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PHYSICAL INTREPRETATION OF MATHEMATICALLY
INVARIANT K(p,P) TYPE EQUATIONS OF STATE FOR
HYDRODYNAMICALLY DRIVEN FLOW
George M. Hrbek
Los Alamos National Laboratory, Los Alamos, NM 87545
In order to apply the power of a full group analysis(') to the problem of an expanding shock in planar,
cylindrical, and spherical geometries, the expression for the shock front position R[t] has been modi-
fied to allow the wave to propagate through a general non-uniform medium. This representation in-
corporates the group parameter ratios as meaningful physical quantities and reduces to the classical
Sedov-Taylor solution for a uniform media.
Expected profiles for the density, particle velocity, and pressure behind a spherically diverging shock
wave are then calculated using the Tait equation of state for a moderate (i.e., 20 t TNT equivalent)
blast load propagating through NaCl. The changes in flow variables are plotted for Mach 1.5
Finally, effects due to variations in the material uniformity are shown as changes in the first deriva-
tive of the bulk modulus (i.e., Ko').
In the companion paper) a general solution to the
1D hydrodynamic shock wave was given in terms
of its group invariance properties.
Self-similar profiles for the reduced density, parti-
cle velocity, and pressure behind the shock were
shown to be explicit functions of the Mach number
at the front, the equation of state, the shock forma-
tion time, and the uniformity of the material ahead
of the shock.
In order to illustrate how the group theoretic
method can be applied to the investigation of real
experiments, this study will consider a moderately
strong (M 1.5), spherically diverging shock wave
propagating through a solid block of NaCl using
the Tait equation of state
DIMENSIONAL ANALYSIS APPLIED TO
THE EXPANDING SHOCK FRONT
The traditional Taylor-Sedov dimensional expres-
sion for the expanding shock front only allows for
an ideal gas, power law non-uniformity.
In order to apply the present work into the non-
uniform regime for a general material, it is neces-
sary to incorporate a velocity dependence into the
expression for the expanding shock front, R[t].
This yields the following ordinary differential
R[t]= (p-ER'[t]/-r1)-'/(J(t - o-) (1)
Where R' [t] is the velocity of the shock front, t is
time, p is the density, E is the energy of the blast, p
is the uniformity of the material ahead of the
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Chen, Bin; Lin, Jung-Fu; Chen, Jiuhua; Zhang, Hengzhong & Zeng, Qiaoshi. Synchrotron-based high-pressure research in materials science, article, June 1, 2016; (digital.library.unt.edu/ark:/67531/metadc928724/m1/2/: accessed September 19, 2018), University of North Texas Libraries, Digital Library, digital.library.unt.edu; crediting UNT Libraries Government Documents Department.