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.

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.

scholarly journals The effect of finite-conductivity Hartmann walls on the linear stability of Hunt’s flow

2017 ◽  
Vol 822 ◽  
pp. 880-891 ◽  
Author(s):  
Thomas Arlt ◽  
Jānis Priede ◽  
Leo Bühler
Keyword(s):  
Magnetic Field ◽  
Linear Stability ◽  
Metal Flow ◽  
Finite Conductivity ◽  
Asymptotic Results ◽  
Square Duct ◽  
Wall Conductivity ◽  
Side Walls ◽  
Conductance Ratio ◽  
The Stability

We analyse numerically the linear stability of fully developed liquid metal flow in a square duct with insulating side walls and thin, electrically conducting horizontal walls. The wall conductance ratio $c$ is in the range of 0.01 to 1 and the duct is subject to a vertical magnetic field with Hartmann numbers up to $\mathit{Ha}=10^{4}$. In a sufficiently strong magnetic field, the flow consists of two jets at the side walls and a near-stagnant core with relative velocity ${\sim}(c\mathit{Ha})^{-1}$. We find that for $\mathit{Ha}\gtrsim 300,$ the effect of wall conductivity on the stability of the flow is mainly determined by the effective Hartmann wall conductance ratio $c\mathit{Ha}.$ For $c\ll 1$, the increase of the magnetic field or that of the wall conductivity has a destabilizing effect on the flow. Maximal destabilization of the flow occurs at $\mathit{Ha}\approx 30/c$. In a stronger magnetic field with $c\mathit{Ha}\gtrsim 30$, the destabilizing effect vanishes and the asymptotic results of Priede et al. (J. Fluid Mech., vol. 649, 2010, pp. 115–134) for ideal Hunt’s flow with perfectly conducting Hartmann walls are recovered.

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.

Direct Numerical Simulations of Magnetic Field Effects on Turbulent Duct Flows

2009 ◽  
Author(s):  
R. Chaudhary ◽  
S. P. Vanka ◽  
B. G. Thomas
Keyword(s):  
Magnetic Field ◽  
Reynolds Number ◽  
Numerical Simulations ◽  
Transient Flow ◽  
Cast Steel ◽  
Three Dimensional ◽  
Numerical Procedure ◽  
Direct Numerical Simulations ◽  
Mhd Equations ◽  
Square Duct

Magnetic fields are crucial in controlling flow in various physical processes of significance. One of these processes, which has significant application of a magnetic field, is continuous casting of steel, where different magnetic field configurations are used to control the turbulent steel flow in the mold to minimize defects in the cast steel. This study has been undertaken to analyze the effect of magnetic field on mean velocities and turbulence parameters in the molten metal flows through a square duct. Direct Numerical Simulations without using a sub-grid scale (SGS) model have been used to characterize the three-dimensional transient flow. The coupled Navier-Stokes-MHD equations have been solved with a three-dimensional fractional-step numerical procedure. Because liquid metals have low magnetic Reynolds number, the induced magnetic field has been neglected and the electric potential method for magnetic field-flow coupling has been implemented. Initially, laminar simulations in a square duct have been performed and results generated were compared with previous series solutions. Next, simulations of a non-MHD flow in a square duct at low Reynolds number were performed and satisfactorily compared with results of a previous DNS study. Subsequently, different levels of a magnetic field were applied to study its effect on the turbulence until the flow completely laminarized. Time-dependent and time-averaged flows have been studied through mean velocities and fluctuations, and power spectrums of instantaneous velocities.

On the stability of plane Poiseuille flow with a finite conductivity in an aligned magnetic field

1968 ◽  
Vol 33 (3) ◽  
pp. 433-443 ◽  
Author(s):  
Sung-Hwan Ko
Keyword(s):  
Magnetic Field ◽  
Differential Equation ◽  
Reynolds Number ◽  
Critical Reynolds Number ◽  
Order Differential Equation ◽  
Three Dimensional ◽  
Sixth Order ◽  
Finite Conductivity ◽  
Two Dimensional ◽  
The Stability

A study is made of the stability of a viscous, incompressible fluid with a finite conductivity flowing between parallel planes in a parallel magnetic field. The general form of the magnetohydrodynamic stability equation is a sixth-order differential equation. The complete sixth-order differential equation is solved numerically as an eigenvalue problem. Stability curves are obtained for a range of values of the magnetic Reynolds number Rm and the Alfvé n number A based on two-dimensional disturbances. It is found that the minimum critical Reynolds number is raised as Rm increases for a given A2 and as A2 increases for a given Rm, respectively. The stability curve closes and finally degenerates to a point which gives the critical value for Rm or A2. Results obtained for two-dimensional disturbances are modified to take into account three-dimensional disturbances. Then the minimum critical Reynolds number where three-dimensional disturbances become apparent is obtained, below which two-dimensional disturbances are the most unstable.

Optimal growth and transition to turbulence in channel flow with spanwise magnetic field

2008 ◽  
Vol 596 ◽  
pp. 73-101 ◽  
Author(s):  
DMITRY KRASNOV ◽  
MAURICE ROSSI ◽  
OLEG ZIKANOV ◽  
THOMAS BOECK
Keyword(s):  
Magnetic Field ◽  
Reynolds Number ◽  
Strong Magnetic Field ◽  
Channel Flow ◽  
Flow Direction ◽  
Three Dimensional ◽  
Optimal Growth ◽  
Transition To Turbulence ◽  
Spanwise Direction ◽  
The Moment

Instability and transition to turbulence in a magnetohydrodynamic channel flow are studied numerically for the case of a uniform magnetic field imposed along the spanwise direction. Optimal perturbations and their maximum amplifications over finite time intervals are computed in the framework of the linear problem using an iterative scheme based on direct and adjoint governing equations. It is shown that, at sufficiently strong magnetic field, the maximum amplification is no longer provided by classical streamwise rolls, but rather by rolls oriented at an oblique angle to the basic flow direction. The angle grows with the Hartmann numberHaand reaches the limit corresponding to purely spanwise rolls atHabetween 50 and 100 depending on the Reynolds number. Direct numerical simulations are applied to investigate the transition to turbulence at a single subcritical Reynolds numberRe= 5000 and various Hartmann numbers. The transition is caused by the transient growth and subsequent breakdown of optimal perturbations, which take the form of one or two symmetric optimal modes (streamwise, oblique or spanwise modes depending onHa) with low-amplitude three-dimensional noise added at the moment of strongest energy amplification. A sufficiently strong magnetic field (Halarger than approximately 30) is found to completely suppress the instability. At smaller Hartmann numbers, the transition is observed but it is modified in comparison with the pure hydrodynamic case.

Nonlinear solutions of modified plane Couette flow in the presence of a transverse magnetic field

1996 ◽  
Vol 307 ◽  
pp. 231-243 ◽  
Author(s):  
M. Nagata
Keyword(s):  
Magnetic Field ◽  
Transverse Magnetic Field ◽  
Critical Reynolds Number ◽  
Three Dimensional ◽  
Transverse Magnetic ◽  
Finite Amplitude ◽  
Parallel Plates ◽  
Electrically Conducting ◽  
Electrically Conducting Fluid ◽  
Nonlinear Solutions

A nonlinear analysis is performed numerically for the motion of an electrically conducting fluid between parallel plates in relative motion when a transverse magnetic field is applied. It is found that steady three-dimensional finite-amplitude solutions exist even when the linear analysis predicts an infinite critical Reynolds number.

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