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
![]() |
(1) |
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
shape
or structure
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
![]() |
(2) |
A comparison with (1) suggests
that , so that
can be interpreted as a dimensionality
or structure
anisotropy
variable.
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
![]() |
(3) |
|
![]() |
(4) |
|
![]() |
(5) |
As for the standard K-epsilon model, the turbulence production is given by
![]() ![]() |
(6) |
and the turbulent viscosity is calculated as
![]() |
(7) |
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
![]() |
(8) |
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) :
![]() |
(9) |
The model term is based
on recent results from direct numerical simulations, and takes the form
![]() |
(10 |
with
![]() |
(11) |
|
![]() |
(12) |
and model constants and
.
The conventional part of the model uses the "standard" values for all constants:
|
(13) |
The new model constants relevant to magnetic effects take the values
|
(14) |
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)