How do you calculate lift force?

Lift L = ½ρv²SC_L, where ρ is air density, v is airspeed, S is wing area, and C_L is the lift coefficient. C_L depends on the airfoil shape and angle of attack.

What is the Reynolds number?

Reynolds number Re = ρvL/μ predicts whether flow is laminar or turbulent. Below ~500,000 flow is typically laminar; above ~1,000,000 it is turbulent. It is dimensionless.

What is Bernoulli principle?

Bernoulli principle states that as fluid velocity increases, pressure decreases. The equation P + ½ρv² + ρgh = constant along a streamline. This explains how wings generate lift.

Mach number M = v/a is the ratio of airspeed to the local speed of sound. M 1 is supersonic, and M > 5 is hypersonic.

Aerodynamics Calculators

Calculate lift, drag, Reynolds number, Mach number, stall speed, and wind loads with CalcREX's aerodynamics toolkit. 8 calculators covering the core equations of fluid dynamics and flight mechanics - each with step-by-step explanations.

From aircraft design to structural wind analysis, these calculators handle the fundamental aerodynamic equations. Scroll down for in-depth topic guides with worked examples, airfoil diagrams, and links to calculators.

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How it works

Aerodynamic Constants

ρ sea level1.225 kg/m³

μ air (15°C)1.81×10⁻⁵ Pa·s

γ air1.4

R air287 J/(kg·K)

Speed of sound (15°C)340.3 m/s

1 knot0.5144 m/s

Aerodynamics - In-Depth Guides with Worked Examples

Master lift, drag, fluid flow, and flight mechanics - the physics of objects moving through air.

Lift & Drag - The Two Fundamental Aerodynamic Forces

How air flowing over a body creates forces perpendicular (lift) and parallel (drag) to the flow direction.

1 The Lift Equation

L = ½ρv²SC_L

ρ = air density (kg/m³), v = airspeed (m/s)S = wing planform area (m²)C_L = lift coefficient (depends on shape & angle of attack)

Airfoil cross-section: The curved upper surface forces air to travel faster (lower pressure), while slower flow beneath maintains higher pressure. The pressure difference creates an upward lift force. The angle of attack (α) amplifies this effect.

2 The Drag Equation

D = ½ρv²SC_D

C_D = drag coefficientTotal drag = parasite drag + induced dragInduced drag ∝ 1/v² (decreases with speed)

Sphere

C_D ≈ 0.47

Flat plate (⊥)

C_D ≈ 1.28

Streamlined body

C_D ≈ 0.04

Car

C_D ≈ 0.25–0.35

Cyclist

C_D ≈ 0.88

Clean aircraft wing

C_D ≈ 0.02–0.04

Worked Example 1

Calculating Lift for Level Flight

Problem: A Cessna 172 weighs 10,900 N, has a wing area of 16.2 m², and flies at 55 m/s at sea level (ρ = 1.225 kg/m³). What C_L is required for level flight?

Level flight: L = W

10900 = ½ × 1.225 × 55² × 16.2 × C_L

10900 = ½ × 1.225 × 3025 × 16.2 × C_L

10900 = 30,024 × C_L

C_L = 0.363

Answer: C_L = 0.363. A typical airfoil can produce C_L up to ~1.6, so 0.363 is comfortable cruise - plenty of margin before stall.
Try the Lift Force Calculator

Worked Example 2

Drag Force and Power at Highway Speed

Problem: A car has frontal area 2.2 m², C_D = 0.30, at 120 km/h (33.3 m/s). What is the aerodynamic drag force and power to overcome it?

Drag and power

D = ½ × 1.225 × 33.3² × 2.2 × 0.30

= ½ × 1.225 × 1108.9 × 0.66

D = 448.5 N

P = D × v = 448.5 × 33.3

P = 14,935 W ≈ 20 hp

Answer: 449 N drag, 20 hp to overcome it. At 160 km/h, drag nearly doubles to ~800 N and power triples to ~47 hp. Drag ∝ v² but power ∝ v³ - speed is expensive.
Try the Drag Force Calculator

Worked Example 3

Lift-to-Drag Ratio

Problem: A glider has L/D = 40 at best glide speed. From 1000 m altitude, how far can it glide (still air)?

