2020 Fundamentals of Fluid Mechanics A Final Exam  
Tuesday December 15th
10:30 — 12:30
INSTRUCTIONS
  NO NOTES OR BOOKS EXCEPT FOR THE FUNDAMENTALS OF FLUID MECHANICS TABLES.
  ONLY A BAREBONE CALCULATOR IS ALLOWED WITHOUT SDCARD SLOTS OR A LARGE SCREEN.
  ALL QUESTIONS HAVE EQUAL VALUE; ANSWER ALL 3 QUESTIONS.
  WRITE YOUR SOLUTIONS IN SINGLE COLUMN FORMAT, WITH ONE STATEMENT FOLLOWING ANOTHER VERTICALLY.
  ALWAYS START YOUR SOLUTIONS USING THE EQUATIONS IN THE TABLES.
  OUTLINE ALL ASSUMPTIONS MADE.
  WRITE YOUR SOLUTIONS NEATLY SO THAT THEY ARE EASY TO READ AND VERIFY.
  USE A PENCIL OR A PEN WITH BLACK INK. DO NOT USE BLUE OR RED INK.
  DON'T WRITE ONE LINE WITH TWO EQUAL SIGNS.
  HIGHLIGHT YOUR ANSWERS USING A BOX.
  SCAN YOUR SOLUTIONS USING CAMSCANNER AND CREATE ONE PDF FILE PER QUESTION. UPLOAD THE TWO PDF FILES ON THE D2L.
12.14.20
Question #1
Consider the following flow in a corner:
figure3.png  ./download/file.php?id=7321&sid=6ff1d717744a47b7a6f2abc54fdf3592  ./download/file.php?id=7321&t=1&sid=6ff1d717744a47b7a6f2abc54fdf3592
The flow velocity in the $x$-direction is described by the expression: $$ u=K\left(x^{(3+a)} - 3 x^{(1+a)} y^2 \right) $$ where $K$ and $a$ are constants. Knowing that the flow is constant density and two-dimensional, do the following:
(a)  determine an expression for $v$, the flow velocity in the $y$-direction
(b)  determine the angle $\theta$
Question #2
You are assigned the task to design a prototype of a new airship. The prototype has a length $L$ of 100 m and its maximum diameter $D$ is of 20 m:
figure4.png  ./download/file.php?id=7322&sid=6ff1d717744a47b7a6f2abc54fdf3592  ./download/file.php?id=7322&t=1&sid=6ff1d717744a47b7a6f2abc54fdf3592
It is wished to estimate the drag force that would act on the prototype when it is at an altitude of 10 km and when the air speed $q_\infty$ is of 5 m/s (at an altitude of 10 km, the air pressure and temperature can be taken equal to 0.26 atm and 223 K respectively). To do so, you decide to build a small scale model and measure its drag in a wind tunnel under dynamically similar conditions, and further use non-dimensional analysis to relate the drag force measured in the model to the one acting on the prototype. For this purpose, do the following:
(a)  Find the non-dimensional numbers associated with this problem, and show how they are related to the Reynolds number and drag coefficient (the criteria of similarity).
(b)  If the length of the model used in the experiment is fixed to 2 m and the air density in the wind tunnel is of $1.2$ kg/m$^3$, determine the diameter of the model airship and determine the wind tunnel air flow speed.
(c)  Knowing that the drag force measured on the model is of 150 N, estimate the drag force acting on the prototype.
Notes: the air viscosity can be taken as $10^{-5}$ kg/ms for both the prototype and the model, and the gas constant for air is $286$ J/kgK.
Question #3
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=7323&sid=6ff1d717744a47b7a6f2abc54fdf3592  ./download/file.php?id=7323&t=1&sid=6ff1d717744a47b7a6f2abc54fdf3592
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.
PDF 1✕1 2✕1 2✕2
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