Global Galerkin Method for Stability Studies in Incompressible CFD and Other Possible Applications

Part of the Computational Methods in Applied Sciences book series (COMPUTMETHODS, volume 50)


In this paper the author reviews methodology of a version of the global Galerkin that was developed and applied in a series of his earlier publications. The method is based on divergence-free basis functions satisfying all the linear and homogeneous boundary conditions. The functions are defined as linear superpositions of the Chebyshev polynomials of the first and second types that are combined in divergence free vectors. The description and explanations of treatment of boundary conditions inhomogeneities and singularities are given. Possible implementation for steady state solvers, path-continuation, stability solvers and straight-forward integration in time are discussed. The most important results obtained using the approach are briefly reviewed and possible future applications are deliberated.


Weighted residual methods Spectral methods Galerkin method Instability Path-continuation 


  1. 1.
    Batina, J., Blancher, S., Amrouche, C., Batchi, M., Creff, R.: Convective heat transfer augmentation through vortex shedding in sinusoidal constricted tube. Int. J. Numer. Meths Heat Fluid Flow 19, 374–395 (2009)CrossRefGoogle Scholar
  2. 2.
    Blackburn, H.M., Lopez, J.M.: Symmetry breaking of the flow in a cylinder driven by a rotating endwall Phys. Fluids 12, 2698–2701 (2000)CrossRefGoogle Scholar
  3. 3.
    Bistrian, D.A., Dragomirescu, I.A., Muntean, S., Topor, M.: Numerical methods for convective hydrodynamic stability of swirling flows. In: Proceedings of the 13th WSEAS International Conference on Systems, pp. 283–288 (2009)Google Scholar
  4. 4.
    Borget, V., Bdéoui, F., Soufiani, A., Le Quéré, P.: The transverse instability in a differentially heated vertical cavity filled with molecular radiating gases. I. Linear stability analysis. Phys. Fluids 13, 1492–1507 (2001)CrossRefGoogle Scholar
  5. 5.
    Borońska, K., Tuckerman, L.S.: Extreme multiplicity in cylindrical Rayleigh-Bénard convection. I. Time dependence and oscillations. Phys. Rev. E 81, 036320 (2010)CrossRefGoogle Scholar
  6. 6.
    Borońska, K., Tuckerman, L.S.: Extreme multiplicity in cylindrical Rayleigh-Bénard convection. II. Bifurcation diagram and symmetry classification. Phys. Rev. E 81, 036321 (2010)CrossRefGoogle Scholar
  7. 7.
    Boyd, J.P.: Chebyshev and Fourier Spectral Methods. Dover Publications, New York (2000)Google Scholar
  8. 8.
    Buffat, M., Le Penven, L.: A spectral fictitious domain method with internal forcing for solving elliptic PDEs. J. Comput. Phys. 230, 2433–2450 (2011)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Canuto, C., Quarteroni, A., Hussaini, M.Y., Zhang, T.A.: Spectral Methods. Fundamentals in Single Domains. Springer, Berlin Heidelberg New York (2006)zbMATHGoogle Scholar
  10. 10.
    Dean, W.R.: ‘Fluid flow in a curved channel, Proc. R. Soc. Lond. Ser. A 121, 402–420 (1928)Google Scholar
  11. 11.
    Dumas, G., Leonard, A.: A divergence free spectral expansion method for three-dimensional flows in spherical gap geometries. J. Comput. Phys. 11, 205–219 (1994)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Erenburg, V., Gelfgat, A.Y., Kit, E., Bar-Yoseph, P.Z., Solan, A.: Multiple states, stability, bifurcations of natural convection in rectangular cavity with partially heated vertical walls. J. Fluid Mech. 492, 63–89 (2003)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Escudier, M.P.: Observation of the flow produced in a cylindrical container by a rotating endwall. Exp. Fluids 2, 189–196 (1984)CrossRefGoogle Scholar
  14. 14.
