Fluid Mechanics Assignment 7 — Boundary Layer  
Instructions
$\xi$ is a parameter related to your student ID, with $\xi_1$ corresponding to the last digit, $\xi_2$ to the last two digits, $\xi_3$ to the last three digits, etc. For instance, if your ID is 199225962, then $\xi_1=2$, $\xi_2=62$, $\xi_3=962$, $\xi_4=5962$, etc. Keep a copy of the assignment — the assignment will not be handed back to you. You must be capable of remembering the solutions you hand in.
05.01.14
Question #1
Starting from the $x$-component of the momentum equation: $$ \rho \left(\frac{\partial u}{\partial t} + u \frac{\partial u}{\partial x} + v \frac{\partial u}{\partial y} + w \frac{\partial u}{\partial z} \right) = -\frac{\partial P}{\partial x} + \mu \frac{\partial^2 u}{\partial x^2} + \mu \frac{\partial^2 u}{\partial y^2} + \mu \frac{\partial^2 u}{\partial z^2} +B_x $$ and from the mass conservation equation: $$ \frac{\partial \rho}{\partial t} + \frac{\partial }{\partial x}(\rho u) + \frac{\partial }{\partial y}(\rho v) + \frac{\partial }{\partial z}(\rho w) = 0 $$ Show that the skin friction coefficient and the thickness of a laminar boundary layer correspond to: $$ C_f=0.647 \cdot {\rm Re}_x^{-0.5} {\rm ~~~~and~~~~} \delta/x=4.64 \cdot {\rm Re}_x^{-0.5} $$ Outline all assumptions. Note: this question is worth double the points awarded to the other questions.
Question #2
A wing with a span of 2 m and a chord of 0.4 m is placed in a wind tunnel where air with a density of 1 kg/m$^3$, a viscosity of $10^{-5}$ kg/ms, and a temperature of $300$ K flows towards the wing with a speed of $(10-0.5\times \xi_1)$ m/s. When the airflow reaches steady-state, you measure a drag force of 0.5 N and a lift force of 6 N acting on the wing. Knowing that the camber is much less than the chord and that the angle of attack is small, determine approximately the percentage of the drag force that is due to skin friction. As well, determine the boundary layer thickness at the trailing edge of the airfoil.
Question #3
Air at atmospheric pressure and at 20$^\circ$C flows over a flat plate at a speed of 4 m/s. What is the velocity parallel to the plate at a perpendicular distance 0.0025 m from the plate and at 0.50 m from the leading edge of the plate?
Question #4
An air stream with a speed of $(5-0.3\times\xi_1)$ m/s and density of $\rho=1.0$ kg/m$^3$ flows parallel to a flat plate with a length of 45 cm and a width of 100 cm. Determine the total drag force on the flat plate and calculate the boundary layer thickness 10 and 45 cm from the leading edge. Take the kinematic viscosity as $15\times 10^{-6}$ m$^2$/s.
Question #5
An air stream with a speed of $50$ m/s and density of $\rho=1.0$ kg/m$^3$ flows parallel to a flat plate with a length of 45 cm and a width of 100 cm. Determine the total drag force on the flat plate and calculate the boundary layer thickness 10 and 45 cm from the leading edge. Take the kinematic viscosity as $15\times 10^{-6}$ m$^2$/s.
Due on Wednesday November 26th
11.19.14
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