Question 1

An aircraft flies at an altitude of 20,000 m. With what speed would it fly if M = 1?
A) 305 m/s
B) 300 m/s
C) 295 m/s
D) 290 m/s

Accepted Answer

$ 295 \mathrm{~m} / \mathrm{s} $
$$
\mathrm{M}=\frac{V}{c}=\frac{V}{\sqrt{k R T}} . \quad 1=\frac{V}{\sqrt{1.4 \times 287 \times 217}} . \quad \therefore V=\underline{295 \mathrm{~m} / \mathrm{s}}
$$

Question 2

$\text {In the region near the wall where $\overline { u ^ { \prime } v ^ { \prime } }$ is constant, if $\bar { u } ( y ) = c _ { 1 } \ln y + c _ { 2 }$ (see Example $7.8$ ), the }$$\text {eddy viscosity $\eta$ is proportional to:}$
A) $\ln y$
B) $y$
C) $y ^ { - 1 }$
D) $y ^ { 2 }$

Accepted Answer

$y$
$\text {Refer to Eq. 7.6.8 and find}$
$$\overline { u ^ { \prime } v ^ { \prime } } = \eta \frac { d \bar { u } } { d y } . \quad \text { Const } = \eta \frac { c _ { 1 } } { y } . \quad \therefore \eta = c y$$

Question 3

$\text {A large dirigible measures $18 \mathrm {~m}$ in diameter and is $100 \mathrm {~m}$ long. The thin boundary layer can}$ $\text {be assumed to be developing on a flat plate with thickness at the leading edge. The drag}$ $\text {contributed by the shear stress due to the boundary layer for a dirigible speed of $10 \mathrm {~m} / \mathrm { s }$ is}$$\text { approximately:}$
A) $\delta = 3.6 \sqrt { v x / U _ { \infty } }$
B) $\delta = 3.3 \sqrt { \nu x / U _ { \infty } }$
C) $\delta = 3 \sqrt { v x / U _ { \infty } }$
D) $\delta = 2.8 \sqrt { v x / U _ { \infty } }$

Accepted Answer

$675 \mathrm {~N}$
$\text {First, let's find $R _ { L } : R _ { L } = U _ { \infty } L / v = 10 \times 100 / 1.5 \times 10 ^ { - 5 } = 6.67 \times 10 ^ { 7 }$. An approximation}$ $\text {to the drag can be found using Eq. 8.6.29:}$
$$F _ { D } = \frac { 1 } { 2 } \rho U _ { \infty } ^ { 2 } L w C _ { f } = \frac { 1 } { 2 } \times 1.2 \times 10 ^ { 2 } \times 100 \times ( 2 \pi \times 9 ) \times 0.073 \left( \frac { 10 \times 100 } { 1.5 \times 10 ^ { - 5 } } \right) ^ { - 0.2 } = \underline { 675 \mathrm {~N} }$$

Question 4

$\text {If the velocity profile in a boundary layer was approximated by the linear distribution}$ $\text {$u = A y + B$, the boundary condition(s) that the profile would be required to satisfy is (are):}$
A) $\text {(1) and (2)}$$\quad $$\quad $$\quad $$\quad $$\quad $$\quad $(1) $\text {$u = 0$ at $y = 0$}$
B)$\text { (1) and (3)}$$\quad $$\quad $$\quad $$\quad $$\quad $$\quad $(2)$\text { $u = U _ { \infty }$ at $y = \delta$}$
C)$\text { (2) and (3)}$$\quad $$\quad $$\quad $$\quad $$\quad $$\quad $(3) $\text {$d u / d y = 0$ at $y = \delta$}$
D)$\text { (1), (2), and (4)}$$\quad $$\quad $$\quad $$\quad $(4)$\text { $d ^ { 2 } u / d y ^ { 2 } = 0$ at $y = 0$}$

Accepted Answer

(1) and (2)
Using (1) and (2) would result in
$u = U _ { \infty } y / \delta$ The other two conditions would not be
used. The error when using this profile would be relatively large, as will be
discovered later.

