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are schematically shown in Fig. 1.45. The graph in which the lift coefficient is
plotted against the drag coe:fficient is called the drag polar. As the Mach number
increases, the drag polars shift to the right and bend forward because at a given
lift coefficient the drag coefficient will be higher as the Mach number increases.
    For a conventional Iow-speed airfoil with a well-rounded leading edge, a de-
tached bow shock wave is formed when the freestream Mach number is in the
supersonic range (Fig. 1.42e). Because of symmetry, a detached shock wave is
 normal to the fiow direction on the line of symmetry. Because the fiow behind the
normal shock wa've is always subsonic, there is a small patch of subsonic fiow
around the leading edge region. The subsonic fiow quickly accelerates to super-
 sonic fiow over the airfoil surface. Somewhere on the upper and lower surfaces, we
 have sonic lines as shown, The fiow over this type of airfoil is one of mixed sub-
 sonic and supersonic fiow. Furthermore, the wave drag is high because a detached
 bow shock wave is formed.
        One way of reducing the wave drag of the wing in supersonic fiow and avoiding
 the formation of a detached bow shock wave ahead of the airfoil is to use a sharp
 leading-edge airfoillike the diamond section ofFig. 1.40. An attached shock wave
REVIEW OF BASIC AERODYNAMIC PRINCIPLES                45
Fig.1.46   Schematic variation ofcrit.icaIMach number with tluckness ratio.
is formed at the leading edge and the flow everywhere is supersonic on the airfoil
                                                              .         -
surface as we have discussed earlier.    .
i.n.    Methods of Postponing Adverse Effects of Co
   There are a number of ways of delaying the adverse effects of compressibil-
ity to higher Mach numbers. This js of considerable importance to the airline
industry because"'even a modest rise in the cruise Mach number in the high sub-
sonic/transonic regime helps to cut down operating costs considerably. Some of
the most commonly used methods are discussed in the following sections.
      Thin airfo17s.    The thinner the airfoil, the smaller the peak local velocitjt on its
surface will be and the higher the critical Mach number wi,ll be as schematically
shown in Fig. 1.46. The use of thin airfoils pushes the critical-Mach number
and the drag divergence number further towards unity. However, a disadvantage
of using very thin airfoils is that their low-speed characteristics, especially the
stalling characteristics, are poor. As we have discussed earlier, thin airfoils exhibit
an abrupt loss oflift at stall, which leads to stability and control problems as we
shall study later in the text.
    Low-aspect ratio wings.   As we know, the lower the aspect ratio, the more
pronounced the indtrced flow effects will be and the lower the peak velocities
on the wing surface will be. As a result, the critical Mach number will increase
with a decrease in aspect ratio as shown schematically in Fig, 1.47. However, the
disadvantage oflow-aspect ratio wings is that they have higher induced drag and
 lower lift curve slopes.
     Supercritica/ a/rfo17s.    For a conventiortal airfoil at Mach numbers above the
critical Mach number, the local region of supersonic flow terminates in a strong
shock wave (Figs. 1.42c and 1.42d) that induces significant flow separation and a
steep rise in the drag coefficient. For a supercritical airfoil12.13 shownrin Fig. 1.48a,
the curvature ofthe middle region of the upper surface is substantially reduced with
46                 PERFORMANCE, STABILITY DYNAMICS, AND CONTROL
A
Fig.1.47   Schematic variation ofcrihcaIMach number with aspect ratio.
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