corresponds to a normal shock, and the weak case,bbmin, represents
the weakest possible oblique shock at that Mach number, which is called
a Mach wave.Mach waves are caused, for example, by very small
nonuniformities on the walls of a supersonic wind tunnel (several can be
seen in Figs. 17–36 and 17–43). Mach waves have no effect on the flow,
since the shock is vanishingly weak. In fact, in the limit, Mach waves are
isentropic. The shock angle for Mach waves is a unique function of the
Mach number and is given the symbol m, not to be confused with the
coefficient of viscosity. Angle mis called the Mach angleand is found by
setting uequal to zero in Eq. 17–46, solving for bm, and taking the
smaller root. We getMach angle: (17–47)Since the specific heat ratio appears only in the denominator of Eq.
17–46,mis independent of k. Thus, we can estimate the Mach number of
any supersonic flow simply by measuring the Mach angle and applying
Eq. 17–47.Prandtl–Meyer Expansion Waves
We now address situations where supersonic flow is turned in the opposite
direction, such as in the upper portion of a two-dimensional wedge at an
angle of attack greater than its half-angle d(Fig. 17–45). We refer to this
type of flow as an expanding flow,whereas a flow that produces an oblique
shock may be called a compressing flow.As previously, the flow changes
direction to conserve mass. However, unlike a compressing flow, an expand-
ing flow does notresult in a shock wave. Rather, a continuous expanding
region called an expansion fanappears, composed of an infinite number of
Mach waves called Prandtl–Meyer expansion waves.In other words, the
flow does not turn suddenly, as through a shock, but gradually—each suc-
cessive Mach wave turns the flow by an infinitesimal amount. Since each
individual expansion wave is isentropic, the flow across the entire expansion
fan is also isentropic. The Mach number downstream of the expansion
increases(Ma 2 Ma 1 ), while pressure, density, and temperature decrease,
just as they do in the supersonic (expanding) portion of a converging–
diverging nozzle.
Prandtl–Meyer expansion waves are inclined at the local Mach angle m,
as sketched in Fig. 17–45. The Mach angle of the first expansion wave is
easily determined as m 1 sin^1 (1/Ma 1 ). Similarly,m 2 sin^1 (1/Ma 2 ),msin^111 >Ma 12856 | ThermodynamicsMa 1dFIGURE 17–43
Still frames from schlieren
videography illustrating the
detachment of an oblique shock from a
cone with increasing cone half-angle d
in air at Mach 3. At (a) d 20 and
(b)d 40 , the oblique shock remains
attached, but by (c) d 60 , the
oblique shock has detached, forming
a bow wave.
Photos by G. S. Settles, Penn State University.
Used by permission.FIGURE 17–44
Shadowgram of a one-half-in diameter
sphere in free flight through air at Ma
1.53. The flow is subsonic behind
the part of the bow wave that is ahead
of the sphere and over its surface back
to about 45. At about 90 the laminar
boundary layer separates through an
oblique shock wave and quickly
becomes turbulent. The fluctuating
wake generates a system of weak
disturbances that merge into the
second “recompression” shock wave.
Photo by A. C. Charters, Army Ballistic Research
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