Discussion on heat transfer correlation in turbulent channel flow imposed wall-normal magnetic field

Heat transfer degradation in high Prandtl number fluid was evaluated via direct numerical simulation (DNS). Target flow fields were fully developed turbulent channel flows imposed a wall-normal magnetic field in the high and low Prandtl number conditions (Pr = 5.25 and 0.025, respectively). Values of the bulk Reynolds number (Re_b = 14,000) and the Hartmann number (Ha = 0-32) were set to be equivalent to those of the previous experimental study by Yokomine et al. The numerical results of the Nusselt number for the high Prandtl number fluid were in good agreement with the experimental results by Yokomine et al. However, the magneto-hydrodynamics (MHD) effect on the heat transfer degradation was considerably larger than the empirical correlation proposed by Blum, particularly in the large interaction parameter range. On the other hand, the DNS results for the low Prandtl number fluid were consistent with the empirical correlation proposed by Blum and the experimental results by Gardner and Lykoudis.Therefore, we proposed a new correlation of the MHD heat transfer in high Prandtl number fluid (Pr = 5.25), and suggested that the empirical correlation proposed by Blum could be recommended for low Prandtl number fluid in the large interaction parameter range.     

Code Validation for Magnetohydrodynamic Buoyant Flow at High Hartmann Number

In liquid metal fusion blanket, the non-uniform volumetric heat deposited by the fusion neutrons leads to the non-uniform density distribution of liquid metal. With the force of gravity, buoyant flows would happen. In the fusion blanket where the magnetic field is up to 4T or even higher and the Hartmann number is ~10⁴, these effects caused by the buoyancy will significantly influence the flow and heat transfer characteristics. In this paper, a module for magnetohydrodynamic (MHD) buoyant flow at high Hartmann number was added to the code MTC. A current density conservative scheme was used to ensure the conservation of current, and the Boussinesq model was used to simulate the buoyancy force. This code was validated by two benchmarks, and the results showed that it can give an accurate simulation for MHD buoyant flows. Main characteristics of buoyancy effects of MHD flows were investigated, and the suppression of buoyant convection by strong magnetic field was studied to understand how the direction of magnetic field and electric conductivity of wall affects the suppression.     

Code validation for the magnetohydrodynamic flow at high Hartmann Number based on unstructured grid

In order to analyze the magnetohydrodynamic (MHD) effect in liquid metal fusion blanket, a parallel and high performance numerical code was developed to study MHD flows at high Hartmann Number based on the unstructured grid. In this code, the induced current and the Lorentz force were calculated with a current density conservative scheme, while the incompressible Navier–Stokes equations with the Lorentz force included as a source term was solved by projection method, a set of method were used to improve the computing performance such as Krylov subspace method and AMG method. To validate this code, three benchmarks of MHD flow at high Hartmann Number were conducted. The first benchmark was the case of Shercliff fully development flow, the second benchmark was the MHD flow in a circular pipe with changing external magnetic field, and the third benchmark was the MHD flow in a pipe with sudden expansion. In these cases the Hartmann Numbers were from 1000 to 6000. The code good computing performance, and numerical results show matched well with the analytical and experimental results.     

Three-dimensional numerical simulation for magnetohydrodynamic duct flows in a staggered grid system

Liquid metal flow in a rectangular duct subjected to a strong external magnetic field is numerically simulated by three-dimensional SIMPLE algorithm on a finite volume structured staggered grid. The current density and the Lorentz force are calculated by solving the electrical potential equation at low magnetic Reynolds numbers and high Hartmann numbers with consistent and conservative scheme. The Crank–Nicolson scheme is used to update the convective and diffusion terms to get a second-order temporal accuracy. The diffusion and convective term is discretised by a second order central difference scheme. The code is verified by simulating Shercliff's case and Hunt's case II with different Hartmann number and Reynolds number. Numerical results have a good agreement with the analytical solution.     

Lattice Boltzmann method for simulation of time-dependent neutral particle transport

In this paper,a novel model is proposed to investigate the neutron transport in scattering and absorbing medium.This solution to the linear Boltzmann equation is expanded from the idea of lattice Boltzmann method (LBM) with the collision and streaming process.The theoretical derivation of lattice Boltzmann model for transient neutron transport problem is proposed for the first time.The fully implicit backward difference scheme is used to ensure the numerical stability,and relaxation time and equilibrium particle distribution function are obtained.To validate the new lattice Boltzmann model,the LBM formulation is tested for a homogenous media with different sources,and both transient and steady-state LBM results get a good agreement with the benchmark solutions.     

