Fundamentals of Fluid Mechanics Assignment 12 — Supersonic Potential Flow  
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.
Question #1
The potential equation for supersonic irrotational flow becomes in the limit of small deflection angle: $$ (M_\infty^2-1)\phi_{xx} - \phi_{yy}=0 $$ Show that the solution to this wave equation corresponds to: $$ \phi(x,y)=f(x+y\sqrt{M_\infty^2-1})+g(x-y\sqrt{M_\infty^2-1}) $$
08.01.19
Question #2
As shown below, a cambered supersonic aerofoil is simulated by an articulated flat plate where the articulated deflections are $2^\circ$ at each step.
thinairfoil2.png  ./download/file.php?id=5375&sid=d536059a11833bcc7a53131e4a7ca8ae  ./download/file.php?id=5375&t=1&sid=d536059a11833bcc7a53131e4a7ca8ae
If the angle of attack for the aerofoil is $\alpha=4^\circ$, determine the lift and drag forces for the aerofoil per unit span using first order linearized theory. Recall that the linearized pressure coefficient collapses to: $$ {C_P}_{f,g}=\mp\frac{ 2 \theta_{\rm defl}}{ \sqrt{M_\infty^2-1}} $$
Question #3
Consider the following airfoil:
Q5.png
It is known that $M_\infty>1$ and that $d<c$ with $c$ the chord of the airfoil. It is also known that the bottom surface of the airfoil for $x>d$ is equal to: $$ y=b-a-\sqrt{b^2-x^2}~~~~{\rm for}~x>d $$ with $b$ and $a$ some constants such that $b>a$ and $a>0$. Determine the lift coefficient $C_{\rm L}$ as a function of $a$, $b$, $c$, and $d$ using linearized theory. Recall that the linearized pressure coefficient collapses to: $$ {C_P}_{f,g}=\mp\frac{ 2 \theta_{\rm defl}}{ \sqrt{M_\infty^2-1}} $$
Question #4
Consider a thin, supersonic airfoil profile expressed by the function $y=-h\left(x/t\right)^{m}$ with $m \ge 1$:
thinairfoil.png  ./download/file.php?id=5877&sid=d536059a11833bcc7a53131e4a7ca8ae  ./download/file.php?id=5877&t=1&sid=d536059a11833bcc7a53131e4a7ca8ae
The leading edge of the profile is tangent to the direction of the oncoming air stream. Using linearized theory,
(a)  Find the expressions for the lift and drag coefficients in terms of $M_\infty$ ,  $h/t$,  and $m$.
(b)  Find an expression for the lift over drag ratio in terms of $h/t$ and $m$. Plot the lift over drag ratio versus $m$ for $t/h=5$ and for $t/h=10$. Consider the range $1 \le m \le 4$.
Recall that the linearized pressure coefficient collapses to: $$ {C_P}_{f,g}=\mp\frac{ 2 \theta_{\rm defl}}{ \sqrt{M_\infty^2-1}} $$
12.05.19
Answers
2.  $8422 \frac{\rm N}{\rm m^2} L$, $806\frac{\rm N}{\rm m^2}L$.
3.  $\frac{2(a-b+\sqrt{b^2-2ab+a^2})}{\sqrt{(2ab-a^2)(M_\infty^2-1)}}$
4.  $\frac{4m^2}{(2m-1)\sqrt{M_\infty^2-1}} \left( \frac{h}{t}\right)^2 \left((\frac{h}{t})^2+1 \right)^{-0.5}$, $\frac{(2m-1)}{m^2} \frac{t}{h}$
Due on Tuesday December 10th at 11:00. Do Questions #1, #2, and #3 only. Question #4 is optional.
Note that the answer for question #3 has been updated.
12.09.19
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