Showing papers on "Hartmann number published in 2002"
TL;DR: In this paper, the authors used the finite difference method to predict the characteristics of hydromagnetic double-diffusive convective flow of a binary gas mixture in a rectangular enclosure with the upper and lower walls being insulated.
110 citations
TL;DR: In this article, the tensorial filtration law in rigid porous media for steady-state slow flow of an electrically conducting, incompressible and viscous Newtonian fluid in the presence of a magnetic field is obtained by upscaling the flow at pore scale.
Abstract: The aim of this work is to investigate the tensorial filtration law in rigid porous media for steady-state slow flow of an electrically conducting, incompressible and viscous Newtonian fluid in the presence of a magnetic field. The seepage law under a magnetic field is obtained by upscaling the flow at the pore scale. The macroscopic magnetic field and electric flux are also obtained. We use the method of multiple-scale expansions which gives rigorously the macroscopic behaviour without any preconditions on the form of the macroscopic equations. For finite Hartmann number, i.e. e [Lt ] Ha [Lt ] e−1, and finite load factor, i.e. e [Lt ] [Kscr ] [Lt ] e−1, where e characterizes the separation of scales, the macroscopic mass flow and electric current are coupled and both depend on the macroscopic gradient of pressure and the electric field. The effective coefficients satisfy the Onsager relations. In particular, the filtration law is shown to resemble Darcy's law but with an additional term proportional to the electric field. The permeability tensor, which strongly depends on the magnetic induction, i.e. Hartmann number, is symmetric, positive and satisfies the filtration analogue of the Hall effect.
72 citations
TL;DR: In this paper, the buoyancy-driven magnetohydrodynamic flow in a liquid-metal filled cubic enclosure was investigated by three-dimensional numerical simulation, where the enclosure was differentially heated at two opposite vertical walls, all other walls being adiabatic, and uniform magnetic field was applied orthogonal to the temperature gradient and to the gravity vector.
64 citations
TL;DR: In this paper, a new asymptotic solution for the modified Mathieu's differential equation was proposed, and the associated eigenvalue problem was solved by obtaining explicit forms of eigenfunctions and related expansion coefficients.
60 citations
TL;DR: In this article, an extensive experimental study is carried out to examine the properties of a quasi-two-dimensional MHD turbulent shear flow, which is characterized by an energy transfer towards the large scales, leading to a relatively small number of large coherent structures.
Abstract: An extensive experimental study is carried out to examine the properties of a quasi-two-dimensional MHD turbulent shear flow. Axisymmetric shear of a mercury layer is enforced by the action of a steady vertical magnetic field and a radial horizontal electric current flowing between a ring set of electrodes and a cylindrical wall. This shear layer is unstable, and the properties of the turbulent flow are studied for a wide range of Hartmann (up to 1800) and Reynolds numbers (up to 106). The mean velocity profiles exhibit a turbulent free shear layer, of thickness larger than that predicted by the laminar theory by two orders of magnitude. The profiles yield the expected linear dependence between the total angular momentum and the electric current when the magnetic field is large enough, but demonstrate a systematic deviation when it is moderate (Ha [lsim ] 250). The quasi-two-dimensional turbulence is characterized by an energy transfer towards the large scales, which leads to a relatively small number of large coherent structures. The properties of these structures result from the competition between the energy transfer and the Joule dissipation within the Hartmann layers. In the intermediate range of wavenumbers (k[lscr ] < k < ki, where k[lscr ] is the integral-length-scale wavenumber and ki the injection wavenumber), the energy spectra exhibit a power law close to k−5/3 when the Joule dissipation is weak and close to k−3 when it is significant. The properties of the turbulent flow in this latter regime depend on only one non-dimensional parameter, the ratio (Ha/Re)(l⊥/h)2 (Ha is the Hartmann number, Re the Reynolds number based on the cell radius, l⊥ a typical transverse scale, and h the layer width).
57 citations
TL;DR: In this article, the problem of hydromagnetic double-diffusive convective flow of a binary gas mixture in a rectangular enclosure with the upper and lower walls being insulated is solved numerically by the finite-difference methodology.
