Fluid Mechanics: Fundamentals and Applications

Yunus Cengel

Chapter 11 FLOW OVER BODIES: DRAG AND LIFT - all with Video Answers

Educators

Chapter Questions

Problem 1

Explain when an external flow is two-dimensional, three-dimensional, and axisymmetric. What type of flow is the flow of air over a car?

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Problem 1

What is terminal velocity? How is it determined?

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Problem 2

What is the difference between the upstream velocity and the free-stream velocity? For what types of flow are these two velocities equal to each other?

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Problem 3

What is the difference between streamlined and blunt bodies? Is a tennis ball a streamlined or blunt body?

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Problem 4

Name some applications in which a large drag is desired.

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Problem 5

What is drag? What causes it? Why do we usually try to minimize it?

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Problem 6

What is lift? What causes it? Does wall shear contribute to the lift?

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During flow over a given body, the drag force, the upstream velocity, and the fluid density are measured. Explain how you would determine the drag coefficient. What area would you use in the calculations?

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Problem 8

During flow over a given slender body such as a wing, the lift force, the upstream velocity, and the fluid density are measured. Explain how you would determine the lift coefficient. What area would you use in the calculations?

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Problem 9

Define the frontal area of a body subjected to external flow. When is it appropriate to use the frontal area in drag and lift calculations?

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Problem 10

Define the planform area of a body subjected to external flow. When is it appropriate to use the planform area in drag and lift calculations?

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Problem 12

What is the difference between skin friction drag and pressure drag? Which is usually more significant for slender bodies such as airfoils?

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Problem 13

What is the effect of surface roughness on the friction drag coefficient in laminar and turbulent flows?

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Problem 14

In general, how does the drag coefficient vary with the Reynolds number at (a) low and moderate Reynolds numbers and $(b)$ at high Reynolds numbers $\left(\operatorname{Re}>10^4\right)$ ?

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Problem 15

Fairings are attached to the front and back of a cylindrical body to make it look more streamlined. What is the effect of this modification on the (a) friction drag, (b) pressure drag, and (c) total drag? Assume the Reynolds number is high enough so that the flow is turbulent for both cases.

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Problem 16

What is the effect of streamlining on (a) friction drag and (b) pressure drag? Does the total drag acting on a body necessarily decrease as a result of streamlining? Explain.

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Problem 17

What is flow separation? What causes it? What is the effect of flow separation on the drag coefficient?

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Problem 19

Which car is more likely to be more fuel-efficient: the one with sharp corners or the one that is contoured to resemble an ellipse? Why?

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Problem 20

Which bicyclist is more likely to go faster: the one who keeps his head and his body in the most upright position or the one who leans down and brings his body closer to his knees? Why?

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Problem 21

The drag coefficient of a car at the design conditions of $1 \mathrm{~atm}, 25^{\circ} \mathrm{C}$, and $90 \mathrm{~km} / \mathrm{h}$ is to be determined experimentally in a large wind tunnel in a full-scale test. The height and width of the car are 1.40 m and 1.65 m , respectively. If the horizontal force acting on the car is measured to be 300 N , determine the total drag coefficient of this car.

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Problem 22

A car is moving at a constant velocity of $80 \mathrm{~km} / \mathrm{h}$. Determine the upstream velocity to be used in fluid flow analysis if (a) the air is calm, (b) wind is blowing against the direction of motion of the car at $30 \mathrm{~km} / \mathrm{h}$, and (c) wind is blowing in the same direction of motion of the car at $50 \mathrm{~km} / \mathrm{h}$.

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Problem 23

The resultant of the pressure and wall shear forces acting on a body is measured to be 700 N , making $35^{\circ}$ with the direction of flow. Determine the drag and the lift forces acting on the body.

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Problem 24

During a high Reynolds number experiment, the total drag force acting on a spherical body of diameter $D$ $=12 \mathrm{~cm}$ subjected to airflow at 1 atm and $5^{\circ} \mathrm{C}$ is measured to be 5.2 N . The pressure drag acting on the body is calculated by integrating the pressure distribution (measured by the use of pressure sensors throughout the surface) to be 4.9 N. Determine the friction drag coefficient of the sphere

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Problem 25

To reduce the drag coefficient and thus to improve the fuel efficiency, the frontal area of a car is to be reduced. Determine the amount of fuel and money saved per year as a result of reducing the frontal area from 18 to $15 \mathrm{ft}^2$. Assume the car is driven $12,000 \mathrm{mi}$ a year at an average speed of $55 \mathrm{mi} / \mathrm{h}$. Take the density and price of gasoline to be $50 \mathrm{lbm} / \mathrm{ft}^3$ and $$\$ 2.20 / \mathrm{gal}$$, respectively; the density of air to be $0.075 \mathrm{lbm} / \mathrm{ft}^3$, the heating value of gasoline to be $20,000 \mathrm{Btu} / \mathrm{bm}$; and the overall efficiency of the engine to be 32 percent.

