CC BY
5
УДК 538.4
SHAMOTA V.P., Doctor of Technical Sciences, Professor (Donetsk Railway Transport Institute)
ORFINIAK E.Yu., Senoir Lecturer (Donetsk Railway Transport Institute)
Turbulent Rotating Magnetohydrodynamic Flow in Circular Channel of Restricted Height
Шамота В.П., д.т.н., профессор (ДОНИЖТ) Орфиняк Е.Ю., старший преподаватель (ДОНИЖТ)
Турбулентное вращательное МГД течение в кольцевой щели конечной длины
Introduction_
The conducting liquid flow produced by the impact of a rotating magnetic field in axisymmetric vessels is a traditional object of magnetohydrodynamic (MHD) research. The broad range of technological MHD force application for intensifying heat-and-mass transfer processes units of metallurgy, foundry manufacture and power engineering. The fact that many technical installations have axial symmetry causes the steady interest in the given issue. Being rather promising in the sphere of soluble technological tasks, the method is based on using a rotating magnetic field in the processes of refining and doping, moving and feeding liquid metals, in reception of high-quality monocrystals.
The main aim of magnetic hydrodynamics of liquid metals study is to get the data, allowing to take into account the influence of electromagnetic forces on liquid metal to improve existing productions and to develop new technological processes. Finally this problem can be reduced to finding out those changes of liquid metals which are turned out to be under influence of electromagnetic forces since the
hydrodynamical parameters transformation substantially determines the characteristics of MHD machines and the course of processes in technological units.
Main aim of the article
Despite wide application of a rotating magnetic field in technological processes, MHD phenomena have been investigated insufficiently up to now. The few works, existing in this sphere, are based on the obtained solutions of one-dimensional stationary problems on MHD rotation which considered the case of liquid streamline without taking into account the impact of induction effects on the flow and the magnetic field. Such quite a rough description does not reflect all the variety of the phenomena in question that arises in conductive liquid under the action of the rotating magnetic field and determining technological suitability of MHD influence.
The main aim of the present article is to obtain the velocity field distribution of liquid along radial and longitudinal axes using a semiempirical model of "external friction" and to compare the results with experimental data.
Theory
Mathematical model of flow produced by rotating magnetic field
The system of the equations describing processes which proceed from
interaction of the conducting liquid and the rotating magnetic field consists of the Maxwell equations written down in the assumption that the flows of displacement and volumetric charges can be neglected.
(1)
,-oUi = ; rotB = /Li/iiy] ; divB = 0 ; divE = 0,
dt
Ohm law:
J = а(Ё + йхв).
(2)
Reynolds equation for turbulent flow of a viscous incompressible liquid with consideration of a ponderomotive force:
du
dt
+
+ (м V) и = -VP + vAu +
(3)
P
P
and the continuity equation
divu = 0.
(4)
In these equations u is the main flow velocity, J is the current density, B is the induction density, E is the electric field strength, ju0 is the magnetic constant, ju is the magnetic capacity of liquid, a is the conductivity of liquid, v is the coefficient of kinematic viscosity of liquid
t = -pui'uj is Reynolds stress tensor.
Since the number of variables included in equations (1) - (4) exceeds the number of the equations, the additional equations are necessary which would allow to connect components of Reynolds stress tensor with average velocity of the flow. The search of
these parities can be carried out by various ways, including the one carried out empirically. The definition of such empirical equations for a particular sphere of problems on the rotary flow of a liquid conductor under the influence of the rotating magnetic field was also one of the tasks in the given research. Thus, at first it was necessary to achieve the qualitative conformity.
