A consistent and conservative scheme for incompressible MHD flows at a low magnetic Reynolds number. Part III: On a staggered mesh

The consistent and conservative scheme developed on a rectangular collocated mesh [M.-J. Ni, R. Munipalli, N.B. Morley, P. Huang, M.A. Abdou, A current density conservative scheme for incompressible MHD flows at a low magnetic Reynolds number. Part I: on a rectangular collocated grid system, Journal of Computational Physics 227 (2007) 174–204] and on an arbitrary collocated mesh [M.-J. Ni, R. Munipalli, P. Huang, N.B. Morley, M.A. Abdou, A current density conservative scheme for incompressible MHD flows at a low magnetic Reynolds number. Part II: on an arbitrary collocated mesh, Journal of Computational Physics 227 (2007) 205–228] has been extended and specially designed for calculation of the Lorentz force on a staggered grid system (Part III) by solving the electrical potential equation for magnetohydrodynamics (MHD) at a low magnetic Reynolds number. In a staggered mesh, pressure (p) and electrical potential (φ) are located in the cell center, while velocities and current fluxes are located on the cell faces of a main control volume. The scheme numerically meets the physical conservation laws, charge conservation law and momentum conservation law. Physically, the Lorentz force conserves the momentum when the magnetic field is constant or spatial coordinate independent. The calculation of current density fluxes on cell faces is conducted using a scheme consistent with the discretization for solution of the electrical potential Poisson equation, which can ensure the calculated current density conserves the charge. A divergence formula of the Lorentz force is used to calculate the Lorentz force at the cell center of a main control volume, which can numerically conserve the momentum at constant or spatial coordinate independent magnetic field. The calculated cell-center Lorentz forces are then interpolated to the cell faces, which are used to obtain the corresponding velocity fluxes by solving the momentum equations. The “conservative” is an important property of the scheme, which can guarantee computational accuracy of MHD flows at high Hartmann number with a strongly non-uniform mesh employed to resolve the Hartmann layers and side layers. 2D fully developed MHD flows with analytical solutions available have been conducted to validate the scheme at a staggered mesh. 3D MHD flows, with the experimental data available, at a constant magnetic field in a rectangular duct with sudden expansion and at a varying magnetic field in a rectangular duct are conducted on a staggered mesh to verify the computational accuracy of the scheme. It is expected that the scheme for the Lorentz force can be employed together with a fully conservative scheme for the convective term and the pressure term [Y. Morinishi, T.S. Lund, O.V. Vasilyev, P. Moin, Fully conservative higher order finite difference schemes for incompressible flow, Journal of Computational Physics 143 (1998) 90–124] for direct simulation of MHD turbulence and MHD instability with good accuracy at a staggered mesh.     

A current density conservative scheme for incompressible MHD flows at a low magnetic Reynolds number. Part II: On an arbitrary collocated mesh

A conservative formulation of the Lorentz force is given here for magnetohydrodynamic (MHD) flows at a low magnetic Reynolds number with the current density calculated based on Ohm’s law and the electrical potential formula. This conservative formula shows that the total momentum contributed from the Lorentz force is conservative when the applied magnetic field is constant. For the case with a non-constant applied magnetic field, the Lorentz force has been divided into two parts: a strong globally conservative part and a weak locally conservative part.The conservative formula has been employed to develop a conservative scheme for the calculation of the Lorentz force on an unstructured collocated mesh. Only the current density fluxes on the cell faces, which are calculated using a consistent scheme with good conservation, are needed for the calculation of the Lorentz force. Meanwhile, a conservative interpolation technique is designed to get the current density at the cell center from the current density fluxes on the cell faces. This conservative interpolation can keep the current density at the cell center conservative, which can be used to calculate the Lorentz force at the cell center with good accuracy. The Lorentz force calculated from the conservative current at the cell center is equivalent to the Lorentz force from the conservative formula when the applied magnetic field is constant, which can conserve the total momentum. We will further prove that the simple interpolation scheme used in the Part I [M.-J. Ni, R. Munipalli, N.B. Morley, P.Y. Huang, M. Abdou, A current density conservative scheme for MHD flows at a low magnetic Reynolds number. Part I. On a rectangular collocated grid system, Journal of Computational Physics, in press, doi:10.1016/j.jcp.2007.07.025] of this series of papers is conservative on a rectangular grid and can keep the total momentum conservative in a rectangular grid.     

液态包层流动通道插件三维MHD效应数值模拟

用直接数值方法对高效液态锂铅包层内的金属流体三维MHD效应进行分析。用投影法对包含洛仑兹力源项的不可压Navier．Stokes方程求解,用相容守恒格式计算电磁力。研究了不同材料的流动通道插件（FCI）对金属磁流体流速、MHD压降和电流流线分布的影响。主要分析了以下三种情况：无FCI插件的通道内的流动状况;加入绝缘材料...     

基于OpenFOAM的投影法磁流体求解器开发与验证

在开源计算流体力学C++工具包OpenFOAM环境下开发了低磁雷诺数条件下的磁流体求解器,并进行了验证。采用投影算法求解动量方程和压力泊松方程;采用非结构网格同位相容守恒算法求解电势泊松方程、感应电流和洛伦兹力;采用边界耦合方法求解流固耦合电势场。通过对均匀磁场下导电方管和导电圆管内的完全发展磁流体层流的数值模拟和解析解的对比,对求解器进行了验证。进一步对非均匀强磁场作用下导电方管和导电圆管内完全发展磁流体层流进行了数值模拟,并与ALEX实验结果进行了比较。数值解和实验结果吻合良好。所开发的求解器可用于复杂结构强磁场作用下磁流体的数值模拟研究。     

交错网格下MHD相容守恒格式的发展

在低磁场雷诺数条件下,基于电势泊松方程,发展了交错网格下可以精确计算电流和洛伦兹力(电磁力)的相容守恒格式。采用压力为变量的原始变量法求解不可压缩Navier-Stokes方程,所计算的电流满足电荷守恒定律,所计算的电磁力满足动量守恒定律。对金属流体在Hartmann数50~5000范围内验证了格式的精确性。交错网格下相容守恒格式的发展为后续MHD稳定性分析、湍流的大涡模拟及直接数值模拟提供很好的选择。     

Consistent projection methods for variable density incompressible Navier–Stokes equations with continuous surface forces on a rectangular collocated mesh

Two consistent projection methods of second-order temporal and spatial accuracy have been developed on a rectangular collocated mesh for variable density Navier–Stokes equations with a continuous surface force. Instead of the original projection methods (denoted as algorithms I and II in this paper), in which the updated cell center velocity from the intermediate velocity and the pressure gradient is not guaranteed solenoidal, the consistent projection methods (denoted as algorithms III and IV) obtain the cell center velocity based on an interpolation from a conservative fluxes with velocity unit on surrounding cell faces. Dependent on treatment of the continuous surface force, the pressure gradient in algorithm III or the sum of the pressure gradient and the surface force in algorithm IV at a cell center is then conducted from the difference between the updated velocity and the intermediate velocity in a consistent projection method. A non-viscous 3D static drop with serials of density ratios is numerically simulated. Using the consistent projection methods, the spurious currents can be greatly reduced and the pressure jump across the interface can be accurately captured without oscillations. The developed consistent projection method are also applied for simulation of interface evolution of an initial ellipse driven by the surface tension and of an initial sphere bubble driven by the buoyancy with good accuracy and good resolution.     

数值模拟磁场作用下气泡在导电流体中运动

在自适应网格上,采用VOF方法捕捉界面,相容守恒格式计算电流及电磁力,发展了金属流体自由界面MHD数值方法。通过数值模拟磁场作用下不同Hartmann数的气泡在导电溶液中的运动和变形,分析磁场对气泡以及流场的影响,同时给出诱导电场和电流的分布。为进一步深入研究冶金及热核聚变相关的金属流体在强磁场作用下的自由界面流打下基础。     

Effect of the induced magnetic field on peristaltic flow of a couple stress fluid

We have analyzed the MHD flow of a conducting couple stress fluid in a slit channel with rhythmically contracting walls. In this analysis we are taking into account the induced magnetic field. Analytical expressions for the stream function, the magnetic force function, the axial pressure gradient, the axial induced magnetic field and the distribution of the current density across the channel are obtained using long wavelength approximation. The results for the pressure rise, the frictional force per wave length, the axial induced magnetic field and distribution of the current density across the channel have been computed numerically and the results were studied for various values of the physical parameters of interest, such as the couple stress parameter γ, the Hartmann number M, the magnetic Reynolds number Rm and the time averaged mean flow rate θ. Contour plots for the stream and magnetic force functions are obtained and the trapping phenomena for the flow field is discussed.     

Analytical and empirical studies on the characteristic resistances of no-insulation GdBCO racetrack pancake coil under various operating currents

This paper presents the effects of the Lorentz force on the electrical behaviors of the no-insulation (NI) GdBCO racetrack pancake (RP) coil without turn-to-turn insulation by performing charging/discharging and sudden-discharging tests. The simulation results of the 2-dimensional finite element method showed that the Lorentz forces were generated towards the center of the winding pack consisting of 60-turn GdBCO CCs, which could enhanced the turn-to-turn contact within the RP coil. The charging/discharging test results indicated that the characteristic resistance (R_c) of the NI RP coil decreased with increasing operating current, which was caused by an increase of the Lorentz force. Further, the results of the sudden-discharging tests exhibited that the Lorentz force was dissipated instantly during the sudden-discharge of the coil, since the operating current immediately decreased to zero. Overall, this study clearly demonstrated that the Lorentz force induced by the operating current exerts the electrical behaviors of an NI RP coil.     

Characterization of the three-dimensional supersonic flow for the MHD generator

A numerical procedure based on a five-wave MHD model associated with non-ideal, low magnetic Reynolds number MHD flows was developed in the present study for analyzing the flow fields in the MHD generator of a MHD bypass scramjet. The numerical procedure is composed of an entropy conditioned scheme for solving the non-homogeneous Navier-Stokes equations, in conjunction with an SOR method for solving the elliptic equation governing the electrical potential. It was found that a separation would take place near the downstream edge of the second electrode, where the local adverse pressure gradient is large, and the core of the flow field is characterized as a 2-D flow due to the Hartmann effects along the direction of the magnetic field. The electric current lines would be increasingly distorted as the magnetic interactive parameter increases, and even induce an eddy current. Induced eddy current was also found in the different cross-sections along the axial direction, all of these would definitely deteriorate the performance of the MHD generator. The cross-sectional M-shape velocity profile found along the axial direction between the insulating walls is responsible for the formation of the vortex flow at the corner of the insulator cross-section, which, in turn, induces the corner eddy current at the corner. A numerical parametric study was also performed, and the computed performance parameters for the MHD generator suggest that, in order to enhance the performance of MHD generator, the magnetic interaction parameter should be elevated.     

导电突扩管中磁流体管流的数值模拟

采用基于OpenFOAM环境自主开发的低磁雷诺数磁流体求解器,对45°和90°突扩矩形管中液态金属流体在受到垂直流向的外加磁场作用时的速度、感应电流、压力的分布及突扩位置处的MHD三维现象进行数值模拟.结果表明：磁场沿突扩方向时,由于无回流涡,45°比90°突扩管在肩部位置速度分布更优.哈特曼数增大,强射流和突扩结构,在突扩肩部位置引发流动的不稳定性.伴随感应电流的不稳定,流动不稳定发展到突扩位置上游.磁场沿垂直突扩方向时感应电流的三维效应显著.哈特曼数增大,MHD压降显著增大.同方向磁场和相同哈特曼数,不同突扩角度的三维无量纲压力梯度无明显差异.     

Hall effect on Hartmann flow and heat transfer of a Burgers' fluid

The effect of Hall current on the steady magnetohydrodynamics (MHD) flow of an electrically conducting, incompressible Burgers' fluid between two parallel electrically insulating infinite planes is studied. The MHD flow is generated by applying constant pressure gradient. An external uniform magnetic field normal to the disks is applied. The disks are kept at two different constant temperatures. Exact solutions are obtained for the governing momentum and energy equations. The effects of Hartmann number M, Reynolds number Re, Prandtl number Pr, Eckert number Ec, pressure gradient dp/dx and Hall parameter η are examined.     

Effects of hall current on MHD Couette flow in a rotating system with arbitrary magnetic field

Author has studied the MHD Couette flow in a rotating environment with non- conducting walls in the presence of an arbitrary magnetic field. The solution in dimensionless form contains four pertinent flow parameters, viz. the Hartmann number, the rotation parameter which is the reciprocal of the Ekman number, the Hall current parameter, and the angle of inclination of the magnetic field to the positive direction of the axis of rotation. An interplay of hydromagnetic force and Coriolis force with an inclusion of Hall current plays a significant role in determining the MHD flow behaviour. The velocity and induced magnetic field distributions are depicted graphically. Also, the numerical results of shear stresses and the rate of mass flows are presented graphically.     

On the calculation of steady-state magnetohydrodynamic flows of liquid metals in circular ducts of a rectangular cross section

A numerical code has been designed to calculate two-dimensional steady-state magnetohydrodynamic (MHD) flows of incompressible conducting fluids (liquid metals) in linear and circular thin-wall ducts of a rectangular cross section. The flows are caused by the Lorentz force J × B that appears when an electric current passes through a fluid placed in a vertical uniform magnetic field. The code is the generalization of the well-known iteration Gauss-Seidel method to the case of a set of elliptical equations. The method proposed can be used to calculate steady-state flows over wide ranges of Hartmann (Ha = 1–10³) and Reynolds (Re = 1–10⁶) numbers.     

