Aerodynamic and physical foundations of Finite Wing Induced Drag.
While an infinite two-dimensional airfoil exhibits zero induced drag, every finite three-dimensional lifting wing must contend with pressure equalization around its tips. The higher static pressure beneath the wing spills around the wingtips toward the low-pressure suction surface above, generating intense trailing vortex filaments that roll up into wingtip vortices. Ludwig Prandtl's lifting-line theory establishes that these trailing vortices induce a downward component of velocity—termed downwash w—along the span. The downwash tilts the local free-stream velocity vector downward by the induced angle of attack alpha_i = w / V_inf. Consequently, the local lift vector tilts rearward by alpha_i, creating an induced drag force component D_i = L * sin(alpha_i) parallel to the oncoming flow. For an elliptical lift distribution, Prandtl showed that the induced drag coefficient satisfies C_Di = C_L^2 / (pi * e * AR), where AR = b^2 / S is the wing aspect ratio and e is the Oswald span efficiency factor. In this laboratory experiment, a family of modular wings with identical NACA 0012 profiles and varying aspect ratios (AR = 2, 3, 4, 5, 6, 8) is tested on an external balance at a fixed lift coefficient C_L = 0.60. Plotting induced drag coefficient C_Di against reciprocal aspect ratio 1/AR demonstrates a strictly linear relationship whose gradient verifies Prandtl's downwash law.
Viva voce questions & analytical theory
How do wingtip vortices generate induced drag physically?
The leakage of high-pressure air from the bottom to the low-pressure top around the wingtips creates twin swirling vortices. These vortices induce a downward velocity field (downwash w) across the wing, which tilts the lift vector rearward, producing a drag component D_i = L * (w / V_inf).
Why is an elliptical planform aerodynamically ideal according to Prandtl's lifting-line theory?
Prandtl proved that an elliptical chord distribution produces a constant downwash velocity w along the entire span. A constant downwash minimizes the kinetic energy discarded into the trailing vortex wake, achieving minimum induced drag with e = 1.0.
How does increasing wing aspect ratio AR improve aircraft aerodynamic efficiency?
Because induced drag scales inversely with aspect ratio (C_Di proportional to 1/AR), high-aspect-ratio wings (such as sailplanes and U-2 aircraft) drastically diminish downwash and induced drag, yielding exceptional lift-to-drag ratios (L/D > 40).
What is the physical function of winglets installed on modern commercial airliners?
Winglets act as vertical endplates that obstruct spanwise cross-flow around the wingtip. By diffusing and displacing the tip vortex core upward and outward, winglets reduce effective downwash, increasing effective aspect ratio and cutting induced drag by 4-6%.
How does the lift curve slope of a finite three-dimensional wing compare to a two-dimensional airfoil?
The lift curve slope of a 3D wing is always lower than the 2D section slope a_0: a = a_0 / (1 + a_0 / (pi * e * AR)). Induced downwash reduces the effective angle of attack alpha_eff = alpha - alpha_i, requiring a higher geometric pitch to achieve the same lift.
What happens to induced drag when an aircraft flies in close ground effect (h / b < 0.2)?
The proximity of the solid ground plane blocks the downward deflection of streamlines and impedes wingtip vortex development. This suppresses downwash, dramatically reducing induced drag and boosting effective lift-to-drag ratio.
Experimental precautions and wind tunnel safety
- Verify that model span does not exceed 75% of test section width to prevent severe solid wall boundary constraint on tip vortices.
- Apply Glauert wind tunnel wall corrections to measured lift and induced drag coefficients for each aspect ratio.
- Ensure the junction between the wing root and mounting balance is aerodynamically faired to eliminate parasitic root separation.
- Keep lift coefficient fixed at exactly C_L = 0.60 for each wing to isolate aspect ratio as the sole independent variable.
Theoretical assumptions & flow limits
- Prandtl lifting-line theory assumes unswept, high-aspect-ratio wings (AR > 4.0); small non-linearities appear at AR = 2.0.
- Square cut-off wingtips create complex secondary separation vortices that slightly reduce Oswald efficiency e.
- Wind tunnel wall interference effects grow larger as wing span approaches tunnel width.
- Assumes steady trailing vortex filament roll-up without atmospheric turbulence dispersion.