Turbulent MHD flow in a cylindrical vessel excited by a misaligned magnetic field

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Turbulent MHD flow in a cylindrical vessel
excited by a misaligned magnetic field
Center for MHD Studies

• A. Kapusta and B. Mikhailovich

Center for MHD StudiesBen-Gurion University of
the NegevBeer-Sheva, Israel
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It is well-known that there are a lot of
difficulties in calculations of MHD turbulent
rotating flows, and certain approximations are
required. In practice, semi-empirical models are
frequently used. One of such models is used in
our presentation. The matter in point is so
called external friction approximation
described in 1 and modified in 2,3, where a
quasi-linear dissipative term analog of the
divergence of stress tensor- appears. In this
case, we can determine azimuthal component of the
mean velocity assuming that all turbulent effects
including the effect of secondary flows on mean
velocity are accounted for through the external
friction coefficient.
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It is defined as
where
is the mean vorticity of the flow,
is an empirical coefficient,
- is a parameter defining the flow structure
,
in the present case,
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Note that the external friction approximation
is based on the results of numerous experimental
data and gives reliable results when calculating
mean velocities of turbulent rotating MHD flows
both under the action of a rotating magnetic
field (RMF) and in homeopolar facilities 4.
Moreover, the model has proved to be applicable
also to the analysis of the behavior of vortical
structures in a turbulent wake behind a bluff
body 5. It is noteworthy that we have managed
to obtain a universal dependence of the angular
velocity of turbulent flow core on the only
dimensionless parameter Q within the frames of
this model
where
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We study the effect of misalignment between the
magnetic field rotation axis Z0 and the symmetry
axis of the cylindrical vessel Z on
two-dimensional structure of the mean turbulent
flow excited by rotating magnetic field (RMF) in
non-inductive approximation using the external
friction model. In practical applications of RMF,
for example, in metallurgical processes, the axis
of the vessel with melt can be shifted with
respect to the inductor axis. Therefore, the
characteristics of the mean flow (its structure)
can be changed depending both on MHD interaction
parameter N and on the eccentricity e. Note that
the eccentricity can also change the structure of
surface waves on the free surface of a rotating
fluid 6.
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If the axes Z0 and Z coincide, mean radial and
azimuthal components of dimensionless
electromagnetic body forces are described by the
following expressions under the condition

(1)
Fig.1. Coordinate systems layout
where p is the number of pole pairs of the
inductor exciting RMF,
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Using the formulas of coordinate and vectorial
components transformation at the transition from
a certain cylindrical coordinate system
to another one we obtain
for parallel Z0 and Z
(2)
(3)
where
(later on we use without upper lines).
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Our next step is to estimate z-component of
that can be written as
(4)
After differentiating (2), (3) and substituting
into (4), we obtain
(5)
where
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In this case a dimensionless equation for
z-component of the velocity vector potential
can be written as follows
(6)
which should be solved under the boundary
conditions
(7)
Neglecting laminar friction in comparison with
turbulent one, we seek the solution of (6)-(7) in
the form of a sum of even and odd (with respect
to j ) solutions
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In the zero approximation, the problem acquires
the form
(8)
and
(9)
where
It follows from the evenness of F (r,j) function
and the operator D with respect to coordinate j
that is also even, and the problem (8)-(9) is
solved in the form
(10)
where
are the roots of the equations for Bessel
functions
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Substituting (10) into (8) and performing the
procedure of Galerkins method, we obtain
(11)
(12)
where
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In the first approximation, the problem can be
written as
(13)
and
It follows from the oddness of the F(r,j)
function with respect to coordinate j that the
problem (13) can be solved in the form
(14)
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Substituting (14) into (13) and performing the
procedure of Galerkins method, and taking into
account the fact that
we obtain
(15)
where
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We have found exact solutions in the cases of
p1 and p2. At p1
(16)
At p2
(17)
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Fig.2 p1,
N10-5, e0.01
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Fig. 3 p1,
N10-5, e0.025
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Fig. 4 p1,
N10-5, e0.3
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Fig. 5 p2,
N10-5, e0.01
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Fig. 6 p2,
N10-5, e0.028
Dresden-2004
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Fig. 7 p2,
N10-5, e0.1
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The results of flow streamlines calculations show
that linear approximation (first term of
solution) gives a rotating flow, which gets
deformed in the core at increasing eccentricity.
The account for nonlinearity changes the flow
pattern. In this case, all effects occur in the
core on the background of mean rotating flow. It
is desirable to perform experimental checking of
the computed results, since the appearance of
such structures may be useful for technological
applications as a possible means of controlling
liquid metal behavior under a rotating magnetic
field.
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References 1. Kapusta, A.B., Levitsky, L.D., On
the universalization of the model of turbulent
magnetohydrodynamic rotation, Magnitnaya
Gidrodinamika, 3 (1991a) 134-136. 2. Kapusta,
A.B., Shamota, V.P., Quasi-laminar and turbulent
flows of a conductive fluid, Magnetohydrodynamics,
32 (1996) 43-49. 3. A. Kapusta, B. Mikhailovich,
Golbraikh, E. 2002. Semiempirical model of
turbulent rotating MHD flows. Proc. 5th Internat.
PAMIR Conf, I-227-230. 4. Branover et al. 2002.
Turbulent MHD rotation of a conducting fluid in a
cylindrical vessel. Proc. 5th Internat. PAMIR
Conf., 2002, I-169-171. 5. Kapusta, A., B.
Mikhailovich, E. Golbraikh, 2000. On the
development of a turbulent vortex in an axial
magnetic field, Proc. 4th International
Conference on Magnetohydrodynamics PAMIR,
627-630. 6. Golbraikh, E., Kapusta, A. and
Mikhailovich, B. 2002. Standing waves on the
surface of a conducting fluid rotating in a
magnetic field. Proc. 9th European Turbulence
Conf., 2002, 885.
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