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Aerodynamics for Engineers

John J. Bertin, Russell Mark Cummings

Chapter 7

Incompressible Flow About Wings of Finite Span - all with Video Answers

Educators

Chapter Questions

03:42

Problem 1

Consider an airplane that weighs $13,500 \mathrm{~N}$ and cruises in level flight at $250 \mathrm{~km} / \mathrm{h}$ at an altitude of $2000 \mathrm{~m}$. The wing has a surface area of $15.0 m^{2}$ and an aspect ratio of $5.8$. Assume that the lift coefficient is a linear function of the angle of attack and that $x_{01}=-1.0^{\circ}$ If the load distribution is elliptic, calculate the value of the circulation in the plane of symmetry $\left(\Gamma_{0}\right)$, the downwash velocity $\left(w_{y 1}\right)$, the induced-drag coefficient $\left(C_{D v}\right)$ the geometric angle of attack $(\alpha)$, and the effective angle of attack $\left(\alpha_{e}\right)$.

Chai Santi
Chai Santi
Numerade Educator
02:55

Problem 2

Consider the case where the spanwise circulation distribution for a wing is parabolic,
$$
\Gamma(y)=\Gamma_{0}\left(1-\frac{y}{s}\right)
$$
If the total lift generated by the wing with the linear circulation distribution is to be equal to the total lift generated by a wing with an elliptic circulation distribution, what is the relation between the $\Gamma_{0}$ values for the two distributions? What is the relation between the induced downwash velocities at the plane of symmetry for the two configurations?

Eduard Sanchez
Eduard Sanchez
Numerade Educator
01:10

Problem 3

When a GA(W)-1 airfoil section (i.e., a wing of infinite span) is at an angle of attack of $5^{\circ}$, the lift coefficient is $0.5$. Using equation $(7.20)$, calculate the angle of attack at which a wing whose aspect ratio is $6.5$ would have to operate to generate the same lift coefficient. What would the angle of attack have to be to generate this lift coefficient for a wing whose aspect ratio is $5.0 ?$

Penny Riley
Penny Riley
Numerade Educator
01:04

Problem 4

Consider a planar wing (i.e., no geometric twist) which has a NACA 0012 section and an aspect ratio of $7.0$. Following Example $7.2$, use a four-term series to represent the load distribution. Compare the lift coefficient, the induced drag coefficient, and the spanwise lift distribution for taper ratios of (a) $0.4$, (b) $0.5$, (c) $0.6$, and (d) $1.0$.

Dominador Tan
Dominador Tan
Numerade Educator
04:07

Problem 5

Consider an airplane that weighs $10,000 \mathrm{~N}$ and cruises in level flight at $185 \mathrm{~km} / \mathrm{h}$ at an altitude of $3.0 \mathrm{~km}$. The wing has a surface area of $16.3 \mathrm{~m}^{2}$, an aspect ratio of $7.52$, and a taper ratio of $0.69$. Assume that the lift coefficient is a linear function of the angle of attack and the airfoil section is a NACA 2412 (see Chapter 6 for the characteristics of this section). The incidence of the root section is $+1.5^{\circ}$; the incidence of the tip section is $-1.5^{\circ}$. Thus, there is a geometric twist of $-3^{\circ}$ (washout). Following Example 7.1, use a four-term series to represent the load distribution and calculate
(a) The lift coefficient $\left(C_{L}\right)$
(b) The spanwise load distribution $\left(C_{l}(y) / C_{L}\right)$
(c) The induced drag coefficient $\left(C_{D v}\right)$
(d) The geometric angle of attack $(a)$

Averell Hause
Averell Hause
Carnegie Mellon University
02:12

Problem 6

Use equation $(7.38)$ to calculate the velocity induced at some point $C(x, y, z)$ by the vortex filament $A B$ (shown in Fig. 7.30); that is, derive equation (7.39a).

Supratim Pal
Supratim Pal
Numerade Educator
02:12

Problem 7

Use equation $(7.38)$ to calculate the velocity induced at some point $C(x, y, 0)$ by the vortex filament $A B$ in a planar wing; that is, derive equation (7.44a).

