The Lid-Driven Cavity

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


The lid-driven cavity is an important fluid mechanical system serving as a benchmark for testing numerical methods and for studying fundamental aspects of incompressible flows in confined volumes which are driven by the tangential motion of a bounding wall. A comprehensive review is provided of lid-driven cavity flows focusing on the evolution of the flow as the Reynolds number is increased. Understanding the flow physics requires to consider pure two-dimensional flows, flows which are periodic in one space direction as well as the full three-dimensional flow. The topics treated range from the characteristic singularities resulting from the discontinuous boundary conditions over flow instabilities and their numerical treatment to the transition to chaos in a fully confined cubical cavity. In addition, the streamline topology of two-dimensional time-dependent and of steady three-dimensional flows are covered, as well as turbulent flow in a square and in a fully confined lid-driven cube. Finally, an overview on various extensions of the lid-driven cavity is given.


Internal flow Singularity Vortex Stability Turbulence Mixing 



We are very grateful to S. Albensoeder, F. Auteri, O. Botella, R. Bouffanais, C.-H. Bruneau, G. Courbebaisse, J. R. Koseff, E. Leriche, J.-C. Loiseau, J. M. Lopez, H. K. Moffatt, J. M. Ottino, A. Povitsky, W. W. Schultz, J. F. Scott and T. W. H. Sheu, who kindly allowed the reproduction of their figures which have been published earlier.


  1. 1.
    Ahmed, M., Kuhlmann, H.C.: Flow instability in triangular lid-driven cavities with wall motion away from a rectangular corner. Fluid Dyn. Res. 44:025501–1–025501–21, 2012MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Aidun, C.K., Triantafillopoulos, N.G., Benson, J.D.: Global stability of a lid-driven cavity with throughflow: flow visualization studies. Phys. Fluids A 3, 2081–2091 (1991)CrossRefGoogle Scholar
  3. 3.
    Åkervik, E., Brandt, L., Henningson, D.S., Hopffner, J., Marxen, O., Schlatter, P.: Steady solutions of the Navier–Stokes equations by selective frequency damping. Phys. Fluids 18, 068102 (2006)CrossRefGoogle Scholar
  4. 4.
    Akyuzlu, K.M.: A numerical and experimental study of laminar unsteady lid-driven cavity flows. In: Proceedings of the ASME 2017 International Mechanical Engineering Congress and Exposition, pp. IMECE2017–70145. ASME (2017)Google Scholar
  5. 5.
    Al-Amiri, A.M.: Analysis of momentum and energy transfer in a lid-driven cavity filled with a porous medium. Int. J. Heat Mass Transf. 43, 3513–3527 (2000)zbMATHCrossRefGoogle Scholar
  6. 6.
    Albensoeder, S.: Zweidimensionale Strömungsmuster in zweiseitig angetriebenen Rechteckbehältern mittels eines Finite-Volumen-Verfahrens (in German). Mathesis, Universität Bremen (1999)Google Scholar
  7. 7.
    Albensoeder, S.: Lineare und nichtlineare Stabilität inkompressibler Strömungen im zweiseitig angetriebenen Rechteckbehälter (in German). Cuvillier, Göttingen (2004)Google Scholar
  8. 8.
    Albensoeder, S., Kuhlmann, H.C.: Linear stability of rectangular cavity flows driven by anti-parallel motion of two facing walls. J. Fluid Mech. 458, 153–180 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Albensoeder, S., Kuhlmann, H.C.: Three-dimensional instability of two counter-rotating vortices in a rectangular cavity driven by parallel wall motion. Eur. J. Mech. B/Fluids 21, 307–316 (2002)zbMATHCrossRefGoogle Scholar
  10. 10.
    Albensoeder, S., Kuhlmann, H.C.: Stability balloon for the double-lid-driven cavity flow. Phys. Fluids 15, 2453–2456 (2003)zbMATHCrossRefGoogle Scholar
  11. 11.
    Albensoeder, S., Kuhlmann, H.C.: Accurate three-dimensional lid-driven cavity flow. J. Comput. Phys. 206, 536–558 (2005)zbMATHCrossRefGoogle Scholar
  12. 12.
    Albensoeder, S., Kuhlmann, H.C.: Nonlinear three-dimensional flow in the lid-driven square cavity. J. Fluid Mech. 569, 465–480 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    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)zbMATHCrossRefGoogle Scholar
  14. 14.
    Albensoeder, S., Kuhlmann, H.C., Rath, H.J.: Multiplicity of steady two-dimensional flows in two-sided lid-driven cavities. Theor. Comput. Fluid Dyn. 14, 223–241 (2001)zbMATHCrossRefGoogle Scholar
  15. 15.
    Alizard, F., Robinet, J.C., Gloerfelt, X.: A domain decomposition matrix-free method for global linear stability. Comput. Fluids 66, 63–84 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Alleborn, N., Raszillier, H., Durst, F.: Lid-driven cavity with heat and mass transport. Int. J. Heat Mass Transf. 42, 833–853 (1999)zbMATHCrossRefGoogle Scholar
  17. 17.
    Anderson, P.D., Galaktionov, O.S., Peters, G.W.M., van de Vosse, F.N., Meijer, H.E.H.: Analysis of mixing in three-dimensional time-periodic cavity flows. J. Fluid Mech. 386, 149–166 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Anderson, P.D., Galaktionov, O.S., Peters, G.W., van de Vosse, F.N., Meijer, H.E.: Chaotic fluid mixing in non-quasi-static time-periodic cavity flows. Int. J. Heat Fluid Flow 21, 176–185 (2000)CrossRefGoogle Scholar
  19. 19.
    Anderson, P.D., Ternet, D., Peters, W.M., Mejer, H.E.H.: Experimental/numerical analysis of chaotic advection in a three-dimensional cavity. Int. Polym. Process. 21, 412–420 (2006)CrossRefGoogle Scholar
  20. 20.
    Aref, H.: Chaotic advection of fluid particles. Philos. Trans. R. Soc. Lond. Ser. A: Phys. Eng. Sci. 333(1631), 273–288 (1990)zbMATHCrossRefGoogle Scholar
  21. 21.
    Arnold, V.I., Kozlov, V.V., Neishtadt, A.I.: Mathematical Aspects of Classical and Celestial Mechanics. Springer (2007)Google Scholar
  22. 22.
    Arnoldi, W.E.: The principle of minimized iterations in the solution of the matrix eigenvalue problem. Quart. Appl. Math. 9, 17–29 (1951)MathSciNetzbMATHCrossRefGoogle Scholar
  23. 23.
    Auteri, F., Parolini, N., Quartapelle, L.: Numerical investigation on the stability of singular driven cavity flow. J. Comput. Phys. 183, 1–25 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  24. 24.
    Auteri, F., Quartapelle, L., Vigevano, L.: Accurate \(\omega \)-\(\psi \) spectral solution of the singular driven cavity problem. J. Comput. Phys. 180, 597–615 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  25. 25.
    Bagheri, S., Åkervik, E., Brandt, L., Hennignson, D.S.: Matrix-free methods for the stability and control of boundary layers. AIAA J. 47, 1057–1068 (2009)CrossRefGoogle Scholar
  26. 26.
    Bajer, K.: Hamiltonian formulation of the equations of streamlines in three-dimensional steady flow. Chaos Solitons Fractals 4, 895–911 (1994)zbMATHCrossRefGoogle Scholar
  27. 27.
    Barkley, D., Henderson, R.D.: Three-dimensional Floquet stability analysis of the wake of a circular cylinder. J. Fluid Mech. 322, 215–241 (1996)zbMATHCrossRefGoogle Scholar
  28. 28.
    Barletta, A., Nield, D.A.: Mixed convection with viscous dissipation and pressure work in a lid-driven square enclosure. Int. J. Heat Mass Transf. 52, 42444253 (2009)zbMATHGoogle Scholar
  29. 29.
    Batchelor, G.K.: On steady laminar flow with closed streamlines at large Reynolds numbers. J. Fluid Mech. 1, 177–190 (1956)MathSciNetzbMATHCrossRefGoogle Scholar
  30. 30.
    Batchelor, G.K.: Small-scale variation of convected quantities like temperature in turbulent fluid part 1. general discussion and the case of small conductivity. J. Fluid Mech. 5, 113–133 (1959)MathSciNetzbMATHCrossRefGoogle Scholar
  31. 31.
    Bayly, B.J.: Three-dimensional instability of elliptical flow. Phys. Rev. Lett. 57, 2160–2163 (1986)MathSciNetCrossRefGoogle Scholar
  32. 32.
    Belhachmi, Z., Bernardi, C., Karageorghis, A.: Spectral element discretization of the circular driven cavity. part iv. the Navier–Stokes equations. J. Math. Fluid Mech. 6, 121–156 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  33. 33.
    Benson, J.D., Aidun, C.K.: Transition to unsteady nonperiodic state in a through-flow lid-driven cavity. Phys. Fluids A 4, 2316–2319 (1992)CrossRefGoogle Scholar
  34. 34.
    Bergamo, L.F., Gennaro, E.M., Theofilis, V., Medeiros, M.A.F.: Compressible modes in a square lid-driven cavity. Aerosp. Sci. Technol. 44, 125–134 (2015)CrossRefGoogle Scholar
  35. 35.
    Beya, B.B., Lili, T.: Three-dimensional incompressible flow in a two-sided non-facing lid-driven cubical cavity. Comptes Rendus Mecanique 336, 863–872 (2008)zbMATHCrossRefGoogle Scholar
  36. 36.
    Bhattacharya, M., Basak, T., Oztop, H.F., Varol, Y.: Mixed convection and role of multiple solutions in lid-driven trapezoidal enclosures. Int. J. Heat Mass Transf. 63, 366–388 (2013)CrossRefGoogle Scholar
  37. 37.
    Billah, M.M., Rahman, M.M., Sharif, U.M., Rahim, N.A., Saidur, R., Hasanuzzaman, M.: Numerical analysis of fluid flow due to mixed convection in a lid-driven cavity having a heated circular hollow cylinder. Int. Commun. Heat Mass Transf. 38, 1093–1103 (2011)CrossRefGoogle Scholar
  38. 38.
