Magnetohydrodynamic Pipe Flow Research Articles

Experimental characterization of coherent states in turbulent magnetohydrodynamic pipe flow

Investigating magnetohydrodynamic (MHD) flows, whether through experimental or numerical study, presents significant challenges. A minimum requirement for experimentally studying MHD flows using optical measurement methods is the use of highly conductive, transparent fluids and strong magnetic fields in close proximity to the flow confinement. In response to these complexities, this work introduces a large-scale pipe rig setup, integrating a magnetic frame structure with a Stereoscopic Particle Image Velocimetry system for MHD flow field characterizations. This allows for the detailed analysis of coherent flow patterns under conditions of fully developed MHD turbulence. In alignment with earlier numerical studies on turbulent MHD pipe flow, our research confirms that low Hartmann number turbulence statistics are minimally affected by the applied magnetic field. Nevertheless, we show that the magnetic field significantly extends the streamwise extent of detected coherent organizational states, while their wave number distribution and morphology largely remain unchanged.

Non-modal stability analysis of magneto-hydrodynamic flow in a single pipe

A complete understanding of the stability of fluid flows under varying magnetic field profiles is imperative for achieving control of plasma and operating fluids in the blankets of future fusion reactors. In this context, the primary objective of this study is to investigate the influence of varying magnetic profiles on the flow regime of a generic fluid, which is representative of both thermonuclear plasma and conductive fluids within a nuclear fusion reactor. To this aim in this work non-modal stability theory is adopted to perform stability analysis of a magneto-hydrodynamic (MHD) flow in an infinite circular pipe in order to study the effects of the magnetic field on the fluid dynamics of the pipe flow. In particular, the effects on the general stability of two magnetic field profiles are compared with the reference case of a pipe Poiseuille flow without magnetic field. Firstly, the classic modal stability technique is employed to study asymptotical stability. Then, non-modal stability analysis is applied to magneto-hydrodynamic pipe flow to study the system's response for a finite time immediately after a perturbation. Fourier–Chebyshev Petrov–Galerkin spectral method is used to compute the eigenvalues and pseudospectra of the weak formulation associated with the linearised system. Investigations on the dependence of spectra and transient growths on the specific magnetic profiles are conducted for different values of perturbation wave numbers. The obtained results show that in general the magnetic field has an effect of stabilization on the system, which depends on the specific magnetic profile considered. In addition, the non-modal stability analysis reveals that the inclusion of the magnetic field mitigates the effects of perturbations also in the short term, a phenomenon that cannot be seen using only modal stability analysis.

On the instability of the magnetohydrodynamic pipe flow subject to a transverse magnetic field

The linear stability of a fully developed liquid–metal magnetohydrodynamic pipe flow subject to a transverse magnetic field is studied numerically. Because of the lack of axial symmetry in the mean velocity profile, we need to perform a BiGlobal stability analysis. For that purpose, we develop a two-dimensional complex eigenvalue solver relying on a Chebyshev–Fourier collocation method in physical space. By performing an extensive parametric study, we show that in contrast to the Hagen–Poiseuille flow known to be linearly stable for all Reynolds numbers, the magnetohydrodynamic pipe flow with transverse magnetic field is unstable to three-dimensional disturbances at sufficiently high values of the Hartmann number and wall conductance ratio. The instability observed in this regime is attributed to the presence of velocity overspeed in the so-called Roberts layers and the corresponding inflection points in the mean velocity profile. The nature and characteristics of the most unstable modes are investigated, and we show that they vary significantly depending on the wall conductance ratio. A major result of this paper is that the global critical Reynolds number for the magnetohydrodynamic pipe flow with transverse magnetic field is Re = 45 230, and it occurs for a perfectly conducting pipe wall and the Hartmann number Ha = 19.7.

