Consider an airplane that weighs 10,000 N and cruises in level flight at 185 km / h at an alti

step 1 We're going to let Brunuli's equation. We're going to apply Brunle's equation, rather. And here

Consider an airplane that weighs 10,000 N and cruises in...

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 $(a)$

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 $(a)$

Key Concepts

Lift Coefficient (C_L)

The lift coefficient is a dimensionless parameter that relates the lift generated by an airfoil or wing to the dynamic pressure and the wing area. It encapsulates the effects of airfoil shape, angle of attack, and flow conditions, and is essential in ensuring that the aerodynamic forces balance the weight in level flight. Its determination is fundamental in aircraft design and performance analysis.

Spanwise Load Distribution

The spanwise load distribution describes how the lift is distributed along the wingspan of an aircraft. It is typically represented by a Fourier series expansion in lifting-line theory to capture variations in local angle of attack, twist (washout), and chord distribution. This distribution is critical for understanding wing efficiency, structural loadings, and the effectiveness of aerodynamic design features.

Induced Drag Coefficient (C_D,v)

The induced drag coefficient quantifies the additional drag created by the vortices at the wing tips, which arise due to the finite span of the wing. It is intrinsically linked to the distribution of lift along the span and the downwash produced by the wing. Minimizing induced drag is a key consideration in aerodynamic design, especially for high-efficiency or low-speed flight regimes.

Geometric Angle of Attack

The geometric angle of attack is the angle between the chord line of the wing (or of a given airfoil section) and the oncoming airflow. Unlike the effective angle of attack, it is determined purely by the wing’s geometry, including any built-in twist or incidence differences along the span. This parameter is essential in predicting the initial aerodynamic performance of the wing before considering flow corrections such as downwash.

Lifting-Line Theory

Lifting-line theory is an aerodynamic model used to approximate the three-dimensional wing effects from a given airfoil section. It involves representing the wing’s circulation distribution along its span via a series expansion (e.g., Fourier series) to analyze both lift and induced drag. This theory is a cornerstone in modern aerodynamics, providing insights into the influence of wing geometry, aspect ratio, and twist on the performance of aircraft wings.

Transcript

00:01 We're going to let brunuli's equation.

00:04 We're going to apply brunle's equation, rather.

00:12 And here we can say that p .1 plus one half row v.

00:17 Sub 1 squared plus row g, h sub 1, plus rather, this would be equal to p sub 2 plus 1 half row v sub 2 squared plus row gh sub 2.

00:35 Here we're going to say that we are applying bernouis equation at the top and the bottom of the wing.

00:46 And so there's not going to be any appreciable height difference.

00:50 So we can eliminate these two terms because of the height difference is going to be very, very small when applying bernoulli's equation at the top and the bottom of the wing.

01:01 And so we can then say that piece of one minus piece of two, the pressure difference.

01:08 Would be equaling to one half times the density of air multiplied by v sub 2 squared minus v sub 1 squared.

01:18 And so this would be equal to the change in pressure.

01:21 And we can say that then finding the velocity sub 2, this would be equal to two times the change in pressure divided by the density row plus v sub 1 squared all to the one half power.

01:37 And for part a, we can solve v .2 would be equaling 2 times 1 ,000 neutons per square meter, divided by the density of air 1 .29 kilograms per cubic meter, plus 60 .0 meters per second quantity squared to the one -half power.

02:09 And so we find v.

02:11 Sub 2 equaling 71 .8 meters per second...

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