0
Research Papers

Characterization of Flow-Magnetic Field Interactions in Magneto-Hydrodynamic Turbulence

[+] Author and Article Information
Jacques C. Richard

Mem. ASME
Senior Lecturer
e-mail: richard@tamu.edu

Gaurav Kumar

Graduate Student
e-mail: gauravkr@neo.tamu.edu
Department of Aerospace Engineering,
Texas A&M University,
College Station, TX 77843-3141

Tamás Kalmár-Nagy

Mem. ASME
Principal Member Research Staff Mitsubishi Electric Research Laboratory,
Cambridge, MA 02139
e-mail: jcnd@kalmarnagy.com

Sharath S. Girimaji

Mem. ASME
Professor
Department of Aerospace Engineering,
Texas A&M University,
College Station, TX 77843-3141
e-mail: girimaji@aero.tamu.edu

1Corresponding author.

Contributed by the Design Engineering Division of ASME for publication in the Journal of Computational and Nonlinear Dynamics. Manuscript received January 4, 2012; final manuscript received December 12, 2012; published online January 25, 2013. Assoc. Editor: D. Dane Quinn.

J. Comput. Nonlinear Dynam 8(3), 031010 (Jan 25, 2013) (12 pages) Paper No: CND-12-1002; doi: 10.1115/1.4023323 History: Received January 04, 2012; Revised December 12, 2012

We examine the complex nonlinear flow-magnetic field dynamics in magneto-hydrodynamic (MHD) turbulence. Using direct numerical simulations (DNS), we investigate the dynamical interactions subject to the influence of a uniform applied background magnetic field. The initial magnetic and kinetic Reynolds numbers (based on Taylor microscale) are 45 and there are no initial magnetic field fluctuations. The sum total of turbulent magnetic and kinetic energies decays monotonically. With time, the turbulent magnetic fluctuations grow by extracting energy from velocity fluctuations. Expectedly, the distribution of energy between kinetic and magnetic fluctuations exhibits large periodic oscillations from the equipartition state due to Alfvén waves. We perform a detailed analysis of the flow-magnetic field coupling and posit a simple model for the energy interchange. Such dynamical analysis can provide the insight required for turbulence control and closure modeling strategies.

FIGURES IN THIS ARTICLE
<>
Copyright © 2013 by ASME
Your Session has timed out. Please sign back in to continue.

