Benchmark Test Problem for Measuring Anomalous Dissipation in Shock
Hydrodynamics Simulations
by
Guy M. Snodgrass
B.S., Computer Science
United States Naval Academy, 1998
Submitted to the Department of Nuclear Engineering and the Department of Electrical
Engineering and Computer Science in Partial Fulfillment of the Requirements for the Degrees of
Master of Science in Nuclear Engineering
and
Master of Science in Electrical Engineering and Computer Science
at the
Massachusetts Institute of Technology
June 2000
©2000 Massachusetts Institute of Technology. All Rights Reserved
Signature of Author .....................
ineering
5,2000
Certified by ...............................
I Molvig
A6sociate Professor
or
C ertified b y ......................................................................................................
klvin W. Drake
nd Engineering
Thesis Reader
A ccepted by .............................................
.................
Chen
dents
TT-
A ccepted by ................................
MASSACHUSETTS !NSITUTE
OF TECHNOLOGY
Arthur C. Smith
Chairman, Departmental Committee on Graduate Students
JUL 3 1 2001
LIBRARIES
LARKER
A
3
Benchmark Test Problem for Measuring Anomalous
Dissipation in Shock Hydrodynamics Simulations
by
Guy M. Snodgrass
Submitted to the Department of Nuclear Engineering and the Department of
Electrical Engineering and Computer Science on May 5, 2000, in partial fulfillment
of the requirements for the degree of Master of Science in Nuclear Engineering and
Master of Science in Electrical Engineering and Computer Science
ABSTRACT
Accurate simulation of strong compressions is a key capability needed in the design of efficient implosion
systems. Anomalous dissipation due to the numerical hydrodynamics scheme can be an important limitation in the
simulations, creating artificial (numerical) heat that reduces compression efficiency. This can inhibit using
simulation to design efficient "adiabatic compression" schemes, for example. One obvious potential source of such
anomalous dissipation is the fractional advection that occurs in Eulerian schemes where a portion of the mass,
momentum and energy of the fluid is transported to neighboring cells. The advection fraction corresponds to what
occurs in the continuum but inevitably spreads the field (mass, momentum, or energy) throughout the destination
cell, effectively causing diffusion. When cell sizes exceed the physical mean free path, which is typically the case,
diffusion produced by this effect will exceed the physical diffusion and anomalous dissipation can result. Various
numerical corrections can be applied to mitigate this effect, but some residual effect is inevitable.
The purpose of this thesis is to define and demonstrate an unclassified benchmark test problem to determine the
extent, if any, of such a dissipation anomaly. The particular cases are to be run using the Radiation Adaptive Grid
Eulerian (RAGE) simulation code, developed at Los Alamos National Laboratory (LANL) in Los Alamos, New
Mexico, USA. This code provides sophisticated strong shock hydrodynamics using an adaptive grid technology.
The physical system devised as part of this thesis is a strong shock implosion modeled by two materials divided into
three regions.
Strong compression can be accomplished by pushing a gas cavity to a smaller volume by means of a heavy
bounding material, usually a spherical shell. In its simplest form, then, the problem is one in which a spherical shell,
moving inward at some high initial velocity compresses the gas (and itself), until pressure buildup reverses the
implosion. Although the actual results in practice depend on properties of the heavy material such as its strength and
equation of state, the numerical anomaly of interest here is more general and can be expected to occur for any heavy
material. By using a particularly simple heavy material we can expect to generate an adiabatic compression problem
whose final state can be determined a priori, and yet can still be expected to show deviations due to anomalous
numerical dissipation. In this way, we can separate the basic numerical issue from the detailed specific (and often
classified) designs. This problem can serve as a benchmark test of any hydro scheme for the existence of anomalous
dissipation, which is demonstrated in this particular test case. The numerical anomaly demonstrated in this thesis
cannot be justified by the hydrodynamics of the system and is an artifact of anomalous dissipation within the
hydrodynamics code.
Results are presented for a variety of simulations using a ID implementation of the spherical adiabatic code
developed as a part of this thesis.
Thesis Supervisor: Kim Molvig
Title: Associate Professor, Nuclear Engineering Department
Thesis Reader: Alvin Drake
Title: Professor of Systems Science and Engineering
4
5
Acknowledgements
This thesis would not have been possible without the research opportunity provided by
my thesis advisor, Professor Kim Molvig. My thesis reader, Professor Al Drake, has been an
excellent source of information and support throughout my course of study at MIT.
Many thanks also go to the team of scientists, engineers, and support staff at the Los
Alamos National Laboratory. Among those with whom I interacted regularly were Mike
Gittings, Bob Weaver, Bob Kares, Beverly Chavez, and Pam Paine. Their help was greatly
appreciated and I hope to work with them again in the future.
I would also like to thank those at MIT who have helped me professionally as well as
personally during my course of study: Clare Egan, Professor Neil Todreas, CAPT Matthew
Lewis - US Army, Richard Williams, Leigh Heyman, Ben Wilson, Petr Swedock, Micah Smith,
and Brendan Beer. Thank you all for putting up with me and for helping nonetheless.
Thanks also to those who provided words of advice, support, and encouragement while
working on my thesis. Camilla Sulak, Professor Eugene Stanley of Boston University Physics
Department, Shelley Fisher, Rachael Stanley, Professor Larry Sulak, Chairman of Physics at
Boston University, and Dean Staton of MIT. All of you helped me look at the brighter side of
life.
A great deal of appreciation goes to my mother, father and family for all of the moral
support and encouragement while at the United States Naval Academy and MIT. Without your
prayers this would not have been possible.
Amanda Johnsen has been a constant source of joy throughout my final year of study.
Her constant encouragement and that of her family has been greatly appreciated. She is perhaps
one of the most intelligent people I have ever met, and I am extremely lucky to have found her.
To those mentioned above as well as all of those who went unmentioned, your help was
immeasurable. God Bless you all.
6
7
Biography of the Author
Guy Marvin Snodgrass was born on June 20, 1976 in Fort Worth, Texas. He graduated
from Grapevine High School, Grapevine, Texas in May of 1994.
He accepted a primary
nomination to attend the United States Naval Academy in Annapolis, Maryland from the
Honorable Joe Barton, United States Congressman, 6th District, Texas as a member of the Class
of 1998.
While at the US Naval Academy he earned numerous academic honors, including
induction into the Upsilon Pi Epsilon Computer Science Honor Society, Golden Key National
Honor Society, Phi Kappa Phi Honor Society (for college seniors in the top six percent of their
class academically), and multiple namings to the Superintendent's and Commandant's Lists (for
superior academic and military performance). In February of 1998 he was selected to join the
ranks of roughly 200 other classmates in service selecting Naval Aviation (Pilot) as his career
choice. He was also awarded the Navy Burke Scholarship for post-baccalaureate study at the
Naval Postgraduate School, Monterey, CA and an Immediate Graduation Education Program
(IGEP) Scholarship for immediate two year post-baccalaureate study at a US university. He
graduated as an Ensign in the top ten percent academically from the US Naval Academy on May
22, 1998.
Having been awarded full tuition at numerous educational institutions he chose to enroll
at the Massachusetts Institute of Technology (MIT) in the Nuclear Engineering department as a
Masters Candidate. While at MIT he petitioned for a Masters Degree from the Department of
Electrical Engineering and Computer Science (awarded for fulfilling the requirements of both
departments).
Upon leaving MIT he plans to begin his Naval Aviation training at NAS Pensacola,
Pensacola, FL. ENS Snodgrass hopes to become an aircraft carrier pilot flying F/A-18 or F-14
fighter aircraft.
8
9
Table of Contents
A B ST RA C T .........................................................................
3
Acknow ledgem ents ............................................................
Biography of the Author ......................................................
5
7
Introd uctio n ..............................................................
1
1.1
1.2
O v erv iew .......................................................................................................................
R esearch Inform ation ................................................................................................
LANL RAGE Project....................................................
2
11
11
12
15
2.1 R A GE O verview ................................................................................................................
2.2 Adaptive Mesh Refinement (AMR) Overview ..............................................................
15
16
Benchmark Test Problem Overview ..............................
19
3
3.1
3.2
4
4.1
4.2
4.3
G eom etrical Setup .....................................................................................................
Physics Derivation of Analytical Test Case..............................................................
20
22
Analytical Comparison of Test Case Results ..................
31
31
Analysis of Primary Test Case (1mm sphere)..........................................................
45
Analysis of Second Test Case Results (4mm shell).................................................
2
shell
density)............................................48
Case
(2.5gm/cm
of
Third
Test
Analysis
Discussion of Numerical Anomaly ................................
5
Appendix A - 1mm Thick Shell Test Case ............................
Appendix B - 1mm Spherical Shell Compression Input
eck ....
53
57
67
10
11
1
Introduction
1.1
Overview
Accurate simulation of strong hydrodynamic compression is a key capability needed in
the design of efficient implosion systems.
Anomalous dissipation due to the numerical
hydrodynamics scheme can be an important limitation in the simulations, creating artificial
(numerical) heat that reduces compression efficiency and inhibits the use of simulations to
design efficient adiabatic compression schemes, for example. One obvious potential source of
such anomalous dissipation is the fractional advection that occurs in Eulerian schemes where a
portion of the mass, momentum and energy of the fluid is transported to neighboring cells. The
advection fraction corresponds to what occurs in the continuum but inevitably spreads the field
(mass, momentum, or energy) throughout the destination cell, effectively causing diffusion.