    Fletcher, C.A.J.: Computational Galerkin Methods. Springer, New York (1984)CrossRefGoogle Scholar
  15. 15.
    Ganske, A., Gebhardt, T., Grossmann, S.: Modulation effects along stability border in Taylor-Couette flow. Phys. Fluids 6, 3823–3832 (1994)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Gelfgat, A.Y.: Instability and oscillatory supercritical regimes of free convection in a laterally heated square cavity, PhD (Cand. Sci.) thesis. Latvian State University, Riga, Latvia (1988)Google Scholar
  17. 17.
    Gelfgat, A.Y., Tanasawa, I.: Numerical analysis of oscillatory instability of buoyancy convection with the Galerkin spectral method. Numer. Heat Transfer. Part A: Applications 25, 627–648 (1994)CrossRefGoogle Scholar
  18. 18.
    Gelfgat, A.Y., Bar-Yoseph, P.Z., Solan, A.: Stability of confined swirling flow with, without vortex breakdown. J. Fluid Mech. 311, 1–36 (1996)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Gelfgat, A.Y., Bar-Yoseph, P.Z., Solan, A.: Steady states, oscillatory instability of swirling flow in a cylinder with rotating top, bottom. Phys. Fluids 8, 2614–2625 (1996)CrossRefGoogle Scholar
  20. 20.
    Gelfgat, A.Y., Bar-Yoseph, P.Z., Yarin, A.L.: On oscillatory instability of convective flows at low Prandtl number. J. Fluids Eng. 119, 823–830 (1997)CrossRefGoogle Scholar
  21. 21.
    Gelfgat, A.Y., Bar-Yoseph, P.Z., Yarin, A.L.: Stability of multiple steady states of convection in laterally heated cavities. J. Fluid Mech. 388, 315–334 (1999)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Gelfgat, A.Y., Bar-Yoseph, P.Z., Yarin, A.L.: Non-Symmetric convective flows in laterally heated rectangular cavities. Int. J. Comput. Fluid Dyn. 11, 261–273 (1999)CrossRefGoogle Scholar
  23. 23.
    Gelfgat, A.Y., Bar-Yoseph, P.Z., Solan, A., Kowalewski, T.: An axisymmetry- breaking instability in axially symmetric natural convection. Int. J. Transp. Phenom. 1, 173–190 (1999)Google Scholar
  24. 24.
    Gelfgat, A.Y.: Different modes of Rayleigh-Bénard instability in two-, three-dimensional rectangular enclosures. J. Comput. Phys. 156, 300–324 (1999)CrossRefGoogle Scholar
  25. 25.
    Gelfgat, A.Y., Bar-Yoseph, P.Z., Solan, A.: Axisymmetry breaking instabilities of natural convection in a vertical Bridgman growth configurations. J. Cryst. Growth 220, 316–325 (2000)CrossRefGoogle Scholar
  26. 26.
    Gelfgat, A.Y.: Two-, three-dimensional instabilities of confined flows: numerical study by a global Galerkin method. Comput. Fluid Dyn. J. 9, 437–448 (2001)Google Scholar
  27. 27.
    Gelfgat, A.Y., Bar-Yoseph, P.Z., Solan, A.: Three-dimensional instability of axisymmetric flow in a rotating lid - cylinder enclosure. J. Fluid Mech. 438, 363–377 (2001)CrossRefGoogle Scholar
  28. 28.
    Gelfgat, A.Y., Bar-Yoseph, P.Z., Solan, A.: Effect of axial magnetic field on three-dimensional instability of natural convection in a vertical Bridgman growth configuration. J. Cryst. Growth 230, 63–72 (2001)CrossRefGoogle Scholar
  29. 29.
    Gelfgat, A.Y., Bar-Yoseph, P.Z.: The effect of an external magnetic field on oscillatory instability of convective flows in a rectangular cavity. Phys. Fluids 13, 2269–2278 (2001)CrossRefGoogle Scholar
  30. 30.
    Gelfgat, A.Y., Yarin, A.L., Bar-Yoseph, P.Z.: Three-dimensional instability of a two-layer Dean flow. Phys. Fluids 13, 3185–3195 (2001)CrossRefGoogle Scholar
  31. 31.
    Gelfgat, A.Y.: Three-dimensionality of trajectories of experimental tracers in a steady axisymmetric swirling flow: effect of density mismatch. Theoret. Comput. Fluid Dyn. 16, 29–41 (2002)CrossRefGoogle Scholar
  32. 32.
    Gelfgat, A.Y.: Stability, slightly supercritical oscillatory regimes of natural convection in a 8:1 cavity: solution of benchmark problem by a global Galerkin method. Int. J. Numer. Meths. Fluids 44, 135–146 (2004)MathSciNetCrossRefGoogle Scholar