Reconstruction of 3D MHD liquid metal velocity from measurements of electric potential on the external surface of a thick-walled pipe

Liquid metal magnetohydrodynamic pipe flow experiments have been performed in a non-uniform strong magnetic field in order to get further insight into the physics of this type of three-dimensional MHD flow and to enlarge the database for validation of numerical tools. In these experiments, in addition to measurements of the axial pressure distribution, the electric potential has been recorded at a large number of axial and circumferential positions by probes distributed around the periphery of the pipe. Since the wall of the pipe is relatively thick, the electric potential varies across the wall thickness, and surface potentials differ from those at the fluid-wall interface. However, the dense population of potential sensors on the pipe surface allows for a reconstruction of data on the fluid-wall interface, from which we can determine components of potential gradient in the fluid in a plane perpendicular to the magnetic field. The latter ones are closely related to the velocity field, because the electric potential may serve as approximate streamfunction of the flow.

Stabilized FEM Solutions of MHD Equations Around a Solid and Inside a Conducting Medium

In this study, the numerical solution of the magnetohydrodynamic (MHD) flow is considered in a circular pipe around a conducting solid and in an insulating or conducting medium. An external magnetic field is applied through axis of the pipe with an angle α with through the x-axis. The mathematical model of the considered physical problem can be defined in terms of coupled MHD equations in the pipe domain and the Laplace equations on the solid and external mediums. The coupled equations are transformed into decoupled inhomogeneous convection-diffusion type equations in order to apply stabilization in the finite element method solution procedure. Obtained stabled solutions for the high values of the problem parameters display the well-known characteristics of the MHD pipe flow.

Stabilized FEM–BEM coupled solution of MHD pipe flow in an unbounded conducting medium

In this study, we have considered the Magnetohydrodynamic (MHD) flow in a pipe with a circular or square cross section around a conducting solid and in an insulating or conducting medium. An external magnetic field is applied with an angle α with the axis of the pipe. The mathematical model of the considered physical problem can be defined in terms of coupled MHD equations in the pipe domain and the Laplace equations on the solid and external mediums. Due to the unbounded external domain and coupled form of the MHD equations, the FEM and BEM coupling approach is used which has many advantages in the view of discretization, computational time and stabilization. In the solution procedure, the fluid domain is discretized by the FEM elements, and the boundaries of the solid and the conducting mediums by the BEM elements. Proposed coupled numerical scheme is tested especially for the high values of the problem parameters and obtained stable solutions are displayed in terms of figures.

Electrical field effect on three-dimensional magnetohydrodynamic pipe flow: a CFD study

Steady, laminar three-dimensional (3D) magnetohydrodynamic (MHD) flow of an electrically conducting fluid in a circular pipe under the both magnetic and electrical field are studied. External magnetic and electrical fields are applied perpendicular to the flow direction and each other while the fluid motion is subjected to constant pressure gradient along axial-direction in the present paper. Fluent 14.0, the finite element software based on the finite volume approach was used to calculate the 3D fluid dynamics and electromagnetic field partial differential equations iteratively. The originality of this work is that, in addition to magnetic field; the effect of electrical field on MHD flows is being examined with help of user defined function (UDF) code. The magnetic field leads to decrease in the velocity of flow, whereas the electrical field applied with magnetic field acted to increase and decrease the velocity of flow depending on the direction of applied external electrical field. The obtained results were depicted graphically and discussed.

A DRBEM solution for MHD pipe flow in a conducting medium

Numerical solutions are given for magnetohydrodynamic (MHD) pipe flow under the influence of a transverse magnetic field when the outside medium is also electrically conducting. Convection–diffusion-type MHD equations for inside the pipe are coupled with the Laplace equation defined in the exterior region, and the continuity requirements for the induced magnetic fields are also coupled on the pipe wall. The most general problem of a conducting pipe wall with thickness, which also has magnetic induction generated by the effect of an external magnetic field, is also solved. The dual reciprocity boundary element method (DRBEM) is applied directly to the whole coupled equations with coupled boundary conditions at the pipe wall. Discretization with constant boundary elements is restricted to only the boundary of the pipe due to the regularity conditions at infinity. This eliminates the need for assuming an artificial boundary far away from the pipe, and then discretizing the region below it. Thus, the computational efficiency of the proposed numerical procedure lies in the solving of small sized systems, as compared to domain discretization methods. Computations are carried out for several values of the Reynolds number Re, the magnetic pressure Rh of the fluid, and the magnetic Reynolds numbers Rm1 and Rm2 of the fluid and the outside medium, respectively. Exact solution of the problem of MHD pipe flow in an insulating medium validates the results of the numerical procedure.