    Blackburn, H.M., Lopez, J.M.: The onset of three-dimensional standing and modulated travelling waves in a periodically driven cavity flow. J. Fluid Mech. 497, 289–317 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  39. 39.
    Blackburn, H.M., Lopez, J.M.: Modulated waves in a periodically driven annular cavity. J. Fluid Mech. 667, 336–357 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  40. 40.
    Blohm, C.: Experimentelle Untersuchung stationärer und zeitabhängiger Strömungen im zweiseitig angetriebenen Rechteckbehälter (in German). Ph.D thesis, University of Bremen (2001)Google Scholar
  41. 41.
    Blohm, C., Kuhlmann, H.C.: The two-sided lid-driven cavity: experiments on stationary and time-dependent flows. J. Fluid Mech. 450, 67–95 (2002)zbMATHCrossRefGoogle Scholar
  42. 42.
    Blohm, C., Albensoeder, S., Kuhlmann, H.C., Broda, M., Rath, H.J.: The two-sided lid-driven cavity: Aspect-ratio dependence of the flow stability. Z. Angew. Math. Mech. 81(Suppl. 3), 781–782 (2001)zbMATHCrossRefGoogle Scholar
  43. 43.
    Bödewadt, U.T.: Die Drehströmung über festem Grunde. Z. Angew. Math. Mech. 20, 241–253 (1940)zbMATHCrossRefGoogle Scholar
  44. 44.
    Boppana, V.B.L., Gajjar, J.S.B.: Global flow instability in a lid-driven cavity. Int. J. Numer. Methods Fluids 62, 827–853 (2010)MathSciNetzbMATHGoogle Scholar
  45. 45.
    Botella, O.: On the solution of the Navier–Stokes equations using Chebyshev projection schemes with third-order accuracy in time. Comput. Fluids 26, 107–116 (1997)zbMATHCrossRefGoogle Scholar
  46. 46.
    Botella, O., Peyret, R.: The Chebyshev approximation for the solution of singular Navier–Stokes problems. In: Numerical Modelling in Continuum Mechanics: Proceedings of the 3rd Summer Conference, pp. 8–11. Prague (1997)Google Scholar
  47. 47.
    Botella, O., Peyret, R.: Benchmark spectral results on the lid-driven cavity flow. Comput. Fluids 27, 421–433 (1998)zbMATHCrossRefGoogle Scholar
  48. 48.
    Botella, O., Peyret, R.: Computing singular solutions of the Navier–Stokes equations with the Chebyshev-collocation method. Int. J. Numer. Methods Fluids 36, 125–163 (2001)zbMATHCrossRefGoogle Scholar
  49. 49.
    Botella, O., Forestier, M.Y., Pasquetti, R., Peyret, R., Sabbah, C.: Chebyshev methods for the Navier-Stokes equations: algorithms and applications. Nonlinear Anal. 47, 4157–4168 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  50. 50.
    Bouffanais, R., Deville, M.O., Fischer, P.F., Leriche, E., Weill, D.: Large-eddy simulation of the lid-driven cubic cavity flow by the spectral element method. J. Sci. Comput. 27, 151–162 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  51. 51.
    Bouffanais, R., Deville, M.O., Leriche, E.: Large-eddy simulation of the flow in a lid-driven cubical cavity. Phys. Fluids 19, 055108–1–055108–20, (2007)zbMATHCrossRefGoogle Scholar
  52. 52.
    Boyling, J.B.: A rigidity result for biharmonic functions clamped at a corner. Z. Angew. Math. Phys. 46, 289–294 (1995)MathSciNetzbMATHCrossRefGoogle Scholar
  53. 53.
    Brandt, A., Livne, O.E.: Multigrid Techniques: 1984 Guide with Applications to Fluid Dynamics, Revised edn. SIAM (2011)Google Scholar
  54. 54.
    Brés, G.A., Colonius, T.: Three-dimensional instabilities in compressible flow over open cavities. J. Fluid Mech. 599, 309–339 (2008)zbMATHCrossRefGoogle Scholar
  55. 55.
    Broer, H.W., Huitema, G.B., Sevryuk, M.B.: Quasi-Periodic Motions in Families of Dynamical Systems: Order Amidst Chaos. Springer, Berlin (2009)Google Scholar
  56. 56.
    Bruneau, C.-H.: Direct Numerical Simulation and Analysis of 2D Turbulent Flows, pp. 33–44. Birkhäuser Basel, Basel, (2007)Google Scholar
  57. 57.
    Bruneau, C.-H., Saad, M.: The 2d lid-driven cavity problem revisited. Comput. Fluids 35, 326–348 (2006)zbMATHCrossRefGoogle Scholar
  58. 58.
    Burggraf, O.R.: Analytical and numerical studies of the structure of steady separated flows. J. Fluid Mech. 24, 113–151 (1966)CrossRefGoogle Scholar
  59. 59.
    Cadou, J.M., Guevel, Y., Girault, G.: Numerical tools for the stability analysis of 2D flows: application to the two- and four-sided lid-driven cavity. Fluid Dyn. Res. 44, 031403–1–021403–12 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  60. 60.
    Cazemier, W., Verstappen, R.W.C.P., Veldman, A.E.P.: Proper orthogonal decomposition and low-dimensional models for driven cavity flows. Phys. Fluids 10, 1685–1699 (1998)CrossRefGoogle Scholar
  61. 61.
    Chandrasekhar, S.: Hydrodynamic and Hydromagnetic Stability. Oxford University Press, Oxford (1961)zbMATHGoogle Scholar
  62. 62.
    Chang, M.-H., Cheng, C.-H.: Predictions of lid-driven flow and heat convection in an arc-shape cavity. Int. Commun. Heat. Mass Transfer 26, 829–838 (1999)CrossRefGoogle Scholar
  63. 63.
    Chatterjee, D.: MHD mixed convection in a lid-driven cavity including a heated source. Numer. Heat Transf. A 64, 235–254 (2013)CrossRefGoogle Scholar
  64. 64.
    Chen, C.-L., Cheng, C.-H.: Numerical prediction of buoyancy-induced periodic flow pattern and heat transfer in a lid-driven arc-shape cavity. Numer. Heat Transf. A 44, 645–663 (2003)CrossRefGoogle Scholar
  65. 65.
    Chen, C.-L., Cheng, C.-H.: Experimental and numerical study of mixed convection and flow pattern in a lid-driven arc-shape cavity. Heat Mass Transf. 41, 58–66 (2004)CrossRefGoogle Scholar
  66. 66.
    Chen, C.-L., Cheng, C.-H.: Numerical simulation of periodic mixed convective heat transfer in a rectangular cavity with a vibrating lid. Appl. Therm. Eng. 29, 2855–2862 (2009)CrossRefGoogle Scholar
  67. 67.
    Chen, C.-L., Cheng, C.-H.: Numerical study of the effects of lid oscillation on the periodic flow pattern and convection heat transfer in a triangular cavity. Int. Commun. Heat Mass Transf. 36, 590–596 (2009)CrossRefGoogle Scholar
  68. 68.
    Chen, C.-L., Chung, Y.-C., Lee, T.-F.: Experimental and numerical studies on periodic convection flow and heat transfer in a lid-driven arc-shape cavity. Int. Commun. Heat Mass Transf. 39, 1563–1571 (2012)CrossRefGoogle Scholar
  69. 69.
    Chen, K.-T., Tsai, C.-C., Luo, W.-J., Chen, C.-N.: Multiplicity of steady solutions in a two-sided lid-driven cavity with different aspect ratios. Theor. Comput. Fluid Dyn. 27, 767–776 (2013)CrossRefGoogle Scholar
  70. 70.
    Chen, K.-T., Tsai, C.-C., Luo, W.-J., Lu, C.W., Chen, C.H.: Aspect ratio effect on multiple flow solutions in a two-sided parallel motion lid-driven cavity. J. Mech. 31, 153–160 (2015). ISSN 1811-8216CrossRefGoogle Scholar
  71. 71.
    Cheng, C.-H., Chen, C.-L.: Buoyancy-induced periodic flow and heat transfer in lid-driven cavities with different cross-sectional shapes. Int. Commun. Heat Mass Transf. 32, 483–490 (2005)CrossRefGoogle Scholar
  72. 72.
    Cheng, C.-Q., Sun, Y.-S.: Existence of invariant tori in three-dimensional measure-preserving mappings. Celest. Mech. Dyn. Astron. 47, 275–292 (1989)MathSciNetzbMATHCrossRefGoogle Scholar
  73. 73.
    Cheng, C.-Q., Sun, Y.-S.: Existence of periodically invariant curves in 3-dimensional measure-preserving mappings. Celest. Mech. Dyn. Astron. 47(3), 293–303 (1989)MathSciNetzbMATHCrossRefGoogle Scholar
  74. 74.
    Cheng, M., Hung, K.C.: Vortex structure of steady flow in a rectangular cavity. Comput. Fluids 35, 1046–1062 (2006)zbMATHCrossRefGoogle Scholar
  75. 75.
    Chiang, T.P., Sheu, W.H.: Numerical prediction of eddy structure in a shear-driven cavity. Comput. Mech. 20, 379–396 (1997)zbMATHCrossRefGoogle Scholar
  76. 76.
    Chiang, T.P., Hwang, R.R., Sheu, W.H.: Finite volume analysis of spiral motion in a rectangular lid-driven cavity. Int. J. Numer. Methods Fluids 23, 325–346 (1996)zbMATHCrossRefGoogle Scholar
  77. 77.
    Chiang, T.P., Hwang, R.R., Sheu, W.H.: On end-wall corner vortices in a lid-driven cavity. J. Fluids Eng. 119, 201–214 (1997)CrossRefGoogle Scholar
  78. 78.
    Chiang, T.P., Sheu, W.H., Hwang, R.R.: Three-dimensional vortex dynamics in a shear-driven rectangular cavity. Int. J. Comput. Fluid Dyn. 8, 201–214 (1997)zbMATHCrossRefGoogle Scholar
  79. 79.
    Chiang, T.P., Sheu, W.H., Hwang, R.R.: Effect of Reynolds number on the eddy structure in a lid-driven cavity. Int. J. Numer. Methods Fluids 26, 557–579 (1998)zbMATHCrossRefGoogle Scholar
  80. 80.
