**1. Introduction**

30 Will-be-set-by-IN-TECH

206 Trends in Electromagnetism – From Fundamentals to Applications

Ziadé, Y.; Wong, M. & Wiart, J. (2008). Reverberation chamber and indoor measurements

*Society International Symposium*, ISBN: 978-1-4244-2041-4, USA, 2008.

for time reversal application, *Proceedings of APS 2008, IEEE Antennas and Propagation*

In conceptual design examples of a fusion reactor power plant, a lithium-bearing blanket in which a great amount of heat is produced is cooled mainly by helium gas, water or liquidmetal lithium (Asada et al. Ed., 2007). The liquid-metal lithium is an excellent coolant having high heat capacity and thermal conductivity and also can breed tritium that is used as fuel of a deuterium-tritium (D-T) fusion reactor. In cooling the blanket, however, the liquidmetal lithium needs to pass through a strong magnetic field that is used to magnetically confine high-temperature reacting plasma in a fusion reactor core. There exists a large magnetohydrodynamic (MHD) pressure drop arising from the interaction between the liquid-metal flow and the magnetic field. In particular, the MHD pressure drop becomes considerably larger in the inlet region or outlet region of the magnetic field than in the fullydeveloped region inside the magnetic field for the reason mentioned later in this chapter.

A three-dimensional calculation is indispensable for the exact calculation of MHD channel flow in the inlet region or outlet region of magnetic field, also as described later in this chapter. There exist a few three-dimensional numerical calculations on the MHD flows in rectangular channels with a rectangular obstacle (Kalis and Tsinober, 1973), with abrupt widening (Itov et al., 1983), or with turbulence promoter such as conducting strips (Leboucher, 1999). All these calculations, however, were carried out for low Hartmann numbers (corresponding to low strength of the applied magnetic field) and low Reynolds number, because of instability problems in numerical calculations.

As to the MHD channel flow in the magnetic-field inlet-region, three-dimensional numerical calculations were conducted for the cases of Hartmann number of ~10 and Reynolds number of ~100 (Khan and Davidson, 1979). The calculations were based on what is called the parabolic approximation, in which the flow and magnetic field effects are assumed to transfer only in the main flow direction. However, the calculations based on parabolic approximation cannot predict exactly the MHD flow in the magnetic-field inlet-region. Were performed full three-dimensional calculations (without any assumptions) on the MHD rectangular-channel flow in the magnetic-field inlet-region (Sterl, 1990). The calculations

Three-Dimensional Numerical Analyses on Liquid-Metal

Fig. 1. Coordinate system for magnetic-field outlet-region.

outlet region (Moreau, 1990).

On the other hand, the induced current in the last section of the outlet region near *z*=*z2* can flow in the positive *x* direction as shown in Fig. 1(b). Thus, a smaller Lorentz force may act in the flow direction and thus a small pressure recovery may occur in this section of the

The induced electric currents in the outlet region, flowing in both *x*- and *z*-directions and in *y*-direction, cannot be calculated by a two-dimensional model. It is also important that a sufficiently large fluid region downstream the magnetic field section is included in a calculation domain. For these reasons, in this study, in order to obtaine mainly the pressure drop quantitatively, the authors have performed three-dimensional numerical calculations on the MHD flow through a circular pipe in the outlet region of the magnetic field, including the region of no magnetic field downstream the magnetic field region. To the authors' knowledge, there have been no numerical calculations or experimental studies on

developed region (Moreau, 1990).

Magnetohydrodynamic Flow Through Circular Pipe in Magnetic-Field Outlet-Region 209

pressure drop also may become considerably larger in the outlet region than in the fully-

were conducted mainly for the ranges of Hartmann numbers from 50 to 70 and Reynolds numbers from 2.5 to 5, and for a smoothly-increasing applied magnetic field. However, these ranges of Reynolds numbers and Hartmann numbers are unrealistic as conditions that appear even in laboratory conditions. The laboratory conditions reach Reynolds numbers up to ~1000 and Hartmann numbers up to ~100 simultaneously.

