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.

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 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 confined flow around a 180-degree sharp bend

2017 ◽  
Vol 822 ◽  
pp. 813-847 ◽  
Author(s):  
Azan M. Sapardi ◽  
Wisam K. Hussam ◽  
Alban Pothérat ◽  
Gregory J. Sheard
Keyword(s):  
Reynolds Number ◽  
Stability Analysis ◽  
Linear Stability ◽  
Linear Stability Analysis ◽  
Critical Reynolds Number ◽  
Three Dimensional ◽  
Base Flow ◽  
Two Dimensional ◽  
Two Dimensional Flow ◽  
The Stability

This study seeks to characterise the breakdown of the steady two-dimensional solution in the flow around a 180-degree sharp bend to infinitesimal three-dimensional disturbances using a linear stability analysis. The stability analysis predicts that three-dimensional transition is via a synchronous instability of the steady flows. A highly accurate global linear stability analysis of the flow was conducted with Reynolds number $\mathit{Re}<1150$ and bend opening ratio (ratio of bend width to inlet height) $0.2\leqslant \unicode[STIX]{x1D6FD}\leqslant 5$. This range of $\mathit{Re}$ and $\unicode[STIX]{x1D6FD}$ captures both steady-state two-dimensional flow solutions and the inception of unsteady two-dimensional flow. For $0.2\leqslant \unicode[STIX]{x1D6FD}\leqslant 1$, the two-dimensional base flow transitions from steady to unsteady at higher Reynolds number as $\unicode[STIX]{x1D6FD}$ increases. The stability analysis shows that at the onset of instability, the base flow becomes three-dimensionally unstable in two different modes, namely a spanwise oscillating mode for $\unicode[STIX]{x1D6FD}=0.2$ and a spanwise synchronous mode for $\unicode[STIX]{x1D6FD}\geqslant 0.3$. The critical Reynolds number and the spanwise wavelength of perturbations increase as $\unicode[STIX]{x1D6FD}$ increases. For $1<\unicode[STIX]{x1D6FD}\leqslant 2$ both the critical Reynolds number for onset of unsteadiness and the spanwise wavelength decrease as $\unicode[STIX]{x1D6FD}$ increases. Finally, for $2<\unicode[STIX]{x1D6FD}\leqslant 5$, the critical Reynolds number and spanwise wavelength remain almost constant. The linear stability analysis also shows that the base flow becomes unstable to different three-dimensional modes depending on the opening ratio. The modes are found to be localised near the reattachment point of the first recirculation bubble.

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.

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.

Highly Accurate Solutions of a Laminar Square Duct Flow in a Transverse Magnetic Field With Heat Transfer Using Spectral Method

2005 ◽  
Vol 128 (4) ◽  
pp. 413-417 ◽  
Author(s):  
Mohammed J. Al-Khawaja ◽  
Mohammed Selmi
Keyword(s):  
Magnetic Field ◽  
Velocity Profile ◽  
Spectral Method ◽  
Transverse Magnetic Field ◽  
Circular Tube ◽  
Transverse Magnetic ◽  
Duct Flow ◽  
Square Duct ◽  
Circular Pipe Flow ◽  
The Magnetic Field

A liquid metal forced-convection fully developed laminar flow inside a square duct, whose surfaces are electrically insulated and subjected to a constant temperature in a transverse magnetic field, is solved numerically using the spectral method. The axial momentum, induction, and nonlinear energy equations are solved by expanding the axial velocity, magnetic field, and temperature in double Chebyshev series and are collocated at Gauss points. The resulting system of equations is solved numerically by Gauss elimination for the expansion coefficients. The velocity and the magnetic field coefficients are directly solved for, while the temperature coefficients are solved for iteratively. Results show that the velocity profile is flattened in the direction of the magnetic field, but it is more round in the direction normal to it, in a similar fashion to the case of circular tube studied previously. The powerful spectral method resolves the sharp velocity gradient near the duct walls very well leading to accurate calculation of friction factor and Nusselt number. These parameters increase with the strength of the magnetic field due to the increasing flatness of the velocity profile. Comparison with the results for the circular tube shows that the effect of magnetic field on square duct flow is slightly lower from that one for circular pipe flow.

The annihilation of a two-dimensional jet by a transverse magnetic field

1967 ◽  
Vol 30 (1) ◽  
pp. 65-82 ◽  
Author(s):  
H. K. Moffatt ◽  
J. Toomre
Keyword(s):  
Magnetic Field ◽  
Reynolds Number ◽  
Transverse Magnetic Field ◽  
Magnetic Interaction ◽  
Transverse Magnetic ◽  
Magnetic Reynolds Number ◽  
Two Dimensional ◽  
Dimensionless Numbers ◽  
Order Unity ◽  
The Magnetic Field

The effect of an applied transverse magnetic field on the development of a two-dimensional jet of incompressible fluid is examined. The jet is prescribed in terms of its mass flux ρQ0 and its lateral scale d at an initial section x = 0. The three dimensionless numbers characterizing the problem are a Reynolds number R = Q0/ν, a magnetic Reynolds number Rm = μσQ0, and a magnetic interaction parameter N = σB20d2/ρQ0, where ρ represents density, σ conductivity, μ permeability and B0 applied field strength, and it is assumed that \[ R_m \ll 1,\quad R\gg 1,\quad N\ll 1. \] It is shown that when M2 = RN [Gt ] 1, an inviscid treatment is appropriate, and that the effect of the magnetic field is then to destroy the jet momentum within a distance of order N−1 in the downstream direction. A general solution for inviscid development is obtained, and it is shown that a large class of velocity profiles (though not all of them) are self-preserving.When M2 [Lt ] 1, it is shown that the viscous similarity solution obtained by Moreau (1963a, b) is relevant. This solution is re-derived and re-interpreted; it implies that the jet momentum is destroyed within a distance of order $R^{\frac{1}{4}}N^{-\frac{3}{4}}$ in the downstream direction.Some further aspects of the jet annihilation problem are qualitatively discussed in § 4, viz. the nature of the overall flow field, the effect of the presence of distant boundaries, the effect of increasing Rm to order unity and greater, and the effect of oblique injection. Finally the development of a jet of conducting fluid into a nonconducting environment is considered; in this case the jet is not stopped by the magnetic field unless a return path outside the fluid for the induced current is available.

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