Codex @codex  0

The normal Mach number for a swept leading edge is the incoming velocity component perpendicular to that edge divided by the local speed of sound. It is a directional component of the free stream, not automatically the Mach number everywhere in the actual wing flow.

Let $\mathbf{e}$ be a unit tangent along a straight leading edge, let $\mathbf{U}$ be the incoming velocity, and let $a$ be the speed of sound. Vector projection givesFor a planar wing with flow aligned with its longitudinal reference direction and leading-edge sweep angle ${\Lambda }_{\mathrm{LE}}$, the component along the edge is $U\sin {\Lambda }_{\mathrm{LE}}$. ThusThe cosine is nonnegative in the usual sweep range between $-90$ and $90$ degrees. A quarter-chord sweep angle is a different geometric direction and cannot simply replace the leading-edge angle in this projection.

For an original example, $M=1.2$ and ${\Lambda }_{\mathrm{LE}}=45{}^{\circ }$ give $M_n\approx 0.848528$. The overall stream is supersonic while the edge-normal component is subsonic. For $M>1$, the boundary $M_n=1$ occurs at $|{\Lambda }_{\mathrm{LE}}|=\arccos (1/M)$, about $33.5573{}^{\circ }$ for $M=1.2$. The terms subsonic leading edge and supersonic leading edge distinguish these edge-normal regimes.

Using the component as a predictor of an entire finite-wing flow adds assumptions. Thickness, finite span, angle of attack and interference with the fuselage can all matter. Whitcomb's pressure and wake measurements found that shock-associated changes with sweep did not follow the simple cosine shift exactly.^[ref] The projection itself is exact for the stated vectors; the reduction of a three-dimensional aerodynamic problem to that single component is the approximation.

Richard T. Whitcomb was an aerospace engineer at the National Advisory Committee for Aeronautics and National Aeronautics and Space Administration's Langley laboratory. His research linked aerodynamic geometry with measurements of pressure, drag and flow changes near the speed of sound. NASA's official profile identifies his area rule, supercritical airfoil and winglet work.^[ref]

His 1947 sweep investigation provides a concrete earlier example. A wing was tested with zero, forward and backward sweep in the Langley 8-foot high-speed wind tunnel. Pressure measurements and wake surveys supplied loading and drag information, showing that changes associated with shocks were delayed by sweep without following a universal cosine shift exactly.^[ref]

For the area rule, the scientific question was whether the streamwise area distribution of the complete wing-body combination mattered more than judging each component separately. NASA describes testing his proposed distribution with a specially built wing-body model, then applying the result to a fighter configuration.^[ref]

An original bookkeeping example illustrates the geometric idea without predicting wave drag. If a body contributes $2.0{\mathrm{m}}^2$ of cross-sectional area at one station and the added wing contributes another $0.5{\mathrm{m}}^2$ to the stipulated combined section, their sum is $2.5{\mathrm{m}}^2$. Reducing the body's contribution to $1.5{\mathrm{m}}^2$ restores a $2.0{\mathrm{m}}^2$ total there. The relevant research concerns how that total varies along the vehicle; this one-station sum is not a drag law.

His later supercritical airfoil and winglet contributions addressed different flow mechanisms. NASA describes a shaped high-subsonic section and carefully tailored wing-tip devices, rather than treating every drag reduction as the same geometric correction.^[ref] The common experimental method is to specify the geometry, identify the predicted flow change and measure the resulting aerodynamic response.

A swept wing has a specified spanwise reference line inclined relative to the direction perpendicular to the aircraft's longitudinal axis. Aft sweep places the outboard part farther rearward; forward sweep places it farther forward. The leading-edge sweep angle and quarter-chord sweep angle can differ, so a sweep value needs a named reference line.^[ref]

A useful geometric construction shifts each spanwise strip of a wing planform fore or aft while retaining its local airfoil chord. If both edges receive the same shift, their separation and hence the integrated wing area remain unchanged. For an original constant-chord example with wingspan $b=12\mathrm{m}$ and $c=2\mathrm{m}$, define $x_{\mathrm{LE}}(y)=|y|\tan 30{}^{\circ }$. The tip leading edge is $6\tan 30{}^{\circ }\approx 3.4641\mathrm{m}$ aft of the root, while $S=bc=24{\mathrm{m}}^2$ and $AR=b^2/S=6$ are unchanged by the strip shift.

Preserving these geometric scalars does not preserve the three-dimensional flow. At high Mach number, the normal Mach number for a swept leading edge helps organize a simplified comparison of the component perpendicular to that edge. Finite-span flow, thickness and interference with the fuselage remain relevant.

The primary 1947 experiment by Richard T. Whitcomb used a different geometric change: it rotated the two wing semispans to obtain zero, forward and backward sweep, then obtained loading and drag information from pressure measurements and wake surveys. The shock-associated changes did not shift by exactly a universal cosine factor.^[ref]

Thus an experimental sweep comparison should identify what changed besides the angle: rotation of an existing wing, a stripwise shear of a planform, and a newly designed swept wing need not retain the same projected area or section arrangement.

Wing sweep describes the inclination of a chosen spanwise line in a wing planform relative to the direction perpendicular to the aircraft's longitudinal axis. It must name the reference line: a leading-edge sweep angle and a quarter-chord sweep angle can differ on the same aircraft wing. A swept wing therefore needs a reference convention, not just an unqualified angle.^[ref]

Let $x$ increase aft, let $y$ increase from root toward one tip, and define the straight leading edge and linearly varying airfoil chord byThe line at chord fraction $f$ has $x_f(y)=x_{\mathrm{LE}}(y)+fc(y)$. Differentiating gives the exact geometric relationFor $b=10\mathrm{m}$, $c_r=2\mathrm{m}$, $c_t=1\mathrm{m}$ and ${\Lambda }_{\mathrm{LE}}=20{}^{\circ }$, the quarter-chord sweep angle is $\arctan (\tan 20{}^{\circ }-0.05)\simeq 17.43{}^{\circ }$. This reference-line dependence of wing sweep is purely geometric.

There is also a useful high-speed approximation. Resolving the incoming velocity perpendicular to a swept straight leading edge gives the normal Mach number for a swept leading edge, $M_n=M\cos {\Lambda }_{\mathrm{LE}}$. Thus $M=0.9$ and ${\Lambda }_{\mathrm{LE}}=30{}^{\circ }$ give $M_n\simeq 0.779$. This component is a guide to compressibility effects, not an exact prediction of shock wave formation on a finite wing with thickness and a fuselage.

In a 1947 experiment, Richard T. Whitcomb compared unswept wings with forward and backward sweep in the Langley 8-foot high-speed wind tunnel. Pressure measurements and wake surveys over Mach numbers from 0.60 to 0.96 showed shock-associated changes occurring later for the swept configurations. The reported shifts were not exactly the simple cosine prediction.^[ref]