**3.4 Comparison with magnetic-field inlet-region**

Figure 7 shows schematically the applied magnetic field in the *y*-direction, the induced currents in the *x*-*z* plane including the directions of Lorentz force and the pressure along the *z*-axis, in the inlet region and the outlet region of the magnetic field. The larger Lorentz force acts and thus the larger pressure drop occurs in the inlet and outlet regions than in the fullydeveloped MHD region for the reason mentioned in Chap. 1. On the other hand, a smaller Lorentz force may act in the flow direction and thus a small pressure recovery may occur in the first section of the inlet region and in the last section of the outlet region also for the reason mentioned in Chap. 1. The pressure drop behavior is not completely symmetric, since the fully- developed non-MHD flow enters the calculation domain in the inlet-region while the fully-developed MHD flow enters the domain in the outlet-region.

Figure 8 presents pressures along *z*-axis for the magnetic-field inlet-region, calculated by the authors and presnted in a previous paper (Kumamaru et al, 2007). The calculation parameters for Fig. 8 are the same as for Fig. 2, except for the Hartmann number change along z-axis. The Hartmann number (relating to the applied magnetic field) is 0 from *z*=0 to *z1*, increases linearly from *z*=*z1* to *z2*, and is 100 from *z*=*z2* to *z0*. The pressure change for the case of *z1*/*z2*=10/12, i.e. a standard case, in the inlet region, indicated by a dotted line, is also compared with the corresponding case in the outlet region in Fig. 3.

The pressure decreases slowly following the drop in a non-MHD laminar flow from *z*=0 to *z* ≈ *z1*. The pressure recovery appears clearly in the region near *z* ≈ *z1*. The pressure decreases more rapidly in the region from *z* ≈ *z1* to *z* ≈ *z2* than in the fully-developed MHD region of *z*>*z2*. The pressure decreases rapidly following the drop of a fully-developed MHD flow in the region of *z*>*z2*.

Figure 9 illustrates induced electric current distribution in the *x-z* plane at *y*=0 for the case of *z1*/*z2*=10/12, i.e. the standard case, in the magnetic-field inlet-region. The distribution in the

Three-Dimensional Numerical Analyses on Liquid-Metal

0

Fig. 9. Induced currents for magnetic-field inlet-region.

region from *z* ≈ 11.5 to *z* ≈ 12, i.e. -Δ*p* ≈ 0.2. The reason is examined later.

MHD fully-developed flow with the flat profile comes into the outlet region.

through the inlet region may become smaller than that through the outlet region.

0.2

0.4

0.6

y

0.8

1

Magnetohydrodynamic Flow Through Circular Pipe in Magnetic-Field Outlet-Region 219

0 5 10 15 20 25

5

j

x

25

j

z

x

inlet region, Fig. 9, and that in the outlet region, Fig. 4, are nearly symmetric. For this reason, the sharp pressure drop in the inlet region from *z* ≈ 11 to *z* ≈ 13, i.e. -Δ*p* ≈ 0.9, agrees nearly with that in the outlet region from *z* ≈ 9 to *z* ≈ 11.5, i.e. -Δ*p* ≈ 0.9. However, the pressure recovery in the inlet region from *z* ≈ 9.5 to *z* ≈ 11, i.e. -Δ*p* ≈ 0.4 is larger than that in the outlet

Figures 10(a), (b) and (c) show calculated velocity *vz* distributions at *z*=10, 11 and 12, respectively, for the case of *z1*/*z2*=10/12, i.e. the standard case, in the magnetic-field inletregion. The velocity profile at *z*=10, shown in Fig. 10(a), still keeps nearly a distribution typical to a non-MHD fully-developed laminar flow with a peak value of ~2. Hereafter, the velocity distribution becoms flatter along the channel axis, i.e. the *z*-axis, as shown in Figs. 10(b) (at *z*=11) and 10(c) (at *z*=12). However, the M-shape profile with extreme flow suppression in the fluid bulk region observed in the outlet region, as shown in Figs. 6(c) and (d), is not seen in the inlet region, as shown in Figs. 10(b) and (c). The reason may be that the non-MHD fully-develloped flow with the parabolic plofile enters the inlet region though the

It is considered that, in addition to the pressure recovery due to the induced current in the positive x-direction, the velocity decrease in the fluid central region results in the pressure increase of Δ*p* ≈ 0.4 in 9.5<*z*<11 of the magnetic-field inlet-region. On the other hand, it can be considered that after the pressure recovery of Δ*p* ≈ 0.4 due to the induced current in the positive x-direction, the pressure decreases by -Δ*p* ≈ 0.3 due to the velocity increase in 11.5<*z*<12 of the magnetic-field outlet-region. From these differences, the pressure drop

Fig. 7. Inlet and outlet regions of magnetic field.

Fig. 8. Pressures along z-axis for magnetic-field inlet-region.

218 Trends in Electromagnetism – From Fundamentals to Applications

Fig. 7. Inlet and outlet regions of magnetic field.

0.0

Fig. 8. Pressures along z-axis for magnetic-field inlet-region.

0 5 10 15 20 25 30

z

p (10/20) p (10/15) p (10/12) p (10/11) p (10/10.5) p (10/10.2) p (10/10.1) p (10/10.05)

z1/z2

0.5

1.0

1.5

2.0

2.5

3.0

3.5

Fig. 9. Induced currents for magnetic-field inlet-region.

inlet region, Fig. 9, and that in the outlet region, Fig. 4, are nearly symmetric. For this reason, the sharp pressure drop in the inlet region from *z* ≈ 11 to *z* ≈ 13, i.e. -Δ*p* ≈ 0.9, agrees nearly with that in the outlet region from *z* ≈ 9 to *z* ≈ 11.5, i.e. -Δ*p* ≈ 0.9. However, the pressure recovery in the inlet region from *z* ≈ 9.5 to *z* ≈ 11, i.e. -Δ*p* ≈ 0.4 is larger than that in the outlet region from *z* ≈ 11.5 to *z* ≈ 12, i.e. -Δ*p* ≈ 0.2. The reason is examined later.

