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Airfoil Lift and Circulation Solution

Formulas & Definitions

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 (F⃗L\vec{F}_L) is directly proportional to the cross product of the free-stream airflow velocity vector (v⃗\vec{v}) and the bound circulation vector (Γ⃗\vec{\Gamma}):

Fβƒ—L=ρ(vβƒ—Γ—Ξ“βƒ—)\vec{F}_L = \rho (\vec{v} \times \vec{\Gamma})
where ρ\rho is the ambient air density.

Consider an aircraft flying through cruising altitude where the air density is ρ=1.2Β kg/m3\rho = 1.2\text{ kg/m}^3. The relative wind velocity flowing horizontally past the wing chord is vβƒ—=⟨150,0,0⟩ m/s\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 Ξ“βƒ—=⟨0,4,0⟩ m2/s\vec{\Gamma} = \langle 0, 4, 0 \rangle\text{ m}^2/\text{s}.

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

Step-by-Step Solution:

  1. Calculate the Cross Product (v⃗×Γ⃗\vec{v} \times \vec{\Gamma}): Set up the 3×33 \times 3 matrix determinant:

    vβƒ—Γ—Ξ“βƒ—=∣i^j^k^15000040∣\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:
    vβƒ—Γ—Ξ“βƒ—=i^(0β‹…0βˆ’0β‹…4)βˆ’j^(150β‹…0βˆ’0β‹…0)+k^(150β‹…4βˆ’0β‹…0)\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)
    vβƒ—Γ—Ξ“βƒ—=0i^βˆ’0j^+600k^=⟨0,0,600⟩ m3/s2\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 (Fβƒ—L\vec{F}_L): Multiply by the fluid density ρ=1.2Β kg/m3\rho = 1.2\text{ kg/m}^3:

    Fβƒ—L=1.2⟨0,0,600⟩=⟨0,0,720⟩ N/m(orΒ 720k^Β N/m)\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+z direction (vertically upward).

Meaning of the Answer & Real-Life Application:

The cross product dictates the orientation of aerodynamic lift: with velocity along +x+x and circulation along +y+y, the right-hand rule requires lift to be directed along +z+z, generating 720Β N720\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.