Airfoil Lift and Circulation Solution

In aerodynamic wing theory, the lift force generated by air flowing past an aircraft wing profile is fundamentally linked to fluid circulation. According to the foundational Kutta-Joukowski Theorem in aerospace engineering, the aerodynamic lift force vector per unit wing span ($\vec{F}_L$) is directly proportional to the cross product of the free-stream airflow velocity vector ($\vec{v}$) and the bound circulation vector ($\vec{\Gamma}$):

where $\rho$ is the ambient air density.

Consider an aircraft flying through cruising altitude where the air density is $\rho = 1.2\text{ kg/m}^3$. The relative wind velocity flowing horizontally past the wing chord is $\vec{v} = \langle 150, 0, 0 \rangle\text{ m/s}$, and the bound circulation around the airfoil profile is oriented along the spanwise direction as $\vec{\Gamma} = \langle 0, 4, 0 \rangle\text{ m}^2/\text{s}$.

1. Compute the cross product $\vec{v} \times \vec{\Gamma}$.
2. Determine the resulting lift force vector per unit span ($\vec{F}_L$) in Newtons per meter ($\text{N/m}$).
3. State the spatial direction in which the lift force acts.

Step-by-Step Solution:

1. Calculate the Cross Product ($\vec{v} \times \vec{\Gamma}$): Set up the $3 \times 3$ matrix determinant:

   $$\vec{v} \times \vec{\Gamma} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ 150 & 0 & 0 \\ 0 & 4 & 0 \end{vmatrix}$$
   Expanding along the top row:
   $$\vec{v} \times \vec{\Gamma} = \hat{i}(0\cdot 0 - 0\cdot 4) - \hat{j}(150\cdot 0 - 0\cdot 0) + \hat{k}(150\cdot 4 - 0\cdot 0)$$
   $$\vec{v} \times \vec{\Gamma} = 0\hat{i} - 0\hat{j} + 600\hat{k} = \langle 0, 0, 600 \rangle\text{ m}^3/\text{s}^2$$

2. Calculate Lift Force Vector per Unit Span ($\vec{F}_L$): Multiply by the fluid density $\rho = 1.2\text{ kg/m}^3$:

   $$\vec{F}_L = 1.2 \langle 0, 0, 600 \rangle = \langle 0, 0, 720 \rangle\text{ N/m} \quad (\text{or } 720\hat{k}\text{ N/m})$$

3. Spatial Direction: The lift force points purely in the positive $+z$ direction (vertically upward).

Meaning of the Answer & Real-Life Application:

The cross product dictates the orientation of aerodynamic lift: with velocity along $+x$ and circulation along $+y$, the right-hand rule requires lift to be directed along $+z$, generating $720\text{ N}$ of upward lift per meter of wing span. Aerospace engineers use the Kutta-Joukowski theorem to estimate aircraft wing performance and design cambered airfoils that generate the required circulation to support aircraft gross weight.