TRANSONIC AIRFOILS

An airfoil operating at high, but subsonic, Mach numbers is pictured in Figure 5.18. If the free-stream Mach number is sufficiently high, the local flow as it progresses back along the upper surface will reach a point where the local Mach is equal to, or greater than, unity. As the flow continues along the concave surface, a region of supersonic flow develops. However, as the flow

m. < i

approaches the trailing edge, it must eventually become subsonic again. As we saw in the preceding section, this can occur only through a shock wave.

If the compression to subsonic flow occurs before the trailing edge, as shown in Figure 5.18a, there is no deflection of the flow as it passes through the wave, so that the shock wave is normal to the flow. As the free-stream Mach number is increased, a similar transonic region will develop on the lower surface, as shown in Figure 5.18b. Immediately behind both normal shocks, the boundary layer will separate because of the strong positive pressure gradients. This results in a loss in lift and a sharp increase in the drag. Increasing still further will result in the shock waves on both the upper and lower surfaces moving to the trailing edge. Here they become oblique shocks in order to turn the flow from both surfaces in the free-stream direction. Because of the developing boundary layer, as the shocks move toward the trailing edge, they assume a bifurcated or A form, as shown. Here, within the boundary layer, compression begins initially through an oblique shock and continues through to a normal shock.

Increasing the Mach number also causes the sonic line, defining the forward extent of the supersonic flow, to move forward. This line, shown dashed in Figure 5.18, is a constant pressure surface along which M = 1.

A detailed treatment of analytical methods for predicting airfoil charac­teristics is beyond the scope of this text. However, an interesting aspect of transonic airfoil behavior, discussed in Reference 5.8, is the limiting Mach number concept, which also leads to a limit on pressure coefficients. This particular reference presents semiempirical methods for estimating two – dimensional and three-dimensional values of CL and CD through the transonic regime.

Combining Equations 5.35 and 5.17b, the pressure, p2, immediately downstream of the normal shock can be written in terms of upstream reservoir pressure, po, and the local Mach number just ahead of the normal shock.

(5.49)

This ratio reaches a maximum value at a Mach number denoted as the limiting Mach number, М, шь and given by,

= 1.483 (y = 1.4)

Laitone argues in Reference 5.9 that the normal shock will be positioned on the surface of a transonic airfoil at the location where the local Mach number equals the limiting Mach number, thus assuring the maximum positive pressure downstream of the shock wave.

This limit on the local Mach number leads to a minimum pressure coefficient that can be attained on an airfoil surface ahead of the shock wave. Cp is defined as

, _ P~P«.

p (1/2 )p„Vj

which can be written as

The ratio of the local pressure to the reservoir pressure, p0, is a function of the local M, according to Equation 5.17b, and decreases monotonically with

M. When the local, M reaches Мц, this ratio attains a minimum value of 0.279. Using Equation 5.17b also to relate the free-stream static pressure, Po„, to po, a limiting value for Cp is obtained as a function of the free-stream Mach number.

This relationship is presented graphically in Figure 5.19. The limiting value of Cp is seen to decrease rapidly in magnitude as М» increases. ^Pcr is also presented on this same figure and is a value of Cp necessary to achieve local sonic flow. The value of M« corresponding to G* is equal to Mcr, the critical Mach number. G, is obtained from Equation 5.51 by setting the local M equal to unity to obtain p/po – The result is identical to Equation 5.52 except for replacing the constant 0.279 by 0.528.

Before discussing the significance of these relationships, let us return to Equation 5.2, which allows us to predict Cp at subsonic Mach numbers based on predictions for incompressible flow. К CPc is the pressure coefficient at a given Mach number, the Prandtl-Glauert correction states that, for the same geometry, Cp at M = 0 will equal /3CPc. Using this scaling relationship, the critical value for the incompressible Cp can be calculated from the com­pressible CPa. This result is also presented in Figure 5.19.

As an example of the use of Figure 5.19, consider the Liebeck airfoil in Figure 3.43. The minimum Cp at M„ = 0 for this airfoil is approximately -2.8. Hence its critical Mach number is estimated from the lower curve of Figure 5.19 to be approximately 0.43. Its limiting Mach number, based on Cp = -2.8, would be 0.57. However, at this Mach number, the Prandtl-Glauert factor, /3, equals 0.82, so that the minimum Cp at this Mach number is estimated to be -3.4. A second iteration on Мім then gives a value of 0.52. Continuing this iterative procedure, a value of Мім = 0.53 is finally obtained.

Next, consider the chordwise pressure distributions presented in Figure

5.20. Here, Cp as a function of chordwise position is presented for the NACA 64A010 airfoil for free-stream Mach numbers of 0.31, 0.71, and 0.85, all at a constant angle of attack of 6.2°. Only the pressure distributions over the upper

0 0.2 0.4 0.6 0.8 1.0 Mach number, Figure 5.19 Limiting and critical pressure coefficients as a function of free – stream Mach number.

surface are shown. The critical Mach number for this airfoil, corresponding to the minimum Cp of approximately -3.0, is approximately Mcr= 0.43. Thus, at М» = 0.31, this airfoil is operating in the subsonic regime. At M„ = 0.71, the flow is transonic and theoretically limited to a Cp of -1.7. Near the nose this value is exceeded slightly. However, the experimental values of Cp are indeed nearly constant and equal to Cp,^, over the leading 30% of the chord. At between the 30 and 40% chord locations, a normal shock compresses the flow,

Figure 5.20 Pressure distributions for NACA 64A010 airfoil.

and the pressure rises over the after 60% to equal approximately the subsonic distribution of Cp over this region. Using the preceding relationships for isentropic flow, the pressure rise across a normal shock, and the limiting Mach number of 1.483, one would expect an increase in Cp of 1.56. The experiment shows a value of around 1.2. This smaller value may be the result of flow separation downstream of the shock.

The results at М» = 0.85 are somewhat similar. Over the forward 35% of the chord, Cp is nearly constant and approximately equal to Cp, imit. Behind the normal shock, at around the 40% chord location, the increase in Cp is only approximately 0.25 as compared to an expected increment of approximately

1.24. Here the separation after the shock is probably more pronounced, as evidenced by the negative Cp values all the way to the trailing edge.

Notice the appreciable reduction in the area under the Cp curve for Moo = 0.85 as compared to M„ = 0.71 because of the difference in the C^, values of the two Mach numbers. This limiting effect on Cp is certainly a contribution to the decrease in Q at the very high subsonic Mach numbers, the other major contributor being shock stall.

Considering the mixed flow in the transonic regime, results such as those shown in Figures 5.3, 5.5, and 5.20, and nonlinear effects such as shock stall and limiting Mach number, the prediction of wing and airfoil characteristics is a difficult task of questionable accuracy. Although the foregoing material may help to provide an understanding of transonic airfoil behavior, one will normally resort to experimental data to determine Q, Cd, and Cm accurately in this operating regime.