= 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.<Wind-tunnel Investigation of Effects of Rear Upper Surface Modification on an NASA Supercritical Airfoil by Charles D. Harris and James A. Blackwell, Jr.>[Introduction, Summary and Concluding Remarks]{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 $\xi=x/c$,
$$
\frac{dp}{dx}=\frac{q_\infty}{c}\frac{dC_p}{d\xi}.
$$
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 $600\,\mathrm{kPa/m}$. 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.<Wind-tunnel Investigation of Effects of Rear Upper Surface Modification on an NASA Supercritical Airfoil by Charles D. Harris and James A. Blackwell, Jr.>[Measurements and Reduction of Data, printed pp. 5–6]{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.
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