Comparative study of finite difference approaches in simulation of magnetohydrodynamic turbulence at low magnetic Reynolds number

Abstract Two approaches to finite difference approximation of turbulent flows of electrically conducting incompressible fluids in the presence of a steady magnetic field are analyzed. One is based on high-order approximations and upwind-biased discretization of the nonlinear term. Another is consistently of the second order and nearly fully conservative in regard of mass, momentum, kinetic energy, and electric charge conservation principles. The analysis is conducted using comparison with high-accuracy spectral direct numerical simulations of channel flows with wall-normal and spanwise magnetic fields. Focus of the analysis is on the quality of finite difference approximation in the situation when the magnetic field leads to significant transformation of the flow structure. In the case of turbulent flows at moderate magnetic fields, the conservative scheme approach offers better stability and equal or higher accuracy than the approach based on the upwind discretization. The conservation property is expected to become increasingly important and even indispensable at stronger magnetic fields.

[1]  O. Zikanov,et al.  Optimal growth and transition to turbulence in channel flow with spanwise magnetic field , 2008, Journal of Fluid Mechanics.

[2]  T. Kunugi,et al.  Direct numerical simulation of turbulent channel flow under a uniform magnetic field for large-scale structures at high Reynolds number , 2006 .

[3]  P. Roberts.  An introduction to magnetohydrodynamics , 1967 .

[4]  Xiaolin Zhong,et al.  High-Order Finite-Difference Schemes for Numerical Simulation of Hypersonic Boundary-Layer Transition , 1998 .

[5]  P. Swarztrauber,et al.  Efficient Fortran subprograms for the solution of separable elliptic partial differential equations , 1979 .

[6]  R. Peyret.  Spectral Methods for Incompressible Viscous Flow , 2002 .

[7]  S. Gavrilakis,et al.  Numerical simulation of low-Reynolds-number turbulent flow through a straight square duct , 1992, Journal of Fluid Mechanics.

[8]  Joel H. Ferziger,et al.  Computational methods for fluid dynamics , 1996 .

[9]  D. Gérard-Varet.  Amplification of small perturbations in a Hartmann layer , 2002 .

[10]  Ramakanth Munipalli,et al.  A current density conservative scheme for incompressible MHD flows at a low magnetic Reynolds number. Part I: On a rectangular collocated grid system , 2007, J. Comput. Phys..

[11]  A. Vorobev,et al.  Anisotropy of magnetohydrodynamic turbulence at low magnetic Reynolds number , 2005 .

[12]  Haecheon Choi,et al.  Magnetohydrodynamic turbulent flow in a channel at low magnetic Reynolds number , 2001, Journal of Fluid Mechanics.

[13]  Axelle Viré,et al.  On discretization errors and subgrid scale model implementations in large eddy simulations , 2009, J. Comput. Phys..

[14]  O. Zikanov,et al.  Large-scale intermittency of liquid-metal channel flow in a magnetic field. , 2008, Physical review letters.

[15]  Mohamed A. Abdou,et al.  Characterization of key magnetohydrodynamic phenomena in PbLi flows for the US DCLL blanket , 2008 .

[16]  O. Zikanov,et al.  Magnetohydrodynamic turbulence in a channel with spanwise magnetic field , 2008, 0809.0964.

[17]  S. Kassinos,et al.  Large-eddy simulations of the turbulent Hartmann flow close to the transitional regime , 2007 .

[18]  P. Davidson.  An Introduction to Magnetohydrodynamics , 2001 .

[19]  L. Leboucher,et al.  Monotone Scheme and Boundary Conditions for Finite Volume Simulation of Magnetohydrodynamic Internal Flows at High Hartmann Number , 1999 .

[20]  Y. Makarov,et al.  Application of magnetic fields in industrial growth of silicon single crystals , 2006 .

[21]  O. Zikanov,et al.  Direct numerical simulation of forced MHD turbulence at low magnetic Reynolds number , 1998, Journal of Fluid Mechanics.

[22]  Adams.  Multigrid software for elliptic partial differential equations: MUDPACK. Technical note , 1991 .

[23]  D. Carati,et al.  Anisotropy of velocity spectra in quasistatic magnetohydrodynamic turbulence , 2008 .

[24]  Ramakanth Munipalli,et al.  A current density conservative scheme for incompressible MHD flows at a low magnetic Reynolds number. Part II: On an arbitrary collocated mesh , 2007, J. Comput. Phys..

[25]  Axelle Viré,et al.  Modeling and discretization errors in large eddy simulations of hydrodynamic and magnetohydrodynamic channel flows , 2011, J. Comput. Phys..

[26]  A. Vorobev,et al.  Smagorinsky constant in LES modeling of anisotropic MHD turbulence , 2008 .

[27]  K. Hanjalic,et al.  A direct-numerical-simulation-based second-moment closure for turbulent magnetohydrodynamic flows , 2004 .

[28]  Brian G. Thomas,et al.  Flow Control with Local Electromagnetic Braking in Continuous Casting of Steel Slabs , 2008 .

[29]  Hiromichi Kobayashi,et al.  Large eddy simulation of magnetohydrodynamic turbulent duct flows , 2008 .

[30]  C. Rhie,et al.  A numerical study of the turbulent flow past an isolated airfoil with trailing edge separation , 1982 .

[31]  Hiromichi Kobayashi,et al.  Large eddy simulation of magnetohydrodynamic turbulent channel flows with local subgrid-scale model based on coherent structures , 2006 .

[32]  T. Boeck,et al.  Numerical study of turbulent magnetohydrodynamic channel flow , 2007, Journal of Fluid Mechanics.

[33]  Thomas Boeck,et al.  Numerical study of the instability of the Hartmann layer , 2004, Journal of Fluid Mechanics.

[34]  Parviz Moin,et al.  Large-eddy simulation of conductive flows at low magnetic Reynolds number , 2004 .

[35]  P. Moin,et al.  Fully Conservative Higher Order Finite Difference Schemes for Incompressible Flow , 1998 .

[36]  C. Rhie,et al.  Numerical Study of the Turbulent Flow Past an Airfoil with Trailing Edge Separation , 1983 .

[37]  C. Mistrangelo,et al.  Numerical investigation of liquid metal flows in rectangular sudden expansions , 2007 .