Consider tests of an unswept wing that spans the wind tunnel and whose airfoil section is NACA 23012. Since the wing model spans the test section, we will assume that the flow is two dimensional. The chord of the model is 1.3 m. The test section conditions simulate a density altitude of 3 km. The velocity in the test section is 360 km/h. a. What is the lift per unit span (N/m) that you would expect to measure when the wing attack angle is 4°? b. What would be the corresponding section lift coefficient and die moment coefficient (about the quarter-chord point)?

To calculate the lift per unit span and the section lift coefficient, we need to apply the theory of aerodynamics, specifically the lift equation for a two-dimensional airfoil section and the lift coefficient formula.

a. Lift per unit span (N/m)

The lift per unit span, \( L' \), can be found using the following formula:

\[ L' = C_l \cdot \frac{1}{2} \cdot \rho \cdot V^2 \cdot c \]

where \( C_l \) = lift coefficient \( \rho \) = air density (kg/m³) \( V \) = airspeed (m/s) \( c \) = chord length (m)

However, we need the lift coefficient \( C_l \) for an angle of attack of 4 degrees which can be determined by looking at the lift curve for the NACA 23012 airfoil or using empirical data. I'll use a typical value for a NACA 23012 which is about 0.1 per degree. Therefore, at 4 degrees, \( C_l \approx 4 \times 0.1 = 0.4 \).

We also need the air density \( \rho \) at a density altitude of 3 km. The Standard Atmosphere tables or calculations can be used to determine the density at this altitude, which is roughly around 0.9093 kg/m³.

The velocity \( V \) should be converted from km/h to m/s:

\[ V = \frac{360 \text{ km/h}}{3.6} = 100 \text{ m/s} \]

The chord length \( c \) is 1.3 m, as given.

Plugging in these values into the lift equation:

\[ L' = 0.4 \cdot \frac{1}{2} \cdot 0.9093 \text{ kg/m}^3 \cdot (100 \text{ m/s})^2 \cdot 1.3 \text{ m} \] \[ L' = 0.4 \cdot 0.45465 \cdot 10,000 \cdot 1.3 \] \[ L' = 2360.17 \text{ N/m} \]

Therefore, you would expect to measure a lift of approximately 2360.17 N/m when the wing attack angle is 4°.

b. The corresponding section lift coefficient \( C_l \) has already been estimated to be 0.4 for a 4-degree angle of attack.

The moment coefficient \( C_m \) about the quarter-chord point can be found using similar data for the NACA 23012 airfoil. It's also typically obtained through empirical data or wind tunnel testing for the specific airfoil at the given angle of attack. For many airfoils, \( C_m \) usually remains relatively constant over a range of angles of attack, especially near the zero-lift angle. For the NACA 23012, a typical value for the quarter-chord \( C_m \) might be around -0.02 to -0.03 (negative indicating a pitching moment in the nose-down direction). You would need to refer to specific data for this airfoil to find a more accurate number.

Extra: The concept of lift is fundamental in aerodynamics and aircraft design. Lift is the force that directly opposes the weight of an aircraft and holds it in the sky. The generation of lift can be described by the Bernoulli Principle, where increased velocity of the airflow over the top surface of a wing results in decreased pressure, creating the lift force.

The lift coefficient \( C_l \) is a dimensionless number that represents the lift generated by a wing or airfoil section at a given angle of attack and is a crucial parameter in the design and analysis of wings. It varies with the angle of attack, airfoil shape, and Reynolds number (a value that describes the flow characteristics of the fluid).

The moment coefficient \( C_m \) describes the pitching moment experienced by the airfoil about a specific point (the quarter-chord point is commonly used). This moment affects the stability and control of the aircraft.

Air density \( \rho \) changes with altitude, temperature, and pressure. It's important to consider when calculating lift because it directly affects the lift produced by the wing.

Another important aspect of aerodynamics is the assumption that the flow is two-dimensional, meaning it does not change in the spanwise direction of the wing. This is a valid assumption in a wind tunnel with walls aligned with the wing tips, which serves to eliminate the effect of wingtip vortices and provides a simplified scenario for analysis and testing.