Comptes Rendus
no. S1
Numerical study of transitions in lid-driven flows in shallow cavities
Tsorng-Whay Pan1   ; Shang-Huan Chiu2  ; Aixia Guo1  ; Jiwen He1
1 Department of Mathematics, University of Houston, Houston, Texas 77204, USA
2 Department of Mathematics, Lehigh University, Bethlehem, PA, 18015, USA
Comptes Rendus. Mécanique, Online first (2023), pp. 1-17.

In this article, three dimensional (3D) lid-driven flow in shallow cavities with a unit square base are studied. The numerical solution of the Navier–Stokes equations modeling incompressible viscous fluid flow in a cavity is obtained via a methodology combining a first order accurate operator-splitting scheme, a L 2 -projection Stokes solver, a wave-like equation treatment of the advection and finite element space approximations. Numerical results of a lid-driven flow in a cubic cavity show a good agreement with those reported in literature. The critical Reynolds numbers (Re cr ) for having flow with increasing of oscillating amplitude (a Hopf bifurcation) in different shallow cavities are obtained and associated oscillating modes are studied.

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DOI : 10.5802/crmeca.166
Mots clés : Lid driven cavity flow, Shallow cavity, Taylor–Görtler-like vortices, Hopf bifurcation, Projection method
Tsorng-Whay Pan 1 ; Shang-Huan Chiu 2 ; Aixia Guo 1 ; Jiwen He 1

1 Department of Mathematics, University of Houston, Houston, Texas 77204, USA
2 Department of Mathematics, Lehigh University, Bethlehem, PA, 18015, USA
Licence : CC-BY 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
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     title = {Numerical study of transitions in lid-driven flows in shallow cavities},
     journal = {Comptes Rendus. M\'ecanique},
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     year = {2023},
     doi = {10.5802/crmeca.166},
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     note = {Online first},
}
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Tsorng-Whay Pan; Shang-Huan Chiu; Aixia Guo; Jiwen He. Numerical study of transitions in lid-driven flows in shallow cavities. Comptes Rendus. Mécanique, Online first (2023), pp. 1-17. doi : 10.5802/crmeca.166.

[1] C. K. Aidun; N. G. Triantafillopoulos; J. D. Benson Global stability of a lid-driven cavity with throughflow: Flow visualization studies, Phys. Fluids A, Volume 3 (1991), pp. 2081-2091 | DOI

[2] P. N. Shankar; M. D. Deshpande Fluid mechanics in the driven cavity, Annu. Rev. Fluid Mech., Volume 32 (2000), pp. 93-136 | DOI | Zbl

[3] O. Goyon High-Reynolds number solutions of Navier–Stokes equations using incremental unknowns, Comput. Methods Appl. Mech. Eng., Volume 130 (1996), pp. 319-335 | DOI | Zbl

[4] F. Auteri; N. Parolini; L. Quartapelle Numerical investigation on the stability of singular driven cavity flow, J. Comput. Phys., Volume 183 (2002), pp. 1-25 | DOI | Zbl

[5] M. Sahin; R. G. Owens A novel fully implicit finite volume method applied to the lid-driven cavity problem. Part II: Linear stability analysis, Int. J. Numer. Meth. Fluids, Volume 42 (2003), pp. 79-88 | DOI | Zbl

[6] C. H. Bruneau; M. Saad The 2D lid-driven cavity problem revisited, Comput. Fluids, Volume 35 (2006), pp. 326-348 | DOI | Zbl

[7] T. Wang; T.-W. Pan; R. Glowinski A comparison of L 2 -projection and H 1 -projection methods for the numerical simulation of incompressible viscous fluid flow: A case study, Chin. J. Eng. Math., Volume 25 (2008), pp. 761-778 | Zbl

[8] Y. Feldman; A. Y. Gelfgat Oscillatory instability of a three-dimensional lid-driven flow in a cube, Phys. Fluids, Volume 22 (2010), 093602 | DOI

[9] A. Liberzon; Y. Feldman; A. Y. Gelfgat Experimental observation of the steady-oscillatory transition in a cubic lid-driven cavity, Phys. Fluids, Volume 23 (2011), 084106 | DOI

[10] K. Anupindi; W. Lai; S. Frankel Characterization of oscillatory instability in lid driven cavity flows using lattice Boltzmann method, Comput. Fluids, Volume 92 (2014), pp. 7-21 | DOI | Zbl

[11] H. C. Kuhlmann; S. Albensoeder Stability of the steady three-dimensional lid-driven flow in a cube ans the supercritical flow dynamics, Phys. Fluids, Volume 26 (2014), 024104 | DOI

[12] R. Iwatsu; J. M. Hyun; K. Kuwahara Analyses of three-dimensional flow calculations in a driven cavity, Fluid Dyn. Res., Volume 6 (1990), pp. 91-102 | DOI

[13] F. Giannetti; P. Luchini; L. Marino Linear stability analysis of three-dimensional lid-driven cavity flow, Atti del XIX Congresso AIMETA di Meccanica Teorica e Applicata, Aras Edizioni Ancona, Italy, 2009

[14] A. J. Chorin; T. J. R. Hughes; M. F. McCracken; J. E. Marsden Product formulas and numerical algorithms, Commun. Pure Appl. Math., Volume 31 (1978), pp. 205-256 | DOI | Zbl