When cell sizes exceed the physical mean free path, which is typically the case, diffusion
produced by this effect will exceed the physical diffusion and anomalous dissipation can result.
Various numerical corrections can be applied to mitigate this effect, but some residual is
inevitable.
The purpose of this thesis is to define and demonstrate an unclassified benchmark test
problem to determine the extent, if any, of such a dissipation anomaly. The particular cases are
12
to be run using the Radiation Adaptive Grid Eulerian (RAGE) hydrodynamics code, developed at
Science Applications International Corporation (SAIC) by Mike Gittings and currently in use at
Los Alamos National Lab (LANL).
This code provides sophisticated strong hydrodynamics
using an adaptive grid technology and copious use of the Message Passing Interface (MPI) for
multiprocessor/multi-machine parallel support. More information on the RAGE hydrodynamics
code is presented in Chapter 2.
1.2
Research Information
Much of the underlying research was performed during a summer internship at Los
Alamos National Laboratory (LANL), one of the Department of Energy's premier national
laboratories.
Spending the entire summer of 1999 with LANL's Applied Theoretical and
Computation Physics (X) Division presented the opportunity to work with excellent people and
superior software tools required to complete this research. The people that collaborated on the
physics design and setup for this project are amongst the best in the areas of simulation,
computational and nuclear science.
The originator of RAGE, Mike Gittings of SAIC, is
currently employed out of LANL and was more than happy to help design some of the early trial
runs of various test simulations, including ID and 2D Sedov blast waves, shock tube tests and
Rayleigh-Taylor mixing.
Bob Weaver, the team leader for RAGE at LANL, was also very
helpful in teaching me how to work with and understand RAGE. Many other people working at
both LANL and MIT helped with this project and they have my most sincere thanks.
The people who collaborated on this project are among the best, and the LANL XDivision is a premier resource in the United States for information, expertise and solutions for a
wide range of nuclear physics and computing issues. It is this expertise that is driving the
13
development of the RAGE hydrodynamics code, the software program used at the Los Alamos
National Laboratory to work on creating strong shock and other hydrodynamic simulations.
Also important are the analytical tools used to extract data from the results, several of which
were created at LANL. WShow, a Microsoft Windows@ compatible software program, was used
to analyze the TEK output files produced by RAGE (graphic xy-plot files).
When more
precision was necessary, Bob Kares' RAGE2ENS conversion program (UNIX OS) was used to
interpret the output from RAGE into a form compatible with CEI's acclaimed EnSight
visualization software. EnSight allows the user to manipulate data to produce plots of material
variables and create animations of the graphical RAGE output. One can load a 2D compression
problem and watch as the shell moves inward towards maximum compression.
Since RAGE itself has been a work in progress, several versions were used. The copy
that was used most often was the 06131999 build of the code. Later, in the design of ID
rectangular grid spaces, other more recent revisions were used, but as no data is included from
those test runs it will not be necessary to document the other versions.
It should be noted that RAGE is a work in progress. As such the code was constantly
evolving with new functionality available only towards the end of this research. Several test
cases were designed that were not fully supported by the current RAGE development code.
However, the cases represented here worked well with the RAGE hydrodynamics code in both
the ID and 2D case.
14
15
2
LANL RAGE Project
Los Alamos National Labs is continuing its ongoing development of several
methods that use computer programs for simulation purposes. Among one of the prominent
programs is the RAGE hydrodynamics code. Developed by Mike Gittings at SAIC, RAGE
(Radiation Adaptive Grid Eulerian) is a one, two and three dimensional multi-material Eulerian
hydrodynamics code for use in solving a variety of high deformation flow of materials problems.
2.1 RAGE Overview
The DNA water shock program, specifically the modeling of underwater explosions in
shallow water, helped influence the development of RAGE. During the development of this test
series numerous one-dimensional calculations were performed to determine what codes or
computational methods would best satisfy the demands of this problem. It was soon shown that
two prominent features would be necessary:
simulation.
second order accuracy and an "adaptive grid"
That is, the simulation code needed to be able to take a predefined cell and, if
necessary, repeatedly divide it into multiple cells for greater accuracy and reduced numerical
dissipation.
16
In the late 1980s an MLG (multi-material, piecewise Linear, Godunov) program was
developed that met these requirements. At that time the MLG was a second order, multi-material
Eulerian hydrodynamics code that was fully vectorized to run on Cray computers. In the MLG
code the adaptive algorithm was designed to allow cells to be subdivided on a cycle-by-cycle and
cell-by-cell basis. This allowed the computational grid to place cells exactly where needed in the
geometrical representation framework. The MLG code has ultimately become what we now
know as RAGE.
2.2 Adaptive Mesh Refinement (AMR) Overview
The RAGE simulation hydrodynamics code uses Adaptive Mesh Refinement (AMR)
techniques to help reduce numerical dissipation and increase resolution in the simulation
'system.' This subsection is intended to provide an overview of AMR.
Typically a ID block has two cells, a 2D block has 4 cells, and a 3D block has 8 cells, as
shown in Figure 2-1.
1 2
IDBlock
3 4
1
2
8
2D Block
X 64
(
1
2
3D Block
Figure 2-1. Grid layout showing the AMR geometry of ID, 2D, and 3D cells.
The code orders and numbers the blocks but the numbering of the blocks is
inconsequential as far as the user is concerned, since the input deck does not reflect the
individual numbering of cells or blocks. All of the AMR and numbering techniques are carried
out by the hydrodynamics code during a simulation cycle.
17
The first step is to define the problem set's grid "domain". The following is a simplified
example of a 4x4 cell space that is cut by a line indicating a region of interest (as in our case with
a metallic shell) or a change in a material property such as density, pressure, etc. The dots within
the cell represent virtual markers that tell the code to subdivide those cells in the next cycle. The
subdivision of cells continues until the line is located only in one cell of the block, or the
maximum level of adaption is reached.
Figure 2-2. Example of 2D grid space bisected by Region of Interest (ROI)
The entire process is computed in parallel, unless only one processor is available, using
the Message Passing Interface (MPI) as first created by Argonne National Lab. The number of
cells per processor is entirely dependant upon the number of processors available. Addressing is
on an internal/external basis. If the cells are located within the same processor the addressing is
local, otherwise the addressing is external for communication between processors. This is the
basic premise for the refinement of the grid space and division of cells amongst processors.
The majority of the code is written using Fortran90 with only a few high-level
instructions written in C.
The high-level C code performs three functions:
decoding the
argument string from the command line, enabling MPI with a different set of arguments
18
(available when run from C), and it calls the controller with the filename. The controller is the
main processing unit, but there are several other modules that handle low-level coding and
physics techniques. When the hydrodynamic program is run the controller steps through these
modules, which perform a variety of functions. The rough pattern of execution is as a cycle.
The subroutines initialize the problem, performs the "restart" process, performs an edit based on
cycle number and time step, cycles through the simulation (t -> t + At ), and performs a
'cleanup'.
During each cycle the program performs hydrodynamic, heat conduction, energy
flow, radiation, and adaption calculations to influence the program's current state and next cycle.
This is how a problem involving a moving substance is simulated: fractional calculations are
performed to solve for the values in each cell for the next time step.
The RAGE hydrodynamic code is a very good package that works extremely well in a
variety of situations. This thesis, however, seeks to demonstrate how the very small numerical
dissipation inherent in hydro-code simulations can actually reduce the effectiveness of a realworld application.
Chapter four describes how a very small dissipative effect can wind up
costing a compression simulation to underpredict the adiabatic value by roughly 50%.
19
3
Benchmark Test Problem Overview
Strong inward compression can be accomplished by pushing a gas cavity to smaller
volume by means of a heavy bounding material, usually a spherical shell. In its simplest form the
problem is one in which a spherical shell, moving inward at some high initial velocity,
compresses the gas (and itself), until pressure build up reverses the implosion. Although the
results in practice depend on properties of the heavy material such as its strength and equation of
state, the numerical anomaly of interest here is more general and can be expected to occur for
any heavy material. By using a particularly simple heavy material we can expect to generate an
adiabatic compression problem whose final state can be determined a priori, and yet can still be
expected to show deviations due to anomalous numerical dissipation. In this way, we can
separate the basic numerical issue from the detailed specific (and often classified) designs. The
problem could serve as a benchmark test of any hydrodynamics scheme for the existence of
anomalous dissipation.
The initial problem is to design and model a strong implosion. To accomplish this we
use two different materials:
a low-density gas and high-density, high-gamma material, also
modeled as a gas. In this case, the model is made of a thin metallic region (approximated by the
high-density, high-gamma (HDHG) material) being driven inward by its high initial velocity.
20
3.1
Geometrical Setup
The initial desired test case is described by a spherical gaseous cavity of low-density gas
(Region III) that would be driven inward by a high-density bounding material (Region II). The
higher-density bounding material would be surrounded by a large region of gas (Region I) at the
same density and at a slightly higher pressure than that of the inner gaseous region.
The
following diagram demonstrates the initial test case schematic:
Geometrical Setup
Region 1 - Heavy bounding material
Region I
(outergas region)
Region IIl - Inner gas region
Figure 3-1. Pictorial representation of gas and material regions
The geometrical layout of the simulation is subdivided into three material regions. The
RAGE hydrodynamics simulation code requires the material boundaries to be defined in
21
inwardly progressing regions.
This is why Region I is the outer most region and not the
innermost.
The premise is to compress the (roughly) atmospheric inner gas
a heavy metal density material (PM =10.0
-3
Pg
= 0.001
3
with
to achieve (20)3 compression of the inner gas
and two to three fold compression of the metal shell.
