• Home
  • Textbooks
  • Introduction to Flight
  • Airfoils, Wings, and Other Aerodynamic Shapes

Introduction to Flight

John David Anderson

Chapter 5

Airfoils, Wings, and Other Aerodynamic Shapes - all with Video Answers

Educators

Chapter Questions

05:40

Problem 1

By the method of dimensional analysis, derive the expression $M=q_{\infty} S c c_{w}$ for the aerodynamic moment on an airfoil, where $c$ is the chord and $c_{m}$ is the moment coefficient.

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
03:37

Problem 2

Consider an infinite wing with a NACA 1412 airfoil section and a chord length of $3 \mathrm{ft}$. The wing is at an angle of attack of $5^{\circ}$ in an airflow velocity of $100 \mathrm{ft} / \mathrm{s}$ at standard sea-level conditions. Calculate the lift, drag, and moment about the quarter-chord per unit span.

Prabhat Tyagi
Prabhat Tyagi
Numerade Educator
03:42

Problem 3

Consider a rectangular wing mounted in a low-speed subsonic wing tunnel. The wing model completely spans the test-section so that the flow "sees" essentially an infinite wing. If the wing has a NACA 23012 airfoil section and a chord of $0.3 \mathrm{~m}$, calculate the lift, drag, and moment about the quarter-chord per unit span when the airflow pressure, temperature, and velocity are $1 \mathrm{~atm}, 303 \mathrm{~K}$, and $42 \mathrm{~m} / \mathrm{s}$, respectively. The angle of attack is $8^{\circ}$.

Chai Santi
Chai Santi
Numerade Educator
View

Problem 4

The wing model in Prob. $5.3$ is pitched to a new angle of attack, where the lift on the entire wing is measured as $200 \mathrm{~N}$ by the wind tunnel force balance. If the wingspan is $2 \mathrm{~m}$, what is the angle of attack?

Rashmi Sinha
Rashmi Sinha
Numerade Educator
View

Problem 5

Consider a rectangular wing with a NACA 0009 airfoil section spanning the test section of a wind tunnel. The test-section airflow conditions are standard sea level with a velocity of $120 \mathrm{mi} / \mathrm{h}$. The wing is at an angle of attack of $4^{\circ}$, and the wind tunnel force balance measures a lift of $29.5 \mathrm{lb}$. What is the area of the wing?

Rashmi Sinha
Rashmi Sinha
Numerade Educator
09:10

Problem 6

The ratio of lift to drag $L D$ for a wing or airfoil is an important aerodynamic parameter, indeed, it is a direct measure of the aerodynamic efficiency of the wing. If a wing is pitched through a range of angle of attack, $L D$ first increases, then goes through a maximum, and then decreases. Consider an infinite wing with an NACA 2412 airfoil. Estimate the maximum value of $L / D$. Assume that the Reynolds number is $9 \times 10^{6}$.

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
03:39

Problem 7

Consider an airfoil in a free stream with a velocity of $50 \mathrm{~m} / \mathrm{s}$ at standard sea-level conditions. At a point on the airfoil, the pressure is $9.5 \times 10^{4} \mathrm{~N} / \mathrm{m}^{2}$. What is the pressure coefficient at this point?

Satpal Satpal
Satpal Satpal
Numerade Educator
View

Problem 8

Consider a low-speed airplane flying at a velocity of $55 \mathrm{~m} / \mathrm{s}$. If the velocity at a point on the fuselage is $62 \mathrm{~m} / \mathrm{s}$, what is the pressure coefficient at this point?

Rashmi Sinha
Rashmi Sinha
Numerade Educator
03:39

Problem 9

Consider a wing mounted in the test-section of a subsonic wind tunnel. The velocity of the airflow is $160 \mathrm{ft} / \mathrm{s}$. If the velocity at a point on the wing is $195 \mathrm{ft} / \mathrm{s}$, what is the pressure coefficient at this point?

Satpal Satpal
Satpal Satpal
Numerade Educator
View

Problem 10

Consider the same wing in the same wind tunnel as in Prob. 5.9. If the test-section air temperature is $510^{\circ} \mathrm{R}$ and the flow velocity is increased to $700 \mathrm{ft} / \mathrm{s}$, what is the pressure coefficient at the same point?

Victor Salazar
Victor Salazar
Numerade Educator
View

Problem 11

Consider a wing in a high-speed wind tunnel. At a point on the wing, the velocity is $850 \mathrm{ft} / \mathrm{s}$. If the test-section flow is at a velocity of $780 \mathrm{ft} / \mathrm{s}$, with a pressure and temperature of $1 \mathrm{~atm}$ and $505^{\circ} \mathrm{R}$, respectively, calculate the pressure coefficient at the point.

