A spectral Galerkin method for the coupled Orr–Sommerfeld and induction equations for free-surface MHD

Giannakis, Dimitrios; Fischer, Paul; Rosner, R.

doi:10.1016/j.jcp.2008.10.016

Cited by 20 publications

(17 citation statements)

References 54 publications

(189 reference statements)

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“…In addition, the behaviour of the hard instability mode, which is the free-surface analogue of the even unstable mode in channel problems (Takashima 1996), will be examined. All numerical work was carried out using a spectral Galerkin method for the coupled OS and induction equations for free-surface MHD (Giannakis et al 2008a).…”

Section: Resultsmentioning

confidence: 99%

“…The resulting formulation will contribute towards a physical interpretation of the results presented in §4, and can also provide consistency checks for numerical schemes (Smith & Davis 1982;Giannakis et al 2008a). In order to keep complexity at a minimum, we restrict attention to two-dimensional normal modes, setting the spanwise wavenumber β equal to zero and assigning an arbitrary length L y to the size of the domain in the y direction.…”

Section: Formulation For Two-dimensional Perturbationsmentioning

confidence: 99%

“…In the first and third-row panels, solid and dashed lines respectively correspond to negative and positive values. As discussed in Giannakis et al (2008a), accurate numerical evaluation of some of the energy transfer rates can be problematic, especially when a large polynomial degree is required for the spectral solution to (2.43) to converge. For this reason, we have not been able to evaluate Γ mech and Γem for Hartmann numbers as large as 10 3 .…”

Section: The Role Of the Steady-state Induced Magnetic Fieldmentioning

confidence: 99%

“…We remark, however, that the complex phase velocity does not cross the Im(c) = 0 axis. Instead, if Pm is allowed to be of order unity, mode M eventually participates in the formation of magnetic eigenvalue branches (Giannakis et al 2008a).…”

Section: Travelling Alfvén Modesmentioning

confidence: 99%

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Instabilities in free-surface Hartmann flow at low magnetic Prandtl numbers

2009

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We study the linear stability of the flow of a viscous electrically conducting capillary fluid on a planar fixed plate in the presence of gravity and a uniform magnetic field, assuming that the plate is either a perfect electrical insulator or a perfect conductor. We first confirm that the Squire transformation for magnetohydrodynamics is compatible with the stress and insulating boundary conditions at the free surface, but argue that unless the flow is driven at fixed Galilei and capillary numbers, respectively parameterising gravity and surface tension, the critical mode is not necessarily two-dimensional. We then investigate numerically how a flow-normal magnetic field, and the associated Hartmann steady state, affect the soft and hard instability modes of free-surface flow, working in the low-magnetic-Prandtl-number regime of laboratory fluids (Pm 10 −4 ). Because it is a critical-layer instability (moderately modified by the presence of the free surface), the hard mode is found to exhibit similar behaviour to the even unstable mode in channel Hartmann flow, in terms of both the weak influence of Pm on its neutral-stability curve, and the dependence of its critical Reynolds number Re c on the Hartmann number Ha. In contrast, the structure of the soft mode's growth-rate contours in the (Re, α) plane, where α is the wavenumber, differs markedly between problems with small, but nonzero, Pm, and their counterparts in the inductionless limit. As derived from large-wavelength approximations, and confirmed numerically, the soft mode's critical Reynolds number grows exponentially with Ha in inductionless problems. However, when Pm is nonzero the Lorentz force originating from the steady-state current leads to a modification of Re c (Ha) to either a sublinearly increasing, or decreasing function of Ha, respectively for problems with insulating and conducting walls. In the former, we also observe pairs of counter-propagating Alfvén waves, the upstream-propagating wave undergoing an instability driven by energy transfered from the steady-state shear to both of the velocity and magnetic degrees of freedom. Movies are available with the online version of the paper.