If we take representative physical
dimensions to be 10cm for the inner gas radius and 0.1cm for the metal shell thickness, then the
initial shell kinetic energy is set by the requirement of 8000 times adiabatic compression of the
inner gas.
Once we have found the adiabatic conditions for the inner gas, the metal shell ratio of
specific heats, y,, can be determined such that it compresses the metal shell to 2.5 times its
initial density, assuming that adiabatic conditions hold throughout the process. To construct the
simulation setup as demonstrated in Figure 3-1 we ascribe specific physical traits to each
material region (density, specific internal energy, initial velocity, etc).
The outer region is composed of a low-density gas (0.001 g/cm 3) of 50.0cm in radius that
is at an initial pressure of 1,000 times atmospheric pressure (1.01x9 dynes/cm2). The second
region is a thin, high-density (10.0 g/cm 3)region of high-gamma (14.7) binding material that
divides Region I and III from each other and was calculated to be able to adiabatically compress
the inner gas region, Region III, inward to a radius of 0.5cm. The third region is the innermost
region. The chart below better describes each region's material properties as set initially in the
RAGE input deck (the input deck is included as Appendix B).
22
Physics Derivation of Analytical Test Case
3.2
Table of Constants
Description
Bounding Material (Region 11) thickness
Radius of inner gas region (Region Ill)
Radius of inner gas and binding material (Region 11)
Radius of outer gas boundary (Region 1)
Density of material 1/Ill
Density of material 11
Pressure of Region I / 11
Temperature of Region I / 11
Variable
Value
6m
0.1 (cm)
10 (cm)
10.1 (cm)
50.0 (cm)
0.001 (g/cm 3)
10 (g/cm 3)
1.01x106 (dynes/cm)
0.025 eV
RQ
Rg+8m
Ro
pg
Pm
Pao
Tqo
Table 3-1. Material values for RAGE setup
The variable values required to initialize the RAGE input deck were calculated using the
physics in this chapter. The physics were also calculated to reach certain adiabatic goals for
compression of the metal and inner gas regions.
Following is a discussion of the physics
involved in this project. For a detailed numerical look at the equations with input properties,
please see Appendix A.
In setting up our equations we first create certain parameters that we would like to
achieve.
One of the first is that we want to solve for an initial velocity that is sufficient to
compress the inner gas region to 0.5cm.
R.
Rf
-
10cm(31
0.5cm
-20.0
(3.1)
As shown in equation (3.1) this decrease in radius corresponds to a 20 times compression of the
inner gas. The metal shell must also have sufficient kinetic energy,
23
Eko
= 4zR
2
UO2
I
(3.2)
to provide the work done in compressing the gas in an adiabatic manner.
This requires the work in compressing the space to be derived, as follows:
W = PdV
(3.3)'
where we express dV in radial geometry:
dV =4)7r 2dr
(3.4)
The pressure at a given radius is adiabatically related to the original gas pressure (Pgo),
initial density (p), final density (po) and ratio of specific heats (y) by equation 3.5:
/
\Y
(3.5)
P(r) = Pgo
Pgo)
Instead of using the density of the material we can convert the equation to require only
the ratio of specific heats, initial gas cavity radius (Rgo), and current radius:
P~r)= P
Rgoy~
P(r)=go
(3.6)
J
Substituting equations (3.6) and (3.4) into (3.3) for pressure and volumetric compression,
respectively, we yield the following result:
W
= R947r2drPgo
0
R(
1
A classical statement of mechanical work.
93
(3.7)
24
Equation series (3.8) walks through the process of solving this integrated form of work.
Upon completion we have an easily solvable equation that relies on constants that we have
already declared for the initial material conditions:
R9
fdrr2-3,
W = PO4 R,
Rf
W
= P 0 4rR3 y
gO
g
r 3- 3y R,
3 -3yR
RP
4
W=
1
R3 P
3W3y
R9
II1
W = 4zR 9 P9 0
3
yI -
-
Rf
The pressure of the gas is a result of the product of the density of the gas and the thermal
velocity,
2
cth
P
(3.9)
2
= PgCth
where: cth2
kT
m
Having previously stated that the metal shell's kinetic energy be sufficient to provide the
work necessary to compress the inner gas we set (3.2) and (3.3) equal to each other:
Eko
=W
(3.10)
Substituting equation (3.2) for initial kinetic energy of the high-density material shell and
equations (3.8) and (3.9) for work needed to perform the compression yields the following
equation:
4irRg O6M
pnumo2 =
R
3PgCth2
(3.11)
25
Rewriting (3.11) to solve for the initial velocity of the metal sphere yields:
U
2
"0
2Rg Pg
3in PMC
~ ---
2
R
3(y-1)
(y -1)K
1
R)
-1]
(3.12)
With the initial conditions stated earlier in the chapter we have all the required variables
to solve equation (3.12) with the exception of the thermal velocity. We also know that if
R
1
= - R
20 g
(3.13)
then the compression factor is:
1 (203(y-1))_1 ~ 100
Y-
1
(3.14)
for a gas region gamma, yg, of 1.5. We also already know that the ratio of the inner gas initial
radius to the initial metal shell thickness is:
Rg /S,
~ 102
(3.15)
and the ratio of the initial gas density (0.001 gm/cm 3 ) to the initial metal region density (10
gm/cm 3 ) is:
4
Pg /PM ~ 10-
(3.16)
Using the above ratios it is reasonable to approximate that the initial metal shell velocity
is equal to the thermal velocity:
uMO
= Cth
(3.17)
In fact, after solving equation (3.12) numerically in Appendix A (A.4) we find that the
initial metal shell velocity is in fact very close to the thermal velocity of the gas particles:
Un
0
=1.0859c,
(3.18)
(
26
An initial inward velocity of the metal is appropriate for the adiabatic calculation if it
meets the following relationship to its Mach number:
M
=
uM2
<jCth
In this case the Mach number is 0.88, less than 1.0, justifying the value of umo for
adiabatic compression.
Finding the relationship between the initial metal shell velocity and thermal velocity is
important. However, RAGE requires the initial gas density, initial gas temperature, and specific
heat at constant volume (c,) to initialize the gas regions. We can therefore use these to compute
the thermal velocity and get initial metal velocity from Equation 3.18.
For an ideal gas the internal energy can be approximated knowing the degrees of freedom
(D), initial gas density, Boltzmann's constant (k = 1.38x10- 23
Joules
), mass of the gas particles
Kelvin
(m), and initial gas temperature (T):
U=
k
D
D
D
P = nkT = p -T
m
2
2
2
= cT
(3.19)
However, internal energy is also equal to the specific heat at constant volume times initial
temperature.
We can therefore use this information to solve (3.19) for the specific heat at
constant volume:
C =
p
2
m
Earlier we stated that the ratio of specific heats, y, for the gas was 1.5.
(3.20)
This was
estimated based on a standard near-ideal gas with four degrees of freedom. The ratio of specific
heats can be calculated knowing the degrees of freedom as in (3.21):
27
D+2
y2
Yg
=(3.21)
D
Earlier we related several constants to solve for the specific heat at constant pressure, but
we can also use several of these values to solve for the thermal velocity:
T =_c,2
(3.22)
Combining equations (3.20) and (3.22) we can solve for the thermal velocity:
Cth
2
2 c
v
D pg
T
(3.23)
Using the result from (3.23) we can solve for the initial metal (and inner gas) radial
velocity, (3.18). The result of that calculation (20,887.89cm/s) is presented in a table at the end
of this chapter.
Now we also want an equation of state for the metal (high-density/high-gamma material)
region such that the compression to achieve the final pressure results in a density compression of
2.5 times the original metal density. If we assume the metal pressure equals the gas pressure at
maximum compression we can model the metal as an ideal 'gas' with a very high ratio of
specific heats.
Pg
=
Pg
=
(20) pgo ~ 8000 pgo
(3.24)
This is an important relationship, because if the simulation test case yields a result of 8.0
gm/cm 3 for the final density of the inner gas we will have experienced adiabatic compression.
Non-adiabatic compression would occur for a variety of reasons, but the result of which would
produce losses that would reduce the effect of compression, therefore lowering the maximum
density of the inner gas region.
28
Using (3.5) we can also solve for our ideal final inner gas pressure when undergoing
adiabatic compression:
Pg
>
(
Pg =P Pg
Pgo)
~Pg (8000).5 ~(7.155x0Pgo
(3.25)
According to (3.25) we can expect to see an increase in the pressure of the inner gas
region at maximum compression on the order of 105 . Assuming that the final pressure of the
metal is equal to the final gas pressure we can say:
Pf =
'"
P1
P
Pg = P.O
(3.26)
Also assuming that RAGE is initialized so that the initial metal and gas region pressures
are equal:
r
H
'
f~
(3.27)
Pgo)
PnO
Solving (3.27) for the ratio of specific heats for the metal yields:
pgo
pg/ lp.,n
In p,
'" =Y In
YgIn
g
In 2.5
(3.28)
Solving (3.28) then only depends upon the ratio of specific heats for the gas (1.5), the
ratio of final gas density to initial gas density (8000), and the ratio of final metal density to initial
metal density (2.5). The ratio of specific heats for the metallic material is therefore 14.7.
The pressure of a region can be equated to the ratio of specific heats, density, and specific
internal energy of that region, as shown in (3.29):
P=(
-
(3.29)
29
RAGE takes the specific internal energy (SIE), not pressure, to initialize a region.