Lattice Boltzmann method for simulation of time-dependent neutral particle transport

In this paper,a novel model is proposed to investigate the neutron transport in scattering and absorbing medium.This solution to the linear Boltzmann equation is expanded from the idea of lattice Boltzmann method (LBM) with the collision and streaming process.The theoretical derivation of lattice Boltzmann model for transient neutron transport problem is proposed for the first time.The fully implicit backward difference scheme is used to ensure the numerical stability,and relaxation time and equilibrium particle distribution function are obtained.To validate the new lattice Boltzmann model,the LBM formulation is tested for a homogenous media with different sources,and both transient and steady-state LBM results get a good agreement with the benchmark solutions.     

Numerical analysis of liquid metal MHD flows through circular pipes based on a fully developed modeling

Magnetohydrodynamics (MHD) laminar flows through circular pipes are studied in this paper by numerical simulation under the conditions of Hartmann numbers from 18 to 10000. The code is developed based on a fully developed modeling and validated by Samad's analytical solution and Chang's asymptotic results. After the code validation, numerical simulation is extended to high Hartmann number for MHD circular pipe flows with conducting walls, and numerical results such as velocity distribution and MHD pressure gradient are obtained. Typical M-type velocity is observed but there is not such a big velocity jet as that of MHD rectangular duct flows even under the conditions of high Hartmann numbers and big wall conductance ratio. The over speed region in Robert layers becomes smaller when Hartmann numbers increase. When Hartmann number is fixed and wall conductance ratios change, the dimensionless velocity is through one point which is in agreement with Samad's results, the locus of maximum value of velocity jet is same and effects of wall conductance ratio only on the maximum value of velocity jet. In case of Robert walls are treated as insulating and Hartmann walls as conducting for circular pipe MHD flows, there is big velocity jet like as MHD rectangular duct flows of Hunt's case 2.     

Numerical analyses on liquid-metal magnetohydrodynamic flow in sudden channel expansion

Three-dimensional numerical calculations have been performed on the magnetohydrodynamic (MHD) flows through a rectangular channel with sudden expansion, particularly in order to estimate the pressure drop through the sudden expansion. The sudden expansion is in the directions both perpendicular and parallel to the applied magnetic field. The Hartmann number, the Reynolds number and the magnetic Reynolds number are set to ～100, ～1000 and ～0.001, respectively, in simulating laboratory conditions. The continuity equation, the momentum equation and the induction equation were solved numerically by the finite difference method as discretization following the MAC method as solution procedure. On the whole, in the sudden expansion in the direction perpendicular to applied magnetic field, the loss coefficient is estimated to be nearly zero or small. In particular, the loss coefficient becomes negative for small aspect ratios. The reason of negative loss coefficient is attributable to decrease in the induced current just upstream of the expansion. On the other hand, in the sudden expansion in the direction of applied magnetic field, all the cases give positive and large loss coefficients, meaning that the pressure drop through the expansion becomes large. In particular, the loss coefficient becomes considerably large when the Hartmann number increases.     

Development of a potential based code for MHD analysis of LLCB TBM

A two dimensional solver is developed for MHD flows with low magnetic Reynolds’ number based on the electrostatic potential formulation for the Lorentz forces and current densities which will be used to calculate the MHD pressure drop inside the channels of liquid breeder based Test Blanket Modules (TBMs). The flow geometry is assumed to be rectangular and perpendicular to the flow direction, with flow and electrostatic potential variations along the flow direction neglected. A structured, non-uniform, collocated grid is used in the calculation, where the velocity (u), pressure (p) and electrostatic potential (?) are calculated at the cell centers, whereas the current densities are calculated at the cell faces. Special relaxation techniques are employed in calculating the electrostatic potential for ensuring the divergence-free condition for current density. The code is benchmarked over a square channel for various Hartmann numbers up to 25,000 with and without insulation coatings by (i) comparing the pressure drop with the approximate analytical results found in literature and (ii) comparing the pressure drop with the ones obtained in our previous calculations based on the induction formulation for the electromagnetic part. Finally the code is used to determine the MHD pressure drop in case of LLCB TBM. The code gives similar results as obtained by us in our previous calculations based on the induction formulation. However, the convergence is much faster in case of potential based code.     

Numerical analyses on liquid-metal magnetohydrodynamic flow in sudden channel contraction

Three-dimensional numerical calculations have been performed on the magnetohydrodynamic (MHD) flows through a rectangular channel with sudden contraction, particularly in order to estimate the pressure drop through the sudden contraction. The sudden contraction is in the directions both perpendicular and parallel to the applied magnetic field. The Hartmann number, the Reynolds number, and the magnetic Reynolds number were set to ～100, ～1000, and ～0.001, respectively, in simulating laboratory conditions. The continuity equation, the momentum equation, and the induction equation were solved numerically. In the sudden contraction in the direction perpendicular to applied magnetic field, the loss coefficient takes a positive value in all the cases performed in this study, contrary to the expectation. This result is in contrast to that in the sudden expansion in the direction perpendicular to applied magnetic field, where the loss coefficient generally takes a negative value due to the MHD effect. In the sudden contraction in the direction of applied magnetic field, the loss coefficient takes a positive and large value in all the cases performed in this study. The loss coefficient generally becomes larger than that in the case of corresponding channel expansion in the direction of applied magnetic field.     