FEM-BEM Coupling for the MHD Pipe Flow in an Exterior Region

In this study, the magnetohydrodynamic (MHD) flow through a circular pipe under the influence of a transverse mag- netic field when the outside medium is also electrically conducting is solved numerically by using FEM-BEM coupling approach. The coupled partial differential equations defined for the interior medium are transformed into homogenous modified Helmholtz equations. For the exterior medium on an infinite region, the Laplace equation is considered for the exterior magnetic field. Unknowns in the equations are also related with the corresponding Dirichlet and Neumann type coupled boundary conditions. Unknown values of the magnetic field on the boundary and for the exterior region are obtained by using BEM, and the unknown velocity and magnetic field inside the pipe are obtained by using SUPG type stabilized FEM. Computations are carried for very high values of magnetic Reynolds numbers Rm1, Reynolds number Re and magnetic pressure Rh of the fluid. The results show that using stabilized method enables us to get stable and accurate numerical approximations consistent with the physical configuration of the problem over rough mesh which also results a cheap computational cost.

Application of homotopy perturbation method to the MHD pipe flow of a fourth grade fluid

This paper applies He's homotopy perturbation method (HPM) to analyze the strongly nonlinear magneto-hydrodynamic (MHD) flow of a fourth grade fluid through a circular pipe. In this method, a homotopy is constructed for the equation. The initial approximations can be freely chosen with possible unknown constants which can be determined by imposing the boundary and initial conditions. The results reveal that HPM is very effective, convenient and quite accurate to both linear and nonlinear problems. It is predicted that HPM can be widely applied in engineering. Some illustrations are presented to show the reliability and simplicity of the method.

On explicit analytic solution for MHD pipe flow of a fourth grade fluid

The goal of this work is the analytical solution of incompressible, magnetohydrodynamic (MHD) flow through a circular pipe for fourth grade fluids. HAM (homotopy analysis method) solution has been obtained for the governing non-linear problem. Results are presented graphically and discussed for the emerging parameters of interest.

Decay bounds for magnetohydrodynamic geophysical flow

This paper establishes exponential decay bounds for steady magnetohydrodynamic geophysical finite pipe flow when homogeneous lateral surface boundary conditions are applied. In the spirit of earlier work of C.O. Horgan, L.T. Wheeler [Spatial decay estimates for the Navier–Stokes equations with application to the problem of entry flow, SIAM J. Appl. Math. 35 (1978) 97–116] and K.A. Ames, L.E. Payne [Decay estimates in steady pipe flow, SIAM J. Math. Anal. 20 (1989) 789–815] the decay to fully developed flow as a function of distance from the entry section is investigated. Here, it is not assumed that the flow is symmetric like in the work of J.C. Song, [Decay estimates for steady magnetohydrodynamic pipe flow, Nonlinear Anal. 54 (2003) 1029–1044] and it is fully developed at the exit section. Using a coupling parameter, we derive an integro-differential inequality that leads to estimates for the “energy” associated with the velocity and magnetic field represented by the difference between the entrance flow and the fully developed flow in a portion of the pipe near the exit section. This paper also indicates how to bound the total energy.

MHD effect on flow structures and heat transfer characteristics of liquid metal–gas annular flow in a vertical pipe

The magnetohydrodynamic (MHD) effect on the flow structures and heat transfer characteristics was studied numerically for a liquid metal–gas annular flow under a transverse magnetic field. The side layers, in which the velocity was increased, appeared near the eastern and western sidewalls in an annular MHD flow as in a single-phase liquid metal MHD flow. Temperature distribution in the liquid film, and the Nusselt number distribution in the angular direction were influenced by the flow structures with the side layers. Consequently heat transfer rate was higher at the eastern/western sidewalls than that at the southern/northern walls. The pressure drop in the MHD annular flow is of the same order of magnitude as in the single-phase MHD pipe flow under similar liquid metal flow condition.