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Problem 26

Reconsider Prob. 11-25E. Using EES (or other) software, investigate the effect of frontal area on the annual fuel consumption of the car. Let the frontal area vary from 10 to $30 \mathrm{ft}^2$ in increments of $2 \mathrm{ft}^2$. Tabulate and plot the results.

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Problem 27

A circular sign has a diameter of 50 cm and is subjected to normal winds up to $150 \mathrm{~km} / \mathrm{h}$ at $10^{\circ} \mathrm{C}$ and 100 kPa . Determine the drag force acting on the sign. Also determine the bending moment at the bottom of its pole whose height from the ground to the bottom of the sign is 1.5 m . Disregard the drag on the pole.

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Problem 28

Wind loading is a primary consideration in the design of the supporting mechanisms of billboards, as evidenced by many billboards being knocked down during high winds. Determine the wind force acting on an 8 -ft-high, 20 - ft -wide billboard due to $90-\mathrm{mi} / \mathrm{h}$ winds in the normal direction when the atmospheric conditions are 14.3 psia and $40^{\circ} \mathrm{F}$.

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Problem 29

Advertisement signs are commonly carried by taxicabs for additional income, but they also increase the fuel cost. Consider a sign that consists of a $0.30-\mathrm{m}$-high, $0.9-\mathrm{m}$-wide, and $0.9-\mathrm{m}$-long rectangular block mounted on top of a taxicab such that the sign has a frontal area of 0.3 m by 0.9 m
from all four sides. Determine the increase in the annual fuel cost of this taxicab due to this sign. Assume the taxicab is driven $60,000 \mathrm{~km}$ a year at an average speed of $50 \mathrm{~km} / \mathrm{h}$ and the overall efficiency of the engine is 28 percent. Take the density, unit price, and heating value of gasoline to be $$0.75 \mathrm{~kg} / \mathrm{L}, \$ 0.50 / \mathrm{L}$$, and $42,000 \mathrm{~kJ} / \mathrm{kg}$, respectively, and the density of air to be $1.25 \mathrm{~kg} / \mathrm{m}^3$.

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Problem 30

It is proposed to meet the water needs of a recreational vehicle (RV) by installing a $2-\mathrm{m}$-long, $0.5-\mathrm{m}$-diameter cylindrical tank on top of the vehicle. Determine the additional power requirement of the RV at a speed of $95 \mathrm{~km} / \mathrm{h}$ when the tank is installed such that its circular surfaces face (a) the front and back and (b) the sides of the RV. Assume atmospheric conditions are 87 kPa and $20^{\circ} \mathrm{C}$.

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Problem 31

At highway speeds, about half of the power generated by the car's engine is used to overcome aerodynamic drag, and thus the fuel consumption is nearly proportional to the drag force on a level road. Determine the percentage increase in fuel consumption of a car per unit time when a person who normally drives at $55 \mathrm{mi} / \mathrm{h}$ now starts driving at $75 \mathrm{mi} / \mathrm{h}$.

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Problem 32

A 4-mm-diameter plastic sphere whose density is $1150 \mathrm{~kg} / \mathrm{m}^3$ is dropped into water at $20^{\circ} \mathrm{C}$. Determine the terminal velocity of the sphere in water.

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Problem 33

During major windstorms, high vehicles such as RVs and semis may be thrown off the road and boxcars off their tracks, especially when they are empty and in open areas. Consider a $5000-\mathrm{kg}$ semi that is 8 m long, 2 m high, and 2 m wide. The distance between the bottom of the truck and the road is 0.75 m . Now the truck is exposed to winds from its side surface. Determine the wind velocity that will tip the truck over to its side. Take the air density to be $1.1 \mathrm{~kg} / \mathrm{m}^3$ and assume the weight to be uniformly distributed.

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Problem 34

An 80-kg bicyclist is riding her $15-\mathrm{kg}$ bicycle downhill on a road with a slope of $12^{\circ}$ without pedaling or braking. The bicyclist has a frontal area of $0.45 \mathrm{~m}^2$ and a drag coefficient of 1.1 in the upright position, and a frontal area of $0.4 \mathrm{~m}^2$ and a drag coefficient of 0.9 in the racing position. Disregarding the rolling resistance and friction at the bearings, determine the terminal velocity of the bicyclist for both positions. Take the air density to be $1.25 \mathrm{~kg} / \mathrm{m}^3$.