References

Alboussière, T., Cardin, P., Debray, F., La Rizza, P., Masson, J., Plunian, F., Ribeiro, A., and Schmitt, D., 2011, “Experimental Evidence of Alfvén Wave Propagation in a Gallium Alloy,” Phys. Fluids, 23, p. 096601. [CrossRef]
Brandenburg, A., 2003, “Computational Aspects of Astrophysical MHD and Turbulence,” Advances in Nonlinear Dynamos, The Fluid Mechanics of Astrophysics and Geophysics, Vol. 9, A. M.Soward, C. A.Jones, and D. W.Hughes, eds., CRC Press, Boca Raton, FL, pp. 269–344.
Brandenburg, A., and Käpylä, P., 2007, “Magnetic Helicity Effects in Astrophysical and Laboratory Dynamos,” New J. Phys., 9, pp. 305–330. [CrossRef]
Mininni, P., Ponty, Y., Montgomery, D., Pinton, J., Politano, H., and Pouquet, A., 2005, “Dynamo Regimes With a Nonhelical Forcing,” Astrophys. J., 626, pp. 853–863. [CrossRef]
Pouquet, A., and Patterson, G., 1978, “Numerical Simulation of Helical Magnetohydrodynamic Turbulence,” J. Fluid Mech., 85(2), pp. 305–323. [CrossRef]
Davidson, P. A., 2001, An Introduction to Magnetohydrodynamics, Cambridge University Press, Cambridge, UK.
Pope, S., 2000, Turbulent Flows, Cambridge University Press, Cambridge, UK.
Davidson, P., 2004, Turbulence: An Introduction for Scientists and Engineers, Oxford University Press, New York.
Roy, R. I. S., Hastings, D. E., and Taylor, S., 1996, “Three-Dimensional Plasma Particle-in-Cell Calculations of Ion Thruster Backflow Contamination,” J. Comput. Phys., 128, pp. 6–18. [CrossRef]
Haas, J. M., and Gallimore, A. D., 2001, “Internal Plasma Potential Profiles in a Laboratory-Model Hall Thruster,” Phys. Plasma, 38(2), pp. 652–660. [CrossRef]
Tarditi, A. G., and Shebalin, J. V., 2003, “Magnetic Nozzle Plasma Exhaust Simulation for the VASIMR Advanced Propulsion Concept,” Proceedings of the 28th International Electric Propulsion Conferenc e.
Riley, B. M., Girimaji, S. S., and Richard, J. C., 2009, “Magnetic Field Effects on Axis-Switching and Instabilities in Rectangular Plasma Jets,” Flow, Turbul. Combust., 82(3), pp. 375–390. [CrossRef]
Macheret, S. O., Shneider, M. N., and Miles, R. B., 2002, “Magnetohydrodynamic Control of Hypersonic Flows and Scramjets Using Electron Beam Ionization,” AIAA J., 40(1), pp. 74–81. [CrossRef]
Biskamp, D., 2003, Magnetohydrodynamic Turbulence, Cambridge University Press, Cambridge, UK.
Balsara, D., and Pouquet, A., 1999, “The Formation of Large-Scale Structures in Supersonic Magnetohydrodynamic Flows,” Phys. Plasmas, 6, pp. 89–100. [CrossRef]
Chen, F. F., 1984, Introduction to Plasma Physics and Controlled Fusion, 2nd ed., Springer, New York.
Christensson, M., Hindmarsh, M., and Brandenburg, A., 2001, “Inverse Cascade in Decaying Three-Dimensional-Magnetohydrodynamic Turbulence,” Phys. Rev. E, 64(5), p. 056405. [CrossRef]
Frisch, U., Pouquet, A., Léorat, J., and Mazure, A., 1975, “Possibility of an Inverse Cascade of Magnetic Helicity in Magnetohydrodynamic Turbulence,” J. Fluid Mech., 68(4), pp. 769–778. [CrossRef]
Knaepen, B., Kassinos, S. C., and Carati, D., 2004, “Magnetohydrodynamic Turbulence at Moderate Magnetic Reynolds Numbers,” J. Fluid Mech., 513, pp. 199–220. [CrossRef]
Knaepen, B., and Moreau, R., 2008, “Magnetohydrodynamic Turbulence at Low Magnetic Reynolds Number,” Ann. Rev. Fluid Mech., 40, pp. 25–45. [CrossRef]
Matthaeus, W. H., Ghosh, S., Oughton, S., and Roberts, D. A., 1996, “Anisotropic Three-Dimensional MHD Turbulence,” J. Geophys. Res. Space Phys., 101(A4), pp. 7619–7629. [CrossRef]
Miller, R. S., Mashayek, F., Adumitoraie, V., and Givi, P., 1996, “Structure of Homogeneous Nonhelical Magnetohydrodynamic Turbulence,” Phys. Plasmas, 3(9), pp. 3304–3317. [CrossRef]
Ponty, Y., Mininni, P. D., Montgomery, D. C., Pinton, J.-F., Politano, H., and Pouquet, A., 2005. “Numerical Study of Dynamo Action at Low Magnetic Prandtl Numbers,”Phys. Rev. Lett., 94(164502), p. 164502. [CrossRef] [PubMed]
Müller, W. C., and Grappin, R., 2005, “Spectral Energy Dynamics in Magnetohydrodynamic Turbulence,” Phys. Rev. Lett., 95, p. 114502. [CrossRef] [PubMed]
Richard, J., Riley, B., and Girimaji, S., 2011, “Magnetohydrodynamic Turbulence Decay Under the Influence of Uniform or Random Magnetic Fields,” J. Fluids Eng., 133, p. 081205. [CrossRef]
Shebalin, J. V., Matthaeus, W. H., and Montgomery, D. C., 1983, “Anisotropy in MHD Turbulence Due to a Mean Magnetic Field,” J. Plasma Phys., 29, pp. 525–547. [CrossRef]
Shebalin, J. V., 2005, “Theory and Simulation of Real and Ideal Magnetohydrodynamic Turbulence,” Discrete Contin. Dyn. Syst., Ser. B, 5(1), p. 153174. [CrossRef]
Shebalin, J., 2009, “Plasma Relaxation and the Turbulent Dynamo,” Phys. Plasmas, 16, p. 072301. [CrossRef]
Yoshizawa, A., Itoh, S.-I., and Itoh, K., 2003, Plasma and Fluid Turbulence: Theory and Modeling, Institute of Physics, London.
