scholarly journals Linear stability of magnetohydrodynamic flow in a perfectly conducting rectangular duct

2012 ◽  
Vol 708 ◽  
pp. 111-127 ◽  
Author(s):  
Jānis Priede ◽  
Svetlana Aleksandrova ◽  
Sergei Molokov
Keyword(s):  
Magnetic Field ◽  
Reynolds Number ◽  
Strong Magnetic Field ◽  
Linear Stability ◽  
Critical Reynolds Number ◽  
Metal Flow ◽  
Transverse Magnetic ◽  
Rectangular Duct ◽  
Instability Modes ◽  
The Magnetic Field

AbstractWe analyse numerically the linear stability of a liquid-metal flow in a rectangular duct with perfectly electrically conducting walls subject to a uniform transverse magnetic field. A non-standard three-dimensional vector stream-function/vorticity formulation is used with a Chebyshev collocation method to solve the eigenvalue problem for small-amplitude perturbations. A relatively weak magnetic field is found to render the flow linearly unstable as two weak jets appear close to the centre of the duct at the Hartmann number $\mathit{Ha}\approx 9. 6. $ In a sufficiently strong magnetic field, the instability following the jets becomes confined in the layers of characteristic thickness $\delta \ensuremath{\sim} {\mathit{Ha}}^{\ensuremath{-} 1/ 2} $ located at the walls parallel to the magnetic field. In this case the instability is determined by $\delta , $ which results in both the critical Reynolds number and wavenumber scaling as ${\ensuremath{\sim} }{\delta }^{\ensuremath{-} 1} . $ Instability modes can have one of the four different symmetry combinations along and across the magnetic field. The most unstable is a pair of modes with an even distribution of vorticity along the magnetic field. These two modes represent strongly non-uniform vortices aligned with the magnetic field, which rotate either in the same or opposite senses across the magnetic field. The former enhance while the latter weaken one another provided that the magnetic field is not too strong or the walls parallel to the field are not too far apart. In a strong magnetic field, when the vortices at the opposite walls are well separated by the core flow, the critical Reynolds number and wavenumber for both of these instability modes are the same: ${\mathit{Re}}_{c} \approx 642{\mathit{Ha}}^{1/ 2} + 8. 9\ensuremath{\times} 1{0}^{3} {\mathit{Ha}}^{\ensuremath{-} 1/ 2} $ and ${k}_{c} \approx 0. 477{\mathit{Ha}}^{1/ 2} . $ The other pair of modes, which differs from the previous one by an odd distribution of vorticity along the magnetic field, is more stable with an approximately four times higher critical Reynolds number.

scholarly journals Linear stability of magnetohydrodynamic flow in a square duct with thin conducting walls

2015 ◽  
Vol 788 ◽  
pp. 129-146 ◽  
Author(s):  
Jānis Priede ◽  
Thomas Arlt ◽  
Leo Bühler
Keyword(s):  
Magnetic Field ◽  
Reynolds Number ◽  
Strong Magnetic Field ◽  
Linear Stability ◽  
Critical Reynolds Number ◽  
Metal Flow ◽  
Three Dimensional ◽  
Transverse Magnetic ◽  
Base Flow ◽  
Square Duct

This study is concerned with the numerical linear stability analysis of liquid-metal flow in a square duct with thin electrically conducting walls subject to a uniform transverse magnetic field. We derive an asymptotic solution for the base flow that is valid for not only high but also moderate magnetic fields. This solution shows that, for low wall conductance ratios $c\ll 1$, an extremely strong magnetic field with Hartmann number $\mathit{Ha}\sim c^{-4}$ is required to attain the asymptotic flow regime considered in previous studies. We use a vector streamfunction–vorticity formulation and a Chebyshev collocation method to solve the eigenvalue problem for three-dimensional small-amplitude perturbations in ducts with realistic wall conductance ratios $c=1$, 0.1 and 0.01 and Hartmann numbers up to $10^{4}$. As for similar flows, instability in a sufficiently strong magnetic field is found to occur in the sidewall jets with characteristic thickness ${\it\delta}\sim \mathit{Ha}^{-1/2}$. This results in the critical Reynolds number and wavenumber increasing asymptotically with the magnetic field as $\mathit{Re}_{c}\sim 110\mathit{Ha}^{1/2}$ and $k_{c}\sim 0.5\mathit{Ha}^{1/2}$. The respective critical Reynolds number based on the total volume flux in a square duct with $c\ll 1$ is $\overline{\mathit{Re}}_{c}\approx 520$. Although this value is somewhat larger than $\overline{\mathit{Re}}_{c}\approx 313$ found by Ting et al. (Intl J. Engng Sci., vol. 29 (8), 1991, pp. 939–948) for the asymptotic sidewall jet profile, it still appears significantly lower than the Reynolds numbers at which turbulence is observed in experiments as well as in direct numerical simulations of this type of flow.

