Two-dimensional and three-dimensional airflow
Overview
Two- and three-dimensional airflow explains why a real wing performs differently from a two-dimensional section. The exam tests tip vortices, downwash, aspect ratio, induced drag and spanwise flow.
Core rules
- In 2-D flow the streamlines stay in one plane; C_L depends on AoA and section shape only.
- On a finite wing, higher pressure below spills around the tips to the lower-pressure upper surface, creating tip vortices and downwash.
- Downwash tilts the local relative airflow downward, reducing the effective AoA and rotating the lift vector rearward; the rearward component is induced drag.
- Induced drag D_i = L²/(½ρV²πb²e), or C_Di = C_L²/(πARe). It is proportional to 1/AR and to 1/V².
- Aspect ratio AR = b²/S. Gliders use high AR to cut induced drag; fighters accept low AR for structural and roll-rate reasons.
- Washout (reduced tip incidence) delays tip stall; winglets and endplates weaken tip vortices.
Worked example
Two wings share the same area and C_L but AR = 6 and AR = 9, e = 0.8. Since D_i ∝ 1/AR, the AR = 9 wing has 6/9 = 0.67 of the induced drag of the AR = 6 wing - a 33 % reduction for the same lift.
Traps
- Induced drag is not constant; it falls as speed rises and dominates at low speed.
- High AR reduces induced drag but increases bending loads and reduces roll rate.
- Ground effect reduces induced drag by suppressing tip vortices; it does not change parasite drag much.