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Problem 35

A wind turbine with two or four hollow hemispherical cups connected to a pivot is commonly used to measure wind speed. Consider a wind turbine with two 8-cm-diameter cups with a center-to-center distance of 25 cm , as shown in Fig. P11-35. The pivot is stuck as a result of some malfunction, and the cups stop rotating. For a wind speed of $15 \mathrm{~m} / \mathrm{s}$ and air density of $1.25 \mathrm{~kg} / \mathrm{m}^3$, determine the maximum torque this turbine applies on the pivot.

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Problem 36

Reconsider Prob. 11-35. Using EES (or other) software, investigate the effect of wind speed on the torque applied on the pivot. Let the wind speed vary from 0 to $50 \mathrm{~m} / \mathrm{s}$ in increments of $5 \mathrm{~m} / \mathrm{s}$. Tabulate and plot the results.

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Problem 37

A 5-ft-diameter spherical tank completely submerged in freshwater is being towed by a ship at $12 \mathrm{ft} / \mathrm{s}$. Assuming turbulent flow, determine the required towing power.

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Problem 38

During steady motion of a vehicle on a level road, the power delivered to the wheels is used to overcome aerodynamic drag and rolling resistance (the product of the rolling resistance coefficient and the weight of the vehicle), assuming the friction at the bearings of the wheels is negligible. Consider a car that has a total mass of 950 kg , a drag coefficient of 0.32 , a frontal area of $1.8 \mathrm{~m}^2$, and a rolling resistance coefficient of 0.04 . The maximum power the engine can deliver to the wheels is 80 kW . Determine (a) the speed at which the rolling resistance is equal to the aerodynamic drag force and (b) the maximum speed of this car. Take the air density to be $1.20 \mathrm{~kg} / \mathrm{m}^3$.

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Problem 39

Reconsider Prob. 11-38. Using EES (or other) software, investigate the effect of car speed on the required power to overcome (a) rolling resistance, (b) the aerodynamic drag, and (c) their combined effect. Let the car speed vary from 0 to $150 \mathrm{~km} / \mathrm{h}$ in increments of $15 \mathrm{~km} / \mathrm{h}$. Tabulate and plot the results.

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Problem 40

A submarine can be treated as an ellipsoid with a diameter of 5 m and a length of 25 m . Determine the power required for this submarine to cruise horizontally and steadily at $40 \mathrm{~km} / \mathrm{h}$ in seawater whose density is $1025 \mathrm{~kg} / \mathrm{m}^3$. Also determine the power required to tow this submarine in air whose density is $1.30 \mathrm{~kg} / \mathrm{m}^3$. Assume the flow is turbulent in both cases.

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Problem 41

An 0.80-m-diameter, $1.2-\mathrm{m}$-high garbage can is found in the morning tipped over due to high winds during the night. Assuming the average density of the garbage inside to be $150 \mathrm{~kg} / \mathrm{m}^3$ and taking the air density to be $1.25 \mathrm{~kg} / \mathrm{m}^3$, estimate the wind velocity during the night when the can was tipped over. Take the drag coefficient of the can to be 0.7 .

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Problem 42

The drag coefficient of a vehicle increases when its windows are rolled down or its sunroof is opened. A sports car has a frontal area of $18 \mathrm{ft}^2$ and a drag coefficient of 0.32 when the windows and sunroof are closed. The drag coefficient increases to 0.41 when the sunroof is open. Determine the additional power consumption of the car when the sunroof is opened at (a) $35 \mathrm{mi} / \mathrm{h}$ and (b) $70 \mathrm{mi} / \mathrm{h}$. Take the density of air to be $0.075 \mathrm{lbm} / \mathrm{ft}^3$.

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Problem 43

What fluid property is responsible for the development of the velocity boundary layer? What is the effect of the velocity on the thickness of the boundary layer?

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Problem 44

What does the friction coefficient represent in flow over a flat plate? How is it related to the drag force acting on the plate?

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Problem 45

Consider laminar flow over a flat plate. How does the local friction coefficient change with position?

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Problem 46

How is the average friction coefficient determined in flow over a flat plate?

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Problem 47

Light oil at $75^{\circ} \mathrm{F}$ flows over a 15 -ft-long flat plate with a free-stream velocity of $6 \mathrm{ft} / \mathrm{s}$. Determine the total drag force per unit width of the plate.

