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 Zh. Vychisl. Mat. Mat. Fiz., 2007, Volume 47, Number 10, Pages 1672–1691 (Mi zvmmf229)

Numerical analysis of the spectrum of the Orr–Sommerfeld problem

S. L. Skorokhodov

Dorodnicyn Computing Center, Russian Academy of Sciences, ul. Vavilova 40, Moscow, 119991, Russia

Abstract: A high-accuracy method for computing the eigenvalues $\lambda_n$ 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 $\mathrm R$ increases is studied in detail. For the Couette flow, it is shown that the eigenvalues $\lambda_n$ regarded as functions of $\mathrm R$ have a countable set of branch points $\mathrm R_k>0$ 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.

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English version:
Computational Mathematics and Mathematical Physics, 2007, 47:10, 1603–1621

Bibliographic databases:

UDC: 519.626

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

Citation in format AMSBIB
\Bibitem{Sko07} \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 \mathnet{http://mi.mathnet.ru/zvmmf229} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2388619} \elib{http://elibrary.ru/item.asp?id=9535225} \transl \jour Comput. Math. Math. Phys. \yr 2007 \vol 47 \issue 10 \pages 1603--1621 \crossref{https://doi.org/10.1134/S096554250710003X} \elib{http://elibrary.ru/item.asp?id=13540425} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-35648966671} 

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This publication is cited in the following articles:
1. O. V. Ilyin, “Stability analysis of the plane Couette flow for a model kinetic equation”, Comput. Math. Math. Phys., 49:5 (2009), 867–880
2. 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
3. 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
4. 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
5. 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
6. 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
7. 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
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