Abstract:
A high-accuracy method for computing the eigenvalues λn and the eigenfunctions of the Orr–Sommerfeld operator is developed. The solution is represented as a combination of power series expansions, and the latter are then matched. The convergence rate of the expansions is analyzed by applying the theory of recurrence equations. For the Couette and Poiseuille flows in a channel, the behavior of the spectrum as the Reynolds number R increases is studied in detail. For the Couette flow, it is shown that the eigenvalues λn regarded as functions of R have a countable set of branch points Rk>0 at which the eigenvalues have a multiplicity of 2. The first ten of these points are presented within ten decimals.
Key words:
Orr–Sommerfeld equation, numerical analysis of the spectrum of the Orr–Sommerfeld equation, Couette flow,
Poiseuille flow, Couette–Poiseuille flow, convergence rate analysis.
Citation:
S. L. Skorokhodov, “Numerical analysis of the spectrum of the Orr–Sommerfeld problem”, Zh. Vychisl. Mat. Mat. Fiz., 47:10 (2007), 1672–1691; Comput. Math. Math. Phys., 47:10 (2007), 1603–1621
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\by S.~L.~Skorokhodov
\paper Numerical analysis of the spectrum of the Orr--Sommerfeld problem
\jour Zh. Vychisl. Mat. Mat. Fiz.
\yr 2007
\vol 47
\issue 10
\pages 1672--1691
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\jour Comput. Math. Math. Phys.
\yr 2007
\vol 47
\issue 10
\pages 1603--1621
\crossref{https://doi.org/10.1134/S096554250710003X}
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Linking options:
https://www.mathnet.ru/eng/zvmmf229
https://www.mathnet.ru/eng/zvmmf/v47/i10/p1672
This publication is cited in the following 20 articles:
S. L. Skorokhodov, N. P. Kuzmina, “Analytical-Numerical Method for Solving the Spectral Problem in a Model of Geostrophic Ocean Currents”, Comput. Math. and Math. Phys., 64:6 (2024), 1240
N. P. Kuzmina, S. L. Skorokhodov, N. V. Zhurbas, D. A. Lyzhkov, “On the Types of Instability of a Geostrophic Current with a Vertical Parabolic Profile of Velocity”, Izv. Atmos. Ocean. Phys., 59:2 (2023), 201
A. D. Nizamova, V. N. Kireev, S. F. Urmancheev, “Influence of Temperature Dependence of Viscosity on the Stability of Fluid Flow in an Annular Channel”, Lobachevskii J Math, 44:5 (2023), 1778
Sen Zou, Chengwen Zhong, Lin Bi, Xianxu Yuan, Zhigong Tang, “A new linear stability analysis approach for microchannel flow based on the Boltzmann Bhatnagar–Gross–Krook equation”, Physics of Fluids, 34:12 (2022)
È. A. Biberdorf, “Algorithm for separating matrix spectrum by an angle”, Comput. Math. Math. Phys., 62:5 (2022), 719–732
Nizamova A.D., Murtazina R.D., Kireev V.N., Urmancheev S.F., “Features of Laminar-Turbulent Transition For the Coolant Flow in a Plane Heat-Exchanger Channel”, Lobachevskii J. Math., 42:9, SI (2021), 2211–2215
Lafitte O., “Unstable Spectrum of a Rayleigh-Benard System With Variable Viscosity”, C. R. Math., 359:9 (2021), 1165–1178
S. L. Skorokhodov, N. P. Kuzmina, “Spectral analysis of small perturbations of geostrophic currents with a parabolic vertical profile of velocity as applied to the ocean”, Comput. Math. Math. Phys., 61:12 (2021), 1966–1979
Kuzmina N.P. Skorokhodov S.L. Zhurbas V N. Lyzhkov D.A., “Effects of Friction and Buoyancy Diffusion on the Dynamics of Geostrophic Oceanic Currents With a Linear Vertical Velocity Profile”, Izv. Atmos. Ocean. Phys., 56:6 (2020), 591–602
S. L. Skorokhodov, N. P. Kuzmina, “On the influence of the beta effect on the spectral characteristics of unstable perturbations of ocean currents”, Comput. Math. Math. Phys., 60:11 (2020), 1900–1912
Kuzmina N.P. Skorokhodov S.L. Zhurbas N.V. Lyzhkov D.A., “Description of the Perturbations of Oceanic Geostrophic Currents With Linear Vertical Velocity Shear Taking Into Account Friction and Diffusion of Density”, Izv. Atmos. Ocean. Phys., 55:2 (2019), 207–217
S. L. Skorokhodov, N. P. Kuzmina, “Spectral analysis of model Couette flows in application to the ocean”, Comput. Math. Math. Phys., 59:5 (2019), 815–835
A. D. Nizamova, V. N. Kireev, S. F. Urmancheev, “Research of eigenfuctions perturbation of the transverse component velocity thermoviscous liquids flow”, Proceedings of the Mavlyutov Institute of Mechanics, 14:2 (2019), 132–137
A. D. Nizamova, V. N. Kireev, S. F. Urmancheev, “Influence of viscosity temperature dependence on the spectral characteristics of the thermoviscous liquids flow stability equation”, Proceedings of the Mavlyutov Institute of Mechanics, 14:1 (2019), 52–58
Kuzmina N.P., Skorokhodov S.L., Zhurbas N.V., Lyzhkov D.A., “On Instability of Geostrophic Current With Linear Vertical Shear At Length Scales of Interleaving”, Izv. Atmos. Ocean. Phys., 54:1 (2018), 47–55
S. L. Skorokhodov, N. P. Kuzmina, “Analytical-numerical method for solving an Orr–Sommerfeld-type problem for analysis of instability of ocean currents”, Comput. Math. Math. Phys., 58:6 (2018), 976–992
M. K. Kerimov, “The theory of regularized traces of operators as applied to approximate computation of eigenvalues and eigenfunctions of fluid dynamics problems”, Comput. Math. Math. Phys., 52:5 (2012), 756–786
Ortiz de Zarate J.M., Sengers J.V., “Hydrodynamic fluctuations in laminar fluid flow. I. Fluctuating Orr-Sommerfeld equation”, J. Stat. Phys., 144:4 (2011), 774–792
O. V. Ilyin, “Stability analysis of the plane Couette flow for a model kinetic equation”, Comput. Math. Math. Phys., 49:5 (2009), 867–880
Ilyin O.V., “Analysis of instability of the plane Couette flow by means of kinetic theory approach”, Rarefied gas dynamics, AIP Conference Proceedings, 1084, 2009, 218–223
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