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A spectral Galerkin method for the coupled Orr-Sommerfeld and induction equations for free-surface MHD. (English) Zbl 1330.76055
Summary: We develop and test spectral Galerkin schemes to solve the coupled Orr-Sommerfeld and induction equations for parallel, incompressible MHD in free-surface and fixed-boundary geometries. The schemes’ discrete bases consist of Legendre internal shape functions, supplemented with nodal shape functions for the weak imposition of the stress and insulating boundary conditions. The orthogonality properties of the basis polynomials solve the matrix-coefficient growth problem, and eigenvalue-eigenfunction pairs can be computed stably at spectral orders at least as large as \(p=3000\) with \(p\)-independent roundoff error. Accuracy is limited instead by roundoff sensitivity due to non-normality of the stability operators at large hydrodynamic and/or magnetic Reynolds numbers (\(Re,Rm\gtrsim 4\times 10^{4}\)). In problems with Hartmann velocity and magnetic-field profiles we employ suitable Gauss quadrature rules to evaluate the associated exponentially weighted sesquilinear forms without error. An alternative approach, which involves approximating the forms by means of Legendre-Gauss-Lobatto quadrature at the \(2p - 1\) precision level, is found to yield equal eigenvalues within roundoff error. As a consistency check, we compare modal growth rates to energy growth rates in nonlinear simulations and record relative discrepancy smaller than \(10^{-5}\) for the least stable mode in free-surface flow at \(Re=3\times 10^{4}\). Moreover, we confirm that the computed normal modes satisfy an energy conservation law for free-surface MHD with error smaller than \(10^{-6}\). The critical Reynolds number in free-surface MHD is found to be sensitive to the magnetic Prandtl number \(Pm\), even at the \(Pm=O(10^{-5})\) regime of liquid metals.

MSC:
76E25 Stability and instability of magnetohydrodynamic and electrohydrodynamic flows
76M10 Finite element methods applied to problems in fluid mechanics
76W05 Magnetohydrodynamics and electrohydrodynamics
65L60 Finite element, Rayleigh-Ritz, Galerkin and collocation methods for ordinary differential equations
76M25 Other numerical methods (fluid mechanics) (MSC2010)
Software:
ARPACK; Eigtool; LAPACK
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References:
[1] Müller, U.; Bühler, L., Magnetofluiddynamics in channels and containers, (2001), Springer Berlin
[2] Abdou, M.A., On the exploration of innovative concepts for fusion chamber technology, Fusion eng. design, 54, 181, (2001)
[3] Brooks, J., Overview of the ALPS program, Fusion sci. technol., 47, 669, (2005)
[4] Shannon, T.E., Conceptual design of the international fusion materials irradiation facility (IFMIF), J. nucl. mater., 258, 106, (1998)
[5] Alexakis, A., On heavy element enrichment in classical novae, Astrophys. J., 602, 931, (2004)
[6] Urpin, V., Instabilities, turbulence, and mixing in the Ocean of accreting neutron stars, Astron. astrophys., 438, 643, (2005)
[7] Balbus, S.A.; Henri, P., On the magnetic Prandtl number behavior of accretion disks, Astrophys. J., 674, 408, (2007)
[8] Kirchner, N.P., Computational aspects of the spectral Galerkin FEM for the orr – sommerfeld equation, Int. J. numer. meth. fluids, 32, 119, (2000) · Zbl 0979.76048
[9] Melenk, J.M.; Kirchner, N.P.; Schwab, C., Spectral Galerkin discretization for hydrodynamic stability problems, Computing, 65, 97, (2000) · Zbl 0967.76074
[10] Shen, J., Efficient spectral-Galerkin method I. direct solvers for the second and fourth order equations using Legendre polynomials, SIAM J. sci. comput., 15, 6, 1489, (1994) · Zbl 0811.65097
[11] Shen, J., Efficient chebyshev – legendre Galerkin methods for elliptic problems, (), 233
[12] D. Giannakis, R. Rosner, P.F. Fischer, Instabilities in free-surface Hartmann flow at low magnetic Prandtl numbers, J. Fluid Mech. (2008), submitted for publication. · Zbl 1183.76737
[13] Nornberg, M.D.; Ji, H.; Peterson, J.L.; Rhoads, J.R., A liquid metal flume for free surface magnetohydrodynamic experiments, Rev. sci. instr., 79, 094501, (2008)
[14] Ji, H.; Fox, W.; Pace, D.; Rappaport, H.L., Study of magnetohydrodynamic surface waves on liquid gallium, Phys. plasmas, 12, 012102, (2005)
[15] N. Katz, Open channel flow of liquid gallium in a transverse magnetic field, Bachelor Thesis, Princeton University, Princeton, NJ, 2004.