Question 5

A traffic flow on a freeway is very dense and it is proposed that possibly a fluid flow could simulate the flow. If it was decided to study the feasibility of the proposal, which flow would Be a possibility?
A) An incompressible flow
B) A compressible flow with M < 1
C) A compressible flow with M > 1
D) A compressible flow with M = 1

Accepted Answer

$ \mathrm{M}>1 $
Consider a primary characteristic of an incompressible flow or a subsonic flow: as the area decreases, the velocity increases. For a supersonic flow, as the area decreases the velocity decreases, as demanded by Eq. 9.3.10. We all know that when a traffic flow encounters a decrease in the number of lanes, the traffic slows down, it does not speed up. So, it would require a compressible flow model with $ M>1 $.

Question 6

The free-stream flow outside a boundary layer is:
A) Either laminar or turbulent, depending on the Reynolds number
B) A smooth flow free of disturbances
C) An irrotational flow
D) Neither laminar nor turbulent

Accepted Answer

Neither laminar nor turbulent
The free-stream flow certainly has fluctuations but they are not turbulence, i.e., the
velocity fluctuations are not correlated: for example,
 $\overline { u ^ { \prime } v ^ { \prime } } = 0$ Any turbulence
would occur inside the boundary layer. The flow outside the boundary layer is
usually irrotational (a potential flow) but it could have rotation (vorticity) if the
oncoming flow possessed velocity gradients.

Question 7

$\text {If an irrotational vortex approximates the air flow in a tornado, the highest possible velocity,}$ $\text {assuming an incompressible flow with $p _ { \text {atm } } = 100 \mathrm { kPa }$, is nearest:}$
A) $400 \mathrm {~m} / \mathrm { s }$
B) $350 \mathrm {~m} / \mathrm { s }$
C) $300 \mathrm {~m} / \mathrm { s }$
D) $250 \mathrm {~m} / \mathrm { s }$

Accepted Answer

$400 \mathrm {~m} / \mathrm { s }$
$\text {Assume the velocity at the extreme outer edges of the tornado is zero and the}$ $\text {pressure is $100 \mathrm { kPa }$. The lowest possible pressure is absolute zero, i.e., $p _ { 2 } = 0 \mathrm { kPa }$.}$$\text { Use Bernoulli's equation and find}$
$$\frac { V _ { 1 } ^ { 2 } } { 2 } + \frac { p _ { 1 } } { \rho } + g L _ { 1 } = \frac { V _ { 2 } ^ { 2 } } { 2 } + \frac { \partial _ { 2 } } { \rho } + g Z _ { 2 } . \quad \frac { 100000 } { 1.2 } = \frac { V _ { 2 } ^ { 2 } } { 2 } . \quad \therefore V _ { 2 } = \underline { 408 \mathrm {~m} / \mathrm { s } }$$
$\text {where standard conditions have been assumed so that $\rho = 1.2 \mathrm {~kg} / \mathrm { m } ^ { 3 }$. This is unrea-}$$\text {sonable since it exceeds the speed of sound, i.e., it would be a compressible flow, so}$$\text { the assumption of an incompressible flow would not be applicable. But, it does }$$\text {illustrate why the velocities are so high near the eye of a tornado.}$

Question 8

In which one of the following flows must compressibility be considered?
A) Flow around a commercial jet at takeoff
B) A category 5 hurricane
C) A Corvette at full-throttle on the Salt Lake Flats
D) Flow around the wing of a commercial aircraft in its approach to an airport

Accepted Answer

Flow around the wing of a commercial aircraft in its approach to an airport
Speeds must be in excess of 100 m/s (360 kph or 240 mph) before compressibility
must be considered in flow around an object. Commercial aircraft take off at speeds
less than 40 m/s, considerably below M = 0.3. Only D
 qualifies.