Glide distance = altitude × L/D

Distance = 1000 × 40

= 40,000 m = 40 km

Every 1 m of altitude trades for 40 m of forward distance. Modern sailplanes achieve L/D of 60+.

Answer: 40 km. L/D is the single most important efficiency metric in aviation. A Boeing 737 has L/D ≈ 17; a Cessna ≈ 8; an F-16 ≈ 9. Higher is better for fuel economy and glide range.

Bernoulli's Principle - Pressure, Velocity & the Origin of Lift

The foundational relationship between fluid speed and pressure - and why it's both powerful and commonly misunderstood.

1 Bernoulli's Equation

P + ½ρv² + ρgh = constant

P = static pressure (Pa)½ρv² = dynamic pressure (kinetic energy/vol)ρgh = hydrostatic pressure (for height changes)

Key insight: Along a streamline, as velocity increases, pressure MUST decrease (and vice versa). This is conservation of energy applied to flowing fluids. Valid for incompressible, inviscid flow along a streamline.

Worked Example 1

Pressure Drop Over a Wing

Problem: Air approaches a wing at 60 m/s. Over the upper surface, it accelerates to 75 m/s. What is the pressure difference? (ρ = 1.225 kg/m³, neglect height changes)

Bernoulli between two points

P₁ + ½ρv₁² = P₂ + ½ρv₂²

ΔP = P₁ − P₂ = ½ρ(v₂² − v₁²)

= ½ × 1.225 × (75² − 60²)

= 0.6125 × (5625 − 3600)

ΔP = 1,240 Pa (≈ 1.2% of atmospheric)

Answer: 1,240 Pa lower on the upper surface. Over a 16 m² wing, this creates 1240 × 16 = 19,840 N of lift - enough for a light aircraft. Even tiny pressure differences over large areas produce large forces.
Try the Bernoulli Calculator

Worked Example 2

Pitot Tube - Measuring Airspeed

Problem: A pitot tube on an aircraft reads total pressure of 103,500 Pa. Static pressure is 101,325 Pa. Air density is 1.225 kg/m³. What is the indicated airspeed?

Dynamic pressure → velocity

q = P_total − P_static = 103,500 − 101,325 = 2,175 Pa

q = ½ρv² → v = √(2q/ρ)

v = √(2 × 2175 / 1.225)

v = 59.6 m/s ≈ 116 knots

Answer: 59.6 m/s (116 knots). This is exactly how airspeed indicators work - they measure the pressure difference between a forward-facing tube (total pressure) and a side port (static pressure).
Try the Dynamic Pressure Calculator

Reynolds Number - Laminar vs. Turbulent Flow

The dimensionless number that determines whether flow is smooth or chaotic - critical for all aerodynamic analysis.

1 Definition & Physical Meaning

Re = ρvL/μ = vL/ν

Re = inertial forces / viscous forcesν = μ/ρ = kinematic viscosity (m²/s)L = characteristic length (chord, diameter, etc.)

Laminar (Re < 5×10⁵)

Smooth, layered flow. Low skin friction but susceptible to separation. Think: honey flowing, or a model aircraft wing.

Turbulent (Re > 10⁶)

Chaotic, mixed flow. Higher friction but resists separation. Think: river rapids, or a full-size aircraft wing.

Boundary layer transition: Flow starts laminar near the leading edge and transitions to turbulent downstream. The turbulent boundary layer is thicker and has higher skin friction, but is more resistant to flow separation.

Worked Example 1

Reynolds Number for a Wing

Problem: A wing has a chord of 1.5 m and flies at 70 m/s at sea level (ρ = 1.225, μ = 1.81 × 10⁻⁵). Is the flow laminar or turbulent?

Calculate Re

Re = ρvL/μ = 1.225 × 70 × 1.5 / 1.81×10⁻⁵

Re = 7.1 × 10⁶ → Turbulent

Well above the transition threshold. Virtually all full-scale aircraft operate in turbulent flow regimes.

