Fluid flow turbulence subjected to a magnetic field will experience magnetic dissipation, or Joule dissipation. While ordinary viscous dissipation acts primarily on the smallest eddies, magnetic dissipation selectively affects turbulent structures depending on their geometrical shape. Isotropic turbulent structures are effectively dissipated, while cigar-shaped structures aligned with the magnetic field lines are only weakly affected. The observed behavior is therefor that turbulence is dissipated in a magnetic field, but also that the turbulent field at the same time becomes more and more two-dimensional, with eddies elongated in the direction of the magnetic field. As these anisotropic eddies are more resistant, the dissipation rate tends to diminish with time.
Theoretical arguments can be used to show that the Joule dissipation, , of turbulent kinetic energy , K, scales as
where is the
inverse of a magnetic timescale, while and are
characteristic turbulent length scales in the directions parallel and
normal, respectively, to the magnetic field (B
is the magnetic flux
density, is the fluid
electric conductivity and the fluid
density). The dependence on the ratio of length scales
clearly demonstrates that the geometrical
of the turbulent
eddies has a decisive role to play in MHD turbulence. The problem is
that this kind of structure information is absent from all conventional
turbulence models, be it eddy-viscosity closures like the k-epsilon
model, or full Reynolds-stress models.
The idea behind the MHD turbulence models discussed here is to introduce a minimum of the structure information necessary to describe MHD turbulence. We do this by defining a new transported turbulence variable called , such that the Joule dissipation of turbulent kinetic energy can be expressed exactly as
A comparison with (1) suggests
that , so that can be interpreted as a dimensionality
In the case of homogeneous turbulence, can be given an exact definition for which an exact transport equation can be derived from the Navier-Stokes equations. All source and sink terms in this equation are rather complicated and must be modeled. There are however plenty of well-proven tools from conventional turbulence theory that can be used to study the properties of these terms. We can for example use linear theory and rapid distortion theory to study the effect of a magnetic field on , and we can use direct numerical simulations (DNS) or large eddy-simulations (LES) to understand nonlinear effects.
The simplest model of this kind is the K-epsilon-alpha model, which is based on the standard K-epsilon model, extended with an equation for and the necessary magnetic sink terms in the K equations. The three transport equations are
As for the standard K-epsilon model, the turbulence production is given by
| , with
and the turbulent viscosity is calculated as
The Joule dissipation term in the K equation is given by (2) and requires no modeling. The corresponding Joule destruction term in the equation is modeled as
where is a new model constant. This is simply a rescaling of (2), in the same logic as both the production and viscous terms in (4) are rescaled versions of the corresponding terms in the K equation (3).
Apart from the conventional transport terms, the proposed alpha equation contains a magnetic destruction term and a "return-to-isotropy" term representing nonlinear inertial effects. The exact alpha equation contains also terms depending on mean shear and strain, but they have been dropped from the model equation (5). The most obvious argument for this is that conventional models neglect the effects of structural effects altogether.
The magnetic destruction term is modeled with a piece-wise polynomial function fitted to exact results from linear theory (assuming a strong magnetic field and negligible nonlinear effects) :
The model term is based on recent results from direct numerical simulations, and takes the form
and model constants and .
The conventional part of the model uses the "standard" values for all constants:
The new model constants relevant to magnetic effects take the values
O. Widlund, S. Zahrai and F. H. Bark (1998), "Development of a Reynolds stress closure for modeling of homogeneous MHD turbulence", Physics of Fluids 10, pp. 1987-1996. (Abstract)
O. Widlund, S. Zahrai and F. H. Bark (2000), "Structure information in rapid distortion analysis and one-point modeling of axisymmetric magnetohydrodynamic turbulence", Physics of Fluids 12, pp. 2609-2620. (Abstract)
O. Widlund (2001), "Modeling anisotropic MHD turbulence in simulations of liquid metal flows", Magnetohydrodynamics 37, pp. 3-12. (Abstract)
O. Widlund, "Using structure information in modeling of magnetohydrodynamic turbulence". TSFP-2, 2nd Int. Symp. on Turbulence and Shear Flow Phenomena, June 2001, Stockholm, Sweden. KTH. (PDF download, 925 KB)
Modeling of magnetohydrodynamic turbulence. Ph.D. thesis, TRITA-MEK 2000:11, KTH, Stockholm, 2000. (Abstract and PDF download)