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Section: Resultsmentioning

confidence: 99%

Section: Formulation For Two-dimensional Perturbationsmentioning

confidence: 99%

Section: The Role Of the Steady-state Induced Magnetic Fieldmentioning

confidence: 99%

Section: Travelling Alfvén Modesmentioning

confidence: 99%

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Instabilities in free-surface Hartmann flow at low magnetic Prandtl numbers

2009

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“…The eigenvalue problem was solved by using a spectral-Galerkin method with numerical integration based on the work of Giannakis, Fischer & Rosner (2009), Camporeale, Canuto & Ridolfi (2012 and Camporeale & Ridolfi (2012b), to which the reader can refer for further details. This method is crucial to obtain the whole spectrum of eigenvalues presented and discussed in § 4.…”

Section: Spectral Solutionmentioning

confidence: 99%

Interplay among unstable modes in films over permeable walls

2013

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The stability of open-channel flows (or film flows) has been extensively investigated for the case of impermeable smooth walls. In contrast, despite its relevance in many geophysical and industrial flows, the case that considers a permeable rather than an impermeable wall is almost unexplored. In the present work, a linear stability analysis of a film falling over a permeable and inclined wall is developed and discussed. The focus is on the mutual interaction between three modes of instability, namely, the well-known free-surface and hydrodynamic (i.e. shear) modes, which are commonly observed in open-channel flows over impermeable walls, plus a new one associated with the flow within the permeable wall (i.e. the porous mode). The flow in this porous region is modelled by the volume-averaged Navier-Stokes equations and, at the wall interface, the surface and subsurface flow are coupled through a stress-jump condition, which allows one to obtain a continuous velocity profile throughout the whole flow domain. The generalized eigenvalue problem is then solved via a novel spectral Galerkin method, and the whole spectrum of eigenvalues is presented and physically interpreted. The results show that, in order to perform an analysis with a full coupling between surface and subsurface flow, the convective terms in the volume-averaged equations have to be retained. In previous studies, this aspect has never been considered. For each kind of instability, the critical Reynolds number (Re c ) is reported for a wide range of bed slopes (θ ) and permeabilities (σ ). The results show that the free-surface mode follows the behaviour that was theoretically predicted by Benjamin and Yih for impermeable walls and is independent of wall permeability. In contrast, the shear mode shows a high dependence on σ : at σ = 0 the behaviour of Re c (θ) recovers the well-known non-monotonic behaviour of the impermeable-wall case, with a minimum at θ ∼ 0.05 • . However, with an increase in wall permeability, Re c gradually decreases and eventually recovers a monotonic decreasing behaviour. At high values of σ , the porous mode of instability also occurs. A physical interpretation of the results is presented on the basis of the interplay between the free-surfaceinduced perturbation of pressure, the increment of straining due to shear with the increase in slope, and the shear stress condition at the free surface. Finally, the † Email address for correspondence: carlo.camporeale@polito.it 528 C. Camporeale, E. Mantelli and C. Manes paper investigates the extent to which Squire's theorem is applicable to the problem presented herein.

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Application of the Jacobi–Davidson method to accurate analysis of singular linear hydrodynamic stability problems

Gheorghiu¹,

Rommes

2012

Numerical Methods in Fluids

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SUMMARY The aim of this paper is to show that the Jacobi–Davidson (JD) method is an accurate and robust method for solving large generalized algebraic eigenvalue problems with a singular second matrix. Such problems are routinely encountered in linear hydrodynamic stability analysis of flows that arise in various areas of continuum mechanics. As we use the Chebyshev collocation as a discretization method, the first matrix in the pencil is nonsymmetric, full rank, and ill‐conditioned. Because of the singularity of the second matrix, QZ and Arnoldi‐type algorithms may produce spurious eigenvalues. As a systematic remedy of this situation, we use two JD methods, corresponding to real and complex situations, to compute specific parts of the spectrum of the eigenvalue problems. Both methods overcome potentially severe problems associated with spurious unstable eigenvalues and are fairly stable with respect to the order of discretization. The real JD outperforms the shift‐and‐invert Arnoldi method with respect to the CPU time for large discretizations. Three specific flows are analyzed to advocate our statements, namely a multicomponent convection–diffusion in a porous medium, a thermal convection in a variable gravity field, and the so‐called Hadley flow. Copyright © 2012 John Wiley & Sons, Ltd.

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A spectral Galerkin method for the coupled Orr–Sommerfeld and induction equations for free-surface MHD

Cited by 20 publications

References 54 publications

Instabilities in free-surface Hartmann flow at low magnetic Prandtl numbers

Instabilities in free-surface Hartmann flow at low magnetic Prandtl numbers

Interplay among unstable modes in films over permeable walls

Application of the Jacobi–Davidson method to accurate analysis of singular linear hydrodynamic stability problems

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