Magnetohydrodynamic flows in a hairpin duct under a magnetic field applied perpendicular to the plane of flow

Abstract

This numerical study examines three-dimensional liquid–metal magnetohydrodynamic flows in a hairpin-shaped electrically-conducting duct with a square cross-section under a uniform magnetic field applied perpendicular to the flow plane. Predicted is detailed information on the fluid velocity, pressure, current, and electric potential in the magnetohydrodynamic duct flows. Higher velocities are observed in the side layers in the inflow and outflow channels, yielding ‘M-shaped’ velocity profiles. More specifically, in the present study the axial velocity in the side layer near the partitioning wall is higher than that near the outer walls because of the current features therein. In the transition segment, a large velocity recirculation is observed at the entrance of the outflow channel (above the end of partitioning wall) caused by the flow separation, yielding complicated distributions of the electric potential and current therein. Cases with different Hartmann numbers are examined. The non-dimensional pressure gradient is smaller in case of higher Hartmann number, while the pressure in each case almost linearly decreases along the main flow direction, except in the transition segment. In the present study, the electromagnetic characteristics of the liquid metal flows are examined in terms of the electro-motive and electric-field components of the current with an aim to analyze the interdependency of the flow variables.

Introduction

The study of magnetohydrodynamic (MHD) duct flows has great relevance to many engineering applications, including the design of electromagnetic pumps, liquid–metal cooling of nuclear reactors, and MHD flowmeters. An increasing interest in the study of MHD phenomena, in particular, relates to the development of fusion reactor, in which plasma is confined by a strong magnetic field [1], [2], [3]. The motion of liquid metal (LM) under a strong magnetic field causes significant MHD effects, which have dramatic impacts on pressure drop, velocity distribution, heat-transfer characteristics, and pumping power required for the cooling system [4]. Therefore, the LM MHD effect is one of the key issues in the design of an optimal LM blanket. An LM blanket can have a complicated piping system to enable the liquid metal to flow through ducts with different geometries.

Numerous experimental studies [5], [6] have been conducted to investigate the characteristics of MHD flows in ducts. Stieglitz [7] experimentally investigated the flow in right-angle bends, in which a flow in a direction perpendicular to the magnetic field is changed to one that is parallel to the field. In addition, a mathematical approach [8], [9] has been performed to analyze the velocity profile in the developed region. Most of the theoretical studies in this field have been performed with an inertialess approximation that assumes a laminar flow in which electromagnetic force is the dominant factors determining the flow and pressure distributions in the liquid metal, except for in thin boundary layers [10]. Asymptotic techniques [11] on MHD flows have been used under the assumption that the flow is fully developed or that the value of the interaction parameter is extremely large. These assumptions may simplify three-dimensional problems into two-dimensional ones, and the inertial effects are neglected. However, when the flow geometry is complex, the inertial effects cannot always be negligible. For example, when the ratio of the Hartmann number to the interaction parameter is large, the inertial effect cannot be neglected [12].

Through the development of computer technology, a numerical solution method based on computational fluid dynamics has become an important and efficient way to analyze MHD flows when inertial terms are not negligible. A number of numerical codes have been developed for investigating LM MHD flows [13], [14], [15], [16], [17], [18]. Ni et al. [16] implemented a consistent and conservative scheme in a three-dimensional parallel code based on a solution method for the electrical potential equation. Zhou et al. [17] built a computational code with structural collocated grids that can predict LM MHD flows in rectangular ducts, and the performance of the code was shown to be very good. Albets-Chico et al. [18] numerically assessed the efficiency of the so-called “core flow approximation” that models liquid–metal flows under the influence of intense magnetic fields. It is shown that their results are in excellent agreement with available experimental measurements.

Three-dimensional numerical works with the use of CFX code have been recently reported. Reimann et al. [19] predicted pressure drop and velocity in components in which a strong three-dimensional electric current occurs, and they explained the multi-channel effects in LM MHD duct flows. In a manifold with three sub-channels of co-current flow, the fully developed velocity profiles were numerically obtained with CFX software; however, the specific conditions for the MHD flow were not described therein. Mistrangelo and Buhler [20] performed numerical analyses of LM MHD flows in a rectangular duct with sudden expansion. These studies showed that the CFX code has high accuracy when Hartmann number is up to 1000.

In addition, some researchers have focused on MHD flows in complex duct geometries. Kalis and Tsinober [21] numerically performed three-dimensional analyses of MHD flows in a channel with a rectangular obstacle. However, the calculations were limited to Hartmann numbers of ∼20 and Reynolds numbers of ∼40. Molokov [22] obtained mathematical solutions for LM flows in manifolds, expansions, contractions, and elbows with insulating duct walls. Arshad et al. [23] numerically performed a detailed study of LM flow in a curved bend under different conditions. It was found that the MHD pressure drop increases as the liquid metal flows increasingly transverse to the magnetic field. Vantieghem et al. [24] numerically investigated laminar MHD pipe flows in a wide parameter range. Morley et al. [25] performed a numerical simulation of MHD flows in a geometry consisting of a single rectangular supply duct and a rectangular manifold, including three sub-channels, in a plane parallel to the applied magnetic field. The formation of the flow imbalance was rationalized with the argument that as the Hartmann number is increased with a fixed Reynolds number, the imbalance is decreased. Mistrangelo [26] performed a numerical study on the three dimensional MHD flows in sudden expansions.

Although many studies on MHD flows in complicated geometries have been recently performed, a detailed investigation on LM MHD flows in a rectangular hairpin duct has rarely been conducted. In this study, three-dimensional MHD flows in a rectangular hairpin duct with parallel inflow and outflow channels under a uniform magnetic field are numerically analyzed using CFX, where the magnetic field is applied in the direction perpendicular to the plane of the inflow and outflow channels. The information of velocity, pressure drop, current density, and electric potential are discussed in detail for LM MHD flows with different Hartmann numbers, with an emphasis on the interdependency among the above flow variables.