Alfvén, H., 1942, “Existence of Electromagnetic-Hydrodynamic Waves,” Nature, 150(3805), pp. 405–406. [CrossRef]
d'Humiéres, D., Ginzburg, I., Krafczyk, M., Lallemand, P., and Luo, L.-S., 2002, “Multiple-Relaxation-Time Lattice Boltzmann Models in Three Dimensions,” Philos. Trans. R. Soc. Lond. A, 220, pp. 437–451. [CrossRef]
Eggels, J. G. M., 1996, “Direct and Large-Eddy Simulation of Turbulent Fluid Flow Using the Lattice-Boltzmann Scheme,” Int. J. Heat Fluid Flow, 17, pp. 307–323. [CrossRef]
Girimaji, S. S., 2007, “Boltzmann Kinetic Equation for Filtered Fluid Turbulence,” Phys. Rev. Lett., 99, p. 034501. [CrossRef] [PubMed]
He, X., and Luo, L.-S., 1997, “A Priori Derivation of the Lattice Boltzmann Equation,” Phys. Rev. E., 55, pp. R6333–R6337. [CrossRef]
He, X., and Luo, L.-S., 1997, “Theory of the Lattice Boltzmann Method: From the Boltzmann Equation to the Lattice Boltzmann Equation a Priori Derivation of the Lattice Boltzmann Equation,” Phys. Rev. E., 56, p. 6811–6817.
Lee, K., Yu, D., and Girimaji, S. S., 2006, “Lattice Boltzmann DNS of Decaying Compressible Isotropic Turbulence With Temperature Fluctuations,” Int. J. Comput. Fluid Dyn., 20(6), pp. 401–413. [CrossRef]
Luo, L.-S., 1998, “Unified Theory of Lattice Boltzmann Models for Nonideal Gases,” Phys. Rev. Lett., 81, pp. 1618–1621. [CrossRef]
Luo, L.-S., and Girimaji, S. S., 2003, “Theory of the Lattice Boltzmann Method: Two-Fluid Model for Binary Mixtures,” Phys. Rev. E., 67, p. 036302. [CrossRef]
Shan, X., and Doolen, G. D., 1993, “Lattice Boltzmann Model for Simulating Flows With Multiple Phases and Components,” Phys. Rev. E., 47(3), pp. 1815–1819. [CrossRef]
Martys, N. S., Shan, X., and Chen, H., 1998, “Evaluation of the External Force Term in the Discrete Boltzmann Equation,” Phys. Rev. E., 58(5), pp. 6855–6857. [CrossRef]
Yu, H., Girimaji, S., and Luo, L., 2005, “Lattice Boltzmann Simulations of Decaying Homogeneous Isotropic Turbulence,” Phys. Rev. E, 71(1), p. 016708. [CrossRef]
Yu, D., and Girimaji, S., 2005, “DNS of Homogenous Shear Turbulence Revisited with the Lattice Boltzmann Method,” J. Turbul.6(6), pp. 1–17. [CrossRef]
Yu, H., Girimaji, S. S., and Luo, L.-S., 2005, “DNS and LES of Decaying Isotropic Turbulence With and Without Frame Rotation using Lattice Boltzmann Method,” J. Comput. Phys., 209(2), pp. 599–616. [CrossRef]
Yu, D., and Girimaji, S. S., 2006, “Direct Numerical Simulations of Homogeneous Turbulence Subject to Periodic Shear,” J. Fluid Mech., 566, pp. 117–151. [CrossRef]
Yu, H., and Girimaji, S. S., 2005, “Near-Field Turbulent Simulations of Rectangular Jets Using Lattice Boltzmann Method,” Phys. Fluids, 17(12), p. 125106. [CrossRef]
Dellar, P., 2002, “Lattice Kinetic Schemes for MHD,” J. Comput. Phys., 179, pp. 95–126. [CrossRef]
Riley, B. M., Richard, J. C., and Girimaji, S. S., 2008, “Assessment of Magnetohydrodynamic Lattice Boltzmann Schemes in Turbulence and Rectangular Jets,” Int. J. Modern Phys. C, Comput. Phys. Phys. Comput., 18(8), pp. 1211–1220. [CrossRef]
Riley, B. M., 2007, “Magnetohydrodynamic Lattice Boltzmann Simulations of Turbulence and Rectangular Jet Flow,” M.S. thesis, Texas A&M University, College Station, TX.
Krafczyk, M., Tölke, J., and Luo, L.-S., 2003, “Large-Eddy Simulations With a Multiple-Relaxation-Time LBE Model,” Int. J. Mod. Phys. B, 17, pp. 33–39. [CrossRef]
Chen, S., and Doolen, G., 2003, “Lattice Boltzmann Method for Fluid Flows,” Ann. Rev. Fluid Mech., 30(1), pp. 329–364. [CrossRef]
Dong, Y., and Sagaut, P., 2008, “A Study of Time Correlations in Lattice Boltzmann-Based Large-Eddy Simulation of Isotropic Turbulence,” Phys. Fluids, 20, p. 035105. [CrossRef]
Premnath, K., Pattison, M., and Banerjee, S., 2009, “Dynamic Subgrid Scale Modeling of Turbulent Flows Using Lattice-Boltzmann Method,” Phys. A: Stat. Mech. Appl., 388(13), pp. 2640–2658. [CrossRef]
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(1), pp. 1–16. [CrossRef]
Gekelman, W., Vincena, S., and Collette, A., 2008, “Visualizing Three-Dimensional Reconnection in a Colliding Laser Plasma Experiment,” IEEE Trans. Plasma Sci., 36(4), pp. 1122–1123. [CrossRef]
Mininni, P., Lee, E., Norton, A., and Clyne, J., 2008, “Flow Visualization and Field Line Advection in Computational Fluid Dynamics: Application to Magnetic Fields and Turbulent Flows,” New J. Phys., 10, p. 125007. [CrossRef]
Zhang, Y., Boehmer, H., Heidbrink, W., McWilliams, R., Leneman, D., and Vincena, S., 2007, “Lithium Ion Sources for Investigations of Fast Ion Transport in Magnetized Plasmas,” Rev. Sci. Instrum., 78, p. 013302. [CrossRef] [PubMed]
Drozdenko, T., and Morales, G., 2001, “Nonlinear Effects Resulting From the Interaction of a Large-Scale Alfvén Wave With a Density Filament,” Phys. Plasmas, 8, pp. 3265–3277. [CrossRef]

