Theoretical and Computational Fluid Dynamics

, Volume 33, Issue 1, pp 59–82 | Cite as

Linear instability of the lid-driven flow in a cubic cavity

  • Alexander Yu. GelfgatEmail author
Original Article


Primary instability of the lid-driven flow in a cube is studied by a linear stability approach. Two cases, in which the lid moves parallel to the cube sidewall or parallel to the diagonal plane, are considered. It is shown that Krylov vectors required for application of the Newton and Arnoldi iteration methods can be evaluated by the SIMPLE procedure. The finite volume grid is gradually refined from \(100^{3}\) to \(256^{3}\) nodes. The computations result in grid converging values of the critical Reynolds number and oscillation frequency that allow for Richardson extrapolation to the zero grid size. Three-dimensional flow and most unstable perturbations are visualized by a recently proposed approach that allows for a better insight into the flow patterns and appearance of the instability. New arguments regarding the assumption that the centrifugal mechanism triggers the instability are given for both cases.


Lid-driven cavity flow Newton method Arnoldi method Linear stability Krylov-subspace iteration SIMPLE 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


Supplementary material

162_2019_483_MOESM1_ESM.avi (3.6 mb)
Supplementary material 1 (avi 3703 KB)
162_2019_483_MOESM2_ESM.avi (3.9 mb)
Supplementary material 2 (avi 3964 KB)
162_2019_483_MOESM3_ESM.avi (3 mb)
Supplementary material 3 (avi 3116 KB)
162_2019_483_MOESM4_ESM.avi (6.3 mb)
Supplementary material 4 (avi 6430 KB)