Let us note that the rotary flow of liquid has a lot of characteristics connected with the periodicity of such flow and its axisymmetry and attributes it to the special class of flows. Therefore, it has appeared to be possible to apply quite a simple "external friction" model for the description of the rotary flow produced by the rotating magnetic field. Suppose that in this model for the forces produced by Reynolds stress tensor there is existence of the following condition:
divr = —pku
(5)
where k is the "external friction" coefficient, dependent on geometrical sizes of the flow area, physical properties of the liquid, frequency of the magnetic field rotation and the average angular velocity of liquid rotation. Such an assumption is justified by the fact that in a big part of experiments on rotary flow of liquid it is possible to allocate a nucleus, occupying the most part of the flow area, and quite
thin boundary layers near the lateral and face surfaces of the vessel. Let us note that in the "external friction" model the dependence between forces produced by Reynolds stress and average flow velocity is not supposed to be linear. Besides the average angular velocity of rotation Q is determined through average velocity u . Moreover as can be seen from formula (5) for Reynolds stress the integrated ratio turns out to induce a choice of the name of the model.
Results and discussion
MHD flow in the annular slot of limited height
Let us suppose that the viscous incompressible conducting liquid is placed in an angular vessel formed by two thin coaxial cylindrical surfaces. This system has radiuses R and R, and height H, and is made of nonmagnetic material.
Fig. 1.
The vessel with liquid is placed in a magnet field inductor,whose the rotational axis coincides with an axis of vessel symmetry (Fig. 1.). The conducting liquid will be set in motion after field rotation like a rotor in an asynchronous engine. It is natural that the velocity of such a movement will be less than the velocity of
the field owing to the presence of the basic energy dissipation of the flow.
We introduce vector potential of the electromagnetic field A using the ratio B = rotA .Then after simple transformations from system (1) - (2) the equation for vector potential easily turns into
AA = /u/u0
ÔA _
--\-u xrotA
dt
(6)
Let us proceed to dimensionless variables and for this purpose we introduce characteristic scales of length R which is
chosen as the external radius of the annular slot, the velocity u0 which is chosen as the velocity of the magnetic field rotation on the surface of the vessel, u0 =a0 R/p (here
®0 is the cyclic frequency of inductive alternating current, p is the number of pole-pairs), magnetic induction bo which
is chosen as the maximal magnetic induction on the external surface of the vessel, time i0 which is chosen as equal to
the ratio of the characteristic size to characteristic velocity, ampere density J = oco0BR , vector electromagnetic
potential A = B/R , volumetric electromagnetic force F0 = JBl P-
Then equation (6) can be transformed to the dimensionless form:
л- -(дй - -
Aa = co\--v x rota
I dt
(7)
Here o=y.jj.oo'0Q R^/p is the relative
frequency having physical sense as a square of the ratio of the characteristic size to the magnetic penetration depth in a liquid conductor.
Let us suppose that the rotating magnetic field generated by the azimuthal
wave of the superficial density of the flow allocates on an external surface of the vessel, and the external space is filled with ideal ferromagnetic substance. Besides we assume that inductor length exceeds many times over the pole-height of the conducting liquid, and owing to this fact we can suppose that the ampere density vector is parallel to forming cylindrical surface.
Next, we can describe the electrodynamic processes in cylindrical coordinate system r,y, z. The axis z of this system is considered to be a symmetry axis of the cylindrical surface. Only one z -component of vector potential is enough to satisfy to the equation:
Aa = œ
da„
v да
dt r dç
(8)
Since on the border of the two matters the tangential component of the magnetic induction changes at the rate of the superficial ampere density, so on the external cylindrical surface the boundary condition for vector potential is:
да„
dr
= eip(t~ç)
(9)
electrodynamic part of the problem under consideration can be solved irrespective of the hydrodynamic part. And for the ideal inductive field the solution is:
a = rpe'p(t-ç)lp .
(10)
Knowing the vector potential it is not easy to determine the density of electromagnetic forces. In non-inductive approximation only azimuthal component of the electromagnetic force has the stationary component and its value can be found by the formula:
fç= r2 p-2 ( r - V )/ 2.
