When an object
or a disturbance moves so fast that fluid particles cannot move
out of the way, the molecular structure of the fluid permits
the relative position of the molecules to move closer together
or to compress. Thus, as the mean free path of the molecules
is shortened, layers of spherical (3-D) disturbances come together
to form a shock wave. A pressure build up or pressure pulse is
created which grows larger and larger until it becomes a shock
wave resulting in a rise in temperature and pressure. In short,
as a disturbance approaches the speed of sound, gas molecules
pack closer and closer together until a region of compressed
fluid results called a shock wave. A shock wave is actually a
wave-front of compressed gas molecules with some unique properties.
Some engineers and researchers define a shock wave as an infinitely thin
discontinuity. Actually, a shock wave is not
discontinuous (i.e. a shock wave is NOT infinitely thin) because a
shock wave has a finite thickness where fluid properties vary
continuously across the region defined by the shock wave. According
to this analysis, shock wave thickness tends to become smaller
as Mach number is increased. For the example presented below
shock wave thickness is approximately 40% smaller at Mach 10
than it is at Mach 1.
SPECIAL NOTE: Contrary to common belief, shock thickness is not
unchanging as a function of Mach number. This analysis
illustrates that shock thickness decreases significantly as Mach
number increases and is not an absolute and unchanging
discontinuity.
John Cipolla
Chief Aerodynamicist, AeroRocket
AeroCFD
Pressure Contours, Cone-Cylinder-Flare, M = 2.81,
AOA = 0.5 degrees
Upstream of a normal shock wave the flow is supersonic,
whereas downstream of a normal shock wave the flow is always
subsonic. In addition, upstream and downstream of a shock wave
the flow is isentropic but the flow is not isentropic in a region
defined by the shock wave. The flow is not isentropic in a shock
wave because friction or shear stress cause the flow to be internally
irreversible. In a shock wave internal irreversibility
(losses) due to friction cause the entropy, ds to be greater
than 0.0. Whereas, before and after a shock wave ds = 0.0.
THICKNESS OF A NORMAL SHOCK
A shock wave has
a finite but very small thickness, dX caused by "packing"
of the molecules during the compression process as the shock
wave moves through a fluid. The density of the fluid in the region
of the shock wave tries to distribute itself evenly during the
passage of the shock wave into undisturbed fluid. The distribution
of density, pressure, etc is governed by fluid viscosity. This
analysis assumes that for a control volume of fluid around the
shock wave, the shear stress, Pyx within the shock wave is of
the same magnitude as the normal stress, DP acting on the shock wave. Please refer to figure
1 below that graphically illustrates the control volume around
the shock wave. Finally, figure 2 illustrates the derivation
of the equations required to compute shock wave thickness as
a function of Mach number plotted in figure 3. Please see the
reference, "Fluid Mechanics", page 845, by R.A. Granger.
Figure 1. Control volume around the shock
Figure 2. Equations for determining shock thickness
EXAMPLE:
The viscosity of
air at 1 atm and 100 degrees F is
n
= 1.80 E-4 ft^2/sec.
In figure 3 the shock wave thickness is plotted as a function
of Mach number. Please see example on page 845 of the book "Fluid
Mechanics" by R.A. Granger where the flow velocity is Mach
= 2.71 (2000 ft/sec) and the resulting shock wave thickness is
9.0 E-8 ft (2.743E-5mm).
Figure 3. Complete MathCAD analysis for shock thickness as a function of Mach number in air.
For more
information please contact AeroRocket.
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