Skip to main content
SpringerLink
Log in

Numerical Implementation of Nonstationary Axisymmetric Problems of an Ideal Incompressible Fluid with a Free Surface

  • Published:
Journal of Applied Mechanics and Technical Physics Aims and scope

Abstract

A fundamentally new unsaturated technique for the numerical solution of the Dirichlet–Neumann problem for the Laplace equation was developed. This technique makes it possible, due to the smoothness of the sought solution of the problem, to take into account the axisymmetric specificity of the problem which is an insurmountable obstacle to any saturated numerical methods, i.e., methods with a leading error term.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. L. V. Ovsyannikov, “The Plane Problem of Unsteady Fluid Motion with Free Boundaries,” in Dynamics of Continuous Media, No. 8 (Inst. of Hydrodynamics, Sib. Branch, USSR Acad. of Sci., 1971), pp. 22–26.

    Google Scholar 

  2. L. V. Ovsyannikov, “Cauchy Problem in the Scale of Banach Spaces,” Tr. Inst. Mat. Steklova 281, 7–15 (2013).

    MathSciNet  Google Scholar 

  3. C. Bardos and E. S. Titi, “Euler Equations for an Ideal Incompressible Fluid,” Usp. Mat. Nauk 62 (3), 3–46 (2007).

    Article  MathSciNet  MATH  Google Scholar 

  4. V. N. Belykh, “On the Evolution of a Finite Volume of Ideal Incompressible Fluid with a Free Surface,” Dokl. Akad. Nauk 473 (6), 650–654 (2017) [Dokl. Phys. 62 (4), 213–217 (2017)]; DOI:10.7868/S0869565217120052.

    MathSciNet  Google Scholar 

  5. K. I. Babenko and V. Yu. Petrovich, “Demonstrative Computations,” Preprint No. 133 (Inst. of Mathematics, USSR Acad. of Sci., Moscow, 1983).

    Google Scholar 

  6. K. I. Babenko, Basics of Numerical Analysis (RKhD, Izhevsk, 2002) [in Russian].

    Google Scholar 

  7. V. N. Belykh, “On Kolmogorov’s ε-Entropy for Compact Sets of Infinitely Differentiable Aperiodic Functions (Babenko’s Problem),” Dokl. Akad. Nauk 482 (2), 125–129 (2018) [Dokl. Math. 98 (2), 416–420 (2018); DOI: 10.1134/S1064562418060066].

    Google Scholar 

  8. V. N. Belykh “ Exterior Axisymmetric Neumann Problem for the Laplace Equation: Numerical Algorithms without Saturation,” Dokl Akad. Nauk 417 (4), 442–445 (2007) [Dokl. Math. 76 (3), 882–885 (2007); DOI: 10.1134/S1064562407060208].

    MATH  Google Scholar 

  9. V. N. Belykh “Particular Features of Implementation of an Unsaturated Numerical Method for the Exterior Axisymmetric Neumann Problem,” Sib. Mat. Zh. 54 (6), 1237–1249 (2013) [Sib. Math. J. 54 (6), 984–993, (2013); DOI: 10.1134/S0037446613060037].

    Article  MathSciNet  MATH  Google Scholar 

  10. V. N. Belykh, “The Problem of Constructing Unsaturated Quadrature Formulae on an Interval,” Mat. Sb. 210 (1), 27–62 (2019); DOI: https://doi.org/10.4213/sm8984 [Sbornik: Mathematics 210 (1), 24–58 (2019); https://doi.org/10.1070/SM8984].

    Article  MathSciNet  MATH  Google Scholar 

  11. N. M. Gu¨nter, Potential Theory and Its Applications to Basic Problems of Mathematical Physics (Gostekhteoretizdat, Moscow, 1953; Unfar, New York, 1967).

    Google Scholar 

  12. V. N. Belykh, “The Problem of Numerical Realization of Integral Operators of Axisymmetric Boundary Value Problems (Algorithms without Saturation),” Ufim. Mat. Zh. 4 (4) 22–37 (2012) [Ufa Math. J. 4 (4), 22–37 (2012); http://matem.anrb.ru/sites/default/files/files/vupe16/Belykh.pdf]

    MathSciNet  Google Scholar 

  13. V. N. Belykh, “Algorithms for Computing Complete Elliptic Integrals and Some Related Functions,” Sib. Zh. Indust. Mat. 15 (2), 21–32 (2012). [J. Appl. Indust. Math. 6 (4), 410–420 (2012), DOI: 10.1134/S1990478912040023]; http://mi.mathnet.ru/sjim722].

    MathSciNet  MATH  Google Scholar 

  14. H. F. Sinwel, “Uniform Approximation of Differentiable Functions by Algebraic Polynomials,” J. Approx. Theory. 32 (1), 1–8 (1981).

    Article  MathSciNet  MATH  Google Scholar 

  15. V. N. Belykh, “On Solving the Problem of an Ideal Incompressible Fluid Flow around Large-Aspect-Ratio Axisymmetric Bodies,” Prikl. Mat. Tekh. Fiz. 47 (5), 56–67 (2006) [J. Appl. Mech. Tech. Phys. 47 (5), 661–670 (2006); https://doi.org/10.1007/s10808-006-0102-4].

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Sobolev Institute of Mathematics, Siberian Branch, Russian Academy of Sciences, Novosibirsk, 630090, Russia

    V. N. Belykh

Authors
  1. V. N. Belykh
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to V. N. Belykh.

Additional information

Original Russian Text © V.N. Belykh.

Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 60, No. 2, pp. 226–237, March–April, 2019.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Belykh, V.N. Numerical Implementation of Nonstationary Axisymmetric Problems of an Ideal Incompressible Fluid with a Free Surface. J Appl Mech Tech Phy 60, 382–391 (2019). https://doi.org/10.1134/S0021894419020196

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0021894419020196

Keywords

Search

Navigation