(11)
The constant component of electromagnetic forces generates the stationary azimuthal flow and the varying part is derived from the wave motion of the liquid whose frequency is twice as high as that of the rotating magnetic field, and periodic change of pressure in the liquid has the same frequency as well.
Let the average flow velocity of one azimuthal component be marked v (r, z) for
which in case of stationary flow we have: \v -Av+Ha2r 2p-p (r - v) = 0. (12)
On the internal cylindrical surface the conditions for the continuous tangential and normal components of the magnetic induction vector must take place.
It is natural that generally equation (9) should be solved with the equations of liquid motion. However, in the case when the relative frequency is quite small co < 1 (non-inductive approximation) the expression in the right part of the equation (9) may be neglected. Then the
Here  is a dimensionless "external friction" coefficient, Haa = BRyffîpv is Hartmann number.
The solution of equation (12) must satisfy the compliance with the conditions on surfaces of the vessel and a symmetry condition:
v = V = V
lr=^ I r=1 lr=^
dv ~t
= 0. (13)
z=0
1
r
Here Sr = RjR0 and Sz = H/2R are dimensionless geometrical parameters. To identify the solution of equation (12) which satisfies boundary conditions (13) we apply Galerkin method. Let us extend v(r, z) in numerical terms on complete systems of functions
W (r ) = J1 (a,r) - J! (a, ) Y (a,r)/ Y (a,) and
cos (ykz ) where a, are solutions of the
equation J (Srx)Y (x)-J (x)Y (Sx) = 0 , J and Y are Bessel functions, rk=(2k -1) nj2Sz . It is easy to ascertain
that the given systems of functions satisfy the given boundary conditions.
v ( r, z )=X VW ( r ) cos (ykr ). (14)
i ,k=1
Setting decomposition (14) into initial equation (12), multiplying the received equality by rWj (r)cos(y,r) and integrating
by radius from Sr to 1 and by z from 0 to S we obtain the following system of the linear algebraic equations for the components of decomposition (14) Vik.
К + fk )VkPAA ^VjQj = 2Ha2a (-if1 SJS:
j=i
(15)
where p2 =A+Ha2a, p = j rWt 2 (r) dr,
Sr
1
S; = j r2 PW ( r ) dr,
Sr
1
Q = j r2p-1W (r )W ( r )dr.
Sr
In the case the number of inductor polepairs is equal to 1 the solution of system (15) can be written in the obvious form:
Vk =
2 Ha2 (-i)k1 S (a2+/k +P2)âIpi ■
(16)
Generally it is necessary to find the solution of this system numerically. In the given work to seek the solution of system (15) when the number of pole-pairs exceeds 1 we use Gauss method.
It has appeared that the "external friction" coefficient determined earlier for flow in a cylindrical vessel [1] is quite suitable for the turbulent flow in an annular slot.
A = QeC2 (ReraQ)1-V(1 -Sr) (17)
Here Re^, =®0 R2/pv is Reynolds number constructed for rotating velocity in the magnetic field on the external surface of the annular slot, Q is average angular velocity in the nucleus of the flow, C = 0,019 , C21 = 10,4, C22 = 12 are empirical constants, s = 0;0,5;1 is an empirical parameter determining flow character. In calculating of the transition point from one mode of the flow to another we use as well as in [2] an angular velocity of liquid rotation with a minimality condition. Thus the average angular velocity included in the formula for "external friction" coefficient is defined by the iterative technique [2].
In fig. 2 the dependence of the average angular flow on the magnetic induction (Hartmann number) and the comparison with the results of [3] are represented. In fig. 3 the dependence of the average velocity of the liquid conductor rotation on height of a pole is shown under
condition of fixed value of the magnetic induction Haa = 9,9 . All represented
measurements were carried out in the inductor on the assumption that the
14 -12 -10 -8 -6 -4 -2 -0 -
0 5 10 15 20
Fig. 2. o - Sr= 0,408, Sz= 2,0 ;
* - 5r = 0,588, S2 = 1,96
frequency of the alternating current is constant and equals to 50 Hz. The values of dimensionless coefficients at the experiments are c = 0,91, = 9,46 • 105.