Answer: Re = 7.1 million (turbulent). Scale model testing must match Re for valid results - this is why wind tunnels use compressed air or special fluids.
Try the Reynolds Number Calculator

Worked Example 2

Scale Model Testing

Problem: You want to test a 1/10 scale model of the wing above in a wind tunnel at the same air conditions. What speed would give matching Reynolds number?

Match Re

Re_model = Re_full → ρv_m × L_m/μ = ρv_f × L_f/μ

v_m = v_f × (L_f/L_m) = 70 × 10

v_m = 700 m/s ≈ Mach 2!

Impractical - this is why wind tunnels use pressurized air (higher ρ) or accept Re mismatch and apply corrections.

Answer: 700 m/s - impossibly fast. This illustrates the "Reynolds number matching problem" in scale testing. Real solutions include cryogenic wind tunnels (dense cold air) or water tunnels (higher ρ and μ).

Mach Number & Compressible Flow - Supersonic & Shock Waves

When air speed approaches and exceeds the speed of sound - the physics changes dramatically.

1 Mach Number & Speed of Sound

M = v/a, where a = √(γRT)

γ = 1.4 (ratio of specific heats for air)R = 287 J/(kg·K), T in KelvinAt 15°C: a = 340.3 m/s = 1,225 km/h

M < 0.8

Subsonic (most prop aircraft)

0.8 < M < 1.2

Transonic (airliners: M ≈ 0.78–0.85)

1.2 < M < 5

Supersonic (fighter jets, Concorde: M 2.04)

M > 5

Hypersonic (re-entry vehicles, scramjets)

Worked Example 1

Mach Number at Altitude

Problem: A Boeing 787 cruises at 250 m/s at 11,000 m altitude where temperature is −56.5°C. What is its Mach number?

Speed of sound at altitude

T = −56.5 + 273.15 = 216.65 K

a = √(1.4 × 287 × 216.65) = √(87,066)

a = 295.1 m/s

M = 250 / 295.1

M = 0.847 (transonic)

Answer: Mach 0.847. The speed of sound drops from 340 m/s at sea level to 295 m/s at cruise altitude because it's colder. Airlines quote "Mach 0.85" rather than km/h because Mach accounts for this.
Try the Mach Number Calculator

Worked Example 2

Shock Wave Mach Cone Angle

Problem: A fighter jet flies at Mach 1.8. What is the half-angle of its Mach cone (shock wave angle)?

Mach cone formula

sin(μ) = 1/M = 1/1.8 = 0.5556

μ = arcsin(0.5556) = 33.7°

The Mach cone trailing behind the jet creates a sonic boom where it intersects the ground.

Answer: 33.7° half-angle. At Mach 1.0 the cone is 90° (flat - the shock is perpendicular). As speed increases, the cone narrows. At Mach 3, the angle is only 19.5°.

Airfoil Design - Shape, Stall & Performance

How wing cross-section geometry determines lift, drag, and stall characteristics.

1 Airfoil Geometry

Chord (c)

Leading edge to trailing edge

Camber

Curvature of the mean line (asymmetry)

Thickness (t/c)

Max thickness as % of chord (typically 12–18%)

Angle of attack (α)

Angle between chord line and freestream

2 Stall - The Lift Limit

V_stall = √(2W / ρSC_Lmax)

Stall occurs at critical angle of attack (~12–18°)Flow separates from upper surface → lift drops sharplyC_Lmax typically 1.2–1.6 (clean wing), up to 3.0 with flaps

Worked Example 1

Stall Speed Calculation

Problem: An aircraft weighs 15,000 N with wing area 20 m² and C_Lmax = 1.5. What is the stall speed at sea level?

Stall speed formula

V_s = √(2 × 15000 / (1.225 × 20 × 1.5))

= √(30000 / 36.75)

V_s = 28.6 m/s = 55.6 knots

Answer: 28.6 m/s (56 knots). With landing flaps (C_Lmax = 2.5), stall speed drops to 22.1 m/s (43 knots). This is why flaps are deployed for landing - they lower the stall speed.
Try the Stall Speed Calculator

Worked Example 2

Effect of Altitude on Stall Speed

Problem: The same aircraft flies at 6000 m altitude where ρ = 0.660 kg/m³. What is the new stall speed?