    Chicheportiche, J., Merle, X., Gloerfelt, X., Robinet, J.-C.: Direct numerical simulation and global stability analysis of three-dimensional instabilities in a lid-driven cavity. Comptes Rendus Mecanique 336, 586–591 (2008)CrossRefGoogle Scholar
  81. 81.
    Chien, W.-L., Rising, H., Ottino, J.M.: Laminar mixing and chaotic mixing in several cavity flows. J. Fluid Mech. 170, 355–377 (1986)CrossRefGoogle Scholar
  82. 82.
    Cohen, N., Eidelman, A., Elperin, T., Kleeorin, N., Rogachevskii, I.: Sheared stably stratified turbulence and large-scale waves in a lid driven cavity. Phys. Fluids 26, 105106–1–105106–16 (2014)CrossRefGoogle Scholar
  83. 83.
    Courbebaisse, G., Bouffanais, R., Navarro, L., Leriche, E., Deville, M.: Time-scale joint representation of DNS and LES numerical data. Comput. Fluids 43, 38–45 (2011)zbMATHCrossRefGoogle Scholar
  84. 84.
    Crouzeix, M., Philippe, B., Sadkane, M.: The Davidson method. SIAM J. Sci. Comput. 15, 62–76 (1994)MathSciNetzbMATHCrossRefGoogle Scholar
  85. 85.
    Davidson, E.R.: The iterative calculation of a few of the lowest eigenvalues and corresponding eigenvectors of large real-symmetric matrices. J. Comput. Phys. 87–94 (1975)MathSciNetzbMATHCrossRefGoogle Scholar
  86. 86.
    Davidson, E.R.: Matrix eigenvector methods. In: Methods in Computational Molecular Physics, pp. 95–113. Springer, Berlin (1983)CrossRefGoogle Scholar
  87. 87.
    Davis, A.M.J., O’Neill, M.E.: Separation in a slow linear shear flow past a cylinder and a plane. J. Fluid Mech. 81, 551–564 (1977)MathSciNetzbMATHCrossRefGoogle Scholar
  88. 88.
    Davis, A.M.J., Smith, S.G.L.: Three-dimensional corner eddies in Stokes flow. Fluid Dyn. Res 46, 015509–1–015509–8 (2014)MathSciNetzbMATHCrossRefGoogle Scholar
  89. 89.
    Davis, A.M.J., O’Neill, M.E., Dorrepaal, J.M., Ranger, K.B.: Separation from the surface of two equal spheres in Stokes flow. J. Fluid Mech. 77, 625–644 (1976)zbMATHCrossRefGoogle Scholar
  90. 90.
    de Vicente, J., Rodríguez, D., Theofilis, V., Valero, E.: Stability analysis in spanwise-periodic double-sided lid-driven cavity flows with complex cross-sectional profiles. Comput. Fluids 43, 143–153 (2011)zbMATHCrossRefGoogle Scholar
  91. 91.
    de Vicente, J., Basley, J., Meseguer-Garrido, F., Soria, J., Theofilis, V.: Three-dimensional instabilities over a rectangular open cavity: from linear stability analysis to experimentation. J. Fluid Mech. 748, 189–220 (2014)CrossRefGoogle Scholar
  92. 92.
    de Vicente, J., Basley, J., Meseguer-Garrido, F., Soria, J., Theofilis, V.: Three-dimensional instabilities over a rectangular open cavity: from linear stability analysis to experimentation – ERRATUM. J. Fluid Mech. 751, 747–748 (2014)CrossRefGoogle Scholar
  93. 93.
    Dean, W.R., Montagnon, P.E.: On the steady motion of viscous liquid in a corner. Proc. Camb. Philos. Soc. 45, 389–394 (1949)MathSciNetzbMATHCrossRefGoogle Scholar
  94. 94.
    Deshpande, M.D., Milton, S.G.: Kolmogorov scales in a driven cavity flow. Fluid Dyn. Res. 22, 359–381 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  95. 95.
    Deville, M., Lê, T.-H., Morchoisne, Y.: Numerical Simulation of 3-D Incompressible Unsteady Viscous Laminar Flows. Notes on Numerical Fluid Mechanics, vol. 36. Vieweg, Braunschweig (1992)Google Scholar
  96. 96.
    Ding, Y., Kawahara, M.: Linear stability of incompressible fluid flow in a cavity using finite element method. Int. J. Numer. Methods Fluids 27, 139–157 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  97. 97.
    Ding, Y., Kawahara, M.: Three-dimensional linear stability analysis of incompressible viscous flows using the finite element method. Int. J. Numer. Methods Fluids 31, 451–479 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  98. 98.
    dos Santos, D.D., Frey, S., Naccache, M.F., de Souza Mendes, P.R.: Numerical approximations for flow of viscoplastic fluids in a lid-driven cavity. J. Non-Newton. Fluid Mech. 166, 667–679 (2011)Google Scholar
  99. 99.
    Drazin, P.G., Reid, W.H.: Hydrodynamic Stability. Cambridge University Press, Cambridge (1981)zbMATHGoogle Scholar
  100. 100.
    Eckmann, J.-P.: Roads to turbulence in dissipative dynamical systems. Rev. Mod. Phys. 53, 643–654 (1981)MathSciNetzbMATHCrossRefGoogle Scholar
  101. 101.
    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)MathSciNetzbMATHCrossRefGoogle Scholar
  102. 102.
    Eloy, C., Le Dizès, S.: Stability of the Rankine vortex in a multipolar strain field. Phys. Fluids 13, 660–676 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  103. 103.
    Erturk, E., Gokcol, O.: Fine grid numerical solutions of triangular cavity flow. Eur. Phys. J. Appl. Phys. 38, 97–105 (2007)CrossRefGoogle Scholar
  104. 104.
    Erturk, E., Corke, T.C., Gökçöl, C.: Numerical solutions of 2-d steady incompressible driven cavity flow at high Reynolds numbers. Int. J. Numer. Methods Fluids 48, 747–774 (2005)zbMATHCrossRefGoogle Scholar
  105. 105.
    Eskandari, M., Nourazar, S.S.: On the time relaxed Monte Carlo computations for the lid-driven micro cavity flow. J. Comput. Phys. 343, 355–367 (2017)MathSciNetCrossRefGoogle Scholar
  106. 106.
    Faure, T., Pastur, L., Lusseyran, F., Fraigneau, Y., Bisch, D.: Three-dimensional centrifugal instabilities development inside a parallelepipedic open cavity of various shape. Exp. Fluids 47, 395–410 (2009)CrossRefGoogle Scholar
  107. 107.
    Faure, T.M., Adrianos, P., Lusseyran, F., Pastur, L.: Visualizations of the flow inside an open cavity at medium range Reynolds numbers. Exp. Fluids 42, 169–184 (2007)CrossRefGoogle Scholar
  108. 108.
    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
  109. 109.
    Feldman, Y., Gelfgat, A.Y.: Oscillatory instability of a three-dimensional lid-driven flow in a cube. Phys. Fluids 22, 093602-1–093602-9 (2010)CrossRefGoogle Scholar
  110. 110.
    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)MathSciNetzbMATHCrossRefGoogle Scholar
  111. 111.
    Fix, G.J., Gulati, S., Wakoff, G.I.: On the use of singular functions with finite element approximations. J. Comput. Phys. 13, 209–228 (1973)MathSciNetzbMATHCrossRefGoogle Scholar
  112. 112.
    Floryan, J.M., Czechowski, L.: On the numerical treatment of corner singularity in the vorticity field. J. Comput. Phys. 118, 222–228 (1995)zbMATHCrossRefGoogle Scholar
  113. 113.
    Fortin, A., Jardak, M., Gervais, J., Pierre, R.: Localization of Hopf bifurcation in fluid flow problems. Int. J. Numer. Methods Fluids 24, 1185–1210 (1997)MathSciNetzbMATHCrossRefGoogle Scholar
  114. 114.
    Franjione, J.G., Leong, C.-W., Ottino, J.M.: Symmetries within chaos: a route to effective mixing. Phys. Fluids A 1, 1772–1783 (1989)MathSciNetCrossRefGoogle Scholar
  115. 115.
    Freitas, C.J., Street, R.L., Findikakis, A.N., Koseff, J.R.: Numerical simulation of three-dimensional flow in a cavity. Int. J. Numer. Methods Fluids 5, 561–575 (1985)zbMATHCrossRefGoogle Scholar
  116. 116.
    Fuchs, L., Tillmark, N.: Numerical and experimental study of driven flow in a polar cavity. Int. J. Numer. Methods Fluids 5, 311–329 (1985)CrossRefGoogle Scholar
  117. 117.
    Garcia, S.: The lid-driven square cavity flow: From stationary to time periodic and chaotic. Commun. Comput. Phys. 2, 900–932 (2007)MathSciNetzbMATHGoogle Scholar
  118. 118.
    Gaskell, P.H., Gürcan, F., Savage, M.D., Thompson, H.M.: Stokes flow in a double-lid-driven cavity with free surface side walls. Proc. Inst. Mech. Eng. 212, 387–403 (1998)Google Scholar
  119. 119.
    Gelfgat, A.Y.: Implementation of arbitrary inner product in the global Galerkin method for incompressible Navier–Stokes equations. J. Comput. Phys. 211, 513–530 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  120. 120.
    Georgiou, G.C., Olson, L.G., Schultz, W.W., Sagan, S.: A singular finite element for Stokes flow: the stick-slip problem. Int. J. Numer. Methods Fluids 9, 1353–1367 (1989)MathSciNetzbMATHCrossRefGoogle Scholar
  121. 121.
    Ghia, U., Goyal, R.K.: Laminar incompressible recirculating flow in a driven cavity of polar cross section. ASME J. Fluids Eng. 99, 774–777 (1977)CrossRefGoogle Scholar
  122. 122.
    Ghia, U., Ghia, K.N., Shin, C.T.: High-Re solutions for incompressible flow using the Navier-Stokes equations and a multigrid method. J. Comput. Phys. 48, 387–411 (1982)zbMATHCrossRefGoogle Scholar
  123. 123.
