Viscous Flow Assignment 4 — Boundary Layer  
Question #1
Starting from the $x$-component of the momentum equation: $$ \rho \left(\frac{\partial u}{\partial t} + u \frac{\partial u}{\partial x} + v \frac{\partial u}{\partial y} + w \frac{\partial u}{\partial z} \right) = -\frac{\partial P}{\partial x} + \mu \frac{\partial^2 u}{\partial x^2} + \mu \frac{\partial^2 u}{\partial y^2} + \mu \frac{\partial^2 u}{\partial z^2} $$ 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.
06.28.16
Question #2
An air stream with a speed of $30$ m/s and density of $\rho=1.0$ kg/m$^3$ flows parallel to a flat plate with a length of 75 cm and a width of 100 cm. Determine the total drag force on the flat plate and calculate the boundary layer thickness 15 and 75 cm from the leading edge. Take the kinematic viscosity as $15\times 10^{-6}$ m$^2$/s.
Question #3
You perform an experiment in which liquid water at a temperature of 27$^\circ$C flows on a flat plate as follows:
figure5.png
The flat plate has a length $L$ of 2 meters, a height $H$ of 2 mm, and a depth $D$ of 1 meter. Knowing that the boundary layer thickness at the trailing edge of the plate is of $\delta=2.76$ cm, estimate as well as possible the total drag force on the plate caused by the flowing water. Please use the following data for liquid water: density $\rho=1000$ kg/m$^3$, viscosity $\mu=10^{-3}$ kg/ms, heat capacity $c=c_p=c_v=4200~$J/kgK, conductivity $k=0.6~$W/mK.
Question #4
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.
Question #5
Consider air flowing on top of a flat plate as follows:
bdrylayer.png
It is known that the length of the plate is $L=19$ cm, that the thickness of the boundary layer at the plate exit at $x=L$ is of $\delta=3$ mm, that the air viscosity is of $\mu=2\times 10^{-5}$ kg/ms, that the air density is of $\rm 1~kg/m^3$. Also, although the freestream velocity is not known precisely, it is certain that it is higher than 12 m/s: $$ u_{\infty} \gt 12~{\rm m/s} $$ Knowing the latter, do the following:
(a)  Determine whether the flow is laminar or turbulent at $x=L$.
(b)  Find the freestream velocity $u_{\infty}$.
(c)  Find total drag force acting on the plate per unit depth in N/m due to friction effects.
10.24.17
Answers
2.  1.27 mm, 11.5 mm, 2.11 N.
3.  0.0246 N, 1.451 N.
4.  0.8 mm, 6.9 mm, 3.52 N.
5.  211 m/s, 13.4 N/m.
Due on Wednesday November 1st at 13:30. Do Questions #3, #4, and #5 only.
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