Therefore, rearranging equation 3.29 allows us to solve for the SIE of the region from the known
value of pressure:
P
e
=
(3.30)
innergas
Since out initial conditions require that the initial pressure in the inner gas region and the
metallic shell be equal, we can solve for the SIE of the metallic shell by simply using the
pressure of the inner gas region with the density (10.0 g/cm 3) and ratio of specific heats (14.7) of
the metal shell region:
P
(3.31)
"innergas
eregionI
The outer gas pressure is related using equation 3.25. This provides an outer gas pressure
that will be equal to the adiabatic final inner gas pressure at maximum compression:
Poutergas =
(3.32)
go
= Pgo (20)
= Pgo (716,000)
A pressure jump of this magnitude would require special treatment at the interface and
with a standard setup RAGE goes unstable numerically. Instead a value of 1,000 times Pgo is
used to help hold the back of the material together. Note that this outer gas pressure is not really
a part of the problem setup, but merely an artifice to eliminate outward expansion of the metal
from its back surface.
Any such backpressure will only enhance compression, in principle
adding to the computed adiabatic density and pressure values.
above, the outer gas specific internal energy is calculated:
Using the same procedure as
30
outergas
region
=
(3.33)
Solving the entire sequence of equations for the desired adiabatic results is included in
Appendix A for your convenience. Also included in this thesis is the RAGE input deck for the
one-millimeter thick metal sphere compression test case, found in Appendix B. Included below
is a list of the important variables solved in this chapter that are subsequently included in the
RAGE input deck and that might change between different simulation test cases:
Adiabatic Value
Variable
Ratio of specific heats (yg)
1.5
Ratio of specific heats (Yin)
14.7
Specific heat at constant volume (cv,)
2.96x10 7
Specific heat at constant volume (cvm)
2.95x 10'
2.02x1 012 erg/g
SIE Region I (outer gas)
7.37x10 3 erg/g
SIE Region II (metal shell)
2.02x10 9 erg/g
SIE Region Ill (inner gas)
0.001 g/cm 3
Density of gas (pg)
10 g/cm 3
Density of metal (pm)
Metal shell velocity (uro)
-2.09x1 03 cm/s
Table 3-2. List of applicable RAGE simulation variables and values
31
4
Analytical Comparison of Test Case
Results
The test cases were run remotely on LANL's theta and tO] remotely accessible computer
clusters. The results were output in two separate forms: TEK plot files, which are black and
white xy-plot representations of the material properties, and DMP files, which are text output
files that RAGE saves to the hard drive at each output time step.
The DMP files can be
converted, using Bob Kares' RAGE2ENS program, into a format that is compatible with CEI's
EnSight 6.2/7.0 analysis toolkit. EnSight allows a user to manipulate data to create various
custom plots and to easily compare the data. The majority of the data contained within this
thesis is provided in TEK format since it is more easily manipulated.
4.1
Analysis of Primary Test Case (1mm sphere)
The first test case was also the benchmark for all other subsequent test cases. Here the
RAGE setup mirrored what has been presented in the previous three chapters. We expect to see
results like those presented in Table 3.1. For example, the initial metal density is 10.0 gm/cm 3,
32
and at maximum compression (an inner gas radius of 0.5cm) the expected result is for the metal
region to compress to 2.5 times its initial value, or 25.0 gm/cm3
The following paragraphs detail the experimental results of the 1mm thick spherical
The major predicators of performance are plots of:
compression test case.
Using the data from these figures the
material, and fractional volume.
temperature,
pressure, density,
determination can be made as to how close the hydrodynamic code comes to achieving adiabatic
compression.
__._._._.
1
. . . ._
x10
10
4
8
3
6
2
4
2
0
0
0
10
Pressure:
20
cycle =
30
0
t
=
40
50
0.OOOE+00
0
10
Pressure:
20
cycle =206332
30
t
=
40
50
1.150E-04
Figure 4-1. Pressure TEK plots for 1mm thick (8m) spherical compression
As Figure 4-1 demonstrates, the final maximum compression occurs at approximately
t=1.15x 10-4 sec.
The right-hand side plot also shows that the final experimental inner gas
pressure does not equal the expected adiabatic result.
If the compression process had been
adiabatic the final inner gas pressure would equal the result of (A.19) which is 7.23x10
dynes/cm 2.
1
Instead, the actual experimental result yielded 4.30x 10" dynes/cm2, which is
approximately 40.5% less than the adiabatic result.
33
10
12
8
10
8
6&
4
4
2
2
0,
0
0
30
20
10
Density:
cycle
=
0
40
50
0
10
20
Density:
cycle =206332
Figure 4-2. Initial and final density for
1mm test case
t = 0.000E+00
30
40
50
t = 1.150E-04
Figure 4-2 displays the initial density condition for the three material regions at
initialization (t = 0.0s).
Remember that initially there are three material regions. The third
region extends to a radius of 10.0cm and has a density of 0.001 g/cm 3 . The second region
extends from 10.0 to 10.1 cm and has a density of 10.0 g/cm 3, which appears as a solid bar in the
left-hand plot. The outermost region extends from 10.1cm to 50.0cm and has the same density
as Region III. The plot on the right-hand side displays the density for the inner gas and metal
sphere at maximum simulated compression. According to the initialization parameters (second
paragraph of Appendix A) the final density for the inner gas would be 8.0 gm/cm 3 and for the
metal sphere is 25.0 g/cm 3 under adiabatic conditions. The simulation results again underpredict
maximum density with an inner gas density of 4.28 g/cm 3 (46.5% lower than adiabatic) at
3
maximum compression. The metal sphere increases its maximum density from 10.0 gm/cm to
12.8 g/cm 3 , (48.8% lower than predicted adiabatic). The analytical density for the inner gas was
calculated based on a maximum compression of the gas to 0.5cm. The expected density for an
adiabatic compression to 0.6cm is:
34
(Rg .
pg
(10.0cm
0.cm) Pgo ~ 4629.63po =
0.6cm)
Pg
=
gf=Rgf
3
4.6 gm / cm
3
(4.1)
The results from the experimental test case show a density very close to this value.
Therefore it is probable that the density has reached near-adiabatic compression.
The temperature plots are useful in this situation for two reasons: the initial conditions
can be easily error checked, and the error present in the pressure and density values might be
attributed to an effect found within the maximum compression plot.
x104
x10
6
6
4
4
2
2
0
0
0
10
tev:
20
cycle =
30
0
40
5(
0
10
20
0.000E+00
tev:
cycle =206332
Figure 4-3. Initial and maximum temperature plots
t =
30
40
50
t = 1.150E-04
The initial plot looks correct, but to ensure accuracy EnSight was used to find the exact
values for temperature in the different regions, as demonstrated in later figures. In Appendix A
we show that for the inner gas the pressure was initially 1.01x106 dynes/cm 2 and the temperature
was 0.025 eV ( 1 / 4 0
h
an eV is approximately room temperature).
The temperature plot at
maximum compression shows that a region of unusually high temperature and low density is
found between 0.6cm and 2.0cm. This heating of the gas could be one reason why the 'hole'
exists within the density plot of Figure 4-2. The temperature will be very useful later in this
section to understand the origin of the non-adiabatic behavior of the simulation.
35
2.0
2.0
1.8
1.8
1.6
1.6
1.4
1.4
1.2
1.2
1.0
1.0
0
10
material:
30
20
cycle =
0
t = 0.
40
50
0
000E+00
10
material:
20
cycle =206332
30
40
50
t = 1.150E-04
Figure 4-4. Material plots at time step 0 and maximum compression
The material plots in figure 4-4 represent the radius of the total sphere that the different
materials occupy. For instance, the material for the inner and outer gasses is represented as
material '1.0'. This reflects the input in the RAGE setup file, as shown in Appendix B, where
material one is set up as the gas in material regions I and III. The low-density gas is therefore set
up as material '1.0' and the high-density gas region (Region II) material is setup as material
'2.0.' The initial plot therefore corresponds exactly with the initial density view: this is because
upon initialization (time step 0) the entire material 2.0 region is located in the high-density
region. This is another good indication that the initial run-time conditions were properly set in
the RAGE input deck.
The material plot at maximum compression reveals something interesting about the
material regions.
The material plot shows an extension of the second material from
approximately 0.6cm to 4.9 cm, a large area. This would indicate that the material has spread 50
fold from its original value of 0.1cm.
The expected spreading of the shell under adiabatic
conditions is 2.86cm as shown by Equation 4.2 when the inner gas is compressed to 0.6cm:
4 (R,03 - Rg 0 3 )Pjo =
(R
3
3
-
Rg 3
p,
(4.2)
36
This represents the spreading of the material assuming constant density at the value, pmf,
corresponding to the adiabatic pressure and conservation of mass. In this case there is a region
of mixed material extending roughly from 0.6cm to 2.5cm which will be discussed later in the
chapter.
The plot in Figure 4-4 will be even more useful after the discussion on the fractional
volume plot for material number two.
1.0
1.0
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0.0
0.0
0
fracvol02
30
20
10
cycle
=
0
t =
0.OOOE+00
40
50
0
10
fracvol02
20
cycle =206332
30
40
50
t = 1.150E-04
Figure 4-5. Fractional volume plots for material #2 (high-density/high-gamma material)
Noticing the previous non-adiabatic behavior in the pressure, density, and temperature
plots the material and fractional volume plots provide a wealth of important information. The
fractional volume plots serve a similar purpose as the material plots, with the distinction that the
material plots reflect the position of a 50% mix of the material. If 50% of material two exists in
the cell, the material plot considers it to be wholly of material two. Conversely, the fractional
volume plot equates a value of '1.0' if the cell at that location along the radius is wholly of
material two, a value of 0.0 if wholly of material one, and a fraction thereof if only partially
material two. For example, at a radius of 1.0cm the fractional volume is approximately 0.5,
...