56 citations
TL;DR: In this article, the authors investigated the buoyant convection in a long vertical rectangular enclosure under the presence of a horizontal uniform magnetic field, where two opposite vertical walls are kept at different temperatures and the other four walls are thermally insulating.
Abstract: The buoyant convection in a long vertical rectangular enclosure under the presence of a horizontal uniform magnetic field is investigated. Two opposite vertical walls are kept at different temperatures and the other four walls are thermally insulating. The main parameters of this study are the Grashoff number Gr, and the Hartmann number Ha. The applied magnetic field is either perpendicular or parallel to the temperature gradient. In both configurations, if it is large enough, it damps out the buoyant flow and ensures a conductive heat flux, but with different scaling laws: Gr/Ha for the perpendicular case, Gr/Ha 2 for the parallel case. This first paper focuses on the fully-established flow, far from the top and the bottom of the enclosure, in the limit of high Ha. Analytical solutions are derived and serve as reference to validate the numerical results. An analytical model of the Hartmann layers describes directly their influence on the core flow, saves important computational resources and yields quite accurate results.
51 citations
TL;DR: In this paper, the authors studied the MHD Couette flow in a rotating environment with non-conducting walls in the presence of an arbitrary magnetic field, and 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 and the Hall current parameter.
Abstract: 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.
47 citations
TL;DR: In this article, the authors considered axisymmetric magnetohydrodynamic motion in a spherical shell driven by rotating the inner boundary relative to the stationary outer boundary of a Couette flow.
Abstract: We consider axisymmetric magnetohydrodynamic motion in a spherical shell driven by rotating the inner boundary relative to the stationary outer boundary – spherical Couette flow. The inner solid sphere is rigid with the same electrical conductivity as the surrounding fluid; the outer rigid boundary is an insulator. A force-free dipole magnetic field is maintained by a dipole source at the centre. For strong imposed fields (as measured by the Hartmann number M), the numerical simulations of Dormy et al. (1998) showed that a super-rotating shear layer (with angular velocity about 50% above the angular velocity of the inner core) is attached to the magnetic field line [Cscr ] tangent to the outer boundary at the equatorial plane of symmetry. At large M, we obtain analytically the mainstream solution valid outside all boundary layers by application of Hartmann jump conditions across the inner- and outer-sphere boundary layers. We formulate the large-M boundary layer problem for the free shear layer of width M−1/2 containing [Cscr ] and solve it numerically. The super-rotation can be understood in terms of the nature of the meridional electric current flow in the shear layer, which is fed by the outer-sphere Hartmann layer. Importantly, a large fraction of the current entering the shear layer is tightly focused and effectively released from a point source at the equator triggered by the tangency of the [Cscr ]-line. The current injected by the source follows the [Cscr ]-line closely but spreads laterally due to diffusion. In consequence, a strong azimuthal Lorentz force is produced, which takes opposite signs either side of the [Cscr ]-line; order-unity super-rotation results on the equatorial side. In fact, the point source is the small equatorial Hartmann layer of radial width M−2/3 ([Lt ]M−1/2) and latitudinal extent M−1/3. We construct its analytic solution and so determine an inward displacement width O(M−2/3) of the free shear layer. We compare our numerical solution of the free shear layer problem with our numerical solution of the full governing equations for M in excess of 104. We obtain excellent agreement. Some of our more testing comparisons are significantly improved by incorporating the shear layer displacement caused by the equatorial Hartmann layer.
46 citations
TL;DR: In this article, the effects of radiation heat transfer on mixed convection through a vertical channel in the presence of transverse magnetic field were examined analytically and the entropy generation characteristics and its dependency on the various dimensionless parameters, i.e., Hartmann number (Ha), Plank number (Pl), Richardson number (Ri), group parameter (Br/Π) etc.
40 citations
TL;DR: In this paper, the steady-state fully developed magnetohydrodynamic (MHD) flow of a conducting fluid through a channel with arbitrary wall conductivity in the presence of a transverse external magnetic field with various inclined angles is considered.