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Problem 48

The local atmospheric pressure in Denver, Colorado (elevation 1610 m ) is 83.4 kPa . Air at this pressure and at $25^{\circ} \mathrm{C}$ flows with a velocity of $6 \mathrm{~m} / \mathrm{s}$ over a $2.5-\mathrm{m} \times 8-\mathrm{m}$ flat plate. Determine the drag force acting on the top surface of the plate if the air flows parallel to the (a) 8 -m-long side and (b) the $2.5-\mathrm{m}$-long side.

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Problem 49

During a winter day, wind at $55 \mathrm{~km} / \mathrm{h}, 5^{\circ} \mathrm{C}$, and 1 atm is blowing parallel to a $4-\mathrm{m}$-high and $10-\mathrm{m}$-long wall of a house. Assuming the wall surfaces to be smooth, determine the friction drag acting on the wall. What would your answer be if the wind velocity has doubled? How realistic is it to treat the flow over side wall surfaces as flow over a flat plate?

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Problem 50

Air at $70^{\circ} \mathrm{F}$ flows over a 10 -ft-long flat plate at $25 \mathrm{ft} / \mathrm{s}$. Determine the local friction coefficient at intervals of 1 ft and plot the results against the distance from the leading edge.

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Problem 51

The forming section of a plastics plant puts out a continuous sheet of plastic that is 1.2 m wide and 2 mm thick at a rate of $15 \mathrm{~m} / \mathrm{min}$. The sheet is subjected to airflow at a velocity of $3 \mathrm{~m} / \mathrm{s}$ on both sides along its surfaces normal to the direction of motion of the sheet. The width of the air cooling section is such that a fixed point on the plastic sheet passes through that section in 2 s . Using properties of air at 1 atm and $60^{\circ} \mathrm{C}$, determine the drag force the air exerts on the plastic sheet in the direction of airflow.

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Problem 52

The top surface of the passenger car of a train moving at a velocity of $70 \mathrm{~km} / \mathrm{h}$ is 3.2 m wide and 8 m long. If the outdoor air is at 1 atm and $25^{\circ} \mathrm{C}$, determine the drag force acting on the top surface of the car.

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Problem 53

The weight of a thin flat plate $50 \mathrm{~cm} \times 50 \mathrm{~cm}$ in size is balanced by a counterweight that has a mass of 2 kg , as shown in Fig. P11-53. Now a fan is turned on, and air at 1 atm and $25^{\circ} \mathrm{C}$ flows downward over both surfaces of the plate with a free-stream velocity of $10 \mathrm{~m} / \mathrm{s}$. Determine the mass of the counterweight that needs to be added in order to balance the plate in this case.

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Problem 54

Consider laminar flow of a fluid over a flat plate. Now the free-stream velocity of the fluid is doubled. Determine the change in the drag force on the plate. Assume the flow to remain laminar.

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Problem 55

Consider a refrigeration truck traveling at $65 \mathrm{mi} / \mathrm{h}$ at a location where the air temperature is at 1 atm and $80^{\circ} \mathrm{F}$. The refrigerated compartment of the truck can be considered to be a 9 - ft -wide, $8-\mathrm{ft}-$ high, and 20 - ft -long rectangular box. Assuming the airflow over the entire outer surface to be turbulent and attached (no flow separation), determine the drag force acting on the top and side surfaces and the power required to overcome this drag.

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Problem 56

Reconsider Prob. 11-55E. Using EES (or other) software, investigate the effect of truck speed on the total drag force acting on the top and side surfaces, and the power required to overcome it. Let the truck speed vary from 0 to $100 \mathrm{mi} / \mathrm{h}$ in increments of $10 \mathrm{mi} / \mathrm{h}$. Tabulate and plot the results.

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Problem 57

Air at $25^{\circ} \mathrm{C}$ and 1 atm is flowing over a long flat plate with a velocity of $8 \mathrm{~m} / \mathrm{s}$. Determine the distance from the leading edge of the plate where the flow becomes turbulent, and the thickness of the boundary layer at that location.

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Problem 58

Repeat Prob. 11-57 for water.

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Problem 59

In flow over cylinders, why does the drag coefficient suddenly drop when the flow becomes turbulent? Isn't turbulence supposed to increase the drag coefficient instead of decreasing it?

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Problem 60

In flow over blunt bodies such as a cylinder, how does the pressure drag differ from the friction drag?

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Problem 61

Why is flow separation in flow over cylinders delayed in turbulent flow?

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Problem 62

A 1.2-in-outer-diameter pipe is to span across a river at a 105 - ft -wide section while being completely immersed in water. The average flow velocity of water is $10 \mathrm{ft} / \mathrm{s}$, and the water temperature is $70^{\circ} \mathrm{F}$. Determine the drag force exerted on the pipe by the river.

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Problem 63

A long 8-cm-diameter steam pipe passes through some area open to the winds. Determine the drag force acting on the pipe per unit of its length when the air is at 1 atm and $5^{\circ} \mathrm{C}$ and the wind is blowing across the pipe at a velocity of $50 \mathrm{~km} / \mathrm{h}$.

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Problem 64

A person extends his uncovered arms into the windy air outside at 1 atm and $60^{\circ} \mathrm{F}$ and $20 \mathrm{mi} / \mathrm{h}$ in order to feel nature closely. Treating the arm as a 2 -ft-long and 3 -indiameter cylinder, determine the drag force on both arms.

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Problem 65

A 6-mm-diameter electrical transmission line is exposed to windy air. Determine the drag force exerted on a $120-\mathrm{m}$-long section of the wire during a windy day when the air is at 1 atm and $15^{\circ} \mathrm{C}$ and the wind is blowing across the transmission line at $40 \mathrm{~km} / \mathrm{h}$.

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Problem 66

Consider $0.8-\mathrm{cm}$-diameter hail that is falling freely in atmospheric air at 1 atm and $5^{\circ} \mathrm{C}$. Determine the terminal velocity of the hail. Take the density of hail to be $910 \mathrm{~kg} / \mathrm{m}^3$.

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Problem 67

A $0.1-\mathrm{mm}$-diameter dust particle whose density is $2.1 \mathrm{~g} / \mathrm{cm}^3$ is observed to be suspended in the air at 1 atm and $25^{\circ} \mathrm{C}$ at a fixed point. Estimate the updraft velocity of air motion at that location. Assume Stokes law to be applicable. Is this a valid assumption?

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Problem 68

Dust particles of diameter 0.05 mm and density $1.8 \mathrm{~g} / \mathrm{cm}^3$ are unsettled during high winds and rise to a height of 350 m by the time things calm down. Estimate how long it will take for the dust particles to fall back to the ground in still air at 1 atm and $15^{\circ} \mathrm{C}$, and their velocity. Disregard the initial transient period during which the dust particles accelerate to their terminal velocity, and assume Stokes law to be applicable.

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Problem 69

A 2 -m-long, 0.2 -m-diameter cylindrical pine $\log \left(\right.$ density $\left.=513 \mathrm{~kg} / \mathrm{m}^3\right)$ is suspended by a crane in the horizontal position. The log is subjected to normal winds of $40 \mathrm{~km} / \mathrm{h}$ at $5^{\circ} \mathrm{C}$ and 88 kPa . Disregarding the weight of the cable and its drag, determine the angle $\theta$ the cable will make with the horizontal and the tension on the cable.

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Problem 70

One of the popular demonstrations in science museums involves the suspension of a ping-pong ball by an upward air jet. Children are amused by the ball always coming back to the center when it is pushed by a finger to the side of the jet. Explain this phenomenon using the Bernoulli equation. Also determine the velocity of air if the ball has a mass of 2.6 g and a diameter of 3.8 cm . Assume air is at 1 atm and $25^{\circ} \mathrm{C}$.

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Problem 71

Why is the contribution of viscous effects to lift usually negligible for airfoils?

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Problem 72

Air is flowing past a symmetrical airfoil at zero angle of attack. Will the (a) lift and (b) drag acting on the airfoil be zero or nonzero?

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Problem 73

Air is flowing past a nonsymmetrical airfoil at zero angle of attack. Will the (a) lift and (b) drag acting on the airfoil be zero or nonzero?

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Problem 74

Air is flowing past a symmetrical airfoil at an angle of attack of $5^{\circ}$. Will the (a) lift and (b) drag acting on the airfoil be zero or nonzero?

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Problem 75

What is stall? What causes an airfoil to stall? Why are commercial aircraft not allowed to fly at conditions near stall?

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Problem 76

Both the lift and the drag of an airfoil increase with an increase in the angle of attack. In general, which increases at a higher rate, the lift or the drag?

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Problem 77

Why are flaps used at the leading and trailing edges of the wings of large aircraft during takeoff and landing? Can an aircraft take off or land without them?

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Problem 78

How do flaps affect the lift and the drag of wings?

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Problem 79

What is the effect of wing tip vortices (the air circulation from the lower part of the wings to the upper part) on the drag and the lift?

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Problem 80

What is induced drag on wings? Can induced drag be minimized by using long and narrow wings or short and wide wings?

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Problem 81

Air is flowing past a spherical ball. Is the lift exerted on the ball zero or nonzero? Answer the same question if the ball is spinning.

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Problem 82

A tennis ball with a mass of 57 g and a diameter of 6.4 cm is hit with an initial velocity of $92 \mathrm{~km} / \mathrm{h}$ and a backspin of 4200 rpm . Determine if the ball will fall or rise under the combined effect of gravity and lift due to spinning shortly after hitting. Assume air is at 1 atm and $25^{\circ} \mathrm{C}$.

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Problem 83

Consider an aircraft, which takes off at $190 \mathrm{~km} / \mathrm{h}$ when it is fully loaded. If the weight of the aircraft is increased by 20 percent as a result of overloading, determine the speed at which the overloaded aircraft will take off.

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Problem 84

Consider an airplane whose takeoff speed is 220 $\mathrm{km} / \mathrm{h}$ and that takes 15 s to take off at sea level. For an airport at an elevation of 1600 m (such as Denver), determine (a) the takeoff speed, (b) the takeoff time, and (c) the additional runway length required for this airplane. Assume constant acceleration for both cases.

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Problem 85

An airplane is consuming fuel at a rate of $5 \mathrm{gal} / \mathrm{min}$ when cruising at a constant altitude of $10,000 \mathrm{ft}$ at constant speed. Assuming the drag coefficient and the engine efficiency to remain the same, determine the rate of fuel consumption at an altitude of $30,000 \mathrm{ft}$ at the same speed.

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Problem 86

A jumbo jet airplane has a mass of about $400,000 \mathrm{~kg}$ when fully loaded with over 400 passengers and takes off at a speed of $250 \mathrm{~km} / \mathrm{h}$. Determine the takeoff speed when the airplane has 100 empty seats. Assume each passenger with luggage is 140 kg and the wing and flap settings are maintained the same.

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Problem 87

Reconsider Prob. 11-86. Using EES (or other) software, investigate the effect of passenger count on the takeoff speed of the aircraft. Let the number of passengers vary from 0 to 500 in increments of 50 . Tabulate and plot the results.

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Problem 88

A small aircraft has a wing area of $30 \mathrm{~m}^2$, a lift coefficient of 0.45 at takeoff settings, and a total mass of 2800 kg . Determine (a) the takeoff speed of this aircraft at sea level at standard atmospheric conditions, (b) the wing loading, and (c) the required power to maintain a constant cruising speed of $300 \mathrm{~km} / \mathrm{h}$ for a cruising drag coefficient of 0.035 .

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Problem 89

A small airplane has a total mass of 1800 kg and a wing area of $42 \mathrm{~m}^2$. Determine the lift and drag coefficients
of this airplane while cruising at an altitude of 4000 m at a constant speed of $280 \mathrm{~km} / \mathrm{h}$ and generating 190 kW of power.

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Problem 90

The NACA 64(1)-412 airfoil has a lift-to-drag ratio of 50 at $0^{\circ}$ angle of attack, as shown in Fig. 11-43. At what angle of attack will this ratio increase to 80 ?

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Problem 91

Consider a light plane that has a total weight of $15,000 \mathrm{~N}$ and a wing area of $46 \mathrm{~m}^2$ and whose wings resemble the NACA 23012 airfoil with no flaps. Using data from Fig. 11-45, determine the takeoff speed at an angle of attack of $5^{\circ}$ at sea level. Also determine the stall speed.

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Problem 92

(8) An airplane has a mass of $50,000 \mathrm{~kg}$, a wing area of $300 \mathrm{~m}^2$, a maximum lift coefficient of 3.2, and a cruising drag coefficient of 0.03 at an altitude of $12,000 \mathrm{~m}$. Determine (a) the takeoff speed at sea level, assuming it is 20 percent over the stall speed, and (b) the thrust that the engines must deliver for a cruising speed of $700 \mathrm{~km} / \mathrm{h}$.

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Problem 93

A 2.4 -in-diameter smooth ball rotating at 500 rpm is dropped in a water stream at $60^{\circ} \mathrm{F}$ flowing at $4 \mathrm{ft} / \mathrm{s}$. Determine the lift and the drag force acting on the ball when it is first dropped in the water.

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Problem 94

An automotive engine can be approximated as a 0.4 m -high, $0.60-\mathrm{m}$-wide, and $0.7-\mathrm{m}$-long rectangular block. The ambient air is at 1 atm and $15^{\circ} \mathrm{C}$. Determine the drag force acting on the bottom surface of the engine block as the car travels at a velocity of $85 \mathrm{~km} / \mathrm{h}$. Assume the flow to be turbulent over the entire surface because of the constant agitation of the engine block.

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Problem 95

Calculate the thickness of the boundary layer during flow over a $2.5-\mathrm{m}$-long flat plate at intervals of 25 cm and plot the boundary layer over the plate for the flow of (a) air, (b) water, and (c) engine oil at 1 atm and $20^{\circ} \mathrm{C}$ at an upstream velocity of $3 \mathrm{~m} / \mathrm{s}$.

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Problem 96

The passenger compartment of a minivan traveling at $60 \mathrm{mi} / \mathrm{h}$ in ambient air at 1 atm and $80^{\circ} \mathrm{F}$ can be modeled as a $3.2-\mathrm{ft}-\mathrm{high}, 6-\mathrm{ft}-\mathrm{wide}$, and 11 -ft-long rectangular box. The airflow over the exterior surfaces can be assumed to be turbulent because of the intense vibrations involved. Determine the drag force acting on the top and the two side surfaces of the van and the power required to overcome it.

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Problem 97

A 1-m-external-diameter spherical tank is located outdoors at 1 atm and $25^{\circ} \mathrm{C}$ and is subjected to winds at $35 \mathrm{~km} / \mathrm{h}$. Determine the drag force exerted on it by the wind.

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Problem 98

A 2-m-high, 4-m-wide rectangular advertisement panel is attached to a $4-\mathrm{m}$-wide, 0.15 -m-high rectangular con-
crete block (density $=2300 \mathrm{~kg} / \mathrm{m}^3$ ) by two $5-\mathrm{cm}$-diameter, 4-m-high (exposed part) poles, as shown in Fig. P11-98. If the sign is to withstand $150 \mathrm{~km} / \mathrm{h}$ winds from any direction, determine (a) the maximum drag force on the panel, (b) the drag force acting on the poles, and (c) the minimum length $L$ of the concrete block for the panel to resist the winds. Take the density of air to be $1.30 \mathrm{~kg} / \mathrm{m}^3$.

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Problem 99

A plastic boat whose bottom surface can be approximated as a $1.5-\mathrm{m}$-wide, $2-\mathrm{m}$-long flat surface is to move through water at $15^{\circ} \mathrm{C}$ at speeds up to $30 \mathrm{~km} / \mathrm{h}$. Determine the friction drag exerted on the boat by water and the power needed to overcome it.

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Problem 100

Reconsider Prob. 11-99. Using EES (or other) software, investigate the effect of boat speed on the drag force acting on the bottom surface of the boat, and the power needed to overcome it. Let the boat speed vary from 0 to $100 \mathrm{~km} / \mathrm{h}$ in increments of $10 \mathrm{~km} / \mathrm{h}$. Tabulate and plot the results.

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Problem 101

A commercial airplane has a total mass of $150,000 \mathrm{lbm}$ and a wing planform area of $1800 \mathrm{ft}^2$. The plane has a cruising speed of $550 \mathrm{mi} / \mathrm{h}$ and a cruising altitude of $38,000 \mathrm{ft}$ where the air density is $0.0208 \mathrm{lbm} / \mathrm{ft}^3$. The plane has double-slotted flaps for use during takeoff and landing, but it cruises with all flaps retracted. Assuming the lift and drag characteristics of the wings can be approximated by NACA 23012, determine (a) the minimum safe speed for takeoff and landing with and without extending the flaps, (b) the angle of attack to cruise steadily at the cruising altitude, and (c) the power that needs to be supplied to provide enough thrust to overcome drag. Take the air density on the ground to be $0.075 \mathrm{lbm} / \mathrm{ft}^3$.

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Problem 102

An 8-cm-diameter smooth ball has a velocity of $36 \mathrm{~km} / \mathrm{h}$ during a typical hit. Determine the percent increase in the drag coefficient if the ball is given a spin of 3500 rpm in air at 1 atm and $25^{\circ} \mathrm{C}$.

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Problem 103

A paratrooper and his 8-m-diameter parachute weigh 950 N . Taking the average air density to be $1.2 \mathrm{~kg} / \mathrm{m}^3$, determine the terminal velocity of the paratrooper.