    Glowinski, R., Guidoboni, G., Pan, T.-W.: Wall-driven incompressible viscous flow in a two-dimensional semi-circular cavity. J. Comput. Phys. 216, 76–91 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  124. 124.
    Gogoi, B.B.: Global 2D stability analysis of the cross lid-driven cavity flow with a streamfunction-vorticity approach. Int. J. Comput. Methods Eng. Sci. Mech. 17, 253–273 (2016)MathSciNetCrossRefGoogle Scholar
  125. 125.
    Golub, H.G., van Loan, H.G.: Matrix Computations. Johns Hopkins University Press (1989)Google Scholar
  126. 126.
    Gómez, F., Paredes, P., Gómez, R., Theofilis, V.: Global stability of cubic and large aspect ratio three-dimensional lid-driven cavities. In: 42nd AIAA Fluid Dynamics Conference and Exhibit, pp. AIAA 2012–3274, New Orleans, Louisiana. AIAA (2012)Google Scholar
  127. 127.
    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
  128. 128.
    Gomilko, A.M., Malyuga, V.S., Meleshko, V.V.: On steady Stokes flow in a trihedral rectangular corner. J. Fluid Mech. 476, 159–177 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  129. 129.
    González, L.M., Ahmed, M., Kühnen, J., Kuhlmann, H.C., Theofilis, V.: Three-dimensional flow instability in a lid-driven isosceles triangular cavity. J. Fluid Mech. 675, 369–696 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  130. 130.
    González, L.M., Ferrer, E., Díaz-Ojeda, H.R.: Onset of three-dimensional flow instabilities in lid-driven circular cavities. Phys. Fluids 29, 064102–1–064102–16 (2017)CrossRefGoogle Scholar
  131. 131.
    Goodier, J.N.: An analogy between the slow motions of a viscous fluid in two dimensions, and systems of plane stress. Lond. Edinb. Dublin Phil. Mag. J. Sci. 17(113), 554–576 (1934)zbMATHCrossRefGoogle Scholar
  132. 132.
    Goodrich, J.W., Gustafson, K., Halasi, K.: Hopf bifurcation in the driven cavity. J. Comput. Phys. 90, 219–261 (1990)MathSciNetzbMATHCrossRefGoogle Scholar
  133. 133.
    Görtler, H.: Über eine dreidimensionale Instabilität laminarer Grenzschichten an konkaven Wänden. Nachrichten von der Akademie der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse 1, 1–26 (1941)Google Scholar
  134. 134.
    Görtler, H.: On the three-dimensional instability of laminar boundary layers on concave walls. Technical Report 1375, National Advisory Committee for Aeronautics (1954)Google Scholar
  135. 135.
    Görtler, H.: Dreidimensionale Instabilität der ebenen Staupunktströmung gegenüber wirbelartigen Strörungen, Fünfzig Jahre Grenzschichtforschung, pp. 304–314. Vieweg, Braunschweig (1955)Google Scholar
  136. 136.
    Grillet, A.M., Yang, B., Khomami, B., Shaqfeh, E.S.G.: Modeling of viscoelastic lid driven cavity flow using finite element simulations. J. Non-Newton. Fluid Mech. 88, 99–131 (1999)zbMATHCrossRefGoogle Scholar
  137. 137.
    Grillet, A.M., Shaqfeh, E.S.G., Khomami, B.: Observations of elastic instabilities in lid-driven cavity flow. J. Non-Newton. Fluid Mech. 94, 15–35 (2000)CrossRefGoogle Scholar
  138. 138.
    Guermond, J.-L., Migeon, C., Pineau, G., Quartapelle, L.: Start-up flows in a three-dimensional rectangular driven cavity of aspect ratio 1:1:2 at Re \(=\) 1000. J. Fluid Mech. 450, 169–199 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  139. 139.
    Gupta, M.M., Manohar, R.P., Noble, B.: Nature of viscous flows near sharp corners. Comput. Fluids 9, 379–388 (1981)zbMATHCrossRefGoogle Scholar
  140. 140.
    Gürcan, F.: Effect of the Reynolds number on streamline bifurcations in a double-lid-driven cavity with free surfaces. Comput. Fluids 32, 1283–1298 (2003)zbMATHCrossRefGoogle Scholar
  141. 141.
    Gürcan, F.: Streamline topologies in Stokes flow within lid-driven cavities. Theor. Comput. Fluid. Dyn. 17, 19–30 (2003)zbMATHCrossRefGoogle Scholar
  142. 142.
    Gürcan, F., Bilgil, H.: Bifurcations and eddy genesis of Stokes flow within a sectorial cavity. Eur. J. Mech. B/Fluids 39, 42–51 (2013)MathSciNetzbMATHCrossRefGoogle Scholar
  143. 143.
    Gustafson, K., Halasi, K.: Cavity flow dynamics at higher Reynolds number and higher aspect ratio. J. Comput. Phys. 70, 271–283 (1987)zbMATHCrossRefGoogle Scholar
  144. 144.
    Habisreutinger, M.A., Bouffanais, R., Leriche, E., Deville, M.O.: A coupled approximate deconvolution and dynamic mixed scale model for large-eddy simulation. J. Comput. Phys. 224, 241–266 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  145. 145.
    Hackbusch, W.: On the multi-grid method applied to difference equations. Computing 20, 291–306 (1978)MathSciNetzbMATHCrossRefGoogle Scholar
  146. 146.
    Hafizi, M.Y.M., Idris, M.S., Ammar, N.M.M.: Study on the behavior of particles in high Reynolds number in semi ellipse lid driven cavity. In: Proceedings of the International Multi-Conference of Engineers and Computer Scientists, vol. 2, 2015Google Scholar
  147. 147.
    Hancock, C., Lewis, E., Moffatt, H.K.: Effects of inertia in forced corner flows. J. Fluid Mech. 112, 315–327 (1981)zbMATHCrossRefGoogle Scholar
  148. 148.
    Hansen, E.B., Kelmanson, M.A.: An integral equation justification of the boundary conditions of the driven-cavity problem. Comput. Fluids 23, 225–240 (1994)MathSciNetzbMATHCrossRefGoogle Scholar
  149. 149.
    Harlow, F.H., Welsh, J.E.: Numerical calculation of the time dependent viscous incompressible flow with free surface. Phys. Fluids 8, 2182–2189 (1965)MathSciNetzbMATHCrossRefGoogle Scholar
  150. 150.
    Hills, C.P., Moffatt, H.K.: Rotary honing: a variant of the Taylor paint-scraper problem. J. Fluid Mech. 418, 119–135 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  151. 151.
    Hopf, E.: Bericht der Math.-Phys. Klasse der Sächsischen Akademie der Wissenschaften zu Leipzig, Abzweigung einer periodischen Lösung von einer stationären Lösung eines Differentialsystems. 94, 1–22 (1942)Google Scholar
  152. 152.
    Hossain, M.S., Bergstrom, D.J., Chen, X.B.: Visualisation and analysis of large-scale vortex structures in three-dimensional turbulent lid-driven cavity flow. J. Turbul. 16, 901–924 (2015)MathSciNetCrossRefGoogle Scholar
  153. 153.
    Huerre, P., Rossi, M.: Hydrodynamic instabilities in open flows. In: Godréche, C., Manneville, P. (eds.), Hydrodynamics and Nonlinear Instabilities, Chapter 2, pp. 81–294. Cambridge University Press, Cambridge (1998)Google Scholar
  154. 154.
    Humphrey, J., Cushner, J., Sudarsan, R., Al-Shannag, M., Herrero, J., Giralt, F.: Experimental and numerical investigation of the shear-driven flow in a toroid of square cross-section. In: Lindborg, E., Johansson, A., Eaton, J., Humphrey, J., Kasagi, N., Leschziner, M., Sommerfeld, M. (eds) 2nd International Symposium on Turbulence and Shear Flow Phenomena, vol. III, p. 351, Stockholm, Sweden. Royal Institute of Technology (2001)Google Scholar
  155. 155.
    Humphrey, J.A.C., Cushner, J., Al-Shannag, M., Herrero, J., Giralt, F.: Shear-driven flow in a toroid of square cross section. ASME J. Fluids Eng. 125, 130–137 (2003)CrossRefGoogle Scholar
  156. 156.
    Idris, M.S., Azwadi, C.S.N., Ammar, N.M.M.; Cubic interpolation profile Navier–Stokes numerical scheme for particle flow behaviour in triangular lid driven cavity. In: 4th International Meeting of Advances in Thermofluids (2012)Google Scholar
  157. 157.
    Inouye, K.: Ecoulement d’un fluide visqueux dans un angle droit. J. de Mécanique 12, 609–628 (1973)zbMATHGoogle Scholar
  158. 158.
    Iooss, G., Joseph, D.D.: Elementary Stability and Bifurcation Theory. Springer (2012)Google Scholar
  159. 159.
    Isaev, S.A., Baranov, P.A., Sudakov, A.G., Mordynsky, N.A.: Numerical analysis of vortex dynamics and unsteady turbulent heat transfer in lid-driven square cavity. Thermophys. Aeromech. 15, 463–475 (2008)CrossRefGoogle Scholar
  160. 160.
    Ishii, K., Adachi, S.: Numerical analysis of 3d vortical cavity flow. Proc. Appl. Math. Mech. 6, 871–874 (2006)CrossRefGoogle Scholar
  161. 161.
    Ishii, K., Adachi, S.: Transition of streamline patterns in three-dimensional cavity flows. Theor. Appl. Mech. Japan 59, 203–210 (2010)Google Scholar
  162. 162.
    Ishii, K., Adachi, S.: Dependence on the aspect ratio of streamline patterns in three-dimensional cavity flows. Theor. Appl. Mech. Japan 60, 51–61 (2011)Google Scholar
  163. 163.
    Ishii, K., Iwatsu, R.: Numerical simulation of the Lagrangian flow structure in a driven cavity. In: Moffatt, H.K., Tsinober, A. (eds.) Topological Fluid Mechanics, pp. 54–63. Cambridge University Press, Cambridge, U.K. (1990)Google Scholar
  164. 164.
