- Email: [email protected]
Numerical analysis for laminar and turbulent liquid-metal flow in a transverse magnetic field
Fusion Engineering and Design North-Holland, Amsterdam
8 (1989)
249-256
249
NUMERICAL ANALYSIS FOR LAMINAR IN A TRANSVERSE MAGNETIC FIELD Minoru Research
TAKAHASHI, Laboratory
Alcira INOUE,
for Nuclear
Reactors,
Tokyo
AND
Masanori Institute
TURBULENT
ARITOMI
of Technology,
LIQUID-METAL
and Mitsuo 0-okayama
2-12-1,
FLOW
MATSUZAKI Meguro-kk
Tokyo,
Japan
A numerical analysis was conducted for mercury and lithium MHD flows in rectangular channels under uniform transverse magnetic fields. In order to couple electromagnetic effects with hydrodynamics, the axial momentum equation and the equations for the axial induced magnetic field in the fluid and wall were solved simultaneously. The suppression of turbulence was taken into account by using the modified k-c model of turbulence. The analytical results agreed well with the experimental results for the velocity profiles in mercury flow and skin friction coefficients in mercury and lithium flows. The M-shaped velocity profile for lithium flows was found, which is expected to be one reason for heat transfer enhancement at the wall parallel to the magnetic field.
1. Introduction From our experimental study on the MHD pressure drop and heat transfer of lithium single-phase and helium-lithium two-phase flows in horizontal rectangular channels under uniform transverse magnetic fields [l], it has been found that as the magnetic field increases, the heat transfer coefficient clearly increases. This is contrary to the experimental result for heat transfer in a pipe flow obtained by Gardner [2]. Thus, in order to find out the reason for this fact, a numerical analysis has been performed. This paper deals with the hydraulic analysis of liquid-metal MHD single-phase flows. A number of analytical studies on laminar liquid-metal MHD duct flows are reviewed by Walker [3]. One of the theoretical models applicable to our lithium flow system is Walker’s one for a constant-area rectangular channel [4] that includes the side layer. However, for more flexible calculation under various conditions, we have chosen a numerical approach using a finite-difference method. The numerical model is tested for the experiment for a mercury flow performed by Reed and Lykoudis [5,6] and then applied to our experiment for a lithium flow.
2. Analysis 2.1. Flow systems and coordinates Fig. 1 shows the channel cross-sections for the mercury and lithium flows. The channel for the mercury flow is electrically insulated with plexiglass, and the uniform magnetic field was applied vertically. The channels for the lithium flow are made of conductive stainless steel and heated from below. The magnetic field was applied uniformly and horizontally. Two types of lithium flow channels were used: Duct A with the height of 10 mm and Duct B with that of 5 mm. The coordinates for the analysis are chosen so that x is in the flow direction, y is perpendicular to the magnetic field, and z is parallel to it. The axial flow developing process from an inlet ordinary turbulent flow to a laminar MHD flow has been dealt with using the k--f model of turbulence. 0920-3796/89/$03.50
0 Elsevier Science Publishers
B.V.
250
M. Takahashi
et al. / Numerical
Non-conbucling
analysis for laminar
and turbulent
liquid-metal/low
wall
2b
r
%I Conducting
wall cm Duct
b=20mm A a=bmm 6 a=2.5mm
Fig. 1. Channel
cross-sections
and coordinate
systems.
2.2. Basic equations It is assumed, small compared negligible. From equation can be
in order to simplify the analysis, that the transverse velocities u and w are negligibly with the axial velocity u, and that the induced magnetic fields, by b,, and b,, are these assumptions, the terms u 3u/Cly, w pu/az, and abJ3x in the axial momentum omitted as
0) where the last term corresponds to Lorentz force expressed by the x-components of the induced magnetic field b,. From Ohm’s law and Maxwell’s equations, the equations for b, are given by [7] a=b 2+2=O
a=b
(inthefluid),
ay2 az2 a=b+ xa=b= 0 (in the conducting x ay2 az2
(2)
wall).
The turbulence damping process due to a magnetic field was simulated turbulence [8] which is modified for MHD turbulence. For the modification, for MHD turbulence [9], Kitamura’s k--E model of MHD turbulence [lo], closure model given by Hanjalic and Launder [ll]. The magnetic Reynolds fluctuations of induced magnetic field due to turbulence were neglected. transport equations for k and z have been derived
by using the k--E model of we followed Deissler’s theory and the ordinary turbulence number is so small that the As a result, the following
(5) cM = C,k=/e.
