# The HFD method for large eddy simulation of MHD turbulence decay

### Abstract

This work deals with the modelling of the Magnetohydrodynamic (MHD) turbulence decay by hybrid finite-difference method (HFDM) combining two different numerical methods: finite-difference and spectral methods. The numerical algorithm of hybrid method solves the Navier-Stokes equations and equation for magnetic field by a finite-difference method in combination with cyclic penta-diagonal matrix, which yields fourth-order accuracy in space and second-order accuracy in time. The pressure Poisson equation is solved by the spectral method. For validation of the developed algorithm the classical problem of the 3-D Taylor and Green vortex flow is considered without considering the magnetic field, and the simulated time-dependent turbulence characteristics of this flow were found to be in excellent agreement with the corresponding analytical solution valid for short times. We also demonstrate that the developed efficient numerical algorithm can be used to simulate the magnetohydrodynamic turbulence decay at different magnetic Reynolds numbers.### References

[1] Abdibekova A., Zhakebayev D., Abdigaliyeva A. and Zhubat K., "Modelling of turbulence energy decay based on hybrid methods,"Engineering Computations 35(5),(2018):1965-77.

[2] Batchelor G. K. "On the spontaneous magnetic field in a conducting liquid in turbulent motion,"Proc. Roy. Soc. A201. 16(1950): 405-16.

[3] Batchelor G. K.,The theory of homogeneous turbulence (Cambridge University Press: 1953)

[4] Batista M., "A Method for Solving Cyclic Block Penta-diagonal Systems of Linear Equations,"ArXiv, March 6, 2008.

Accessed March 14, 2008. http://arxiv.org/abs/0803.0874.

[5] Brachet M.E. , Meiron D. I., Orszag S. A., Nickel B. G., Morf R.H. and Frisch U., "Small-scale structure of the

Taylor-Green vortex"Fluid Mechanics 130(1983): 411-52.

[6] Brachet M.E., "Direct simulation of three - dimensional turbulence in the Taylor - Green vortex,"Fluid Dynamics Research 8(1991): 1-8.

[7] Burattini P., Zikanov O. and Knaepen B., "Decay of magnetohydrodynamic turbulence at low magnetic Reynolds number"Fluid Mechanics 657(2010): 502-38.

[8] DeBonis J.R., "Solutions of the Taylor-Green vortex problem using high - resolution explicit finite difference methods" (paper presented at 51st AIAA Aerospace Sciences Meeting including the New Horizons Forum and Aerospace Exposition, Dallas, Texas, January 07-10, 2013).

[9] Hossain M., "Inverse energy cascades in three dimensional turbulence,"Physics of Fluids B: Plasma Physics 3(2)(1991): 511-14.

[10] Frisch U., Turbulence. The legacy of A.N. Kolmogorov (Cambridge University Press: 1995)

[11] Ievlev V. M., The method of fractional steps for solution of problems of mathematical physics (Moscow: Science Nauka: 1975) [12] Kampanis N.A. and Ekaterinaris J.A., "A staggered grid, high-order accurate method for the incompressible Navier- Stokes equations,"Computational Physics 215(2006): 589-613.

[13] Kim J. and Moin P., "Application of a fractional - step method to incompressible Navier- Stokes equations,"Computational Physics 59(1985): 308-23.

[14] Knaepen B., Kassinos S. and Carati D., "Magnetohydrodynamic turbulence at moderate magnetic Reynolds number"Fluid Mechacincs 513(3)(2004): 199-220.

[15] Kolmogorov A. N., "Local structure of turbulence in an incompressible fluid at very high Reynolds numbers"Dokladi Akademii Nauk USSR 30(1941): 299-303.

[16] Moatt H. K., "On the suppression of turbulence by a uniform magnetic field,"Fluid Mechanics 28(1967): 571-592.

[17] Monin A. C., Yaglom A. M., Statistical fluid mechanics( Cambridge: MIT Press: 1975)

[18] Navon M., "Pent: A periodic pentadiagonal systems solver,"Communications in applied numerical methods 3(1987): 63-9.

[19] Schumann U., "Numerical simulation of the transition from three- to two-dimensional turbulence under a uniform magnetic field,"Fluid Mechanics 74(1976):31-58.

[20] Sirovich L., Smith L., Yakhot V., "Energy spectrum of homogeneous and isotropic turbulence in far dissipation range,"Physical Review Letters 72(3)(1994): 344-47.

[21] Taylor G.I. and Green A.E., "Mechanism of production of small eddies from large ones,"Proceedings of the royal society, Mathematics and physical sciences 158(895)(1937): 499-521.

[22] Vorobev A., Zikanov O., Davidson P. and Knaepen B., "Anisotropy of magnetohydrodynamic turbulence at low magnetic Reynolds number,"Physics of Fluids 17( 2005): 125105-1-125105-12.

[23] Zhakebayev D., Zhumagulov B. and Abdibekova A., "The decay of MHD turbulence depending on the conductive properties of the environment,"Magnetohydrosynamics 50(2)(2014): 121-38.

