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Journal of applied science in the thermodynamics and fluid mechanics
Vol. 5, No. 2/2011, ISSN 1802-9388
CFD SOLUTION of MHD
*Kateřina HORÁKOVÁ, **Karel FRAŇA
*Technical University of Liberec
Studentská 2, 461 17, Liberec, Czech Republic
Phone:+ 420 485 353434 Email: [email protected]
**Technical University of Liberec
Studentská 2, 461 17, Liberec, Czech Republic
Phone:+ 420 485 353436 Email: [email protected]
This part of study deals with description of unsteady melt flow Lorentz forces in the container with cylindrical
shape. This melt is driven by rotating magnetic field and solution of magnetohydrodynamics is made in the
commercial software Ansys Fluent (version 13). Taylor number of this flow is 10 6, so magnetic induction is
0,004478 T. Contours of Lorentz forces in the container are compared with analytic formula results.
Keywords: MHD; Lorentz forces; Ansys Fluent
1 INTRODUCTION
A theory of flow behaviour under magnetic field effect is called magnetohydrodynamics (MHD).
First mentions about MHD appeared in relation to astrophysics and geophysics. In the fifties the interest in
MHD focused especially to plasma physics and thermonuclear fusion control. The interest in MHD was
extended to industry later. Now the magnetic field is used in technical practice e.g. in metallurgy
for controlled solidification of the melt, for suppressing of the natural convection thereby increasing
homogeneity of the melt. Rotating magnetic field generates eddy flow in electric conductive melt. This effect
is used to e.g. for non-contact electromagnetic stirring of the melt in metallurgy and for crystal growing,
when rotating magnetic field homogenize of varied metal alloy and fine metal. The melt flow positively affects
metallographical structure of casts. For more details about electromagnetic stirring see [1], [2].
This processes is possible to simulate by commercial software Ansys Fluent (version 13). Using MHD
module for computing MHD processes is allowed from version 12 of Ansys Fluent.
The Magnetohydrodynamics (MHD) module is provided as an additional module with the standard ANSYS
FLUENT licensed software. The ANSYS FLUENT MHD model allows analysing the behaviour of electrically
conducting fluid flow under the influence of constant (DC) or oscillating (AC) electromagnetic fields.
The externally-imposed magnetic field may be generated either by selecting simple built-in functions or by
importing a user supplied data file. [3]. Equations of MHD are solved through user-defined scalar (UDS)
transport equations. Other model-related variables such as the external magnetic field data, current density,
Lorentz force and Joule heat are stored as user-defined memory (UDM) variables [3]. Computing of MHD
in Fluent is possible by two different methods – solving magnetic induction equation or scalar potential
equation. Both methods are solved by scalar transport equations. Solution by magnetic induction uses two
or three transport equations (2D or 3D solution). Solution by electric potential uses only one transport
equation.
MHD module is relatively new and some computing limits and other parts (solidification, simulation of free
surface etc.) will be added to upgraded version. Some computing limits are: computing program is set for the
melt with sufficient conductivity, it can be set up only magnetic induction value (no current density), it is
assumed that frequency is low etc.
2 PROBLEM FORMULATION
The subject of this work is a description of unsteady melt inside the cylindrical container and her Lorentz
forces. The flow of the melt is driven by rotating magnetic field which increases homogeneity of the melt and
suppresses the natural convection.
Lorentz forces computed in software Ansys Fluent were compared with results of analytic formula for Lorentz
forces in the cylindrical container [5,6].
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Journal of applied science in the thermodynamics and fluid mechanics
Vol. 5, No. 2/2011, ISSN 1802-9388
2.1 Model in Gambit
MHD processes were simulated by commercial software Ansys Fluent. The cylindrical model and the grid
were created in software Gambit. The grid has c. 70 000 elements. Boundary conditions whole container
were set to wall. Model is displayed in figure 1. A shape of the container is displayed in figure 2.
Figure 1: Grid of the cylindrical container
Figure 2: Shape of the container
2.2 Model in Fluent
The result mesh was export to Fluent (version 13), grid scale was checked (diameter of the container
is 0,03 m, height of the container is 0,03 m as well) and unsteady calculation was set. Turbulence model was
chosen DES (for later comparison with cuboid container – computing by NSFEM3D [4]) with RANS model
Spalart-Allmaras. Then MHD module was unloaded and method by magnetic induction was chosen. Solution
of MHD equations was added on Lorentz forces computing. Values of density, electric conductivity and
viscosity were set up as well as on NSFEM3D (for later comparison with cuboid container) - ρ = 6361 kg. m-3,
σ = 3,3.106 S·m−1, and ν =3,4.10-7 m2. s-1. Walls of the container were chosen as insulated walls, external
magnetic field as AC, frequency of the field ϖ = 439 s-1and magnitude of the field in x and y direction
B0 = 0,004478 T. Everything is the same as in computing program NSFEM3D. The same values were set
up to analytic formula of Lorentz forces. This analytical formula was deduced in my earlier work [5,6].
In [3] is written that convergence of the computing is poor and slow, so very small time step was chosen.
In the beginning of the computing the time step was 10-4 s, at the end 10-3 s. Total runtime was 102 s.
2.3 Postprocessing and comparing
Auxiliary planes were created for displaying results. Lorentz forces is possible displaying in Fluent but only
in Cartesian coordinate system (no Cylindrical system). That is the reason why Fx and Fy were monitored.