(6)
M. Takahashi Table
et al. / Numerical
analysis
for laminar
and turbulent
liquid-metal flow
251
1
The values
of the empirical
constants
in the k-c
model
of turbulence
. 1.55
0.09
2.0
0.8
The last terms in eqs. (4) and (5) represent constants used here are presented in table 1.
the additional
1.0
MHD
1.0
1.3
effect. The values of the empirical
2.3. Boundary conditions Since the flow characteristics are symmetrical at the center planes of the channel, it is necessary to calculate the flow only in one quarter of the cross-section. The boundary conditions are that the gradients of u, b,, k, and e are zero at the symmetry planes; no slip condition; the continuity of normal current density and b, at the inside surface of the channels; and b, = 0 at z = b and at the outside surface of the channel. By eliminating the convection and diffusion terms in eqs. (4) and (5), the boundary values for k and c in the vicinity of the wall can be derived as k, = C;1’2u*2/{ ep = {(GC,
-
1 + C,( C, - C,)/( C,>/(C2
-
C,))(&k,h).
C,C, - C,)} 1’2,
(7) (8)
2.4. Solution procedure The parabolic equations of eqs. (l)-(5) were solved using the marching integral method of the finite-difference method. One quarter of the channel cross-section was divided into 22 X 22 cells, where the mesh sizes in the viscous sublayer near the wall were taken much finer than the others.
3. Results and discussion 3. I. Results for mercury flow Fig. 2 shows the results for the non-dimensional velocity profiles u+ of the mercury flow in the center plane, at y = a, parallel to the magnetic field. It is found that the calculated profiles agree well with the experimental results obtained by Reed and Lykoudis [5,6]. It can be seen that the Hartman flattening is simulated reasonably well. Fig. 3 shows the calculated turbulence kinetic energy in the same plane. It is found that turbulence is damped significantly by the magnetic field. This result means that the turbulence effect can be neglected except near the entrance of the magnetic field. Fig. 4 shows the experimental and analytical results for the skin friction coefficient Cr. It is found that the present analysis provides a better prediction than the Hartman flow correlation. However, the local maximum or “hump” in the turbulent region could not be simulated successfully. 3.2. Resultsfor lithium ji’ow Figs. 5 and 6 show the calculated velocity profiles for lithium flows in Duct A and Duct B, respectively. It is found that the M-shaped velocity profiles, well known as the side layer [3], are formed in the plane
M. Takahashi
252
et al. / Numerical
analysis for laminar
and turbulent
liquid-metal flow
-t
0
I
10’
100
10’
10’
10.
z*
Fig. 2. Non-dimensional
velocity
profiles
in mercury
flow.
I
I
Non-conducting
wall
Mercury
3--
ol 0
Fig. 3. Calculated
turbulence
kinetic flows.
energy
I
I
2.1
profiles
Ha=0
Duct
v) 1. ‘E
1.5
Conducting
2
k
1.’
0.
I
0.1
I
0.2’
velocity
a
HaIRe coefficient
10
12
wall
I
in mercury
A I I
272
543 815 109c I
I I
I
0.3
0.4
y/2a profiles
0.5
0 I
x=400mm y=a 0.1
0.2
zl2b in the planes of y = a and P = b for lithium
0.3
14
x10-
wall
x=400mm
Fig. 5. Calculated
6
I
Conducting Lithium u=O.Wm/s Re=14800
2.1
4
Fig. 4. Skin friction
in mercury
I
2
0.4
flow in Duct
0.5
A.
flow.
M. Takahashi
ei al. / Numerical
analysis for laminar
and turbulent
liquid-metal flow
253
Max.l.19mls
U (0.b)
h.0)
y
(0.0) x=Omm
(0.b)
(a.0)
Y
(0.0) x=200mm
U
U (0.b)
(O,b)
x=400mm
(O,b)
x=600mm Duct
Fig. 6. Calculated
00=0.4T velocity profiles
0
x=600mm Ha=302,Re=7367
for lithium
flow in Duct
B.
perpendicular to the magnetic field, i.e. z = constant. On the other hand, the analytical results show the Hartmann flattening in the velocity profile in the plane parallel to the magnetic field, i.e. y = constant. Figs. 7 and 8 show the analytical results for the induced magnetic field and induced current density vectors. The y-component of the induced current that causes the Lorentz force is larger in the core than near the bottom wall, so that the local MHD pressure loss is higher in the core than near the bottom wall. This is shown more clearly in fig. 9, in which the magnitudes o friction, acceleration, and MHD losses are
b
r=o <
Fig. 7. Calculated
vectors
0.5 Almm’
of induced current flow in Duct A.
density
Duct
z=b
for lithium
Fig.
