Lec05: Introduction to Drag Polar, 3D Lift Curve Slope and the Concept of Stalling Velocity

Main Lecture: Finite-Wing Aerodynamics

Induced Angle of Attack and Induced Drag

Wingtip vortices create downwash, which tilts the local airflow and reduces the angle of attack the wing effectively experiences. The effective angle of attack is the geometric angle minus the induced angle:

[ \alpha_{\text{effective}} = \alpha_G - \alpha_i ]

Because the lift vector is tilted relative to the free-stream direction, it has a component that acts as induced drag. For small induced angles:

[ D_i \approx L\alpha_i ]

For an ideal elliptical lift distribution, the induced angle is related to lift coefficient and aspect ratio. Real wings generally depart from this ideal, so the lecture uses a span-efficiency factor (e), typically below 1. The lecturer gives a range of approximately 0.85–0.95 for subsonic wings.

The resulting induced-drag coefficient is:

[ C_{D_i} = \frac{C_L^2}{\pi e\,AR} = K C_L^2, \qquad K = \frac{1}{\pi e\,AR} ]

Induced drag therefore increases with the square of the lift coefficient and decreases as aspect ratio increases.

Drag Polar

Total drag is treated as the sum of zero-lift (profile) drag and induced drag:

[ C_D = C_{D0} + K C_L^2 ]

A graph of (C_D) against (C_L) is called the drag polar. The lecture presents it as a key summary of an aircraft’s aerodynamic drag characteristics.

Finite-Wing Lift-Curve Slope

A finite wing has a smaller lift-curve slope than an ideal two-dimensional airfoil:

[ C_{L_\alpha} < c_{l_\alpha} ]

The induced angle means a finite wing needs a greater geometric angle of attack to produce the same lift coefficient. The relation given is:

[ C_{L_\alpha} = \frac{c_{l_\alpha}} {1+\dfrac{57.3\,c_{l_\alpha}}{\pi e\,AR}} ]

This form applies when the lift-curve slopes are expressed per degree. If they are expressed per radian, the equivalent form omits the (57.3) conversion factor.

For a wing with infinite aspect ratio, the induced angle tends to zero and the finite-wing slope approaches the two-dimensional slope. For a cambered airfoil, the lecturer explains that the zero-lift angle of attack is unchanged between the finite- and infinite-wing curves because the induced angle is zero at zero lift.

Example: Cessna 172

Using a span of 11 m and a wing area of 16.2 m², the lecturer calculates an aspect ratio of approximately 7.46. With (e = 0.95) and a two-dimensional slope of (2\pi) per radian, the example gives a finite-wing lift-curve slope of about 0.0854 per degree.

The example illustrates that a finite wing produces less lift per degree of angle-of-attack change than an idealized two-dimensional airfoil.

Stalling Speed and High-Lift Devices

Stalling Speed

At the stall angle of attack, the wing reaches its maximum lift coefficient, (C_{L_{\max}}). For straight-and-level flight, using (L = W), the lecture gives:

[ V_{\text{stall}} = \sqrt{\frac{2W}{\rho S C_{L_{\max}}}} ]

A higher (C_{L_{\max}}) means a lower stalling speed. The lecturer favors a lower stalling speed because it permits slower takeoff and landing, while noting that operational safety margins are also needed.

Increasing Maximum Lift

Trailing-edge flaps increase the wing’s effective camber, which increases lift. Their deflection also creates a virtual increase in angle of attack. Flaps shift the lift curve so that lift is higher at a given angle of attack and the zero-lift angle becomes more negative. The lecturer says the lift-curve slope itself does not change, although the stall angle of attack may decrease.

Some flaps deflect downward; others also translate rearward, as is common on passenger aircraft. The lecture gives an approximate increase in maximum lift of up to a factor of two for these arrangements.

Leading-edge slats further increase effective camber and can raise maximum lift. The lecturer says the increase can reach roughly a factor of three, depending on the configuration.

These devices raise (C_{L_{\max}}), thereby reducing stalling speed.

Separate Literary Segment

The subtitles switch abruptly after the lecture to a short literary reading. In the tale “Appointment in Samarra,” a merchant’s servant sees Death in Baghdad and flees to Samarra. The merchant later learns that Death was surprised to see the servant in Baghdad because their appointment was that evening in Samarra.

Speakers and Sources Featured

• Aerodynamics lecturer: Unnamed instructor presenting the lecture on drag polar, finite-wing lift-curve slope, stalling speed, and high-lift devices.
• Swayam Prabha/channel ident: A brief opening slogan is heard; no individual speaker is identified.
• Literary-snippet narrator: An unnamed presenter who reads the Appointment in Samarra story and says the adaptation is close to one by Somerset Maugham.