V-n diagram and flight envelope

Why it matters

The V-n diagram is the contract between aerodynamicists, structural designers and pilots: it states, for every speed, the load factors the aircraft must be able to carry without permanent deformation. Airworthiness codes require it, structural sizing is done at its corners, and the speeds printed on the airspeed indicator (manoeuvring speed, never-exceed speed) come from it. It brings together stall, load factor and turn performance from the previous topics.

Key ideas

What is plotted. Load factor n = L/W on the vertical axis against equivalent airspeed V_E on the horizontal axis, for a given weight and configuration. Using EAS makes the aerodynamic boundaries independent of altitude (until Mach effects appear), because lift depends on ½ρ₀V_E².

Stall (aerodynamic) boundaries. The largest lift the wing can produce at a speed is ½ρ₀V_E²·S·C_L,max, so n_stall = ρ₀·V_E²·C_L,max / (2·W/S) — a parabola through the origin. The aircraft physically cannot exceed it: pulling harder simply stalls the wing. The negative side uses the most negative lift coefficient C_L,min (inverted flight or a push-over), giving a smaller parabola below the axis. The 1-g stall speed is where the positive parabola crosses n = 1: V_s = √(2(W/S)/(ρ₀·C_L,max)).

Structural limits.

• Positive limit load factor n₁ (e.g. about 2.5–3.8 for transports, 3.8 for normal-category light aircraft, 6 for aerobatic aircraft — take the value from the applicable airworthiness code).
• Negative limit load factor n₂ (e.g. −1 for transports, −0.4·n₁ for normal-category light aircraft).
• Ultimate load factor = 1.5 × limit load factor (factor of safety). At limit load there must be no permanent deformation; at ultimate load the structure must not fail (for a short time).

Speed limits.

• Design dive speed V_D: the vertical right-hand boundary; the highest speed the structure, flutter and control margins are cleared for. The never-exceed speed V_NE is set below it.
• Design cruise speed V_C: used for the gust cases.

Corner points.

• Point A — the positive stall parabola meets n₁. Its speed is the design manoeuvring speed V_A = V_s·√n₁. Below V_A a full, abrupt control input stalls the wing before it can overstress the structure; above V_A it can break the aircraft. V_A is also the corner speed for turning: maximum turn rate and minimum turn radius occur here.
• The negative stall parabola meets n₂ at V = V_s,neg·√|n₂|, where V_s,neg uses |C_L,min|. The envelope is closed by the lines n = n₁ from A to V_D, the vertical line at V_D, and n = n₂ back to the negative stall curve (codes allow a sloped line on the negative side between V_C and V_D).

Gust envelope. A sharp vertical gust of speed U changes the wing's angle of attack by about U/V and produces an increment Δn = K_g·ρ₀·U_de·V_E·a / (2·W/S), where a is the lift-curve slope (per radian), U_de is the design gust velocity (in EAS) and K_g is a gust alleviation factor depending on the mass ratio (take K_g and U_de from the airworthiness code). Gust lines radiate from n = 1 at V = 0. Light, low-wing-loading aircraft can see gust loads that exceed their manoeuvre limits, so the final design envelope is the combination of both.

Altitude and weight. In EAS the stall parabola does not change with altitude, but in TAS all speeds rise by 1/√σ. A heavier aircraft has a higher wing loading, so its stall parabola is flatter (higher V_s and V_A) and gust increments are smaller.

Formulas

n = L / W

n_stall = ρ₀·V_E²·C_L,max / (2·W/S) (positive stall boundary)

n_stall,neg = ρ₀·V_E²·C_L,min / (2·W/S) (negative stall boundary, C_L,min < 0)

V_s = √(2·(W/S) / (ρ₀·C_L,max))

V_A = V_s·√n₁ (design manoeuvring speed)

V_s(n) = V_s·√n (stall speed at load factor n)

n_ult = 1.5·n_limit

Δn = K_g·ρ₀·U_de·V_E·a / (2·W/S) (gust load factor increment)

Symbols: L lift, W weight (N); n load factor; V_E equivalent airspeed (m/s); ρ₀ = 1.225 kg/m³; S wing area (m²); W/S wing loading (N/m²); C_L,max, C_L,min maximum positive and negative lift coefficients; n₁, n₂ positive and negative limit load factors; V_s 1-g stall speed (EAS); V_A manoeuvring speed (m/s); U_de design gust velocity (m/s, EAS); a lift-curve slope (1/rad); K_g gust alleviation factor (from the code).

Worked examples

Example 1 (standard): corner points of a light-aircraft envelope. Given: W = 10 000 N, S = 16 m², C_L,max = 1.6, C_L,min = −0.8, n₁ = 3.8, n₂ = −0.4·n₁ = −1.52.

1. Wing loading W/S = 625 N/m².
2. V_s = √(2(W/S)/(ρ₀C_L,max)) = √(1250/1.96) = 25.3 m/s.
3. V_A = V_s√n₁ = 25.25 × √3.8 = 49.2 m/s (EAS).
4. Negative 1-g stall speed with |C_L,min| = 0.8: √(1250/0.98) = 35.7 m/s; the negative corner is at 35.71 × √1.52 = 44.0 m/s.
5. Ultimate load factors: 1.5 × 3.8 = 5.7 and 1.5 × (−1.52) = −2.28.

Example 2 (GATE level): gust load at cruise. Given: the same aircraft at V_E = 60 m/s meets a design gust U_de = 15.24 m/s; lift-curve slope a = 5.0 per rad; K_g = 0.80 (given from the code formula for this aircraft's mass ratio).

1. Δn = K_g·ρ₀·U_de·V_E·a/(2·W/S) = 0.80 × 1.225 × 15.24 × 60 × 5.0/(2 × 625).
2. Numerator = 4480.6; denominator = 1250; Δn = 3.58.
3. Up-gust: n = 1 + 3.58 = 4.58; down-gust: n = 1 − 3.58 = −2.58. Both exceed the manoeuvre limits (3.8 and −1.52), so for this light aircraft the gust case, not the manoeuvre case, sizes the wing at this speed.

Common mistakes

• Plotting the V-n diagram in TAS and then wondering why it changes with altitude; use EAS.
• Writing V_A = V_s·n₁ instead of V_s·√n₁.
• Treating the stall parabola as a structural limit: it is aerodynamic — the aircraft cannot exceed it, but it can exceed n₁.
• Forgetting the factor of safety: limit load is what the aircraft should ever see; ultimate = 1.5 × limit.
• Assuming flying below V_A protects against any combination of control inputs; it covers a single full input in one axis, not repeated or combined inputs.

For GATE AE

Expect numericals for stall speed, manoeuvring speed and the stall-limited load factor at a given speed, questions on the shape and corner points of the V-n diagram, limit versus ultimate loads, and gust-load-factor increments with given constants. Practise sketching the envelope from W/S, C_L,max, C_L,min, n₁, n₂ and V_D.

Quick check

1. What curve forms the positive stall boundary of the V-n diagram?
2. If V_s = 30 m/s and n₁ = 4, what is V_A?
3. What is the ultimate load factor for n₁ = 2.5?
4. Why is EAS used on the speed axis?

Answers: 1. A parabola, n ∝ V_E². 2. 60 m/s. 3. 3.75. 4. It makes the aerodynamic boundaries independent of altitude.