Hostname: page-component-848d4c4894-wg55d Total loading time: 0 Render date: 2024-05-03T15:21:09.577Z Has data issue: false hasContentIssue false
  • English
  • Français

Numerical simulation of the transition from three- to two-dimensional turbulence under a uniform magnetic field

Published online by Cambridge University Press:  29 March 2006

U. Schumann
U. Schumann
Affiliation:
Kernforschungszentrum Karlsruhe, Institut für Reaktorentwicklung 75 Karlsruhe, Postfach 3640, West Germany
Get access Rights & Permissions [Opens in a new window]

Abstract

The transition of homogeneous turbulence from an initially isotropic three-dimensional to a quasi-two-dimensional state is simulated numerically for a conducting, incompressible fluid under a uniform magnetic field B0. The magnetic Reynolds number is assumed to be small, so that the induced fluctuations of the magnetic field are small compared with the imposed magnetic field B0, and can be computed from a quasi-static approximation. If the imposed magnetic field is strong enough, all variations of the flow field in the direction of B0 are damped out. This effect is important e.g. in the design of liquid-metal cooling systems for fusion reactors, and the properties of the final state are relevant to atmospheric turbulence. An extended version of the code of Orszag & Patterson (1972) is used to integrate the Navier-Stokes equations for an incompressible fluid. The initial hydrodynamic Reynolds number is 60. The magnetic interaction number N is varied between zero and 50. Periodic boundary conditions are used. The resolution corresponds to 323 points in real space. The full nonlinear simulations are compared with otherwise identical linear simulations; the linear results agree with the nonlinear ones within 3% for about one-fifth of the large-scale turnover time. This departure is a consequence of the return-to-equilibrium tendencies caused mainly by energy transfer towards high wavenumbers. The angular energy transfer and the energy exchange between different components are smaller, and become virtually zero for large values of N. For N ≈ 50 we reach a quasi-two-dimensional state. Here, the energy transfer towards high wavenumbers is reduced for the velocity components perpendicular to B0 but relatively increased for the component parallel to B0. The overall behaviour is more similar to three-than to purely two-dimensional turbulence. This finding is of great importance for turbulence models of the atmosphere. The realization of a purely two-dimensional state does not seem to be possible for decaying turbulence. The magnetic field causes highly intensified pressure fluctuations, which contribute to the redistribution of the anisotropic Lorentz forcing.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 74 , Issue 1 , 9 March 1976 , pp. 31 - 58
Copyright
© 1976 Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Batchelor, G. K. 1959 The Theory of Homogeneous Turbulence. Cambridge University Press.
Herring, J. R. 1974 Approach of axisymmetric turbulence to isotropy Phys. Fluids, 17, 859–872.Google Scholar
Herring, J. R., Orszag, S. A., Kraichnan, R. H. & Fox, D. G. 1974 Decay of two-dimensional homogeneous turbulence J. Fluid Mech. 66, 417–444.Google Scholar
Kit, L. G. & Tsinober, A. B. 1971 Possibility of creating and investigating two-dimensional turbulence in a strong magnetic field Magn. Girodin. 7, 27–34.Google Scholar
Kraichnan, R. H. 1964 Decay of isotropic turbulence in the direct interaction approximation Phys. Fluids, 7, 1030–1048.Google Scholar
Launder, B. E. 1975 On the effects of a gravitational field on the turbulent transport of heat and momentum J. Fluid Mech. 67, 569–581.Google Scholar
Lehnert, B. 1955 The decay of magneto-turbulence in the presence of a magnetic field and Coriolis force Quart. Appl. Math. 12, 321–341.Google Scholar
Leith, C. E. 1968 Two-dimensional eddy viscosity coefficients. Proc. WMO-IUGG Symp. on Numerical Weather Prediction, Tokyo, vol. 1, pp. 1–44.Google Scholar
Lilly, D. K. 1965 On the computational stability of numerical solutions of time-dependent nonlinear geophysical fluid dynamics problems Mon. Wea. Rev. 93, 11–26.Google Scholar
Lilly, D. K. 1971 Numerical simulation of developing and decaying two-dimensional turbulence J. Fluid Mech. 45, 395–415.Google Scholar
Moffatt, H. K. 1967 On the suppression of turbulence by a uniform magnetic field J. Fluid Mech. 28, 571–592.Google Scholar
Moreau, R. 1968 On magnetohydrodynamic turbulence. Proc. Symp. on Turbulence of Fluids and Plasmas, Polytechnic Institute of Brooklyn, pp. 359–372.
Orszag, S. A. & Patterson, G. S. 1972 Numerical simulation of three-dimensional homogeneous isotropic turbulence Phys. Rev. Lett. 28, 76–79.Google Scholar
Pleshanov, A. S. & Tseskis, A. L. 1973 Two-dimensional turbulence in a magnetic field Magn. Girodin. 9, 137–139.Google Scholar
Pouquet, A. & Patterson, G. S. 1976 Numerical simulation of helical magnetohydro-dynamics turbulence. Submitted to J. Fluid Mech.Google Scholar
Reynolds, W. C. 1973 Recent advances in the computation of turbulent flows Adv. in Chem. Engng 9, 193–246.Google Scholar
Roberts, P. H. 1967 An Introduction to Magnetohydrodynamics. New York: American Elsevier.
Rotta, J. 1951 Statistische Theorie nichthomogener Turbulenz Z. Phys. 129, 547–572.Google Scholar
Schumann, U. 1975 Subgrid scale model for finite difference simulations of turbulent flows in plane channels and annuli J. Comp. Phys. 18, 376–404.Google Scholar
Schumann, U. & Herring, J. R. 1976 Axisymmetric homogeneous turbulence: a comparison of direct spectral simulations with the direct interaction approximation. Submitted to J. Fluid Mech.Google Scholar
Schumann, U. & Patterson, G. S. 1976a Numerical study of pressure and velocity fluctuations in nearly isotropic turbulence. Submitted to J. Fluid Mech.Google Scholar
Schumann, U. & Patterson, G. S. 1976b Numerical study of the return of axisymmetric. turbulence to isotropy. Submitted to J. Fluid Mech.Google Scholar
Volkov, A. V. 1973 The effects of a magnetic field on the turbulence behind a grating Magn. Girodin. 9, 26–810.Google Scholar