Three-dimensional (3D) modeling of magneto-inertial fusion (MIF) is at a nascent stage of development. A suite of test cases relevant to plasma liner formation and implosion is presented to present the community with some exact solutions for verification of hydrocodes pertaining to MIF confinement concepts. MIF is of particular interest to fusion research, as it may lead to the development of smaller and more economical reactor designs for power and propulsion. The authors present simulated test cases using a new smoothed particle hydrodynamic (SPH) code called SPFMax. These test cases consist of a total of six problems with analytical solutions that incorporate the physics of radiation cooling, heat transfer, oblique-shock capturing, angular-momentum conservation, and viscosity effects. These physics are pertinent to plasma liner formation and implosion by merging of a spherical array of plasma jets as a candidate standoff driver for MIF. An L2 norm analysis was conducted for each test case. Each test case was found to converge to the analytical solution with increasing resolution, and the convergence rate was on the order of what has been reported by other SPH studies.

References

References
1.
Lindemuth
,
I.
, and
Siemon
,
R.
,
2009
, “
The Fundamental Parameter Space of Controlled Thermonuclear Fusion
,”
Am. J. Phys.
,
77
(
5
), pp.
407
416
.
2.
Cassibry
,
J.
,
Cortez
,
R.
,
Stanic
,
M.
,
Watts
,
A.
,
Seidler
,
W.
,
Adams
,
R.
,
Statham
,
G.
, and
Fabisinski
,
L.
,
2015
, “
Case and Development Path for Fusion Propulsion
,”
J. Spacecr. Rockets
,
52
(
2
), pp.
595
611
.
3.
Lindemuth
,
I. R.
, and
Kirkpatrick
,
R. C.
,
1983
, “
Parameter Space for Magnetized Fuel Targets in Inertial Confinement Fusion
,”
Nucl. Fusion
,
23
(
3
), pp.
263
284
.
4.
Kirkpatrick
,
R. C.
,
Lindemuth
,
I. R.
, and
Ward
,
M. S.
,
1995
, “
Magnetized Target Fusion: An Overview
,”
Fusion Technol.
,
27
(
3
), pp.
201
214
.
5.
Miernik
,
J.
,
Statham
,
G.
,
Fabisinski
,
L.
,
Maples
,
C.
,
Adams
,
R.
,
Polsgrove
,
T.
,
Fincher
,
S.
,
Cassibry
,
J.
,
Cortez
,
R.
,
Turner
,
M.
, and
Percy
,
T.
,
2013
, “
Z-Pinch Fusion-Based Nuclear Propulsion
,”
Acta Astronaut.
,
82
(
2
), pp.
173
182
.
6.
Adams
,
R.
,
Alexander
,
R.
,
Chapman
,
J.
,
Fincher
,
J.
,
Philips
,
S.
,
Polsgrove
,
A.
,
Wayne
,
T.
,
Patton
,
B.
,
Statham
,
G.
,
White
,
S.
, and
Thio
,
Y.
,
2003
, “
Conceptual Design of In-Space Vehicles for Human Exploration of the Outer Planets
,” National Aeronautics and Space Administration, Washington, DC, Report No.
2003-212691
.
7.
Slutz
,
S. A.
,
Herrmann
,
M. C.
,
Vesey
,
R. A.
,
Sefkow
,
A. B.
,
Sinars
,
D. B.
,
Rovang
,
D. C.
,
Peterson
,
K. J.
, and
Cuneo
,
M. E.
,
2010
, “
Pulsed-Power-Driven Cylindrical Liner Implosions of Laser Preheated Fuel Magnetized With an Axial Field
,”
Phys. Plasmas
,
17
(
5
), p.
056303
.
8.
Slutz
,
S. A.
, and
Vesey
,
R. A.
,
2012
, “
High-Gain Magnetized Inertial Fusion
,”
Phys. Rev. Lett.
,
108
(
2
), p. 025003.
9.
Gomez
,
M. R.
,
Slutz
,
S. A.
,
Sinars
,
D.
,
Hahn
,
K. D.
,
Hansen
,
S. B.
,
Harding
,
E. C.
,
Knapp
,
P. F.
,
Schmit
,
P. F.
,
Jennings
,
C. A.
,
Awe
,
T. J.
,
Geissel
,
M.
,
Rovang
,
D.
