Deck 6: Dimensional Analysis and Similitude

Question 1

A single bar of soap has a small light attached. The bar is released in a river and a snapshot is taken every second from a blimp which is stationary over the river. The snapshots are all
Superimposed on one large picture. The line formed by connecting the dots is:
(A) A streamline
(B) A streakline
(C) A pathline
(D) A line that has not been defined

Answer 1

A SINGLE BAR OF SOAP HAS A SMALL LIGHT ATTACHED. THE BAR IS RELEASED IN A RIVER AND A SNAPSHOT IS
taken every second from a blimp which is stationary over the river. The snapshots are all
superimposed on one large picture. The line formed by connecting the dots is:
(D) A line that has not been defined
The line does not satisfy any of the definitions of a streamline, a streakline, or a
pathline.

Question 2

$\text {The velocity vector at a point in a fluid flow is given by \(5 \mathbf { i } + 12 \mathbf { j }\). The unit vector which is }$$\text {perpendicular to the streamline passing through the point is:}$
(A) $( - 12 \mathbf { i } + 5 \mathbf { j } ) / 13$
(B) $( - 12 \mathbf { i } = 5 \mathbf { j } ) / 13$
(C) $( 5 \mathbf { i } + 12 \mathbf { j } ) / 13$
(D) $( - 5 \mathbf { i } + 12 \mathbf { j } ) / 13$

Answer 2

A
$( - 12 \mathbf { i } + 5 \mathbf { j } ) / 13$
By definition, the velocity vector is tangent to the streamline, so the unit vector is perpendicular to the velocity vector so that $\mathbf { V } \cdot \mathbf { n } = 0$ . The velocity vector has only $x$ and $y$ -components so the unit vector will have only $x$ - and $y$ -components. We have
$$\mathbf { V } \cdot \mathbf { n } = ( 5 \mathbf { i } + 12 \mathbf { j } ) \cdot \left( n _ { x } \mathbf { i } + n _ { y } \mathbf { j } \right) = 0 \text { or } 5 n _ { x } + 12 n _ { y } = 0$$
The unit vector requires that $n _ { x } ^ { 2 } + n _ { y } ^ { 2 } = 1$ giving two equations and two unknowns:
$$\left( - 12 n _ { y } / 5 \right) ^ { 2 } + n _ { y } ^ { 2 } = 1 . \quad \therefore n _ { y } = 5 / 13 \quad \text { and } \quad n _ { x } = - \frac { 12 } { 5 } n _ { y } = - 12 / 13$$
The unit vector is written as $\mathbf { n } = ( - 12 \mathbf { i } + 5 \mathbf { j } / ) 13$ . The unit vector $( 12 \mathbf { i } - 5 \mathbf { j } / ) 13$ in the opposite direction would also be normal to the streamline.

Question 3

If a sensor were placed in the flow of Problem 4, it would rotate about the z-axis at what angular velocity at (1, −1)?

A) 5 rad/s
B) 4 rad/s
C) 3 rad/s
D) 2 rad/s

Answer 3

3 rad/s

Question 4

$\text {The velocity vector in a particular flow field is given by \(\mathbf { V } = 2 x ^ { 2 } y \mathbf { i } - 4 y ^ { 2 } x \mathbf { j } \mathrm { m } / \mathrm { s }\). The acceleration at \(( 1 , - 1 )\) is:}$
(A) $16 \mathbf { i } \mathrm { m } / \mathrm { s } ^ { 2 }$
(B) $16 \mathbf { i } - 40 \mathbf { j } \mathrm { m } / \mathrm { s } ^ { 2 }$
(C) $- 24 \mathrm { j } \mathrm { m } / \mathrm { s } ^ { 2 }$
(D) 0

Question 5

In a steady flow in a long pipe, such as the Alaska oil pipe line, the Eulerian description of the velocity field would express the velocity V in the pipe as:
(A) V(t)
(B) V(r)
(C) V(r, t)
(D) V(r, z)