倾斜非均匀磁场下导电方管磁流体管流的数值模拟研究

热核聚变反应堆液态金属包层应用中的一个重要问题是液态金属在导电管中流动和强磁场相互作用产生的额外的磁流体动力学压降.这种磁流体动力学压降远远大于普通水力学压降.美国阿贡国家实验室ALEX研究小组,对非均匀磁场下导电管中液态金属磁流体动力学效应进行了实验研究,其实验结果成为液态金属包层数值验证的标准模型之一.液态金属包层在应用中会受到不同方向的磁场作用,本文以ALEX的非均匀磁场下导电方管中液态金属管流实验中的一组参数为基础,保持哈特曼数、雷诺数和壁面电导率不变,采用三维直接数值模拟的方法,研究了外加磁场与侧壁之间的倾角对导电方管内液态金属流动的速度、电流和压降分布的影响.研究结果表明:沿流向相同横截面上的速度、电流以及压力分布均随磁场的倾斜而同向旋转.倾斜磁场均匀段,横截面上的高速区位于平行磁场方向的哈特曼层和平行层交叉位置,压力梯度随磁场倾角的增大先增大后减小.倾斜磁场递减段,在三维磁流体动力学效应作用下,横截面上的高速射流位置向垂直磁场方向偏移.磁场递减段的三维磁流体动力学压降随磁场倾角的增大而增大.随磁场倾斜,截面上的射流峰值逐渐减小,二次流增强,引发层流向湍流的转捩.      

The effect of magnetic field on two-phase liquid metal flow

In this paper, the basic equations of two-phase liquid metal flow in a magnetic field are derived, and specifically, two-phase liquid metal MHD flow in a rectangular channel is studied, and the expressions of velocity distribution of liquid and gas phases and the ratioK ₀ of the pressure drop in two-phase MHD flow to that in single-phase are derived. Results of calculation show that the ratioK ₀ is smaller than unity and decreases with increasing void fraction and Hartmann number because the effective electrical conductivity in the two-phase case decreases. The Project is supported by the National Natural Science Foundation of China.     

Velocity measurements in liquid sodium by means of ultrasound Doppler velocimetry

A successful application of ultrasound Doppler velocimetry in liquid sodium flows is described. To obtain sufficient Doppler signals, different problems had to be solved: the transmission of the ultrasonic beam through the channel wall made of stainless steel, the acoustic coupling between the transducer and the channel wall, and the wetting of the inner surface of the wall by the liquid metal, respectively. A sodium flow in a square duct exposed to a transverse magnetic field is investigated. In accordance with the existing knowledge about MHD channel flows, we found that the velocity profiles modified to a M-shape owing to the effect of an inhomogeneous magnetic field. Received: 12 June 2001/Accepted: 27 October 2001     

Theory of a magnetohydrodynamic flow regulator for liquid metal

A theory is developed for unsteady flow of liquid metal in a de MHD flow regulator channel for small magnetic Reynolds numbers. It is shown that it is possible to use the quasistationary approximation for calculating the integral flow parameters.MHD methods, along with their applications in direct transformation of thermal energy into electrical energy, in recent years have been ever more widely used in nuclear energy and metallurgy for transporting and for measuring the parameters of liquid metal flows [1, 2]. Complete sealing, operation over a wide temperature range, and simplicity of control and automation are among the unquestionable advantages of all MHD devices.The existing theory of MHD devices is limited primarily to the stationary regimes of operation of electromagnetic pumps and magnetic flow-meters.The most characteristic operating regimes for the MHD regulator are the transient regimes, which are defined primarily by the hydrodynamics of the liquid metal flow.The present article is devoted to the study of the unsteady flow of liquid metal in the channel of a dc MHD flow regulator with independent excitation. The magnetic Reynolds are low (R _m =U ₀ b1).     

