jabishop@uokmax.ecn.uoknor.edu (Jonathan A Bishop) (04/24/91)
From: jabishop@uokmax.ecn.uoknor.edu (Jonathan A Bishop)
Several recent articles have discussed shock waves and possible damage
caused by shock waves generated by supersonic aircraft. Some of the
information presented was inaccurate, and I am posting this in an attempt
to clarify.
Someone commented that the angle of a shock generated by an aircraft is
45 degrees divided by the Mach number.
Actually, a better approximation is given by:
asin(1/M) where M is the Mach number. Even this is a large scale
approximation based on assuming that the object is a point generating
a single disturbance. In reality, the shape of the shock very close to the
aircraft is complex and depends on many factors. It may be attached to the
body or be detached; also, each major aircraft component generates a separate
shock which interferes with other shocks.
The original post asked where the energy of the shock comes from, and one
reply suggested that sound energy builds up in the shock wave. Actually,
energy does not really accumulate in a shock; the energy of the air ahead
of the shockwave is the same as that of the air behind. This can be proved
by demonstrating that the stagnation temperature does not change across the
shock. (Stagnation temperature is the temperature the air would have if it
was isentropically brought to rest, or stagnated. Stagnation pressure is
also important; in actuality, a pitot tube measures stagnation pressure.
The customary "dynamic pressure" measurement is a low Mach number
approximation of the stagnation pressure.)
However, the pressure does change across the shock; it is much greater behind
the shock. It is this pressure change which causes damage. This pressure
gain is costly in that the entropy of the air behind the shock is much
greater than that of the air in front of the shock. This entropy jump is
required by the Second Law of Thermodynamics.
This entropy jump also results in a large stagnation pressure loss across
the shock. This isn't of tremendous concern to aircraft in flight, but it
is a major factor in the design of supersonic wind tunnels. It is easy to
accelerate air to supersonic speeds through a nozzle. Except in a thin
boundary layer, this process is isentropic. A favorable pressure gradient
is also created which forces the viscous boundary layer to adhere to the
wall. After the test section, the air must be slowed by a diffuser. This
is difficult because the air is now facing an adverse pressure gradient. The
boundary layer will thicken, effectively narrowing the tunnel. The air will
then slow down through a series of shocks. The objective of diffuser design
is to make these shocks as weak as possible in order to minimize stagnation
pressure loss since such loss must be restored by pumps before the air can
be recirculated through the nozzle again.
--------
jabishop@uokmax.ecn.uoknor.edudeichman@cod.nosc.mil (Shane D. Deichman) (04/29/91)
From: deichman@cod.nosc.mil (Shane D. Deichman) >From: jabishop@uokmax.ecn.uoknor.edu (Jonathan A Bishop) >Someone commented that the angle of a shock generated by an aircraft is >45 degrees divided by the Mach number. >Actually, a better approximation is given by: > asin(1/M) where M is the Mach number. I realized (after my original posting) that I goofed on my Trig. Actually, the correct answer should be arctan(1/M), since tan(angle) is equal to the quotient of the "opposite" and the "adjacent." The sound energy will diffuse orthogonally at a Mach number of 1.0000 (by definition) to the aircraft's direction of travel. <very insightful lesson on Thermodynamics and wind tunnel design deleted> -shane