In fusion reactor conditions, the Reynolds number and the Hartmann number reach ~104 and ~104, respectively, the channel walls are electrically-conducting, the magnetic field changes in steps at the inlet or the outlet, and the flow changes from non-MHD turbulent flow to MHD laminar flow. However, because of instability problems in numerical calculations, it is quite difficult to obtain three-dimensional numerical solutions on MHD flows in the magnetic-field inlet-region or outlet-region even in the laboratory conditions that reach Reynolds numbers up to ~1000 and Hartmann numbers up to ~100 simultaneously.

Within the present limit of computer performance, the authors have already performed full three-dimensional calculations on the MHD flow through a circular pipe in the magneticfield inlet-region, in simulating typical laboratory conditions (Kumamaru et al., 2007). In the calculations, the Hartmann number and the Reynolds number are ~100 and ~1000, respectively, the channel walls are electrically-insulating, the applied magnetic field changes in steps, and a laminar non-MHD flow enters the calculation domain. In this study, full three-dimensional calculations are performed on the MHD flow through a circular pipe in the magnetic-field outlet-region for the same conditions as for the magnetic-field inletregion.

Figure 1 shows schematically the coordinate system, the applied magnetic field and the induced electric currents, together with the directions of Lorentz force, in the outlet region of the magnetic field. The applied magnetic field is imposed in the *y* direction, having a constant value for *z*=0~*z1*, a linear decrease from *z*=*z1*~*z2*, and a value of zero for *z*=*z2*~*z0*, as shown in Fig. 1(a).

In the region of fully-developed MHD flow near *z*=*0*, the induced electric current which is produced by the vector product of flow velocity and applied magnetic field flows in the negative *x* direction as shown in Figs. 1(b) and 1(c1). The induced current returns by passing through regions very near the walls (in an *x-y* plane at the same *z*) where the flow velocity is nearly zero, in the case of insulating walls. (The induced current can also pass through the walls in the case of conducting walls.) The induced current loop has a relatively large electrical resistance, since the current needs to flow in the thin regions near the walls. The Lorentz force which is caused by the vector product of induced current and applied magnetic field acts in the negative *z* direction and produces a large pressure drop.

In the outlet region of magnetic field from *z* ≈ *z1* to *z* ≈ *z2*, the induced electric current flows in the negative *x* direction, as was the case of fully-developed region, as shown in Fig. 1(b).

However, the induced current can pass through the large region downstream the magnetic field section (in an *x-z* plane with the same *y*) where no magnetic field or small magnetic field is applied. The electric resistance in this region is much smaller than the resistance in the thin region near the walls mentioned above. Hence, the induced current becomes larger in the outlet region than in the fully-developed region. The Lorentz force and thus the 208 Trends in Electromagnetism – From Fundamentals to Applications

were conducted mainly for the ranges of Hartmann numbers from 50 to 70 and Reynolds numbers from 2.5 to 5, and for a smoothly-increasing applied magnetic field. However, these ranges of Reynolds numbers and Hartmann numbers are unrealistic as conditions that appear even in laboratory conditions. The laboratory conditions reach Reynolds numbers up

In fusion reactor conditions, the Reynolds number and the Hartmann number reach ~104 and ~104, respectively, the channel walls are electrically-conducting, the magnetic field changes in steps at the inlet or the outlet, and the flow changes from non-MHD turbulent flow to MHD laminar flow. However, because of instability problems in numerical calculations, it is quite difficult to obtain three-dimensional numerical solutions on MHD flows in the magnetic-field inlet-region or outlet-region even in the laboratory conditions that reach Reynolds numbers up to ~1000 and Hartmann numbers up to ~100

Within the present limit of computer performance, the authors have already performed full three-dimensional calculations on the MHD flow through a circular pipe in the magneticfield inlet-region, in simulating typical laboratory conditions (Kumamaru et al., 2007). In the calculations, the Hartmann number and the Reynolds number are ~100 and ~1000, respectively, the channel walls are electrically-insulating, the applied magnetic field changes in steps, and a laminar non-MHD flow enters the calculation domain. In this study, full three-dimensional calculations are performed on the MHD flow through a circular pipe in the magnetic-field outlet-region for the same conditions as for the magnetic-field inlet-