Validation of numerical solvers for liquid metal flow in a complex geometry in the presence of a strong magnetic field

Following the magnetohydrodynamic (MHD) code validation and verification proposal by Smolentsev et al. (Fusion Eng Des 100:65–72, 2015), we perform code to code and code to experiment comparisons between two computational solvers, FLUIDYN and HIMAG, which are presently considered as two of the prospective CFD tools for fusion blanket applications. In such applications, an electrically conducting breeder/coolant circulates in the blanket ducts in the presence of a strong plasma-confining magnetic field at high Hartmann numbers, \(\textit{Ha}\) (\(\textit{Ha}^2\) is the ratio between electromagnetic and viscous forces) and high interaction parameters, \(\textit{N}\) (\(\textit{N}\) is the ratio of electromagnetic to inertial forces). The main objective of this paper is to provide the scientific and engineering community with common references to assist fusion researchers in the selection of adequate computational means to be used for blanket design and analysis. As an initial validation case, the two codes are applied to the classic problem of a laminar fully developed MHD flows in a rectangular duct. Both codes demonstrate a very good agreement with the analytical solution for \(\textit{Ha}\) up to 15, 000. To address the capabilities of the two codes to properly resolve complex geometry flows, we consider a case of three-dimensional developing MHD flow in a geometry comprising of a series of interconnected electrically conducting rectangular ducts. The computed electric potential distributions for two flows (Case A) \(\textit{Ha}=515\), \(\textit{N}=3.2\) and (Case B) \(\textit{Ha}=2059\), \(\textit{N}=63.8\) are in very good agreement with the experimental data, while the comparisons for the MHD pressure drop are still unsatisfactory. To better interpret the observed differences, the obtained numerical data are analyzed against earlier theoretical and experimental studies for flows that involve changes in the relative orientation between the flow and the magnetic field.     

磁约束核聚变关键能量转换部件的磁流体力学研究

简述了磁约束热核聚变反应堆关键能量转换部件包层相关的流体力学及传热学研究背景, 重点介绍了强磁场作用下的金属流体流动与传热问题的实验及数值研究进展.     

Simulation of operation modes of a centrifugal conductive magnetohydrodynamic pump

A mathematical model of a centrifugal conductive magnetohydrodynamic (MHD) pump that calculates the distributions of velocity, current density, and pressure along the channel is developed. The viscous forces in the original system of MHD equations are taken into account on the basis of the known square law of the drag for a turbulent flow in a pipe, generalized for the case of plane flows in a channel. Dependences of the drag coefficient on the main governing parameters (metal flow rate, current intensity, and intensity of magnetic induction), which provide the agreement of the calculated and experimental data on the pressure at the pump outlet for different operation modes, are obtained. It is shown that these dependences have a universal character and the proposed model can be used to design pumps of this type and to manage their operation in production industry.     

流向磁场作用下圆柱绕流的直接数值模拟

绕流是托卡马克装置中液态包层内常见的流动形态,对流场与热量分布有着重要的影响.本文通过直接数值模拟(DNS),研究了不同磁场强度下$Re=3900$的圆柱绕流,分析了磁场强度对于湍流尾迹的影响.无磁场情况下,直接数值模拟的结果与前人的实验及模拟结果吻合很好.圆柱下游的尾迹中,随着流向距离的增大, 流向速度剖面逐渐从U型进化呈V型, 并慢慢趋于平缓,这表明尾迹中的流动结构受圆柱影响逐渐减小.圆柱后方两侧的剪切层中,由于Kelvin-Helmholtz不稳定性的影响,可以清晰地看到小尺度剪切层涡的脱落.通过对无磁场的计算结果施加流向磁场,本文计算了哈特曼数($Ha$)分别为20, 40和80的工况,以研究磁场效应对于湍流的影响.结果表明磁场较弱时,流动依然呈三维湍流状态.随着磁场增强, 近圆柱尾流区受磁场抑制明显,回流区被拉长,剪切层失稳位置向下游转移.圆柱后方的涡结构由于受到竖直方向洛伦兹力的挤压作用,随着哈特曼数的增加尾迹区域逐渐变窄.相比于无磁场情况的涡结构,由于磁场的耗散作用,相应的涡结构尺度变小.该研究不仅扩展了现有磁场下湍流运动的参数范围,对于液态包层的设计及安全运行同样具有重要的理论指导意义和工程应用价值.      

