Problem Statement
An aeroplane weighing 22500N has a wing area of 22.5m² and span of 12m. What is the lift coefficient if it travels at 320 km/hr in the horizontal direction? Also compute the value of circulation and angle of attack measured from zero lift axis.
Given Data
Solution Approach
To solve this problem, we need to:
- Calculate the lift coefficient (CL) using the lift equation
- Determine the angle of attack using the relationship between lift coefficient and angle of attack
- Calculate the circulation using the Kutta-Joukowski theorem
Calculations
Lift Coefficient Calculation
Step 1: In steady horizontal flight, the lift force equals the weight of the aircraft. We can use the lift equation to find the lift coefficient:
Rearranging to solve for CL:
Step 2: Substituting the known values:
Step 3: For a thin airfoil, the lift coefficient is related to the angle of attack by:
Rearranging to solve for θ:
Step 4: The circulation can be calculated using:
Where c is the mean chord length (wing area divided by wingspan):
Substituting:
Lift Coefficient (CL) = 0.2095
Angle of Attack (θ) = 1.911°
Circulation (Γ) = 17.46 m²/s
Detailed Explanation
Lift Coefficient
The lift coefficient is a dimensionless parameter that relates the lift force generated by an airfoil to the fluid density, velocity, and reference area. In level flight, the lift force must equal the weight of the aircraft, which allows us to calculate the lift coefficient directly. The calculated value of 0.2095 is typical for an aircraft in cruise configuration.
Angle of Attack
The angle of attack is the angle between the oncoming air flow and the chord line of the airfoil. For thin airfoil theory, there is a linear relationship between the lift coefficient and the angle of attack measured from the zero-lift angle. The calculated angle of 1.911° is relatively small, which is typical for efficient cruising flight where the aircraft seeks to minimize drag.
Circulation
Circulation is a fundamental concept in aerodynamics that quantifies the rotational flow field around the airfoil. According to the Kutta-Joukowski theorem, the lift per unit span is directly proportional to the circulation around the airfoil and the fluid density and velocity. The calculated circulation of 17.46 m²/s represents the strength of the vortex system created by the wing.
Physical Interpretation
In horizontal flight, the aircraft is in equilibrium with the lift force balancing the weight. The wings generate this lift through a combination of factors:
- The shape of the airfoil creates pressure differences between the upper and lower surfaces
- The angle of attack determines how much the airfoil deflects the incoming airflow
- The circulation around the wing is directly related to the generation of lift
Engineering Significance
These parameters are crucial for aircraft design and performance analysis:
- The lift coefficient helps engineers optimize wing design for different flight phases
- The angle of attack is monitored during flight to avoid stalling conditions
- Understanding circulation aids in analyzing wing efficiency and designing high-performance airfoils
This problem demonstrates how fundamental principles of fluid dynamics and aerodynamics can be applied to analyze the performance of an aircraft in steady flight conditions.
