Compressibility Drag: MDiv
Since MDiv is 2 to 4% above Mcc (we shall see that the '2 to 4%' is dependent on wing sweepback angle), we can predict the drag rise Mach number, MDiv if we can predict Mcc. If we can identify the pressure drop or more conveniently the local pressure coefficient, Cp , required on an airfoil to accelerate the flow locally to exactly the speed of sound, measured or calculated crest pressures can be used to determine the freestream Mach numbers at which M= 1.0 at the crest. If p is the pressure at a point on an airfoil of an unswept wing, the pressure coefficient is
The Cp may be expressed in terms of the local and freestream Mach numbers. Under the assumption of adiabatic flow:
By definition, when local Mach number M= 1.0 , Cp = Cp*, the critical pressure coefficient. Thus,
Here is a simple calculator that provides Cp* given a value for freestream Mach number using these equations.
A graph of this equation is shown in figure 6. If the Cp at the crest is known, the value of M0 for which the speed of sound occurs at the crest can be immediately determined. The above discussion applies to unswept wings and must be modified for wings with sweepback.
Figure 6. Variation of Pressure Coefficient at the Crest on a Modern Peaky Airfoil, t/c = 0.104, Re - 14.5 Million
It will be noted from Figure 6 that the airfoil information required is Cpcrest versus M. In Figure 6, typical wind tunnel airfoil crest Cp variations with M are shown for several angles of attack. Mcc occurs when the Cpcrest versus M curve for a given angle of attack intersects the curve of Cp* versus M. A few percent above this speed, the abrupt drag rise will start at MDiv. The approximate relationship between MDiv and Mcc is given in the next section.
If the airfoil pressure distribution is calculated by one of various complex theoretical methods at M = 0, the value of the crest Cp can be plotted versus M0 using the Prandtl-Glauert approximation:
or the somewhat more involved Karman-Tsien relationship:
The value of Cp at the crest is an important design characteristic of high speed airfoils. In general, Cpcrest at a given CL is dependent upon the thickness ratio (ratio of the maximum airfoil thickness to the chord) and the shape of the airfoil contour.
We have been describing a method of predicting Mcc which is useful in evaluating a particular airfoil design and in understanding the nature of the process leading to the occurrence of significant additional drag on the wing. Often in an advanced design process the detailed airfoil pressure distribution is not available. The airfoil is probably not even selected. It is still possible to closely estimate the Mcc from Figure 7. This graph displays Mcc as a function of airfoil mean thickness ratio t/c and CL. It is based on studies of the Mcc of various airfoils representing the best state of the art for conventional 'Peaky' type airfoils typical of all existing late model transport aircraft. The significance of the term 'peaky' is discussed in the chapter on airfoils. Use of the chart assumes that the new aircraft will have a well developed peaky airfoil and that the upper surface of the wing is critical for compressibility drag rise. Implied in the latter assumption is a design that assures that elements other than the wing, i.e. fuselage, nacelles, etc., have a higher Mdiv than the wing. Up to design Mach numbers greater than .92 to .94 this is attainable. Furthermore, it is assumed that the lower surface of the wing is not critical. This assumption is always valid at the normal cruise lift coefficients but may not be true at substantially lower lift coefficients. Here the wing twist or washout designed to approach elliptical loading at cruise and to avoid first stalling at the wing tips, may lead to very low angles of attack on the outer wing panel. The highest Cpcrest may then occur on the lower surface, a condition not considered in developing figure 7. Thus the chart may give optimistic values of Mcc at lift coefficients more than 0.1 to 0.15 below the design cruise lift coefficients.
Figure 7 Crest Critical Mach Number vs. CL and t/c for a Family of Peaky Airfoil Sections
Figure 7 does not apply directly to airfoils with pressure distributions that look significantly different from the peaky airfoil family. Modern supercritical airfoils, discussed in later chapters, can achieve higher drag divergence Mach numbers than those suggested by the figure. Although the performance of such airfoil families is often a closely guarded company secret, the effect can be approximated by adding an increment to the value of Mcc shown in the figure. A very aggressive supercritical section might achieve a drag divergence Mach number increment of 0.06, while more typically the increment is 0.03 to 0.04 above the peaky sections.
The form below includes a fit to the above data:
A recent paper by Harbeck and Jameson applied optimal design to create a families of shock free transonic airfoils. These sections are not practical because many of them behave very badly when they are operated at all off design. Nevertheless, the results provide an upper bound of what is possible for transonic section thickness, CL, and Mach trade-offs. The figure below shows the results of this study. Also shown is a curve based on the peaky section Mcc results above, marked up by a factor of about 14%. Given that Mdiv is about 2% greater than Mcc for these sections, optimized shock free sections appear possible at Mach numbers about 12% higher than the peaky sections. More practical, yet still rather extreme supercritical sections are possible with about 8-10% higher Mach number than the peaky sections, which is consistent with the previous estimate of about a 0.06 Mach number improvement for advanced sections.
Figure 8. Shock-free airfoil designs (from Harbek and Jameson) and comparison with crest critical Mach number for peaky sections.
Note also, that it would not be necessary to fit result for a large family of airfoils. We could design a family of 10% thick sections at various CL's and then use transonic similarity to construct the variation. Figure 9 shows how the entire Mach vs. thickness curve at CL=0 can be computed from a single airfoil design result. It suggests that our fit may be a bit conservative for very thick sections. This approach may permit slightly better extrapolation of the transonic data, but is a bit more combersome to implement.
Figure 9. Comparison of simple curve fit and transonic similarity rule.