Predictions of axisymmetric and two-dimensional impinging turbulent jets

The k − turbulence model and a version of a second-moment closure, modified to include the effect of pressure reflections from a solid surface, have been used as the basis of predictions of the flow that results from the orthogonal impingement of circular and two-dimensional (2-D) jets on a flat surface. Comparison of model predictions has been made with velocity measurements obtained in the stagnation and wall jet regions of the impinging flows. Results, in general, confirm the superiority of the Reynolds stress transport equation model for predicting mean and fluctuating velocities within the latter regions of such flows. In particular, modifications to the second-moment closure to account for the influence of the surface in distorting the fluctuating pressure field away from the wall successfully predict the damping of normal-to-wall velocity fluctuations throughout the impinging flows. In contrast, results derived from the eddy-viscosity-based approach do not, in general, accurately reproduce experimental observations.     

Steady flow of a liquid metal in a rectangular MHD channel

The known experimental studies of steady flows of a liquid metal in magnetohydrodynamic (MHD) channels of rectangular section [1–4] were performed only for a few values of the Reynolds number, which does not permit a clear delineation of the fundamental governing laws of the flow in the zone of transition from laminar to turbulent flow. In addition, the study of turbulent MHD flows has been limited to two-dimensional channels.Below we present some results of experimental studies of the effect of a transverse magnetic field on the resistance coefficient for mercury flow in an MHD channel with side ratio 1 to 2.5. The choice of a channel with this side ratio was dictated by the need for studies of the intermediate case between flows in two-dimensional and square channels, which differ significantly from one another because of the different effect of the walls parallel to the magnetic field. In our studies, for each value of the Hartmann number the investigations were made for 30–50 values of the Reynolds number.Notation B₀ flux density of the applied magnetic field - M Hartmann number - R Reynolds number - _tm resistance factor of turbulent MHD flow - _* critical value of the resistance factor - geometric parameter of channel - the component of resistance factor in ordinary hydrodynamics due to pulsations - normed function - electric conductivity of metal - viscosity of metal - R₀ shydraulic radius - N smagnetic field parameter     

DIRECT NUMERICAL SIMULATIONS ON THE TURBULENT FLOW PAST A CONFINED CIRCULAR CYLINDER WITH THE INFLUENCE OF THE STREAMWISE MAGNETIC FIELDS1)

绕流是托卡马克装置中液态包层内常见的流动形态,对流场与热量分布有着重要的影响.本文通过直接数值模拟(DNS),研究了不同磁场强度下$Re=3900$的圆柱绕流,分析了磁场强度对于湍流尾迹的影响.无磁场情况下,直接数值模拟的结果与前人的实验及模拟结果吻合很好.圆柱下游的尾迹中,随着流向距离的增大, 流向速度剖面逐渐从U型进化呈V型, 并慢慢趋于平缓,这表明尾迹中的流动结构受圆柱影响逐渐减小.圆柱后方两侧的剪切层中,由于Kelvin-Helmholtz不稳定性的影响,可以清晰地看到小尺度剪切层涡的脱落.通过对无磁场的计算结果施加流向磁场,本文计算了哈特曼数($Ha$)分别为20, 40和80的工况,以研究磁场效应对于湍流的影响.结果表明磁场较弱时,流动依然呈三维湍流状态.随着磁场增强, 近圆柱尾流区受磁场抑制明显,回流区被拉长,剪切层失稳位置向下游转移.圆柱后方的涡结构由于受到竖直方向洛伦兹力的挤压作用,随着哈特曼数的增加尾迹区域逐渐变窄.相比于无磁场情况的涡结构,由于磁场的耗散作用,相应的涡结构尺度变小.该研究不仅扩展了现有磁场下湍流运动的参数范围,对于液态包层的设计及安全运行同样具有重要的理论指导意义和工程应用价值.     

Non equilibrium gas-liquid flows in variable cross section ducts

The behaviour of two-phase high velocity flows in variable cross section ducts was investigated using a one-dimensional numerical model developed for the study of the annular flow configuration. Heat, mass, and momentum transfer between the phases during the flow were considered. The validation of the calculation procedure was made with some experimental data for the air-water couple, while the main application concerned the evaluation of momentum transfer from an expanding gas to an entrained liquid stream in droplet form. A liquid metal-gas flow was considered to simulate the process taking place in a plant where electrical power is generated by a liquid metal flowing in a magnetic field (MHD). The effectiveness of energy and momentum transfer between the liquid and the gas phase during the expansion was evaluated and the influence of nozzles with different convergence angles was investigated.     