Abstract: In this paper, we consider the problem of the steady-state fully developed magnetohydrodynamic (MHD) flow of a conducting fluid through a channel with arbitrary wall conductivity in the presence of a transverse external magnetic field with various inclined angles. The coupled governing equations for both axial velocity and induced magnetic field are firstly transformed into decoupled Poisson-type equations with coupled boundary conditions. Then the dual reciprocity boundary element method (DRBEM) [20] is used to solve the Poisson-type equations. As testing examples, flows in channels of three different crosssections, rectangular, circular and triangular, are calculated. It is shown that solutions obtained by the DRBEM with constant elements are accurate for Hartmann number up to 8 and for large conductivity parameters comparing to exact solutions and solutions by the finite element method (FEM).
TL;DR: The linear marginal instability of an axisymmetric magnetohydrodynamics Taylor-Couette flow of infinite vertical extension is considered and those vertical wave numbers for which the characteristic Reynolds number is minimum are considered.
Abstract: The linear marginal instability of an axisymmetric magnetohydrodynamics Taylor-Couette flow of infinite vertical extension is considered. We are only interested in those vertical wave numbers for which the characteristic Reynolds number is minimum. For hydrodynamically unstable flows minimum Reynolds numbers exist even without a magnetic field, but there are also solutions with smaller characteristic Reynolds numbers for certain weak magnetic fields. The magnetic field, therefore, destabilizes the rotating flow by the so-called magnetorotational instability (MRI). The MRI, however, can only exist for hydrodynamically unstable flow if the magnetic Prandtl number, Pr, is not too small. For too small magnetic Prandtl numbers (and too strong magnetic fields) only the well-known magnetic suppression of the Taylor-Couette instability can be found. The MRI is even more pronounced for hydrodynamically stable flows. In this case we can always find a magnetic field amplitude where the characteristic Reynolds number is minimum. These critical values are computed for different magnetic Prandtl numbers and for three types of geometry (small, medium, and wide gaps between the rotating cylinders). In all cases the minimum Reynolds numbers are running with 1/Pr for small enough Pr so that the critical Reynolds numbers may easily exceed values of ${10}^{6}$ for the magnetic Prandtl number of sodium ${(10}^{\ensuremath{-}5})$ or gallium ${(10}^{\ensuremath{-}6}).$ The container walls are considered either electrically conducting or insulating. For insulating walls with small and medium-size gaps between the cylinders (i) the critical Reynolds number is smaller, (ii) the critical Hartmann number is higher, and (iii) the Taylor vortices are longer in the direction of the rotation axis. For wider gaps the differences in the results between both sets of boundary conditions become smaller and smaller.
TL;DR: In this paper, two linear stability analyses for an electrically conducting liquid contained in a vertical cylinder with a thermally insulated vertical wall and with isothermal top and bottom walls are presented.
Abstract: This paper presents two linear stability analyses for an electrically conducting liquid contained in a vertical cylinder with a thermally insulated vertical wall and with isothermal top and bottom walls. There is a steady uniform vertical magnetic field. The first linear stability analysis involves a hybrid approach which combines an analytical solution for the Hartmann layers adjacent to the top and bottom walls with a numerical solution for the rest of the liquid domain. The second linear stability analysis involves an asymptotic solution for large values of the Hartmann number. Numerically accurate predictions of the critical Rayleigh number can be obtained for Hartmann numbers from zero to infinity with the two solutions presented here and a previous numerical solution which gives accurate results for small values of the Hartmann number.
TL;DR: In this article, the behavior of an insulating particle suspended in a liquid metal and subject to the influence of locally uniform electric and magnetic fields (E, B) is considered, and the general form of these bilinear relationships is determined, with particular attention to three types of particle symmetry: (i) isotropy; (ii) axisymmetry; and (iii) orthotropy.