Victor Salazar
Victor Salazar
Numerade Educator
03:39

Problem 12

If the test-section flow velocity in Prob. $5.11$ is reduced to $100 \mathrm{ft} / \mathrm{s}$, what will the pressure coefficient become at the same point on the wing?

Satpal Satpal
Satpal Satpal
Numerade Educator
01:22

Problem 13

Consider an NACA 1412 airfoil at an angle of attack of $4^{\circ}$. If the free-stream Mach number is $0.8$, calculate the lift coefficient.

Nicholas Mogoi
Nicholas Mogoi
Numerade Educator
03:37

Problem 14

An NACA 4415 airfoil is mounted in a high-speed subsonic wind tunnel. The lift coefficient is measured as $0.85$. If the test-section Mach number is $0.7$, at what angle of attack is the airfoil?

Prabhat Tyagi
Prabhat Tyagi
Numerade Educator
01:28

Problem 15

Consider an airfoil at a given angle of attack, say $\alpha_{1}$. At low speeds, the minimum pressure coefficient on the top surface of the airfoil is $-0.90$. What is the critical Mach number of the airfoil?

Chai Santi
Chai Santi
Numerade Educator
01:42

Problem 16

Consider the airfoil in Prob. $5.15$ at a smaller angle of attack, say $\alpha_{2}$. At low speeds, the minimum pressure coefficient is $-0.65$ at this lower angle of attack. What is the critical Mach number of the airfoil?

Chai Santi
Chai Santi
Numerade Educator
02:00

Problem 17

Consider a uniform flow with a Mach number of 2 . What angle does a Mach wave make with respect to the flow direction?

Jeff Vermeire
Jeff Vermeire
Numerade Educator
00:22

Problem 18

Consider a supersonic missile flying at Mach $2.5$ at an altitude of $10 \mathrm{~km}$ (see Fig. P5.18). Assume that the angle of the shock wave from the nose is approximated by the Mach angle (this is a very weak shock). How far behind the nose of the vehicle will the shock wave impinge upon the ground? (Ignore the fact that the speed of sound, and hence the Mach angle, changes with altitude.)

Dading Chen
Dading Chen
Numerade Educator
02:51

Problem 19

The wing area of the Lockheed F-104 straight-wing supersonic fighter is approximately $210 \mathrm{ft}^{2}$. If the airplane weighs $16,000 \mathrm{lb}$ and is flying in level
flight at Mach $2.2$ at a standard altitude of $36,000 \mathrm{ft}$, estimate the wave drag on the wings.

Chai Santi
Chai Santi
Numerade Educator
07:54

Problem 20

Consider a flat plate at an angle of attack of $2^{\circ}$ in a Mach $2.2$ airflow. (Mach $2.2$ is the cruising Mach number of the Concorde supersonic transport.) The length of the plate in the flow direction is $202 \mathrm{ft}$, which is the length of the Concorde. Assume that the free-stream conditions correspond to a standard altitude of $50,000 \mathrm{ft}$. The total drag on this plate is the sum of wave drag and skin friction drag. Assume that a turbulent boundary layer exists over the entire plate. The results given in Ch. 4 for skin friction coefficients hold for incompressible flow only; there is a compressibility effect on $C_{f}$ such that its value decreases with increasing Mach number. Specifically, at Mach $2.2$ assume that the $C_{f}$ given in Ch. 4 is reduced by 20 percent.
a. Given all the preceding information, calculate the total drag coefficient for the plate.
$b$. If the angle of attack is increased to $5^{\circ}$, assuming that $C_{f}$ stays the same, calculate the total drag coefficient.
c. For these cases, what can you conclude about the relative influence of wave drag and skin friction drag?

Chai Santi
Chai Santi
Numerade Educator
01:40

Problem 21

The Cessna Cardinal, a single-engine light plane, has a wing with an area of $16.2 \mathrm{~m}^{2}$ and an aspect ratio of $7.31$. Assume that the span efficiency factor is
0.62. If the airplane is flying at standard sea-level conditions with a velocity of
$251 \mathrm{~km} / \mathrm{h}$, what is the induced drag when the total weight is $9800 \mathrm{~N}$ ?

Adnan Gill
Adnan Gill
Numerade Educator
01:57

Problem 22

For the Cessna Cardinal in Prob. 5.21, calculate the induced drag when the velocity is $85.5 \mathrm{~km} / \mathrm{h}$ (stalling speed at sea level with flaps down).