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Problem 104

A 17,000-kg tractor-trailer rig has a frontal area of $9.2 \mathrm{~m}^2$, a drag coefficient of 0.96 , a rolling resistance coefficient of 0.05 (multiplying the weight of a vehicle by the rolling resistance coefficient gives the rolling resistance), a bearing friction resistance of 350 N , and a maximum speed of $110 \mathrm{~km} / \mathrm{h}$ on a level road during steady cruising in calm weather with an air density of $1.25 \mathrm{~kg} / \mathrm{m}^3$. Now a fairing is installed to the front of the rig to suppress separation and to streamline the flow to the top surface, and the drag coefficient is reduced to 0.76 . Determine the maximum speed of the rig with the fairing.

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Problem 105

Stokes law can be used to determine the viscosity of a fluid by dropping a spherical object in it and measuring the terminal velocity of the object in that fluid. This can be done by plotting the distance traveled against time and observing when the curve becomes linear. During such an experiment a 3 -mm-diameter glass ball ( $\rho=2500 \mathrm{~kg} / \mathrm{m}^3$ ) is dropped into a fluid whose density is $875 \mathrm{~kg} / \mathrm{m}^3$, and the terminal velocity is measured to be $0.12 \mathrm{~m} / \mathrm{s}$. Disregarding the wall effects, determine the viscosity of the fluid.

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Problem 106

During an experiment, three aluminum balls ( $\rho_s$ $=2600 \mathrm{~kg} / \mathrm{m}^3$ ) having diameters 2,4 , and 10 mm , respectively, are dropped into a tank filled with glycerin at $22^{\circ} \mathrm{C}\left(\rho_f\right.$ $=1274 \mathrm{~kg} / \mathrm{m}^3$ and $\left.\mu=1 \mathrm{~kg} / \mathrm{m} \cdot \mathrm{s}\right)$. The terminal settling velocities of the balls are measured to be $3.2,12.8$, and $60.4 \mathrm{~mm} / \mathrm{s}$, respectively. Compare these values with the velocities predicted by Stokes law for drag force $F_D$ $=3 \pi \mu D V$, which is valid for very low Reynolds numbers ( $\operatorname{Re} \ll 1)$. Determine the error involved for each case and assess the accuracy of Stokes law.

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Problem 107

Repeat Prob. 11-106 by considering the general form of Stokes law expressed as $F_D=3 \pi \mu D V+$ $(9 \pi / 16) \rho_s V^2 D^2$.

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Problem 108

A small aluminum ball with $D=2 \mathrm{~mm}$ and $\rho_s$ $=2700 \mathrm{~kg} / \mathrm{m}^3$ is dropped into a large container filled with oil at $40^{\circ} \mathrm{C}\left(\rho_f=876 \mathrm{~kg} / \mathrm{m}^3\right.$ and $\left.\mu=0.2177 \mathrm{~kg} / \mathrm{m} \cdot \mathrm{s}\right)$. The Reynolds number is expected to be low and thus Stokes law for drag force $F_D=3 \pi \mu D V$ to be applicable. Show that the variation of velocity with time can be expressed as $V$ $=(a / b)\left(1-e^{-b l}\right)$ where $a=g\left(1-\rho_f / \rho_s\right)$ and $b=18 \mu /\left(\rho_s D^2\right)$. Plot the variation of velocity with time, and calculate the time it takes for the ball to reach 99 percent of its terminal velocity.

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Problem 109

Engine oil at $40^{\circ} \mathrm{C}$ is flowing over a long flat plate with a velocity of $4 \mathrm{~m} / \mathrm{s}$. Determine the distance $x_{\mathrm{cr}}$ from the leading edge of the plate where the flow becomes turbulent, and calculate and plot the thickness of the boundary layer over a length of $2 x_{\mathrm{cr}}$.

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Problem 110

Write a report on the history of the reduction of the drag coefficients of cars and obtain the drag coefficient data for some recent car models from the catalogs of car manufacturers.

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Problem 111

Write a report on the flaps used at the leading and trailing edges of the wings of large commercial aircraft. Discuss how the flaps affect the drag and lift coefficients during takeoff and landing.

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Problem 112

Large commercial airplanes cruise at high altitudes (up to about $40,000 \mathrm{ft}$ ) to save fuel. Discuss how flying at high altitudes reduces drag and saves fuel. Also discuss why small planes fly at relatively low altitudes.

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Problem 113

Many drivers turn off their air conditioners and roll down the car windows in hopes of saving fuel. But it is claimed that this apparent "free cooling" actually increases the fuel consumption of the car. Investigate this matter and write a report on which practice will save gasoline under what conditions.

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Problem 189

What is drafting? How does it affect the drag coefficient of the drafted body?

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