Figures 10(a), (b) and (c) show calculated velocity *vz* distributions at *z*=10, 11 and 12, respectively, for the case of *z1*/*z2*=10/12, i.e. the standard case, in the magnetic-field inletregion. The velocity profile at *z*=10, shown in Fig. 10(a), still keeps nearly a distribution typical to a non-MHD fully-developed laminar flow with a peak value of ~2. Hereafter, the velocity distribution becoms flatter along the channel axis, i.e. the *z*-axis, as shown in Figs. 10(b) (at *z*=11) and 10(c) (at *z*=12). However, the M-shape profile with extreme flow suppression in the fluid bulk region observed in the outlet region, as shown in Figs. 6(c) and (d), is not seen in the inlet region, as shown in Figs. 10(b) and (c). The reason may be that the non-MHD fully-develloped flow with the parabolic plofile enters the inlet region though the MHD fully-developed flow with the flat profile comes into the outlet region.

It is considered that, in addition to the pressure recovery due to the induced current in the positive x-direction, the velocity decrease in the fluid central region results in the pressure increase of Δ*p* ≈ 0.4 in 9.5<*z*<11 of the magnetic-field inlet-region. On the other hand, it can be considered that after the pressure recovery of Δ*p* ≈ 0.4 due to the induced current in the positive x-direction, the pressure decreases by -Δ*p* ≈ 0.3 due to the velocity increase in 11.5<*z*<12 of the magnetic-field outlet-region. From these differences, the pressure drop through the inlet region may become smaller than that through the outlet region.

Three-Dimensional Numerical Analyses on Liquid-Metal

**4. Conclusion** 

field region.

region.

**5. References** 

Magnetohydrodynamic Flow Through Circular Pipe in Magnetic-Field Outlet-Region 221

Three-dimensional numerical analyses have been performed on liquid-metal magnetohydrodynamic (MHD) flow through a circular pipe in the outlet region of magnetic

a. Along the flow axis, i.e. the circular pipe axis, the pressure decreases steeply as a fullydeveloped MHD flow, drops more sharply in the magnetic-field outlet-region, and

b. If examined in detail, in the magnetic-field outlet-region, after the pressure drops most sharply, it recovers once and thereafter it drops sharply again outside the magnetic-

c. The first sharp pressure drop and temporary pressure recovery are due to the formation of induced current loop which circulates in passing in the region downstream the magnetic-field region. The second sharp pressure drop is attributable to the change in

d. The distribution of velocity in main flow direction changes from a flat profile of a fullydeveloped MHD flow, to an M-shaped profile and finally to a parabolic profile of a

e. The total pressure drop through the magnetic-field outlet-region becomes larger than the corresponding drop through the magnetic-field inlet-region. The main reason may be that the difference in velocity profile change between the outlet region and the inlet

Aitov, T.N., Kalyutik, A.I. & Tananaev, A.V. (1983). Numerical Analysis of Three-

Asada, C. et al. Ed. (2007). *Handbook of Nuclear Engineering*, Ohmsha, Ltd., ISBN 978-4-274-

Kalis, K.E. & Tsinober, A.B. (1973). Numerical Analysis of Three Dimensional MHD Flow Problems, *Magnetohydrodynamics*, Vol. 2, pp. 175-179, ISSN 0024-998x Khan, S. and Davidson, J. N. (1979). Magneto-hydrodynamic Coolant Flows in Fusion Reactor Blankets, *Annals of Nuclear Energy*, Vol. 6, pp. 499-509, ISSN 0306-4549 Kumamaru, H. & Fujiwara, Y. (1999). Pressure Drops of Magnetohydrodynamic Flows in

Kumamaru, H., Shimoda, K. & Itoh, K. (2007). Three-Dimensional Numerical Calculations

Leboucher, L. (1999). Monotone Scheme and Boundary Conditions for Finite Volume

Lielausis, O. (1975). Liquid-Metal Magnetohydrodynamics, *Atomic Energy Review*, Vol. 13,

*Magnetohydrodynamics*, Vol. 19, pp. 223-229, ISSN 0024-998x

*on Nuclear Engineering (ICONE-7)*, Tokyo, Japan, April 1999

*of Computational Physics*, Vol. 150, pp. 181-198, ISSN 0021-9991

Dimensional MHD-Flow in Channels with Abrupt Change of Cross Section,

Rectangular Channel with Small Aspect Ratio and Circular Pipe for Very-Large Hartmann Numbers, *Proceedings of JSME/ ASME/SFEN 7th International Conference* 

on Liquid-Metal Magnetohydrodynamic Flow through Circular Pipe in Magnetic-Field Inlet-Region, *J. of Nuclear Science and Technology*, Vol. 44, No. 5, pp. 714-722,

Simulation of Magnetohydrodynamic Internal Flows at High Hartmann Number, *J.* 

field. The following conclusions have been obtained from the calculation results.

finally decreases slowly as a normal fully-developed non-MHD flow.

velosity distribution outside the magnetic-field region.

fully-developed non-MHD flow.

20443-2, Tokyo, Japan [In Japanese]

ISSN 0022-3131 & 1881-1248

pp. 527-581, ISSN 0004-7112

Fig. 10. Velocity distribution for magnetic-field inlet-region.
