Fluids Lecture 7 Notes

Elliptical Lift Distribution

Definition and Lift Calculation

• Concept: Elliptical spanwise circulation distribution.
• Formula: ( \Gamma(y) = \Gamma_0 \sqrt{1 - (\frac{2y}{b})^2} )
• Overall Lift: Calculated by integrating lift/span distribution ( L(y) = \rho V \Gamma(y) ).
  • Integrates to ( L = \pi \rho V \Gamma_0 b / 4 ).
  • Can use integral tables or note the area under an ellipse is ( \pi/4 ) times the area of the enclosing rectangle.

Downwash Calculation

• Trailing Vortex Sheet Strength: ( \gamma(y) = \frac{d\Gamma}{dy} = \frac{4 \Gamma_0}{b^2} y \sqrt{1 - (\frac{2y}{b})^2} )
• Downwash at Location (y_o):
  • Use trigonometric substitution for integration.
  • Result: ( w(\theta_o) = \frac{\Gamma_0}{2b} ).
  • Uniform downwash over wing span with elliptical circulation.
  • Sharp upwash near tips, but doesn't affect flow angles seen by wing.
• Expression for Downwash: ( w = \frac{2L}{\rho V b^2} ).

Induced Angles

• Uniform Induced Angles Across Span: ( i = \frac{w}{V} = \frac{L}{\pi \rho V^2 b^2} ).
• Overall Lift Coefficient: ( C_L = \frac{L}{\frac{1}{2} \rho V^2 S} ).
• Induced Angle Expression: ( i = \frac{S C_L}{b^2} = \frac{C_L}{AR} ).

Induced Drag

• Consistent Tilt of Lift Vectors: Induces drag along the span.
• Induced Drag Formula: ( D_i = i L ).
  • Substituting ( i ) yields: ( D_i = \frac{(L/b)^2}{\pi \rho V^2} ).
• Dimensionless Relation: ( C_{Di} = \frac{C_L^2}{AR \pi} )._

Total Wing Drag

• Overall Wing Drag: Combination of profile and induced drag.
  • Formula: ( D = D_p + D_i ) or ( C_D = C_{Dp} + C_{Di} ).
• Profile Drag Coefficient: Chord-weighted average of local ( c_d(y) ).
  • Simplification: Assume ( c_d(y) ) constant for average chord.
  • Formula: ( C_{Dp} = \frac{1}{S} \int_{-b/2}^{b/2} c_d(y) c(y) dy ).
  • Overall Drag Coefficient: ( C_D(C_L; Re_{avg}) = c_d(C_L; Re_{avg}) + \frac{C_L^2}{AR \pi} ).

Graphical Analysis

• CD(CL) Polar Plot: Shows relation at one Reynolds number and two aspect ratios (AR = 20 and AR = 10).
• Key Observations:
  1. Maximum lift/drag ratio ( (C_L/C_D)_{max} ) decreases with decreasing AR.
  2. CL for maximum ( (C_L/C_D) ) decreases as AR decreases, requiring faster flight for smaller aspect ratios to achieve best range/duration._