[15] R. Glowinski; G. Guidoboni; T.-W. Pan Wall-driven incompressible viscous flow in a two-dimensional semi-circular cavity, J. Comput. Phys., Volume 216 (2006), pp. 79-91 | DOI | Zbl

[16] T.-W. Pan; R. Glowinski A projection/wave-like equation method for the numerical simulation of incompressible viscous fluid flow modeled by the Navier–Stokes equations, Comput. Fluid Dyn. J., Volume 9 (2009), pp. 28-42

[17] E. J. Dean; R. Glowinski A wave equation approach to the numerical solution of the Navier–Stokes equations for incompressible viscous flow, C. R. Acad. Sci. Paris Sér. I, Volume 325 (1997), pp. 783-791 | DOI | Zbl

[18] J. Hao; T.-W. Pan; R. Glowinski; D. D. Joseph A fictitious domain/distributed Lagrange multiplier method for the particulate flow of Oldroyd-B fluids: A positive definiteness preserving approach, J. Non-Newtonian Fluid Mech., Volume 156 (2009), pp. 95-111 | DOI | Zbl

[19] A. J. Chorin A numerical method for solving incompressible viscous flow problems, J. Comput. Phys., Volume 2 (1967), pp. 12-26 | DOI

[20] A. J. Chorin Numerical solution of the Navier–Stokes equations, Math. Comput., Volume 23 (1968), pp. 341-354

[21] R. Temam Sur l’approximation des équations de Navier–Stokes par la méthode des pas fractionnaires (I), Arch. Rat. Mech. Anal., Volume 32 (1969), pp. 135-153 | DOI | Zbl

[22] R. Temam Sur l’approximation des équations de Navier–Stokes par la méthode des pas fractionnaires (II), Arch. Rat. Mech. Anal., Volume 33 (1969), pp. 377-385 | DOI | Zbl

[23] G. I. Marchuk Splitting and alternating direction methods, Handbook of Numerical Analysis (P. G. Ciarlet; J.-L. Lions, eds.), Volume I, North-Holland, Amsterdam, 1990 | Zbl

[24] S. Turek A comparative study of time-stepping techniques for the incompressible Navier–Stokes equations: from fully implicit nonlinear schemes to semi-implicit projection methods, Int. J. Numer. Math. Fluids, Volume 22 (1996), pp. 987-1011 | DOI | Zbl

[25] M. Marion; R. Temam Navier–Stokes equations, Handbook of Numerical Analysis (P. G. Ciarlet; J.-L. Lions, eds.), Volume VI, North-Holland, Amsterdam, 1998

[26] R. Glowinski Finite element methods for incompressible viscous flow, Handbook of Numerical Analysis (P. G. Ciarlet; J.-L. Lions, eds.), Volume IX, North-Holland, Amsterdam, 2003 | Zbl

[27] Splitting Methods in Communication, Imaging, Science, and Engineering (R. Glowinski; S. J. Osher; W. Yin, eds.), Springer, Cham, Switzerland, 2017 | DOI | Zbl

[28] M. O. Bristeau; R. Glowinski; J. Périaux Numerical methods for the Navier–Stokes equations. Applications to the simulation of compressible and incompressible viscous flow, Comput. Phys. Rep., Volume 6 (1987), pp. 73-187 | DOI

[29] R. Glowinski; T.-W. Pan Numerical Simulation of Incompressible Viscous Flow: Methods and Applications, De Gruyter, Berlin/Boston, 2022 | DOI

[30] S. Fujima; M. Tabata; Y. Fukasawa Extension to three-dimensional problems of the upwind finite element scheme based on the choice up- and downwind points, Comput. Meth. Appl. Mech. Eng., Volume 112 (1994), pp. 109-131 | Zbl

[31] H. C. Ku; R. S. Hirsh; T. D. Taylor A pseudospectral method for solution of the three-dimensional incompressible Navier–Stokes equations, J. Comput. Phys., Volume 70 (1987), pp. 439-462 | DOI | Zbl

[32] T. P. Chiang; W. H. Sheu; R. R. Hwang Effect of Reynolds number on the eddy structure in a lid-driven cavity, Int. J. Numer. Methods Fluids, Volume 26 (1998), pp. 557-579 | DOI | Zbl

[33] A. K. Prasad; J. R. Koseff Reynolds number and end-wall effects on a lid-driven cavity flow, Phys. Fluids A: Fluid Dyn., Volume 1 (1989), pp. 208-218 | DOI

[34] C. Y. Perng; R. L. Street Three-dimensional unsteady flow simulations: Alternative strategies for a volume-averaged calculation, Int. J. Numer. Math. Fluids, Volume 9 (1989), pp. 341-362 | DOI

[35] C. M. Teixeira Digital physics simulation of lid-driven cavity flow, Int. J. Mod. Phys. C, Volume 8 (1997), pp. 685-696 | DOI

[36] S. Albensoeder; H. C. Kuhlmann; H. J. Rath Three-dimensional centifugal-flow instabilities in a lid-driven-cavity problem, Phys. Fluids, Volume 13 (2001), pp. 121-135 | DOI | Zbl

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