Question

For the NACA 2412 airfoil, the lift coefficient and moment coefficient about the quarter-chord at ?6? angle of attack are ?0.39 and ?0.045, respectively. At 4? angle of attack, these coefficients are 0.65 and ?0.037, respectively. Calculate the location of the aerodynamic center.

Short answer

The aerodynamic center is approximately at 0.2808 from the leading edge.

Step-by-step solution

Understand the Given Values

We have two sets of data for the NACA 2412 airfoil: 1. At an angle of attack of \( 6^\circ \), the lift coefficient \( C_{l6} = 0.39 \) and the moment coefficient \( C_{m6} = -0.045 \).2. At an angle of attack of \( 4^\circ \), the lift coefficient \( C_{l4} = 0.65 \) and the moment coefficient \( C_{m4} = -0.037 \). We need to determine how these coefficients change with respect to the angle of attack to find the aerodynamic center.

Calculate Change in Moment Coefficient

Calculate the difference in the moment coefficients between the two angles:\[\Delta C_m = C_{m4} - C_{m6} = (-0.037) - (-0.045) = 0.008\]

Calculate Change in Lift Coefficient

Calculate the change in the lift coefficient between the two angles:\[\Delta C_l = C_{l4} - C_{l6} = 0.65 - 0.39 = 0.26\]

Determine Change in Angle of Attack

Calculate the change in angle of attack between the two cases:\[\Delta \alpha = 4^\circ - 6^\circ = -2^\circ = -2 \times \frac{\pi}{180} = -\frac{\pi}{90} \text{ radians}\]Since we are working with coefficients, the change in angle is not directly used in finding the aerodynamic center location, but it's important to recognize it's the independent variable alteration.

Calculate Slope of Moment Coefficient

The change in moment coefficient with respect to the lift coefficient is given by:\[\frac{\Delta C_m}{\Delta C_l} = \frac{0.008}{0.26}\]

Apply Moment Coefficient Equation

Using the equation for the moment about the quarter-chord, we have:\[C_{m_{ac}} = C_{m_{cc}} + (x_{ac} - 0.25)C_{l}\]Where:- \( C_{m_{ac}} \) is the moment coefficient at the aerodynamic center (which does not change with \( C_{l} \)), and- \( C_{m_{cc}} \) is the moment coefficient at the quarter-chord line (given).Using: \[x_{ac} = 0.25 + \frac{\Delta C_m}{\Delta C_l} = 0.25 + \frac{0.008}{0.26}\]Calculate for \( x_{ac} \), the aerodynamic center.

Compute Final Result for Aerodynamic Center Location

Substitute and calculate:\[x_{ac} = 0.25 + \frac{0.008}{0.26} = 0.25 + 0.03077 \approx 0.2808\]Thus, the aerodynamic center is located at approximately 0.2808 from the leading edge.

Key concepts

Lift Coefficient

The lift coefficient, denoted as \( C_l \), is a crucial factor in determining the aerodynamic performance of an airfoil. It represents the lift force produced by a wing or airfoil per unit area compared to the pressure of the wind against it. The formula for calculating lift is given by:\[C_l = \frac{L}{0.5 \times \rho \times V^2 \times A}\]where:

• \( L \) is the lift force.
• \( \rho \) is the air density.
• \( V \) is the velocity of the airflow.
• \( A \) is the wing area.

The lift coefficient is vital in understanding how wings generate lift under varying conditions. It allows engineers to predict how changes in design or operational parameters (like angle of attack) will affect lift. A higher lift coefficient indicates better lift generating potential, which is critical for aircraft during takeoff and landing. In the context of the NACA 2412 airfoil exercise, the lift coefficients given at different angles of attack help determine the airfoil's aerodynamic properties.

Moment Coefficient

The moment coefficient, symbolized as \( C_m \), quantifies the pitching moment acting on an airfoil. It provides insight into the stability and control of the aircraft. This coefficient is defined relative to a reference point, commonly the quarter-chord line—a point located at 25% of the chord length from the leading edge.To find the moment coefficient about the aerodynamic center using the change in lift coefficient, we apply the relation:\[C_{m_{ac}} = C_{m_{cc}} + (x_{ac} - 0.25)C_{l}\]Where:

• \( C_{m_{ac}} \) is the moment coefficient at the aerodynamic center.
• \( C_{m_{cc}} \) is the moment coefficient at the quarter-chord line (known from measurements).
• \( x_{ac} \) is the position of the aerodynamic center.

This formula helps determine the position of the aerodynamic center by analyzing changes in the lift and moment coefficients as the angle of attack varies. For the NACA 2412, the calculation reveals how far the center is from the leading edge, impacting stability insight.

NACA Airfoils

NACA airfoils are a series of aerofoil shapes developed by the National Advisory Committee for Aeronautics (NACA). These airfoils are defined by a series of digits, each describing specific geometric features of the airfoil's shape. For instance, the NACA 2412 airfoil used in the exercise has:

• "24" indicating the maximum camber is 2% of the chord and located 40% back from the leading edge.
• "12" representing the maximum thickness of the airfoil is 12% of the chord.

NACA airfoils were designed to provide a high degree of lift while maintaining manageable drag characteristics. The shape is crucial for efficiently handling air flows, helping aircraft achieve desired performance levels. These airfoils still serve as a foundational reference in the design and analysis of modern aerodynamics, allowing engineers to build on tested and reliable data.

Angle of Attack

The angle of attack (\( \alpha \)) is the angle between the chord line of an airfoil and the oncoming airflow. It's a fundamental parameter affecting both lift and drag, as it determines the relative wind flow over the wing.Increasing the angle of attack improves lift up to a point. However, if it becomes too steep, it can cause a stall, drastically reducing lift. Understanding the angle of attack helps in optimizing performance during various phases of flight.In the exercise for the NACA 2412 airfoil, the angle of attack changes from \( 6^\circ \) to \( 4^\circ \). This modification affects both the lift and moment coefficients, providing insight into the airfoil's aerodynamic characteristics. Monitoring how the coefficients vary with changing angles of attack aids in calculating the aerodynamic center, a key point affecting the stability and control of the airfoil.