.....
....
37
signifying that that at a distance of 1cm from the center of the sphere there are equal parts of
material one and material two.
The real usefulness of the plots is in the ability to use a tool such as Adobe PhotoshopTM
to overlay the fractional volume plot with the material plot at time step zero and also at
maximum compression, as in Figure 4-6.
1.0
Fractional volume of material 2 (discontinalty)
0.8
0.6
0.4
Material plot for material #2 (discontialty)
0.2
0.0
0
10
20
30
40
50
Figure 4-6. Plot of material and fractional volume
The overlay of the material and fractional volume plots make apparent that at maximum
compression the two do not overlap completely. What this reveals is that mixing of the two
working fluids is occurring at the leading and trailing edge of the high-gamma material sphere as
it implodes.
One of the most important side effects is that the mixing makes it extremely
difficult to determine where the material two inner boundary ends. The material plot indicates
that the inner bound of material two is at approximately 0.6cm. The fractional volume plot,
however, reveals that at a 50% level of material two the fractional volume boundary is at the
material boundary, but to reach a cell that contains all material two means moving further from
38
the center to approximately 1.75cm. The difference in final compression radius makes a very
large difference in the expected final density of both the inner gas and metal (high-gamma
material) shell.
Variable
Initial value
Expected value
Actual value
Under
prediction
Inner gas radius (R.)
Metal shell thickness (8m)
Inner gas density (p.)
10.0 cm
0.1 cm
0.001 gm/cm 3
0.5 cm
n/a
8.0 gm/cm 3
0.6 cm
4.0 cm
4.28 gm/cm 3
20%
n/a
46.50%
12.8 gm/cm 3
25.0 gm/cm 3
10 gm/cm 3
Metal shell density (pm)
2
Inner gas pressure (P.) 1.01x106 dynes/cm2 7.23x1011 dynes/cm 2 4.3x1011 dynes/cm
2.75x1011 dynes/cm 2
n/a
Meta region pressure (Pm) 1.01x106 dynes/cm2
Table 4-1. Change in values for test case variables
48.80%
40.50%
n/a
The data in Table 4-1 presents an overview of the initial values, the expected values at
maximum
compression
(adiabatic),
the
actual
experimental
result,
and
the
error.
Underprediction was calculated by:
Underprediction=
jExpected - Actualj
xl 100%
(4.3)
Expected
With the exception of inner gas radius, almost all of the underpredictions hover around
50%, a consistent result that indicates the experiment results from the RAGE hydrodynamics
code are approximately half of the analytical adiabatic expectation.
This behavior, lack of
adiabatic compression, can be explained by several mechanisms.
First, the use of any Eulerian hydrodynamics code with a rectangular cell space will have
a hard time arriving at the exact adiabatic answer. The reason for this lies in the linear method of
travel for a vector as well as the possibilities for numerical anomalies in the mass, momentum,
and energy of the material traveling between cell spaces. In a simulation code the cells are of
finite size, which causes the mass, momentum, and energy of the adjacent cells to prematurely
'rush'
into the adjacent cell.
Despite the adaptive mesh algorithm found in the RAGE
39
hydrodynamics code the transport of mass, momentum, and energy will not be perfect. This will
be expressed in greater detail shortly.
Another reason for underprediction from the adiabatic case is the material interface. This
simulation was planned using physics that took a high-gamma/high-density material as the
binding spherical material. In the real application the binding material would be more likely to
compress and increase in density rather than spread as quickly as seen here. One reason for the
spreading and anomalous results is that both materials were initialized as gases (as the simulation
represents them). This could potentially allow the two materials to interface and rapidly begin to
mix as the second region (the thin metallic shell) moves inward. This mixing unbalances the
results providing two regions, one directly in front of Region II and one on the trailing edge of
Region II, that are a mixture of gas materials one and two.
This mixing creates a new
heterogeneous gas that is a combination of the first two. This reasoning and that of the previous
paragraph can be shown in the following sequence of plots produced in EnSight:
Frame 00 (Initial Conditions)
Temperature
Density
Pressure
Temperature
0
9
10
93
Distance (cm)
10.5-
Figure 4-7. Initial conditions plot as produced using CEI's EnSight
40
In Figure 4-7 the initial conditions are presented using a plot illustrating the initial metal
shell density (10.0 g/cm 3), temperature of the outer gas region (= 6,000eV), and pressure
2
9
(1.01x10 6 dynes/cm 2 in the inner region and metal shell and 1.01x10 dynes/cm in the outer gas
region). The following sequence of plots shows the progression of the shell compression as it
nears maximum compression (which occurs in time step 23).
Frame 02
Temperature
Density
Pressure
Temperature
Density
Pressure
0
9
9.5
10
Distance (cm)
10.5
Figure 4-8. Time step #02 after initiating simulation run
Figure 4-8 shows the plot of temperature, pressure and density at the second time output
time step (RAGE was configured in the input file to output plots at constant time intervals, which
is every 5.0x10-6 seconds, as shown in Appendix B). At this frame we see the pressure gradient
progressing through the metal sphere, which is represented here by the area of high density. The
curvature of the pressure plot in the metallic region indicates expansive behavior in the metal
except for the leading edge, which appears compressive.
41
The curvature of the pressure gradient can typically be characterized as compressive,
expansive, or neutral. A straight line (i.e. no curvature) indicates a neutral pressure profile and
the material is not expanded or compressed in a hydrodynamic manner. If the pressure profile is
concave relative to the direction of movement this indicates a compressive profile. The material
that is in this region would therefore tend to compress. Finally, if the pressure profile is convex
(bowing outward) relative to the direction of movement this indicates an expansive profile and
the material is likely to expand in thickness.
Figure 4-8 then makes sense because the metal has to expand to conserve mass as it is
compressed inward because the surface area of the inner and outer surfaces of the shell are
reducing. This holds true unless the density rises appreciably in the metal region, which is what
we expect to see based on our criteria that the density of the metal increase to 25.0 g/cm3.
Time step #06
Temperature
Density
Pressure
Temperature
Deinsity
Pressure
0
8.4
8.6
8.8
9.0
9.2
Distance (cm)
9.4
9.6
Figure 4-9. Time step #6 plot of density, pressure, and temperature
42
Figure 4-9 illustrates the relative positioning of the three curves after six plots have been
produced (t = 3.Ox 0-5 seconds). The plot shows the progression inward of the metal shell,
which has now spread to approximately 0.2cm in thickness.
Time step #20
~
_
Temperature
-Density
Pressure
Pressure
Temperature
Density
0
3.5
4
4.5
Distance (cm)
5
5.5
Figure 4-10. Time step #20 plot
Figure 4-10 shows a steep rise in the pressure despite several scaling changes (numerical
values for the y-axis are not included to prevent confusion between the three material variables
as EnSight renders them unreadable). The simulation has now proceeded for 1.Ox 104 seconds.
The pressure profile in Figure 4-11 now exhibits a maximum within the high density metal
region indicationg an outward force on the back half of the metal while still accelerating inward
on the inward half. This is the beginning of pressure gradient reversal that will signal the
reflection of the incoming shell. It is also a highly expansive form of pressure profile.
Figure 4-11 demonstrates that as the compression process nears maximum compression
the pressure plot reverses, effectively causing the shell to slow down and bounce after maximum
compression. The plots also illustrate that there is no hydrodynamic reason for the inner surface
43
of the metal to spray inward towards the inner gas. The two materials are mixing only slightly,
and as we will describe soon this causes large problems with our particular compression test
case.
Time step #22
Density
Pressure
Temperature
0
0
1
3
2
Distance (cm)
4
Figure 4-11. Plot of time step #22
The plot also shows a rise in the temperature of the region between 0.8cm and 2.1cm.
This corresponds with the region of gas/metal mixture residing between the compressed inner
gas and the inner surface of the metal shell. The temperature 'spike' will be even more evident in
Figure 4-12, which illustrates the profile and relative values of temperature, pressure, and density
at maximum compression.
The change in time between time step #22 and time step #23 is the same as all other
frames. One important detail to visually recognize is that the inner gas region radius changes
significantly between the two frames but the inner radius of the metal shell is nearly constant.
44
This indicates that there is some sort of buffer that is keeping the metal shell from continuing the
compression inward, and is perhaps causing premature reflection.
Time step #23 (max)
Pressure
Temperature
Density
0
0
1
2
Distance (cm)
3
4
Figure 4-12. Plot of temperature, density, and pressure at maximum compression
The reason for not achieving adiabatic behavior can be attributed to the region of high
temperature just behind the inner gas region, which indicates a process that has taken place that
hinders compression. This was hinted at earlier in this subsection when the issues of numerical
anomaly and the rectangular grid were cited as potential non-adiabatic compression. This region
corresponds to the mixed region of gas in the interface boundary between the inner surface of the
Region II sphere and the Region III inner gas. As the plots illustrates, in conjunction with the
earlier fractional volume TEK plots, is that as the shell compresses inward some of the metal on
the inner surface of the shell is ejected into the inner gas region due to extremely small numerical
artifacts. This forms a new 'hybrid' mixed region that maintains a low density similar to that of
the inner gas region but mixes other properties of the metal and gas, such as the ratio of specific
45
heats, Ym and yg. This hybridization causes the new region to be nearly incompressible. In this
case an attempt to compress the mixed gas region results not in a pressure and density rise but
instead in a spike in the temperature.