B
Bo=0.4T
Ha=302
x=400mm
(Lithium)
Re=7367
8. Calculated induced magnetic field and vectors of induced current density for lithium flow in Duct B.
254
M. Takahashi
et al. / Numerical
Conducting
wall
Lithium
Re=14600
analysis/or
Duct
laminar
and
turbulent
liquid-metal/low
A
Ha-815
? 27 0.5 kz t
APMHD t
ow
Fig. 9. Calculated
ratios
of friction,
acceleration
and MHD
losses to total pressure flow in Duct A.
5 z12b
loss
in the planes
of z = b and
y = LI for lithium
compared with each other. This characteristic feature in current density profiles might possibly yield the M-shaped velocity profile. Fig. 10 shows the results of the skin friction coefficients for lithium flows in Duct B. It is found that the analytical results agree relatively well with our experimental ones. The enhancement of heat transfer found from our lithium experiment may possibly be caused by the wall jet in the side layer predicted by the present analysis. The effect of the enhancement of turbulence reported by Branover and Sukoriansky [12] has not been checked here, which may be a future research subject.
--
10-2-o
4
-
2.4x101 1.2X10’
4 HaIRe
Fig. 10. Skin friction
coefficient
for lithium
flow in Duct
B.
h4. Takahashi
et al. / Numerical
analysis for laminar
and rurbulenl
liquid-metal
flow
255
4. Conclusions Numerical analysis was conducted for mercury and lithium flows in horizontal rectangular channels under the uniform transverse magnetic field, where the magnetic Reynolds number was neglected. The developing processes from an ordinary turbulent flow to a laminar MHD one were simulated using the k-c model of turbulence modified for MHD turbulence. The conclusions are as follows: (1) The analytical results for the velocity profiles of mercury flow agree well with the experimental ones obtained by Reed [5] and Reed and Lykoudis [6]. The Hartmann flattening was well simulated by the present analysis. (2) The analytical results for the skin friction coefficients of mercury and lithium flows also agree reasonably well with the experimental ones obtained by Reed [5], Reed and Lykoudis [6], and the authors. However, the local maximum of the skin friction coefficient in the turbulent region of mercury flow could not be simulated successfully. (3) The present analysis has predicted the M-shaped velocity profiles for the lithium flow, which is known as the side layer. Even if neglecting the enhancement of turbulence suggested by Branover and Sukoriansky [12], the heat transfer enhancement at the wall parallel to the magnetic field may possibly be caused by the high convection heat transfer due to the wall jet in the side layer.
Nomenclature a *o b b,, $7 6, C ct c l-49 De
Ha j k kv P Re u U+ u* 0,
w
X
Y7 = Z+ E CM K PO
V P
CD
half of the channel height in the z-direction, applied magnetic flux density, half of the channel width in the y-direction, x-, y- and z-components of induced magnetic flux density, wall conductance ratio (= u,,,6,Jafb), skin friction coefficient (= (Ap/AL)ab/( a + b)(1/2)pu2), empirical constants in the k-c model of turbulence, hydraulic diameter ( = 4ab/( a + b)), Hartmann number ( = B,, De/F, electrical current density, turbulence kinetic energy, distance between inside wall surface and boundary point for k and e, pressure, Reynolds number (= u/De/v), axial velocity, dimensionless axial velocity ( = U/U * ), frictional velocity (= J7,/p), velocities in the y- and z-directions, axial coordinate, coordinates perpendicular to and parallel to magnetic field, respectively, dimensionless distance from channel wall ( = zu */v), turbulence dissipation rate, eddy diffusivity, Karman’s constant, magnetic permeability, kinematic viscosity, fluid density,
M. Takahashi
256
et al. / Numerical
analysis for laminar
and rurbulent
liquid-metal
flow
turbulent Prandtl number for k and C, respectively, electrical conductivity in fluid and wall, respectively, wall shear stress, wall thickness.
