Question

Consider an NACA 2412 airfoil with a 2-m chord in an airstream with a velocity of 50 m/s at standard sea level conditions. If the lift per unit span is 1353 N, what is the angle of attack?

Short answer

The angle of attack for the NACA 2412 airfoil is approximately 4.54 degrees.

Step-by-step solution

Identify Known Values

We begin by identifying the known values from the problem statement. The problem gives us:- Chord length, \(c = 2\) meters.- Free stream velocity, \(V_{ ext{inf}} = 50\) m/s.- Lift per unit span, \(L' = 1353\) N.- Air density at standard sea level, \(\rho = 1.225\, \text{kg/m}^3\).

Use Lift Equation

The lift per unit span is given by the equation:\[ L' = \frac{1}{2} \rho V_{ ext{inf}}^2 c C_l \]where \(C_l\) is the lift coefficient. We need to rearrange this equation to find \(C_l\).\[ C_l = \frac{2L'}{\rho V_{ ext{inf}}^2 c} \]

Calculate Lift Coefficient

Substitute the known values into the rearranged equation for \(C_l\):\[ C_l = \frac{2 \times 1353}{1.225 \times 50^2 \times 2} \]Calculate the result to find \(C_l\).

Relate Lift Coefficient to Angle of Attack

For the NACA 2412 airfoil, the lift coefficient \(C_l\) is related to the angle of attack \(\alpha\) (in degrees) by:\[ C_l = C_{l_0} + \left(\frac{dC_l}{d\alpha}\right) \alpha \]where \(C_{l_0} = 0.2\) and \(\frac{dC_l}{d\alpha} = 0.11\) deg\(^{-1}\).

Solve for Angle of Attack

Substitute the value of \(C_l\) obtained from Step 3 into the lift coefficient equation:\[ C_l = 0.2 + 0.11 \alpha \]Rearrange to solve for \(\alpha\):\[ \alpha = \frac{C_l - 0.2}{0.11} \]

Compute Angle of Attack

Using the value of \(C_l\) from Step 3, calculate the angle of attack \(\alpha\). This will give the angle of attack in degrees.

Key concepts

Angle of Attack

The angle of attack is a very important concept in aerodynamics. It is the angle between the wing's chord line and the oncoming airflow, or relative wind. This angle influences the lift generated by an airfoil. The lift increases with the angle of attack up to a certain point, known as the critical angle. Beyond this, lift begins to decrease due to airflow separation, leading to a stall. When considering an NACA airfoil, such as the NACA 2412, the angle of attack (\(\alpha\)) is calculated based on the relationship between the lift coefficient and the aerodynamic characteristics of the airfoil. By understanding and calculating the angle of attack, engineers can design wings to achieve optimal performance during different phases of flight.

Lift Coefficient

The lift coefficient (\(C_l\)) is a dimensionless number that describes the lift produced by an airfoil at a given angle of attack and speed. It is a crucial factor because it allows us to predict how much lift a wing will generate under various conditions. For the NACA 2412 airfoil, there is a specific relationship between the lift coefficient and the angle of attack, typically expressed as: \[C_l = C_{l_0} + \left(\frac{dC_l}{d\alpha}\right) \alpha\]Where \(C_{l_0}\) is the lift coefficient at zero angle of attack and \(\frac{dC_l}{d\alpha}\) is the lift curve slope, measuring how lift changes with the angle of attack.Knowing how to calculate \(C_l\) from this relationship is essential since it forms the basis for determining how an airfoil behaves under different flight conditions.

Aerodynamic Lift

Aerodynamic lift is the force acting upward on an airfoil. It is a result of the air pressure differences created over the surface of a wing as air flows over it. This force enables an aircraft to rise above the ground and is influenced by several factors. To maintain flight, the lift must counteract the weight of the aircraft. Factors affecting lift include:

• The shape and size of the airfoil
• The airspeed over the wings
• The air density, affected by altitude and temperature
• The angle of attack, which changes the lift coefficient

Understanding aerodynamic lift involves not only comprehending these factors but also how they interrelate to sustain an aircraft in flight. By applying lift equations, one can calculate the necessary lift to keep a plane airborne, ensuring safety and efficiency in aviation.