基于格子Boltzmann方法的反应堆核-热-流耦合模拟

反应堆堆芯内部存在多种不同物理场之间的相互作用和反馈,对其准确模拟需要考虑这些物理过程之间的耦合。为了降低堆芯核 热 流耦合模拟的实现难度,消除不同物理场之间的外部插值过程,本文构建了核 热 流耦合模拟的格子Boltzmann方法（LBM）,将中子输运（包括SN方程、SP3方程以及扩散方程）、考虑燃料流动效应的缓发中子先驱核守恒方程以及流动传热方程统一到相似的LBM格式下,采用统一的LBM碰撞 迁移过程进行求解,有效降低了堆芯多物理耦合模拟的实现难度。计算结果表明：本文建立的核 热 流耦合LBM模型对不同雷诺数下的流动效应均能准确模拟,同时温度反馈在高温熔盐堆低速流动条件下有较为明显的影响,不能忽略;提高堆芯熔盐流速能够有效地展平功率及温度分布。     

MHD Stability Analysis and Flow Controls of Liquid Metal Free Surface Film Flows as Fusion Reactor PFCs

Numerical and experimental investigation results on the magnetohydrodynamics(MHD) film flows along flat and curved bottom surfaces are summarized in this study. A simplified modeling has been developed to study the liquid metal MHD film state, which has been validated by the existing experimental results. Numerical results on how the inlet velocity(V), the chute width(W) and the inlet film thickness(d0) affect the MHD film flow state are obtained. MHD stability analysis results are also provided in this study. The results show that strong magnetic fields make the stable V decrease several times compared to the case with no magnetic field,especially small radial magnetic fields(Bn) will have a significant impact on the MHD film flow state. Based on the above numerical and MHD stability analysis results flow control methods are proposed for flat and curved MHD film flows. For curved film flow we firstly proposed a new multi-layers MHD film flow system with a solid metal mesh to get the stable MHD film flows along the curved bottom surface. Experiments on flat and curved MHD film flows are also carried out and some firstly observed results are achieved.     

Simulation of Bubble Motion under Gravity by Lattice Boltzmann Method

We describe the numerical simulation results of bubble motion under gravity by the lattice Boltzmann method(LBM), which assumes that a fluid consists of mesoscopic fluid particles repeating collision and translation and a multiphase interface is reproduced in a self-organizing way by repulsive interaction between different kinds of particles. The purposes in this study are to examine the applicability of LBM to the numerical analysis of bubble motions, and to develop a three-dimensional version of the binary fluid model that introduces a free energy function. We included the buoyancy terms due to the density difference in the lattice Boltzmann equations, and simulated single- and two-bubble motions, setting flow conditions according to the Eötvös and Morton numbers. The two-dimensional results by LBM agree with those by the Volume of Fluid method based on the Navier-Stokes equations. The three-dimensional model possesses the surface tension satisfying the Laplace's law, and reproduces the motion of single bubble and the two- bubble interaction of their approach and coalescence in circular tube. These results prove that the buoyancy terms and the 3D model proposed here are suitable, and that LBM is useful for the numerical analysis of bubble motion under gravity.     

Laminar pipe flow at the entrance into transverse magnetic field

High-resolution numerical simulations are conducted to analyze transformation of a liquid metal flow in a pipe at the entrance into a transverse magnetic field. The case of laminar flow, perfectly insulating pipe walls, and Hartmann number up to 200 is considered. The simulations reveal detailed structure of velocity and electric current fields and distribution of forces with particular attention given to the flow with an M-shaped velocity profile. They also establish criteria for accurate computations of laminar magnetohydrodynamic flows in strong non-uniform magnetic fields.     

3D thermo-fluid MHD simulation of single straight channel flow in LLCB TBM

India is developing lead lithium cooled ceramic breeder (LLCB) blanket for its DEMO fusion reactor. The mock-up blanket (TBM), using this concept, will be tested in ITER for its tritium breeding and high-grade heat extraction efficiency. In this TBM, pressurized helium is used to remove the heat from first wall, top and bottom plates of TBM. The Pb–Li is used to extract heat from the breeder zones. The flow of Pb–Li with average velocity 0.1 m/s inside the channel can be significantly modified due to MHD effects, which arise because of the presence of strong toroidal magnetic field. A numerical approach is established to capture this flow modification at higher Hartmann numbers (≥20,000). As a validation part of the developed code, MHD phenomenon is studied in 2-D square geometry and numerically obtained velocity profile is compared with available Hunt's analytical results. Thermo-fluid MHD analysis using this code, has been carried out for single rectangular duct of LLCB TBM. The heat transfer has been studied by keeping hot breeders at both sides of the flow channel. The results suggest modification in steady state MHD velocity profile as the liquid flows along the flow length. However, the temperature in various zone remains well within the maximum allowable limit.     