,
Chandler
,
G. A.
,
Cooper
,
G. W.
,
Cuneo
,
M. E.
,
Harvey-Thompson
,
A. J.
,
Herrmann
,
M. C.
,
Hess
,
M. H.
,
Johns
,
O.
,
Lamppa
,
D. C.
,
Marin
,
M. R.
,
McBride
,
R. D.
,
Peterson
,
K. J.
,
Porter
,
J. L.
,
Robertson
,
G. K.
,
Rochau
,
G. A.
,
Ruiz
,
C. L.
,
Savage
,
M. E.
,
Smith
,
I. C.
,
Stygar
,
W. A.
, and
Vesey
,
R. A.
,
2014
, “
Experimental Demonstration of Fusion-Relevant Conditions in Magnetized Liner Inertial Fusion
,”
Phys. Rev. Lett.
,
113
(
15
), p.
155003
.
10.
Thio
,
Y.
,
Kirkpatrick
,
R.
,
Knapp
,
C. E.
,
Wysocki
,
F.
,
Parks, P.
, and
Schmidt, G.
,
1999
, “
Magnetized Target Fusion in a Spheroidal Geometry With Standoff Drivers
,”
Current Trends in International Fusion Research—Second Symposium
, Ottawa, ON, Canada, pp. 113–132.
11.
Hsu
,
S. C.
,
Awe
,
T. J.
,
Brockington
,
S.
,
Case
,
A.
,
Cassibry
,
J. T.
,
Kagan
,
G.
,
Messer
,
S. J.
,
Stanic
,
M.
,
Tang
,
X.
,
Welch
,
D. R.
, and
Witherspoon
,
F. D.
,
2012
, “
Spherically Imploding Plasma Liners as a Standoff Driver for Magnetoinertial Fusion
,”
IEEE Trans. Plasma Sci.
,
40
(
5
), pp.
1287
1298
.
12.
Samulyak
,
R.
,
Parks
,
P.
, and
Wu
,
L.
,
2010
, “
Spherically Symmetric Simulation of Plasma Liner Driven Magnetoinertial Fusion
,”
Phys. Plasmas
,
17
(
9
), p. 092702.
13.
Kim
,
H.
,
Samulyak
,
R.
,
Zhang
,
L.
, and
Parks
,
P.
,
2012
, “
Influence of Atomic Processes on the Implosion of Plasma Liners
,”
Phys. Plasmas
,
19
(
8
), p.
082711
.
14.
Hopkins
,
P. F.
,
2015
, “
A New Class of Accurate, Mesh-Free Hydrodynamic Simulation Methods
,”
Mon. Not. R. Astron. Soc.
,
450
(
1
), pp.
53
110
.
15.
Cassibry
,
J.
,
Cortez
,
R.
,
Hsu
,
S.
, and
Witherspoon
,
F.
,
2009
, “
Estimates of Confinement Time and Energy Gain for Plasma Liner Driven Magnetoinertial Fusion Using an Analytic Self-Similar Converging Shock Model
,”
J. Plasma Phys.
,
16
(
11
), p.
112707
.
16.
Cassibry
,
J. T.
,
2004
, “
Numerical Modeling Studies of a Coaxial Plasma Accelerator as a Standoff Driver for Magnetized Target Fusion
,” Doctoral dissertation, University of Alabama in Huntsville, Huntsville, AL.
17.
Knapp
,
C. E.
, and
Kirkpatrick
,
R. C.
,
2014
, “
Possible Energy Gain for a Plasma-Liner-Driven Magneto-Inertial Fusion Concept
,”
Phys. Plasmas
,
21
(
7
), p. 070701.
18.
Cassibry
,
J.
,
Stanic
,
M.
, and
Hsu
,
S.
,
2013
, “
Ideal Hydrodynamic Scaling Relations for a Stagnated Imploding Spherical Plasma Liner Formed by an Array of Merging Plasma Jets
,”
Phys. Plasmas
,
20
(
3
), p.
032706
.
19.
Santarius
,
J. F.
,
2012
, “
Compression of a Spherically Symmetric Deuterium-Tritium Plasma Liner Onto a Magnetized Deuterium-Tritium Target
,”
Phys. Plasmas
,
19
(
7
), p.
072705
.
20.
Awe
,
T.
,
Adams
,
C. S.
,
Davis
,
J. S.
,
Hanna
,
D. S.
,
Hsu
,
S. C.
, and
Cassibry
,
J. T.
,
2011
, “
One-Dimensional Radiation-Hydrodynamic Scaling Studies of Imploding Spherical Plasma Liners
,”
Phys. Plasmas
,
18
(
7
), p.
072705
.
21.
Davis
,
J. S.
,
Hsu
,
S. C.
,
Golovkin
,
I. E.
,
MacFarlane
,
J. J.
, and
Cassibry
,
J. T.
,
2012
, “
One-Dimensional Radiation-Hydrodynamic Simulations of Imploding Spherical Plasma Liners With Detailed Equation-of-State Modeling
,”
Phys. Plasmas
,
19
(
10
), p. 102701.
22.
Cassibry
,
J.
,
Stanic
,
M.
,
Hsu
,
S. C.
,
Abarzhi
,
S. I.
, and
Witherspoon
,
F. D.
,
2012
, “
Tendency of Spherically Imploding Plasma Liners Formed by Merging Plasma Jets to Evolve Toward Spherical Symmetry
,”
Phys. Plasmas
,
19
(
5
), p.
052702
.
23.
Liu
,
G.
, and
Liu
,
M.
,
2003
,
Smoothed Particle Hydrodynamics: A Meshfree Particle Method
, World Scientific, Singapore.
24.
Monaghan
,
J.
,
2005
, “
Smoothed Particle Hydrodynamics
,”
Rep. Prog. Phys.
,
68
(
8
), pp.
1703
1759
.
25.
Hsu
,
S. C.
,
Langendorf
,
S. J.
,
Yates
,
K. C.
,
Dunn
,
J. P.
,
Brockington
,
S.
,
Case
,
A.
,
Cruz
,
E.
,
Witherspoon
,
F. D.
,
Gilmore
,
M. A.
,
Cassibry
,
J. T.
,
Samulyak
,
R.
,
Stoltz
,
P.
,
Schillo
,
K.
,
Shih
,
W.
,
Beckwith
,
K.
, and
Thio
,
Y. C. F.
,
2017
, “
Experiment to Form and Characterize a Section of a Spherically Imploding Plasma Liner
,”
IEEE Trans. Plasma Sci.
,
PP
(
99
), pp.
1
11
.
26.
Rodriguez
,
M. A.
, and
Cassibry
,
J. T.
,
2017
, “
A 3-D Smoothed-Particle Hydrodynamics Model of Electrode Erosion
,”
IEEE Trans. Plasma Sci.
,
45
(
11
), pp.
3030
3037
.
27.
Schillo
,
K. J.
,
Cassibry
,
J. T.
,
Rodriguez
,
M.
, and
Thompson
,
S.
,
2016
, “
Test Suite for Hydrodynamic Modeling for Plasma Driven Magneto-Inertial Fusion
,”
AIAA
Paper No. 2016-4686.
28.
MacFarlane
,
J. J.
,
Golovkin
,
I. E.
, and
Woodruff
,
P. R.
,
2006
, “
HELIOS-CR—A 1-D Radiation-Magnetohydrodynamics Code With Inline Atomic Kinetics Modeling
,”
J. Quant. Spectrosc. Radiat. Transfer
,
99
(
1–3
), pp.
381
397
.
29.
Fatehi
,
R.
, and
Manzari
,
M. T.
,
2011
, “
Error Estimation in Smoothed Particle Hydrodynamics and a New Scheme for Second Derivatives
,”
Comput. Math. Appl.
,
61
(
2
), pp.
482
498
.
30.
Bonet
,
J.
, and
Lok
,
T. S. L.
,
1996
, “
Variational and Momentum Preservation Aspects of Smooth Particle Hydrodynamic Formulation
,”
Comput. Methods Appl. Mech. Eng.
,
180
(
1–2
), pp.
97
115
.
31.
Watkins
,
S. J.
,
Bhattal
,
A. S.
,
Francis
,
N.
,
Turner
,
J. A.
, and
Whitworth
,
A. P.
,
1996
, “
A New Prescription for Viscosity in Smoothed Particle Hydrodynamics
,”
Astron. Astrophys.
,
119
, pp.
177
187
.
32.
Jeong
,
J. H.
,
Jhon
,
M. S.
,
Halow
,
J. S.
, and
Osdol
,
J. V.
,
2003
, “
Smoothed Particle Hydrodynamics: Applications to Heat Conduction
,”
Comput. Phys. Commun.
,
153
(
1
), pp.
71
84
.
33.
Morris
,
J. P.
,
Patrick
,
J. F.
, and
Zhu
,
Y.
,
1997
, “
Modeling Low Reynolds Number Incompressible Flows Using SPH
,”
J. Comput. Phys.
,
136
(
1
), pp.
214
226
.
34.
Castor
,
J.
,
2004
,
Radiation Hydrodynamics
,
Cambridge University Press
,
Cambridge, UK
, p.
368
.
35.
Shampine
,
L. F.
, and
Reichelt
,
M. W.
,
1997
, “
The MATLAB ODE Suite
,”
SIAM J. Sci. Comput.
,
18
(
1
), pp.
1
22
.
36.
VanSant
,
J. H.
,
1983
, “
Conduction Heat Transfer Solutions
,” Lawrence Livermore National Laboratory, CA, Report No. UCRL-52863-Rev.1.
37.
Zhu
,
Q.
,
Hernquist
,
L.
, and
Li
,
Y.
,
2015
, “
Numerical Convergence in Smoothed Particle Hydrodynamics
,”
Astrophys. J.
,
800
(
6
), pp. 1–13.
38.
Anderson
,
J. D.
,
1991
,
Fundamentals of Aerodynamics
,
2nd ed.
,
McGraw-Hill
,
New York
.
39.
Liska
,
R.
, and
Wendroff
,
B.
,
2003
, “
Comparison of Several Different Schemes on 1D and 2D Test Problems for the Euler Equations
,”
SIAM J. Sci. Comput.
,
25
(
3
), pp.
995
1017
.
40.
Balsara
,
D.
,
2001
, “
Divergence-Free Adaptive Mesh Refinement for Magnetohydrodynamics
,”
J. Comput. Phys.
,
174
(
2
), pp.
614
648
.
41.
Merritt
,
E. C.
,
Moser
,
A. L.
,
Hsu
,
S. C.
,
Loverich
,
J.
, and
Gilmore
,
M.
,
2013
, “
Experimental Characterization of the Stagnation Layer Between Two Obliquely Merging Supersonic Plasma Jets
,”
Phys. Rev. Lett.
,
111
(
8
), p.
085003
.
42.
Merritt
,
E. C.
,
Moser
,
A. L.
,
Hsu
,
S. C.
,
Adams
,
C. S.
,
Dunn
,
J. P.
,
Holgado
,
A. M.
, and
Gilmore
,
M. A.
,
2014
, “
Experimental Evidence for Collisional Shock Formation Via Two Obliquely Merging Supersonic Plasma Jets
,”
Phys. Plasmas
,
21
(
5
), p. 055703.
43.
Anderson
,
J.
,
2003
,
Modern Compressible Flow
,
3rd ed.
,
McGraw-Hill
,
New York
, Chap. 4.
44.
Hsu
,
S. C.
,
Langendorf
,
S. J.
,
Dunn
,
J. P.
,
Yates
,
K. C.
,
Gilmore
,
M. A.
,
Witherspoon
,
F. D.
,
Brockington
,
S.
,
Case
,
A.
,
Cruz
,
E.
, and
Thio
,
Y. C. F.
,
2017
, “
Characterizing an Octant of a Spherically Imploding Plasma Liner as an MIF Driver
,”
Bull. Am. Phys. Soc.
,
62
(12), p. BAPS.2017.DPP.T07.3
45.
Taylor
,
G. I.
, and
Green
,
A. E.
,
1937
, “
Mechanism of the Production of Small Eddies From Larger Ones
,”
Proc. R. Soc. London. Series A
,
158
(
895)
, pp.
499
521
.
46.
DeBonis
,
J. R.
,
2013
, “
Solutions of the Taylor-Green Vortex Problem Using High-Resolution Explicit Finite Difference Methods
,”
AIAA
Paper No. 2013-0382.
47.
Kajzer
,
A.
,
Pozorski
,
J.
, and
Szwec
,
K.
,
2014
, “
Large-Eddy Simulations of 3D Taylor-Green Vortex: Comparison of Smoothed Particle Hydrodynamics, Lattice Boltzmann and Finite Volume Methods
,”
J. Phys.: Conf. Ser.
,
530
(1), p. 012019.
You do not currently have access to this content.