This inherently points to a region of non-adiabatic
compression caused by a very small numerical anomaly. Normally the mixing of the gas regions
would be a small issue on the whole, but in this case the temperature is related by:
(4-4)
T oc p '-
Despite the low density of the compressed region (compared to the metal shell) the attempt to
compress this material at such a high gamma number (14.7) results in a large rise in the
temperature.
This is a very good indication that the incompressible region cannot be explained due to
hydrodynamic causes.
4.2
Analysis of Second Test Case Results (4mm shell)
The second test case arose from the desire to attempt to construct a test case that would
yield a simulation that produces results closer to the adiabatic analytical case. The test case and
input (reference Appendix C and Appendix D for physics and input file, respectively) for this test
case are almost the same as for the 1mm thick shell compression case. The difference is that the
shell thickness was increased by a factor of four, in the hopes that the thicker shell boundary
would result in greater adiabatic compression.
In order to adapt the physics of the new
simulation case to remain adiabatic the initial metal shell velocity is decreased by
1
,f4
1
--
2
46
which still yields a final compression radius of 0.5cm, a 8000x compression of the inner gas
density, and a 2.5x compression of the metal shell density.
x10 8
Mavim unm
X1 10
Initial Condition
10
Ciompression
12
10
8
8
6
6
4
4
2
2
0
0
Pressure:
30
20
10
0
:
cycle
=
0
t
=
40
0.000E+00
Pressure:
30
20
10
0
50
:
t
cycle =378832
=
40
50
2.050E-04
Figure 4-13. Pressure plots for 4mm thick shell simulation
Figure 4-7 shows the pressure plots for the 4mm thick shell case at the initiation of the
run and at the point of maximum compression, t = 2.05x10- 4 s. The expected adiabatic result is
7.23x 101 dynes/cm2 (for compression of inner gas to 0.5cm), the maximum compression
pressure for the inner gas from the 1mm shell is 4.3x10 1 dynes/cm 2 , and the pressure at
maximum compression for the 4mm thick spherical shell is 1.2x10 1 dynes/cm 2.
10
15
8
10
6
4
5
2
-----
0
0
30
20
10
Density:
:
cycle =
0 t
=
0.OOOE+00
40
50
0-0
30
20
10
Density:
:
cycle =378832
t
=
2.050E-04
Figure 4-14. Density plots for 4mm thick shell simulation test case
40
50
47
Figure 4-8 shows mixed results. The inner gas density at maximum compression has
fallen to approximately 2.25 gm/cm3. However, the metal density error has improved, with the
density rising from around 12.0 gm/cm3 to a little over 15.0 gm/cm 3 . The redesign of the 1mm
thick shell test case to a 4mm thick shell test case was to try to arrive at experimental results that
are closer to the adiabatic case in Appendix A. The main predicator of this is the density of the
metal shell, thus we can claim a significant improvement (an 8% reduction in error). An even
thicker shell case might prove beneficial but would move away from real-world cases.
4
x1__x10
Initial Condition
..
4
6
6
4
4
2
2
0
0
0
10
tev:
20
cycle
=
30
0
t
=
0.OOOE+00
40
50
_6_._._._._
0
10
tev:
Maximu1iii I C.Imnpresion
20
cycle =378832
30
t
=
_10
40
50
2.050E-04
Figure 4-15. Temperature plots for 4mm thick shell
The temperature plots represented in Figure 4-9 still show a spike in the temperature of
the mixed material region at a radius of approximately 1.5cm. The temperature of this spike,
however, has decreased from a previous value of 6.0ev in the 1mm thick shell case to 3.9eV in
this case. This reduction, coupled with the reduction of error in the metal density at maximum
compression, indicated an increase in the adiabatic performance of this simulation test case.
48
initial
itj
im
Condiliml
1.0
1.0
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0.0
0.0
0
cycle
40
30
20
10
fracvo102
0
t
=
0.000E+00
50
fracvol02
30
20
10
0
p (osjprc.sion
cycle =378832
t
=
50
40
2.050E-04
Figure 4-16. Fractional volume of 4mm thick shell
The fractional volume plot is shown to illustrate how the shell spread more in this case than the
previous 1mm thick shell case. A large part of this is the mixing of the two working fluids and
the fact that the initial shell was 4 times thicker than in the previous case. The overall results,
however, do not show more than a marginal improvement in the under-prediction of this second
case.
Variable
Initial value
Expected value
Actual value
Ratio
Inner gas radius (R.)
Metal shell thickness (m)
10.0 cm
0.4 cm
0.5 cm
n/a
0.7 cm
5.0 cm
40%
n/a
Inner gas density (pq)
0.001 gm/cm 3
8.0 gm/cm 3
2.25 gm/cm 3
71.80%
Metal shell density (pm)
Inner gas pressure (P)
Metal region pressure (Pm)
gm/cm 3
gm/cm 3
gm/cm 3
40.00%
15.0
25.0
10
2
1.01 x1 06 dynes/cm2 7.23x1 011 dynes/cm 1.2x1 011 dynes/cm2 83.40%
n/a
1.1x1011 dynes/cm2
n/a
1.01x106 dynes/cm 2
Table 4-2. Comparison of expected and actual values for 4mm thick shell simulation
4.3
Analysis of Third Test Case (2.5gm/cm2 shell density)
The third test case is merely an extension of the premise upon which the second test case
was based. The multiple test cases were also designed to isolate any hydrodynamic reasons in
the problem initialization that might preclude an adiabatic response. Therefore, in an attempt to
49
get results even closer to adiabatic the second test case was revised to restore the original initial
metal shell velocity. To compensate for the 4 times increase in metal shell thickness and the
restoration of the initial metal velocity the density of the metal shell was reduced by four,
yielding 2.5 gm/cm3 . The results from this third case should be approximately equal to that of
the second case.
x10 10
x10
10
15
8
10
6
4
2
0
0
0
10
Pressure:
30
20
cycle =
0
t
=
40
50
0.OOOE+00
Figure 4-17. Pressure of the
0
10
Pressure:
20
cycle =187055
30
40
50
t = 1.050E-04
density, 4x thickness test case
The pressure plot for the third test case yields values close to those from the second test
case.
Some error is to be expected in the taking of measurements from the TEK plot and
EnSight program plot. The cavity between the compressed inner gas region and the metal shell
is more readily apparent in this plot of the pressure at maximum compression than in the plot for
the second test case. One reason for this is the possibility of an interim plot not being produced
that would have a higher pressure. The plots are produced at constant time step intervals, so if
the time step of 1.05x10 4 seconds is not the actual "maximum compression" the plot is missing.
The deviation would be slight, however.
In the plots below, Figure 4-18, we also get very similar results as in the second test case.
The original inner gas density is 0.001 gm/cm 3 and the final density is 2.0 g/cm 3.
This
50
corresponds very closely will the result from the second test case (ratios are almost identical).
The metal density is also similar in ratio, 40.48% compared to 40.0%. The initial metal density
in this case was 2.5 g/cm 3 as opposed to the previous two test cases where it was 10.0 g/cm 3.
Initial
.o.pr.sion
Condition
Mn
xim.m.
4
2.5
3
2.0
1.5
2
1.0
1
0.5
0
0.0
0
30
20
10
Density:
:
cycle =
0
t
=
40
50
Density:
Figure 4-18. Density of the
x10
6
4
4
2
2
0
0
10
20
:
cycle =
30
0
t =
40
0.000E+00
Figure 4-19. Temperature of the
40
30
50
t = 1.050E-04
cycle =187055
50
M
4
6
tev:
:
density, 4x thick shell compression test case
Initial Condition
x10 4
20
10
0
0.000E+00
0
10
tev:
20
:
nmpres ion
mxLIifII
C
cycle =187055
30
t = 1.050E-04
density, 4x thick shell compression test case
40
50
51
nit itt
C( ld it iq(
1.0
1.0
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0.0
0.0
fracvol02
30
20
10
0
:
cycle
=
0
t
=
, , Na %in um
"q,
40
50
0
0. OOOE+00
10
fracvol02
Figure 4-20. Fractional volume of the
C:ompryepsin
30
20
:
cycle
=187055
...
t
=
40
50
1.050E-04
density, 4x thick shell test case
The results of the third case are very similar in underprediction to the second case. This helps
prove that the reasons cited in section 4.1 for underprediction are not based on a bad initial
geometry.
Variable
Initial value
Expected value
Actual value
Ratio
40%
0.7 cm
0.5 cm
10.0 cm
Inner gas radius (R.)
n/a
5.0 cm
0.4 cm
n/a
Metal shell thickness (8m)
3
3
3
75.00%
2.0 gm/cm
8.0 gm/cm
0.001 gm/cm
Inner gas density (pg)
40.48%
3.72 gm/cm 3
6.25 gm/cm 3
2.5 gm/cm 3
Metal shell density (pm)
2 1.08x1 011 dynes/cm 2 85.06%
2
011
dynes/cm
7.23x1
1.01 x1 06 dynes/cm
Inner gas pressure (P.)
n/a
1.5x1 011 dynes/cm 2
n/a
Metal region pressure (Pm) 1.01 x1 06 dynes/cm2
Table 4-3. Expected and actual results from third test case
52
53
5
Discussion of Numerical Anomaly
The benchmark test problems used in this thesis were devised as a test for the predictive
capability of simulation methods as applied to solve strong compression problems. The specific
case first considered was that of a heavy (metal) shell filled with gas and driven inward by a
large initial velocity sufficient to compress the inner gas and metal shell resulting in a desired
density increase of 8000 times in the inner gas and 2.5 times in the metal shell. A precise
physics simulation in one dimension (and therefore without mixing) would be expected to
achieve a result very near these values, given the setup described above. When this test case was
run using the RAGE hydrodynamics code, large departures from adiabatic behavior were
observed, in spite of having a solver methodology in RAGE with very small numerical errors.