scholarly journals Linear stability of Hunt's flow

2010 ◽  
Vol 649 ◽  
pp. 115-134 ◽  
Author(s):  
JĀNIS PRIEDE ◽  
SVETLANA ALEKSANDROVA ◽  
SERGEI MOLOKOV
Keyword(s):  
Magnetic Field ◽  
Reynolds Number ◽  
Linear Stability ◽  
Critical Reynolds Number ◽  
Transverse Magnetic ◽  
Base Flow ◽  
Maximal Velocity ◽  
Square Duct ◽  
Stream Function Formulation ◽  
The Magnetic Field

We analyse numerically the linear stability of the fully developed flow of a liquid metal in a square duct subject to a transverse magnetic field. The walls of the duct perpendicular to the magnetic field are perfectly conducting whereas the parallel ones are insulating. In a sufficiently strong magnetic field, the flow consists of two jets at the insulating walls and a near-stagnant core. We use a vector stream function formulation and Chebyshev collocation method to solve the eigenvalue problem for small-amplitude perturbations. Due to the two-fold reflection symmetry of the base flow the disturbances with four different parity combinations over the duct cross-section decouple from each other. Magnetic field renders the flow in a square duct linearly unstable at the Hartmann number Ha ≈ 5.7 with respect to a disturbance whose vorticity component along the magnetic field is even across the field and odd along it. For this mode, the minimum of the critical Reynolds number Rec ≈ 2018, based on the maximal velocity, is attained at Ha ≈ 10. Further increase of the magnetic field stabilizes this mode with Rec growing approximately as Ha. For Ha > 40, the spanwise parity of the most dangerous disturbance reverses across the magnetic field. At Ha ≈ 46 a new pair of most dangerous disturbances appears with the parity along the magnetic field being opposite to that of the previous two modes. The critical Reynolds number, which is very close for both of these modes, attains a minimum, Rec ≈ 1130, at Ha ≈ 70 and increases as Rec ≈ 91Ha1/2 for Ha ≫ 1. The asymptotics of the critical wavenumber is kc ≈ 0.525Ha1/2 while the critical phase velocity approaches 0.475 of the maximum jet velocity.

Liquid-metal flow in a rectangular duct with a strong non-uniform magnetic field

1984 ◽  
Vol 139 ◽  
pp. 309-324 ◽  
Author(s):  
John C. Petrykowski ◽  
John S. Walker
Keyword(s):  
Magnetic Field ◽  
Liquid Metal ◽  
Metal Flow ◽  
Velocity Profiles ◽  
Transverse Magnetic ◽  
Secondary Flows ◽  
Rectangular Duct ◽  
Liquid Metal Flow ◽  
Displacement Thickness ◽  
The Magnetic Field

Liquid-metal flows in rectangular ducts having electrically insulating tops and bottoms and perfectly conducting sides and in the presence of strong, polar, transverse magnetic fields are examined. Solutions are presented for the boundary layers adjacent to the sides that are parallel to the magnetic field. Overshoots in the radial velocity profiles show that the side layers have zero displacement thickness and do not perturb the inviscid core. Very weak secondary flows involve four significant vortices, as reflected in the polar velocity profiles.

scholarly journals Linear Stability Analysis of Liquid Metal Flow in an Insulating Rectangular Duct under External Uniform Magnetic Field

Fluids ◽  
2019 ◽  
Vol 4 (4) ◽  
pp. 177 ◽  
Author(s):  
Tagawa
Keyword(s):  
Magnetic Field ◽  
Stability Analysis ◽  
Liquid Metal ◽  
Linear Stability ◽  
Linear Stability Analysis ◽  
Uniform Magnetic Field ◽  
Metal Flow ◽  
Rectangular Duct ◽  
Liquid Metal Flow ◽  
External Uniform Magnetic Field

Linear stability analysis of liquid metal flow driven by a constant pressure gradient in an insulating rectangular duct under an external uniform magnetic field was carried out. In the present analysis, since the Joule heating and induced magnetic field were neglected, the governing equations consisted of the continuity of mass, momentum equation, Ohm’s law, and conservation of electric charge. A set of linearized disturbance equations for the complex amplitude was decomposed into real and imaginary parts and solved numerically with a finite difference method using the highly simplified marker and cell (HSMAC) algorithm on a two-dimensional staggered mesh system. The difficulty of the complex eigenvalue problem was circumvented with a Newton—Raphson method during which its corresponding eigenfunction was simultaneously obtained by using an iterative procedure. The relation among the Reynolds number, the wavenumber, the growth rate, and the angular frequency was successfully obtained for a given value of the Hartmann number as well as for a direction of external uniform magnetic field.

Control of Vortex Shedding From a Bluff Body Using Imposed Magnetic Field

2006 ◽  
Vol 129 (5) ◽  
pp. 517-523 ◽  
Author(s):  
Sintu Singha ◽  
K. P. Sinhamahapatra ◽  
S. K. Mukherjea
Keyword(s):  
Magnetic Field ◽  
Reynolds Number ◽  
Steady Flow ◽  
Transverse Magnetic Field ◽  
Vortex Shedding ◽  
Bluff Body ◽  
Reynolds Numbers ◽  
Transverse Magnetic ◽  
Staggered Grid ◽  
The Magnetic Field

The two-dimensional incompressible laminar viscous flow of a conducting fluid past a square cylinder placed centrally in a channel subjected to an imposed transverse magnetic field has been simulated to study the effect of a magnetic field on vortex shedding from a bluff body at different Reynolds numbers varying from 50 to 250. The present staggered grid finite difference simulation shows that for a steady flow the separated zone behind the cylinder is reduced as the magnetic field strength is increased. For flows in the periodic vortex shedding and unsteady wake regime an imposed transverse magnetic field is found to have a considerable effect on the flow characteristics with marginal increase in Strouhal number and a marked drop in the unsteady lift amplitude indicating a reduction in the strength of the shed vortices. It has further been observed, that it is possible to completely eliminate the periodic vortex shedding at the higher Reynolds numbers and to establish a steady flow if a sufficiently strong magnetic field is imposed. The necessary strength of the magnetic field, however, depends on the flow Reynolds number and increases with the increase in Reynolds number. This paper describes the algorithm in detail and presents important results that show the effect of the magnetic field on the separated wake and on the periodic vortex shedding process.

scholarly journals Two-dimensional nonlinear travelling waves in magnetohydrodynamic channel flow

2014 ◽  
Vol 760 ◽  
pp. 387-406 ◽  
Author(s):  
Jonathan Hagan ◽  
Jānis Priede
Keyword(s):  
Magnetic Field ◽  
Linear Stability ◽  
Channel Flow ◽  
Travelling Waves ◽  
Transverse Magnetic ◽  
Travelling Wave ◽  
Linear Stability Theory ◽  
Two Dimensional ◽  
Order Of Magnitude ◽  
The Magnetic Field

AbstractThis study is concerned with the stability of a flow of viscous conducting liquid driven by a pressure gradient in the channel between two parallel walls subject to a transverse magnetic field. Although the magnetic field has a strong stabilizing effect, this flow, similarly to its hydrodynamic counterpart – plane Poiseuille flow – is known to become turbulent significantly below the threshold predicted by linear stability theory. We investigate the effect of the magnetic field on two-dimensional nonlinear travelling-wave states which are found at substantially subcritical Reynolds numbers starting from $\mathit{Re}_{n}=2939$ without the magnetic field and from $\mathit{Re}_{n}\sim 6.50\times 10^{3}\mathit{Ha}$ in a sufficiently strong magnetic field defined by the Hartmann number $\mathit{Ha}$. Although the latter value is a factor of seven lower than the linear stability threshold $\mathit{Re}_{l}\sim 4.83\times 10^{4}\mathit{Ha}$, it is still more than an order of magnitude higher than the experimentally observed value for the onset of turbulence in magnetohydrodynamic (MHD) channel flow.

Magnetohydrodynamic channel flows with weak transverse magnetic fields

2014 ◽  
Vol 372 (2020) ◽  
pp. 20130344
Author(s):  
A. P. Rothmayer
Keyword(s):  
Magnetic Field ◽  
Reynolds Number ◽  
High Reynolds Number ◽  
Hartmann Number ◽  
Plane Channel ◽  
Transverse Magnetic ◽  
Unsteady Viscous Flow ◽  
High Reynolds ◽  
Magnetohydrodynamic Channel ◽  
The Magnetic Field

Magnetohydrodynamic flow of an incompressible fluid through a plane channel with slowly varying walls and a magnetic field applied transverse to the channel is investigated in the high Reynolds number limit. It is found that the magnetic field can first influence the hydrodynamic flow when the Hartmann number reaches a sufficiently large value. The magnetic field is found to suppress the steady and unsteady viscous flow near the channel walls unless the wall shapes become large.