The leading monitoring quantity is Lorentz forces (these values were compared with analytic formula), but
other values is possible to display – instantaneous and mean velocity, induced magnetic induction (which
is neglected in analytic formula), current density etc. In figure 3 contours of Lorentz forces in x direction and
in figure 4 contours of Lorentz forces in y direction are displayed. Blue colour represents minima of Lorentz
forces, red colour maxima of Lorentz forces.
Absolute values of Lorentz forces in x and y directions were approximately the same (only reflected images).
AC magnetic field and the same value of magnetic induction in x and y direction were the reason of this.
Units of Lorentz forces were N.m-3. Absolute values of Lorentz forces were located near the lateral walls and
in the direction of container axis this absolute values falling.
Lorentz forces from analytic formula is:
2
 =
f rot
∞
⋅B0⋅⋅r
J  m ⋅r 
sinh mi⋅z sinh mi⋅ H −z 
2
⋅[1− ∑ 2 1 i
⋅
]
2
r i−1  mi −1⋅J 1  mi 
sinh mi⋅H 
2
.(1)
Journal of applied science in the thermodynamics and fluid mechanics
Vol. 5, No. 2/2011, ISSN 1802-9388
Figure 3: Contours of Lorentz forces in x direction from Figure 4: Contour of Lorentz forces in y direction from
Ansys Fluent
Ansys Fluent
Lorentz forces from analytic formula are displayed in figure 5 and 6 – in x and y direction. Blue colour
represents minima of Lorentz forces, red colour maxima of Lorentz forces. These forces were performed by
software MathCad. Lorentz forces from analytic formula have a little bit higher absolute values. Shape
of Lorentz forces contours is very similar – contours from analytic formula are little bit flatter. Differences are
caused probably because of some limitations of Fluent and simplifications in inferring of analytic formula.
Fluent computed with induced magnetic induction and Lorentz forces in z direction are not zero. The shape
of contours from Fluent shows as if container had been a little bit smaller (smaller container high). Shape
of Lorentz forces contours is flatter with increasing container high – for more details see [7].
Figure 5: Contours of Lorentz forces in x direction
from analytic formula
Figure 6: Contours of Lorentz forces in y direction from
analytic formula
Vectors of Lorentz forces are displayed in figure 7 and 8 (x and y direction). In figure 7 there are vectors
of Lorentz forces colored by Fx values, in figure 8 colored by Fy values. Fluent computing forces in z
direction as well. These forces are smaller than forces in x and y direction.
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Journal of applied science in the thermodynamics and fluid mechanics
Vol. 5, No. 2/2011, ISSN 1802-9388
Figure 7: Vectors of Lorentz forces coloured by Fx
Figure 8: Vectors of Lorentz forces coloured by Fy
3 CONCLUSIONS
This part of my work deals with description of unsteady melt flow Lorentz forces in the container with
cylindrical shape. Results of Lorentz forces from commercial software Ansys Fluent were compared with
Lorentz forces from analytic formula. The high of the cylindrical container is the same as radius of container.
Lorentz forces were displayed as forces in x and y directions. Lorentz forces from analytic formula were
performed by software MathCad. These forces were performed by software MathCad. Lorentz forces from
analytic formula have a little bit higher absolute values. Shape of Lorentz forces contours is very similar –
contours from analytic formula are are little bit flatter. Differences are caused probably some limitations
of Fluent and simplifications in inferring of analytic formula. Fluent computed with induced magnetic induction
and Lorentz forces in z direction are not zero. The shape of contours from Fluent shows situation when
container had been a little bit smaller (smaller container high). Shape of Lorentz forces contours is flatter with
increasing container high – for more details see [7].
ACKNOWLEDGEMENTS
This work was financially supported by the research project MSM 4674788501.
REFERENCES
[1] DAVIDSON, P., A.: An Introduction to Magnetohydrodynamics, Edinburgh: Cambridge University Press,
2001
[2] DOLEŽAL, I. - MUSIL, L,.: Modern industrial technology, GA ČR 102/03/0047, Praha, 2003
(from: k315.feld.cvut.cz/coupled_problems/Publikace/...kovy/.../paper1.pdf)
[3] anon.: ANSYS FLUENT 12.0 Magnetohydrodynamics (MHD) Module Manual, April 2009
[4] FRAŇA K.; STILLER J.; The Finite Element Method for Simulations of Magnetically Driven Flows,
International journal of mathematics and computers in simulation, Issue 3, Vol. 1, 2007, pp. 300-306
[5] HORÁKOVÁ, K. - FRAŇA, K.: The effect of rotating magnetic field, In XXIX Setkání kateder mechaniky
tekutin a termomechaniky, Rožnov pod Radhoštěm, Česká republika, květen 23-25, 2010,
Ostrava:VŠB-TU Ostrava, 2010
[6] HORÁKOVÁ K.; FRAŇA K.; Lorentz forces in the container, World Academy Of Science, Engineering
And Technology (WASET), Issue 59, Venice Italy, November 2011
[7] HORÁKOVÁ K.; FRAŇA K.; The effect of Lorentz forces parameters In Experimental fluid mechanics,
Liberec, Česká republika, listopad 25-27, 2009 Liberec: Technical University of Liberec, 2009
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