Figure 1 shows schematically the coordinate system, the applied magnetic field and the induced electric currents, together with the directions of Lorentz force, in the outlet region of the magnetic field. The applied magnetic field is imposed in the *y* direction, having a constant value for *z*=0~*z1*, a linear decrease from *z*=*z1*~*z2*, and a value of zero for *z*=*z2*~*z0*,

In the region of fully-developed MHD flow near *z*=*0*, the induced electric current which is produced by the vector product of flow velocity and applied magnetic field flows in the negative *x* direction as shown in Figs. 1(b) and 1(c1). The induced current returns by passing through regions very near the walls (in an *x-y* plane at the same *z*) where the flow velocity is nearly zero, in the case of insulating walls. (The induced current can also pass through the walls in the case of conducting walls.) The induced current loop has a relatively large electrical resistance, since the current needs to flow in the thin regions near the walls. The Lorentz force which is caused by the vector product of induced current and applied

In the outlet region of magnetic field from *z* ≈ *z1* to *z* ≈ *z2*, the induced electric current flows in the negative *x* direction, as was the case of fully-developed region, as shown in Fig. 1(b). However, the induced current can pass through the large region downstream the magnetic field section (in an *x-z* plane with the same *y*) where no magnetic field or small magnetic field is applied. The electric resistance in this region is much smaller than the resistance in the thin region near the walls mentioned above. Hence, the induced current becomes larger in the outlet region than in the fully-developed region. The Lorentz force and thus the

magnetic field acts in the negative *z* direction and produces a large pressure drop.

to ~1000 and Hartmann numbers up to ~100 simultaneously.

simultaneously.

region.

as shown in Fig. 1(a).

pressure drop also may become considerably larger in the outlet region than in the fullydeveloped region (Moreau, 1990).

Fig. 1. Coordinate system for magnetic-field outlet-region.

On the other hand, the induced current in the last section of the outlet region near *z*=*z2* can flow in the positive *x* direction as shown in Fig. 1(b). Thus, a smaller Lorentz force may act in the flow direction and thus a small pressure recovery may occur in this section of the outlet region (Moreau, 1990).

The induced electric currents in the outlet region, flowing in both *x*- and *z*-directions and in *y*-direction, cannot be calculated by a two-dimensional model. It is also important that a sufficiently large fluid region downstream the magnetic field section is included in a calculation domain. For these reasons, in this study, in order to obtaine mainly the pressure drop quantitatively, the authors have performed three-dimensional numerical calculations on the MHD flow through a circular pipe in the outlet region of the magnetic field, including the region of no magnetic field downstream the magnetic field region. To the authors' knowledge, there have been no numerical calculations or experimental studies on

Three-Dimensional Numerical Analyses on Liquid-Metal

kinematic viscosity and

*v*

only *y*-component as a given function of *z*, i.e. *Ha*(*z*).

θ=0 and

θ

 π ξ, η, ζ

θ

conditions on the induced magnetic fields at the symmetry plane of

 θ

 θπ

small Hartmann numbers, it has been confirmed that the conditions are given by:

θ π

> θ

=

curvilinear coordinate system (

nondimensional coordinates.)

θ

condition is adopted at

π

 θ

Here, *r*,

direction,

respectively:

system (*r*,

θ

ν

ν

Magnetohydrodynamic Flow Through Circular Pipe in Magnetic-Field Outlet-Region 211

== = , , *v a v a z z Re Ha B a Rm <sup>o</sup>*

ν*m* (=1/

( ) ( ) 1 1 <sup>2</sup> *<sup>p</sup> <sup>t</sup> Re Re* <sup>∂</sup> + ⋅∇ = −∇ + ∇ + ∇ × × <sup>∂</sup>