  33. 33.
    Gelfgat, A.Y., Bar-Yoseph, P.Z.: Multiple solutions, stability of confined convective, swirling flows—a continuing challenge. Int. J. Numer. Meth. Heat, Fluid Flow 14, 213–241 (2004)MathSciNetCrossRefGoogle Scholar
  34. 34.
    Gelfgat, Y.M., Gelfgat, A.Y.: Experimental, numerical study of rotating magnetic field driven flow in cylindrical enclosures with different aspect ratios. Magnetohydrodynamics 40, 147–160 (2004)Google Scholar
  35. 35.
    Gelfgat, A.Y.: On three-dimensional instability of a traveling magnetic field driven flow in a cylindrical container. J. Crystal Growth 279, 276–288 (2005)CrossRefGoogle Scholar
  36. 36.
    Gelfgat, A.Y.: Implementation of arbitrary inner product in global Galerkin method for incompressible Navier-Stokes equation. J. Comput. Phys. 211, 513–530 (2006)MathSciNetCrossRefGoogle Scholar
  37. 37.
    Gelfgat, A.Y.: Stability of convective flows in cavities: solution of benchmark problems by a low-order finite volume method. Int. J. Numer. Meths. Fluids 53, 485–506 (2007)CrossRefGoogle Scholar
  38. 38.
    Gelfgat, A.Y.: Three-dimensional instability of axisymmetric flows: solution of benchmark problems by a low-order finite volume method. Int. J. Numer. Meths. Fluids 54, 269–294 (2007)MathSciNetCrossRefGoogle Scholar
  39. 39.
    Gelfgat, A.Y.: Visualization of three-dimensional incompressible flows by quasi-two-dimensional divergence-free projections. Comput. Fluids 97, 143–155 (2014)MathSciNetCrossRefGoogle Scholar
  40. 40.
    Godeferd, F.S., Lollini, L.: Direct numerical simulations of turbulence with confinement and rotation. J. Fluid Mech. 393, 257–308 (1999)CrossRefGoogle Scholar
  41. 41.
    Grants, I., Gerbeth, G.: Stability of axially symmetric flow driven by a rotating magnetic field in a cylindrical cavity. J. Fluid Mech. 431, 407–426 (2001)CrossRefGoogle Scholar
  42. 42.
    Gresho, P.M., Sani, R.L.: On pressure boundary conditions for the incompressible Navier-Stokes equations. Int. J. Numer. Meth. Fluids 7, 1111–1145 (1987)CrossRefGoogle Scholar
  43. 43.
    Hassard, B.D., Kazarinoff, N.D., Wan, Y.H.: Theory and Applications of Hopf Bifurcation. London Mathematical Society Lecture Note Series, vol. 41 (1981)Google Scholar
  44. 44.
    Holte, S.: Numerical experiments with a three-dimensional model of an enclosed basin. Cont. Shelf Res. 2, 301–315 (1983)CrossRefGoogle Scholar
  45. 45.
    Iwatsu, R.: Numerical study of flows in a cylindrical container with rotating bottom and top free surface. J. Phys. Soc. Jpn. 74(333), 344 (2005)zbMATHGoogle Scholar
  46. 46.
    Kerr, R.: Rayleigh number scaling in numerical convection. J. Fluid Mech. 310, 139–179 (1996)CrossRefGoogle Scholar
  47. 47.
    Marques, F., Lopez, J.M.: Precessing vortex breakdown mode in an enclosed cylinder flow. Phys. Fluids 13, 1679–1682 (2001)MathSciNetCrossRefGoogle Scholar
  48. 48.
    Marques, F., Lopez, J.M., Shen, J.: Mode interactions in an enclosed swirling flow: a double Hopf bifurcation between azimuthal wavenumbers 0 and 2. J. Fluid Mech. 455, 263–281 (2002)MathSciNetCrossRefGoogle Scholar
  49. 49.
    Marques, F., Gelfgat, A.Y., Lopez, J.M.: Tangent double Hopf bifurcation in a differentially rotating cylinder flow. Phys. Rev. E 68, 06310-1–06310-13 (2003)Google Scholar
  50. 50.
    Mason, J.C., Handscomb, D.C.: Chebyshev Polynomials. CRC Press, London (2003)zbMATHGoogle Scholar
  51. 51.
    Meseguer, A., Mellibovsky, F.: On a solenoidal Fourier–Chebyshev spectral method for stability analysis of the Hagen–Poiseuille flow, Appl. Numer. Math., 920–938 (2007)MathSciNetCrossRefGoogle Scholar
  52. 52.