References [l] [2] [3] [4] [5] [6] [7] [8] [9] [lo] [ll] [12]
A. Inoue, M. Aritomi, M. Takahashi, M. Matsuzaki, Y. Narita and K. Yano, Studies on MHD pressure drop and heat transfer of hehum-Iithium annular mist flow, 23th Nat. Heat Transfer Symp. Japan, B241 (1986). pp. 253-255 [in Japanese]. R.A. Gardner, K.L. Uherka and P.S. Lykoudis. Influence of a transverse magnetic field on forced convection Liquid metal heat transfer, AIAA J. (May 1966) 848. J.S. Walker, Single- and multi-phase flows in an electromagnetic field: laminar duct flows in strong magnetic fields (American Inst. Aeronaut. Astronaut., Inc., 1985, pp. 3-16. J.S. Walker, Magnetohydrodynamic flows in rectangular ducts with thin conducting walls: Part I. Constant-area and variable-area ducts with strong uniform magnetic fields, J. De Mecanique 20 (1981) 79. C.B. Reed, An investigation of shear turbulence in the presence of magnetic fields, Ph.D. thesis, Purdue University (1976). C.B. Reed and P.S. Lykoudis, The effect of a transverse magnetic field on shear turbulence, J. Fluid Mech. 89 (1978) 147. G.W. Sutton and A. Sherman, Engineering Magnetohydrodynamics (McGraw-Hill, 1965). W.P. Jones and B.E. Launder, The prediction of laminarization with a twoequation model of turbulence. Int. J. Heat Mass Transfer 15 (1972) 301. R.G. Deissler, Magneto-fluid dynamic turbulence with a uniform imposed magnetic field, Phys. Fluid 9 (1963) 1250. K. Kitamura and M. Hirata, Turbulent heat and momentum transfer for electrically conducting field flowing in two-dimensional channel under transverse magnetic field, 5th Int. Heat Transfer Conf. 3 M-18 (1978). pp. 159-164. K. HanjaIic and B.E. Launder, Contribution towards a Reynolds-stress closure for low-Reynolds-number turbulence, J. Fluid Mech. 74 (1976) 593. H. Branover and S. Sukoriansky, Enhancement of turbulence in a magnetic field, 5th Beer-Sheva Seminar on Magnetohydrodynamic Flows and Turbulence, Jerusalem (1987).
8 (1989)
249-256
249
NUMERICAL ANALYSIS FOR LAMINAR IN A TRANSVERSE MAGNETIC FIELD Minoru Research
TAKAHASHI, Laboratory
Alcira INOUE,
for Nuclear
Reactors,
Tokyo
AND
Masanori Institute
TURBULENT
ARITOMI
of Technology,
LIQUID-METAL
and Mitsuo 0-okayama
2-12-1,
FLOW
MATSUZAKI Meguro-kk
Tokyo,
Japan
A numerical analysis was conducted for mercury and lithium MHD flows in rectangular channels under uniform transverse magnetic fields. In order to couple electromagnetic effects with hydrodynamics, the axial momentum equation and the equations for the axial induced magnetic field in the fluid and wall were solved simultaneously. The suppression of turbulence was taken into account by using the modified k-c model of turbulence. The analytical results agreed well with the experimental results for the velocity profiles in mercury flow and skin friction coefficients in mercury and lithium flows. The M-shaped velocity profile for lithium flows was found, which is expected to be one reason for heat transfer enhancement at the wall parallel to the magnetic field.
1. Introduction From our experimental study on the MHD pressure drop and heat transfer of lithium single-phase and helium-lithium two-phase flows in horizontal rectangular channels under uniform transverse magnetic fields [l], it has been found that as the magnetic field increases, the heat transfer coefficient clearly increases. This is contrary to the experimental result for heat transfer in a pipe flow obtained by Gardner [2]. Thus, in order to find out the reason for this fact, a numerical analysis has been performed. This paper deals with the hydraulic analysis of liquid-metal MHD single-phase flows. A number of analytical studies on laminar liquid-metal MHD duct flows are reviewed by Walker [3]. One of the theoretical models applicable to our lithium flow system is Walker’s one for a constant-area rectangular channel [4] that includes the side layer. However, for more flexible calculation under various conditions, we have chosen a numerical approach using a finite-difference method. The numerical model is tested for the experiment for a mercury flow performed by Reed and Lykoudis [5,6] and then applied to our experiment for a lithium flow.