[24] Zikanov O. and Thess A., "Direct numerical simulation of forced MHD turbulence at low magnetic Reynolds

number,"Fluid Mechanics 358(1998): 299333.

[2] Batchelor G. K. "On the spontaneous magnetic field in a conducting liquid in turbulent motion,"Proc. Roy. Soc. A201. 16(1950): 405-16.

[3] Batchelor G. K.,The theory of homogeneous turbulence (Cambridge University Press: 1953)

[4] Batista M., "A Method for Solving Cyclic Block Penta-diagonal Systems of Linear Equations,"ArXiv, March 6, 2008.

Accessed March 14, 2008. http://arxiv.org/abs/0803.0874.

[5] Brachet M.E. , Meiron D. I., Orszag S. A., Nickel B. G., Morf R.H. and Frisch U., "Small-scale structure of the

Taylor-Green vortex"Fluid Mechanics 130(1983): 411-52.

[6] Brachet M.E., "Direct simulation of three - dimensional turbulence in the Taylor - Green vortex,"Fluid Dynamics Research 8(1991): 1-8.

[7] Burattini P., Zikanov O. and Knaepen B., "Decay of magnetohydrodynamic turbulence at low magnetic Reynolds number"Fluid Mechanics 657(2010): 502-38.

[8] DeBonis J.R., "Solutions of the Taylor-Green vortex problem using high - resolution explicit finite difference methods" (paper presented at 51st AIAA Aerospace Sciences Meeting including the New Horizons Forum and Aerospace Exposition, Dallas, Texas, January 07-10, 2013).

[9] Hossain M., "Inverse energy cascades in three dimensional turbulence,"Physics of Fluids B: Plasma Physics 3(2)(1991): 511-14.

[10] Frisch U., Turbulence. The legacy of A.N. Kolmogorov (Cambridge University Press: 1995)

[11] Ievlev V. M., The method of fractional steps for solution of problems of mathematical physics (Moscow: Science Nauka: 1975) [12] Kampanis N.A. and Ekaterinaris J.A., "A staggered grid, high-order accurate method for the incompressible Navier- Stokes equations,"Computational Physics 215(2006): 589-613.

[13] Kim J. and Moin P., "Application of a fractional - step method to incompressible Navier- Stokes equations,"Computational Physics 59(1985): 308-23.

[14] Knaepen B., Kassinos S. and Carati D., "Magnetohydrodynamic turbulence at moderate magnetic Reynolds number"Fluid Mechacincs 513(3)(2004): 199-220.

[15] Kolmogorov A. N., "Local structure of turbulence in an incompressible fluid at very high Reynolds numbers"Dokladi Akademii Nauk USSR 30(1941): 299-303.

[16] Moatt H. K., "On the suppression of turbulence by a uniform magnetic field,"Fluid Mechanics 28(1967): 571-592.

[17] Monin A. C., Yaglom A. M., Statistical fluid mechanics( Cambridge: MIT Press: 1975)

[18] Navon M., "Pent: A periodic pentadiagonal systems solver,"Communications in applied numerical methods 3(1987): 63-9.

[19] Schumann U., "Numerical simulation of the transition from three- to two-dimensional turbulence under a uniform magnetic field,"Fluid Mechanics 74(1976):31-58.

[20] Sirovich L., Smith L., Yakhot V., "Energy spectrum of homogeneous and isotropic turbulence in far dissipation range,"Physical Review Letters 72(3)(1994): 344-47.

[21] Taylor G.I. and Green A.E., "Mechanism of production of small eddies from large ones,"Proceedings of the royal society, Mathematics and physical sciences 158(895)(1937): 499-521.

[22] Vorobev A., Zikanov O., Davidson P. and Knaepen B., "Anisotropy of magnetohydrodynamic turbulence at low magnetic Reynolds number,"Physics of Fluids 17( 2005): 125105-1-125105-12.

[23] Zhakebayev D., Zhumagulov B. and Abdibekova A., "The decay of MHD turbulence depending on the conductive properties of the environment,"Magnetohydrosynamics 50(2)(2014): 121-38.

[24] Zikanov O. and Thess A., "Direct numerical simulation of forced MHD turbulence at low magnetic Reynolds

number,"Fluid Mechanics 358(1998): 299333.

Published

2018-12-21

How to Cite

ABDIBEKOVA, Aigerim; ZHAKEBAYEV, D. B..
The HFD method for large eddy simulation of MHD turbulence decay.

**Journal of Mathematics, Mechanics and Computer Science**, [S.l.], v. 99, n. 3, p. 53-77, dec. 2018. ISSN 1563-0277. Available at: <http://bm.kaznu.kz/index.php/kaznu/article/view/519>. Date accessed: 20 jan. 2019.
Section

Applied Mathematics

Keywords
Magnetohydrodynamics, Taylor-Green vortex problem, hybrid finite difference method, spectral method, turbulence decay, Large eddy simulation