Advertisement

Computational Aspects of a Time Evolution Scheme for Incompressible Boussinesq Navier-Stokes in a Cylinder

  • Damián CastañoEmail author
  • María Cruz Navarro
  • Henar Herrero
Conference paper
Part of the Lecture Notes in Computational Science and Engineering book series (LNCSE, volume 119)

Abstract

In this work we show some computational aspects of the implementation of a three dimensional spectral time evolution scheme for incompressible Boussinesq Navier-Stokes including rotation effects in a cylinder with a primitive variable formulation. The scheme is a second-order time-splitting method combined with pseudo-spectral Fourier Chebyshev in space. To deal with the singularity at the origin a radial expansion is considered in the diameter of the cylinder. The order expansion in the radial coordinate gets doubled. We develop a matrix processing that combines the use of the parity of the fields and the discretization functions to cancel half of the terms in the matrix reducing the radial dimension to the original one.

Notes

Acknowledgements

This work was partially supported by the Research Grant GI20163529 (UCLM) and MTM2015-68818-R MINECO (Spanish Government), which includes RDEF funds.

References

  1. 1.
    C.K. Batchelor, An Introduction to Fluid Dynamics (Cambridge University Press, Cambridge, 1967)zbMATHGoogle Scholar
  2. 2.
    J. Boyd, Chebyshev and Fourier Spectral Methods (Dover, New York, 2001)zbMATHGoogle Scholar
  3. 3.
    C. Canuto, M.Y. Hussain, A. Quarteroni, T.A. Zang, Spectral Methods in Fluid Dynamics (Springer, Berlin, 1988)CrossRefzbMATHGoogle Scholar
  4. 4.
    D. Castaño, M.C. Navarro, H. Herrero, Secondary whirls in thermoconvective vortices developed in a cylindrical annulus locally heated from below. Commun. Nonlinear Sci. Numer. Simul. 28(1–3), 201–209 (2015)CrossRefGoogle Scholar
  5. 5.
    D. Castaño, M.C. Navarro, H. Herrero, Evolution of secondary whirls in thermoconvective vortices: strengthening, weakening and disappearance in the route to chaos. Phys. Rev. E (2016). doi:10.1103/PhysRevE.93.013117Google Scholar
  6. 6.
    D. Castaño, M.C. Navarro, H. Herrero, Double vortices and single-eyed vortices in a rotating cylinder non-homogeneously heated. Comp. and Math. with Appl. 73, 2238–2257 (2017)CrossRefzbMATHGoogle Scholar
  7. 7.
    S. Chandrasekhar, Hydrodynamic and Hydromagnetic Stability (Dover Publications, New York, 1981)Google Scholar
  8. 8.
    R. Chokri, B. Brahim, Three-dimensional natural convection of molten Lithium in a differentially heated rotating cubic cavity about a vertical ridge. Powder Technol. 291, 97–109 (2006)CrossRefGoogle Scholar
  9. 9.
    K.A. Emanuel, Divine Wind (Oxford University Press, Oxford, 2005)Google Scholar
  10. 10.
    B. Fornberg, A Practical Guide to Pseudospectral Methods (Cambridge University Press, Cambridge, 1998)zbMATHGoogle Scholar
  11. 11.
    K. Goda, A multistep technique with implicit difference schemes for calculating two- and three-dimensional cavity flows. J. Comput. Phys. 30, 76–95 (1979)CrossRefzbMATHGoogle Scholar
  12. 12.
    P. Gresho, On the theory of semi-implicit projection methods for viscous incompressible flow and its implementation via finite element method that also introduces a nearly consistent mass matrix. Int. J. Numer. Meths. Fluids 11, 587–620 (1990)CrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    H. Herrero, A.M. Mancho, On pressure boundary conditions for thermoconvective problems. Int. J. Numer. Methods Fluids 39, 391–402 (2002)CrossRefzbMATHGoogle Scholar
  14. 14.
    S. Hugues, A. Randriamampianina, An improved projection scheme applied to pseudospectral methods for the incompressible Navier-Stokes equations. Int. J. Numer. Methods Fluids 28, 501–521 (1998)CrossRefzbMATHMathSciNetGoogle Scholar
  15. 15.
    C.L. Jordan, Marked changes in the characteristics of the eye of intense typhoons between the deeping and filling stages. J. Meteorol. 18, 779–789 (1961)CrossRefGoogle Scholar
  16. 16.
    G.E. Karniadakis, M. Israeli, S.A. Orsaq, High order splitting methods for the incompressible Navier-Stokes equations. J. Comput. Phys. 97, 414–443 (1991)CrossRefzbMATHMathSciNetGoogle Scholar
  17. 17.
    A.M. Mancho, H. Herrero, J. Burguete, Primary instabilities in convective cells due to nouniform heating. Phys. Rev. E 56, 2916–2923 (1997)CrossRefGoogle Scholar
  18. 18.
    I. Mercader, O. Batiste, A. Alonso, An efficient spectral code for incompressible flows in cylindrical geometries. Comput. Fluids 39, 215–224 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  19. 19.
    M.C. Navarro, A.M. Mancho, H. Herrero, Instabilities in Buoyant flows under localized heating. Chaos Interdisciplinary J. Nonlinear Sci. 17, 023105-1-12 (2007)Google Scholar
  20. 20.
    Lord Rayleigh, on convective currents in a horizontal layer of fluid when the temperature is on the under side. Phil. Mag. 32, 529–46 (1916)Google Scholar
  21. 21.
    P.C. Sinclair, The lower structure of dust devils. J. Atmos. Sci. 30, 1599–1619 (1973)CrossRefGoogle Scholar
  22. 22.
    L.N. Trefethen, Spectral Methods in Matlab (SIAM, Philadelfia, 2000)CrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • Damián Castaño
    • 1
    • 2
    Email author
  • María Cruz Navarro
    • 2
  • Henar Herrero
    • 2
  1. 1.Instituto de Matemática Aplicada a la Ciencia y la Ingeniería (IMACI)Universidad de Castilla-La ManchaCiudad RealSpain
  2. 2.IMACIUniversidad de Castilla-La ManchaCiudad RealSpain

Personalised recommendations