Three-Dimensional Numerical Calculations on Liquid-Metal Magneto-hydrodynamic Flow through Circular Pipe in Magnetic-Field Inlet-Region

Three-dimensional numerical calculations have been performed on liquid-metal magnetohydrodynamic (MHD) flows through a circular pipe in the inlet region of the applied magnetic field, including a sufficient calculation region upstream in the magnetic field section. The continuity equation, the momentum equation including the Lorentz force term, and the induction equation derived from basic equations in the electromagnetism have been solved numerically. Along the flow axis (i.e., the channel axis), the pressure decreases slightly as a normal non-MHD flow, increases once, thereafter, decreases sharply, and finally decreases as a fully-developed MHD flow. The sharp decrease in the pressure, resulting in a large pressure drop in the inlet region is due to the increase in the induced electric current in this region compared with that in the fully-developed region. The velocity distribution changes from a parabolic profile of a laminar non-MHD flow to a profile with peaks near the walls parallel to the magnetic field, and finally to a flat profile of a fully-developed MHD flow.     

An Eulerian MHD model for the analysis of magnetic flux compression by expanding diamagnetic fusion plasma sphere

The process of magnetic flux compression (MFC) inside a solenoid by expanding diamagnetic plasma sphere produced by an inertial fusion micro-explosions and its application as a direct energy conversion scheme to convert a part of plasma kinetic energy into pulsed electrical energy has been recently reported [1]. For a detailed analysis of this concept, an Eulerian multi-material MHD model is developed using magnetic vector potential formulation for electro-magnetic field calculations and classical volume-of-fluid method for material interface tracking. The diffusion term in the magnetic induction equation is solved implicitly while the advection terms are computed using a second-order MUSCL scheme. An iteration procedure using ADI scheme is used for the free space field calculation. In this paper, we describe the details of the new MHD model, its validation against the semi-analytical solutions (for magnetic Reynolds number ?1) of magnetic convective-diffusion equations and application to explore the concept of MFC by expanding plasma sphere. The simulation results show that the algorithm is capable of handling complex plasma dynamics inside the MFC system. Also, the results indicate the development and the evolution of MRT like instability near the stagnation point. The magnetic field diffusion into the plasma during the expansion phase is found to be negligible.     

Free convection effects on MHD Stokes problem for a vertical plate

An exact analysis is made of the effects of mass transfer and free convection currents on MHD Stokes' (Rayleigh's) problem for the flow of an electrically conducting, incompressible, viscous fluid past an impulsively started vertical plate, under the action of a transversely applied magnetic field. The heat due to viscous and Joule dissipation and the induced magnetic field is neglected. During the course of the discussion, the effects of heating (Gr < 0, Gr = Grashof number) or cooling (Gr > 0) of the plate by the free convection currents, Gr_m (modified Grashof number), Sc (Schmidt number) and M (Hartmann number) and the velocity and skin-friction are studied.     

Eurofer corrosion by the flow of the eutectic alloy Pb-Li in the presence of a strong magnetic field

A new investigation of the Eurofer 97 corrosion by the MHD flow of the liquid eutectic alloy Pb-17Li is presented. The experimental data previously obtained in Riga are confirmed and an attempt to model this phenomenon is presented. The model is based on a thermodynamic analysis of the dissolution and electro-dissolution mechanisms, leading to a relevant boundary condition at the liquid-solid interface. Then, analyzing the MHD flow, guiding ideas and scaling laws are derived for the dissolution rate of the Hartmann wall. The results obtained in the regime, where the solid wall is assumed to remain planar, allow determining a plausible value for an important non-dimensional number, the dissolution number Di. A linear analysis leads to predictions on the mechanism responsible for the formation of streaks imbedded within the Hartmann layer and associated with the wall shape disturbance, as well as for the selection of the unstable mode. It is found that this mechanism is related to an additional contribution due to the electric current, based on an electro-dissolution number Ed.     

Effects of mass transfer and free convection currents on MHD Stokes' problem for a vertical plate

An exact analysis is made of the effects of mass transfer and free convection currents on MHD Stokes' (Rayleigh's) problem for the flow of an electrically conducting, incompressible, viscous fluid past an impulsively started vertical plate, under the action of a transversely applied magnetic field. The heat due to viscous and Joule dissipation and the induced magnetic field is neglected. During the course of the discussion, the effects of heating (Gr < 0, Gr = Grashof number) or cooling (Gr > 0) of the plate by the free convection currents, Gr_m (modified Grashof number), Sc (Schmidt number) and M (Hartmann number) and the velocity and skin-friction are studied.