In addition to the wave drag, additional drags due to viscous and finite-span effects, considered in the preceding chapter, act on a real wing.ДЇigure 7-7. The magnitude of the supersonic wave drag is independent of the value of the viscosity, although the energy spent in overcoming this drag is finally dissipated through viscous effects within the shock waves. The supersonic wave drag is analogous to the gravity wave drag experienced by a ship moving at a speed greater than the velocity of surface gravity waves, in which a system of bow waves is carried with the ship. The dвЂ™Alembert paradox therefore does not apply in a supersonic flow. The drag in (15.70) is the wave drag experienced by a body in a supersonic stream, and exists even in an inviscid flow. The flow in FigureВ 15.27 does leave the trailing edge smoothly. However, in a supersonic flow it is not necessary to invoke the Kutta condition (discussed in the preceding chapter) to determine the magnitude of the circulation. Note that the flow in FigureВ 15.27 does have a circulation because the velocities at the upper and lower surfaces are parallel but have different magnitudes. The expression for the lift coefficient should be compared to the incompressible expression C L = 2 ПЂО± derived in the preceding chapter. These expressions do not hold at transonic speeds M в€ћ в†’ 1, when the process of linearization used here breaks down. It is, however, not important to distinguish between shock and expansion waves in FigureВ 15.27, because the linearized theory treats them the same way, except for the sign of the pressure changes they produce. The streamlines above the plate turn through an angle О± by expanding through an expansion fan, downstream of which they become parallel to the plate with a pressure p 2 p в€ћ. The streamlines ahead of these waves are straight. AtВ the leading edge there is a weak expansion fan above the top surface and a weak oblique shock below the bottom surface.

To see how the lift and drag of a thin body in a supersonic stream can be estimated, consider a flat plate inclined at a small angle О± to a horizontal stream ( FigureВ 15.27).

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