Question 9

A tall cell-phone tower is 12 cm in diameter over much of the height. A wind blows at 25 m/s. The frequency of the vortices being shed from the tower is nearest:
A) 44 Hz
B) 33 Hz
C) 22 Hz
D) 0 Hz

Accepted Answer

$0 \mathrm {~Hz}$
$\text {The temperature of the air is not given so assume standard conditions. The Reynolds}$$\text { number is}$
$$\operatorname { Re } = \frac { V D } { v } = \frac { 25 \times 0.12 } { 1.5 \times 10 ^ { - 5 } } = 2 \times 10 ^ { 5 }$$
$\text {Refer to Fig. $8.10$ and observe that this Reynolds number is too high for vortices to be}$ $\text {shed so $f = \underline { 0 \mathrm {~Hz} }$. There would be a large steady separated region behind the tower.}$ $\text {The temperature would be insignificant since the Reynolds number would be too}$$\text { large for any wind temperature. Vortices would be shed at a much lower velocity.}$

Question 10

To determine boundary-layer characteristics on a body, for example the boundary-layer thickness, we must first know the potential-flow solution. The information used from the
Potential-flow solution is:
A) The lift and drag acting on the body
B) The pressure and velocity at the boundary of the body
C) The shearing stress acting on the boundary of the body
D) The normal and shearing stress acting on the boundary of the body

Accepted Answer

he pressure and velocity at the boundary of the body
The velocity at the outer edge of the boundary layer is assumed to be the velocity at
the wall obtained from the potential flow solution. The pressure is needed since the
Navier-Stokes equation, which must be solved to obtain the solution to the
boundary-layer problem, includes the pressure gradient.

Question 11

A bomb blast sends a shock wave through the 20 $$20 ^ { \circ } \mathrm { C }$$ C atmosphere. At a certain location, the shock wave is travelling at M = 3. The induced velocity behind the wave is nearest:
A) 760 m/s
B) 620 m/s
C) 540 m/s
D) 410 m/s

Accepted Answer

A
 $760 \mathrm {~m} / \mathrm { s }$
$\text {At $\mathrm { M } = 3$ the normal-shock flow Table D.2 gives $\mathrm { M } _ { 2 } = 0.4752$ and $T _ { 2 } = 293 \times 2.679$}$ $\text {$= 785 \mathrm {~K}$. Hence,}$
$$V _ { 2 } = M _ { 2 } \sqrt { k R T _ { 2 } } = 0.4752 \sqrt { 1.4 \times 287 \times 785 } = 267 \mathrm {~m} / \mathrm { s }$$
$\text {This is the velocity behind a stationary shock wave. Superpose the velocity}$ $\text {$V _ { 1 } = 3 \sqrt { 1.4 \times 287 \times 293 } = 1029 \mathrm {~m} / \mathrm { s }$ and find}$
$$V = 1029 - 267 = \underline { 762 \mathrm {~m} / \mathrm { s } }$$
$\text {Such high induced velocities obviously cause severe damage.}$

Question 12

A smooth 10-cm-diameter sphere weighs 6 N. If it is dropped in 5 $5 ^ { \circ } \mathrm { C }$ C water, its terminal velocity will be nearest:
A) 0.66 m/s
B) 0.95 m/s
C) 6.5 m/s
D) 10.5 m/s

Accepted Answer

$0.66 \mathrm {~m} / \mathrm { s }$
$\text {When the sphere reaches terminal velocity, sum the forces acting on the sphere:}$
$$\begin{aligned}
W = F _ { B } + F _ { D } & = \gamma _ { w } \frac { 4 } { 3 } \pi R ^ { 3 } + \frac { 1 } { 2 } \rho A V ^ { 2 } C _ { D } \
6 & = 9810 \times \frac { 4 } { 3 } \times \pi \times 0.05 ^ { 3 } + \frac { 1 } { 2 } \times 1000 \times \pi \times 0.05 ^ { 2 } \times V ^ { 2 } \times 0.5
\end{aligned}$$
$\text {We have selected $C _ { D } = 0.5$ from Fig. $8.9$ assuming $2 \times 10 ^ { 4 } < \operatorname { Re } < 2 \times 10 ^ { 5 }$. With this}$$\text { assumption we find $V = 0.663 \mathrm {~m} / \mathrm { s }$. Now check the Reynolds number to determine if }$$\text {it's in the proper range:}$
$$\operatorname { Re } = \frac { V D } { v } = \frac { 0.663 \times 0.1 } { 1.519 \times 10 ^ { - 6 } } = 4.37 \times 10 ^ { 4 } . \quad \therefore \mathrm { OK }$$

