In this work, the gas-kinetic method (GKM) is enhanced with resistive and Hall magnetohydrodynamics (MHD) effects. Known as MGKM (for MHD–GKM), this approach incorporates additional source terms to the momentum and energy conservation equations and solves the magnetic field induction equation. We establish a verification protocol involving numerical solutions to the one-dimensional (1D) shock tube problem and two-dimensional (2D) channel flows. The contributions of ideal, resistive, and Hall effects are examined in isolation and in combination against available analytical and computational results. We also simulate the evolution of a laminar MHD jet subject to an externally applied magnetic field. This configuration is of much importance in the field of plasma propulsion. Results support previous theoretical predictions of jet stretching due to magnetic field influence and azimuthal rotation due to the Hall effect. In summary, MGKM is established as a promising tool for investigating complex plasma flow phenomena.

References

References
1.
Richard
,
J. C.
,
Riley
,
B. M.
, and
Girimaji
,
S. S.
,
2011
, “
Magnetohydrodynamic Turbulence Decay Under the Influence of Uniform or Random Magnetic Fields
,”
ASME J. Fluids Eng.
,
133
(
8
), p.
081205
.10.1115/1.4003985
2.
Shebalin
,
J. V.
,
2014
, “
Temperature and Entropy in Ideal Magnetohydrodynamic Turbulence
,”
ASME J. Fluids Eng.
,
136
(
6
), p.
060901
.10.1115/1.4025674
3.
Chatterjee
,
D.
,
Chatterjee
,
K.
, and
Mondal
,
B.
,
2012
, “
Control of Flow Separation Around Bluff Obstacles by Transverse Magnetic Field
,”
ASME J. Fluids Eng.
,
134
(
9
), p.
091102
.10.1115/1.4007316
4.
Tordella
,
D.
,
Belan
,
M.
,
Massaglia
,
S.
,
De Ponte
,
S.
,
Mignone
,
A.
,
Bodenschatz
,
E.
, and
Ferrari
,
A.
,
2011
, “
Astrophysical Jets: Insights Into Long-Term Hydrodynamics
,”
New J. Phys.
,
13
, p.
043011
.10.1088/1367-2630/13/4/043011
5.
Lorzel
,
H.
, and
Mikellides
,
P. G.
,
2010
, “
Three-Dimensional Modeling of Magnetic Nozzle Processes
,”
AIAA J.
,
48
(
7
), pp.
1494
1503
.10.2514/1.J050123
6.
Xisto
,
C. M.
,
Páscoa
,
J. C.
, and
Oliveira
,
P. J.
,
2013
, “
Numerical Modeling of Electrode Geometry Effects on a 2D Self-Field MPD Thruster
,”
ASME
Paper No. V001T01A051.10.1115/IMECE2013-63144
7.
Mikellides
,
P. G.
,
Turchi
,
P. J.
, and
Roderick
,
N. F.
,
2000
, “
Applied-Field Magnetoplasmadynamic Thrusters, Part 1: Numerical Simulations Using the MACH2 Code
,”
J. Propul. Power
,
16
(
5
), pp.
887
893
.10.2514/2.5656
8.
Mikellides
,
I. G.
,
Mikellides
,
P. G.
,
Turchi
,
P. J.
, and
York
,
T. M.
,
2002
, “
Design of a Fusion Propulsion System-Part 2: Numerical Simulation of Magnetic-Nozzle Flows
,”
J. Propul. Power
,
18
(
1
), pp.
152
158
.10.2514/2.5911
9.
Liao
,
W.
,
Peng
,
Y.
,
Luo
,
L.-S.
, and
Xu
,
K.
,
2008
, “
Modified Gas-Kinetic Scheme for Shock Structures in Argon
,”
Prog. Comput. Fluid Dyn.
,
8
(
1–4
), pp.
97
108
.10.1504/PCFD.2008.018082
10.
Xu
,
K.
,
Martinelli
,
L.
, and
Jameson
,
A.
,
1995
, “
Gas-Kinetic Finite Volume Methods, Flux-Vector Splitting, and Artificial Diffusion
,”
J. Comput. Phys.
,
120
(
1
), pp.
48
65
.10.1006/jcph.1995.1148
11.
Xu
,
K.
,
2001
, “
A Gas-Kinetic Scheme for the Navier–Stokes Equations and its Connection With Artificial Dissipation and the Godunov Method
,”
J. Comput. Phys.
,
171
(
1
), pp.
289
335
.10.1006/jcph.2001.6790
12.
Kerimo
,
J.
, and
Girimaji
,
S.
,
2007
, “
Boltzmann-BGK Approach to Simulating Weakly Compressible 3D Turbulence: Comparison Between Lattice Boltzmann and Gas Kinetic Methods
,”
J. Turbul.
,
8
(
46
).10.1080/14685240701528551
13.
Kumar
,
G.
,
Girimaji
,
S.
, and
Kerimo
,
J.
,
2013
, “
WENO-Enhanced Gas-Kinetic Scheme for Direct Simulations of Compressible Transition and Turbulence
,”
J. Comput. Phys.
,
243
, pp.
499
523
.10.1016/j.jcp.2012.10.005
14.
Xu
,
K.
, and
Tang
,
H.
,
2000
, “
A High-Order Gas-Kinetic Method for Multidimensional Ideal Magnetohydrodynamics
,”
J. Comput. Phys.
,
165
(
1
), pp.
69
88
.10.1006/jcph.2000.6597
15.
Xu
,
K.
,
1999
, “
Gas-Kinetic Theory-Based Flux Splitting Method for Ideal Magnetohydrodynamics
,”
J. Comput. Phys.
,
153
(
2
), pp.