Same formula, different ρ

V_s = √(2 × 15000 / (0.660 × 20 × 1.5))

= √(30000 / 19.8)

V_s = 38.9 m/s = 75.6 knots (36% higher!)

Stall speed ratio: √(1.225/0.660) = 1.362 - stall speed increases as √(1/ρ).

Answer: 38.9 m/s (76 knots) - 36% higher than sea level. This is why aircraft must fly faster at altitude and why high-altitude airports have longer runways. Thin air = less lift at same speed.

Wind Engineering - Structural Loads, Buildings & Design

How wind creates forces on buildings, bridges, and structures - and how engineers design for safety.

1 Wind Pressure on Structures

F = ½ρv²C_DA

Same drag equation, applied to buildingsC_D for buildings ≈ 1.0–2.0Design codes use gust factors and exposure categories

v² means danger grows fast: Doubling wind speed from 50 km/h to 100 km/h quadruples the force. A Category 5 hurricane (250 km/h) exerts 25× more force than a 50 km/h breeze.

2 The Beaufort Scale (Selected)

Force 6: 39–49 km/h

Strong breeze - umbrellas difficult

Force 8: 62–74 km/h

Gale - twigs break, walking impeded

Force 10: 89–102 km/h

Storm - trees uprooted, structural damage

Force 12: >118 km/h

Hurricane - devastating damage

Worked Example 1

Wind Force on a Building Face

Problem: A building face is 15 m wide × 40 m tall. Wind speed is 130 km/h (36.1 m/s). C_D = 1.3, ρ = 1.225 kg/m³. What is the total wind force?

Calculate force

A = 15 × 40 = 600 m²

F = ½ × 1.225 × 36.1² × 1.3 × 600

= ½ × 1.225 × 1303.2 × 780

F = 623,000 N = 623 kN ≈ 63.5 tonnes force

Answer: 623 kN (63.5 tonnes) pushing on one face. This creates an enormous overturning moment at the base. The structural foundation must resist this - which is why skyscrapers have massive concrete cores.
Try the Wind Load Calculator

Worked Example 2

Solar Panel Wind Loading

Problem: A rooftop solar array has 20 panels, each 1.7 m × 1.0 m, tilted at 30°. Design wind speed is 90 km/h (25 m/s). C_D for tilted flat plate ≈ 1.2. Will the mounting bolts (rated 500 N each, 4 per panel) hold?

Force per panel

Projected area = 1.7 × 1.0 × sin(30°) = 0.85 m²

F_panel = ½ × 1.225 × 25² × 1.2 × 0.85

= 0.6125 × 625 × 1.02

F_panel = 390 N per panel

Bolt capacity: 4 × 500 = 2000 N

Safety factor = 2000/390 = 5.1× ✓

Answer: 390 N per panel with safety factor 5.1× - the bolts hold. However, engineers apply gust factors (typically 1.5–2.5×), which reduces the effective margin. Always use local building codes for real designs.
Try the Wind Load Calculator

Worked Example 3

Hurricane vs. Normal Wind Comparison

Problem: Compare the wind pressure at 50 km/h (normal), 120 km/h (strong storm), and 250 km/h (Cat 5 hurricane).

Dynamic pressure q = ½ρv²

50 km/h (13.9 m/s): q = ½ × 1.225 × 13.9² = 118 Pa

120 km/h (33.3 m/s): q = ½ × 1.225 × 33.3² = 680 Pa

250 km/h (69.4 m/s): q = ½ × 1.225 × 69.4² = 2,953 Pa

Ratio: 1× : 5.8× : 25×

Answer: Hurricane force is 25× stronger than a normal breeze. At 2,953 Pa over a 10 m² surface, that's 29,530 N (~3 tonnes). This is why hurricane building codes require impact-resistant windows and reinforced roofing.
Try the Dynamic Pressure Calculator