2. Analysis 2.1. Flow systems and coordinates Fig. 1 shows the channel cross-sections for the mercury and lithium flows. The channel for the mercury flow is electrically insulated with plexiglass, and the uniform magnetic field was applied vertically. The channels for the lithium flow are made of conductive stainless steel and heated from below. The magnetic field was applied uniformly and horizontally. Two types of lithium flow channels were used: Duct A with the height of 10 mm and Duct B with that of 5 mm. The coordinates for the analysis are chosen so that x is in the flow direction, y is perpendicular to the magnetic field, and z is parallel to it. The axial flow developing process from an inlet ordinary turbulent flow to a laminar MHD flow has been dealt with using the k--f model of turbulence. 0920-3796/89/$03.50
0 Elsevier Science Publishers
B.V.
250
M. Takahashi
et al. / Numerical
Non-conbucling
analysis for laminar
and turbulent
liquid-metal/low
wall
2b
r
%I Conducting
wall cm Duct
b=20mm A a=bmm 6 a=2.5mm
Fig. 1. Channel
cross-sections
and coordinate
systems.
2.2. Basic equations It is assumed, small compared negligible. From equation can be
in order to simplify the analysis, that the transverse velocities u and w are negligibly with the axial velocity u, and that the induced magnetic fields, by b,, and b,, are these assumptions, the terms u 3u/Cly, w pu/az, and abJ3x in the axial momentum omitted as
0) where the last term corresponds to Lorentz force expressed by the x-components of the induced magnetic field b,. From Ohm’s law and Maxwell’s equations, the equations for b, are given by [7] a=b 2+2=O
a=b
(inthefluid),
ay2 az2 a=b+ xa=b= 0 (in the conducting x ay2 az2
(2)
wall).
The turbulence damping process due to a magnetic field was simulated turbulence [8] which is modified for MHD turbulence. For the modification, for MHD turbulence [9], Kitamura’s k--E model of MHD turbulence [lo], closure model given by Hanjalic and Launder [ll]. The magnetic Reynolds fluctuations of induced magnetic field due to turbulence were neglected. transport equations for k and z have been derived
by using the k--E model of we followed Deissler’s theory and the ordinary turbulence number is so small that the As a result, the following
(5) cM = C,k=/e.
(6)
M. Takahashi Table
et al. / Numerical
analysis
for laminar
and turbulent
liquid-metal flow
251
1
The values
of the empirical
constants
in the k-c
model
of turbulence
. 1.55
0.09
2.0
0.8
The last terms in eqs. (4) and (5) represent constants used here are presented in table 1.
the additional
1.0
MHD
1.0
1.3
effect. The values of the empirical
2.3. Boundary conditions Since the flow characteristics are symmetrical at the center planes of the channel, it is necessary to calculate the flow only in one quarter of the cross-section. The boundary conditions are that the gradients of u, b,, k, and e are zero at the symmetry planes; no slip condition; the continuity of normal current density and b, at the inside surface of the channels; and b, = 0 at z = b and at the outside surface of the channel. By eliminating the convection and diffusion terms in eqs. (4) and (5), the boundary values for k and c in the vicinity of the wall can be derived as k, = C;1’2u*2/{ ep = {(GC,
-
1 + C,( C, - C,)/( C,>/(C2
-
C,))(&k,h).
C,C, - C,)} 1’2,
(7) (8)
2.4. Solution procedure The parabolic equations of eqs. (l)-(5) were solved using the marching integral method of the finite-difference method. One quarter of the channel cross-section was divided into 22 X 22 cells, where the mesh sizes in the viscous sublayer near the wall were taken much finer than the others.
3. Results and discussion 3. I. Results for mercury flow Fig. 2 shows the results for the non-dimensional velocity profiles u+ of the mercury flow in the center plane, at y = a, parallel to the magnetic field. It is found that the calculated profiles agree well with the experimental results obtained by Reed and Lykoudis [5,6]. It can be seen that the Hartman flattening is simulated reasonably well. Fig. 3 shows the calculated turbulence kinetic energy in the same plane. It is found that turbulence is damped significantly by the magnetic field. This result means that the turbulence effect can be neglected except near the entrance of the magnetic field. Fig. 4 shows the experimental and analytical results for the skin friction coefficient Cr. It is found that the present analysis provides a better prediction than the Hartman flow correlation. However, the local maximum or “hump” in the turbulent region could not be simulated successfully. 3.2. Resultsfor lithium ji’ow Figs. 5 and 6 show the calculated velocity profiles for lithium flows in Duct A and Duct B, respectively. It is found that the M-shaped velocity profiles, well known as the side layer [3], are formed in the plane
M. Takahashi
252
et al. / Numerical
analysis for laminar
and turbulent
liquid-metal flow
-t
0
I
10’
100
10’
10’
10.
z*
Fig. 2. Non-dimensional
velocity
profiles
in mercury
flow.