Wall functions for numerical modeling of laminar MHD flows

A general wall function treatment is presented for the numerical modeling of laminar magnetohydrodynamic (MHD) flows. The wall function expressions are derived analytically from the steady-state momentum and electric potential equations, making use only of local variables of the numerical solution. No assumptions are made regarding the orientation of the magnetic field relative to the wall, nor of the magnitude of the Hartmann number, or the wall conductivity. The wall functions are used for defining implicit boundary conditions for velocity and electric potential, and for computing mass flow and electrical currents in near wall-cells. The wall function treatment was validated in a finite volume formulation, and compared with an analytic solution for a fully developed channel flow in a transverse magnetic field. For the case with insulating walls, a uniform 20×20 grid, and Hartmann numbers Ha={10,30,100}, the accuracy of pressure drop and wall shear stress predictions was {1.1%,1.6%,0.5%}, respectively. Comparable results were obtained also with conducting Hartmann walls. The accuracy of predicted pressure drop and wall shear stress was essentially independent of the resolution of the Hartmann layers. When applied also to the parallel walls, the wall functions reduced the errors by a factor two to three. The wall functions can be implemented in any general flow solver, to allow accurate predictions at reasonable cost even for complex geometries and nonuniform magnetic fields.     

Confined viscoplastic flows with heterogeneous wall slip

The steady, pressure-driven flow of a Herschel-Bulkley fluid in a microchannel is considered, assuming that different power-law slip equations apply at the two walls due to slip heterogeneities, allowing the velocity profile to be asymmetric. Three different flow regimes are observed as the pressure gradient is increased. Below a first critical pressure gradient G ₁, the fluid moves unyielded with a uniform velocity, and thus, the two slip velocities are equal. In an intermediate regime between G ₁ and a second critical pressure gradient G ₂, the fluid yields in a zone near the weak-slip wall and flows with uniform velocity near the stronger-slip wall. Beyond this regime, the fluid yields near both walls and the velocity are uniform only in the central unyielded core. It is demonstrated that the central unyielded region tends towards the midplane only if the power-law exponent is less than unity; otherwise, this region rends towards the weak-slip wall and asymmetry is enhanced. The extension of the different flow regimes depends on the channel gap; in particular, the intermediate asymmetric flow regime dominates when the gap becomes smaller than a characteristic length which incorporates the wall slip coefficients and the fluid properties. The theoretical results compare well with available experimental data on soft glassy suspensions. These results open new routes in manipulating the flow of viscoplastic materials in applications where the flow behavior depends not only on the bulk rheology of the material but also on the wall properties.     

3-D numerical modelling of greenwater loading on fixed ship-shaped FPSOs

This work is focused on the assessment of greenwater overtopping onto fixed ship-shaped FPSO models using three dimensional (3-D) Computational Fluid Dynamics (CFD) simulations. Good agreement between the numerical results and published experimental data from Barcellona et al. (2003) indicates that 3-D CFD is an effective tool which may be used to assess greenwater associated with an incident wave group. Different wall-sided bow shapes are investigated numerically, and the results provide insight into how the bow shape influences the evolution of on-deck flows, and the horizontal force on a vertical wall spanning the full deck width for incident waves approaching normal to the bow. It is found that the horizontal force, thus, the horizontal momentum of greenwater flows, is a result of the combined action of water-front velocities and the corresponding water volume impacting on the structures. For the bow shapes considered in this study, although clear differences in on-deck flow are observed for different bow shape, differences in force on the vertical wall are relatively small. Simulations for a 2-D vertical rectangular box that has the same longitudinal section as the 3-D model FPSOs give somewhat similar results to 3-D predictions along the centre-line. This similarity may be due partly to the fact that increases in on-deck flow velocity due to increased freeboard exceedance in the 2-D simulations compensates for a lack of focusing of the on-deck flow observed in the 3-D simulations. This finding has implications for understanding how computationally cheaper 2-D greenwater simulations relate to more realistic 3-D greenwater events.     

