Trailing edge

The trailing edge is the rear region of an airfoil where the upper and lower surfaces approach or meet. The airfoil chord connects its chosen section reference to the airfoil leading edge. A mathematical profile may end at a sharp point, but a specified family can retain a small finite airfoil trailing-edge thickness.^[ref]

For the symmetric thickness envelope of a NACA four-digit airfoil, let $\xi =x/c$ and let $\eta$ be the maximum thickness-to-chord ratio. One standard form isHere $y_t$ is half the thickness, so the full terminal thickness is $2y_t(1)$. This polynomial leaves a finite rear gap.^[ref] Substitution givesThus the original example $c=0.20\mathrm{m}$, $\eta =0.12$ has $t_{\mathrm{TE}}=0.504\mathrm{mm}$. Replacing the final coefficient $0.1015$ by $0.1036$ makes the endpoint thickness zero. That is a different geometric specification, not a numerical rounding of the same profile.

For a sharp edge, the Kutta condition selects an attached-flow solution with smooth departure and finite rear-edge velocity, helping determine circulation and lift. It does not require fluid parcels passing above and below the airfoil to arrive simultaneously. For a finite-thickness edge, a model must also represent the rear region and its wake; simply imposing a sharp-edge condition does not remove that geometric difference.

Airfoil trailing-edge thickness measures the separation of the upper and lower surfaces at a specified terminal reference. A sharp ideal trailing edge has zero thickness. A finite-edge model needs both the endpoint convention and the measurement direction; these choices matter when comparing coordinates or manufacturing tolerances.

For directional measurement of trailing-edge thickness, choose terminal points ${\mathbf{r}}_u,{\mathbf{r}}_l$ and a unit normal vector $\mathbf{n}$. The reported projected gap isFor a NACA four-digit airfoil built with normal offsets, the full endpoint separation has length $2y_t(1)$. Its component normal to the airfoil chord is $2y_t(1)|\cos {\theta }_{\mathrm{TE}}|$, while their chordwise separation is $2y_t(1)|\sin {\theta }_{\mathrm{TE}}|$. Comparing a full normal gap to a projected gap silently changes the quantity being measured.^[ref]

For a symmetric section with $c=0.20\mathrm{m}$ and $\eta =0.12$, the standard polynomial gives $t_{\mathrm{TE}}=0.504\mathrm{mm}$ and $t_{\mathrm{TE}}/c=0.00252$. The same absolute $0.504\mathrm{mm}$ gap on a $0.40\mathrm{m}$ section would have only half that normalized thickness. Absolute and normalized tolerances answer different questions.

Measurement uncertainty also depends on shared coordinate errors. If signed endpoint heights give $t=y_u-y_l$, thenAn identical coordinate-origin shift cancels from the difference, whereas independent endpoint errors do not. Stipulated independent $10\mu \mathrm{m}$ standard uncertainties give $u_t=14.14\mu \mathrm{m}$. A common translation alone contributes zero. This is a covariance calculation, not evidence that every instrument error is common.

Directional measurement of trailing-edge thickness specifies which component of the gap between two terminal surface points is reported. A full endpoint separation, a chord-normal projection and a mean-line-normal thickness can be different quantities. The convention is part of the geometric measurement, not an adjustment to an aerodynamic result.

For selected endpoints ${\mathbf{r}}_u,{\mathbf{r}}_l$, write $\Delta \mathbf{r}={\mathbf{r}}_u-{\mathbf{r}}_l$ and choose a unit vector $\mathbf{n}$ in the measurement direction. ThenThe last inequality is the Cauchy-Schwarz inequality. Equality requires the measurement direction to align with the endpoint gap, apart from its sign. If that alignment is off by an angle $\varphi$, the reading is $g|\cos \varphi |$.

For an original coordinate example, let $\Delta \mathbf{r}=(0.30,0.40)\mathrm{mm}$. The full gap is $0.50\mathrm{mm}$; a vertical measurement gives $0.40\mathrm{mm}$ and a horizontal one gives $0.30\mathrm{mm}$. The direction $(3/5,4/5)$ recovers the full $0.50\mathrm{mm}$. A separate one-millimetre gap measured with $20{}^{\circ }$ misalignment gives $0.939693\mathrm{mm}$, about six percent low.

Changing coordinate frames is harmless when both the endpoints and direction are transformed consistently. For a rotation matrix $Q$, $(Q\Delta \mathbf{r})\cdot (Q\mathbf{n})=\Delta \mathbf{r}\cdot \mathbf{n}$. Using the page's vertical direction after rotating the profile, however, changes the measurement direction.

The NACA four-digit airfoil equations place the two terminal points on opposite sides of the airfoil mean camber line along its normal. When that mean line slopes at the endpoint, the gap has both chordwise and transverse components.^[ref] A profile-image or coordinate comparison should therefore identify the terminal-point pairing, the chord reference and the unit measurement direction. Interpolating both surfaces at the same horizontal coordinate is not automatically the same endpoint measurement.