Figures

Grahic Jump Location
Fig. 1

Kinetic and Magnetic Energy Decay with eddy turnover time, τ. (B = 0: solid; N = 0.0: _. _. ; N = 0.3: ... ; N = 0.05: - - -) [25].

Grahic Jump Location
Fig. 2

Evolution of EM, EK, Et and Lorentz work with τ. (a) N = 0:3; (b) N = 0:05 (solid: Kinetic Energy; _._. : Total Energy; ... : Lorentz work, - - - : Magnetic Energy) [25].

Grahic Jump Location
Fig. 3

Evolution of helicities with τ (a) N = 0:3; (b) N = 0:05 (solid: Cross Helicity; ... : Kinetic Helicity, - - - : Magnetic Helicity) [25]

Grahic Jump Location
Fig. 4

Evolution of helicities (scaled to kinetic plus magnetic energy) with τ. (a) N = 0:3; (b) N = 0:05 (solid: Cross Helicity; ... : Kinetic Helicity, - - - : Magnetic Helicity) [25]

Grahic Jump Location
Fig. 5

Kinetic and magnetic energy (normalized with total energy) vs. eddy turnover time

Grahic Jump Location
Fig. 6

Comparison of computational data with proposed model

Grahic Jump Location
Fig. 7

Kinetic and Magnetic Energy plot from the model

Grahic Jump Location
Fig. 8

Evolution of EM/Et and EK/Et with τ. (solid:Kinetic Energy; - - - : Magnetic Energy)

Grahic Jump Location
Fig. 9

The Lorentz force at (a) τ = 0:12, (b) τ = 0:18, (c) τ = 0:2 and (d) τ = 0:25

Grahic Jump Location
Fig. 10

Current density progression indicating propagation of Alfvén wavesas also seen in other studies [1] (a) τ = 0:12, (b) τ = 0:138, (c) τ = 0:158, (d) τ = 0:18, (d) τ = 0:2 and (e) τ = 0:25

Grahic Jump Location
Fig. 11

Streamlines at eddy turnover times, (a) τ = 0.18 and (b) τ = 0.25

Grahic Jump Location
Fig. 12

Vorticity field “streamlines” at eddy turnover times, (a) τ = 0.18 and (b) τ = 0.25

Grahic Jump Location
Fig. 13

Magnetic field “streamlines” at eddy turnover times, (a) τ = 0.18 and (b) τ = 0.25

Grahic Jump Location
Fig. 14

Magnitude of the vorticity at (a) τ = 0:12, (b) τ = 0:18, (c) τ = 0:2 and (d) τ = 0:25

Tables

Errata

Discussions

Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In