  1. 1.
    Shankar, P.N., Deshpande, M.D.: Fluid mechanics in the driven cavity. Ann. Rev. Fluid Mech. 32, 93–136 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Kuhlmann, H.C., Romano F.: The lid-driven cavity. In: Computational Modelling of Bifurcations and Instabilities in Fluid Dynamics. In: A. Gelfgat (ed.) Springer, Berlin (2018)Google Scholar
  3. 3.
    Deshmuck, R., McNamara, J.J., Liang, Z., Kolter, J.Z., Abhijit, G.: Model order reduction using sparse coding exemplified for the lid-driven cavity. J. Fluid Mech. 808, 189–223 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Kalita, J.C., Gogoi, B.B.: A biharmonic approach for the global stability analysis of 2D incompressible viscous flows. Appl. Math Model. 40, 6831–6849 (2016)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Nuriev, A.N., Egorov, A.G., Zaitseva, O.N.: Bifurcation analysis of steady-state flows in the lid-driven cavity. Fluid Dyn. Res. 48, 061405 (2016)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Babu, V., Korpela, S.A.: Numerical solution of the incompressible, three-dimensional Navier–Stokes equations. Comput. Fluids 23, 675–691 (1994)CrossRefzbMATHGoogle Scholar
  7. 7.
    Albensoeder, S., Kuhlmann, H.C.: Accurate three-dimensional lid-driven cavity flow. J. Comput. Phys. 206, 536–558 (2006)CrossRefzbMATHGoogle Scholar
  8. 8.
    Liberzon, A., Feldman, Y., Gelfgat, A.Y.: Experimental observation of the steady—oscillatory transition in a cubic lid-driven cavity. Phys. Fluids 23, 084106 (2011)CrossRefGoogle Scholar
  9. 9.
    Feldman, Y., Gelfgat, A.Y.: On pressure-velocity coupled time-integration of incompressible Navier–Stokes equations using direct inversion of Stokes operator or accelerated multigrid technique. Comput. Struct. 87, 710–720 (2009)CrossRefGoogle Scholar
  10. 10.
    Feldman, Y., Gelfgat, A.Y.: Oscillatory instability of a 3D lid-driven flow in a cube. Phys. Fluids 22, 093602 (2010)CrossRefGoogle Scholar
  11. 11.
    Hammami, F., Ben-Cheikh, N., Campo, A., Ben-Beya, B., Lili, T.: Prediction of unsteady states in lid-driven cavities filled with an incompressible viscous fluid. Int. J. Mod. Phys. C 23, 1250030 (2012)CrossRefzbMATHGoogle Scholar
  12. 12.
    Mynam, M., Pathak, A.D.: Lattice Boltzmann simulation of steady and oscillatory flows in lid-driven cubic cavity. Int. J. Mod. Phys. C 24, 1350005 (2013)CrossRefGoogle Scholar
  13. 13.
    Chang, H.W., Hong, P.Y., Lin, L.S., Lin, C.A.: Simulations of flow instability in three dimensional deep cavities with multi relaxation time lattice Boltzmann method on graphic processing units. Comput. Fluids 88, 866–871 (2013)CrossRefzbMATHGoogle Scholar
  14. 14.
    Kuhlmann, H.C., Albensoeder, S.: Stability of the steady three-dimensional lid-driven flow in a cube and the supercritical flow dynamics. Phys. Fluids 26, 024104 (2014)CrossRefGoogle Scholar
  15. 15.
    Anupindi, K., Lai, W., Frankel, S.: Characterization of oscillatory instability in lid driven cavity flows using lattice Boltzmann method. Comput. Fluids 92, 7–21 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Loiseau, J.C., Robinet, J.C., Leriche, E.: Intermittency and transition to chaos in the cubical lid-driven cavity flow. Fluid Dyn. Res. 48, 061421 (2016)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Gómez, F., Gómez, R., Theofilis, V.: On three-dimensional global linear instability analysis of flows with standard aerodynamics codes. Aerosp. Sci. Technol. 32, 223–234 (2014)CrossRefGoogle Scholar
  18. 18.
    Lopez, J.M., Welfert, B.D., Wu, K., Yalim, J.: Transitions to complex dynamics in the cubic lid-driven cavity. Phys. Rev. Fluids 2, 074401 (2017)CrossRefGoogle Scholar
  19. 19.
    Povitsky, A.: High-incidence 3-D lid-driven cavity flow. AIAA Paper, 2847 (2001)Google Scholar
  20. 20.
    Povitsky, A.: Three-dimensional flow in cavity at yaw. Nonlinear Anal. Theory Methods Appl. 63, e1573–e1584 (2005)CrossRefzbMATHGoogle Scholar
  21. 21.
    Feldman, Y., Gelfgat, A.Y.: From multi- to single-grid CFD on massively parallel computers: numerical experiments on lid-driven flow in a cube using pressure-velocity coupled formulation. Comput. Fluids 46, 218–223 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Feldman, Y.: Theoretical analysis of three-dimensional bifurcated flow inside a diagonally lid-driven cavity. Theor. Comput. Fluid Dyn. 29, 245–261 (2015)CrossRefGoogle Scholar
  23. 23.
    Gulberg, Y., Feldman, Y.: On laminar natural convection inside multi-layered spherical shells. Int. J. Heat Mass Transf. 91, 908–921 (2015)CrossRefGoogle Scholar
  24. 24.
    Gelfgat, A.Y.: Visualization of three-dimensional incompressible flows by quasi-two-dimensional divergence-free projections. Comput. Fluids 97, 143–155 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Gelfgat, A.Y.: Visualization of three-dimensional incompressible flows by quasi-two-dimensional divergence-free projections in arbitrary flow regions. Theor. Comput. Fluid Dyn. 30, 339–348 (2016)CrossRefGoogle Scholar
  26. 26.
    Patankar, S.V.: Numerical Heat Transfer and Fluid Flow. Taylor & Francis, London (1980)zbMATHGoogle Scholar
  27. 27.
    van der Vorst, H.: Iterative Krylov Methods for Large Linear Systems. Cambridge Univ Press, Cambridge (2003)CrossRefzbMATHGoogle Scholar
  28. 28.
    Bayly, B.J.: Three-dimensional centrifugal-type instabilities in inviscid two-dimensional flows. Phys. Fluids 31, 56–64 (1988)CrossRefzbMATHGoogle Scholar
  29. 29.
    Lanzerstorfer, D., Kuhlmann, H.C.: Global stability of the two-dimensional flow over a backward-facing step. J. Fluid Mech. 693, 1–27 (2012)CrossRefzbMATHGoogle Scholar
  30. 30.
    Albensoeder, S., Kuhlmann, H.C., Rath, H.J.: Three-dimensional centrifugal-flow instabilities in the lid-driven-cavity problem. Phys. Fluids 13, 121–135 (2001)CrossRefzbMATHGoogle Scholar
  31. 31.