> *t Rm Rm* <sup>∂</sup> = ∇× × + ∇ <sup>∂</sup>

Superscript \* is omitted to simplify the description in Eqs. (6) through (8) and in the following description. Note that the Hartmann number *Ha* is a given (known) vector having

The coordinate system is transformed from the Cartesian coordinate system (*x*, *y*, *z*) to the

cross-section in the future, and is thereafter transformed into the cylindrical coordinate

(Note that the inner surface of the wall corresponds to *r*=1 (*x*=1 or *y*=1) in the

As the boundary condition on the flow velocities, the inflow boundary condition is adopted at the flow inlet, i.e. at *z*=0, by fixing a fully-developed MHD flow velocity (Kumamaru & Fujiwara, 1999). The outflow boundary condition is given at the flow outlet, i.e. at *z*=*z0*, by fixing the reference pressure. No-slip condition is given at the wall and the symmetry

fields, ∂ ∂= *B*/ 0 *z* and *B*=0 are specified at the flow inlet and the flow outlet, respectively. The former reflects the situation that the induced current does not change in the *z-*direction at the flow inlet in a fully-developed MHD flow region, and the latter represents that no induced current exists at the flow outlet in a fully-developed non-MHD flow region. At the wall, *B*=0 is specified assuming that the walls are electrically insulating (nonconducting). The boundary

intuitively clear. Hence, by performing a calculation for the whole cross section in the case of

( ) ( )( ) ( ) ( ) ( )

− =− + − = + − = + /2: *A AB BC C* / 2 /2 , /2 /2 , /2 /2 , (9b)

θθθθ

 π

= − =− − = − =− 0: ( ) ( ), ( ) ( ), ( ) ( ) *A AB BC C* , (9a)

 θπ

symmetry, the numerical calculations are carried out for the region of 0<*r*<1 and 0<

, *z*) as a special case of the curvilinear coordinate system. Considering the

( ) 1 1 <sup>2</sup>

Nondimensional numbers *Re*, *Ha* and *Rm* are Reynolds number, Hartmann number and magnetic Reynolds number, respectively. The final nondimensional basic equations become

σ

η

, *z* are coordinates in the cylindrical coordinate systen; *<sup>z</sup> v* mean velocity in *z*-

σμ

 ν

*v v v B Ha* , (7)

*<sup>B</sup> v Ha B* . (8)

) in order to deal with a channel with an arbitrary flow

=π/2. As the boundary condition on the induced magnetic

θ=0 and

 θ

> θ

θ

 π  θ

=π/2 are not

*m*

∇⋅ = *v* 0 , (6)

. (5)

θ<π/2.

) magnetic kinematic viscosity.

the MHD flow through a circular pipe in the magnetic-field outlet-region. In this study, calculation results on the magnetic-field outlet-region have also been compared with authors' calculation results on the magnetic-field inlet-region.

## **2. Numerical analyses**

Numerical calculations are performed for an MHD flow in a circular pipe with an inner radius of *a*, shown in Fig. 1. A fully-developed MHD laminar flow enters the calculation domain at *z*=0, and a fully-developed non-MHD flow leaves the domain at *z*=*z0*. The applied magnetic field is imposed in the *y* direction as shown in Fig. 1(a), as was stated previously.

The basic equations which describe a liquid-metal MHD flow are the continuity equation, the momentum equation and the induction equation. The equations are expressed respectively by:

$$\nabla \cdot \mathbf{v} = 0 \,, \tag{1}$$

$$
\rho \left[ \frac{\partial \mathbf{v}}{\partial t} + (\mathbf{v} \cdot \nabla) \mathbf{v} \right] = -\nabla p + \eta \nabla^2 \mathbf{v} + \frac{1}{\mu} (\nabla \times \mathbf{B}) \times \mathbf{B}\_{\mathbf{O}} \,. \tag{2}
$$

$$\frac{\partial \mathbf{B}}{\partial t} = \nabla \times \left(\mathbf{v} \times \mathbf{B}\_{\mathbf{o}}\right) + \frac{1}{\sigma \mu} \nabla^{2} \mathbf{B} \,. \tag{3}$$