Of primary interest was to analyze the results to understand how this comes about.
First note that the underprediction anomalies observed were of two forms. The low
density gas in the inner region did not quite achieve adiabatic compression density, but this could
be accounted for by a small discrepancy in the minimum radius achieved at maximum
compression (0.6cm instead of 0.5cm). This can be considered a relatively small anomaly. The
metal shell, however, did not come close to achieving anything near the adiabatic density for the
54
pressures that were actually achieved. For the high gamma "metal" gas we expect the density
compression ratio, given a pressure ratio of Pf/Pi to be
"_ =
=
400,0001147 = 2.41
(5.1)
This is much higher than the observed ratio of 1.28 and can be considered a large
anomaly.
The following is our interpretation of the cause of this anomaly. The metal 'gas' is
observed to eject a very small fraction of material from the inner surface that propagates inward
in advance of the imploding shell. Such material cannot be accounted for by hydrodynamic
forces since the pressure profile is tending to compress the heavy material at the inner surface
(steeper pressure gradients inside the metal).
Therefore, we attribute this ejection of heavy
material to numerical errors of the sort that would be expected in a Eulerian hydrodynamics code
scheme without an interface model.
This new region of hybrid low-density, high-gamma
material could aptly be referred to as the "numerical metal gas." The numerical errors of this
kind seen in RAGE are extremely small, on the order of
10
-4 of
the metal density. Nonetheless,
in this particular situation the effect can be dramatic because the quantity of this "numerical
metal gas" is still comparable in density to the fill gas in the inner gas region and is almost
incompressible. Thus when the entire inner gas region, now comprised of the original fill gas
and this "numerical inner gas", are pushed inward the original inner gas portion compresses,
essentially adiabatically, while the numerical metal gas, because of its high gamma, mostly heats
up to achieve the ambient pressure conditions with very little density compression.
These
properties cause it to make up a large high pressure region extending outward from the fill gas to
'real' metal shell, effectively prohibiting compression of the shell to the adiabatic final state.
55
This problem tends to be inherent in these types of hydrodynamic codes.
This
numerical evaluation of the simulation and physics of the adiabatic case are important in
expressing how a seemingly small error can lead to very large losses in strong compression
simulations.
It is hoped that this thesis can be used to further increase the reliability of the
numerical algorithms found in the RAGE hydrocode.
56
57
Appendix A - 1mm Thick Shell Test Case
The data in this appendix covers the complete physics work-up of the spherical
compression benchmark.
The compression benchmark test reflects a spherical geometry of
50cm in radius split into three regions: Region I, Region II and Region III. Region II is a
spherical shell extending from a radius of 10cm to 10.1cm, effectively bisecting Region I and
Region III. The analytical calculations are performed to solve for the initial simulation quantities
for the RAGE hydrodynamic code from the given geometrical setup.
To arrive at the initial values we calculate based on a desired density compression,
p,, /p,o0 = 2.5, desired radial compression of the gas, Ri,,,,,,,_f /Rinnergas-o = 20, and the desired
density compression of the inner gas region, Pgj /Po = 8000.
Since the input variables are
solved for with these requirements in mind, these are our analytical results. If the RAGE input
deck is set up correctly we can use these values are performance indicators: a failure to reach the
level of compression as set forth above will help to indicate the presence of numerical dissipation
within the geometrical scheme or hydrodynamics code.
The initial constants are geometrically set for inner gas radius, Rg , the thickness of the
Region II shell, J,,, the outer radius of the Region II shell, R,,, the initial density of the inner
58
(Region III) and outer (Region I) gas regions, p, , and the initial density of the thin highdensity/high-gamma material, o,,,. The constants are shown below:
R6 = 10cm
6,, ~0.1cm
RM =R9 +5 ,, =10.1cm
Pg =10-3
g3
Scm3
PM =10 g3
cm
The following series of equations were presented in chapter X along with a fairly detailed
explanation. The following equation/calculation sequence provides the steps performed to arrive
at the remaining information required by the RAGE hydrodynamics code.
4TR2g "? PunM
EkO
S41r(0cm)2
(A.1)
2
(0.1cm)-1 10
2
3
u ,O2
cm3
= (628.32 ,,0 2
Having solved for the initial kinetic energy of the high-density/high-gamma material
shell we are left with a single unknown, the initial shell inward velocity, umo. Using the given
geometrical constants we can solve for the following equation:
2
u,,O~
2R
Pg
3 im PM
2
ch
1
-1)
R
-
(A.2)
59
Given that system is setup for four degrees of freedom, D = 4, we can solve for the ratios
of specific heats, or gamma number, of the gas, y g.
(A.3)
1.5
=
YD+2
2
As stated above the initial radius of the internal gas cavity is 10cm. To achieve the
desired radial compression of the gas (20 times) we must set the final inner gas cavity radius,
Rf , to 0.5cm.
Having solved for all of our constants we plug them into (x.2) which yields the following
results:
3
2
3 0.1cm 10
= 2 (1010-4
3
1
)10
(0.5)
0.5
2
210cm 10-
th
3(.5-1)
(A.4)
(2)[(89.44)-1]
2h
1.1792cth 2
The result is very close to the u,
0
cth relationship that was estimated in chapter X.
However, as before, this still leaves us with another unsolved variable,
ch,
which is the thermal
velocity. Solving for cth is performed by first calculating specific heat at constant volume, c,.
To do so requires an assumption about the working fluid, which in this case is taken to be
primarily N 14
D
k
T = cT
Pg
2
m
(A.5)
Using (x.5), as set out in Chapter 5, allows us to divide out temperature, T, from the
equation leaving the desired quantity c, on one side.
60
Cg
D
k
2P
m
(A.6)
Solving the above equation requires the Boltzmann constant, k, the mass of N2 , the
density of the inner gas region, p, , and the degree of freedom, D. The Boltzmann constant, k, is
a constant which describes the relationship between temperature and kinetic energy for
molecules in an ideal gas, whose value is equal to 1.380622
x
Joules
10-23 Kelvin
As noted above the
molecular weight of N 2 is required, since it is our working fluid. To solve for this value we use
the following conversion:
28.1 gm
N' 4 =
= 4.66x10 2 1 gm N 14
mol
6.022x10
23
atom
atom
mol
The above conversion gave us the weight of nitrogen in grams/atom. However, we are
looking at a single N14 atom so then the mass of N2 is therefore 9.332x10-23 gm. We now have
all the necessary components to solve for the value of c,. The RAGE hydrodynamics code uses
variables with units as recorded in Appendix X. Since RAGE expects the Boltzmann constant in
erg we perform the following conversion:
eV
1.38x10-
23
7
Joules 10 ergs
Kelvin 1Joule
1Kelvin
10- eV
erg
1.
3 8 x0_12
eV
We can now solve for c,:
cV =
D
2
p
k
-(
m
(A.7)
61
2
1.38x10-1 erg
eV
' 9.332x 0-2 gm
=4A(10-31
2
=2.96x10
7
erg
eV
*
cm
I
3
Having solved the relation for c, we can solve for the specific internal energy of the
inner gas region, Region III, by multiplying c, by the Region III temperature. The initial inputs
for Region III are based on standard temperature and pressure (STP). In this case this means that
the temperature is equal to 0.025eV and the pressure is equal to 14.7psi.
(A.8)
SIE = cT
=
(2.96xl0)
=7.4x10
5
erg
cm3
From our knowledge that the density of the inner gas, P, , = lOe-3 gm3 we can solve for
cm
specific internal energy in the units that RAGE required,
gm
. The result of the conversion is a
specific internal energy result of SIE = 7.4x10 8 erg. With all of the above calculations we can
gm
solve for the thermal speed of the gas,
Cth2
2
Cth2
-
2c
(A.9)
c T
D pg
=2 7.4x108 erg
4
gm
=
3.7x10 8 erg
gm
62
Now holding all of our previous unknowns we solve for the initial metal velocity and
initial kinetic energy necessary to meet our compression guidelines:
(1792)c,
u,,o2 =L
=
=
2
(A.10)
(1.17923.7xlo8)
20,887.89 cm
s
Eko
= (628.32X4.363xlo8)
= 2.74x10"1 ergs
gm
Thus the initial kinetic energy (of the high gamma spherical material region) necessary to
ergs
compress the inner gas region to a radius of 0.5cm is 2.74x10"
gm
If we say that metal pressure at maximum compression equals gas pressure we can model
the metallic region as an ideal gas with very high ratio of specific heats, y. Necessary to do so,
however, is our desired relationship between the initial and final inner gas densities. The final
expected gas density is dependant upon the ratio of the initial and final gas radii as follows:
R g JP~O
P -+~Pgf=;
(A.ll)
10cm
g)
0.5cm
=(8000)(go)
Pg
-+)Pf = P
go
~ Pgo(800)
~(7.155x10)Pgo
(A.12)
Pgo,
Using the above assumption about metal and gas pressures at final compression we make the
following statement:
63
Pf
= Pgf = P0
f
j
(A.13)
We wish to set the initial inner region gas pressure to equal the initial high-density gas region
pressure. In doing so we also wish to compress the high-density/high-gamma material to two
point five times (2.5) times its original density. To do so we must set the two pressures equal
and solve for
,:
m0
innergasO
r
PMf
PMO
Pinnergas-f
(A.14)
Pinnergas-0
r, InMf
Yg
In Pinnergas-f
PmO
Pinnergas-0
n
innergas-f
Pinnergas-0
Pmf
In8000
In 2.5
=14.71
This provides the high gamma number material for our multi-material equation of state.