Abstract: The behaviour of an insulating particle suspended in a liquid metal and subject to the influence of locally uniform electric and magnetic fields (E, B) is considered. The electric field drives a current J which is perturbed by the presence of the particle, and the resulting Lorentz force drives a flow. It is assumed that both the Reynolds number and the Hartmann number based on particle size are small. If the particle is fixed, it experiences a force and couple that are each bilinear in J and B; if it is freely suspended, then it moves with translational velocity U and angular velocity Q each similarly bilinear in J and B. The general form of these bilinear relationships is determined, with particular attention to three types of particle symmetry: (i) isotropy; (ii) axisymmetry; and (iii) orthotropy
TL;DR: In this article, the stabilizing effects of a constant horizontal magnetic field on the flow in a heated planar liquid metal layer were investigated, and the results revealed a similar behaviour in both cases: small variation of the thresholds (critical Grashof number, Gr c ) and of the wavelengths at small Ha ( Ha is the Hartmann number proportional to the intensity of the magnetic field) before an increase leading to asymptotic linear variations at large Ha.
TL;DR: In this article, the buoyancy-driven magnetohydrodynamic flow in a cubic enclosure was investigated by three-dimensional numerical simulation, and the effects of increasing the Hartmann number on the complex threedimensional flow and current pattern were laid on the effects.
Abstract: The buoyancy‐driven magnetohydrodynamic flow in a cubic enclosure was investigated by three‐dimensional numerical simulation. The enclosure was volumetrically heated by a uniform power density and cooled along two opposite vertical walls, all remaining walls being adiabatic. A uniform magnetic field was applied orthogonally to the gravity vector and to the temperature gradient. The Prandtl number was 0.0321 (characteristic of Pb–17Li at 300°C), the Rayleigh number was 104, and the Hartmann number was made to vary between 0 and 2×103. The steady‐state Navier–Stokes equations, in conjunction with a scalar transport equation for the fluid's enthalpy and with the Poisson equation for the electrical potential, were solved by a finite volume method using a purposely modified CFD code and a computational grid with 643 nodes in the fluid. Emphasis was laid on the effects of increasing the Hartmann number on the complex three‐dimensional flow and current pattern.
TL;DR: In this article, the authors considered convective contamination of self-diffusion experiments with an applied magnetic field using a two-dimensional axisymmetric model, where constant, uniform, and an additional non-uniform heat fluxes are imposed along the sidewall of the cylinder while constant heat loss occurs through the top and bottom.
Abstract: Convective contamination of self-diffusion experiments with an applied magnetic field is considered using a two-dimensional axisymmetric model. Constant, uniform, and an additional non-uniform heat fluxes are imposed along the sidewall of the cylinder while constant heat loss occurs through the top and bottom. In this model, due to a very small thermal Peclet number, convective heat transfer is neglected, and the flow is steady and inertialess. Time-dependent concentration is solved for various values of the mass Peclet number, Pe(sub m), (the ratio between the convective transport rate and the diffusive transport rate) and different magnetic field strengths represented by the Hartmann number Ha. Normalized values of these diffusivities vs. effective Pe(sub m) are presented for different imposed temperature profiles. In all cases, the diffusivity value obtained through the simulated measurement increases as the effective Pe(sub m) increases. The numerical results suggest that an additional periodic flux, or hot and cold spots, can significantly decrease the convective contamination in our geometry.
TL;DR: In this article, conditions for similarity solutions are determined for arbitrary stable and unstable thermal environment stratification using an implicit, iterative, tri-diagonal finite-difference method, and the results are found to be in excellent agreement.
Abstract: The problem of steady, laminar, natural convection flow along a vertical permeable surface immersed in a thermally stratified environment in the presence of magnetic-field and heat-absorption effects is studied numerically. Conditions for similarity solutions are determined for arbitrary stable and unstable thermal environment stratification. Numerical solution of the similarity equations is performed using an implicit, iterative, tri-diagonal finite-difference method. Comparison with previously published work is performed and the results are found to be in excellent agreement. The effects of Hartmann number, heat-absorption coefficient, and the wall mass-transfer parameter on the velocity and temperature profiles as well as the skin-friction coefficient and Nusselt number are presented graphically and discussed. It is found that both the magnetic-field and heat-absorption effects eliminate the occurrence of the fluid backflow and temperature deficit in the outer part of the boundary layer predicted for t...
