center for mhd studies turbulent mhd flow in a cylindrical vessel excited by a misaligned magnetic...

Center for MHD Studies

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

A. Kapusta and A. Kapusta and B. MikhailovichB. Mikhailovich

Center for MHD StudiesBen-Gurion University of the Negev

Beer-Sheva, Israel

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It is well-known that there are a lot of difficulties in calculations It is well-known that there are a lot of difficulties in calculations of MHD turbulent rotating flows, and certain approximations of MHD turbulent rotating flows, and certain approximations are required. In practice, semi-empirical models are frequently are required. In practice, semi-empirical models are frequently used. One of such models is used in our presentation. used. One of such models is used in our presentation. The matter in point is so called “external” friction The matter in point is so called “external” friction approximation described in [1] and modified in [2,3], where a approximation described in [1] and modified in [2,3], where a quasi-linear dissipative term –analog of the divergence of stress quasi-linear dissipative term –analog of the divergence of stress tensor- appears.tensor- appears.In this case, we can determine azimuthal component of the mean In this case, we can determine azimuthal component of the mean velocity assuming that all turbulent effects including the effect of velocity assuming that all turbulent effects including the effect of secondary flows on mean velocity are accounted for through the secondary flows on mean velocity are accounted for through the “external” friction coefficient.“external” friction coefficient.

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It is defined asIt is defined as

CRe 1

z,

wherewhere

0

1

rotV r is the mean vorticity of the flow,is the mean vorticity of the flow,

is an empirical coefficient, is an empirical coefficient,

C

Re R02

, z

Z0R0,

in the present case, in the present case,

- is a parameter defining the flow structure , - is a parameter defining the flow structure ,

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0.

0,12, 1


Note that the “external” friction approximation is based on the Note that the “external” friction approximation is based on the results of numerous experimental data and gives reliable results results of numerous experimental data and gives reliable results when calculating mean velocities of turbulent rotating MHD when calculating mean velocities of turbulent rotating MHD flows both under the action of a rotating magnetic field (RMF) flows both under the action of a rotating magnetic field (RMF) and in homeopolar facilities [4]. Moreover, the model has and in homeopolar facilities [4]. Moreover, the model has proved to be applicable also to the analysis of the behavior of proved to be applicable also to the analysis of the behavior of vortical structures in a turbulent wake behind a bluff body [5]. vortical structures in a turbulent wake behind a bluff body [5]. It is noteworthy that we have managed to obtain a universal It is noteworthy that we have managed to obtain a universal dependence of the angular velocity of turbulent flow core on the dependence of the angular velocity of turbulent flow core on the only dimensionless parameter only dimensionless parameter QQ within the frames of this model within the frames of this model

wherewhere

QNzC

,

N Ha2

Re, Ha B0R0

.

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We study the effect of misalignment between the magnetic field We study the effect of misalignment between the magnetic field rotation axis Zrotation axis Z0 0 and the symmetry axis of the cylindrical vessel and the symmetry axis of the cylindrical vessel Z on two-dimensional structure of the mean turbulent flow Z on two-dimensional structure of the mean turbulent flow excited by rotating magnetic field (RMF) in non-inductive excited by rotating magnetic field (RMF) in non-inductive approximation using the “external” friction model.approximation using the “external” friction model.

In practical applications of RMF, for example, in metallurgical In practical applications of RMF, for example, in metallurgical processes, the axis of the vessel with melt can be shifted with processes, the axis of the vessel with melt can be shifted with respect to the inductor axis. Therefore, the characteristics of the respect to the inductor axis. Therefore, the characteristics of the mean flow (its structure) can be changed depending both on mean flow (its structure) can be changed depending both on MHD interaction parameter MHD interaction parameter NN and on the eccentricity and on the eccentricity

Note that the eccentricity can also change the structure of Note that the eccentricity can also change the structure of surface waves on the free surface of a rotating fluid [6].surface waves on the free surface of a rotating fluid [6].

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If the axes ZIf the axes Z0 0 and Z coincide, and Z coincide, mean radial and azimuthal mean radial and azimuthal components of dimensionless components of dimensionless electromagnetic body forces are electromagnetic body forces are described by the following described by the following expressions under the condition expressions under the condition 0 R0

2 1 :

fr 0, f S2p

r02p1 , (1)

where p is the number of pole pairs of the inductor exciting RMF, S 1 .

