Numerical simulation of MHD flows in a coupled U-turn rectangular duct with different wall conductance ratios

Fusion Engineering and Design 149 (2019) 111334

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Numerical simulation of MHD flows in a coupled U-turn rectangular duct with different wall conductance ratios

T

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Yanli Wang1, Jie Mao , Hao Wang, Mingliang Jin School of Mechanical Engineering, Hangzhou Dianzi University, Hangzhou 310018, China

A R T I C LE I N FO

A B S T R A C T

Keywords: Magnetohydrodynamic Coupled U-turn rectangular duct Wall conductance ratio Pressure drop

Liquid metal flow in a conducting coupled U-turn rectangular duct subjected to an external uniform magnetic field is numerically analyzed using our in-house MHD solver developed in OpenFOAM software. The coupled Uturn duct is the structure used in self-cooled liquid-metal blankets, which includes the duct wall, partition wall, inflow channel, outflow channel, and connection channel. The characteristics of the fluid velocity, induced electric current, electric potential, and pressure distribution depending on the wall conductance ratio of the duct wall and the partition wall are investigated. As the partition wall divides the inflow and outflow, the threedimensional induced electric currents close loops at the end of the partition wall in the connection channel. When the duct wall is insulating and the partition wall is weak conducting, the three-dimensional electric current is significant in the connection channel and the start of the outflow channel. The three-dimensional electric current results in the pressure fluctuation. The flow forms a visible primary vortex at the end of the partition wall because of a strong adverse pressure gradient. The pressure along the flow direction has a significant fluctuation near the connection channel. When the duct wall is weak conducting, high-velocity jets form in the side layers near the partition wall and the duct wall. The velocity distribution is a strong asymmetrical Mtype profile in the U-turn rectangular duct with conducting duct walls. The normalized electric potential distribution in the different walls along the magnetic field direction is independent of the Reynolds number when the Reynolds number and the Hartmann number is moderate. When the Hartmann number and Reynolds number is high, the normalized electric potential varies with the Reynolds number. The conductance ratio of the duct wall has a decisive effect on the pressure gradient. The pressure drop with conducting duct wall is two orders of magnitude higher than that of cases with an insulating duct wall. As the duct wall is insulating, the wall conductance ratio of the partition wall does not affect the pressure gradient remarkably. While the duct wall is conducting, the increase of the wall conductance ratio of the partition wall results in an increase of the pressure drop significantly.

1. Introduction The study of magnetohydrodynamic (MHD) flow is significant in thermonuclear reactors, metallurgy, medicine, and other fields [1–4]. The liquid breeder blanket is a critical component in the thermonuclear reactor, which plays a vital role in energy transmission, radiation, and tritium breeding. The moving liquid metal interacts with a strong magnetic field and induces an electric current. The induced electric current interacting with the vertical magnetic field generates a Lorentz force which redistributes the flow and transforms the flow from turbulent to laminar state at a certain condition. Furthermore, it results in a tremendous MHD pressure drop that is much higher than that in the

hydraulic flow [5]. The wall conductance ratio of the rectangular duct has a decisive effect on the distribution of the velocity and the MHD pressure drop [6]. In practical engineering applications, complex geometry structures are necessary, such as a sudden expansion, single- or multi-channel bends, and a coupled U-turn rectangular duct [7]. Dhinakaran et al. [8] and Kim [9] carried out a numerical simulation study of the sudden expansion duct with different expansion structures. After the sudden expansion, the flow state changes. Yan et al. [10] investigated the flow state of the fluid in a right-angled curved duct under different Hartmann numbers and different wall conductance ratios using CFX. He found that the velocity recirculation region decreases and the pressure

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Corresponding author. E-mail address: [email protected] (J. Mao). 1 Present address: College of Astronautics, Nanjing University of Aeronautics and Astronautics, Nanjing, 210016, China. https://doi.org/10.1016/j.fusengdes.2019.111334 Received 23 April 2019; Received in revised form 13 August 2019; Accepted 20 September 2019 0920-3796/ © 2019 Elsevier B.V. All rights reserved.

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Fig. 1. The geometric model of the coupled U-turn rectangular duct.

metal flow in a coupled U-turn rectangular duct has been simulated using a three-dimensional model with a fully developed assumption at the outlet of the duct. The effects of the wall conductance ratio of the duct wall and the partition wall on the liquid metal flow are analyzed in detail.

gradient increases with the wall conductance ratio increasing. U-turn duct is one kind of self-cooling liquid metal blanks structure [11], in which the flow state is changed by the influence of the geometric model. Molokov et al. [12] analyzed MHD flow in a U-bend duct regarding the toroidal concepts of self-cooled liquid-metal blankets. The results show that the flow characteristic of the metal fluid in the duct is very sensitive to the wall conductance ratio and the aspect ratio of the toroidal duct cross-section. In the connection channel, the high-velocity jets in the side layer of the inlet section are mixed, and the redistribution of the flow in the duct must be considered in the study of heat transfer characteristics. He et al. [13] and Yang et al. [14] investigated the U-shaped ducts under a uniform magnetic field with different wall conductance ratios by optimizing the structure of the geometric model using the OpenFOAM and CFX respectively. The inflow duct and outflow duct are separated and connected by a connecting rectangular duct. Therefore, the wall conductance ratios of different parts of the solid walls are uniform and change synchronously. The results show that the velocity recirculates in the right-angle segment and the pressure gradient increases with the wall conductance ratio increasing. Xiao et al. [15] studied the conducting U-turn duct with two Hartmann numbers (Ha = 1000 and 300) and a constant wall conductance ratio by using the CFX. There is a distinct vortex at the end of the partition wall. The characteristics of MHD flow in multiple channels under a magnetic field with different parameters have been studied by Luo et al. [16]. The wall conductance ratios of the solid wall also remained the same. The pressure gradient in the first outflow channel is the largest, and the flow state of the metal fluid in the duct is quite complex. Zhang et al. [17] studied four kinds of coupling MHD rectangular duct flows with different magnetic field directions, same or opposite velocity direction by using a two-dimensional fully developed model. The Hartmann number and the wall conductance ratio are fixed at Ha = 1000 and Cw = 0.01. When the velocity is opposite in the duct and the magnetic field is perpendicular to both the flow and the coupling wall, the MHD pressure drop is much higher than that of the other three cases. Valls et al. [18] used the OpenFOAM to analyze the volumetric heating and magnetic field in the U-bend of electrically insulating or perfectly conducting walls. In conclusion, most of the known studies of the MHD flow in a Ubend duct has independent inflow and outflow channel. The coupled Uturn MHD duct flow does not discuss the effect of the different wall conductance ratios on the MHD pressure drop and velocity distribution. In practical engineering applications, a wholly insulating or completely conductive condition is challenging to achieve. In this paper, liquid

2. Governing equations Considering a liquid metal flow subjected to a strong, uniform external magnetic field with low magnetic Reynolds number, the governing equations are:

∇⋅u = 0,

(1)

p ∂u 1 + u⋅∇u = −∇ + υ∇2 u + (J × B0), ∂t ρ ρ

(2)

J=σf (−∇ϕ + u × B0),

(3)

∇2 ϕ = ∇⋅(u × B0),

(4)

The induced electrical potential equation in solid walls is:

∇2 ϕw = 0.

(5)

Where u , J, B0, t, υ, ρ , p, and ϕ are the velocity, induced electric current, the applied magnetic field, time, kinematic viscosity, density of the liquid metal, pressure, and electric potential, respectively. Several dimensionless parameters are used in this paper. The Reynolds number Re = Um a/ υ represents the ratio of inertial to viscous forces. The interaction parameter N = σf aB02 ρUm provides a measure for the ratio of the Lorentz forces to the inertial force. The Hartmann number Ha = B0 a σf ρυ = Re⋅N , its square represents the ratio of the Lorentz forces to the viscous forces. Here Um is the average flow velocity, a is the characteristic length, B0 is the external magnetic field intensity, and σf is the electrical conductivity of the liquid metal. Another crucial dimensionless parameter is the wall conductance ratio C = σw tw / σf a, where tw and σw are the thickness and the electric conductivity of the solid walls, respectively. 3. Numerical model 3.1. Geometric model We investigated the liquid metal flows in a coupled U-turn rectangular duct using numerical simulation in this paper. The geometric 2

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σf = 3.3074 × 106 (Ωm)−1. Cases with different wall conductance ratios are listed in Table 1. The no-slip boundary condition is applied at the liquid-solid wall interface. The average velocity is provided at the inlet, and the velocity is set as a zero gradient at the outlet. The pressure at the inlet and outlet is set as a zero gradient and a fixed value of zero, respectively. The electric potential is ∂ϕ / ∂n=0 at the inlet, outlet, and outside surfaces of the duct wall. Here n is the normal direction of the wall. The electric potential and the induced electric current are coupled at the interface of the fluid and the solid. At the liquid-wall interface, ϕf = ϕw , σf (∂ϕf / ∂n)=σw (∂ϕw / ∂n) . The subscripts f and w denote the fluid region and the solid wall, respectively. The governing equations are solved using our in-house MHD solver developed in the open-source CFD environment of OpenFOAM. The solver has been validated and verified [19]. Consistent and conservative schemes on a rectangular collocated grid developed by Ni et al. [20] are used to solve the equations for the electric current, the electric potential, and the Lorentz force. The second-order backward scheme is used to discrete the time. The Gauss linear scheme is used for convective terms and diffusion terms. The convergence tolerances for u , p, and ϕ , are set as 1e-8, 1e-7, and 1e-8, respectively. In Case 1 to Case 8, B0 = 1.264T , Um = 9.1 × 10-3m/s , The corresponding dimensionless parameters are Re=1000, Ha=2000 .

Table 1 Cases with different wall conductance ratios.

C1 C2

Case 1

Case 2

Case 3

Case 4

Case 5

Case 6

Case 7

Case 8

0 0

0 0.007

0 0.07

0 1

0.07 0

0.07 0.007

0.07 0.07

0.07 1

model is shown in Fig. 1. The dimensions of the inlet and outlet crosssection of the fluid flow are 2a × 2a, the length is 10a , and the thickness of the solid wall is tw = 0.1a. The half-length of the inlet and outlet duct cross-section parallel to the applied magnetic field is defined as the characteristic length a=0.0439m . The coupled U-turn rectangular duct consists of an outside duct wall, a partition wall, an inflow channel, an outflow channel, and a connection channel. Throughout this paper, the non-dimensional wall conductance ratio of the outside duct wall and the partition wall are defined as C1 and C2 , respectively. The external uniform magnetic field is applied in the y-direction. The average velocity is applied at the inlet. 3.2. Parameters and boundary condition In this study, the liquid metal is Ga68In20Sn12 whose physical parameters are ρ = 6363 kg/m3 , υ = 4.0 × 10−7 m2/s , and

Fig. 2. The colored contours of the velocity component u and the streamlines (y/a = 0). 3

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4. Results and discussions

CDEF is shown in Fig. 4. The values in the outflow channel are absolute values to provide a precise comparison with the velocity distribution in the inflow channel. It should be noted that there are small velocity jets (u max Um = 1.039) at the side layers both in the inflow channel and outflow channel in Case 1, whose duct wall and partition wall are both insulating. When the partition wall is weakly conducting, the jets decrease (u max Um = 1.03) in the inflow channel and almost disappear in the outflow channel. In general, there is no velocity jets and the velocity distribution is flat in MHD rectangular duct flow with insulating walls [22]. The small velocity jets forms as the flow is developing in insulating rectangular duct. The electric current on the plane at x / a = 3 is presented in Fig. 5. It shows that the electric current in the core in the inflow channel and the outflow channel is straight lines and perpendicular to the magnetic field. In the side layers, part of lines protrude to the centre y / a = 0 a little and then turns to y-direction to close the loops. The protruding lines has a short part parallel to the y-direction, which results in a decreases of the Lorentz force. Therefore, a small velocity jet forms. The velocity component u and w distribution along the AB line is shown in Fig. 6. It indicates that the velocity component u increases and then decreases to zero. The positive and negative signs mean that the fluid is flowing away from the partition wall and toward the partition wall. It is consistent with the colored velocity contours in Fig. 2. The distribution of the velocity component w that is the streamwise direction in the connection channel shows that there is a strong velocity jet near the partition wall. Since the fluid is pushed toward the partition wall in the connection channel, it results in a higher velocity component w in Case 1 (wmax Um = 4.89) than the corresponding values in the duct with weakly conducting partition wall. When C1 = 0 and C2 increases from 0.007 to 1, the velocity distribution along CDEF and AB does not vary significantly. The colored electric potential (Elpot) contours and the electric current streamline distribution in the duct are shown in Fig. 7. In all cases, the U-turn structure and the insulating duct wall triggers the three-dimensional current loops with different cores and opposite directions in the connection channel and the start of the outflow channel. The three-dimensional effects result in the current streamlines bending even at x / a = 4 (Case 2-Case 4)in the outflow channel. Moreover, the opposite electric current direction around the current core generates contrary Lorentz force. The currents both in the inflow channel and in the outflow channel flow to the partition wall on the plane y / a = 0 . They close through the side layers as the wall is insulating (Case 1) or through both the side layers and the outside conducting duct wall (Case 2-Case 4). It distributes symmetrically and typically as the distribution

In MHD rectangular duct flow, the fluid can be divided into three parts [21]: the core, the Hartmann layers, and the side layers. The Hartmann layers are perpendicular to the direction of the magnetic field and their thickness is O (Ha−1) . The side layers are parallel to the direction of the magnetic field and their thickness is O (Ha−1/2) . As the walls are electrically insulating, the induced electric current closes its loops through the Hartmann layers and the side layers. As the walls are conducting, the induced electric current forms closed loops through the Hartmann layers, the side layers, and the conducting walls. In general, there must be 4∼5 cells in the Hartmann layers and 10∼15 cells in the side layers. The grid independence test is attached at the end of the paper. We present the results according to the wall conductance ratios of the duct wall and the partition wall. In Section 4.1, the duct wall is insulating. In Section 4.2, the results with the weak conducting duct wall are presented and analyzed. 4.1. Insulating duct wall Fig. 2 presents the colored contours of the streamwise velocity and the streamlines on plane y / a = 0 . The solid wall is fully insulating in Fig. 2(a) (Case 1 C1 = C2 = 0 ), the streamline turns from the connection channel section to the outflow channel continuously. The negative sign means that the streamwise direction in the outflow channel is the reverse of the x-axis. It shows that the u-component in 3 ≤ x / a ≤ 7 is almost uniform in the inflow channel and the outflow channel respectively. The maximum values of the velocity on the plane locate at the two sides of the end part of the partition wall. This is caused by the flow turning around through a 90° angle of the partition wall. There is a quite small vortex at 7.8 ≤ x / a ≤ 8 and 0 ≤ z / a ≤ 0.2 . As the duct wall is insulating and the partition wall is conducting, the maximum velocity in the outflow channel is much lower than that in the inflow channel as shown in Fig. 2(b)–(d). The streamlines form a vortex near the end of the partition wall after they turn to the outflow channel. The maximum velocity in the inflow channel still locates at the end of the partition wall, while it locates at the outside of the vortex in the outflow channel. The high-velocity area in the outflow channel is much larger than that in the inflow channel. Fig. 3 presents the position where the results are obtained. The line CDEF is chosen because the streamlines are almost parallel both in the inflow channel and the outflow channel. The line AB is chosen to show the velocity component u and w distribution at the turning position. The normalized velocity component u/ Um distribution along the line

Fig. 3. Location schematic diagram (y/a = 0). 4

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Fig. 4. The velocity distribution along the line CDEF.

