Chapter 9, Problem 1P

The roof of a minivan is approximated as a horizontal flat plate. Plot the length of the laminar boundary layer as a function of minivan speed, V, as the minivan accelerates from 10 mph to 90 mph.

To determine

The plot of the length of the laminar boundary versus minivan speed.

Explanation of Solution

Given:

The initial speed of the minivan $\left(V_1\right)$ is $10\text{mph}$.

The final speed of the minivan $\left(V_2\right)$ is $20\text{mph}$.

Calculation:

The Reynolds number for the laminar boundary layer is $5\times {10}^5$.

Refer the appendix, Table A.9 “Properties of Air at Atmospheric Pressure (U.S. Customary Units)” for the temperature of $68°\text{F}$.

  $\begin{array}{c}\mu =3.379\times {10}^{-7}\text{ lbf}\cdot {\text{s/ft}}^2\times 5\\ \rho =0.00234{\text{ slugs/ft}}^3\end{array}$

Calculate the expression for the length of the laminar boundary layer using the Reynolds number $\left(\mathrm{Re}\right)$.

  $\begin{array}{l}\mathrm{Re}=\frac{\rho VL}{\mu }\\ L=\frac{\mathrm{Re}\mu }{\rho V}\end{array}$

Substitute the known values in the above Equation.

  $\begin{array}{c}L=\frac{3.379\times {10}^{-7}\text{ lbf}\cdot {\text{s/ft}}^2\times 5\times {10}^5}{0.00234{\text{ slugs/ft}}^3\times V}\\ =\frac{80.342}{V}\end{array}$        (I)

Use the spread sheet and equation (I), calculate the length of minivan for various speed range from 10 to 90 mph.

$V_1\left(\text{mph}\right)$	$V_1\left(\text{ft/s}\right)$	$L\left(\text{ft}\right)$
10	14.6667	5.477851
20	29.3334	2.738926
30	44.0001	1.82595
40	58.6668	1.369463
50	73.3335	1.09557
60	88.0002	0.912975
70	102.6669	0.78255
80	117.3336	0.684731
90	132.0003	0.60865

Use the above table values; plot the length of the laminar boundary layer as a function of minivan speed as shown in the figure below: