Abstract
This study presents a continuation method to calculate flow bifurcation in a two-sided lid-driven cavity with different aspect ratios for anti-parallel motion. In anti-parallel motion, the top and bottom walls of the cavity move in opposite directions simultaneously, while the two walls both moving to the right give parallel motion at the same speed. Comprehensive bifurcation diagrams of the cavity flows with different aspect ratios of the cavities are derived via Keller’s continuation method, and linear- stability analysis is used to identify the nature of the various flow solutions. The Reynolds number (1 ≤ Re ≤ 1,200) is used as the continuation parameter to trace the solution curves. In anti-parallel motion, the evolution of the bifurcation diagrams in cases with different aspect ratios (1 ≤ AR ≤ 2.5) is illustrated. Two stable symmetric flows and one stable asymmetric flow are identified, and the existent regions of the stable flows in the aspect ratios and Reynolds numbers are distinguished. The newly found asymmetric flow state can be obtained at a high aspect ratio and a low Reynolds number.
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Ahlman D., Soderlund F., Jackson J., Kurdila A., Shyy W.: Proper orthogonal decomposition for time-dependent lid-driven cavity flows. Numer. Heat Transf. Part B Fundam 42, 285–306 (2002)
Albensoeder, S., Kuhlmann, H.C., Rath, H.J.: Multiplicity of steady two dimensional flows in two-sided lid-driven cavities. Theor. Comp. Fluid Dyn. 14, 223–241 (2001)
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)
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)
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)
Alleborn N., Raszillier H., Durst F.: Lid-driven cavity with heat and mass transport. Int. J. Heat Mass Transf. 42, 833–853 (1999)
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)
Croce G., Comini G., Shyy W.: Incompressible flow and heat transfer computations using a continuous pressure equation and nonstaggered grids. Numer. Heat Transf. Part B Fundam 38, 291–307 (2000)
Keller H.B.: Numerical solution of bifurcation and nonlinear eigenvalue problems. In: Rabinowitz, P. (ed.) Applications of Bifurcation Theory, pp. 359–384. Academic Press, New York (1977)
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)
Kuhlmann H.C., Wanschura M., Rath H.J.: Elliptic instability in two-sided lid-driven cavity flow. Eur. J. Mech. B Fluids 17, 561–569 (1998)
Luo W.J., Yang R.J.: Multiple fluid flow and heat transfer solutions in a two-sided lid-driven cavity. Int. J. Heat Mass Transf. 50, 2394–2405 (2007)
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)
Pan F., Acrivos A.: Steady flows in rectangular cavities. J. Fluid Mech. 28, 643–655 (1967)
Perumal D.A., Dass A.K.: Multiplicity of steady solutions in two-dimensional lid-driven cavity flows by Lattice Boltzmann Method. Comput. Math. Appl. 61, 3711–3721 (2011)
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)
Yang R.J., Luo W.J.: Flow bifurcations in a thin gap between two rotating Spheres. Theor. Comp. Fluid Dyn. 16, 115–131 (2002)
Sorensen D.C.: Implicit application of polynomial filters in a k-step Arnoldi method. SIAM J. Matrix Anal. Appl. 13, 357–367 (1992)
Saad Y.: Numerical Methods for Large Eigenvalues Problems. Halsted Press, New York (1992)
Tekić M., Radenović J.B., Lukić N.Lj., Popović S.S.: Lattice Boltzmann simulation of two-sided lid-driven flow in a staggered cavity. Int. J. Comput. Fluid Dyn. 24(9), 383–390 (2010)
Waheed M.A.: Mixed convective heat transfer in rectangular enclosures driven by a continuously moving horizontal plate. Int. J. Heat Mass Transf. 52, 5055–5063 (2009)
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Chen, KT., Tsai, CC., Luo, WJ. et al. Multiplicity of steady solutions in a two-sided lid-driven cavity with different aspect ratios. Theor. Comput. Fluid Dyn. 27, 767–776 (2013). https://doi.org/10.1007/s00162-013-0296-z
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DOI: https://doi.org/10.1007/s00162-013-0296-z