in the rectangular duct on the plane x / a = 3 in Case 1 to Case 3. However, the electric current fluctuates and bends in Case 4 affected by the strong three-dimensional current. The electric potential is the highest at the partition wall and the lowest at the outside duct wall on the plane y / a = 0 . The electric potential difference does not change significantly with C2 increase. The pressure drop is a critical problem in the application of a liquid metal blanket. As the vortex locates near the partition wall, two lines along the flow direction are defined to discuss the pressure distribution. Both lines are parallel to the duct wall. Line 1 (L1, M1-N1-P1-Q1) is 0.05a distance to the partition wall and Line 2 (L2, M2-N2-P2-Q2) locates at the center as shown in Fig. 3. The total length of the line L1 and L2 is defined as L, which is L = 0.7085 m for L1 and L = 0.878 m for L2. The pressure variation along the two lines are presented in Fig. 8. It shows that the pressure distribution is straight lines at upstream in the inflow channel (0 ≤ L/ a ≤ 5, 0 ≤ x / a ≤ 5) and downstream in the outflow channel along L1 (11.25 ≤ L/ a ≤ 16.25, 0 ≤ x / a ≤ 5) and L2 (16.25 ≤ L/ a ≤ 20 , 0 ≤ x / a ≤ 3.75). The linear distribution of the pressure along the flow direction indicates that the flow is fully developed and not influenced by the three-dimensional effects. The lines in the outflow channel almost coincide for all cases along both L1 and L2. In the inflow channel the pressure is a little bit higher of Case 1 with completing insulating walls than those of the other three cases with a conducting partition wall. The pressure distribution fluctuates near the connection channel at 5 ≤ L/ a ≤ 11.25, 5 ≤ x / a ≤ 8.05 along L1. When the solid wall is completely insulating, the maximum dimensionless pressure gradient in the connection channel is 80 times greater than that in the inflow and outflow channels along L1. The pressure fluctuates significantly in the section 5 ≤ L/ a ≤ 16.25 along L2. There are more peaks and valleys along the line L2 as C2 ≠ 0 . The dimensionless pressure gradient is defined as K = −p σf Um B02 .The total dimensionless average pressure gradient along the flow direction is K a . It is defined as Ki in the inflow channel (0 ≤ L/ a ≤ 5) , K o in the outflow channel along the line L1 (11.25 ≤ L/ a ≤ 16.5) and the line L2 (16.25 ≤ L/ a ≤ 20) . The pressure gradient in the inflow channel and outflow channel does not calculate according to the duct structure in order to avoid the effects of the pressure fluctuation near and in the connection channel. All results are shown in Table 2. The dimensionless pressure gradient difference is defined as:

Fig. 5. The electric current streamlines at x /a= 3 for Case 1.

Dj = (K j − K jR ) K jR *100%. 5

(6)

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Fig. 6. The distribution of velocity components u and w along the line AB.

significantly in the coupled U-turn duct. To study the formation mechanism of the vortex at the end of the partition wall in the outflow channel, we take Case 3 (C1 = 0, C2 = 0.07) as the benchmark model to study the effects of the Hartmann number and Reynolds number on the size of the vortex region. The cases shown in Table 3 are simulated. Fig. 9 is the colored contours of the u-component and the

Here, j = a, i,o which corresponds to the value of K in different parts, R is defined as the reference case. Here Case 1 is chosen as the reference case. The results are presented in Table 2. It can be seen from Table 2 that the maximum difference of the pressure gradient compared with the corresponding part of Case 1 is only 5.628%. In conclusion, when the outside duct wall is insulating, the conductivity of the partition wall does not affect the total pressure drop along the flow direction

Fig. 7. The colored Elpot contours and the electric current streamlines on the plane. 6

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Fig. 8. The pressure distribution at different locations along the flow direction, (a) L1, (b) L2.

minimum values of u Um within the vortex region increase significantly. The value of u Um in the area of vortex changes little as the Hartmann number continues increasing to 15000. The influence of Reynolds number on the vortex region at the end of the partition wall in the outflow channel is investigated at Ha = 5000 as shown in Fig. 9(b) and (e)-(h). The vortex region contracts firstly and then expands remarkably. More velocity vortices appear near the end of the partition wall. The maximum velocity still locates in the vortex region and moves closer to the partition wall. The distribution of the high- and low-velocity zones separates obviously with the increasing of the Reynolds number. Fig. 10 presents the pressure distribution along the line L1 with different Hartmann numbers. Dimensionless pressure gradient K and relative difference D are listed in Table 4. The dimensionless pressure gradient Ki in the inflow channel is calculated in 0 ≤ L/ a ≤ 3.75. Case 3 is chosen as the reference case used to calculate D defined in Eq. (6). At Re = 1000, the pressure fluctuation amplitude and range along the flow direction in the connection channel increases with the Hartmann number increasing. Moreover, the pressure instability diffuses into the inflow channel and the outflow channel at Ha = 15000 . The maximum dimensionless pressure gradient is 325 times greater than that of the entire duct at Ha = 15000. The sharp increase and decrease of the pressure along the streamwise will cause extra mechanical stress of the partition wall. The dimensionless pressure gradient decreases with the Hartmann number increasing. The pressure distribution along the L1 also fluctuates violently with the increase of the Reynolds number as shown in Fig. 11. The pressure varies strongly at the end of the partition wall. The pressure distribution has several local peaks and valleys at Re = 15000. The instability expands to the outlet of the outflow channel. There is a significant pressure drop at the end of the partition wall at Re = 10000, and its dimensionless pressure gradient is 182 times greater than that of the whole duct. There is a tremendous adverse pressure gradient which will produce strong mechanical stress at the end of the partition wall. This must be considered in practical engineering design. The dimensionless pressure gradient K and the relative difference D

Table 2 The dimensionless pressure gradient and the relative difference for C1 = 0. C2 L1

L2

0

0.007

0.07

1

K a ∗ 10 4

6.022

5.897

5.918

5.983

Ki ∗ 10 4

5.187

5.007

5.050

5.479

K o ∗ 10 4 Da Di Do

5.196

5.038

5.079

5.315

K a ∗ 10 4

— — — 4.833

−2.087 −3.475 −3.042 4.738

−1.730 −2.633 −2.239 4.734

−0.651 5.628 2.299 4.767

Ki ∗ 10 4

5.061

5.254

4.933

5.112

K o ∗ 10 4 Da Di Do

5.182

5.116

5.063

5.077

— — —

−1.969 −3.434 −1.292

−2.066 −2.531 −2.315

−1.369 1.010 −2.044

streamlines within the vortex region formed at the end of the partition wall in the outflow channel with different dimensionless parameters. The larger vortex is anticlockwise, while the smaller vortex (Fig. 9(a) and (b)) at the end of the partition wall is clockwise. The size of the vortex region at the end of the partition wall varies with Hartmann numbers and Reynolds numbers. The size of the vortex is defined by the streamlines where the velocity recirculates along the streamlines. To illustrate the size of each vortex region more clearly, the range of the vortex region presented in Fig. 9 is different. When Re = 1000, Ha increases from 2000 to 5000, the vortex region decreases significantly as shown in Fig. 9(a)–(b). When Ha increases furtherly from 5000 to 10000, the vortex size does not vary significantly. As Ha increases to 15000, the vortex size decreases a little compared with that at Ha = 10000. The vortex center continues to approach the partition wall in the z-axis. The smaller vortex (Fig. 9(a) and (b)) at the end of the partition wall in the outflow channel disappears (Fig. 9(c) and (d)) with the increase of the external magnetic field intensity. However, the small vortex at the end of the partition wall in the connection channel gradually increases and moves upward (Fig. 9(c) and (d)). When the Hartmann number increases to Ha = 5000, the maximum and Table 3 Cases with different dimensionless parameters based on Case 3 (C1 = 0, C2 = 0.07).

Re Ha

Case 3

Case 3-1

Case 3-2

Case 3-3

Case 3-4

Case 3-5

Case 3-6

Case 3-7

1000 2000

1000 5000

1000 10000

1000 15000

2000 5000

5000 5000

10000 5000

15000 5000

7

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Fig. 9. The colored contours of the velocity component u and the streamlines with the different dimensionless parameters in the vortex region (y/a = 0).

Fig. 10. The pressure distribution along the streamwise direction along the line L1.

Fig. 11. The pressure distribution along the flow direction near the partition wall.

Table 4 The dimensionless pressure gradients and relative difference at Re = 1000.

Table 5 The dimensionless pressure gradients and relative difference at Ha = 5000.

Ha L1

2000

5000

10000

15000

Re

K a ∗ 10 4

5.918

2.316

1.138

0.756

L1

Ki ∗ 10 4

5.050

2.002

1.030

0.456

K o ∗ 10 4 Da Di Do

5.079

2.067

1.133

0.599

— — —

−60.873 −60.364 −59.302

−80.766 −79.603 −77.691

−87.220 −90.980 −88.213

are listed in Table 5. The results show that the increase of Reynolds number causes an increase of the average pressure gradient along the line L1 compared with the values of Case 3. However, it should be noted that the pressure gradient in the inflow channel and outflow channel varies little (≤1.769%) as the Reynolds number increases from 1000 to

1000

2000

50000

10000

15000

K a ∗ 10 4

20316

2.360

2.603

2.574

2.795

Ki ∗ 10 4

2.102

1.981

2.012

2.025

2.907

K o ∗ 10 4 Da Di Do

2.063

2.061

2.097

2.026

1.821

— — —

1.903 1.509 −0.102

12.418 0.025 1.647

11.176 0.656 −1.769

20.696 44.525 11.728

10000. At Re = l5000 , the pressure gradient grows 44.525% and 11.728% in the inflow channel and outflow channel respectively. In the case with Ha = 5000, Re = 15000, there are a quite big vortex and some small vortex near the partition wall in the outflow channel. The negative pressure gradient vector − ∇ (p ρ) , the Lorentz force 8

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Fig. 12. Parameters in the area of the vortex, (a) the colored contours of the pressure gradient with negative pressure gradient vector, (b) the colored contours of the velocity with Lorentz force vector, (c) the colored contours of the velocity with the resultant of the Lorentz force and the negative pressure gradient.

staggered distribution of − (dp /dx ) is consistent with the pressure fluctuation distribution along the flow direction. The Lorentz force is generally damping the flow with direction reverse to the flow. In Fig. 9, the primary vortex is anticlockwise and flows along x-axis direction near the partition wall, while the small vortex is clockwise and flows along negative x-axis direction near the partition wall. Referring to the vortex region, the vortex core in Fig. 9(b) and comparing the vector distribution in Fig. 12 (1)(a), it can be found that the pressure gradient projection along the x-axis in primary vortex region points to the positive x-axis, while the Lorentz force points to the negative x-axis. The resultant of the pressure gradient and the Lorentz force points to the negative x-axis, which is consistent with the direction of the pressure

vector and the resultant vector are presented in Fig. 12. It should be noted that the vector length rulers are not the same in Fig. 12(a), (b) and (c). The vector is adjusted to show the direction and distribution clearly. The Lorentz force is defined as Lforce = (J × B ) ρ while J=σ (−∇ϕ + u × B ) . As the component of the induced potential gradient is small, the direction of the induced current is mainly determined by the velocity and the external magnetic field. As a result, the Lorentz force is determined by the velocity and the direction of the external magnetic. In the outflow channel − (dp /dx ) < 0 is favorable and − (dp /dx ) > 0 is adverse. From Fig. 12 (1)(a), we could see that the negative pressure gradient has a favorable and adverse part. The 9

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Fig. 12. (continued)

distribution on the cross-section of the inflow and outflow channel is an asymmetrical M-type. The velocity jets in the side layer near the partition wall are higher than that near the duct wall in both channels. The maximum value of the velocity jets in the side layers near the partition wall increases along the streamwise direction in the inflow channel and decreases in the outflow channel. The colored contour of the velocity component u and the streamlines on the plane ( y / a = 0 ) are presented in Fig. 14. In all four cases, the colored contour of the velocity component u shows that there are velocity jets in the layers parallel to the magnetic field. The maximum velocity locates at the end of the partition wall in the inflow channel. The maximum value of the velocity u decreases with an increase of C2. When the duct wall is conducting and the partition wall is insulating, the streamline distribution is similar for all four cases. Because of the non-slip boundary condition and the high-velocity gradient at the liquid-solid wall interface, the streamlines become tight near the walls for all cases. At the start of the inflow channel and the end of the outflow channel ( x / a < 5), the streamlines are direct lines parallel to the x-direction. The streamlines bend to the outflow channel from the end of the inflow channel through the connection channel. Some streamlines turn back a little and then return to the outflow direction near the partition wall in Case 5 to Case 7 but do not form an obvious vortex (5 < x / a < 8 ) . When C2 is much higher than C1 (Case 8) as shown in Fig. 14(d), the streamlines turn smoothly to the outflow channel without visible reverse flow. The formation of the reverse flow disappearing with the increase of C2 will be explained in the later part related to the electric current. Moreover, it should be noted that there still a quite small vortex and almost invisible at the end of the partition wall in the outflow channel for all four cases. The distribution of the normalized velocity component u Um along

gradient at the core of the primary vortex. The direction of the resultant vector in the small vortex region is an agreement with the Lorentz force pointing to x-axis direction. It means that the adverse pressure gradient is dominant. In the case with Re = 1000 , Ha = 10000 , the smaller vortex attached the primary vortex near the end of the partition wall disappears as shown in Fig. 9(c). Fig. 12 (2) shows that the pressure gradient in the primary vortex zone is adverse and its projection on the x-axis is positive. The marked zone in the circle is the location of the smaller vortex. In the marked zone, the direction of the x-component part of the negative pressure gradient and the Lorentz force is negative and positive, while the resultant of the vector is almost positive. It means that in this zone the Lorentz force is dominant. Therefore, there is no obvious contract force vector and the smaller vortex disappears. In Fig. 12 (3)–(5), the distribution of the negative pressure gradient, the Lorentz force and the resultant vector shows a similar distribution as Fig. 12 (1). As a result, there are both a primary vortex and attached inverse smaller vortex. It can be considered that the primary vortex is mainly generated by the strong adverse pressure gradient near the partition wall, while the small vortex at the end of the partition wall attached to the primary vortex is caused by the action of Lorentz force. The maximum value of u/ Um locates the position where the positive and negative vector of the resultant meet. 4.2. Conducting duct wall In this section, the result of Case 5 to Case 8 is presented. Threedimensional distribution of the velocity component u on different crosssections of Case 7 (C1 = C2 = 0.07 ) is shown in Fig. 13. The velocity 10

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Fig. 13. The three-dimensional velocity distribution (Case 7, C1 = C2 = 0.07 ).

Fig. 14. The colored contour of the velocity component u and the streamlines (y/a = 0). 11

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Fig. 15. The velocity distribution along the line CDEF.

zero. The profile of the velocity w is also asymmetrical M-type. There are high-velocity jets at the end of the partition wall. The maximum wmax Um = 21.09 dimensionless velocity is in Case 6 (C1 = 0.07, C2 = 0.007 ). The jets in the connection channel are even stronger than that in the inflow and outflow channels. However, the intensity of the jet near the duct wall is weak and increases with the increase of C2. The colored electric potential Elpot contours and the electric current streamlines on the plane y /a= 0 are shown in Fig. 17. The electric current is almost parallel upward in the inflow channel and downward in the outflow channel ( x / a < 7 ). As C2 ≠ 0 , the electric current converges and closes loops in the partition wall throughout the duct. The current in the connection channel also flows and converges through (C2 ≠ 0 , Fig. 17(b) to (d)) or around the end of the partition wall (C2 = 0 , Fig. 17(a)). The density of the electric currents is strong at the end of the partition wall. The strong electric current zone near the partition wall at the start of the outflow channel moves to the end of the partition wall as C2 increases. When C2 = 1, the electric current lines between the connection channel and the partition wall is almost straight lines and does not bend obviously. The stronger electric current interacting with the magnetic field generates a stronger Lorentz force. The location

