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National Advisory Committee for Aeronautics, Technical Notes - A Velocity Correction Formula for the Calculation of Transonic Mach Number Distributions Over Diamond Shaped Airfoils

A velocity-correction formula is proposed for the purpose of
calculating, from the known Mach number distribution for a diamond-
shaped airfoil at a stream Mach number of 1.0, Mach number distributions
on the same airfoil at speeds from a Mach number of about 0.8 to shocks
attachment Mach number. The time required to calculate these additional
Mach number distributions is small in comparison with the time required
by rigorous methods. The accuracy of the results for stream Mach numbers
near 1.0 is of the same order as the accuracy of the known Mach number
distribution. Mbreover, the results tend to become exact as the stream
Mach number is increased toward that for shock attachment. An expression
for the rate of change of local Mach number with stream Mach number is
derived and an explicit equation for the drag coefficient as a function
of stream Mach number and thickness ratio is given.

The pressure distribution for a diamond-shaped airfoil at a stream
Mach number of 1.0 has been calculated by Guderley and Yoshihara
(reference 1). Calculations for a similar airfoil at four speeds between
Mach number 1.0 and the shock-attachment Mach number have been performed
by Vincenti and Wagoner (reference 2). According to reference 2,
similar calculations have been made by Cole at slightLy subsonic speeds.
These rigorous results combined with reliable experimental results
provide sufficient information for checking the accuracy of an approximate
velocity-correction formula proposed for calculating Mach number distri-
butions on a diamond-shaped airfoil. The concepts involved should
facilitate the calculations for other shapes.

The Justification for the proposed velocity—correction formula is
based upon its good agreement with existing rigorous calculations.
Moreover, it is in accord with the general transonic similarity rule,
and with the special form of that rule for a stream Mach number of 1.0.
Also, it agrees with.Guderley's result that the local Mach numbers are
constant for small departures of the stream Mach number from 1.0.

The present paper gives the derivation of the method of calculation
and its application for the determination of the Mach number distributions
over a lO-percent-thick diamond-shaped airfoil in the Mach number range
0.80 g.Mm g 1.30. The results are compared with the earlier results
of'Vincenti and.Wagoner. >A brief discussion is given of the movement
of the shock wave along the airfoil surface as the stream Mach number
is decreased below 1.0.