Question

The question is often asked: Can an airfoil fly upside-down? To answer this, make the following calculation. Consider a positively cambered airfoil with a zero-lift angle of ?3?. The lift slope is 0.1 per degree. (a) Calculate the lift coefficient at an angle of attack of 5?. (b) Now imagine the same airfoil turned upside-down, but at the same 5? angle of attack as part (a). Calculate its lift coefficient. (c) At what angle of attack must the upside-down airfoil be set to generate the same lift as that when it is right-side-up at a 5? angle of attack?

Short answer

(a) 0.8, (b) 0.8, (c) 11°.

Step-by-step solution

Understanding the Lift Coefficient Formula

The lift coefficient, denoted as \( C_L \), can be calculated using the formula \( C_L = a(\alpha - \alpha_0) \), where \( a \) is the lift slope, \( \alpha \) is the angle of attack, and \( \alpha_0 \) is the zero-lift angle.

Calculating Lift Coefficient Right-Side-Up

For an angle of attack \( \alpha = 5^\circ \) and a zero-lift angle \( \alpha_0 = -3^\circ \), the lift slope \( a \) is 0.1 per degree. Substitute into the formula: \[ C_L = 0.1(5 - (-3)) = 0.1 \times 8 = 0.8 \].

Calculating Lift Coefficient Upside-Down

When the airfoil is upside-down, the effective angle of attack changes to \( \alpha = 5^\circ - 2\alpha_0 \) because the zero-lift angle is now positive. For the upside-down airfoil, \( \alpha_0 = 3^\circ \) (opposite sign of right-side-up). Calculate: \[ C_L = 0.1(5 + 3) = 0.1 \times 8 = 0.8 \].

Determining Angle of Attack for Same Lift Upside-Down

To find the angle \( \alpha_{up} \) for the upside-down airfoil to generate the same lift, equate the lift coefficients: \[ 0.8 = 0.1(\alpha_{up} - 3) \]. Solving for \( \alpha_{up} \): \[ 0.8 = 0.1\alpha_{up} - 0.3 \] \[ 0.8 + 0.3 = 0.1\alpha_{up} \] \[ 1.1 = 0.1\alpha_{up} \] \[ \alpha_{up} = 11^\circ \].

Key concepts

Lift Coefficient

The lift coefficient, often represented by the symbol \( C_L \), is a number that describes the lifting capability of an airfoil, or wing, in a specific flow of air. It relates the lift force created by the airfoil to the dynamic pressure of the airflow and the area of the wing. The higher the \( C_L \), the more lift is generated at a given airspeed and wing area.

The formula to calculate the lift coefficient is \( C_L = a(\alpha - \alpha_0) \), where:

• \( a \) represents the lift slope, typically constant for small angles of attack.
• \( \alpha \) is the angle of attack, which is the angle between the oncoming air or relative wind and a reference line on the airfoil.
• \( \alpha_0 \) is the zero-lift angle, the angle at which the airfoil generates no lift.

Understanding this relationship helps in predicting how changes in angle of attack affect lift, essential for analyzing flight performance and stability.

Angle of Attack

The angle of attack, denoted as \( \alpha \), is a fundamental concept in aerodynamics. It is defined as the angle between the chord line of an airfoil and the oncoming airflow. This angle is crucial because it directly influences the lift generated by the wing.

As the angle of attack increases, the lift increases up to a certain point known as the critical angle or stall angle. Beyond this point, lift decreases as the wing stalls. The angle of attack can be changed by adjusting the pitch of the aircraft.

For instance, in our exercise, the airfoil’s angle of attack is set at \( 5^\circ \). When the airfoil is upside down, the angle of attack still initially operates at \( 5^\circ \), but effective components change due to its orientation related to airflow and the zero-lift angle shift.

Cambered Airfoil

A cambered airfoil has a curved shape designed to optimize lift even at relatively low angles of attack. It is distinct from a symmetric airfoil that is straight along its chord line. The camber, or curvature, allows the airfoil to produce more lift than a flat or symmetrical airfoil, especially at lower speeds.

In a positively cambered airfoil, the mean camber line, which is the imaginary line equidistant from the upper and lower surfaces of the wing, is above the chord line. This configuration usually results in a positive lift coefficient even when the angle of attack is zero or slightly negative, meaning the wing can generate lift more efficiently.

• Benefits: Improves lift-to-drag ratio and enhances performance during takeoff and landing.
• Trade-offs: Lacks the same performance while inverted, typically creating less lift upside-down unless adjusted for such conditions.

Zero-Lift Angle

The zero-lift angle, noted as \( \alpha_0 \), is a significant aerodynamic parameter indicating the angle of attack at which an airfoil produces no lift. For cambered airfoils, this angle is usually not zero and might be negative, due to their inherent curvature favoring lift.

In our exercise, the zero-lift angle is \(-3^\circ \), so the cambered airfoil naturally produces lift even when the angle of attack is slightly negative. This property makes cambered airfoils efficient for normal flight conditions.

When inverted, this zero-lift angle changes effectively, i.e., it becomes positive for computational purposes since the airfoil is flipped upside down, requiring adjustment in the angle of attack to achieve the same lift as when right-side up.