A fluid exerts a distributed stress over an airfoil surface. The complete aerodynamic loading is represented not only by a resultant force but also by a moment about a stated reference point.

Consider a two-dimensional section of chord $c$, analyzed per unit span. Let

• $p$ be the local pressure;
• $\tau_w$ be the local wall-shear magnitude;
• $\mathbf n$ be the outward unit normal from the body into the fluid;
• $\mathbf t$ be a chosen surface-tangent direction;
• $ds$ be an element of surface arc length.

The force of the fluid on the body per unit span is

$$\boxed{d\mathbf F'=(-p\mathbf n+\tau_w\mathbf t),ds},$$

with the sign of the shear term chosen so that $\tau_w\mathbf t$ points in the actual tangential traction direction.

Resultant aerodynamic force

Integrating around the section contour gives

$$\boxed{\mathbf R'=\oint_Cd\mathbf F'}.$$

Relative to the free-stream velocity, decompose this resultant into

• drag $D'$, parallel to the free stream;
• lift $L'$, perpendicular to the free stream.

The primes denote force per unit span.

Pressure is often the dominant contributor to lift. Wall shear can be much smaller locally than pressure while still making an important contribution to drag because pressure-drag contributions can strongly cancel on a streamlined body.

Pitching moment about a reference point

Choose a reference point with position $\mathbf r_{ref}$. The aerodynamic moment per unit span is

$$\boxed{\mathbf M'{ref} =\oint_C(\mathbf r-\mathbf r{ref})\times d\mathbf F'}.$$

For a two-dimensional airfoil, the relevant scalar component is the pitching moment. In this unit, positive pitching moment is defined as nose up.

Changing the reference point does not change $L'$ or $D'$, but it generally changes the reported pitching moment.

Thin-airfoil loading approximation

For a thin airfoil at small angle of attack, the pressure difference between lower and upper surfaces supplies most of the lift and pitching moment. Let

$$\Delta p(x)=p_{lower}(x)-p_{upper}(x).$$

Then, to leading order,

$$\boxed{L'\approx\int_0^c\Delta p(x),dx}.$$

About the leading edge, an upward force acting farther aft produces a nose-down contribution under the stated convention, so

$$\boxed{M'_{LE}\approx-\int_0^cx,\Delta p(x),dx}.$$

The lift is the area under the pressure-difference distribution; the pitching moment is its first moment about the chosen reference.

Shifting the moment reference

The general vector rule for changing the moment reference from point $A$ to point $B$ is

$$\boxed{\mathbf M'_B =\mathbf M'_A-(\mathbf r_B-\mathbf r_A)\times\mathbf R'}.$$

For two reference points on the chord, using the aerodynamic nose-up sign convention and the small-angle approximation in which the normal force is approximately lift,

$$\boxed{M'_B \approx M'_A+L'(x_B-x_A)}.$$

Thus moving the reference point aft increases the reported nose-up moment by the lift times the reference shift.

Nondimensional section coefficients

Define the free-stream dynamic pressure

$$\boxed{q_\infty=\frac12\rho U_\infty^2}.$$

For a two-dimensional section, use chord $c$ as the reference area per unit span and as the moment reference length:

$$\boxed{c_l=\frac{L'}{q_\infty c}},$$

$$\boxed{c_d=\frac{D'}{q_\infty c}},$$

$$\boxed{c_{m,ref}=\frac{M'{ref}}{q\infty c^2}}.$$

The reference-point transformation becomes

$$\boxed{c_{m,B} \approx c_{m,A}+c_l\frac{x_B-x_A}{c}}.$$

A moment coefficient is therefore incomplete unless its reference point and sign convention are known.

Worked example

Suppose a two-dimensional section has

$$c_l=0.80$$

and a pitching-moment coefficient about the leading edge

$$c_{m,LE}=-0.24.$$

What is the moment coefficient about the quarter-chord point?

Here

$$\frac{x_B-x_A}{c}=\frac14.$$

Therefore

$$c_{m,c/4} =c_{m,LE}+\frac14c_l$$

$$=-0.24+\frac14(0.80) =-0.24+0.20 =\boxed{-0.04}.$$

The physical aerodynamic loading has not changed. Only the point about which its rotational effect is reported has changed.

Force coefficients summarize the integrated aerodynamic force; the pitching-moment coefficient retains the equally important information about where the distributed loading acts relative to the chosen reference.