On the stability of certain nonparallel flows of a viscous incompressible liquid in a channel

The stability of very simple nonparallel flows of a viscous incompressible liquid in an infinite plane channel described by the exact solutions of the Navier-Stokes equations is studied. Such solutions are realized between two parallel porous plates when the liquid (or gas) is forced in at one wall and drawn out at the same velocity at the other, with a steady flow of liquid along the channel. In this case the transverse velocity component is constant, and the profile of the longitudinal velocity component is independent of the longitudinal con-ordinate x, being an asymmetric function of the transverse coordinate y. A study of the hydrodynamic stability then reduces to the solution of an equation differing from the Orr-Sommerfeld equation by virtue of the presence of additional terms containing the transverse velocity component of the main flow. By numerically solving both this equation and the ordinary Orr-Sommerfeld equation and comparing the corresponding results for various inflowing Reynolds numbers R₀=v₀h/ (v₀ is the inflow velocity, h is the width of the channel), the effect of the nonparallel and asymmetrical nature of such flows on their stability is discussed.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 125–129, July–August, 1970.     

Analysis on creeping channel flows of compressible fluids subject to wall slip

Creeping channel flows of compressible fluids subject to wall slip are widely encountered in industries. This paper analyzes such flows driven by pressure in planar as well as circular channels. The analysis elucidates unsteady flows of Newtonian fluids subject to the Navier slip condition, followed by steady flows of viscoplastic fluids, in particular, Herschel–Bulkley fluids and their simplifications including power law and Newtonian fluids, that slip at wall with a constant coefficient or a coefficient inversely proportional to pressure. Under the lubrication assumption, analytical solutions are derived, validated, and discussed over a wide range of parameters. Analysis based on the derived solutions indicates that unsteadiness alters cross-section velocity profiles. It is demonstrated that compressibility of the fluids gives rise to a concave pressure distribution in the longitudinal direction, whereas wall slip with a slip coefficient that is inversely proportional to pressure leads to a convex pressure distribution. Energy dissipation resulting from slippage can be a significant portion in the total dissipation of such a flow. A distinctive feature of the flow is that, in case of the pressure-dependent slip coefficient, the slip velocity increases rapidly in the flow direction and the flow can evolve into a pure plug flow at the exit.     

Analytical solution of magnetohydrodynamic sink flow

An analytical solution to the famous Falkner-Skan equation for the magnetohydrodynamic (MHD) flow is obtained for a special case, namely, the sink flow with a velocity power index of −1. The solution is given in a closed form. Multiple solution branches are obtained. The effects of the magnetic parameter and the wall stretching parameter are analyzed. Interesting velocity profiles are observed with reversal flow regions even for a stationary wall. These solutions provide a rare case of the Falkner-Skan MHD flow with an analytical closed form formula. They greatly enrich the analytical solution for the celebrated Falkner-Skan equation and provide better understanding of this equation.     

Comparison of magnetic resonance concentration measurements in water to temperature measurements in compressible air flows

Magnetic resonance imaging (MRI) measurements in liquid flows provide highly detailed 3D mean velocity and concentration data in complex turbulent mixing flow applications. The scalar transport analogy is applied to infer the mean temperature distribution in high speed gas flows directly from the MRI concentration measurements in liquid. Compressibility effects on turbulent mixing are known to be weak for simple flows at high subsonic Mach number, and it was not known if this would hold in more complex flows characteristic of practical applications. Furthermore, the MRI measurements are often done at lower Reynolds number than the compressible application, although both are generally done in fully turbulent flows. The hypothesis is that the conclusions from MRI measurements performed in water are transferable to high subsonic Mach number applications. The present experiment is designed to compare stagnation temperature measurements in high speed airflow (M = 0.7) to concentration measurements in an identical water flow apparatus. The flow configuration was a low aspect ratio wall jet with a thick splitter plate producing a 3D complex downstream flow mixing the wall-jet fluid with the mainstream flow. The three-dimensional velocity field is documented using magnetic resonance velocimetry in the water experiment, and the mixing is quantified by measuring the mean concentration distribution of wall-jet fluid marked with dissolved copper sulfate. The airflow experiments are operated with a temperature difference between the main stream and the wall jet. Profiles of the stagnation temperature are measured with a shielded thermocouple probe. The results show excellent agreement between normalized temperature and concentration profiles after correction of the temperature measurements for the effects of energy separation. The agreement is within 1 % near the edges of the mixing layer, which suggests that the mixing characteristics of the large scale turbulence structures are the same in the two flows.