Magnetohydrodynamic flow in rectangular ducts

1965 ◽  
Vol 21 (4) ◽  
pp. 577-590 ◽  
Author(s):  
J. C. R. Hunt
Keyword(s):  
Magnetic Field ◽  
Exact Solutions ◽  
Boundary Layers ◽  
Hartmann Number ◽  
Transverse Magnetic ◽  
Rectangular Duct ◽  
Conducting Liquid ◽  
The Core ◽  
Thin Walls ◽  
The Magnetic Field

The paper presents an analysis of laminar motion of a conducting liquid in a rectangular duct under a uniform transverse magnetic field. The effects of the duct having conducting walls are investigated. Exact solutions are obtained for two cases, (i) perfectly conducting walls perpendicular to the field and thin walls of arbitrary conductivity parallel to the field, and (ii) non-conducting walls parallel to the field and thin walls of arbitrary conductivity perpendicular to the field.The boundary layers on the walls parallel to the field are studied in case (i) and it is found that at high Hartmann number (M), large positive and negative velocities of order MVc are induced, where Vc is the velocity of the core. It is suggested that contrary to previous assumptions the magnetic field may in some cases have a destabilizing effect on flow in ducts.

The effect of a transverse magnetic field on shear turbulence

1978 ◽  
Vol 89 (1) ◽  
pp. 147-171 ◽  
Author(s):  
Claude B. Reed ◽  
Paul S. Lykoudis
Keyword(s):  
Magnetic Field ◽  
Reynolds Number ◽  
Friction Coefficient ◽  
Aspect Ratio ◽  
Transverse Magnetic ◽  
Skin Friction Coefficient ◽  
Reynolds Stresses ◽  
Mean Velocity ◽  
The Mean ◽  
The Magnetic Field

Turbulence measurements under the influence of a transverse magnetic field have been made at Purdue University's Magneto-Fluid-Mechanic Laboratory in a high aspect ratio channel. The Reynolds number range covered was 25000 ≤ Re 282000; the geometry and experimental conditions were such that the experiment approximated turbulent Hartmann flow. The aspect ratio of the channel was 5·8:1, its walls were electrically insulated and the working fluid was mercury. Measurements in the presence of a magnetic field were made of the skin friction coefficient, the mean velocity profiles, the turbulence intensity profiles (both u’ and v’) and the Reynolds stress profiles.A sudden change in the damping of the Reynolds stresses was manifested by a ‘hump’ in the curves of Cf versus M/Re taken with the Reynolds number held constant. This ‘hump’ occurs as a gentle rise and sudden drop to the Hartmann laminar line of the Cf data. Close examination of the $\overline{u^{\prime}v^{\prime}}$ data near the wall confirms this behaviour, indicating that the turbulent contribution to the shear stress is the controlling factor in this behaviour of Cf. The Reynolds stresses were completely suppressed to zero at high values of the magnetic field, though the turbulence intensities of u’ and v’ were not. The Reynolds stress data are fundamental in revealing the mechanisms which are at work during the suppression of turbulence by a magnetic field.It was also found that at high magnetic fields, when most of the turbulence was damped, the skin friction coefficient fell below the values predicted by Hartmann's (1937) laminar solution for high values of M/Re. This result was linked to the presence of ‘M-shaped’ velocity profiles in the direction perpendicular to both the magnetic field and the mean velocity vector. The presence of ‘M-shaped’ profiles has not previously been linked to a reduction in Cf.

Hydromagnetic instability of a free shear layer at small magnetic Reynolds numbers

1971 ◽  
Vol 49 (1) ◽  
pp. 21-31 ◽  
Author(s):  
Kanefusa Gotoh
Keyword(s):  
Magnetic Field ◽  
Reynolds Number ◽  
Shear Layer ◽  
Critical Reynolds Number ◽  
Three Dimensional ◽  
Two Dimensional ◽  
Free Shear Layer ◽  
Electrically Conducting ◽  
The Stability ◽  
The Magnetic Field

The effect of a uniform and parallel magnetic field upon the stability of a free shear layer of an electrically conducting fluid is investigated. The equations of the velocity and the magnetic disturbances are solved numerically and it is shown that the flow is stabilized with increasing magnetic field. When the magnetic field is expressed in terms of the parameter N (= M2/R2), where M is the Hartmann number and R is the Reynolds number, the lowest critical Reynolds number is caused by the two-dimensional disturbances. So long as 0 [les ] N [les ] 0·0092 the flow is unstable at all R. For 0·0092 < N [les ] 0·0233 the flow is unstable at 0 < R < Ruc where Ruc decreases as N increases. For 0·0233 < N < 0·0295 the flow is unstable at Rlc < R < Ruc where Rlc increases with N. Lastly for N > 0·0295 the flow is stable at all R. When the magnetic field is measured by M, the lowest critical Reynolds number is still due to the two-dimensional disturbances provided 0 [les ] M [les ] 0·52, and Rc is given by the corresponding Rlc. For M > 0·52, Rc is expressed as Rc = 5·8M, and the responsible disturbance is the three-dimensional one which propagates at angle cos−1(0·52/M) to the direction of the basic flow.

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