    Ishii, K., Ota, C., Adachi, S.: Streamlines near a closed curve and chaotic streamlines in steady cavity flows. Proc. IUTAM 5, 173–186 (2012)CrossRefGoogle Scholar
  165. 165.
    Ismael, M.A.: Numerical solution of mixed convection in a lid-driven cavity with arc-shaped moving wall. Eng. Comput. 43, 869–891 (2016)Google Scholar
  166. 166.
    Iwatsu, R., Hyun, J.M.: Three-dimensional driven-cavity flows with a vertical temperature gradient. Intl J. Heat Mass Transf. 38, 3319–3328 (1995)zbMATHCrossRefGoogle Scholar
  167. 167.
    Iwatsu, R., Ishii, K., Kawamura, T., Kuwahara, K., Hyun, J.M.: Numerical simulation of three-dimensional flow structure in a driven cavity. Fluid Dyn. Res. 5, 173–189 (1989)CrossRefGoogle Scholar
  168. 168.
    Iwatsu, R., Hyun, J.M., Kuwahara, K.: Mixed convection in a driven cavity with a stable vertical temperature gradient. Intl J. Heat Mass Transf. 36, 1601–1608 (1993)CrossRefGoogle Scholar
  169. 169.
    Jana, S.C., Metcalfe, G., Ottino, J.M.: Experimental and numerical studies of mixing in complex Stokes flow: the vortex mixing flow and multicellular cavity flow. J. Fluid Mech. 269, 199–246 (1994)MathSciNetCrossRefGoogle Scholar
  170. 170.
    Jana, S.C., Tjahjadi, M., Ottino, J.M.: Chaotic mixing of viscous fluids by periodic changes in geometry: baffled cavity flow. AIChE J. 40, 1769–1781 (1994)CrossRefGoogle Scholar
  171. 171.
    Jeong, J., Hussain, F.: On the identification of a vortex. J. Fluid Mech. 285, 69–94 (1995)MathSciNetzbMATHCrossRefGoogle Scholar
  172. 172.
    Jordan, S.A., Ragab, S.A.: On the unsteady and turbulent characteristics of the three-dimensional shear-driven cavity flow. J. Fluids Eng. 116, 439–449 (1994)CrossRefGoogle Scholar
  173. 173.
    Jordi, B.E., Cotter, C.J., Sherwin, S.J.: An adaptive selective frequency damping method. Phys. Fluids 27, 094104–1–094104–8 (2015)CrossRefGoogle Scholar
  174. 174.
    Joseph, D.D.: The convergence of biorthogonal series for biharmonic and Stokes flow edge problems part I. SIAM J. Appl. Math. 33, 337–347 (1977)MathSciNetzbMATHCrossRefGoogle Scholar
  175. 175.
    Joseph, D.D., Sturges, L.: The convergence of biorthogonal series for biharmonic and Stokes flow edge problems: part II. SIAM J. Appl. Math. 34, 7–26 (1978)MathSciNetzbMATHCrossRefGoogle Scholar
  176. 176.
    Kandemir, I., Kaya, A.M.: Molecular dynamics simulation of compressible hot/cold moving lid-driven microcavity flow. Microfluid Nanofluid 12, 509–520 (2012)CrossRefGoogle Scholar
  177. 177.
    Kawaguti, M.: Numerical solution of the Navier-Stokes equations for the flow in a two-dimensional cavity. J. Phys. Soc. Jap. 16, 2307–2315 (1961)MathSciNetzbMATHCrossRefGoogle Scholar
  178. 178.
    Keiller, R.A., Hinch, E.J.: Corner flow of a suspension of rigid rods. J. Non-Newton. Fluid Mech. 40, 323–335 (1991)zbMATHCrossRefGoogle Scholar
  179. 179.
    Keller, H.B.: Numerical Solution of Bifurcation and Nonlinear Eigenvalue Problems, pp. 359–384. Academic Press, New York (1977)Google Scholar
  180. 180.
    Kelmanson, M.A.: An integral equation method for the solution of singular slow flow problems. J. Comput. Phys. 51, 139–158 (1983)zbMATHCrossRefGoogle Scholar
  181. 181.
    Kelmanson, M.A.: Modified integral equation solution of viscous flows near sharp corners. Comput. Fluids 11, 307–324 (1983)zbMATHCrossRefGoogle Scholar
  182. 182.
    Kelmanson, M.A.: Solution of nonlinear elliptic equations with boundary singularities by an integral equation method. J. Comput. Phys. 56, 244–258 (1984)zbMATHCrossRefGoogle Scholar
  183. 183.
    Kelmanson, M.A., Lonsdale, B.: Eddy genesis in the double-lid-driven cavity. Q. J. Mech. Appl. Math. 49, 635–655 (1996)zbMATHCrossRefGoogle Scholar
  184. 184.
    Kelmanson, M.A., Lonsdale, B.: Annihilation of boundary singularities via suitable Green’s functions. Comput. Math. Appl. 29, 1–7 (1995)MathSciNetzbMATHCrossRefGoogle Scholar
  185. 185.
    Kelvin, Lord: Vibrations of a columnar vortex. Phil. Mag. 10, 155–168 (1880)zbMATHCrossRefGoogle Scholar
  186. 186.
    Kerstin, J., Wood, R.T.: On the stability of two-dimensional stagnation flow. J. Fluid Mech. 44, 461–479 (1970)zbMATHCrossRefGoogle Scholar
  187. 187.
    Khanafer, K.: Comparison of flow and heat transfer characteristics in a lid-driven cavity between flexible and modified geometry of a heated bottom wall. Int. J. Heat Mass Transf. 78, 1032–1041 (2014)CrossRefGoogle Scholar
  188. 188.
    Khanafer, K.M., Al-Amiri, A.M., Pop, I.: Numerical simulation of unsteady mixed convection in a driven cavity using an externally excited sliding lid. Eur. J. Mech. B/Fluids 26, 669–687 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  189. 189.
    Khorasanizade, S., Sousa, J.M.M.: A detailed study of lid-driven cavity flow at moderate Reynolds numbers using incompressible SPH. Int. J. Numer. Methods Fluids 76, 653–668 (2014)MathSciNetCrossRefGoogle Scholar
  190. 190.
    Kneib, F., Faug, T., Nicolet, G., Eckert, N., Naaim, M., Dufour, F.: Force fluctuations on a wall in interaction with a granular lid-driven cavity flow. Phys. Rev. E 96, 042906–1–042906–15 (2017)Google Scholar
  191. 191.
    Knoll, D.A., Keyes, D.E.: Jacobian-free Newton-Krylov methods: a survey of approaches and applications. J. Comput. Phys. 193, 357–397 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  192. 192.
    Kondratiev, V.A.: Asymptotics of solutions of the Navier-Stokes equation in a neighbourhood of a corner point. Prikl. Math. Mekh. 31, 119–123 (1967)Google Scholar
  193. 193.
    Koseff, J.R., Street, R.L.: Visualization studies of a shear driven three-dimensional recirculating flow. J. Fluids Eng. 106, 21–29 (1984)CrossRefGoogle Scholar
  194. 194.
    Koseff, J.R., Street, R.L.: On endwall effects in a lid-driven cavity flow. J. Fluids Eng. 106, 385–389 (1984)CrossRefGoogle Scholar
  195. 195.
    Koseff, J.R., Street, R.L.: The lid-driven cavity flow: a synthesis of qualitative and quantitative observations. J. Fluids Eng. 106, 390–398 (1984)CrossRefGoogle Scholar
  196. 196.
    Koseff, J.R., Street, R.L., Gresho, P.M., Upson, C.D., Humphrey, J.A.C., To, W.-M.: A three-dimensional lid-driven cavity flow: Experiment and simulation. In: Taylor, C. (ed) Proceedings of the 3rd International Conference on Numerical Methods in Laminar and Turbulent Flow, pp. 564–581, Swansea. Pineridge Press (1983)Google Scholar
  197. 197.
    Koseff, J.R., Prasad, A.K., Perng, C., Street, R.L.: Complex cavities: Are two dimensions sufficient for computation? Phys. Fluids A 2, 619–622 (1990)CrossRefGoogle Scholar
  198. 198.
    Koseff, R.J., Street, R.L.: Circulation structure in a stratified cavity flow. J. Hydraul. Eng. 111, 334–354 (1985)CrossRefGoogle Scholar
  199. 199.
    Kosinski, P., Kosinska, A., Hoffmann, A.C.: Simulation of solid particles behaviour in a driven cavity flow. Powder Technol. 191, 327–339 (2009)CrossRefGoogle Scholar
  200. 200.
    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, 024104–1–024104–11 (2014)CrossRefGoogle Scholar
  201. 201.
    Kuhlmann, H.C., Wanschura, M., Rath, H.J.: Flow in two-sided lid-driven cavities: non-uniqueness, instabilities, and cellular structures. J. Fluid Mech. 336, 267–299 (1997)MathSciNetzbMATHCrossRefGoogle Scholar
  202. 202.
    Kuhlmann, H.C., Romanò, F., Wu, H., Albensoeder, S.: Particle-motion attractors due to particle-boundary interaction in incompressible steady three-dimensional flows. In: Ivey, G., Zhou, T., Jones, N., Draper, S. (eds) The 20th Australasian Fluid Mechanics Conference, p. 102, Paper no. 449. Australasian Fluid Mechanics Society (2016)Google Scholar
  203. 203.
    Lanczos, C.: An iteration method for the solution of the eigenvalue problem of linear differential and integral operators. J. Res. Natl Bur. Stand. 45, 255–282 (1950)MathSciNetCrossRefGoogle Scholar
  204. 204.
    Landahl, M.T.: Wave breakdown and turbulence. SIAM J. Appl. Math. 28, 735–756 (1975)zbMATHCrossRefGoogle Scholar
  205. 205.
    Larchevêque, L., Sagaut, P., Lê, T.-H., Comte, P.: Large-eddy simulation of a compressible flow in a three-dimensional open cavity at high Reynolds number. J. Fluid Mech. 516, 265–301 (2004)zbMATHCrossRefGoogle Scholar
  206. 206.
    Leong, C.W., Ottino, J.M.: Experiments on mixing due to chaotic advection in a cavity. J. Fluid Mech. 209, 463–499 (1989)MathSciNetCrossRefGoogle Scholar
  207. 207.