п y inn
0 12 3
Fig. 3.
o - 5r= 0,235, * - 5r= 0,408, • - 8r= 0,588
To be convinced of the approximation of the approached solutions, we carried out the comparison of the estimations in the introduced algorithms with the exact solution obtained on the assumption that the heterogeneity of the velocity distribution along the vertical axis was neglected. The results of the comparison are shown in fig. 4.
Velocity distribution of slot radius in section z=0 submitting curve 1 corresponds to the exact solution for the flow nucleus. . Note that despite some differences between two curves, we managed to obtain quite a good approximation of the solutions that testifies to the correctness of the used method for the close results.
Fig. 4.
Fig. 5.
In fig. 4 the velocity distributions of the liquid flow along the radius are shown under assumption that Sr = 0,235 ,
S2 = 0,784 , C = 0,91 , Rec = 9,46• 105 ,
Haa = 11,3. Here r*=(r-Sr)/(1-Sr) . For
comparison in fig.4 the measurements executed by means of Poirier's two-channel probe [4] at the same values of dimensionless parameters are given. In fig. 5 the average flow velocity distributions of the liquid conductor with height of a slot in two sections o - r = 0,5 and * - r = 0,75 are represented. The accounts and measurements were carried out under the same values of dimensionless parameters of interaction.
Conclusions
To summarize, it is suggested that the introduced measurements and the flow velocity of the liquid conductor produced by the rotating magnetic field have assumed the existence of a flow nucleus in which the average velocity distribution along length is homogeneous and linear along the radius. This nucleus occupies the most part of the flow area and the boundary layers of the lateral and face surfaces of the vessel are quite thin. When the magnetic induction increases then, on the contrary, a boundary layer decreases.
It is necessary to notice that the considered process can be useful for all technical spheres of human activity consuming any products of the mentioned fields.
References:
1. Kapusta, A. B. Turbulent Rotating MHD Flows in Axially Symmetric Vessels / A. B. Kapusta, V. P. Shamota // Transfer Phenomena in Magnetohydrodynamics and
Electroconducting Flows. Proceedings. France. - 1997. - № 2. - P. 88-94.
2. Kapusta A. B. Using the "External Friction" Model for Describing Quasilaminar and Turbulent Flows Caused by a Rotating Magnetic Field / A. B. Kapusta, V. P. Shamota // Progress in Fluid Flow Research. Turbulence and Applied MHD. AIAA Virginia, USA. -1998. - № 182. - P. 771-778.
3. Shamota V. P. Flow of a Conducting Liquid in Annular Slots Produced by a Rotating Magnetic Field / V.P. Shamota // Magnetohydrodynamics. -1997. - № 1(33). - P. 56-59.
4. Poirier Y. Contribution a l'etude experimentale de la magnetodinamique des liquides / Y. Poirier // Alger. Sciences Physiques. - 1960. - № 1(VI). - P. 5-101.
Abstracts
The article deals with the turbulent flow of liquid conductor caused by a rotating electromagnetic field in an annular channel of limited length. The velocity field distribution of liquid along radial and longitudinal axes is obtained using a semiempirical model of "external friction". The obtained results are compared with experimental data.
Key words: rotating electromagnetic field, conductive liquid, annular channel, turbulent flow.
В работе рассматривается турбулентное течение жидкого проводника в кольцевом канале ограниченной высоты, создаваемое вращающимся электромагнитным полем. С помощью полуэмпирической модели «внешнего трения» найдено распределение поля скоростей жидкости по радиальной и продольной осям. Полученные результаты сравниваются с экспериментальными данными.
Ключевые слова: вращающееся электромагнитное поле, проводящая жидкость, кольцевой канал, турбулентное течение.