the line CDEF and AB is shown in Figs. 15 and 16 . The results indicate that the velocity distribution is an asymmetrical M-type profile in all cases. The velocity jets intensity near the partition wall decreases with C2 increase from 0 to 1. In Case 5 and Case 6, the jets near the two sides of the partition wall are higher in the outflow channel than those in the inflow channel. The variation trend of the velocity jets near the partition wall and near the duct wall with C2 is opposite. The velocity jets near the partition wall decrease, while the jets near the duct wall increase with an increase of C2. In these four cases, the maximum velocity appears at the side of the partition wall in the outflow channel of Case 5 (C1 = 0.07, C2 = 0) . Its dimensionless velocity jet value is 20.837, which is 4.9 times greater than that in Case 8. The induced electric current can only close its loops through the side layers at the side near the partition wall as it is electrically insulating in Case 5. Therefore, the electric current is parallel to the external magnetic field, which results in a smaller Lorentz force and higher velocity jets near the side of the partition wall. As C2 increases, the current closes the loops through the conducting wall and the side layers. As a result, the jets near the wall decrease. The distribution of the velocity components u and w along the line AB is shown in Fig. 16. In the connection channel, the flow turns from the x-direction to the z-direction. The velocity u fluctuates and drops to

Fig. 16. The distribution of the velocity components u and w along the line AB. 12

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Fig. 17. The colored contours of Elpot and the electric current streamlines on the plane y/a = 0.

Fig. 18. The pressure distribution along the flow direction. 13

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is used to evaluate the flux in the channel as shown in [23–25]. In these three papers, the normalized electric potential is independent of the Reynolds number. We presented the electric potential distribution in different walls at x a = 3 in Case 7 (C1 = 0.07, C2 = 0.07). The coordinate ξ in Fig. 19 is defined as ξ = y a + 2.3i (i = 0, 1, 2) , where i = 0, 1, 2 corresponds to z/a = −2.1, z/a = 0, and z/a = 2.1, respectively. One more case’s results (C1 = 0.07, C2 = 0.07, Re = 10000) are added. Fig. 19 shows that the normalized electric potential distribution with different Reynolds numbers is coincident perfectly at each wall as the Reynolds number varies from 1000 to 10000. The electric potential distribution is parabolic. Furthermore, the distribution and the value are almost the same at the bottom wall (z/a = −2.1) and the upper wall (z/a = 2.1). The direction of the parabolic distribution of the electric potential is reverse at different walls. It is positive on the partition wall and negative on the duct walls because the electric potential decreases from the maximum positive value on the partition wall to the negative value on the duct walls as shown in Fig. 17. The electric potential difference is higher on the partition wall than that on the bottom and upper duct walls. The reason is that the electric current from the inflow channel and the outflow channel converges to the partition wall, which results in a stronger electric current. The electric potential distribution along the magnetic field with different wall conductance ratios of the duct walls at x/a = 3 is also presented in Fig. 19. It shows that the electric potential distribution is still parabolic in general. However, the electric potential distribution on the partition wall has an obvious boundary effect. The electric potential distributes flat in Case 6 (C1 = 0.07, C2 = 0.007) and has a sharp decrease in Case 8 (C1 = 0.07, C2 = 1) . The variation of the electric potential at the two side boundaries is caused by the partition wall merging with the duct wall, where the wall conductance ratio of C1 and C2 varies greatly. The increasing of C2 with constant C1 results in the electric potential difference decrease on the partition wall and increase slightly on the side wall in the inflow and outflow channel. In order to check and verified whether the normalized electric potential is independent of the Reynolds number when Hartmann number and Reynolds number are high enough. The electric potential distribution along the magnetic field direction in the wall at z / a = 0 , x / a = 3 for the subcases C1 = 0 , C2 = 0.07 , Ha = 5000 , Re = 1000 , 2000, 5000,10000, and 15000 is presented in Fig. 20. It shows that the normalized electric potential in the partition wall depends on the Reynolds number at Ha = 5000 . When Re ≤ 10000 , the normalized electric potential lines cluster together but do not coincide with each other. As Re = 15000 , the normalized electric potential is much lower than that in the other four cases. It can be concluded that when the Hartmann number and Renolds number are medium (here Ha = 2000, Re ≤ 10000 ), the normalized electric potential is independent of the Reynolds number as shown in Fig. 19. This is because

Table 6 The dimensionless pressure gradient and the relative differences for C1 = 0.07. C2 L1