Fig.1. Coordinate systems layoutFig.1. Coordinate systems layout

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Using the formulas of coordinate and vectorial components Using the formulas of coordinate and vectorial components transformation at the transition from a certain cylindrical transformation at the transition from a certain cylindrical coordinate system to another one we obtain coordinate system to another one we obtain for parallel Zfor parallel Z00 and Z and Z :

(2)

(later on we use without upper lines).

where

r0, 0,z0 r, , zfr 1

2pr2 2rcos 2p1sin,

f 1

2pr cosr2 2rcos 2p1, (3)

r rR, R, V0r0 Vr constr,

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Our next step is to estimate z-component of that can Our next step is to estimate z-component of that can be written as:be written as:

where

(4)

After differentiating (2), (3) and substituting into (4), we obtain:

rotf

rotzf1rr f r

fr.

(5)

r, F p2

rFr cosp 1rr cos2 2sin2,F r2 2 r cos 2.

rotzf

Sp

r, ,

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In this case a dimensionless equation for z-component of the In this case a dimensionless equation for z-component of the velocity vector potential can be written as follows velocity vector potential can be written as follows

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 ) solutions:

(6)

which should be solved under the boundary conditions:

(7)

r 1 0, rr 1

0.

12r

r

r

1

Re2

Nrotzf

,

Re:

0 1.

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In the zero approximation, the problem acquires the form:In the zero approximation, the problem acquires the form:

It follows from the evenness of (r,function and the operator with respect to coordinate that is also even, and the problem (8)-(9) is solved in the form:

(8)

and (9)

where

where

2

r21r

r

142r2

2

2,

Q21 4

Q 1.

0 NS

r, 0r 1 0,

0

are the roots of the equations for Bessel functions:

(10)

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kn , k

Jnkn 0, J0k 0.

0 k 1

k0J0krn 1 kn1cos nJnknr,


Substituting (10) into (8) and performing the procedure of Substituting (10) into (8) and performing the procedure of Galerkin’s method, we obtain:Galerkin’s method, we obtain:

where

kn1 A Ikn1

kn2Jn12kn,A

4NS

,

Ikn1 0

10

1

r, rJnknrcos 2n r.

(12)

(11)

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k0 A Ik0

k2J12k,Ik0

0

10

1

r, rJ0kr r,


In the first approximation, the problem can be written as:In the first approximation, the problem can be written as:

It follows from the oddness of the (r,function with respect to coordinate that the problem (13) can be solved in the form:

and

(13)

(14)

1 1

2r 0

0 r

0 r

0

1

r, 1r 1 0.

1 k 1

n 1

kn1sin2 n Jnknr.

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Substituting (14) into (13) and performing the procedure of Substituting (14) into (13) and performing the procedure of Galerkin’s method, and taking into account the fact thatGalerkin’s method, and taking into account the fact that

we obtain (15)

0 NS

r,

kn1 A2 Ikn2kn1

4kn2J2n1kn,where Ikn2

0

10

1

r r, Jnknrsin2n r.

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We have found exact solutions in the cases of We have found exact solutions in the cases of p=1 p=1 and and p=2.p=2.

AtAt p=1: p=1:

At p=2:

(16)

p 1 A41 r2

2r lnr cos

A2

4r 1sin

A2

32r lnr 1

81 2lnr sin2.

A4

r2lnr 1 r2

4sin

A56r21 r2

12 2r2lnr 1

8sin2.

p 2 A2

21 r2 r

4lnr 1 r2

42cos

(17)

A 4NS

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Fig.2 p=1, N=10^-5, Fig.2 p=1, N=10^-5,

0

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Fig. 3 p=1, N=10^-5, Fig. 3 p=1, N=10^-5,

0

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0

Fig. 4 p=1, N=10^-5, Fig. 4 p=1, N=10^-5,

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Fig. 5 p=2, N=10^-5, Fig. 5 p=2, N=10^-5,

0

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Dresden-2004

0

Fig. 6 p=2, N=10^-5, Fig. 6 p=2, N=10^-5,

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Fig. 7 p=2, N=10^-5, Fig. 7 p=2, N=10^-5,

0

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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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References1. 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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