Formula sheet

Aerodynamics Formula Sheet

Aerodynamics formulas for dynamic pressure, lift, drag, Reynolds number, Mach number, pressure coefficient, drag polar, induced drag, stall speed, and vortex shedding.

Branch: Aerospace Engineering
Content: Formula sheet
Updated: May 29, 2026
Revision: v1.0.4 · reviewed

This formula sheet collects common first-pass relationships used in introductory aerodynamics and aircraft performance estimates. Most formulas assume steady flow, consistent reference geometry, and a clearly defined freestream condition. They are useful for early sizing, sanity checks, wind-tunnel planning, and interpreting CFD or test data.

Always state reference area, reference length, coordinate system, sign convention, Reynolds number, Mach number, and configuration when reporting aerodynamic coefficients.

Dynamic pressure

Dynamic pressure:

\displaystyle q=\frac{1}{2}\rho V^2

where $\rho$ is fluid density and $V$ is freestream speed relative to the body.

Aerodynamic force coefficients generally use dynamic pressure:

\displaystyle C=\frac{\text{force or moment normalized quantity}}{q \times \text{reference quantity}}

Because $q$ scales with $V^2$ , aerodynamic loads increase rapidly with speed.

Lift

Lift equation:

L=qSC_L

Lift coefficient:

\displaystyle C_L=\frac{L}{qS}

Approximate linear lift curve before stall:

C_L \approx C_{L0}+C_{L_\alpha}\alpha

where $\alpha$ is angle of attack. Use radians when applying lift-curve slope values expressed per radian.

For steady level flight:

L\approx W

where $W$ is aircraft weight. This is an approximation; manoeuvres, climb, descent, acceleration, and gusts change the force balance.

Drag

Drag equation:

D=qSC_D

Drag coefficient:

\displaystyle C_D=\frac{D}{qS}

Simple parabolic drag polar:

C_D=C_{D0}+kC_L^2

where $C_{D0}$ is zero-lift drag and $kC_L^2$ represents induced drag in a simplified form.

Lift-to-drag ratio:

\displaystyle \frac{L}{D}=\frac{C_L}{C_D}

In steady level flight:

T\approx D

where $T$ is thrust required. This relation changes during climb, acceleration, and manoeuvre.

Induced drag

For a finite wing, a common induced-drag estimate is:

\displaystyle C_{D_i}=\frac{C_L^2}{\pi e AR}

where:

$e$ is Oswald efficiency factor;
$AR$ is aspect ratio.

Aspect ratio:

\displaystyle AR=\frac{b^2}{S}

where $b$ is wingspan and $S$ is wing reference area.

Induced drag is most important at high lift coefficient, such as takeoff, climb, turn, and slow flight. It is not the only drag source and should not be confused with total drag.

Stall speed

For steady level flight at maximum lift coefficient:

\displaystyle V_s=\sqrt{\frac{2W}{\rho S C_{L,max}}}

With load factor $n$ :

V_{s,n}=V_s\sqrt{n}

This explains why stall speed increases in a turn or manoeuvre with higher load factor. Real stall speed also depends on configuration, thrust effects, contamination, icing, gusts, Reynolds number, Mach number, and control inputs.

Reynolds number

Reynolds number:

\displaystyle Re=\frac{\rho V L}{\mu}

or:

\displaystyle Re=\frac{VL}{\nu}

where:

$L$ is characteristic length;
$\mu$ is dynamic viscosity;
$\nu$ is kinematic viscosity.

Reynolds number affects boundary-layer transition, separation, skin friction, stall, wake behaviour, and scale effects. Wind-tunnel tests should document how model Reynolds number compares with full-scale Reynolds number.

Mach number

Mach number:

\displaystyle M=\frac{V}{a}

Speed of sound for an ideal gas:

a=\sqrt{\gamma RT}

where $\gamma$ is ratio of specific heats, $R$ is gas constant, and $T$ is absolute temperature.

Mach number controls compressibility effects. Low-speed incompressible assumptions are often reasonable below about $M=0.3$ , but geometry and accuracy requirements matter.

Pressure coefficient

Pressure coefficient:

\displaystyle C_p=\frac{p-p_\infty}{q_\infty}

where $p$ is local static pressure, $p_\infty$ is freestream static pressure, and $q_\infty$ is freestream dynamic pressure.

For incompressible inviscid flow along a streamline, a simplified relation is:

\displaystyle C_p=1-\left(\frac{V}{V_\infty}\right)^2

This relation should not be applied blindly to viscous separated flow, compressible flow, shocks, or regions where the assumptions behind Bernoulli’s equation fail.