Question 13

River water flows around circular 20-cm-diameter pier at 0.01 m/s. the flow is:
A) Creeping flow
B) Flow with a wake
C) Stokes flow
D) Low-Reynolds-number flow

Accepted Answer

$\text { Flow with a wake}$
$\text {Calculate the Reynolds number:}$
$$\operatorname { Re } = \frac { V D } { v } = \frac { 0.01 \times 0.2 } { 10 ^ { - 6 } } = 2000$$
$\text {Obviously this is not a creeping flow (a Stokes flow or a low-Reynolds-number flow).}$$\text { A cylinder or a sphere must be very small before a creeping flow is observed. Even}$ $\text {though the water temperature could be different from $20 ^ { \circ } \mathrm { C }$, the kinematic viscosity}$ $\text {could not be such that the $\operatorname { Re } < 5$.}$

Question 14

Select the incorrect statement for an inviscid flow.
A) The drag on a body in an inviscid flow is zero
B) The lift on a streamlined body can be approximated by neglecting viscous effects
C) The velocity distribution in the wake of a body can be approximated by neglecting viscous effects
D) The normal velocity component at the boundary of a body in an inviscid flow is zero

Accepted Answer

The velocity distribution in the wake of a body can be approximated by neglecting
viscous effects
Viscous effects are responsible for the velocity distribution in the wake region. We
have not attempted to approximate the velocity distribution in the wake of a body.
That is beyond the scope of an introductory or even an intermediate fluids course.

Question 15

$\text {A line source of strength $2 \mathrm {~m} ^ { 2 } / \mathrm { s }$ is superposed with a velocity of $8 \mathrm {~m} / \mathrm { s }$ producing along-}$$\text {slender body that resembles the one shown. The thickness $h$ of the body is nearest:}$
A) $15 \mathrm {~cm}$ 
B) $20 \mathrm {~cm}$
C) $25 \mathrm {~cm}$
D) $30 \mathrm {~cm}$

Accepted Answer

$25 \mathrm {~cm}$
$\text {Superposition of the uniform flow function and the source function provides}$
$$\psi = 8 y + \frac { 2 } { 2 \pi } \theta = 8 r \sin \theta + \frac { \theta } { \pi }$$
$\text {The streamline at $\theta = \pi$ splits at the stagnation point so its value remains the same at }$$\text {$\psi = 8 r \sin \pi + \pi / \pi = 1$ along the top of the body ( $\psi =$ const along a streamline). But,}$$\text { using the equation for $\psi$, we see that $\psi = 0$ along the positive $x$-axis where $\theta = 0$. So, }$$\text {the thickness of the body is}$
$$q = U \times ( h / 2 ) = \Delta \psi . \quad \therefore 8 ( h / 2 ) = 1 . \quad \therefore h = \underline { 0.25 \mathrm {~m} }$$
$\text {using the result of Example 8.9.}$
$\text {Actually, at a large distance from the origin the velocity in the flow would be 8 }$$\text {$\mathrm { m } / \mathrm { s }$ so, the thickness of the body would be $q = h V$ or $2 = h \times 8$ so that $h = \underline { 0.25 \mathrm {~m} }$.}$

Question 16

$\text {If the straight-line approximation $u = U _ { \infty } y / \delta$ to the velocity profile is used, the boundary-}$$\text {layer thickness for a zero-pressure gradient flow on a flat plate would be approximated to be:}$
A) $15 \mathrm {~cm}$
B) $20 \mathrm {~cm}$
C) $25 \mathrm {~cm}$
D) $30 \mathrm {~cm}$