    Leray, J.: Etude de diverses equations integrales non lineaires et de quelques problemes que pose l’Hydrodynamique. J. Math. Pures et Appl. 12, 1–82 (1933)zbMATHGoogle Scholar
  208. 208.
    Leriche, E.: Direct numerical simulation in a lid-driven cubical cavity at high Reynolds number by a Chebyshev spectral method. J. Sci. Comput. 27, 335–345 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  209. 209.
    Leriche, E., Gavrilakis, S.: Direct numerical simulation of the flow in a lid-driven cubical cavity. Phys. Fluids 12, 1363–1376 (2000)zbMATHCrossRefGoogle Scholar
  210. 210.
    Leriche, E., Labrosse, G.: High-order direct Stokes solvers with or without temporal splitting: numerical investigations of their comparative properties. SIAM J. Sci. Comput. 22, 1386–1410 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  211. 211.
    Leriche, E., Labrosse, G.: Are there localized eddies in the trihedral corners of the Stokes eigenmodes in cubical cavity? Comput. Fluids 43, 98–101 (2011)zbMATHCrossRefGoogle Scholar
  212. 212.
    Li, M., Tang, T.: Steady viscous flow in a triangular cavity by efficient numerical techniques. Comput. Math. Appl. 31, 55–65 (1996)MathSciNetzbMATHCrossRefGoogle Scholar
  213. 213.
    Liberzon, A., Feldman, Y., Gelfgat, A.Y.: Experimental observation of the steady-oscillatory transition in a cubic lid-driven cavity. Phys. Fluids 23, 084106–1–084106–7 (2011)CrossRefGoogle Scholar
  214. 214.
    Lin, L.-S., Chen, Y.-C., Lin, C.-A.: Multi relaxation time lattice Boltzmann simulations of deep lid driven cavity flows at different aspect ratios. Comp. Fluids 45, 233–240, (2011)zbMATHCrossRefGoogle Scholar
  215. 215.
    Liu, C.H., Joseph, D.D.: Stokes flow in conical trenches. SIAM J. Appl. Math. 34, 286–296 (1978)MathSciNetzbMATHCrossRefGoogle Scholar
  216. 216.
    Liu, M., Muzzio, F.J., Peskin, R.L.: Quantification of mixing in aperiodic chaotic flows. Chaos Solitons Fractals 4, 869–893 (1994)zbMATHCrossRefGoogle Scholar
  217. 217.
    Liu, M., Peskin, R.L., Muzzio, F.J., Leong, C.W.: Structure of the stretching field in chaotic cavity flows. AIChE J. 40, 1273–1286 (1994)CrossRefGoogle Scholar
  218. 218.
    Liu, Q., Gómez, F., Theofilis, V.: Linear instability analysis of incompressible flow over a cuboid cavity. Procedia IUTAM 14, 511–518 (2015)CrossRefGoogle Scholar
  219. 219.
    Loiseau, J.-C.: Analyse de la stabilité globale et de la dynamique d’écoulements tridimensionnels (Dynamics and global stability analyses of three-dimensional flows). PhD thesis, l’École Nationale Supérieur d’Arts et Métiers (2014)Google Scholar
  220. 220.
    Loiseau, J.C., Robinet, J.C., Leriche, E.: Intermittency and transition to chaos in the cubical lid-driven cavity flow. Fluid Dyn. Res. 061421–1–061421–11 (2016)Google Scholar
  221. 221.
    Lopez, J.M., Welfert, B.D., Wu, K., Yalim, J.: Transition to complex dynamics in the cubic lid-driven cavity. Phys. Rev. Fluids 2, 074401–1–074401–23 (2017)Google Scholar
  222. 222.
    Ma, H., Ruth, D.: A new scheme for vorticity computations near a sharp corner. Comput. Fluids 23, 23–38 (1994)MathSciNetzbMATHCrossRefGoogle Scholar
  223. 223.
    Malhotra, C.P., Weidman, P.D., Davis, A.M.J.: Nested toroidal vortices between concentric cones. J. Fluid Mech. 522, 117–139 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  224. 224.
    Marcus, P.S., Tuckerman, L.S.: Simulation of flow between concentric rotating spheres. part 2. transitions. J. Fluid Mech. 185, 31–65 (1987)zbMATHCrossRefGoogle Scholar
  225. 225.
    Maull, D.J., East, L.F.: Three-dimensional flow in cavities. J. Fluid Mech. 16, 620–632 (1963)zbMATHCrossRefGoogle Scholar
  226. 226.
    McIlhany, K.L., Mott, D., Oran, E., Wiggins, S.: Optimizing mixing in lid-driven flow designs through predictions from Eulerian indicators. Phys. Fluids 23, 082005–01–082005–13 (2011)CrossRefGoogle Scholar
  227. 227.
    Mezić, I., Wiggins, S.: On the integrability and perturbation of three-dimensional fluid flows with symmetry. J. Nonlinear Sci. 4, 157–194 (1994)MathSciNetzbMATHCrossRefGoogle Scholar
  228. 228.
    Migeon, C.: Details on the start-up development of the Taylor–Gortler like vortices inside a square-section lid-driven cavity for \(1,000 \le {R}e \le 3,200\). Exp. Fluids 33, 594–602 (2002)CrossRefGoogle Scholar
  229. 229.
    Migeon, C., Texier, A., Pineau, G.: Effects of lid-driven cavity shape on the flow establishment phase. J. Fluids Struct. 14, 469–488 (2000)CrossRefGoogle Scholar
  230. 230.
    Migeon, C., Pineau, G., Texier, A.: Three-dimensionality development inside standard parallelepipedic lid-driven cavities at \({R}e=1000\). J. Fluids Struct. 17, 717–738 (2003)CrossRefGoogle Scholar
  231. 231.
    Moffatt, H.K.: Viscous and resistive eddies near a sharp corner. J. Fluid Mech. 18, 1–18 (1964)zbMATHCrossRefGoogle Scholar
  232. 232.
    Moffatt, H.K.: Viscous eddies near a sharp corner. Arch. Mech. Stosow. 16, 365–372 (1964)MathSciNetzbMATHGoogle Scholar
  233. 233.
    Moffatt, H.K.: Singularities in Fluid Dynamics and their Resolution, vol. 1973. Lecture Notes in Mathematics, pp. 157–166. Springer, Berlin (2001)Google Scholar
  234. 234.
    Moffatt, H.K., Duffy, B.R.: Local similarity solutions and their limitations. J. Fluid Mech. 96, 299–313 (1980)zbMATHCrossRefGoogle Scholar
  235. 235.
    Moffatt, H.K., Mak, V.: Corner singularities in three-dimensional Stokes flow. In: Durban, D., Pearson, J.R.A. (eds) Symposium on non-linear singularities in deformation and flow, pp. 21–26, Netherland. IUTAM, Kluwer Academic Publishers (1999)zbMATHCrossRefGoogle Scholar
  236. 236.
    Mohamad, A.A., Viskanta, R.: Transient low Prandtl number fluid convection in a lid-driven cavity. Numer. Heat Transf. A 19, 187–205 (1991)CrossRefGoogle Scholar
  237. 237.
    Mohamad, A.A., Viskanta, R.: Flow and heat transfer in a lid-driven cavity filled with a stably stratified fluid. Appl. Math. Model. 19, 465–472 (1995)zbMATHCrossRefGoogle Scholar
  238. 238.
    Moore, D.W., Saffman, P.G.: The instability of a straight vortex filament in a strain field. Proc. R. Soc. Lond. A 346, 413–425 (1975)MathSciNetzbMATHCrossRefGoogle Scholar
  239. 239.
    Neary, M.D., Stephanoff, D.: Shear-layer-driven transition in a rectangular cavity. Phys. Fluids 30, 2936–2946 (1987)CrossRefGoogle Scholar
  240. 240.
    Nobile, E.: Simulation of time-dependent flow in cavities with the additive-correction multigrid method, part I: mathematical formulation. Numer. Heat Transf. B 30, 341–350 (1996)CrossRefGoogle Scholar
  241. 241.
    Noor, D.Z., Kanna, P.R., Chern, M.-J.: Flow and heat transfer in a driven square cavity with double-sided oscillating lids in anti-phase. Int. J. Heat Mass Transf. 52, 3009–3023 (2009)zbMATHCrossRefGoogle Scholar
  242. 242.
    Nuriev, A.N., Egorov, A.G., Zaitseva, O.N.: Bifurcation analysis of steady-state flows in the lid-driven cavity. Fluid Dyn. Res. 48, 061405-1–061405-15 (2016)MathSciNetCrossRefGoogle Scholar
  243. 243.
    Ohmichi, Y., Suzuki, K.: Compressibility effects on the first global instability mode of the vortex formed in a regularized lid-driven cavity flow. Comput. Fluids 145, 1–7 (2017)MathSciNetzbMATHCrossRefGoogle Scholar
  244. 244.
    Ottino, J.M., Leong, C.W., Rising, H., Swanson, P.D.: Morphological structures produced by mixing in chaotic flows. Nature 333, 419–425 (1988)CrossRefGoogle Scholar
  245. 245.
    Ottino, J.M., Muzzio, F.J., Tjahjadi, M., Franjione, J.G., Jana, S.C., Kusch, H.A.: Chaos, symmetry, and self-similarity: exploiting order and disorder in mixing processes. Science 257, 754–760 (1992)CrossRefGoogle Scholar
  246. 246.
    Oztop, H.F.: Combined convection heat transfer in a porous lid-driven enclosure due to heater with finite length. Int. Commun. Heat Mass Transf. 33, 772–779 (2006)CrossRefGoogle Scholar
  247. 247.
    Oztop, H.F., Dagtekin, I.: Mixed convection in two-sided lid-driven differentially heated square cavity. Int. J. Heat Mass Transf. 47, 1761–1769 (2004)zbMATHCrossRefGoogle Scholar
  248. 248.
    Oztop, H.F., Varol, A.: Combined convection in inclined porous lid-driven enclosures with sinusoidal thermal boundary condition on one wall. Prog. Comput. Fluid Dyn. 9, 127–131 (2009)CrossRefGoogle Scholar
  249. 249.