[16] Orszag, S.A., Accurate solution of the orr – sommerfeld stability equation, J. fluid mech., 50, 689, (1971) · Zbl 0237.76027
[17] Dongarra, J.J.; Straughan, B.; Walker, D.W., Chebyshev tau-QZ algorithm methods for calculating spectra of hydrodynamic stability problems, Appl. numer. math., 22, 399, (1996) · Zbl 0867.76025
[18] De Bruin, G.J., Stability of a layer of liquid flowing down an inclined plane, J. eng. math., 8, 3, 259, (1974) · Zbl 0279.76025
[19] Smith, M.K.; Davis, S.H., The instability of sheared liquid layers, J. fluid mech., 121, 187, (1982) · Zbl 0491.76045
[20] L.W. Ho, A Legendre spectral element method for simulation of incompressible unsteady viscous free-surface flows, Ph.D. Thesis, Massachusetts Institute of Technology, Cambridge, MA, 1989.
[21] Potter, M.C.; Kutchey, J.A., Stability of plane Hartmann flow subject to a transverse magnetic field, Phys. fluids, 16, 11, 1848, (1973) · Zbl 0273.76032
[22] Lingwood, R.J.; Alboussiere, T., On the stability of the Hartmann layer, Phys. fluids, 11, 8, 2058, (1999) · Zbl 1147.76446
[23] Dahlburg, R.B.; Zang, T.A.; Montgomery, D.; Hussaini, M.Y., Viscous, resistive magnetohydrodynamic stability computed by spectral techniques, Proc. natl. acad. sci. USA, 80, 5798, (1983)
[24] Takashima, M., The stability of the modified plane Poiseuille flow in the presence of a transverse magnetic field, Fluid dyn. res., 17, 293, (1996) · Zbl 1051.76562
[25] Takashima, M., The stability of the modified plane Couette flow in the presence of a transverse magnetic field, Fluid. dyn. res., 22, 105, (1998) · Zbl 1051.76563
[26] Drazin, P.G.; Reid, W.H., Hydrodynamic stability, (2004), Cambridge University Press Cambridge · Zbl 0513.76031
[27] McFadden, G.B.; Murray, B.T.; Boisvert, R.F., Elimination of spurious eigenvalues in the Chebyshev tau spectral method, J. comput. phys., 91, 228, (1990) · Zbl 0717.65063
[28] Straughan, B.; Walker, D.W., Two very accurate and efficient methods for computing eigenvalues and eigenfunctions in porous convection problems, J. comput. phys., 127, 1, 128, (1996) · Zbl 0858.76064
[29] Dawkins, P.T.; Dunbar, S.R.; Douglass, R.W., The origin and nature of spurious eigenvalues in the spectral tau method, J. comput. phys., 149, 441, (1998) · Zbl 0924.65077
[30] Schwab, C., p- and hp-finite element methods, numerical mathematics and scientific computation, (1998), Clarendon Press Oxford
[31] Hill, A.A.; Straughan, B., A Legendre spectral element method for eigenvalues in hydrodynamic stability, J. comp. appl. math., 193, 363, (2006) · Zbl 1092.65065
[32] Hill, A.A.; Straughan, B., Linear and non-linear stability thresholds for thermal convection in a box, Math. meth. appl. sci., 29, 2123, (2006) · Zbl 1104.76050
[33] Di Prima, R.C.; Habetler, G.J., A completeness theorem for non-selfadjoint eigenvalue problems in hydrodynamic stability, Arch. rat. mech. anal., 34, 218, (1969) · Zbl 0181.54703
[34] Trefethen, L.N., Hydrodynamic stability without eigenvalues, Science, 261, 578, (1993) · Zbl 1226.76013
[35] Trefethen, L.N.; Embree, M., Spectra and pseudospectra: the behavior of nonnormal matrices and operators, (2005), Princeton University Press Princeton · Zbl 1085.15009
[36] Schmid, P.J., A study of eigenvalue sensitivity for hydrodynamic stability operators, Theoret. comput. fluid dyn., 4, 227, (1993) · Zbl 0782.76036
[37] Mack, L.M., A numerical study of the temporal eigenvalue spectrum of the Blasius boundary layer, J. fluid mech., 73, 3, 497, (1976) · Zbl 0339.76030
[38] Reddy, S.C.; Schmid, P.J.; Henningson, D.S., Pseudospectra of the orr – sommerfeld operator, SIAM J. appl. math, 53, 1, 15, (1993) · Zbl 0778.34060
[39] Mach, R., Orthogonal polynomials with exponential weight in a finite interval and application to the optical model, J. math. phys., 25, 7, 2186, (1984)
