Chord-normal aerodynamic force ($N$)

In the airfoil section plane, the chord-normal aerodynamic force $N$ is the component of an aerodynamic force perpendicular to the airfoil chord. The chord-axial aerodynamic force $A$ points along the chord, chosen positive aft. These body directions differ from the flow-relative lift and drag axes.^[ref]

Choose $z$ toward the upper surface and let the incoming flow direction have body components $(\cos \alpha ,\sin \alpha )$. Resolving the same vector in the two orthonormal bases gives the airfoil force-axis transformation,The rotation matrix preserves the squared Euclidean norm: $A^2+N^2=D^2+L^2$. An original example with $A=20\mathrm{N}$, $N=100\mathrm{N}$ and $\alpha =30{}^{\circ }$ gives $D\simeq 67.32\mathrm{N}$ and $L\simeq 76.60\mathrm{N}$. Calling the 100-newton normal reading lift would confuse two different components. The transformation is kinematic; it does not assert that a particular wing produces those loads at that angle.

For a pressure-only reconstruction, the chord-normal component is $N_p^{\prime }=\int (p_l-p_u)dx$ per unit span when upper and lower surfaces are graphs over the chord. The total $N$ can also include viscous stress. A force balance resolving body axes and a pressure reconstruction therefore need matched axes and a stated load decomposition before they are compared.

The airfoil force-axis transformation expresses a section-plane aerodynamic force in chord-fixed axes or flow-fixed axes. Choose $x$ aft along the airfoil chord, $z$ toward the upper surface, and a flow direction with body components $(\cos \alpha ,\sin \alpha )$. Taking positive lift along $(-\sin \alpha ,\cos \alpha )$ givesHere $A$ and $N$ are the chord-axial aerodynamic force and chord-normal aerodynamic force. The convention for angle and axis direction must be stated before applying the signs. NASA's definitions make drag parallel and lift perpendicular to the flow.^[ref]

The rotation matrix is orthogonal: $R^{T}R=I$. Its inverse is therefore its transpose, givingIt also preserves $A^2+N^2=D^2+L^2$, a useful numerical check on a conversion. The two component pairs need not individually agree.

An original conversion uses measured $D=10$ N and $L=100$ N at $\alpha =10{}^{\circ }$. It gives $A\simeq -7.517$ N and $N\simeq 100.217$ N. Rotating these values back recovers the original flow-axis loads. Setting $A=D$ because the angle appears small would miss the term $L\sin \alpha$, which is large compared with the small drag in this example.

Angle uncertainty can likewise mix the large lift component into inferred drag. Holding body-axis readings fixed,For the example, an angular error of $0.1{}^{\circ }\simeq 0.001745$ radians contributes approximately 0.175 N to inferred drag. This is a coordinate-sensitivity calculation, not a prediction of the aerodynamic change caused by physically rotating the model. Changing its actual incidence can change the underlying load as well as the axes.

The same rotation applies to the lift coefficient, drag coefficient, chord-normal force coefficient and chord-axial force coefficient only when they share the same pressure and area normalization.

The chord-normal force coefficient normalizes the chord-normal aerodynamic force:The reference dynamic pressure, wing area, body axis and sign convention accompany the coefficient. For sectional force per unit span, use $c_n=N^{\prime }/(qc)$ with airfoil chord $c$. Mixing total force and sectional normalization gives different units.

With stipulated $q=500\mathrm{Pa}$, $S=0.40{\mathrm{m}}^2$ and $N=100\mathrm{N}$, $C_N=0.50$. Reporting the same force with reference area $0.20{\mathrm{m}}^2$ gives $C_N=1.0$. This reference-area scaling of aerodynamic coefficients changes a reported number without changing the measured load.

When chord-axial force coefficient $C_A=A/(qS)$ uses the same references, the airfoil force-axis transformation gives
For $C_A=0.10$, $C_N=0.50$ and $\alpha =30{}^{\circ }$, the drag coefficient is approximately $0.3366$ and the lift coefficient is $0.3830$. Neither equals the normal coefficient.

Normalization does not remove measurement error. At fixed mass density and reference area, a small fractional speed error contributes approximately twice that fractional error to $q$, and hence the opposite fractional error to $C_N$ if $N$ is otherwise fixed. The usefulness of a dimensionless coefficient depends on retaining the measured references and their measurement uncertainty.