    Feldman, Y.: Direct numerical simulation of transitions and supercritical regimes in confined three-dimensional recirculating flows, Ph.D. Thesis, Tel-Aviv University (2010)Google Scholar
  32. 32.
    Roache, P.J.: Perspective: a method for uniform reporting of grid refinement studies. J. Fluids Eng. 116, 405–413 (1994)CrossRefGoogle Scholar
  33. 33.
    Sorensen, D.C.: Implicit application of polynomial filters in a k-step Arnoldi method. SIAM J. Matrix Anal. Appl. 13, 357–385 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Scott, J.A.: An Arnoldi code for computing selected eigenvalues of sparse real unsymmetric matrices. ACM Trans. Math. Softw. 21, 432–475 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Edwards, W.S., Tuckerman, L.S., Friesner, R.A., Sorensen, D.C.: Krylov methods for the incompressible Navier–Stokes equations. J. Comput. Phys. 110, 82–102 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Tuckerman, L.S., Barkley, D.: Bifurcation analysis for time-steppers. In: Doedel, K., Tuckerman, L. (eds.) Numerical Methods for Bifurcation Problems and Large-Scale Dynamical Systems. IMA Volumes in Mathematics and Its Applications, vol. 119, pp. 453–466. Springer, New York (2000)Google Scholar
  37. 37.
    Tuckerman, L.S., Bertagnolio, F., Daube, O., Le Quéré, P., Barkley, D.: Stokes preconditioning for the inverse Arnoldi method. In D. Henry, A. Bergeon, Vieweg Göttingen (eds.) Continuation Methods for Fluid Dynamics (Notes on Numerical Fluid Dynamics, 74), pp. 241–255 (2000)Google Scholar
  38. 38.
    Gelfgat, A.Y.: Krylov-subspace-based steady state and stability solvers for incompressible flows: replacing time steppers and generation of initial guess. In: A. Gelfgat (ed.) Computational Modelling of Bifurcations and Instabilities in Fluid Dynamics. Springer, 2018 (to appear)Google Scholar
  39. 39.
    Chorin, A.J.: Numerical solution of the Navier–Stokes equations. Math. Comput. 22, 745–762 (1968)MathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    Vitoshkin, H., Gelfgat, A.Y.: On direct inverse of Stokes, Helmholtz and Laplacian operators in view of time-stepper-based Newton and Arnoldi solvers in incompressible CFD. Commun. Comput. Phys. 14, 1103–1119 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    Lynch, R.E., Rice, J.R., Thomas, D.H.: Direct solution of partial differential equations by tensor product methods. Numer. Math. 6, 185–199 (1964)MathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    Gelfgat, A.Y.: Stability of convective flows in cavities: solution of benchmark problems by a low-order finite volume method. Intl. J. Num. Methods Fluids 53, 485–506 (2007)CrossRefzbMATHGoogle Scholar
  43. 43.
    Gelfgat, A.Y.: Implementation of arbitrary inner product in global Galerkin method for incompressible Navier–Stokes equation. J. Comput. Phys. 211, 513–530 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  44. 44.
    Poliashenko, M., Aidun, C,K.: A direct method for computation of simple bifurcations. J. Comput. Phys 121, 246–260 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  45. 45.
    Gervais, J.J., Lemelin, D., Pierre, R.: Some experiments with stability analysis of discrete incompressible flows in the lid-driven cavity. Int. J. Numer. Meth. Fluids 24, 477–492 (1997)CrossRefzbMATHGoogle Scholar
  46. 46.
    Fortin, A., Jardak, M., Gervais, J.J., Pierre, R.: Localization of Hopf bifurcations in fluid flow problems. Int. J. Numer. Meth. Fluids 24, 1185–1210 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  47. 47.
    Auteri, F., Parolini, N., Quartapelle, L.: Numerical investigations on the stability of singular driven cavity flow. J. Comput. Phys. 183, 1–25 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  48. 48.
    Peng, Y.F., Shiau, Y.H., Hwang, R.R.: Transition in a 2-D lid-driven cavity flow. Comput. Fluids 32, 337–352 (2003)CrossRefzbMATHGoogle Scholar
  49. 49.
    Abouhamza, A., Pierre, R.: A neutral stability curve for incompressible flows in a rectangular driven cavity. Math. Comput. Model. 38, 141–157 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  50. 50.
    Cadou, J.M., Potier-Ferry, M., Cochelin, B.: A numerical method for the computation of bifurcation points in fluid mechanics. Eur. J. Mech. B/Fluids 25, 234–254 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  51. 51.
    Sahin, M., Owens, R.G.: A novel fully-implicit finite volume method applied to the lid-driven cavity problem. Part II. Linear stability analysis. Int. J. Numer. Meth. Fluids 42, 79–88 (2003)CrossRefzbMATHGoogle Scholar
  52. 52.
    Boppana, V.B.L., Gajjar, J.S.B.: Global flow instability in a lid-driven cavity. Int. J. Numer. Meth. Fluids 62, 827–853 (2010)MathSciNetzbMATHGoogle Scholar
  53. 53.
    Tiesinga, G., Wubs, F.W., Veldman, A.E.P.: Bifurcation analysis of incompressible flow in a driven cavity by the Newton–Picard method. J. Comput. Appl. Math. 140, 751–772 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  54. 54.
    Kalita, J.C., Gogoi, B.B.: A biharmonic approach for the global stability analysis of 2D incompressible viscous flows. Appl. Math. Model. 40, 6831–6849 (2016)MathSciNetCrossRefGoogle Scholar
  55. 55.
    Gelfgat, A.Y., Molokov, S.: Quasi-two-dimensional convection in a 3D laterally heated box in a strong magnetic field normal to main circulation. Phys. Fluids 23, 034101 (2011)CrossRefGoogle Scholar
  56. 56.
    Brès, C.A., Colonius, T.: Three-dimensional instabilities in compressible flow over open cavities. J. Fluid Mech. 599, 309–339 (2008)CrossRefzbMATHGoogle Scholar
  57. 57.
    Barkley, D., Gomes, G., Gabriela, M., Henderson, D.: Three-dimensional instability in flow over a backward-facing step. J. Fluid Mech 473, 167–190 (2002)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

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

Personalised recommendations