334
352
.10.1006/jcph.1999.6280
16.
Tang
,
H.
,
Xu
,
K.
, and
Cai
,
C.
,
2010
, “
Gas-Kinetic BKG Scheme for Three Dimensional Magnetohydrodynamics
,”
Numer. Math.: Theory Meth. Appl.
,
3
(
4
), pp.
387
404
.10.4208/nmtma.2010.m9007
17.
Fuchs
,
F.
,
Mishra
,
S.
, and
Risebro
,
N.
,
2009
, “
Splitting Based Finite Volume Schemes for Ideal MHD Equations
,”
J. Comput. Phys.
,
228
(
3
), pp.
641
660
.10.1016/j.jcp.2008.09.027
18.
Huba
,
J.
,
2003
, “
Hall Magnetohydrodynamics—A Tutorial
,”
Space Plasma Simulation
,
Springer
, Berlin, pp.
166
192
.
19.
Brackbill
,
J.
, and
Barnes
,
D.
,
1980
, “
The Effect of Nonzero ·Bon the Numerical Solution of the Magnetohydrodynamic Equations
,”
J. Comput. Phys.
,
35
(
3
), pp.
426
430
.10.1016/0021-9991(80)90079-0
20.
Tóth
,
G.
,
2000
, “
The ·B=0Constraint in Shock-Capturing Magnetohydrodynamics Codes
,”
J. Comput. Phys.
,
161
(
2
), pp.
605
652
.10.1006/jcph.2000.6519
21.
Sutton
,
G.
, and
Sherman
,
A.
,
1965
,
Engineering Magnetohydrodynamics
,
McGraw-Hill
,
New York
.
22.
Amano
,
R.
,
Xu
,
Z.
, and
Lee
,
C.-H.
,
2007
, “
Numerical Simulation of Supersonic MHD Channel Flows
,”
ASME 2007 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference
,
American Society of Mechanical Engineers
, pp.
669
676
.
23.
Attia
,
H. A.
,
2005
, “
Unsteady MHD Couette Flow and Heat Transfer Between Parallel Porous Plates With Exponential Decaying Pressure Gradient
,”
ASME 2005 International Mechanical Engineering Congress and Exposition
,
American Society of Mechanical Engineers
, pp.
901
905
.
24.
Jiang
,
F.
,
Oliveira
,
M. S.
, and
Sousa
,
A. C.
,
2005
, “
SPH Simulations for Turbulence Control of Magnetohydrodynamic Poiseuille Flow
,”
ASME 2005 Fluids Engineering Division Summer Meeting
,
American Society of Mechanical Engineers
, pp.
385
394
.
25.
Midya
,
C.
,
Layek
,
G.
,
Gupta
,
A.
, and
Mahapatra
,
T. R.
,
2003
, “
Magnetohydrodynamic Viscous Flow Separation in a Channel With Constrictions
,”
ASME J. Fluids Eng.
,
125
(
6
), pp.
952
962
.10.1115/1.1627834
26.
Smith
,
T.
, and
Paul
,
P.
,
1979
, “
Radiative Transfer in Hartmann MHD Flow
,”
ASME J. Heat Transfer
,
101
(
3
), pp.
502
506
.10.1115/1.3451017
27.
Liu
,
R.
,
Vanka
,
S. P.
, and
Thomas
,
B. G.
,
2014
, “
Particle Transport and Deposition in a Turbulent Square Duct Flow With and Imposed Magnetic Field
,”
ASME J. Fluids Eng.
,
136
(
12
), p.
121201
.10.1115/1.4027624
28.
Sato
,
H.
,
1961
, “
The Hall Effect in the Viscous Flow of Ionized Gas Between Parallel Plates Under Transverse Magnetic Field
,”
J. Phys. Soc. Jpn.
,
16
(
7
), pp.
1427
1433
.10.1143/JPSJ.16.1427
29.
Brio
,
M.
, and
Wu
,
C.
,
1988
, “
An Upwind Differencing Scheme for the Equations of Ideal Magnetohydrodynamics
,”
J. Comput. Phys.
,
75
(
2
), pp.
400
422
.10.1016/0021-9991(88)90120-9
30.
Ebersohn
,
F.
,
2012
, “
Gas Kinetic Study of Magnetic Field Effects on Plasma Plumes
,” Master's thesis, Texas A&M University, College Station, TX.
31.
Srinivasan
,
B.
,
2010
, “
Numerical Methods for Three-Dimensional Magnetic Confinement Configurations Using Two-Fluid Plasma Equations
,” Ph.D. thesis, University of Washington, Seattle, WA.
32.
Shumlak
,
U.
, and
Loverich
,
J.
,
2003
, “
Approximate Riemann Solver for the Two-Fluid Plasma Model
,”
J. Comput. Phys.
,
187
(
2
), pp.
620
638
.10.1016/S0021-9991(03)00151-7
33.
Araya
,
D.
,
2011
, “
Resistive MHD Simulations of Laminar Round Jets With Applications to Magnetic Nozzle Flows
,” Master's thesis, Texas A&M University, College Station, TX.
34.
Davidson
,
P.
,
1995
, “
Magnetic Damping of Jets and Vortices
,”
J. Fluid Mech.
,
229
, pp.
153
185
.10.1017/S0022112095003466
35.
Gerwin
,
R.
,
Markin
,
G.
,
Sgro
,
A.
, and
Glasser
,
A.
,
1990
, “
Characterization of Plasma Flow Through Magnetic Nozzles
,” Los Alamos National Laboratory, Technical Report No. AL-TR-89-092.
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