TL;DR: In this article, a variational principle is established for three-dimensional MHD equations with high Hartmann number and a special technique is proposed to incorporate the no-slip condition and far distance boundary condition as natural boundary conditions.
01 Oct 2002-Metallurgical and Materials Transactions B-process Metallurgy and Materials Processing Science
TL;DR: In this article, an approach to the computational simulation of three-dimensional nonlinear magnetohydrodynamic (MHD) flow in two-phase systems is proposed, where the electromagnetic field in the conducting fluid, including the particle, is represented using the current-vector potential (T) and reduced magnetic scalar potential (Ψ) to avoid the discontinuity of the electric field at the fluid-particle interface.
Abstract: The magnetohydrodynamic (MHD) flow around a suspended particle in a liquid metal subjected to electric and magnetic fields can affect the force exerted by the applied electromagnetic field on the particle. In this article, a novel approach to the computational simulation of three-dimensional nonlinear MHD flow in two-phase systems is proposed. The electromagnetic field in the conducting fluid, including the particle, is represented using the current-vector potential (T) and reduced magnetic scalar potential (Ψ) to avoid the discontinuity of the electric field at the fluid-particle interface. To avoid the solution of the electromagnetic field in free space and to account exactly for the electromagnetic field interactions with the fluid and the particle, the electric and magnetic fields are specified at the boundary of the fluid-flow domain using Ampere’s law. This formulation permits the numerical solution of the coupled electromagnetic and fluid-flow equations on a common mesh. The discretized equations are derived using a finite-element formulation, and an iterative procedure is described for the efficient solution of these equations. This method is used to investigate the electromagnetic and fluid-flow phenomena in electromagnetic separation of a nonconducting spherical particle in crossed uniform electric and magnetic fields at intermediate Hartmann numbers. The computed results show that the magnetic field has no effect on either the velocity field or the net force on the particle when the Hartmann number is less than 1. Beyond this threshold value of the Hartmann number, the velocity decreases almost linearly with increasing magnetic-field strength. The damping of the flow by the magnetic field manifests itself in a reduction of the separation force, even though it is relatively small for this system.
01 Jan 2002
TL;DR: In this article, the effect of thermal and mass diffusion on unsteady free convection flow of an incompressible and electrically conducting fluid subjected to constant heat flux and accelerated boundary motion in presence of constant magnetic field was investigated.
Abstract: This paper concerns with the effect of thermal and mass diffusion on unsteady free convection flow of an incompressible and electrically conducting fluid subjected to constant heat flux and accelerated boundary motion in presence of constant magnetic field. An exact solution has been obtained for species concentration. temperature and velocity variables. The fluid velocity and skin friction have been computed for some saturated liquids. The results are discussed with respect to buoyancy ratio parameter (N) and Hartmann number (m).
16 Sep 2002
TL;DR: In this article, the Hartmann number and the interaction parameter for small and medium-size MHD flows in circular ducts in nonuniform magnetic fields are studied with reference to liquid metal blankets and divertors of fusion reactors.
Abstract: Magnetohydrodynamic (MHD) flows in circular ducts in nonuniform magnetic fields are studied with reference to liquid metal blankets and divertors of fusion reactors. Flows in small and medium size reactors are characterized by moderate and low values of the Hartmann number ({approx}50-2000) and the interaction parameter ({approx}0.1-1000). The validity of the high-Hartmann number flow model for the intermediate range is discussed and the results of theoretical and experimental investigations are presented.
06 Mar 2002
TL;DR: In this paper, the authors studied magnetohydrodynamic flows in insulated circular ducts in nonuniform magnetic fields with reference to liquid metal blankets and divertors of fusion reactors.