    Oztop, H.F., Zhao, Z., Yu, B.: Fluid flow due to combined convection in lid-driven enclosure having a circular body. Int. J. Heat Fluid Flow 30, 886–901 (2009)CrossRefGoogle Scholar
  250. 250.
    Oztop, H.F., Zhao, Z., Yu, B.: Conduction-combined forced and natural convection in lid-driven enclosures divided by a vertical solid partitionstar, open. Int. Commun. Heat Mass Transf. 36, 661–668 (2009)CrossRefGoogle Scholar
  251. 251.
    Pakdel, P., McKinley, G.H.: Elastic instability and curved streamlines. Phys. Rev. Lett. 77, 2459–2462 (1996)CrossRefGoogle Scholar
  252. 252.
    Pakdel, P., McKinley, G.H.: Cavity flows of elastic liquids: purely elastic instabilities. Phys. Fluids 10, 1058–1070 (1998)CrossRefGoogle Scholar
  253. 253.
    Pakdel, P., Spiegelberg, S.H., McKinley, G.H.: Cavity flows of elastic liquids: two-dimensional flows. Phys. Fluids 9, 3123–3140 (1997)CrossRefGoogle Scholar
  254. 254.
    Pan, F., Acrivos, A.: Steady flows in rectangular cavities. J. Fluid Mech. 28, 643–655 (1967)CrossRefGoogle Scholar
  255. 255.
    Parlett, B.N.: The Symmetric Eigenvalue Problem. Prentice-Hall, Englewood Cliffs, NJ (1980)zbMATHGoogle Scholar
  256. 256.
    Pasquim, B.M., Mariani, V.C.: Solutions for incompressible viscous flow in a triangular cavity using cartesian grid method. Comput. Model. Eng. Sci. 35, 113–132 (2008)MathSciNetzbMATHGoogle Scholar
  257. 257.
    Patel, D.K., Das, M.K.: LES of incompressible turbulent flow inside a cubical cavity driven by two parallel lids moving in opposite direction. Int. J. Heat Mass Transf. 67, 1039–1053 (2013)CrossRefGoogle Scholar
  258. 258.
    Patel, D.K., Das, M.K., Roy, S.: LES of turbulent flow in a cubical cavity with two parallel lids moving in opposite direction. Int. J. Heat Mass Transf. 72, 37–49 (2014)CrossRefGoogle Scholar
  259. 259.
    Peng, Y.-F., Shiau, Y.-H., Hwang, R.R.: Transition in a 2-D lid-driven cavity flow. Comput. Fluids 32, 337–352 (2003)zbMATHCrossRefGoogle Scholar
  260. 260.
    Peplinski, A., Schlatter, P., Fischer, P.F. Henningson, D.S. Stability tools for the spectral-element code Nek5000: application to jet-in-crossflow. In: Spectral and High Order Methods for Partial Differential Equations-ICOSAHOM 2012, pp. 349–359. Springer, Berlin (2014)zbMATHGoogle Scholar
  261. 261.
    Peplinski, A., Schlatter, P., Henningson, D.S.: Global stability and optimal perturbation for a jet in cross-flow. Eur. J. Mech. B/Fluids 49, 438–447 (2015)MathSciNetCrossRefGoogle Scholar
  262. 262.
    Pierrehumbert, R.T.: Universal short-wave instability of two-dimensional eddies in an inviscid fluid. Phys. Rev. Lett. 57, 2157–2159 (1986)CrossRefGoogle Scholar
  263. 263.
    Poliashenko, M., Aidun, C.K.: A direct method for computation of simple bifurcations. J. Comput. Phys. 121, 246–260 (1995)MathSciNetzbMATHCrossRefGoogle Scholar
  264. 264.
    Povitsky, A.: Three-dimensional flow in cavity at yaw. Technical Report NASA/CR-2001-211232, ICASE Report No. 2001–31, ICASE, NASA Langley Research Center, ICASE, Hampton, Virginia (2001)Google Scholar
  265. 265.
    Povitsky, A.: Three-dimensional flow in cavity at yaw. Nonlinear Analysis 63, e1573–e1584 (2005)zbMATHCrossRefGoogle Scholar
  266. 266.
    Povitsky, A.: Three-dimensional flow with elevated helicity in driven cavity by parallel walls moving in perpendicular directions. Phys. Fluids 29, 083601-1–083601-11 (2017)MathSciNetCrossRefGoogle Scholar
  267. 267.
    Prandtl, L.: Über Flüssigkeitsbewegung bei sehr kleiner Reibung. In: Verhdlg. III Intern. Math.-Kongr., pp. 484–491, Leipzig. Teubner (1904)Google Scholar
  268. 268.
    Prasad, A.K., Koseff, J.R.: Reynolds number and end-wall effects on a lid-driven cavity flow. Phys. Fluids A 1, 208–218 (1989)CrossRefGoogle Scholar
  269. 269.
    Prasad, A.K., Koseff, J.R.: Combined forced and natural convection heat transfer in a deep lid-driven cavity flow. Int. J. Heat Fluid Flow 17, 460–467 (1996)CrossRefGoogle Scholar
  270. 270.
    Ramanan, N., Homsy, G.M.: Linear stability of lid-driven cavity flow. Phys. Fluids 6, 2690–2701 (1994)zbMATHCrossRefGoogle Scholar
  271. 271.
    Rao, P., Duggleby, A., Stremler, M.A.: Mixing analysis in a lid-driven cavity flow at finite Reynolds numbers. ASME J. Fluids Eng. 134, 041203-1–041203-8 (2012)CrossRefGoogle Scholar
  272. 272.
    Rayleigh, L.: On the dynamics of revolving fluids. In: Scientific Papers VI, pp. 447–453. Cambridge University Press, Cambridge (1920)Google Scholar
  273. 273.
    Riedler, J., Schneider, W.: Viscous flow in corner regions with a moving wall and leakage of fluid. Acta Mech. 48, 95–102 (1983)zbMATHCrossRefGoogle Scholar
  274. 274.
    Rockwell, D.D., Naudascher, E.E.: Review – self-sustaining oscillations of flow past cavities. ASME J. Fluids Eng. 100, 152–165 (1978)CrossRefGoogle Scholar
  275. 275.
    Romanò, F., Kuhlmann, H.C.: Numerical investigation of the interaction of a finite-size particle with a tangentially moving boundary. Int. J. Heat Fluid Flow 62(Part A), 75–82 (2016)CrossRefGoogle Scholar
  276. 276.
    Romanò, F., Kuhlmann, H.C.: Particle-boundary interaction in a shear-driven cavity flow. Theor. Comput. Fluid Dyn. 31, 427–445 (2017)CrossRefGoogle Scholar
  277. 277.
    Romanò, F., Albensoeder, S., Kuhlmann, H.C.: Topology of three-dimensional steady cellular flow in a two-sided anti-parallel lid-driven cavity. J. Fluid Mech. 826, 302–334 (2017)MathSciNetCrossRefGoogle Scholar
  278. 278.
    Rossiter, J.E.: Wind tunnel experiments on the flow over rectangular cavities at subsonic and transonic speeds. Technical Report 64037, Royal Aircraft Establishment (1964)Google Scholar
  279. 279.
    Safdari, A., Kim, K.C.: Lattice Boltzmann simulation of solid particles behavior in a three-dimensional lid-driven cavity flow. Comput. Math. Appl. 68, 606–621 (2014)MathSciNetzbMATHCrossRefGoogle Scholar
  280. 280.
    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. Methods Fluids 42, 79–88 (2003)zbMATHCrossRefGoogle Scholar
  281. 281.
    Sano, O., Hasimoto, H.: Three-dimensional Moffatt-type eddies due to a Stokeslet in a corner. J. Phys. Soc. Japan 48, 1763–1768 (1980)MathSciNetzbMATHCrossRefGoogle Scholar
  282. 282.
    Schimmel, F., Albensoeder, S., Kuhlmann, H.: Stability of thermocapillary-driven flow in rectangular cavities. Proc. Appl. Math. Mech. 5, 583–584 (2005)CrossRefGoogle Scholar
  283. 283.
    Schneider, T.M., Gibson, J.F., Lagha, M., Lillo, F.D., Eckhardt, B.: Laminar-turbulent boundary in plane Couette flow. Phys. Rev. E 78, 037301-1–037301-4 (2008)Google Scholar
  284. 284.
    Schreiber, R., Keller, H.B.: Driven cavity flows by efficient numerical techniques. J. Comput. Phys. 49, 310–333 (1983)zbMATHCrossRefGoogle Scholar
  285. 285.
    Schultz, W.W., Lee, N.Y., Boyd, J.P.: Chebyshev pseudosprectral method of viscous flows with corner singularities. J. Sci. Comput. 4, 1–24 (1989)zbMATHCrossRefGoogle Scholar
  286. 286.
    Schumack, M.R., Schultz, W.W., Boyd, J.P.: Spectral method solution of the Stokes equations on nonstaggered grids. J. Comput. Phys. 94, 30–58 (1991)MathSciNetzbMATHCrossRefGoogle Scholar
  287. 287.
    Scott, J.F.: Moffatt-type flows in a trihedral cone. J. Fluid Mech. 725, 446–461 (2013)MathSciNetzbMATHCrossRefGoogle Scholar
  288. 288.
    Scriven, L.E., Sternling, C.V.: The Marangoni effects. Nature 187, 186–188 (1960)CrossRefGoogle Scholar
  289. 289.
    Serrin, J.: On the stability of viscous fluid motions. Arch. Ration. Mech. Anal. 3, 1–13 (1959)MathSciNetzbMATHCrossRefGoogle Scholar
  290. 290.
    Shankar, P.N.: On Stokes flow in a semi-infinite wedge. J. Fluid Mech. 422, 69–90 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  291. 291.
    Shankar, P.N.: Moffatt eddies in the cone. J. Fluid Mech. 539, 113–135 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  292. 292.
    Shankar, P.N.: Slow Viscous Flows. Imperial College Press, London (2007)CrossRefGoogle Scholar
  293. 293.
    Shankar, P.N., Deshpande, M.D.: Fluid mechanics in the driven cavity. Annu. Rev. Fluid Mech. 32, 93–136 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  294. 294.
