Mechanics of Fluids

340 Boundary layers, wakes and other shear layers
8.10 EFFECT OF COMPRESSIBILITY ON DRAG
Our study of drag has so far been concerned only with flow of a fluid in
which the density is constant throughout, and the drag coefficient has been a
function of Reynolds number only. However, when the velocity approaches
that of sound in the fluid, the drag coefficient becomes a function also of
Mach number, that is, the ratio of the fluid velocity (relative to the body) to
the velocity of sound in the fluid.
Drag is always the result of shear forces and pressure differences, but
when compressibility effects are significant the distribution of these quantities
round a given body differs appreciably from that in flow at constant
density. The abrupt rise of pressure that occurs across a shock wave (see
Section 11.5) is particularly important. This is not simply because the difference
of pressure between front and rear of the body is thereby affected. The
adverse pressure gradient produced by the shock wave thickens the boundary
layer on the surface, and encourages separation. The problem, however,
is complicated by the interaction between shock waves and the boundary
layer. Pressure changes cannot be propagated upstream in supersonic flow;
in the boundary layer, however, velocities close to the surface are subsonic,
so the pressure change across the shock wave can be conveyed upstream
through the boundary layer. The result is to make the changes in quantities
at the surface less abrupt. At high Mach numbers, heat dissipation as a result
of skin friction causes serious rises of temperature and further complication
of the problem.
At high values of Mach number the Reynolds number may be high enough
for viscous effects to be relatively unimportant. But although a valuable
simplification of problems is achieved if effects of either compressibility or
viscosity may be neglected, it must be remembered that there is no velocity at
which the effects of compressibility begin or those of viscosity cease, and in
many situations the effects governed by Reynolds number and Mach number
are of comparable significance.
With a completely non-viscous fluid (which would produce no skinfriction
and no separation of the flow from the boundary) there would be (in
the absence of lift) zero drag in subsonic flow. In supersonic flow, however,
the change of pressure across a shock wave would produce a drag even with
a non-viscous fluid. This drag is known as wave drag.
The drag coefficient of a given body rises sharply as the Mach number M
of the oncoming flow approaches 1.0 (Fig. 8.23). For a blunt body, for which
the position of separation is fixed by its shape, the skin-friction is small, and
CD continues to rise beyond M = 1, as a result of shock wave effects at or
near the front of the body. These effects make the largest contribution to the
total drag, so, for supersonic flow, streamlining the rear of such a body has
little effect on the total drag. The greatest reduction of drag is achieved by
making the nose of the body a sharp point (see Fig. 8.23).
For other bodies the position of separation is closely associated with the
shock phenomena. A shock wave first appears on the surface at the position
of maximum velocity, and separation occurs close behind it. There is thus
a sharp rise of CD. An increase of Mach number, however, shifts the shock