Here, *v* is velocity vector, *p* pressure, *B* induced magnetic field vector and *t* time; *B0* is applied magnetic field vector, and ρ is density, η viscosity, μ magnetic permeability and σ electric conductivity. The vector *B* is an induced magnetic field produced by the induced electric current, and is treated as an unknown variable together with the velocity *v* and the pressure *p*. The induced electric current *j* can be calculated by the Ampere equation *j* = ∇× ( ) 1 μ ( ) *B* from *B*. The third term in the right-hand side of Eq. (2) represents the Lorentz force. The induction equation, i.e. Eq. (3), is derived from Maxwell's equations and Ohm's law in electromagnetism.

The basic equations are expressed in nondimensional forms by introducing the following nondimensional variables (indicated by superscript \*) and nondimentional numbers:

$$\begin{array}{ll} \mathbf{t}^\* = \frac{\mathbf{t}}{a \sqrt{\overline{\upsilon}\_Z}}, & \mathbf{r}^\* = \frac{\mathbf{r}}{a}, & \mathbf{z}^\* = \frac{\mathbf{z}}{a}, \\\\ \boldsymbol{\upsilon}\_r^\* = \frac{\boldsymbol{\upsilon}\_r}{\overline{\upsilon}\_Z}, & \boldsymbol{\upsilon}\_\theta^\* = \frac{\overline{\upsilon}\_\theta}{\overline{\upsilon}\_Z}, & \boldsymbol{\upsilon}\_Z^\* = \frac{\overline{\upsilon}\_Z}{\overline{\upsilon}\_Z}, & \boldsymbol{p}^\* = \frac{\boldsymbol{p}}{\rho \overline{\upsilon}\_Z^2}, \\\\ \boldsymbol{B}\_r^\* = \frac{\overline{\boldsymbol{B}}\_r}{\overline{\upsilon}\_Z \mu \sqrt{\sigma \eta \eta}}, & \boldsymbol{B}\_\theta^\* = \frac{\overline{\boldsymbol{B}}\_\theta}{\overline{\upsilon}\_Z \mu \sqrt{\sigma \eta \eta}}, & \boldsymbol{B}\_z^\* = \frac{\overline{\boldsymbol{B}}\_z}{\overline{\upsilon}\_Z \mu \sqrt{\sigma \eta \eta}}. \end{array}$$

210 Trends in Electromagnetism – From Fundamentals to Applications

the MHD flow through a circular pipe in the magnetic-field outlet-region. In this study, calculation results on the magnetic-field outlet-region have also been compared with

Numerical calculations are performed for an MHD flow in a circular pipe with an inner radius of *a*, shown in Fig. 1. A fully-developed MHD laminar flow enters the calculation domain at *z*=0, and a fully-developed non-MHD flow leaves the domain at *z*=*z0*. The applied magnetic field is imposed in the *y* direction as shown in Fig. 1(a), as was stated

The basic equations which describe a liquid-metal MHD flow are the continuity equation, the momentum equation and the induction equation. The equations are expressed

( ) ( ) <sup>2</sup> <sup>1</sup>

<sup>∂</sup> + ⋅∇ = −∇ + ∇ + ∇ × ×

<sup>∂</sup> =∇× × + ∇ <sup>∂</sup>

is density,

Here, *v* is velocity vector, *p* pressure, *B* induced magnetic field vector and *t* time; *B0* is

electric conductivity. The vector *B* is an induced magnetic field produced by the induced electric current, and is treated as an unknown variable together with the velocity *v* and the pressure *p*. The induced electric current *j* can be calculated by the Ampere equation

The basic equations are expressed in nondimensional forms by introducing the following

= == \* \*\* , , / *t rz t rz av a a <sup>z</sup>*

 = = == \*\* \*\* , , , 2 *vvv <sup>p</sup> r z vv vp r z vvv zzz vz*

θ

μ ση

=== \*\*\* , , *BBB r z BBB r z vvv zzz*

nondimensional variables (indicated by superscript \*) and nondimentional numbers:

θ

θ

θ

μ ση

 η

( ) <sup>1</sup> <sup>2</sup>

η

 ( ) *B* from *B*. The third term in the right-hand side of Eq. (2) represents the Lorentz force. The induction equation, i.e. Eq. (3), is derived from Maxwell's equations and

σμ

viscosity,

μ

*<sup>v</sup> v v v BBo* , (2)

*<sup>B</sup> vB B <sup>o</sup>* . (3)

μ

,

ρ

μ ση

,

*<sup>p</sup> <sup>t</sup>*

*t*

ρ

 

ρ

applied magnetic field vector, and

Ohm's law in electromagnetism.

∂

∇⋅ = *v* 0 , (1)

magnetic permeability and

, (4)

σ

authors' calculation results on the magnetic-field inlet-region.

**2. Numerical analyses** 

previously.

respectively by:

*j* = ∇× ( ) 1 μ

$$RRe = \frac{\overline{\upsilon}\_{z}a}{\nu}, \quad Ha = B\_{0}a\sqrt{\frac{\sigma}{\eta}}, \quad Rm = \frac{\overline{\upsilon}\_{z}a}{\nu\_{m}}.\tag{5}$$

Here, *r*, θ, *z* are coordinates in the cylindrical coordinate systen; *<sup>z</sup> v* mean velocity in *z*direction, ν kinematic viscosity and ν*m* (=1/σμ) magnetic kinematic viscosity. Nondimensional numbers *Re*, *Ha* and *Rm* are Reynolds number, Hartmann number and magnetic Reynolds number, respectively. The final nondimensional basic equations become respectively:

$$\nabla \cdot \mathbf{v} = 0 \,, \tag{6}$$

$$\frac{\partial \mathbf{v}}{\partial t} + (\mathbf{v} \cdot \nabla)\mathbf{v} = -\nabla p + \frac{1}{Re}\nabla^2 \mathbf{v} + \frac{1}{Re}(\nabla \times \mathbf{B}) \times \mathbf{H}\mathbf{a},\tag{7}$$

$$\frac{\partial \mathbf{B}}{\partial t} = \frac{1}{Rm} \nabla \times (\mathbf{v} \times \mathbf{H} \mathbf{a}) + \frac{1}{Rm} \nabla^2 \mathbf{B} \,. \tag{8}$$

Superscript \* is omitted to simplify the description in Eqs. (6) through (8) and in the following description. Note that the Hartmann number *Ha* is a given (known) vector having only *y*-component as a given function of *z*, i.e. *Ha*(*z*).

The coordinate system is transformed from the Cartesian coordinate system (*x*, *y*, *z*) to the curvilinear coordinate system (ξ, η, ζ) in order to deal with a channel with an arbitrary flow cross-section in the future, and is thereafter transformed into the cylindrical coordinate system (*r*, θ, *z*) as a special case of the curvilinear coordinate system. Considering the symmetry, the numerical calculations are carried out for the region of 0<*r*<1 and 0<θ<π/2. (Note that the inner surface of the wall corresponds to *r*=1 (*x*=1 or *y*=1) in the nondimensional coordinates.)

As the boundary condition on the flow velocities, the inflow boundary condition is adopted at the flow inlet, i.e. at *z*=0, by fixing a fully-developed MHD flow velocity (Kumamaru & Fujiwara, 1999). The outflow boundary condition is given at the flow outlet, i.e. at *z*=*z0*, by fixing the reference pressure. No-slip condition is given at the wall and the symmetry condition is adopted at θ=0 and θ=π/2. As the boundary condition on the induced magnetic fields, ∂ ∂= *B*/ 0 *z* and *B*=0 are specified at the flow inlet and the flow outlet, respectively. The former reflects the situation that the induced current does not change in the *z-*direction at the flow inlet in a fully-developed MHD flow region, and the latter represents that no induced current exists at the flow outlet in a fully-developed non-MHD flow region. At the wall, *B*=0 is specified assuming that the walls are electrically insulating (nonconducting). The boundary conditions on the induced magnetic fields at the symmetry plane of θ=0 and θ=π/2 are not intuitively clear. Hence, by performing a calculation for the whole cross section in the case of small Hartmann numbers, it has been confirmed that the conditions are given by:

$$\mathcal{A}\theta = 0 \text{: } A(-\theta) = -A(\theta), \\ B(-\theta) = B(\theta), \\ C(-\theta) = -C(\theta) \text{,} \tag{9a}$$

$$\theta = \pi \,/\, 2 \,\, : \, \begin{aligned} \theta &= \pi \,/\, 2 \,\, : \\ \theta &= \text{3-2} \end{aligned}$$

$$\text{A} \left(\pi \,/\, 2 - \theta \right) = -\text{A} \left(\pi \,/\, 2 + \theta \right), \text{B} \left(\pi \,/\, 2 - \theta \right) = \text{B} \left(\pi \,/\, 2 + \theta \right), \text{C} \left(\pi \,/\, 2 - \theta \right) = \text{C} \left(\pi \,/\, 2 + \theta \right)' \quad \text{(9b)}$$

Three-Dimensional Numerical Analyses on Liquid-Metal

0.0

Fig. 2. Pressures along *z*-axis for *z1*/*z2*=10/20 to 10/10.05.

0.0

Fig. 3. Pressures along *z*-axis for *z1*/*z2*=10/12.

explained in Sec. 3.4.

0.5

1.0

1.5

2.0

2.5

3.0

3.5

0.5

1.0

1.5

2.0

2.5

3.0

3.5

Magnetohydrodynamic Flow Through Circular Pipe in Magnetic-Field Outlet-Region 213

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 10 15 20 25 30

z

Outlet-Region Inlet-Region

p (10/12) p (10/12)

z1/z2

0 5 10 15 20 25

z

The small pressure recovery, which was also pointed out in Chap. 1 and again will be expained in Sec. 3.2, is observed in the region near *z* ≈ *z2* for the cases of *z1*/*z2*=10/12~10/10.05. The pressure drop appears again outside the magnetic-field region. This may be due to rapid change in velocity distibution in this region, which will be

The pressure drops in the fully-developed region of *z*<*z1*, -Δ*p*/Δ*z*, are almost the same for all the cases. The pressure drops agree with a value calculated numerically by the authors for the fully-developed MHD flow, -Δ*p*/Δ*z* ≈ 0.123 (Kumamaru and Fujiwara, 1999), and also agree nearly with a value predicted by Schercliff's theoretical approximate equation, - Δ*p*/Δ*z* ≈ 0.118 (Schercliff, 1956; Lielausis, 1975), for the case of *Ha*=100 and *Re*=1000. As

where *A*, *B* and *C* are the *x*, *y* and *z* components of *B*, respectively.

The discretization of the equations is carried out by the finite difference method. The calculations are performed using a non-uniform expanding 15 x 15 x 30 grid with grid elements closely spaced near the channel wall of *r*=1 and the region between or around *z*=*z1* and *z*=*z2*. The first-order accurate upwind differencing is adopted for the fluid convection terms in Eq. (7). The solution procedure follows the MAC method that is widely used in numerical calculations.

Even for the fully-developed region, it is difficult to obtain a stable numerical solution for large Hartmann numbers (Kumamaru & Fujiwara, 1999). In the present three-dimensional calculations, stable numerical solutions have been obtained for Hartmann numbers up to 100 and Reynolds numbers up to 1000 by applying the following means or procedures. (1) The grids are arranged closely near the wall of *r*=1, i.e. at *r*=0.0, … , 0.95, 0.97, 0.99, 0.995, 1.0, on referring to a velocity profile of the classical Hartmann flow, i.e. fully-developed MHD flow in infinite parallel plates (Kumamaru and Fujiwara, 1999). (2) Simultaneous linear equations on the pressure, i.e. Poisson equation, are solved not by the iterative method but by the elimination method. (3) First, a solution is obtained for *Re* (Reynolds number) of 0.01, and thereafter *Re* is increased gradually to a final value, i.e. 1000.