I
I
Non-conducting
wall
Mercury
3--
ol 0
Fig. 3. Calculated
turbulence
kinetic flows.
energy
I
I
2.1
profiles
Ha=0
Duct
v) 1. ‘E
1.5
Conducting
2
k
1.’
0.
I
0.1
I
0.2’
velocity
a
HaIRe coefficient
10
12
wall
I
in mercury
A I I
272
543 815 109c I
I I
I
0.3
0.4
y/2a profiles
0.5
0 I
x=400mm y=a 0.1
0.2
zl2b in the planes of y = a and P = b for lithium
0.3
14
x10-
wall
x=400mm
Fig. 5. Calculated
6
I
Conducting Lithium u=O.Wm/s Re=14800
2.1
4
Fig. 4. Skin friction
in mercury
I
2
0.4
flow in Duct
0.5
A.
flow.
M. Takahashi
ei al. / Numerical
analysis for laminar
and turbulent
liquid-metal flow
253
Max.l.19mls
U (0.b)
h.0)
y
(0.0) x=Omm
(0.b)
(a.0)
Y
(0.0) x=200mm
U
U (0.b)
(O,b)
x=400mm
(O,b)
x=600mm Duct
Fig. 6. Calculated
00=0.4T velocity profiles
0
x=600mm Ha=302,Re=7367
for lithium
flow in Duct
B.
perpendicular to the magnetic field, i.e. z = constant. On the other hand, the analytical results show the Hartmann flattening in the velocity profile in the plane parallel to the magnetic field, i.e. y = constant. Figs. 7 and 8 show the analytical results for the induced magnetic field and induced current density vectors. The y-component of the induced current that causes the Lorentz force is larger in the core than near the bottom wall, so that the local MHD pressure loss is higher in the core than near the bottom wall. This is shown more clearly in fig. 9, in which the magnitudes o friction, acceleration, and MHD losses are
b
r=o <
Fig. 7. Calculated
vectors
0.5 Almm’
of induced current flow in Duct A.
density
Duct
z=b
for lithium
Fig.
B
Bo=0.4T
Ha=302
x=400mm
(Lithium)
Re=7367
8. Calculated induced magnetic field and vectors of induced current density for lithium flow in Duct B.
254
M. Takahashi
et al. / Numerical
Conducting
wall
Lithium
Re=14600
analysis/or
Duct
laminar
and
turbulent
liquid-metal/low
A
Ha-815
? 27 0.5 kz t
APMHD t
ow
Fig. 9. Calculated
ratios
of friction,
acceleration
and MHD
losses to total pressure flow in Duct A.
5 z12b
loss
in the planes
of z = b and
y = LI for lithium
compared with each other. This characteristic feature in current density profiles might possibly yield the M-shaped velocity profile. Fig. 10 shows the results of the skin friction coefficients for lithium flows in Duct B. It is found that the analytical results agree relatively well with our experimental ones. The enhancement of heat transfer found from our lithium experiment may possibly be caused by the wall jet in the side layer predicted by the present analysis. The effect of the enhancement of turbulence reported by Branover and Sukoriansky [12] has not been checked here, which may be a future research subject.
--
10-2-o
4
-
2.4x101 1.2X10’
4 HaIRe
Fig. 10. Skin friction
coefficient
for lithium
flow in Duct
B.
