Equation (12.15) is very handy for estimating the lift and wave drag for thin supersonic airfoils, such as sketched in Figure 12.3. When applying Equation (12.15) to any surface, one can follow a formal sign convention for 9, which is different for regions of left-running waves (such as above the airfoil in Figure 12.3) than for regions of right-running waves (such as below the airfoil in Figure 12.3). This sign convention is
developed in detail in Reference 21. However, for our purpose here, there is no need to be concerned about the sign associated with в in Equation (12.15). Rather, keep in mind that when the surface is inclined into the freestream direction, linearized theory predicts a positive Cp. For example, points A and В in Figure 12.3 are on surfaces inclined into the freestream, and hence CPiA and CP. B are positive values given by
In contrast, when the surface is inclined away from the freestream direction, linearized theory predicts a negative Cp. For example, points C and D in Figure 12.3 are on surfaces inclined away from the freestream, and hence Cp c and C Ptp are negative values, given by
In the above expressions, в is always treated as a positive quantity, and the sign of Cp is determined simply by looking at the body and noting whether the surface is inclined into or away from the freestream.
With the distribution of Cp over the airfoil surface given by Equation (12.15), the lift and drag coefficients, c; and c(i, respectively, can be obtained from the integrals given by Equations (1.15) to (1.19).
Let us consider the simplest possible airfoil, namely, a flat plate at a small angle of attack a as shown in Figure 12.4. Looking at this picture, the lower surface of the
——————- c———————— «J
………………. ………… Ї….. ……………………
Figure 1 2.4 A flat plate at angle of attack in a supersonic flow.
plate is a compression surface inclined at the angle a into the freestrearn. and from Equation (12.15),
Since the surface inclination angle is constant along the entire lower surface, Cpj is a constant value over the lower surface. Similarly, the top surface is an expansion surface inclined at the angle a away from the freestrearn, and from Equation (12.15),
Cp u is constant over the upper surface. The normal force coefficient for the flat plate can be obtained from Equation (1.15):
cn = – f (Cpj — Cp u) dx [12.18]
Substituting Equations (12.16) and (12.17) into (12.18), we obtain
4a 1 Ґ 4a
Cn = . : – ax = , :
The axial force coefficient is given by Equation (1.16):
Q/ — ~ j (.Cp, u Epj’) dy c J LE
However, the flat plate has (theoretically) zero thickness. Hence, in Equation (12.20), dy = 0, and as a result, ca = 0. This is also clearly seen in Figure 12.4; the pressures act normal to the surface, and hence there is no component of the pressure force in the x direction. From Equations (1.18) and (1.19),
ci = cn cos a — ca sin a cd = c„ sin a + ca cos a
and, along with the assumption that a is small and hence cos a ~ 1 and sin a % a, we have
cd — c„a + ca [12.22]
Substituting Equation (12.19) and the fact that ca = 0 into Equations (12.21) and
(12.22) , we obtain
Equations (12.23) and (12.24) give the lift and wave-drag coefficients, respectively, for the supersonic flow over a flat plate. Keep in mind that they are results from linearized theory and therefore are valid only for small a.
For a thin airfoil of arbitrary shape at small angle of attack, linearized theory gives an expression for q identical to Equation (12.23); that is,
Cl = 7^=i
Within the approximation of linearized theory, q depends only on a and is independent of the airfoil shape and thickness. However, the same linearized theory gives a wave – drag coefficient in the form of
Cd = ^r=i(a2 + 8c + 8f)
where gc and g, are functions of the airfoil camber and thickness, respectively. For more details, see References 25 and 26.
Example 12.1 | Using linearized theory, calculate the lift and drag coefficients for a flat plate at a 5° angle of attack in a Mach 3 flow. Compare with the exact results obtained in Example 9.10.
a = 5° = 0.087 rad
From Equation (12.23),
From Equation (12.24),
The results calculated in Example 9.10 for the same problem are exact results, utilizing the exact oblique shock theory and the exact Prandtl-Meyer expansion-wave analysis. These results were
(exact results from Example 9.10
Note that, for the relatively small angle of attack of 5°, the linearized theory results are quite accurate—to within 1.6 percent.
Figure 12.5 Transonic area ruling for the F-16. Variation af normal cross-sectional area as a function of location along the fuselage axis.
1. Using the results of linearized theory, calculate the lift and wave-drag coefficients for an infinitely thin flat plate in a Mach 2.6 freestream at angles of attack of
(a) a = 5° (b)a = 15° (c) a = 30°
Compare these approximate results with those from the exact shock-expansion theory obtained in Problem 9.13. What can you conclude about the accuracy of linearized theory in this case?
2. For the conditions of Problem 12.1, calculate the pressures (in the form of p/Poo) on the top and bottom surfaces of the flat plate, using linearized theory. Compare these approximate results with those obtained from exact shock-expansion theory in Problem 9.13. Make some appropriate conclusions regarding the accuracy of linearized theory for the calculation of pressures.
3. Consider a diamond-wedge airfoil such as shown in Figure 9.24, with a half-angle є = 10°. The airfoil is at an angle of attack a = 15° to a Mach 3 freestream. Using linear theory, calculate the lift and wave-drag coefficients for the airfoil. Compare these approximate results with those from the exact shock-expansion theory obtained in Problem 9.14.