[40] Ciarlet, P.G., Basic error estimates for elliptic problems, (), 17 · Zbl 0875.65086
[41] Deville, M.O.; Fischer, P.F.; Mund, E.H., High-order methods for incompressible fluid flow, Cambridge monographs on applied and computational mathematics, vol. 9, (2002), Cambridge University Press Cambridge · Zbl 1007.76001
[42] Banerjeee, U.; Osborn, J.E., Estimation of the effect of numerical integration in finite element eigenvalue approximation, Numer. math., 56, 735, (1990)
[43] Babuska, I.; Osborn, J., Eigenvalue problems, (), 641 · Zbl 0875.65087
[44] Shercliff, J.A., A textbook of magnetohydrodynamics, (1965), Pergamon Press Oxford · Zbl 0134.22101
[45] Stuart, J.T., On the stability of viscous flow between parallel planes in the presence of a co-planar magnetic field, Proc. roy. soc. A, 221, 1145, 189, (1954) · Zbl 0055.20709
[46] Batchelor, G.K., An introduction to fluid dynamics, (1967), Cambridge University Press Cambridge · Zbl 0152.44402
[47] Hof, A., Onset of oscillatory convection in molten gallium, J. fluid mech., 515, 391, (2004) · Zbl 1060.76546
[48] Hardy, S.C., The surface tension of liquid gallium, J. cryst. growth, 71, 602, (1985)
[49] Kolevzon, V., Anomalous temperature dependence of the surface tension and capillary waves at a liquid gallium surface, J. phys.: condens. matter, 11, 8785, (1999)
[50] Adams, R.A.; Fournier, J.J.F., Sobolev spaces, Pure and applied mathematics, vol. 140, (2003), Elsevier Science Oxford · Zbl 0347.46040
[51] Ho, L.W.; Patera, A.T., Variational formulation of three-dimensional viscous free-surface flows: natural imposition of surface tension boundary conditions, Int. J. numer. meth. fluids, 13, 691, (1991) · Zbl 0739.76057
[52] Moler, C.B.; Stewart, G.W., An algorithm for generalized matrix eigenproblems, SIAM J. numer. anal., 10, 2, 241, (1973) · Zbl 0253.65019
[53] Anderson, E., LAPACK users’ guide, (1999), SIAM Philadelphia · Zbl 0755.65028
[54] Lehoucq, R.B.; Sorensen, D.C.; Yang, C., Arpack user’s guide: solution of large-scale eigenvalue problems with implicitly restarted Arnoldi methods, (1998), SIAM Philadelphia · Zbl 0901.65021
[55] Abramowitz, M.; Stegun, I.A., Handbook of mathematical functions, (1971), Dover New York · Zbl 0515.33001
[56] Davis, P.J.; Rabinowitz, P., Methods of numerical integration, (2007), Dover New York · Zbl 0154.17802
[57] Lin, C.C., The theory of hydrodynamic stability, (1955), Cambridge University Press Cambridge · Zbl 0068.39202
[58] Yih, C.S., Stability of liquid flow down an inclined plane, Phys. fluids, 6, 3, 321, (1963) · Zbl 0116.19102
[59] Yih, C.S., Fluid mechanics, (1969), McGraw-Hill New York
[60] Lock, R.C., The stability of the flow of an electrically conducting fluid between parallel planes under a transverse magnetic field, Proc. roy. soc. London A, 233, 1192, (1955)
[61] Drazin, P.G., Stability of parallel flow in a parallel magnetic field, J. fluid mech., 8, 130, (1960) · Zbl 0091.19901
[62] Craik, A.D.D., Wave interactions and fluid flows, (1985), Cambridge University Press · Zbl 0172.56201
[63] Fornberg, B., A practical guide to pseudospectral methods, Cambridge monographs on applied and computational mathematics, vol. 1, (1996), Cambridge University Press Cambridge
[64] E. Rønquist, Optimal spectral element methods for the unsteady three-dimensional incompressible Navier-Stokes equations, Ph.D. Thesis, Massachusetts Institute of Technology, Cambridge, MA, 1988.
[65] Fischer, P.F., An overlapping Schwarz method for spectral element solution of the incompressible navier – stokes equations, J. comput. phys., 133, 84-101, (1997) · Zbl 0904.76057
[66] Gordon, W.; Hall, C., Transfinite element methods: blending-function interpolation over arbitrary curved element domains, Numer. math., 21, 109-129, (1973) · Zbl 0254.65072
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