Induced electric current-based formulation in computations of low magnetic Reynolds number magnetohydrodynamic flows

We use the induced electric current as the main electromagnetic variable to compute low magnetic Reynolds number magnetohydrodynamic (MHD) flows. The equation for the induced electric current is derived by taking the curl of the induction equation and using Ampère’s law. Boundary conditions on the induced electric current are derived at the interface between the liquid and the thin conducting wall by considering the current loop closing in the wall and the adjacent liquid. These boundary conditions at the liquid–solid interface include the Robin boundary condition for the wall-normal component of the current and an additional equation for the wall potential to compute the tangential current component. The suggested formulation (denominated j-formulation) is applied to three common types of MHD wall-bounded flows by implementing the finite-difference technique: (i) high Hartmann number fully developed flows in a rectangular duct with conducting walls; (ii) quasi-two-dimensional duct flow in the entry into a magnet; and (iii) flow past a magnetic obstacle. Comparisons have been performed against the traditional formulation based on the induced magnetic field (B-formulation), demonstrating very good agreement.     

MHD from a Microscopic Concept and Onset of Turbulence in Hartmann Flow

We derive higher order magneto-hydrodynamic （MHD） equations from a microscopic picture using pro-jection and perturbation formalism. In an application to Hartmann flow we find velocity profiles flattening towards the center at the onset of turbulence in hydrodynamic limit. Comparison with the system under the effect of a uniform magnetic field yields difference in the onset of turbulence consistent with observations, showing that the presence of magnetic field inhibits onset of instability or turbulence. The laminar-turbulent transition is demonstrated in a phase transition plot of the development in time of the relative average velocities vs. Reynolds number showing a sharp increase of the relative average velocity at the transition point as determined by the critical Reynolds number.     

A representation of curved boundaries for the solution of the Navier–Stokes equations on a staggered three-dimensional Cartesian grid

A method is presented for representing curved boundaries for the solution of the Navier–Stokes equations on a non-uniform, staggered, three-dimensional Cartesian grid. The approach involves truncating the Cartesian cells at the boundary surface to create new cells which conform to the shape of the surface. We discuss in some detail the problems unique to the development of a cut cell method on a staggered grid. Methods for calculating the fluxes through the boundary cell faces, for representing pressure forces and for calculating the wall shear stress are derived and it is verified that the new scheme retains second-order accuracy in space. In addition, a novel “cell-linking” method is developed which overcomes problems associated with the creation of small cells while avoiding the complexities involved with other cell-merging approaches. Techniques are presented for generating the geometric information required for the scheme based on the representation of the boundaries as quadric surfaces. The new method is tested for flow through a channel placed oblique to the grid and flow past a cylinder at Re=40 and is shown to give significant improvement over a staircase boundary formulation. Finally, it is used to calculate unsteady flow past a hemispheric protuberance on a plate at a Reynolds number of 800. Good agreement is obtained with experimental results for this flow.     

Magnetohydrodynamic Natural Convection in a Rotating Enclosure

Magnetohydrodynamic natural convection heat transfer in a rotating, differentially heated enclosure is studied numerically in this article. The governing equations are in velocity, pressure and temperature formulation and solved using the staggered grid arrangement together with MAC method. The governing parameters considered are the Hartmann number, 0≤$Ha$≤70, the inclination angle of the magnetic field, 0$^◦$≤$θ$≤90$^◦$, the Taylor number, 8.9×10$^4$≤$Ta$≤1.1×10$^6$ and the centrifugal force is smaller than the Coriolis force and the both forces were kept below the buoyancy force. It is found that a sufficiently large Lorentz force neutralizes the effect of buoyancy, inertial and Coriolis forces. Horizontal or vertical direction of the magnetic field is the most effective in reducing the global heat transfer.     

Calculation of the electric potential and Lorentz force acting on a locally ionized region in a magnetohydrodynamic flow placed in a nonuniform magnetic field

An analytical solution to electrodynamic equations for the electric potential in a locally ionized magnetohydrodynamic (MHD) flow in the nonuniform magnetic field produced by a straight-line conductor is found. Analytical formulas are obtained to evaluate the volume density of the Lorentz force and the integral Lorentz force acting on the locally ionized region of the MHD flow. It is shown that the MHD action on the locally ionized flow in the nonuniform magnetic field can be used to control the elevating force as well as the ratio of the elevating force to the drag force.