Aspect ratio — effect on induced drag — Physics notes

WHAT is being asked

WHY it matters: In cruise, induced drag is often 30–40% of total drag. Cutting it via geometry directly cuts fuel burn — which is why sailplanes and high-altitude drones have enormous wingspans.

WHY does a finite wing make drag? (first principles)

Figure — Aspect ratio — effect on induced drag

HOW to derive the induced-drag formula

Formula Step-by-step derivation

Step 1 — Geometry of the tilt. The local lift $L$ is rotated by the induced angle $\alpha_i$ . Its backward (drag) component: $D_i = L\sin\alpha_i \approx L\,\alpha_i \quad(\alpha_i \text{ small})$ Why this step? Drag is always along the freestream; the lift vector's projection onto that direction is the induced drag.

Step 2 — Express in coefficients. Divide by $q S$ where $q=\tfrac12\rho V^2$ : $C_{D,i} = C_L\,\alpha_i$ Why? Coefficients strip out $\rho, V, S$ so geometry stands alone.

Step 3 — Find $\alpha_i$ from lifting-line theory. For an elliptical lift distribution the downwash is uniform across the span, and Prandtl's lifting-line result gives $\alpha_i = \frac{C_L}{\pi AR}$ Why elliptical? It is the distribution that, for a given lift, makes downwash constant — and constant downwash is the minimum-drag case (proved by calculus of variations). It is the natural benchmark.

Step 4 — Combine Steps 2 and 3. $\boxed{C_{D,i} = \frac{C_L^2}{\pi\, AR}}$ Why this is the headline: induced drag $\propto C_L^2$ (so it explodes at low speed / high lift) and $\propto 1/AR$ (so a long thin wing kills it).

Step 5 — Real (non-elliptical) wings. Introduce the span efficiency $e\le 1$ : $C_{D,i} = \frac{C_L^2}{\pi\, e\, AR}$ Elliptical wing: $e=1$ . Rectangular: $e\approx 0.7$ – $0.9$ .

Worked examples

Common mistakes

Flashcards

Define aspect ratio in terms of span and area

AR=b^2/S

(and

=b/c

for constant chord)

Induced-drag coefficient (elliptical wing)

C_{D,i}=C_L^2/(\pi AR)

Full induced-drag coefficient with efficiency

C_{D,i}=C_L^2/(\pi e AR)

Why is induced drag zero for an infinite wing?

No tips ⇒ no trailing vortices ⇒ no downwash ⇒ lift stays vertical

What is downwash and what does it do?

Downward velocity from tip vortices that tilts local flow down by

\alpha_i

, leaning lift backward

Induced angle for elliptical loading

\alpha_i=C_L/(\pi AR)

If you double AR at fixed

C_L,e

, induced drag changes how?

Halves (∝ 1/AR)

C_L

triples, induced drag changes how?

×9 (∝

C_L^2

)

Span efficiency

e

of an elliptical wing

e=1

(the optimum)

Why isn't high AR always chosen?

Extra wetted/parasite drag, weight, flutter — optimum balances induced vs parasite drag

Is induced drag viscous or inviscid?

Inviscid — energy stored in trailing vortices

Recall Feynman: explain to a 12-year-old

Imagine pushing a wide spoon flat through bathtub water. Near the edges of the spoon, water curls around and makes little whirlpools that trail behind. Making those whirlpools takes energy, and that energy is a kind of "drag" pulling the spoon back. If you use a long skinny ruler instead of a wide spoon, only the two tiny ends make whirlpools — much less of the ruler wastes energy. That's why airplanes that want to glide far (gliders) have very long, thin wings: long-and-thin = fewer wasted whirlpools = less drag.

Connections

Lifting-line theory (Prandtl) — origin of $\alpha_i=C_L/(\pi AR)$
Trailing vortices & downwash — physical cause
Drag polar — $C_D = C_{D,0} + C_L^2/(\pi e AR)$
Parasite drag — the competing term that caps useful $AR$
Wingtip devices (winglets) — raise effective $AR$ / $e$
Elliptical lift distribution — the minimum-drag benchmark
Glide ratio & L/D max — set by the induced/parasite balance

Concept Map

Hinglish (regional understanding)

Intuition Hinglish mein samjho

Dekho, jab wing finite hota hai (yaani tips hote hain), to neeche ka high pressure air tip ke around ghoom ke upar low pressure side me chala jaata hai. Isse trailing vortices ban'te hain jo peeche air ko neeche dhakelte hain — isko downwash kehte hain. Downwash ki wajah se wing ko aane wali hawa thodi neeche ki taraf tilt lagti hai, aur lift hamesha local flow ke perpendicular hota hai, to lift thoda peeche ki taraf jhuk jaata hai. Yeh peeche wala component hi induced drag hai. Important baat: yeh viscous friction nahi hai — yeh energy hai jo vortices me chhod di gayi.

Ab aspect ratio $AR = b^2/S$ — matlab wing kitna lamba aur patla hai. Long-thin wing (glider) me sirf chhote tips problem karte hain, baaki poora wing clean lift deta hai. Stubby wing (fighter) me tip-effect poore wing ko kharab karta hai. Formula yaad rakho: $C_{D,i} = C_L^2/(\pi e AR)$ . Do cheezein notice karo — $C_L^2$ upar hai (slow flight me $C_L$ zyada chahiye, to induced drag phat jaata hai, 3x lift = 9x drag), aur $AR$ neeche hai (AR double karo to induced drag aadha).

Lekin trap mat phaso: high AR hamesha best nahi. Lamba patla wing ka wetted area zyada, weight zyada, flutter risk zyada — yeh sab parasite drag aur structural problems badhate hain. Isliye real design me induced vs parasite drag ka balance karke optimum AR choose karte hain. Glider slow chalta hai, $C_L$ high, isliye high AR; fighter fast chalta hai, $C_L$ low, isliye low AR. Bas yahi core intuition hai — exam aur real engineering dono me kaam aata hai.

Aspect ratio — effect on induced drag

WHAT is being asked

WHY does a finite wing make drag? (first principles)

HOW to derive the induced-drag formula

Worked examples

Common mistakes

Flashcards

Connections

Concept Map

Hinglish (regional understanding)

Go deeper — visual, from zero

Test yourself — Compressible Flow & Aerodynamics

Connections