    Shankar, P.N., Nikiforovich, E.I.: Slow mixed convection in rectangular containers. J. Fluid Mech. 471, 203–217 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  295. 295.
    Shatrov, V., Mutschke, G., Gerbeth, G.: Three-dimensional linear stability analysis of lid-driven magnetohydrodynamic cavity flow. Phys. Fluids 15, 2141–2151 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  296. 296.
    Shen, J.: Hopf bifurcation of the unsteady regularized driven cavity flow. J. Comput. Phys. 95, 228–245 (1991)zbMATHCrossRefGoogle Scholar
  297. 297.
    Sheu, T.W.H., Tsai, S.F.: Flow topology in a steady three-dimensional lid-driven cavity. Comput. Fluids 31, 911–934 (2002)zbMATHCrossRefGoogle Scholar
  298. 298.
    Sidik, N.A.C., Attarzadeh, S.M.R.: An accurate numerical prediction of solid particle fluid flow in a lid-driven cavity. Int. J. Mech. 5, 123–128 (2011)Google Scholar
  299. 299.
    Siegmann-Hegerfeld, T.: Wirbelinstabilitäten und Musterbildung in geschlossenen Rechteckbehältern mit tangential bewegten Wänden (in German). PhD thesis, TU Wien (2010)Google Scholar
  300. 300.
    Siegmann-Hegerfeld, T., Albensoeder, S., Kuhlmann, H.C.: Two- and three-dimensional flows in nearly rectangular cavities driven by collinear motion of two facing walls. Exp. Fluids 45, 781–796 (2008)CrossRefGoogle Scholar
  301. 301.
    Siegmann-Hegerfeld, T., Albensoeder, S., Kuhlmann, H.C.: Three-dimensional flow in a lid-driven cavity with width-to-height ratio of 1.6. Exp. Fluids 54, 1526–1–1526–10 (2013)Google Scholar
  302. 302.
    Sleijpen, G.L.G., van der Vorst, H.A.: A Jacobi–Davidson iteration method for linear eigenvalue problems. SIAM Rev. 42, 267–293 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  303. 303.
    Sousa, R., Poole, R., Afonso, A., Pinho, F., Oliveira, P., Morozov, A., Alves, M.: Lid-driven cavity flow of viscoelastic liquids. J. Non-Newton. Fluid Mech. 234, 129–138 (2016)MathSciNetCrossRefGoogle Scholar
  304. 304.
    Spasov, Y., Herrero, J., Grau, F.X., Giralt, F.: Linear stability analysis and numerical calculations of the lid-driven flow in a toroidally shaped cavity. Phys. Fluids 15, 134–146 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  305. 305.
    Stremler, M.A., Chen, J.: Generating topological chaos in lid-driven cavity flow. Phys. Fluids 19, 103602-1–103602-6 (2007)zbMATHCrossRefGoogle Scholar
  306. 306.
    Symm, G.T.: Treatment of Singularities in the Solution of Laplace’s Equation by an Integral Equation Method. National Physical Laboratory, Division of Numerical Analysis and Computing (1973)Google Scholar
  307. 307.
    Tang, L.Q., Cheng, T., Tsang, T.T.H.: Transient solutions for three-dimensional lid-driven cavity flows by a least-squares finite element method. Int. J. Numer. Methods Fluids 21, 413–432 (1995)zbMATHCrossRefGoogle Scholar
  308. 308.
    Taylor, G.I.: Similarity solutions of hydrodynamic problems. In: Aeronautics and Astronautics (Durand Anniversary Volume), pp. 21–28. Pergamon (1960)Google Scholar
  309. 309.
    Taylor, G.I.: On scraping viscous fluid from a plane surface. In: Batchelor, G.K (ed), The Scientific Papers of Sir Geoffrey Ingram Taylor (1962)Google Scholar
  310. 310.
    Teixeira, C.M.: Digital physics simulations of lid-driven cavity flow. Int. J. Mod. Phys. C 8, 685–696 (1997)CrossRefGoogle Scholar
  311. 311.
    Theofilis, V.: Globally unstable basic flows in open cavities. In: 6th AIAA/CEAS Aeroacoustics Conference, pp. AIAA 2000–1965, Reston, VA. AIAA (2000)Google Scholar
  312. 312.
    Theofilis, V.: Global linear instability. Annu. Rev. Fluid Mech. 43, 319–352 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  313. 313.
    Theofilis, V., Colonius, T.: An algorithm for the recovery of 2- and 3-D BiGlobal instabilities of compressible flow over 2-D open cavities. In: 33rd Fluid Dynamics Conference and Exhibit, vol. 39, pp. AIAA 2003–4143, Reston, VA. AIAA (2003)Google Scholar
  314. 314.
    Theofilis, V., Duck, P.W., Owen, J.: Viscous linear stability analysis of rectangular duct and cavity flows. J. Fluid Mech. 505, 249–286 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  315. 315.
    Thiffeault, J.-L., Gouillart, E., Dauchot, O.: Moving walls accelerate mixing. Phys. Rev. E 84, 036313-1–036313-8 (2011)Google Scholar
  316. 316.
    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)MathSciNetzbMATHCrossRefGoogle Scholar
  317. 317.
    Tiwari, R.K., Das, M.K.: Heat transfer augmentation in a two-sided lid-driven differentially heated square cavity utilizing nanofluids. Intl J. Heat Mass Transf. 50, 2002–2018 (2007)zbMATHCrossRefGoogle Scholar
  318. 318.
    Torrance, K., Davis, R., Eike, K., Gill, P., Gutman, D., Hsui, A., Lyons, S., Zien, H.: Cavity flows driven by buoyancy and shear. J. Fluid Mech. 51, 221–231 (1972)CrossRefGoogle Scholar
  319. 319.
    Tranter, C.J.: The use of the Mellin transform in finding the stress distribution in an infinite wedge. Q. J. Mech. Appl. Math. 1, 125–130 (1948)MathSciNetzbMATHCrossRefGoogle Scholar
  320. 320.
    Tsorng, S.J., Capart, H., Lai, J.S., Young, D.L.: Three-dimensional tracking of the long time trajectories of suspended particles in a lid-driven cavity flow. Exp. Fluids 40, 314–328 (2006)CrossRefGoogle Scholar
  321. 321.
    Tsorng, S.J., Capart, H., Lo, D.C., Lai, J.S., Young, D.L.: Behaviour of macroscopic rigid spheres in lid-driven cavity flow. Int. J. Multiph. Flow 34, 76–101 (2008)CrossRefGoogle Scholar
  322. 322.
    van Lenthe, J.H., Pulay, P.: A space-saving modification of Davidson’s eigenvector algorithm. J. Comput. Chem. 11, 1164–1168 (1990)MathSciNetCrossRefGoogle Scholar
  323. 323.
    Vandeven, H.: Family of spectral filters for discontinous problems. J. Sci. Comput. 6, 159–192 (1991)MathSciNetzbMATHCrossRefGoogle Scholar
  324. 324.
    Verstappen, R., Wissink, J.G., Veldman, A.E.E.: Direct numerical simulation of driven cavity flows. Appl. Sci. Res. 51, 377–381 (1993)zbMATHCrossRefGoogle Scholar
  325. 325.
    Verstappen, R., Wissink, J.G., Cazemier, W., Veldman, A.E.P.: Direct numerical simulations of turbulent flow in a driven cavity. Future Gener. Comput. Syst. 10, 345–350 (1994)CrossRefGoogle Scholar
  326. 326.
    Vogel, M.J., Hirsa, A.H., Lopez, J.M.: Spatio-temporal dynamics of a periodically driven cavity flow. J. Fluid Mech. 478, 197–226 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  327. 327.
    Wahba, E.M.: Multiplicity of states for two-sided and four-sided lid driven cavity flows. Comput. Fluids 38, 247–253 (2009)zbMATHCrossRefGoogle Scholar
  328. 328.
    Wakiya, S.: Axisymmetric flow of a viscous fluid near the vertex of a body. J. Fluid Mech. 78, 737–747 (1976)zbMATHCrossRefGoogle Scholar
  329. 329.
    Waleffe, F.: On the three-dimensional instability of strained vortices. Phys. Fluids A 2, 76–80 (1990)MathSciNetzbMATHCrossRefGoogle Scholar
  330. 330.
    Wesseling, P.: A Robust and Efficient Multigrid Method, vol. 960, pp. 614–630. Springer, Berlin (1982)Google Scholar
  331. 331.
    Xu, B., Gilchrist, J.F.: Shear migration and chaotic mixing of particle suspensions in a time-periodic lid-driven cavity. Phys. Fluids 22, 053301-1–053301-7 (2010)zbMATHCrossRefGoogle Scholar
  332. 332.
    Yang, X., Forest, M.G., Mullins, W., Wang, Q.: 2-D lid-driven cavity flow of nematic polymers: an unsteady sea of defects. Soft Matter 6, 1138–1156 (2010)CrossRefGoogle Scholar
  333. 333.
    Zang, Y., Street, R.L., Koseff, J.R.: A dynamic mixed subgrid-scale model and its application to turbulent recirculating flows. Phys. Fluids A 5, 3186–3196 (1993)zbMATHCrossRefGoogle Scholar
  334. 334.
    Zang, Y., Street, R.L., Koseff, J.R.: A non-staggered grid, fractional step method for time-dependent incompressible Navier–Stokes equations in curvilinear coordinates. J. Comput. Phys. 114, 18–33 (1994)MathSciNetzbMATHCrossRefGoogle Scholar
  335. 335.
    Zhou, Y.C., Patnaik, B.S.V., Wan, D.C., Wei, G.W.: DSC solution for flow in a staggered double lid driven cavity. Int. J. Numer. Methods Eng. 57, 211–234 (2003)zbMATHCrossRefGoogle Scholar
  336. 336.
    Znaien, J., Speetjens, M.F.M., Trieling, R.R., Clercx, H.J.H.: Observability of periodic lines in three-dimensional lid-driven cylindrical cavity flows. Phys. Rev. E 85, 066320–1–066320–14 (2012)Google Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Institute of Fluid Mechanics and Heat Transfer, TU WienViennaAustria

Personalised recommendations