    Moser, R.D., Moin, P., Leonard, A.: A spectral numerical method for the Navier-Stokes equations with applications to Taylor-Couette flow. J. Comput. Phys. 52, 524–544 (1983)MathSciNetCrossRefGoogle Scholar
  53. 53.
    Nore, C., Tuckerman, L.S., Daube, O., Xin, S.: The 1:2 mode interaction in exactly counter-rotating von Karman swirling flow. J. Fluid Mech. 477, 51–88 (2003)MathSciNetCrossRefGoogle Scholar
  54. 54.
    Orszag, S.A.: Galerkin approximations to flows within slabs, spheres, and cylinders. Phys. Rev. Lett. 26, 1100–1103 (1971)CrossRefGoogle Scholar
  55. 55.
    Orszag, S.A.: Accurate solution of the Orr-Sommerfeld stability equation. J. Fluid Mech. 50, 689–703 (1971)CrossRefGoogle Scholar
  56. 56.
    Pasquarelli, F.: Domain decomposition for spectral approximation to Stokes equations via divergence-free functions. Appl. Umr. Math. 8, 493–551 (1991)MathSciNetGoogle Scholar
  57. 57.
    Paszkowski, S.: Numerical Applications of Chebyshev Polynomials and Series. PWN, Warsaw (in Polish) (1975)zbMATHGoogle Scholar
  58. 58.
    Picardo, J.R., Garg, P., Pushpavanam, S.: Centrifugal instability of stratified two-phase flow in a curved channel. Phys. Fluids 27, 054106 (2015)CrossRefGoogle Scholar
  59. 59.
    Rempfer, D.: On boundary conditions for incompressible Navier-Stokes problems. Appl. Mech. Rev. 59, 107–125 (2006)CrossRefGoogle Scholar
  60. 60.
    Rubinov, A., Erenburg, V., Gelfgat, A.Y., Kit, E., Bar-Yoseph, P.Z., Solan, A.: Three-dimensional instabilities of natural convection in a vertical cylinder with partially heated sidewalls. J. Heat Transfer 126, 586–599 (2004)CrossRefGoogle Scholar
  61. 61.
    Sørensen, J.N., Naumov, I., Mikkelsen, R.: Experimental investigation of three-dimensional flow instabilities in a rotating lid-driven cavity. Exp. Fluids 41, 425–440 (2006)CrossRefGoogle Scholar
  62. 62.
    Sørensen, J.N., Gelfgat, A.Y., Naumov, I., Mikkelsen, R.: Experimental and numerical results on three-dimensional instabilities in a rotating disk–tall cylinder flow. Phys. Fluids 21, 054102 (2009)CrossRefGoogle Scholar
  63. 63.
    Suslov, S.A., Paolucci, S.: A Petrov-Galerkin method for flows in cavities: enclosure of aspect ratio 8. Int. J. Numer. Meths. Fluids 40, 999–1007 (2002)CrossRefGoogle Scholar
  64. 64.
    Tuckerman, L.S., Langham, J., Willis, A.: Stokes preconditioning in Channelflow and Openpipeflow for steady states and traveling waves. In: < this book> (2018)Google Scholar
  65. 65.
    Uhlmann, M., Nagata, M.: Linear stability of flow in an internally heated rectangular duct. J. Fluid Mech. 551, 387–404 (2006)CrossRefGoogle Scholar
  66. 66.
    Wan, X., Yu, H.: Dynamic-solver–consistent minimum action method: With an application to 2D Navier-Stokes equations. J. Comput. Phys. 331, 209–226 (2017)MathSciNetCrossRefGoogle Scholar
  67. 67.
    Yahata, H.: Stability analysis of natural convection in vertical cavities with lateral heating. J. Phys. Soc. Jpn. 68, 446–460 (1999)CrossRefGoogle Scholar
  68. 68.
    Yahata, H.: Stability analysis of natural convection evolving along a vertical heated plate. J. Phys. Soc. Jpn. 70, 11–130 (2001)CrossRefGoogle Scholar
  69. 69.
    Yang, W.M.: Stability of viscoelastic fluids in a modulated gravitational field. Int. J. Heat Mass Transfer 40, 1401–1410 (1997)CrossRefGoogle Scholar
  70. 70.
    Yueh, C.S., Weng, C.I.: Linear stability analysis of plane Couette flow with viscous heating. Phys. Fluids 8, 1802–1813 (1996)CrossRefGoogle Scholar
  71. 71.
    Zebib, A.: A Chebyshev method for the solution of boundary value problems. J. Comput. Phys. 53, 443–455 (1984)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2019

Authors and Affiliations

  1. 1.School of Mechanical Engineering, Faculty of EngineeringTel-Aviv UniversityTel-AvivIsrael

Personalised recommendations