Using the Prandtl-Glauert rule, calculate the lift coefficient for an NACA 2412 airfoil at 5° angle of attack in a Mach 0.6 free stream. (Refer to Fig. 4.5 for the original airfoil data.)
The Correct Answer and Explanation is:
To calculate the compressible lift coefficient using the Prandtl-Glauert rule, we follow these steps:
Given:
- Airfoil: NACA 2412
- Angle of attack, α=5∘\alpha = 5^\circ
- Freestream Mach number: M∞=0.6M_\infty = 0.6
Step 1: Get the incompressible lift coefficient CL0C_{L_0}
From standard airfoil data (see Figure 4.5 in Anderson’s Fundamentals of Aerodynamics):
For the NACA 2412 at low subsonic incompressible conditions: CL0=0.11⋅α=0.11⋅5=0.55C_{L_0} = 0.11 \cdot \alpha = 0.11 \cdot 5 = 0.55
(This is a common linear approximation: CL≈0.11⋅αC_L \approx 0.11 \cdot \alpha, where α\alpha is in degrees.)
Step 2: Apply the Prandtl-Glauert rule
The Prandtl-Glauert correction gives the compressible lift coefficient CLC_L as: CL=CL01−M∞2C_L = \frac{C_{L_0}}{\sqrt{1 – M_\infty^2}} CL=0.551−0.62=0.551−0.36=0.550.64=0.550.8=0.6875C_L = \frac{0.55}{\sqrt{1 – 0.6^2}} = \frac{0.55}{\sqrt{1 – 0.36}} = \frac{0.55}{\sqrt{0.64}} = \frac{0.55}{0.8} = 0.6875
✅ Final Answer:
CL=0.6875\boxed{C_L = 0.6875}
✍️ Explanation
The lift coefficient, CLC_L, of an airfoil characterizes how much lift the airfoil generates relative to the dynamic pressure and wing area. For subsonic flows, especially below Mach 0.7, air is treated as compressible. The Prandtl-Glauert rule provides a method to adjust the incompressible lift coefficient to account for compressibility effects.
We begin with the baseline lift coefficient under incompressible conditions for the NACA 2412 airfoil. Standard airfoil data shows that the lift curve slope is approximately 0.11 per degree for small angles of attack. At 5°, this gives an incompressible lift coefficient of 0.55.
However, in a compressible flow (e.g., Mach 0.6), pressure disturbances travel slower relative to the flow speed, leading to amplified aerodynamic forces. The Prandtl-Glauert rule compensates for this by dividing the incompressible lift coefficient by the square root of 1−M21 – M^2, where MM is the freestream Mach number. This correction increases the lift coefficient due to the enhanced pressure gradients associated with compressibility.
Applying the rule for Mach 0.6 gives a correction factor of 1/0.64=1.251 / \sqrt{0.64} = 1.25, increasing the lift coefficient from 0.55 to approximately 0.6875. This shows a 25% increase in lift due to compressibility. It’s important to note that the Prandtl-Glauert rule is only valid for subsonic speeds below Mach 0.7–0.8 and small angles of attack, as it doesn’t account for nonlinear effects like shock waves or flow separation.
In summary, the Prandtl-Glauert rule provides a quick and reasonable way to estimate compressible lift effects for thin airfoils in low subsonic flow.
