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Research Papers: Multiphase Flows

Magneto-Gas Kinetic Method for Nonideal Magnetohydrodynamics Flows: Verification Protocol and Plasma Jet Simulations

[+] Author and Article Information
Daniel B. Araya

Aerospace Laboratories,
California Institute of Technology,
Pasadena, CA 91125
e-mail: daniel.b.araya@gmail.com

Frans H. Ebersohn

Aerospace Engineering Department,
University of Michigan,
Ann Arbor, MI 48109
e-mail: febersohn@gmail.com

Steven E. Anderson

Aerospace Engineering Department,
Texas A&M University,
College Station, TX 77843
e-mail: ande.ste88@gmail.com

Sharath S. Girimaji

Professor
Aerospace Engineering Department,
Texas A&M University,
College Station, TX 77843
e-mail: girimaji@tamu.edu

1Corresponding author.

Contributed by the Fluids Engineering Division of ASME for publication in the JOURNAL OF FLUIDS ENGINEERING. Manuscript received September 5, 2014; final manuscript received March 2, 2015; published online April 28, 2015. Assoc. Editor: Shizhi Qian.

J. Fluids Eng 137(8), 081302 (Aug 01, 2015) (11 pages) Paper No: FE-14-1492; doi: 10.1115/1.4030067 History: Received September 05, 2014; Revised March 02, 2015; Online April 28, 2015

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.

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Figures

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Fig. 1

Diagram of Hall term subcycling implementation

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Fig. 2

Time per iteration and equivalent time per iteration with subcycling

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Fig. 3

Speedup with subcycling

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Fig. 4

Computations of MHD Couette flow. Profiles shown are the normalized velocity, U/Uwall (left), and induced magnetic field, Bx/Bz (right), versus normalized channel height, Z/h.

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Fig. 5

Computations of Hartmann flow. Profiles shown are the normalized velocity, U/Uwall (left), and induced magnetic field, Bx/Bz (right), versus the normalized channel height, 2Z/h.

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Fig. 6

Computations of Hall Hartmann flow. Comparison is made with the semi-analytical results of Sato [28]. Profiles shown are the normalized streamwise velocity, U/Umass (left), and spanwise velocity, V/Vmass (right), versus the normalized channel height, 2Z/h, for α = 5.

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Fig. 7

Setup of the 1D MHD shock tube

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Fig. 8

Plasma density for an MHD shock tube as computed by MGKM. Comparison is made with the ideal MHD results of Brio and Wu [29].

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Fig. 9

MGKM computations for an MHD shock tube. Viscous/resistive results shown are the velocities parallel (x∧) and transverse (y∧) to the tube length (U and V), as well as the transverse magnetic field (By), and the pressure (P) at the same time instant as in Fig. 8. Comparison is made with the ideal MHD results of Brio and Wu [29].

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Fig. 10

Plasma density for the Hall MHD shock tube as computed by MGKM. Comparison is shown with the ideal Hall MHD results of Srinivasan [31] for nondimensional rL/L=6.7×10-4.

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Fig. 11

Plasma density for the Hall MHD shock tube as computed by MGKM. Comparison is shown with the ideal Hall MHD results of Shumlak [32] for nondimensional rL/L=3×10-3.

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Fig. 12

Schematic of the computational domain for jet simulations

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Fig. 13

Schematic of the applied magnetic field produced by a current loop

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Fig. 14

Schematic of jet stretching from Ref. [34] along Bz, with flow reversal in the X–Y plane and outflow near the inlet

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Fig. 15

Schematic of current streamlines near jet inlet from Ref. [34]

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Fig. 16

Contours of the normalized streamwise velocity, U/Uin, in the Y–Z plane at X/D = 3.09 (left column) and X/D = 9.34 (right column) as computed by MGKM. The top row shows the jet without an external magnetic field applied, and the bottom row shows jet stretching in the direction parallel to the applied magnetic field, i.e., Bz.

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Fig. 17

Contours of the normalized streamwise velocity, U/Uin, at several slices in the total computational domain as computed by MGKM. Flow reversal is observed in X–Y plane.

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Fig. 18

Contours of normalized streamwise velocity, U/Uin, as computed by MGKM. Also shown are streamlines and vectors (not to scale) of the U–V components of velocity.

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Fig. 19

Contours of normalized streamwise velocity, U/Uin, as computed by MGKM. Also shown are streamlines and vectors (not to scale) of the JyJz components of current density.

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Fig. 20

Schematic of the axial currents, Jaxial, due to Hall physics that produce azimuthal forces, FJ×B, on the plasma jet

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Fig. 21

Contours of normalized axial (x∧) current density, J/Jmax, and azimuthal (θ∧) velocity, Vθ/Vθmax, as computed by MGKM with and without the Hall term

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Fig. 22

Normalized volume average of ∇·B versus Δt (s)

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Fig. 23

Normalized volume average of ∇·B versus Δx/L

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