Subsonic Flow at High Mach Numbers

Since weak pressure disturbances propagate at the speed of sound, the time that a fluid particle ahead of a moving body is influenced by the pressure field around the body is proportional to the difference between the acoustic velocity, a, and the speed of the body, V. As V increases to a (i. e., as М» approaches unity), the fluid is displaced less and less ahead of the body. Thus the streamline pattern around an airfoil and hence its pressure distribution can be expected to change with M„, even though the flow is subsonic everywhere.

As long as the flow remains entirely subsonic, the effect of M* on airfoil characteristics can be estimated by the use of a factor, p, where /3 is defined as

p = Vl-Mo»2 (5.1)

P is known as the Prandtl-Glauert compressibility correction factor.

In a later, more complete treatment of p, it will be noted that the local pressure coefficient at a given point on an airfoil in subsonic compressible flow, CPc, is related to the pressure coefficient in incompressible flow, CPi, by

u5k и	(5.2)
It can easily be shown that
Ci — fl (Cpt — CPu) dx Jo and	(5.3)
Cm, = — J X(Cp, — Cpu) dx Jo	(5.4)

where x is the dimensionless distance along the airfoil chord and the sub­scripts l and и refer to lower and upper surfaces, respectively. Thus, it follows from Equations 5.2, 5.3, and 5.4 that the lift and moment coefficients for compressible flow are related to those for incompressible flow in a manner similar to Equation 5.2.

, =9l

ІС p

Cm,

P

Notice from the use of Equations 3.11 and 3.12 that neither the center of pressure, Xcp, nor the location of the aerodynamic center, Xac, varies with Mach number in the purely subsonic regime.

Obviously, the lift curve slope, Cio, also obeys Equation 5.5.

(5.7)

This relationship is presented graphically in Figure 5.1 together with the corresponding supersonic relationship, which will be discussed later. However, it must be used with caution. First, the theoretical basis on which it rests is valid only f<?r Mach numbers less than critical. Second, by comparison with experiment, the ratio QJG, is overestimated by Equation 5.5 in some cases and underestimated for others, depending on the airfoil geometry.

Reference 5.4 presents data on nine different airfoils at Mach numbers up to 1.0. These airfoils vary in thickness, design lift coefficient, and thickness distribution; they are illustrated in Figure 5.2. Pressure distribution*’ measurements were made in order to determine lift and pitching moment, and wake surveys were taken for determination of drag. Unfortunately, it is difficult to generalize on the data, and they are too voluminous to present here. A sample of the data is presented in Figure 5.3 for the 64A009 airfoil (taken from Ref. 5.4). The normal force coefficient, C„, is defined as the force normal to the chord line (obtained by integrating the normal pressure around the airfoil contour) divided by the product of the free-stream dynamic

d

c

Figure 5.2 Airfoil profiles.

pressure and the airfoil chord. Cd is the usual drag coefficient and is com­posed of the skin friction drag and the component of C„ in the drag direction. The lift coefficient is slightly less than C„ and can be obtained from

Q = C„ cos a (5.8)

Estimated critical Mach numbers are indicated by arrows in Figure 5.3 and were obtained from calculated graphs found in Reference 3.13. An example of such a graph is presented in Figure 5.4a and 5.4b. The results of Figure 5.4a apply approximately to the airfoils of Figure 5.3 and were used to obtain the Mcr values shown there. The 64Axxx airfoils are similar to the 64-xxx airfoils except that the rear portion of the 64Axxx airfoils are less curved than the corresponding surfaces of the 64-xxx airfoils.

Observe that the thinner symmetrical airfoils, as one might guess, have the higher critical Mach numbers at а С/ of zero. However, the rate at which M„ decreases with С/ is greater for the thinner airfoils. Thus, the thicker airfoils become relatively more favorable as C increases. As shown in Figure 5.4b, camber results in shifting the peak Mcr to the right. As a function of

thickness, the curves for the cambered airfoils are similar in appearance to those for the symmetrical airfoils.

It can be seen from Figure 5.3 that Equation 5.5 holds in a qualitative sense. At a given angle of attack, the lift coefficient increases with Mach number; however, the increase is not as great as Equation 5.5 predicts. For example, at an angle of attack of 6° and a Mach number of 0.3, C, is equal to 0.51. Therefore, at this same angle of attack, one would predict a C( of 0.61 at a Mach number of 0.6. Experimeiitally, however, Q equals only 0.57 at the higher M value.

Figure 5.5, also based on the data of Reference 5.4, presents the variation with Mach number of the slope of the normal force coefficient curve for 4, 6, 9, and 12% thick airfoils. The theoretical variation of C„a with M, matched to the experiment at an M of 0.3, is also included. Again, the Glauert correction is seen to be too high by comparison to the experimental results. Contrary to these observations, Reference 5.2 states that Equation 5.2 underestimates the effect of Mach number and presents a comparison between theory and experiment for a 4412 airfoil to substantiate the statement.

Reference 5.6 presents a graph similar to Figure 5.5 for symmetrical airfoils varying in thickness from 6 to 18%. The results are somewhat similar except that, at the lower Mach numbers, below approximately 0.8, the trend of C/a with thickness is reversed. Both graphs show Qa continuing to increase with a Mach number above the critical Mach number. Unlike Figure 5.5, the

Figure 5.5 Effect of Mach number on the slope of the normal force coefficient angle-of-attack curve (C„ = 0).

results presented in Reference 5.6 show a closer agreement with the Prandtl – Glauert factor for the lower thickness ratios.

Reference 5.5 is a voluminous collection of data pertaining to aircraft and missiles. Subsonic and supersonic data are given for airfoil sections, wings, bodies, and wing-body combinations. Any practicing aeronautical engineer should be aware of its existence and have access to the wealth of material contained therein. In Section 4 of this reference, the Prandtl-Glauert factor is used up to the critical Mach number. Isolated examples given in this reference using j8 show reasonably good agreement with test results. Thus, in the absence of pliable data, it is recommended that the Prandtl-Glauert com­pressibility correction be used, but with caution, keeping in mind dis­crepancies such as those shown in Figure 5.5.

From Figures 5.3 and 5.5 it is interesting to note that nothing drastic happens to the lift or drag when the critical Mach number is attained. Indeed, the lift appears to increase at a faster rate with Mach number for M values higher than M„. Only when M„ is exceeded by as much as 0.2 to 0.4 does the normal force coefficient drop suddenly with increasing M„. The same general behavior is observed for Cd, except that the increments in above Mcr where the Cd curves suddenly bend upward are somewhat less than those for the breaks in the G, curves.

The value of М» above which Q increases rapidly with Mach number is known as the drag-divergence Mach number. A reliable determination of this number is of obvious importance in estimating the performance of an airplane such as a jet transport, designed to operate at high subsonic Mach numbers.