Abstract: Magnetohydrodynamic flows in insulated circular ducts in nonuniform magnetic fields are studied with reference to liquid metal blankets and divertors of fusion reactors. Particular emphasis is made on C-MOD. The ducts are supposed to be straight, while the gradient of the magnetic field to be inclined by an angle {alpha} to the duct axis. The results are presented for the values of the Hartmann numbers, Ha, of 1000 and 100. Three-dimensional pressure drop, development length, three-dimensional length and nonuniformities of the velocity profiles have been evaluated. It has been shown that for Ha = 1000 the three-dimensional effects are of considerable importance, while for Ha = 100 they may be neglected.
04 Nov 2002
TL;DR: In this paper, an asymptotic, high Hartmann number model has been used to study a combined effect of surface tension, non-uniform magnetic field, gravity and inertia.
Abstract: Steady, two-dimensional flow of a liquid metal jet pouring vertically down from a nozzle in the presence of crossed magnetic and electric fields has been investigated. The magnetic field is supposed to have a single component transverse to the flow. An asymptotic, high Hartmann number model has been used to study a combined effect of surface tension, nonuniform magnetic field, gravity and inertia. Relations have been obtained for a jet issuing from a duct, pouring into a draining duct, pouring from one duct into another, and that in a liquid bridge. The results show that the jet becomes thicker if the field increases along the flow and thinner if it decreases. It has also been shown that for gradually varying fields characteristic for the divertor region of both C-MOD and NSTX tokamaks, inertial effects are negligible for N > 10, where N is the interaction parameter. Thus, provided the jet remains stable, the inertialess flow model is expected to give good results even for relatively low magnetic fields and high jet velocity. Surface tension plays a crucial role in shaping the jet profile at the nozzle. Partial flooding of the nozzle walls is predicted. Finally, proposals have been made tomore » investigate a possibility of using an axisymmetric curtain along the perimeter of the bottom of a tokamak as an alternative to the film- or jet-divertors, or to use a system of plane liquid metal sheets.« less
01 Jan 2002
TL;DR: In this article, the authors considered axisymmetric magnetohydrodynamic motion in a spherical shell driven by rotating the inner boundary relative to the stationary outer boundary { spherical Couette flow}.
Abstract: We consider axisymmetric magnetohydrodynamic motion in a spherical shell driven by rotating the inner boundary relative to the stationary outer boundary { spherical Couette flow. The inner solid sphere is rigid with the same electrical conductivity as the surrounding fluid; the outer rigid boundary is an insulator. A force-free dipole magnetic eld is maintained by a dipole source at the centre. For strong imposed elds (as measured by the Hartmann number M), the numerical simulations of Dormy et al. (1998) showed that a super-rotating shear layer (with angular velocity about 50% above the angular velocity of the inner core) is attached to the magnetic eld line C tangent to the outer boundary at the equatorial plane of symmetry. At large M, we obtain analytically the mainstream solution valid outside all boundary layers by application of Hartmann jump conditions across the inner- and outer-sphere boundary layers. We formulate the large-M boundary layer problem for the free shear layer of width M 1=2 containingC and solve it numerically. The super-rotation can be understood in terms of the nature of the meridional electric current flow in the shear layer, which is fed by the outer-sphere Hartmann layer. Importantly, a large fraction of the current entering the shear layer is tightly focused and eectively released from a point source at the equator triggered by the tangency of the C-line. The current injected by the source follows theC-line closely but spreads laterally due to diusion. In consequence, a strong azimuthal Lorentz force is produced, which takes opposite signs either side of the C-line; order-unity super-rotation results on the equatorial side. In fact, the point source is the small equatorial Hartmann layer of radial width M 2=3 ( M 1=2 ) and latitudinal extent M 1=3 . We construct its analytic solution and so determine an inward displacement width O(M 2=3 ) of the free shear layer. We compare our numerical solution of the free shear layer problem with our numerical solution of the full governing equations for M in excess of 10 4 . We obtain excellent agreement. Some of our more testing comparisons are signicantly improved by incorporating the shear layer displacement caused by the equatorial Hartmann layer.