Moment coefficient

Pitching moment coefficient about a chosen reference point:

\displaystyle C_M=\frac{M}{qSc}

where $M$ is pitching moment and $c$ is reference chord.

Rolling and yawing moment coefficients commonly use wingspan:

\displaystyle C_l=\frac{L_{roll}}{qSb}

\displaystyle C_n=\frac{N}{qSb}

Moment coefficients are meaningful only when the reference point, axes, signs, and reference dimensions are stated.

Center of pressure

For a resultant aerodynamic force and moment about a reference point, the center of pressure location can be inferred from moment balance in simple two-dimensional cases. If lift dominates and pitching moment is measured about a reference point:

\displaystyle x_{cp}-x_{ref}=-\frac{M_{ref}}{L}

The sign depends on the moment convention. In practice, aerodynamic center and moment coefficients are often more useful than center of pressure because center of pressure can move strongly with angle of attack and become ill-defined when lift is small.

Vortex shedding

Strouhal number:

\displaystyle St=\frac{f_sD}{U}

Shedding frequency:

\displaystyle f_s=St\frac{U}{D}

where $f_s$ is shedding frequency, $D$ is characteristic body width, and $U$ is flow speed.

Vortex shedding can drive vibration if $f_s$ approaches a structural natural frequency. The Strouhal number depends on geometry, Reynolds number, surface roughness, blockage, and flow conditions.

Mass flow rate

For one-dimensional flow through an area:

\dot{m}=\rho VA

where $\dot{m}$ is mass flow rate, $V$ is average velocity normal to the area, and $A$ is flow area.

This is useful for inlets, ducts, wind tunnels, cooling flows, and propulsion estimates. Compressible or nonuniform flow may require area integration:

\dot{m}=\int_A \rho \mathbf{V}\cdot \mathbf{n}\,dA

Wind-tunnel scaling checks

A wind-tunnel test should usually document:

\displaystyle Re_{model}=\frac{\rho_m V_m L_m}{\mu_m}

and:

\displaystyle M_{model}=\frac{V_m}{a_m}

Compare with full-scale values:

\displaystyle Re_{full}=\frac{\rho_f V_f L_f}{\mu_f}

\displaystyle M_{full}=\frac{V_f}{a_f}

Matching both Reynolds number and Mach number can be difficult in ordinary tunnels. If they cannot both be matched, the test plan should explain which effects are matched, corrected, or accepted as uncertainty.

Mini example: lift and stall speed

An aircraft has:

W=12\,000\ \text{N}

S=16\ \text{m}^2

C_{L,max}=1.6

At sea-level density:

\rho=1.225\ \text{kg/m}^3

Stall speed estimate:

\displaystyle V_s=\sqrt{\frac{2W}{\rho S C_{L,max}}}

Substitute:

\displaystyle V_s=\sqrt{\frac{2(12\,000)}{(1.225)(16)(1.6)}}

V_s=27.7\ \text{m/s}

At load factor $n=2$ :

V_{s,n}=27.7\sqrt{2}=39.2\ \text{m/s}

This is an idealized estimate. Real performance requires configuration, thrust, weight variation, manoeuvre margin, certification basis, airspeed calibration, and stall behaviour.

Mini example: drag estimate

Suppose:

\rho=1.0\ \text{kg/m}^3

V=80\ \text{m/s}

S=10\ \text{m}^2

C_D=0.035

Dynamic pressure:

\displaystyle q=\frac{1}{2}(1.0)(80)^2=3200\ \text{Pa}

Drag:

D=qSC_D=(3200)(10)(0.035)=1120\ \text{N}

If steady level flight is assumed, thrust required is approximately:

T\approx1120\ \text{N}

This estimate is only as good as the drag coefficient and reference condition. Configuration, Mach number, Reynolds number, trim, cooling flow, and interference effects can change the result.

Common cautions

Do not compare aerodynamic coefficients without checking reference area and coordinate convention. Do not use low-speed incompressible formulas when compressibility, shocks, or wave drag matter. Do not treat two-dimensional airfoil coefficients as full-aircraft data. Do not assume wind-tunnel results are full scale without Reynolds number, Mach number, blockage, support, and transition checks. Do not use CFD results without mesh, model, boundary-condition, and validation evidence.

REF

Disciplines

Aerodynamics Formula Sheet

Dynamic pressure

Lift

Drag

Induced drag

Stall speed

Reynolds number

Mach number

Pressure coefficient

Moment coefficient

Center of pressure

Vortex shedding

Mass flow rate

Wind-tunnel scaling checks

Mini example: lift and stall speed

Mini example: drag estimate

Common cautions

See also