Decay estimates for steady magnetohydrodynamic pipe flow

This paper establishes exponential decay bounds for steady magnetohydrodynamic pipe flow when homogeneous lateral surface boundary conditions are applied. In the spirit of earlier work of Ames and Payne (SIAM J. Math. Anal. 20 (1989) 789) and Horgan and Wheeler (SIAM J. Appl. Math. 35 (1978) 97), the decay to fully developed flow as a function of distance from the entry section is investigated. Here, it is not assumed that the flow is fully developed at the exit section. Energy inequalities are derived that lead to estimates for the “energy” associated with the velocity and magnetic field represented by the difference between the entrance flow and the fully developed flow in a portion of the pipe near the exit section. This paper also indicates how to bound the total energy.

Finite element analysis of magnetohydrodynamic pipe flow

This paper deals with magnetohydrodynamic flows in pipes with arbitrary cross-section and arbitrary wall conductivities under the influence of a transverse magnetic field. The equations studied are of considerable practical interest since they model induction flow meters, electromagnetic pumps, magnetic propulsion devices and generators and have application in medicine, in power generation and in nuclear reactor technology. We are primarily interested in the questions of existence, uniqueness and finite element approximation of solutions to the equations of steady-state magnetohydrodynamics with arbitrary boundary conditions which describe this phenomenon. We derive error estimates for the approximate solution and show that one obtains optimal accuracy when using the finite element method to construct approximate solutions. We illustrate these results with some numerical experiments.

Magnetohydrodynamic pipe flow in ducts with partial circular ring cross section and annular cross section

In this paper we use the Green function method to solve the problem of steady one dimensional flow of an incompressible viscous, electrically conducting fluid through a pipe with partial circular ring cross section and one with annular cross section, in the presence of an applied transverse uniform magnetic field. We obtain analytic solutions and carry out some numerical calculations of the velocity distribution and induced magnetic field.

Magnetohydrodynamic pipe flow in a duct with sector cross section

The authors use the Green function method to solve the problem of steady one-dimensional flow of an incompressible, viscous, electrically conducting fluid through a pipe with sector cross section, in the presence of an applied transverse uniform magnetic field. They obtain an analytic solution in this case which other methods have not given before.

Single-phase and two-phase magnetohydrodynamic pipe flow

Experimental results are presented for a range of measured temperatures and other parameters of vertically downward flows, both single-phase (sodium) and two-phase (sodium-nitrogen), in a conducting-wall pipe in the presence of a transverse magnetic field. Existing MHD theory predicted, to within experimental error, all single-phase pressure differences for magnetic interaction parameter values of up to approximately 100, beyond which the single-phase normalized resistance coefficients were noticeably lower than the laminar-flow predictions. The magnetic interaction parameter at which such deviation occurred was governed by the conductivity ratio. Two-phase pressure differences were obtained across a range of void fractions, approximately 0.3-0.8, where two distinct flow regimes were encountered. For those two regimes, the normalized resistance coefficients of pressure difference were predicted to within experimental error by the corresponding two-phase MHD pressure-difference models. In half of the two-phase cases examined, decreases were observed in normalized resistance coefficients at high values of the magnetic interaction parameter, a trend similar to that found in single-phase flow. The wall-voltage profiles of single-phase flows were symmetric with respect to the center of the applied magnetic field region; two-phase wall-voltage profiles were asymmetric because of the expansion of the gaseous nitrogen along the length of the test section. The influence of temperature and other system parameters upon pressure differences and wall voltages, and the possible effect of ‘ M-shaped’ velocity profiles in the two types of flow are discussed.

Magnetohydrodynamic pipe flow in nonuniform, axisymmetric fields

The perturbation of Poiseuille pipe flow by nonuniform, axisymmetric magnetic fields with weak axial gradients is treated theoretically for small magnetic Reynolds numbers and finite Hartmann and Reynolds numbers. Numerical examples for pipe flow through a finite length magnet solenoid are given. The results indicate that separated flow may develop in the fringing magnetic fields accompanied by appreciable local static pressure gradients and high local current densities.

A Note on Dual Extremum Principles in Magnetohydrodynamic Pipe Flow

A Note on Dual Extremum Principles in Magnetohydrodynamic Pipe Flow