e9c31fcc3c1d3983a8e1  ./download/file.php?id=5802&t=1&sid=a882b88f8912e9c31fcc3c1d3983a8e1
The desired height of the bump is $R/2$, where $R$ is the cylinder radius. Starting from the fundamental solutions outlined in the tables, do the following:
(a)  Determine the distance $h$.
(b)  Determine the maximum velocity on the bump, $u_\max$. Note: you have to prove that the location you use to determine the velocity $u_{\rm max}$ is a maximum.
Question #3
Far from a non-lifting body that is emitting fluid (e.g., from a jet engine),
A10Q3.png  ./download/file.php?id=5806&sid=a882b88f8912e9c31fcc3c1d3983a8e1  ./download/file.php?id=5806&t=1&sid=a882b88f8912e9c31fcc3c1d3983a8e1
we can approximate the velocity potential as $$ \phi \approx U r \cos \theta + \frac{Q}{2\pi} \ln r + \frac{\cal D}{2 \pi r}\cos \theta ~~~~{\rm for~}r\rightarrow\infty $$ where $U$ is freestream velocity, $Q$ is the mass-injection rate and $\cal D$ is an effective doublet strength. Do the following:
(a)  Verify that $$ P-P_\infty \approx - \frac{\rho Q U }{2 \pi r}\cos \theta ~~~~{\rm as~}r\rightarrow \infty $$
(b)  Using a circular control volume of radius $r$ show that: $$ \int_S \rho \vec{v} (\vec{v}\cdot\vec{n}) dS = \frac{3}{2}\rho Q U \vec{i} $$ and that $$ \int_S(P-P_\infty)\vec{n}dS = -\frac{1}{2}\rho Q U \vec{i} $$
(c)  Use the results of part (b) to determine the force on the body.
Question #4
Starting from the streamfunction of a source/sink, derive the streamfunction for a doublet: $$ \psi=-\frac{\cal D}{2\pi r} \sin(\theta) + \textrm{constant} $$
11.14.19
Answers
1.  $\frac{8}{3} \rho U^2 R$
2.  $1.5~R$, $\frac{5}{4} U$.
Due on Tuesday November 26th at 11:00. Do Questions #1, #2, #3. Question #4 is optional.
11.21.19
Note that there was a mistake in the answers for Problem #2. This has been corrected.
11.25.19
Note that Assignment #10 is now due on Tuesday December 3rd at 11:00.
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