2020 Fundamentals of Fluids Mechanics B Final Exam  
Tuesday May 12th 2020
10:30 — 12:30
INSTRUCTIONS
  USE FUNDAMENTALS OF FLUID MECHANICS TABLES THAT WERE DISTRIBUTED.
  ALL QUESTIONS HAVE EQUAL VALUE; ANSWER ALL 3 QUESTIONS.
  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.
04.22.20
Question #1
(a)  Recall the Reynold's solution to the boundary layer thickness: $$ \frac{\delta}{x}=3.68 ~\textrm{Re}_x^{-0.5} $$ Recall the Blasius solution to the boundary layer thickness: $$ \frac{\delta}{x}=4.9 ~\textrm{Re}_x^{-0.5} $$ Both are theoretical solutions to the same problem: a boundary layer on a flat plate. Explain why there is a large difference between the two.
(b)  The momentum thickness of the boundary layer corresponds to the height of the inflow that is responsible for the momentum within the boundary layer, as follows:
A8Q3.png  ./download/file.php?id=6185&sid=6186f82cb41335cfc1cf762941266ea9  ./download/file.php?id=6185&t=1&sid=6186f82cb41335cfc1cf762941266ea9
Find an expression for $\delta_m$ that is applicable to a constant pressure boundary layer over a flat plate in which the density is not constant but the viscosity is constant. Simplify the expression as much as possible.
(c)  For the case of a constant density and constant viscosity boundary layer, find an expression for $\delta_m/\delta$ and simplify as much as possible. Outline all assumptions.
Question #2
Consider a fluid flowing on top of a flat plate as follows:
F2020Q2.png  ./download/file.php?id=6277&sid=6186f82cb41335cfc1cf762941266ea9  ./download/file.php?id=6277&t=1&sid=6186f82cb41335cfc1cf762941266ea9
Starting from the Kelvin's circulation theorem: $$ \frac{d\Gamma}{dt} = \sum_{i=1}^3 \oint_C \left( \frac{\mu}{\rho} (\vec{\nabla}\cdot\vec{\nabla})v_i \right) d\xi_i $$ Calculate the rate of change of the circulation $\frac{d\Gamma}{dt}$ within the contour $C$ for $H=2 \delta_{x=L}$ with $\delta$ being the thickness of the boundary layer. Express $d \Gamma/dt$ as a function of the free stream fluid properties $u_\infty$, $\rho$, and $\mu$ and the dimensions $L$ and $H$, and simplify the expression as much as possible. Hint: the rate of change of $\Gamma$ is not zero.
05.03.20
Question #3
Consider water flowing around a cylinder as follows:
F2020Q3.png  ./download/file.php?id=6270&sid=6186f82cb41335cfc1cf762941266ea9  ./download/file.php?id=6270&t=1&sid=6186f82cb41335cfc1cf762941266ea9
The water temperature is of 27$^\circ$C, its density 1000 kg/m$^3$, its viscosity $0.001$ kg/ms, the velocity far from the cylinder $u_\infty$ is of 0.5 m/s, and the cylinder diameter $D$ is of 0.01 m. It is decided to obtain a steady-state solution of the wake by neglecting pressure gradients and streamwise viscous stresses within the momentum equation. Thus, the $x$ momentum equation reduces to: $$ \rho u \frac{\partial u}{\partial x} + \rho v \frac{\partial u}{\partial y} = \mu \frac{\partial^2 u}{\partial y^2} $$ Do the following:
(a)  Use order of magnitude analysis and find the condition when the term $\mu \frac{\partial^2 u}{\partial x^2}$ can be ignored in the $x$ momentum equation. The condition should apply not only to the wake far away from the cylinder but also to the wake near the cylinder. The condition derived thus can not involve the distance to the cylinder (the $x$ location).
(b)  Using the condition found in (a) and the freestream flow properties estimate the error that originates from neglecting the term $\mu \frac{\partial^2 u}{\partial x^2}$ in the $x$ momentum equation. Such an error should not depend on $x$.
(c)  Starting from the exact solution to the wake in the far field derived in class: $$ \Psi=\frac{F_{\rm D} {\rm erf}(\eta)}{2 \rho L u_\infty} $$ with $$ \eta=y \sqrt{\frac{u_\infty}{4 \nu x}} $$ derive an expression for $\frac{\partial^2 u}{\partial x^2}$ at $y=0$ and another expression for $\frac{\partial^2 u}{\partial y^2}$ at $y=0$.
(d)  Using the expressions found in (c), calculate the error due to neglecting the term $\mu \frac{\partial^2 u}{\partial x^2}$ in the $x$ momentum equation at $y=0$ for $x=2D,~5D,~20D$. Is the error found in (b) similar to the one found in (c)? Explain why there is a discrepancy.
05.07.20
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