Accepted Answer

$\delta = 3 \sqrt { v x / U _ { \infty } }$
$\text {Substitute into the von Karman integral equation:}$
\[\begin{array} { c } 
\tau _ { 0 } = \frac { d } { d x } \int _ { 0 } ^ { \delta } \rho U _ { \infty } \frac { y } { \delta } \left( U _ { \infty } - U _ { \infty } \frac { y } { \delta } \right) d y \
= \rho U _ { \infty } ^ { 2 } \frac { d } { d x } \int _ { 0 } ^ { \delta } y \left( 1 - \frac { y } { \delta } \right) d y = \rho U _ { \infty } ^ { 2 } \frac { d } { d x } \left( \frac { \delta ^ { 2 } } { 2 } - \frac { \delta ^ { 2 } } { 3 } \right) = \rho U _ { \infty } ^ { 2 } \frac { \delta } { 3 } \frac { d \delta } { d x } \
\text { But, } \quad \tau _ { 0 } = \left. \mu \frac { \partial u } { \partial y } \right| _ { y = 0 } = \mu \frac { U _ { \infty } } { \delta } . \quad \text { So, } \mu \frac { U _ { \infty } } { \delta } = \rho U _ { \infty } ^ { 2 } \frac { \delta } { 3 } \frac { d \delta } { d x } \
\therefore \delta ^ { 2 } d \delta = \frac { 3 v } { U _ { \infty } } d x . \text { Integrate, using } \delta = 0 \text { at } x = 0 \text { and obtain } \delta = 3 \sqrt { v x / U _ { \infty } } .
\end{array}\]

Question 17

$\text {If the velocity potential in an inviscid flow is given by $\phi = A y$, where $\mathrm { A }$ is a constant, the}$ $\text {stream function is:}$
A) $- A y ^ { 2 } + B$
B) $A x ^ { 2 } + B$
C) $A y + B$
D) $- A x + B$

Accepted Answer

$- A x + B$
$\text {Equation (8.5.11) relates the velocity potential function and the stream function:}$
$$\frac { \partial \psi } { \partial x } = - \frac { \partial \phi } { \partial y } = - A . \quad \therefore \psi ( x , y ) = \int - A d x = - A x + f ( y )$$
$\text {Then $\frac { \partial \psi } { \partial y } = \frac { d f } { d y }$ which must, by Eq. 8.5.11, equal $\frac { \partial \phi } { \partial x } = 0$.}$$\text { $\therefore \frac { d f } { d y } = 0$ so $f = \mathrm { const } = B$ and $\psi = \underline { - A x + B }$. Most often, we let $B = 0$.}$

Question 18

A farmer is using a tank of 20 $20 ^ { \circ } \mathrm { C }$ C nitrogen pressurized to 540 kPa absolute. It exits the tank out a hose. The hose snaps and the farmer is hit with the expanding nitrogen. Air is primarily
Nitrogen so assuming it is air flowing with no losses, the temperature of the exiting nitrogen is
Nearest: 
A) $ -120^{\circ} \mathrm{C} $
B) $ -90^{\circ} \mathrm{C} $
C) $ -70^{\circ} \mathrm{C} $
D) $ -50^{\circ} \mathrm{C} $

Accepted Answer

$ -90^{\circ} \mathrm{C} $
The pressure ratio between the exit and the tank is $ p_{e} / p_{0}=100 / 540=0.185 $. From the isentropic flow Table D.1 we find, at $ \mathrm{M}=1.76 $, that
$$
\frac{T}{T_{0}}=\frac{T}{293}=0.6175=181 \mathrm{~K} \text { or } \underline{-92^{\circ} \mathrm{C}} \text { or }-134^{\circ} \mathrm{F}
$$
Such a low temperature would require that the farmer be treated for "burns." The temperature of the nitrogen in the tank is hardly a factor since it would undoubtedly be in the range of $ 10^{\circ} \mathrm{C}<T<35^{\circ} \mathrm{C} $.