h4. Takahashi
et al. / Numerical
analysis for laminar
and rurbulenl
liquid-metal
flow
255
4. Conclusions Numerical analysis was conducted for mercury and lithium flows in horizontal rectangular channels under the uniform transverse magnetic field, where the magnetic Reynolds number was neglected. The developing processes from an ordinary turbulent flow to a laminar MHD one were simulated using the k-c model of turbulence modified for MHD turbulence. The conclusions are as follows: (1) The analytical results for the velocity profiles of mercury flow agree well with the experimental ones obtained by Reed [5] and Reed and Lykoudis [6]. The Hartmann flattening was well simulated by the present analysis. (2) The analytical results for the skin friction coefficients of mercury and lithium flows also agree reasonably well with the experimental ones obtained by Reed [5], Reed and Lykoudis [6], and the authors. However, the local maximum of the skin friction coefficient in the turbulent region of mercury flow could not be simulated successfully. (3) The present analysis has predicted the M-shaped velocity profiles for the lithium flow, which is known as the side layer. Even if neglecting the enhancement of turbulence suggested by Branover and Sukoriansky [12], the heat transfer enhancement at the wall parallel to the magnetic field may possibly be caused by the high convection heat transfer due to the wall jet in the side layer.
Nomenclature a *o b b,, $7 6, C ct c l-49 De
Ha j k kv P Re u U+ u* 0,
w
X
Y7 = Z+ E CM K PO
V P
CD
half of the channel height in the z-direction, applied magnetic flux density, half of the channel width in the y-direction, x-, y- and z-components of induced magnetic flux density, wall conductance ratio (= u,,,6,Jafb), skin friction coefficient (= (Ap/AL)ab/( a + b)(1/2)pu2), empirical constants in the k-c model of turbulence, hydraulic diameter ( = 4ab/( a + b)), Hartmann number ( = B,, De/F, electrical current density, turbulence kinetic energy, distance between inside wall surface and boundary point for k and e, pressure, Reynolds number (= u/De/v), axial velocity, dimensionless axial velocity ( = U/U * ), frictional velocity (= J7,/p), velocities in the y- and z-directions, axial coordinate, coordinates perpendicular to and parallel to magnetic field, respectively, dimensionless distance from channel wall ( = zu */v), turbulence dissipation rate, eddy diffusivity, Karman’s constant, magnetic permeability, kinematic viscosity, fluid density,
M. Takahashi
256
et al. / Numerical
analysis for laminar
and rurbulent
liquid-metal
flow
turbulent Prandtl number for k and C, respectively, electrical conductivity in fluid and wall, respectively, wall shear stress, wall thickness.
References [l] [2] [3] [4] [5] [6] [7] [8] [9] [lo] [ll] [12]
A. Inoue, M. Aritomi, M. Takahashi, M. Matsuzaki, Y. Narita and K. Yano, Studies on MHD pressure drop and heat transfer of hehum-Iithium annular mist flow, 23th Nat. Heat Transfer Symp. Japan, B241 (1986). pp. 253-255 [in Japanese]. R.A. Gardner, K.L. Uherka and P.S. Lykoudis. Influence of a transverse magnetic field on forced convection Liquid metal heat transfer, AIAA J. (May 1966) 848. J.S. Walker, Single- and multi-phase flows in an electromagnetic field: laminar duct flows in strong magnetic fields (American Inst. Aeronaut. Astronaut., Inc., 1985, pp. 3-16. J.S. Walker, Magnetohydrodynamic flows in rectangular ducts with thin conducting walls: Part I. Constant-area and variable-area ducts with strong uniform magnetic fields, J. De Mecanique 20 (1981) 79. C.B. Reed, An investigation of shear turbulence in the presence of magnetic fields, Ph.D. thesis, Purdue University (1976). C.B. Reed and P.S. Lykoudis, The effect of a transverse magnetic field on shear turbulence, J. Fluid Mech. 89 (1978) 147. G.W. Sutton and A. Sherman, Engineering Magnetohydrodynamics (McGraw-Hill, 1965). W.P. Jones and B.E. Launder, The prediction of laminarization with a twoequation model of turbulence. Int. J. Heat Mass Transfer 15 (1972) 301. R.G. Deissler, Magneto-fluid dynamic turbulence with a uniform imposed magnetic field, Phys. Fluid 9 (1963) 1250. K. Kitamura and M. Hirata, Turbulent heat and momentum transfer for electrically conducting field flowing in two-dimensional channel under transverse magnetic field, 5th Int. Heat Transfer Conf. 3 M-18 (1978). pp. 159-164. K. HanjaIic and B.E. Launder, Contribution towards a Reynolds-stress closure for low-Reynolds-number turbulence, J. Fluid Mech. 74 (1976) 593. H. Branover and S. Sukoriansky, Enhancement of turbulence in a magnetic field, 5th Beer-Sheva Seminar on Magnetohydrodynamic Flows and Turbulence, Jerusalem (1987).