Off-design pressure recovery of a supercritical airfoil

Off-design pressure recovery of a supercritical airfoil describes how the surface pressure returns from a low-pressure region toward its downstream value when the supercritical airfoil operates away from its intended flow and loading condition. The changed recovery can move or strengthen a shock wave, create a second acceleration and compression region, or impose a different adverse pressure gradient on the viscous boundary layer. A contour optimized at one Mach number and incidence need not give the same behavior elsewhere.

A primary NASA investigation examined the second upper-surface velocity peak near the rear part of a supercritical section. It changed the rear upper contour to reduce that peak while keeping trailing-edge thickness fixed. Tests covered freestream Mach numbers $0.60$–$0.81$; the reported modification could improve an off-design feature while increasing drag at the design condition.^[ref] The observation is a tradeoff between tested contours and conditions, not a rule that every reduction of every secondary peak must increase drag.

The pressure measurement is a spatial profile. For $C_p=(p-p_{\infty })/q_{\infty }$ and dimensionless chord coordinate $\xi =x/c$,For an original example, a rise from $C_p=-0.8$ to $-0.3$ over $\Delta \xi =0.02$ at $q_{\infty }=12\mathrm{kPa}$ and $c=0.50\mathrm{m}$ gives $\Delta p=6\mathrm{kPa}$ over $10\mathrm{mm}$, an average gradient of $600kPa/m$. A pressure-tap spacing of $0.05c$ would be wider than that entire recovery region. Its measured difference could locate a large change only coarsely; it could not resolve the transition's detailed shape. This motivates pressure-gradient resolution across an airfoil shock.

The NASA apparatus used flush pressure orifices with denser coverage near strong gradients, together with a downstream rake measuring wake total and static pressures. Forces and moments were inferred by pressure integration, while drag was inferred separately from wake measurements.^[ref] A surface-pressure improvement and a lower integrated drag are separate observations and should be compared directly.

A proposed test holds the manufactured contour and transition treatment fixed while varying Mach number and angle of attack, then repeats for the modified contour. Where the scientific comparison requires equal lift, adjust incidence to match lift rather than comparing only equal geometric angles. Record surface pressure, wake drag and repeatability, and mark the transition strips and tunnel correction convention.

At transonic speed, an incompressible conversion from $C_p$ to local velocity cannot characterize a shock reliably. An abrupt measured recovery is consistent with a shock-associated compression, but sparse taps alone do not establish the instantaneous shock structure or the extent of separated flow. Complementary flow visualization or near-wall measurements can test those interpretations. The directly established result remains the measured pressure distribution and force or wake response under the reported conditions.