L2

0

0.007

0.07

1

K a ∗ 102

4.628

4.783

5.456

6.071

Ki ∗ 102

4.232

4.066

4.808

5.436

K o ∗ 102 Da Di Do

3.942

4.386

5.036

5.458

K a ∗ 102

— — — 3.828

2.175 −3.943 11.272 3.917

17.902 13.579 27.773 4.469

31.188 28.420 38.479 4.915

Ki ∗ 102

4.500

4.442

5.029

5.467

K o ∗ 102 Da Di Do

4.063

4.371

4.994

5.428

— — —

2.319 −1.296 7.583

16.731 11.760 22.933

28.372 21.475 33.615

variation of the strong electric currents can explain the backward velocity streamlines near the partition wall in Fig. 14. The electric potential is high at the partition wall and low at the duct wall. As C2 increases, the value of the high electric potential decreases and the low electric potential decreases. The absolute electric potential difference also decreases a little. Moreover, the high electric potential zone near the partition wall expands. The distribution and variation tendency is consistent with those in Case 1 to Case 4. The pressure variation along the line L1 (M1-N1-P1-Q1) and L2 (M2-N2-P2-Q2) defined in Fig. 3 are presented in Fig. 18. The pressures gradient and the relative difference defined in Section 4.1 are presented in Table 6. Case 5 (C1 = 0.07, C2 = 0) is selected as the reference case in this section. The pressure distribution along the streamwise direction fluctuates near the connection channel along the line L1. However, the fluctuation amplitude and range are much smaller than those in Cases with C1 = 0. There is no adverse pressure gradient along the line L2. As the duct wall is weakly conducting, K a and K o increase with the increase of C2 along both lines. The pressure gradient in the outflow channel K o rises more. The maximum pressure gradient difference was 38.479% along the line L1 and 33.615% along the line L2 in the outflow channel. There is a little drop of Ki at C2 = 0.007 . The total average pressure gradient K a is bigger along the line L1 and smaller than Ki and K o along the line L2. This is caused by the pressure fluctuation near the connection channel. Comparing to the pressure gradient in the Cases with insulating duct wall as shown in Table 2, the dimensionless pressure gradient along the flow direction with conducting duct wall is two orders of magnitude higher than that with insulating duct wall. Furthermore, the partition wall conductance ratio influences the pressure gradient much stronger as the duct wall is weak conducting. The electric potential distribution along the magnetic field direction

Fig. 19. The electric potential distribution along the magnetic field direction in the walls ( x / a = 3).

Fig. 20. The electric potential distribution along the magnetic field direction in the walls at z / a = 0 , x / a = 3. 14

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Table A1 Three grid systems distribution. Total Grid number

Mesh 1 Mesh 2 Mesh 3

Cell number in boundary layers

6

2.85∙10 4.416∙106 7.29∙106

Hartmann layer

Side layer

2 4 6

18 22 26

Fig. A1. The absolute value of the streamwise velocity distribution at the center line on the y–z plane in the outflow channel with the different mesh systems at x / a = 1 (Case 7 C1 = 0.07, C2 = 0.07, Re = 1000, Ha = 2000 ).

the flow is laminar and the normalized field is not influenced by the limited variation of the Reynolds number. However, the higher the Hartmann number and the Reynolds number, the more instability of the field. Although we use a laminar model to carry out all cases simulating, we must point out that the flow is not quite stable when Ha = 5000 and Re = 15000 as shown the pressure distribution in Fig. 11. As a result, the field varies with the Reynolds number. The electric potential distribution along the magnetic field depends on the Reynolds number. Such dependence on the Reynolds number should be considered in the experimental study.

the wall conductance ratio of the partition wall does not affect the pressure gradient in the duct much. The vortex region changes with the Hartmann number and the Reynolds number. The higher the Hartmann number, the smaller the vortex is. The vortex becomes smaller as Re increases from 1000 to 5000. After that, it becomes expanding. It is almost full of the start of the outflow channel as Re=l5000. The pressure fluctuation strengthens with the increase of the Hartmann number and the Reynolds number. The dimensionless pressure gradient decreases with the increase of the Hartmann number. The primary vortex is generated because of the dominant strong adverse pressure gradient near the partition wall, and the small vortex at the end of the partition wall is generated by the dominant of Lorenz force. When the duct wall is weak conducting, there are reverse flows in the outflow channel near the partition wall but no visible vortex forms. The velocity profile is asymmetrical M-type. The average pressure gradient in the duct with the conducting duct wall is two orders of magnitude higher than that of the duct with insulating duct walls. Furthermore, the partition wall conductance ratio influences the average pressure gradient along the flow direction significantly. In conclusion, the effect of the partition wall conductance ratio and the duct wall conductance ratio is different on the velocity distribution and the pressure drop.

5. Conclusions In this paper, we have numerically simulated the liquid metal flows in a coupled U-turn rectangular duct with different wall conductance ratio using an in-house solver developed using OpenFOAM. When the walls are thoroughly insulating, the velocity profile has small jets in the side layer because of the flow is developing and the electric current bends to y-direction a little. The small jets disappear when the partition wall is weak conducting. The average pressure gradient with conducting partition wall is a little lower compared to the fully insulating wall. Three-dimensional electric current and current vortex with different cores assemble in the connection channel and the start of the outflow channel. The closed electric current loops generates opposite Lorentz force. Therefore, a strong and wide range fluctuation of the pressure distribution in and near the connection channel forms along the flow direction. There is a strong adverse pressure gradient near the partition wall which will cause substantial mechanical stress on the wall. When the duct wall is electrically insulating and the partition wall is weakly conducting, there is a vortex at the end of the partition wall in the outflow channel. When the duct wall is insulating,

Acknowledgments This work was supported by the National Natural Science Foundation of China (NSFC) under grants No. 11375049 and 11572107 and the National Magnetic Confinement Fusion Science Program of China under grant No. 2014GB125003. The authors thank the reviewers’ detailed, precise and constructive comments.

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Appendix A. Grid independence test It is necessary to verify the independence of the grids to ensure calculation accuracy. Case 7 (C1 = C2 = 0.07 , Re = 1000, Ha = 2000 ) is used to perform the grid independence verification test. Three grid systems with different grid distributions in the Hartmann layer and the side layer used in the simulation are listed in Table A1. The normalized streamwise velocity distribution along the center lines on the y-zplane in the outflow channel at x/a = 1 of the three grid systems is shown in Fig. A1. It shows that the velocity distribution is almost coincident. The maximum difference of the maximum value of the velocity jet in the side layer based on Mesh 1, Mesh 2, and Mesh 3 is only 1.06%. Considering the calculation time and accuracy, Mesh 2 is selected for all cases. The grid is also refined at the end of the partition wall and the fluid region near the solid walls along the xdirection.

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