Narayan Hari
Narayan Hari
Numerade Educator
01:04

Problem 23

Consider a finite wing with an area and aspect ratio of $21.5 \mathrm{~m}^{2}$ and 5 , respectively (this is comparable to the wing on a Gates Learjet, a twin-jet executive transport).
Assume that the wing has a NACA 65-210 airfoil, a span efficiency factor of $0.9$, and a profile drag coefficient of $0.004$. If the wing is at a $6^{\circ}$ angle of attack, calculate $C_{L}$ and $C_{D}$.

Dominador Tan
Dominador Tan
Numerade Educator
02:55

Problem 24

During the $1920 \mathrm{~s}$ and early $1930 \mathrm{~s}$, the NACA obtained wind tunnel data on different airfoils by testing finite wings with an aspect ratio of 6 . These data were then "corrected" to obtain infinite-wing airfoil characteristics. Consider such a finite wing with an area and aspect ratio of $1.5 \mathrm{ft}^{2}$ and 6 , respectively, mounted in a wind tunnel where the test-section flow velocity is $100 \mathrm{ft} / \mathrm{s}$ at standard sea-level conditions. When the wing is pitched to $\alpha=-2^{\circ}$, no lift is measured. When the wing is pitched to $\alpha=10^{\circ}$, a lift of $17.9 \mathrm{lb}$ is measured. Calculate the lift slope for the airfoil (the infinite wing) if the span effectiveness factor is $0.95$.

Penny Riley
Penny Riley
Numerade Educator
09:10

Problem 25

A finite wing of area $1.5 \mathrm{ft}^{2}$ and aspect ratio of 6 is tested in a subsonic wind tunnel at a velocity of $130 \mathrm{ft} / \mathrm{s}$ at standard sea-level conditions. At an angle of attack of $-1^{\circ}$, the measured lift and drag are 0 and $0.181 \mathrm{lb}$, respectively.
At an angle of attack of $2^{\circ}$, the lift and drag are measured as $5.0$ and $0.23 \mathrm{lb}$,
respectively. Calculate the span efficiency factor and the infinite-wing lift slope.

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
01:05

Problem 26

Consider a light, single-engine airplane such as the Piper Super Cub. If the maximum gross weight of the airplane is $7780 \mathrm{~N}$, the wing area is $16.6 \mathrm{~m}^{2}$, and the maximum
lift coefficient is $2.1$ with flaps down, calculate the stalling speed at sea level.

Narayan Hari
Narayan Hari
Numerade Educator
01:28

Problem 27

The airfoil on the Lockheed F-104 straight-wing supersonic fighter is a thin,
symmetric airfoil with a thickness ratio of $3.5$ percent. Consider this airfoil in a flow at an angle of attack of $5^{\circ}$. The incompressible lift coefficient for the airfoil is given approximately by $c_{1}=2 \pi \alpha$, where $\alpha$ is the angle of attack in radians. Estimate the airfoil lift coefficient for $(a) M=0.2,(b) M=0.7$, and $(c) M=2.0$.

Chai Santi
Chai Santi
Numerade Educator
03:59

Problem 28

The whirling-arm test device used in 1804 by Sir George Cayley is shown in Figure 1.7. Cayley was the first person to make measurements of the lift on inclined surfaces. In his 1804 notebook, he wrote that on a flat surface moving through the air at $21.8 \mathrm{ft} / \mathrm{s}$ at $3^{\circ}$ angle of attack, a lift force of 1 ounce was measured. The flat surface was a $1 \mathrm{ft}$ by $1 \mathrm{ft}$ square. Calculate the lift coefficient for this condition. Compare this measured value with that predicted by the expression for lift coefficient for a flat-plate airfoil in incompressible flow given by $c_{1}=2 \pi \alpha$, where $\alpha$ is in radians. What are the reasons for the differences in the two results? (See Anderson, A History of Aerodynamics and Its Impact on Flying Machines, Cambridge University Press, 1997, pp. 68-71, for a detailed discussion of this matter.)

James Kiss
James Kiss
Numerade Educator
View

Problem 29

Consider a finite wing at an angle of attack of $6^{\circ}$. The normal and axial force coefficients are $0.8$ and $0.06$, respectively. Calculate the corresponding lift and drag coefficients. What comparison can you make between the lift and normal force coefficients?

Rashmi Sinha
Rashmi Sinha
Numerade Educator
01:04

Problem 30

Consider a finite wing with an aspect of ratio of 7; the airfoil section of the wing is a symmetric airfoil with an infinite-wing lift slope of $0.11$ per degree. The lift-todrag ratio for this wing is 29 when the lift coefficient is equal to $0.35$. If the angle of attack remains the same and the aspect ratio is simply increased to 10 by adding extensions to the span of the wing, what is the new value of the lift-to-drag ratio?
Assume that the span efficiency factors $e=e_{1}=0.9$ for both cases.

Dominador Tan
Dominador Tan
Numerade Educator
View

Problem 31

Consider a flat plate oriented at a $90^{\circ}$ angle of attack in a low-speed incompressible flow. Assume that the pressure exerted over the front of the plate (facing into the flow) is a constant value over the front surface, equal to the stagnation pressure. Assume that the pressure exerted over the back of the plate is also a constant value, but equal to the free-stream static pressure. (In reality, these assumptions are only approximations to the real flow over the plate. The pressure over the front face is neither exactly constant nor exactly equal to the stagnation pressure, and the pressure over the back of the plate is neither constant nor exactly equal to the free-stream pressure. The preceding approximate model of the flow, however, is useful for our purpose here.) Note that the drag is essentially all pressure drag; due to the $90^{\circ}$ orientation of the plate, skin friction drag is not a factor. For this model of the flow, prove that the drag coefficient for the flat plate is $C_{D}=1$.

Ivan Kochetkov
Ivan Kochetkov
Numerade Educator
01:01

Problem 32

In some aerodynamic literature, the drag of an airplane is couched in terms of the "drag area" instead of the drag coefficient. By definition, the drag area, $f$, is the area of a flat plate at $90^{\circ}$ to the flow that has a drag force equal to the drag of the airplane. As part of this definition, the drag coefficient of the plate is assumed to be equal to 1 , as shown in Prob. $5.31$. If $C_{D}$ is the drag coefficient of the airplane based on wing planform area $S$, prove that $f=C_{D} S$.

Hast Aggarwal
Hast Aggarwal
Numerade Educator
02:31

Problem 33

One of the most beautifully streamlined airplanes ever designed is the North American P-51 Mustang shown in Fig. 4.46. The Mustang has one of the lowest minimum drag coefficients of any airplane in history: $C_{D}=0.0163$. The wing planform area of the Mustang is $233 \mathrm{ft}^{2}$. Using the result from Prob. $5.32$, show that the drag area for the Mustang is $3.8 \mathrm{ft}^{2}$; that is, drag on the whole $\mathrm{P}-51$
airplane is the same as the drag on a flat plate perpendicular to the flow of an area of only $3.8 \mathrm{ft}^{2}$.

Chai Santi
Chai Santi
Numerade Educator
09:10

Problem 34

Consider an NACA 2412 airfoil in a low-speed flow at zero degrees angle of attack and a Reynolds number of $8.9 \times 10^{\circ}$. Calculate the percentage of drag from pressure drag due to flow separation (form drag). Assume a fully turbulent boundary layer over the airfoil. Assume that the airfoil is thin enough that the skin-friction drag can be estimated by the flat-plate results discussed in Ch. 4 .

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
01:22

Problem 35

Repeat Problem $5.34$, assuming that the airfoil is at an angle of attack of 6 degrees. What does this tell you about the rapid increase in $c_{d}$ as the angle of attack of the airfoil is increased?

Nicholas Mogoi
Nicholas Mogoi
Numerade Educator
01:32

Problem 36

Returning to the conditions of Problem $5.34$, where the boundary layer was assumed to be fully turbulent, let us now consider the real situation where the boundary layer starts out as laminar, and then makes a transition to turbulent somewhere downstream of the leading edge. Assume a transition Reynolds number of 500,000 . For this case, calculate the percentage of drag that is due to flow separation (form drag).

Dominador Tan
Dominador Tan
Numerade Educator
02:55

Problem 37

Here we continue in the vein of Probs. $5.34$ to $5.36$, except we examine a thicker airfoil and look at the relative percentages of skin friction and pressure drag for a thicker airfoil. Estimate the skin friction drag coefficient for the NACA 2415 airfoil in low-speed incompressible flow at $\operatorname{Re}=9 \times 10^{6}$ and zero angle of attack for $(a)$ a laminar boundary layer, and $(b)$ a turbulent boundary layer. Compare the results with the experimentally measured section drag coefficient given in App. D for the NACA 2415 airfoil. What does this tell you about the relative percentages of pressure drag and skin friction drag on the airfoil for each case?

Penny Riley
Penny Riley
Numerade Educator
02:55

Problem 38

In reality, the boundary layer on the airfoil discussed in Prob. $5.37$ is neither fully laminar nor fully turbulent. The boundary layer starts out as laminar, and then transitions to turbulent at some point downstream of the leading edge (see the discussion in Sec. 4.19). Assume that the critical Reynolds number for transition is 650,000 . Calculate the skin friction drag coefficient on the NACA 2415 airfoil, and compare your result with the experimental section drag coefficient in App. D. Note: You will find from the answer to this problem that 86 percent of the airfoil section drag coefficient is due to skin friction and 14 percent due to pressure drag from flow separation. Comparing this answer with the result of Prob. $5.36$, which pertains to a thinner airfoil, we find that the pressure drag is a higher percentage for the thicker airfoil. However, for airfoils in general, the pressure drag is still a small percentage of the total drag. This drag breakdown is somewhat typical for airfoils at small angles of attack. By intent, the streamlined shape of airfoils results in small pressure drag, typically on the order of 15 percent of the total drag.

Penny Riley
Penny Riley
Numerade Educator
09:10

Problem 39

This problem examines the cause and effect of a lower Re on airfoil drag. Repeat Prob. $5.38$, except for $\operatorname{Re}=3 \times 10^{6}$. Comment on how and why Re affects the drag. Note: From the answer to this question, you will see that the lower Re results in a higher percentage of skin friction drag than found at the higher Re in Prob. $5.38$, and hence a lower percentage of pressure drag on the airfoil section.

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
View

Problem 40

The lift and drag measured on an aerodynamic body mounted at a five -degree angle of attack in a wind tunnel are 100 and $145 \mathrm{lb}$, respectively. Calculate the corresponding normal and axial forces.

Rashmi Sinha
Rashmi Sinha
Numerade Educator
02:18

Problem 41

Otto Lilienthal (see Section 1.5) carried out a series of aerodynamic measurements on small model wings using first a whirling arm and later stationary models. He tested both flat plate wings and wings with a thin curved (cambered) airfoil. For the flat plate, the resultant aerodynamic force was always inclined behind the perpendicular to the plate, with an axial force always oriented in the backward direction. However, his data for the cambered airfoil showed that at some angles of attack the resultant aerodynamic force was inclined ahead of the perpendicular to the chord line; for these cases the axial force was oriented in the forward direction. Lilienthal called this forward component a "pushing component" and cited its existence as evidence of the superiority of cambered airfoils. (For more historical detail on this matter, see Anderson, A History of Aerodynamics, Cambridge University Press, 1998.) With the above information as background, show that the aerodynamic condition that results in a forward-facing axial force is
$$
L / D>\cot \alpha
$$
where $\alpha$ is the angle of attack.

Chai Santi
Chai Santi
Numerade Educator
View

Problem 42

The airfoil data in Appendix D were obtained in the NACA two-dimensional Low Turbulence Pressure Tunnel at the NACA Langley Memorial Laboratory. This facility went into operation in Spring 1941. The tunnel was especially designed for airfoil testing, with a test section $3 \mathrm{ft}$ wide and $7.5 \mathrm{ft}$ high. The wing models spanned the entire test section of width $3 \mathrm{ft}$, so that the flow over the model was essentially two-dimensional. The chord length of the models was $2 \mathrm{ft}$. When the tunnel became operational, the four- and five-digit airfoil series, originally tested in older tunnels, were retested in the new tunnel. The Low Turbulence Pressure Tunnel is most noted, however, as the testing facility for the NACA Laminar Flow Airfoils. Consider a series of tests where the tunnel is pressurized to $3 \mathrm{~atm}$, the temperature of the airstream in the test section is $60^{\circ} \mathrm{F}$, and the flow velocity is $160 \mathrm{mi} / \mathrm{h}$. A wing with an NACA 2412 airfoil is mounted in the tunnel at an angle of attack such that the section lift coefficient is $0.2$.
a. What is the angle of attack of the model?
b. What is the total drag force on the model?

Rashmi Sinha
Rashmi Sinha
Numerade Educator
View

Problem 43

The wing model in Problem $5.42$ is replaced with a model with an NACA 64-210 Laminar Flow Airfoil at the same conditions as in Problem 5.42.
a. What is the angle of attack of the model?
b. What is the total drag force on the model?

Rashmi Sinha
Rashmi Sinha
Numerade Educator
01:32

Problem 44

For the NACA 2412 airfoil in problem $5.42$, what percentage of the total drag is due to skin friction drag? Assume the skin friction drag on the airfoil is essentially that for a flat plate. Also, assume that the boundary layer over the model is turbulent and incompressible.

Dominador Tan
Dominador Tan
Numerade Educator
01:32

Problem 45

For the Laminar Flow NACA 64-210 airfoil in Problem 5.43, what percentage of the total drag is due to skin friction? Assume that the skin friction drag on the airfoil is essentially that for a flat plate, and that the boundary layer is laminar and incompressible.

Dominador Tan
Dominador Tan
Numerade Educator