This high-gamma/high-density material is the thin spherical shell that is set up as region II. We
want to set external pressure to a high enough number to help hold together the outer edge of the
high-gamma/high-density material region while still attempting to keep any over-pressure shock
waves to a minimum (if not eliminated entirely). Using the results for initial thin shell velocity
and density of the thin shell region we can calculate the Region I pressure required:
64
P
(A.15)
= !puO2
1,
r
4.363x10l 0, ergj
=S1o 10 gm
2(
=
gm
CM3
2.182xlO 9 ergs
As noted earlier, initial conditions have the inner gas region set to STP, or standard
temperature and pressure. In this case, the initial temperature of the inner gas region (T) is set to
0.025eV. Likewise, the initial inner gas region pressure is set to atmospheric pressure at sea
level, or 14 .7psi. Due to the units required by RAGE in the input deck, we must convert the
pressure to either dynes/cm2 or ergs/cm3 (both are equivalent).
Po =14.7psi
=
101325Pa
=
N
101325 N
=1O1325,~Qd~e
M2
=
10-5N
2
104CM2
6
1.01x10 dynes
2
cm2
RAGE does not allow the pressure of a region to be set directly in the input deck.
Instead, the specific internal energy (SIE) of the region is required and the pressure calculated
internally. Having solved for initial region III pressure we know use equation x.x from chapter 3
to solve for the region III SIE.
P = g-1J(A.16)
9(
P
e innergas =
innerga
(A.17)
65
6
1.01x10
ergs
CM3
)
cm3
= 2.02x10,
erg
gm
Having stated previously that in our initial setup the region III (inner region) gas pressure
equals the region II (high-gamma/high-density) pressure we can solve for the SIE of region II.
e,
)
=
innergas
regonh - 0(
1.01x10
6
(A.18)
ergs
CM3
gm
10 cm
(14.7 -1
=7.37x10
3
erg
gm
Solving for the pressure in region I (surrounding gas) is based on the desired 20 times
compression of the inner gas, to a radius of 0.5cm. As such, solving for the outer gas pressure
evolves from equation x.x in chapter X:
Pouergas
r
Rgo
= Pgo (20)"
= Pgo (716,000)
= 7.23x10' ergs
CM3
Now adapting (x. 16) for Region I we yield the following results:
(A.19)
66
Polutergas
eregionI
(A.20)
"''"
(A2=
7.23x10" ergs
CM3
(1.5 -1
1.45x0 1
10_3 gM
cm3
5
ergs
gm
67
Appendix B - 1 mm Spherical Shell Compression Input Deck
pname =
! problem name for output files
"ld-r220a"
! CONTROL
setup problem if < 1, otherwise, restart cycle
maximum cycle for run
maximum time for run
dump frequency in time
dump frequency in cycles
short edit frequency in cycles
allows the first time step to be reduced
kread = -l
ncmax = 500000
tmax = 6.0e-4
tedit = 5.0e-6
ncedit = 0
modcyc = 0
dtpct = 0.0
shortmodcyc = 50
uselast = .true.
! HYDRO
hydro-version = 1
! PLOTS
grid =
.true.
onenum = 6
onetype(l) =
'prs',
'rho',
'tev',
'mat','vOl','v02'
! SETUP
numfine = 4
!number of cells
cell dimension
imxset = 500
dxset = 0.1
sphere =
!
.true.
geometry
MATERIALS
keos = 0
gamma law gas eos
nummat = 2
number of materials
= 0.5
= 2.96e7
gamma -
matdef(16,2) = 13.7
matdef(30,2) = 2.95e8
gamma -
matdef (16,1)
matdef (30,1)
!
GRID ADAPTION
sizemat(2) = 0.003125
1 of gas
! cv
1 of metal
number of "tek" plots
types of tek plots
68
!
REGIONS
! number of regions
numreg = 3
typreg(1)
matreg(1)
rhoreg(1)
siereg(l)
=
=
=
=
1
1
0.001
2.02e12
circle
material of region 1
density of outer gas region
SIE of outer gas region (I)
typreg(2) = 1
diareg(1,2) = 20.2
matreg(2) = 2
rhoreg(2) = 10.0
siereg(2) = 7.37e3
drdreg(2) = -2.09e3
rdreg(2) = 0
circle
outer diameter of metal region
use 'metal' material
3
density = 10.0 g/cm
SIE of Region II (metal shell)
Initial metal velocity inward
typreg(3) =
diareg(1,3)
matreg(3) =
rhoreg(3) =
siereg(3) =
drdreg(3) =
circle
Outer diameter of inner gas region (R=10.Ocm)
use 'gas' material
3
density = 0.001 g/cm
SIE of Region I
Initial inward velocity of Region III
1
= 20.0
1
0.001
2.02e9
-2.09e3
(R=10.1cm)
69
Appendix C - 4mm Spherical Shell Compression Input Deck
pname =
"ld-r100"
! problem name for output files
! CONTROL
setup problem if < 1, otherwise, restart cycle
maximum cycle for run
kread = -1
ncmax = 500000
tmax = 6.0e-3
tedit = 5.0e-6
ncedit = 0
modcyc = 0
dtpct = 0.0
shortmodcyc = 50
uselast = .true.
maximum time for run
dump frequency in time
dump frequency in cycles
short edit frequency in cycles
allows the first time step to be reduced
! HYDRO
hydro-version = 1
! PLOTS
grid = .true.
onenum = 6
onetype(l) = 'prs',
!
number of "tek" plots
'rho',
'mat','v01','v02'
'tev',
SETUP
numfine = 4
imxset = 500
dxset = 0.1
number of cells
cell dimension
sphere =
geometry
.true.
! MATERIALS
gamma law gas eos
keos = 0
nummat = 2
number of materials
= 0.5
= 2.96e7
gamma
matdef(16,2) = 13.7
matdef(30,2) = 2.95e8
gamma
matdef (16,1)
matdef (30,1)
GRID ADAPTI ON
sizemat(l) =
sizemat(2) =
sizecell =
sizebnd(2)
sizerho(2)
!
REGIONS
0.0125
0.003125
0.05
= 0.0125
= 100
-
1
-
1
! cv
70
numreg = 3
typreg(l) =
matreg(l) =
rhoreg(l) =
siereg(l) =
! drdreg(l)
number of regions
1
1
circle
material of region 1
0.001
2.02e12
= 0.0
circle
typreg(2) = 1
diareg(1,2) = 20.8
matreg(2) = 2
rhoreg(2) = 10.0
siereg(2) = 7.37e3
drdreg(2) = -1.045e3
rdreg(2) = 0
typreg(3) =
diareg(1,3)
matreg(3) =
rhoreg(3) =
siereg(3) =
drdreg(3) =
1
= 20.0
1
0.001
2.02e9
-1.045e3
circle
71
Appendix D - Low Density Shell Compression Input Deck
pname =
"id-r100"
! problem name for output files
! CONTROL
kread = -1
ncmax = 500000
tmax = 6.0e-3
tedit = 5.0e-6
ncedit = 0
modcyc = 0
dtpct = 0.0
shortmodcyc = 50
uselast = .true.
setup problem if < 1, otherwise, restart cycle
maximum cycle for run
maximum time for run
dump frequency in time
dump frequency in cycles
short edit frequency in cycles
allows the first time step to be reduced
! HYDRO
hydro version = 1
! PLOTS
grid = .true.
onenum = 6
onetype(l) = 'prs',
'rho',
'tev',
'mat','v01','v02'
! SETUP
numfine = 4
imxset = 500
dxset = 0.1
number of cells
cell dimension
sphere =
geometry
!
.true.
MATERIALS
keos = 0
nummat = 2
! gamma law gas eos
! number of materials
matdef(16,1) = 0.5
matdef(30,1) = 2.96e7
gamma
cv
matdef(16,2) = 13.7
matdef(30,2) = 2.95e8
gamma
GRID ADAPTION
sizemat(l) = 0.0125
sizemat(2) = 0.003125
sizecell = 0.05
sizebnd(2) = 0.0125
sizerho(2) = 100
! REGIONS
-
1 of gas
1 of metal
cv of metal
number of "tek" plots
types of plots
72
number of regions
numreg = 3
typreg(l) =
matreg(l) =
rhoreg(l) =
siereg(l) =
! drdreg(l)
circle
material of region 1
1
1
0.001
2.02e12
=
0.0
typreg(2) = 1
diareg(1,2) = 20.8
matreg(2) = 2
rhoreg(2) = 2.5
siereg(2) = 7.37e3
drdreg(2) = -2.09e3
rdreg(2) = 0
circle
typreg(3) =
diareg(1,3)
matreg(3) =
rhoreg(3) =
siereg(3) =
drdreg(3) =
circle
1
= 20.0
1
0.001
2.02e9
-2.09e3