Supratim Pal
Supratim Pal
Numerade Educator
01:46

Problem 8

Calculate the downwash velocity at the CP of panel 1 induced by the horseshoe vortex of panel 4 of the starboard wing for the flow configuration of Example $7.2$.

Saketh Meka
Saketh Meka
Numerade Educator
01:04

Problem 9

Following the VLM approach used in Example 7.4, calculate the lift coefficient for a swept wing. The wing has an aspect ratio of 8 , a taper ratio of unity (i.e., $c_{r}=c_{t}$ ), and an uncambered section (i.e., it is a flat plate). Since the taper ratio is unity, the leading edge, the quarter-chord line, the three-quarter chord line, and the trailing edge all have the same sweep, $45^{\circ}$. How does the lift coefficient for this aspect ratio (8) compare with that for an aspect ratio of 5 (i.e., that computed in Example 7.4)? Is this consistent with our knowledge of the effect of aspect ratio (e.g., Fig. 7.10)?

Dominador Tan
Dominador Tan
Numerade Educator
01:55

Problem 10

Following the VLM approach used in Example 7.4, calculate the lift coefficient for a swept wing. The wing has an aspect ratio of 5 , a taper ratio of $0.5$ (i.e., $c_{t}=0.5 c_{r}$ ), an uncambered section, and the quarter chord is swept $45^{\circ}$. Since the taper ratio is not unity, the leading edge, the quarter-chord line, the three-quarter-chord line, and the trailing edge have different sweep angles. This should be taken into account when defining the coordinates of the horseshoe vortices and the control points.

Chai Santi
Chai Santi
Numerade Educator
01:51

Problem 11

Following the VLM approach used in Example 7.4, calculate the lift coefficient for the forward swept wing of Fig. $7.24 \mathrm{~b}$. The quarter chord is swept forward $45^{\circ}$, the aspect ratio is $3.55$, and the taper ratio $0.5$. The airfoil section (perpendicular to the quarter chord) is a NACA 64A112. For this airfoil section $\alpha_{o 2}=-0.94^{\circ}$ and $C_{l, \alpha}=6.09$ per radian. For purposes of applying the no-flow boundary condition at the control points, assume that the wing is planar. Prepare a graph of the lift coefficient. How does this compare with that of Fig. $7.24$ ?

Chai Santi
Chai Santi
Numerade Educator
01:10

Problem 12

Following the VLM approach used in Example 7.4, calculate the lift coefficient for a delta wing whose aspect ratio is $1.5$. What is the sweep angle of the leading edge? The fact that the quarter-chord and the three-quarter-chord lines have different sweeps should be taken into account when defining the coordinates of the horseshoe vortices and the control points. How do the calculated values for the lift coefficient compare with the experimental values presented in Fig. 7.45?

Penny Riley
Penny Riley
Numerade Educator
01:10

Problem 13

Use equation (7.61) to calculate the lift coefficient as a function of the angle of attack for a flat delta wing with sharp leading edges. The delta wing has an aspect ratio of $1.0$. Compare the solution with the data of Fig. $7.44$.

Penny Riley
Penny Riley
Numerade Educator
01:10

Problem 14

Use equation (7.62) to calculate the induced drag for a flat delta wing with sharp leading edges. The delta wing has an aspect ratio of $1.0$. Compare the solution with the data of Fig. $7.47 .$

Penny Riley
Penny Riley
Numerade Educator
03:43

Problem 15

Assume that the wing area of an airplane is proportional to the square of the wing span and the volume, and thus the weight, is proportional to the cube of the wing span (this is the square-cube law). Find the wing loading of the aircraft as a function of wing span, $b$. Using these relationships explain why very large aircraft (like the Boeing 747 or the Airbus 380 ) have to fly with very large wing loadings. If you wanted to re-design an existing aircraft that weighed $82,000 \mathrm{lbs}$ with a wing span of $65 \mathrm{ft}$, what wing span would you need to add an additional $20 \%$ to the weight?

Ricajoy Montero
Ricajoy Montero
Numerade Educator