Question 19

The drag on a blunt object is due primarily to:
A) The relatively high pressure on the front of the object
B) The wake region
C) The shear stress acting on the front and rear of the object
D) The separated region

Accepted Answer

The separated region Even on a streamlined object, there is a relatively high pressure on the front. The wake does not exist except downstream of an object so it doesn't account for the drag.
It is the existence of a separated region in which a relatively low pressure combined with the relatively high pressure over the front that results in the drag.

Question 20

A cyclist, traveling 12 m/s, wears a helmet that is close to being a 36-cm-diameter sphere. The drag force acting on that helmet is approximately:
A) 1.8 N
B) 2.6 N
C) 3.2 N
D) 3.8 N

Accepted Answer

$1.8 \mathrm {~N}$
$\text {The Reynolds number, assuming standard air, is}$
$$\operatorname { Re } = \frac { V D } { v } = \frac { 12 \times 0.36 } { 1.5 \times 10 ^ { - 5 } } = 2.86 \times 10 ^ { 5 }$$
$\text {Using the drag coefficient of $C _ { D } = 0.2$ from Fig. $8.9$, the drag force is approximately}$
$$F _ { D } = \frac { 1 } { 2 } \rho V ^ { 2 } A C _ { D } = \frac { 1 } { 2 } 1.2 \times 12 ^ { 2 } \times \pi \times 0.18 ^ { 2 } \times 0.2 = \underline { 1.76 \mathrm {~N} }$$

Quiz 15: Fluid Dynamics and Aerodynamics

An aircraft flies at an altitude of 20,000 m. With what speed would it fly if M = 1?

$\text {In the region near the wall where \overline { u ^ { \prime } v ^ { \prime } } is constant, if \(\bar { u } ( y ) = c _ { 1 } \ln y + c _ { 2 }\) (see Example \(7.8\) ) , the }$ $\text {eddy viscosity \eta is proportional to:}$

$\text {If the velocity profile in a boundary layer was approximated by the linear distribution}$ $\text {u = A y + B, the boundary condition(s) that the profile would be required to satisfy is (are) :}$

A traffic flow on a freeway is very dense and it is proposed that possibly a fluid flow could simulate the flow. If it was decided to study the feasibility of the proposal, which flow would Be a possibility?

The free-stream flow outside a boundary layer is:

$\text {If an irrotational vortex approximates the air flow in a tornado, the highest possible velocity,}$ $\text {assuming an incompressible flow with p _ { \text {atm } } = 100 \mathrm { kPa }, is nearest:}$

In which one of the following flows must compressibility be considered?

A tall cell-phone tower is 12 cm in diameter over much of the height. A wind blows at 25 m/s. The frequency of the vortices being shed from the tower is nearest:

To determine boundary-layer characteristics on a body, for example the boundary-layer thickness, we must first know the potential-flow solution. The information used from the Potential-flow solution is:

A bomb blast sends a shock wave through the 20 $20 ^ { \circ } \mathrm { C }$ C atmosphere. At a certain location, the shock wave is travelling at M = 3. The induced velocity behind the wave is nearest:

A smooth 10-cm-diameter sphere weighs 6 N. If it is dropped in 5 $5 ^ { \circ } \mathrm { C }$ C water, its terminal velocity will be nearest:

River water flows around circular 20-cm-diameter pier at 0.01 m/s. the flow is:

Select the incorrect statement for an inviscid flow.

$\text {A line source of strength 2 \mathrm {~m} ^ { 2 } / \mathrm { s } is superposed with a velocity of \(8 \mathrm {~m} / \mathrm { s }\) producing along-}$ $\text {slender body that resembles the one shown. The thickness h of the body is nearest:}$

$\text {If the straight-line approximation u = U _ { \infty } y / \delta to the velocity profile is used, the boundary-}$ $\text {layer thickness for a zero-pressure gradient flow on a flat plate would be approximated to be:}$

$\text {If the velocity potential in an inviscid flow is given by \phi = A y, where \(\mathrm { A }\) is a constant, the}$ $\text {stream function is:}$

The drag on a blunt object is due primarily to:

A cyclist, traveling 12 m/s, wears a helmet that is close to being a 36-cm-diameter sphere. The drag force acting on that helmet is approximately: