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Chaotic Lid-Driven Square Cavity Flows at Extreme Reynolds Numbers

Published online by Cambridge University Press:  03 June 2015

Salvador Garcia*
Instituto de Ciencias Físicas y Matemáticas, Universidad Austral de Chile, Casilla 567, Valdivia, Chile
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This paper investigates the chaotic lid-driven square cavity flows at extreme Reynolds numbers. Several observations have been made from this study. Firstly, at extreme Reynolds numbers two principles add at the genesis of tiny, loose counterclockwise- or clockwise-rotating eddies. One concerns the arousing of them owing to the influence of the clockwise- or counterclockwise currents nearby; the other, the arousing of counterclockwise-rotating eddies near attached to the moving (lid) top wall which moves from left to right. Secondly, unexpectedly, the kinetic energy soon reaches the qualitative temporal limit’s pace, fluctuating briskly, randomly inside the total kinetic energy range, fluctuations which concentrate on two distinct fragments: one on its upper side, the upper fragment, the other on its lower side, the lower fragment, switching briskly, randomly from each other; and further on many small fragments arousing randomly within both, switching briskly, randomly from one another. As the Reynolds number Re → ∞, both distance and then close, and the kinetic energy fluctuates shorter and shorter at the upper fragment and longer and longer at the lower fragment, displaying tall high spikes which enlarge and then disappear. As the time t → ∞ (at the Reynolds number Re fixed) they recur from time to time with roughly the same amplitude. For the most part, at the upper fragment the leading eddy rotates clockwise, and at the lower fragment, in stark contrast, it rotates counterclockwise. At Re=109 the leading eddy — at its qualitative temporal limit’s pace — appears to rotate solely counterclockwise.

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Copyright © Global Science Press Limited 2014

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[1]Barragy, E. and Carey, G. F.Stream function-vorticity driven cavity solution using p finite elements. Comput. & Fluids, 26(5):453–468, 1997.CrossRefGoogle Scholar
[2]Bruneau, C.-H. and Jouron, C.An efficient scheme for solving steady incompressible Navier-Stokes equations. J. Comput. Phys., 89(2):389–413, 1990.CrossRefGoogle Scholar
[3]Bruneau, C.-H. and Saad, M.The 2D lid-driven cavity problem revisited. Comput. & Fluids, 35(3):326–348, 2006.CrossRefGoogle Scholar
[4]Chai, Z.-H., Shi, B.-C. and Zheng, L.Simulating high Reynolds number flow in two-dimensional lid-driven cavity by multi-relaxation-time lattice Boltzmann method. Chinese Phys., 15(8):1855–1863, 2006.Google Scholar
[5]Chen, M., Miranville, A., and Temam, R.Incremental unknowns in finite differences in three space dimensions. Comput. Appl. Math., 14(3):219–252, 1995.Google Scholar
[6]Chen, M. and Temam, R.The incremental unknown method I.Appl. Math. Lett., 4(3):73–76, 1991.Google Scholar
[7]Chen, M. and Temam, R.The incremental unknown method II. Appl. Math. Lett., 4(3):77–80, 1991.Google Scholar
[8]Chen, M. and Temam, R.Incremental unknowns for solving partial differential equations. Numer. Math., 59(3):255–271, 1991.CrossRefGoogle Scholar
[9]Chen, M. and Temam, R.Incremental unknowns for convection-diffusion equations. Appl. Numer. Math., 11(5):365–383, 1993.CrossRefGoogle Scholar
[10]Chen, M. and Temam, R.Incremental unknowns in finite differences: Condition number of the matrix. SIAM Matrix, J.Anal. Appl., 14(2):432–455, 1993.Google Scholar
[11]Chen, M. and Temam, R.Nonlinear Galerkin method in the finite difference case and wavelet-like incremental unknowns. Numer. Math., 64(3):271–294, 1993.CrossRefGoogle Scholar
[12]Concus, P., Golub, G. H., and O’Leary, D. P.A generalized conjugate gradient method for the numerical solution of elliptic partial differential equations. Sparse Matrix Computations. Academic Press, 1976. Bunch, J. R. and Rose, D. J. (Eds.).CrossRefGoogle Scholar
[13]Constantin, P.A few results and open problems regarding incompressible fluids. Notices Amer. Math. Soc., 42(6):658–663, 1995.Google Scholar
[14]Demmel, J. W., Eisenstat, S. C., Gilbert, J. R., Li, X. S., and Liu, J. W. H.A supernodal approach to sparse partial pivoting. SIAM Matrix, J.Analysis and Applications, 20(3):720–755, 1999.Google Scholar
[15]Dijkstra, H. A., Wubs, F. W., Cliffe, A. K., Doedel, E., Dragomirescu, I. F., Eckhardt, B., Yu, A.Gelfgat, , Hazel, A. L., Lucarini, V., Salinger, A. G., Phipps, E. T., Sanchez-Umbria, J., Schutte-laars, H., Tuckerman, L. S., and Thiele, U.Numerical bifurcation methods and their application to fluid dynamics: analysis beyond simulation. Commun. Comput. Phys., 15:1–45, 2014.CrossRefGoogle Scholar
[16]Erturk, E., Corke, T. C., and Gökçö, C. Numerical solutions of 2-D steady incompressible driven cavity flow athigh Reynolds numbers. Internat. J. Numer. Methods Fluids, 48(7):747–774, 2005.CrossRefGoogle Scholar
[17]Eugene, C.Wayne. Vortices and two-dimensional fluid motion. Notices Amer. Math. Soc., 58(1):10–19, 2011.Google Scholar
[18]Frigo, M. and Johnson, S. G.The design and implementation of FFTW3. Proc. IEEE, 93(2):216–231, 2005.CrossRefGoogle Scholar
[19]Garcia, S.The matricial framework for the incremental unknowns method. Numer. Funct. Anal. Optim., 14(1 & 2):25–44, 1993.CrossRefGoogle Scholar
[20]Garcia, S.Numerical study of the incremental unknowns method. Numer. Methods Partial Differential Equations, 10(1):103–127, 1994.CrossRefGoogle Scholar
[21]Garcia, S.Higher-order incremental unknowns, hierarchical basis, and nonlinear dissipative evolutionary equations. Appl. Numer. Math., 19(4):467–494, 1996.CrossRefGoogle Scholar
[22]Garcia, S.Algebraic conditioning analysis of the incremental unknowns preconditioner. Appl. Math. Modelling, 22(4–5):351–366, 1998.CrossRefGoogle Scholar
[23]Garcia, S.Incremental unknowns for solving the incompressible Navier-Stokes equations. Math. Comput. Simulation, 52(5–6):445–489,2000.CrossRefGoogle Scholar
[24]Garcia, S.Incremental unknowns and graph techniques in three space dimensions. Appl. Numer. Math., 44(3):329–365,2003.CrossRefGoogle Scholar
[25]Garcia, S.The lid-driven square cavity flow: From stationary to time periodic and chaotic. Commun. Comput. Phys., 2(5):900–932, 2007.Google Scholar
[26]Garcia, S.Hopf bifurcations, drops in the lid-driven square cavity flow. Adv. Appl. Math. Mech., 1(4):546–572, 2009.Google Scholar
[27]Garcia, S.Aperiodic, chaotic lid-driven square cavity flows. Math. Comput. Simulation, 81(9):1741–1769, 2011.CrossRefGoogle Scholar
[28]Garcia, S. and Tone, F.Incremental unknowns and graph techniques with in-depth refinement. Int. J. Numer. Anal. Model., 4(2):143–177, 2007.Google Scholar
[29]Ghia, U., Ghia, K. N., and Shin, C. T.High-Re solutions for incompressible flow using the Navier-Stokes equations and a multigrid method. J. Comput. Phys., 48(3):387–411, 1982.CrossRefGoogle Scholar
[30]Glowinski, R.Finite Element Methods for the Numerical Simulation of Incompressible Viscous Flow. Introduction to the Control of the Navier-Stokes Equations, volume 28 of Lectures in Applied Mathematics. American Mathematical Society, 1991.Google Scholar
[31]Goyon, O.High-Reynolds number solutions of Navier-Stokes equations using incremental unknowns. Comput. Methods Appl. Mech. Engrg., 130(3–4):319–335, 1996.CrossRefGoogle Scholar
[32]Harlow, F. H. and Welch, J. E.Numerical calculation of time-dependent viscous incompressible flow of fluid with free surface. Phys. Fluids, 8(12):2182–2189, 1965.CrossRefGoogle Scholar
[33]Ito, K., Li, Z., and Qiao, Z.The sensitivity analysis for the flow past obstacles problem with respect to the Reynolds number. Adv. Appl. Math. Mech., 4:21–35, 2012.CrossRefGoogle ScholarPubMed
[34]Kim, J. and Moin, P.Application of a fractional-step method to incompressible Navier-Stokes equations. J. Comput. Phys., 59(2):308–323, 1985.CrossRefGoogle Scholar
[35]Li, M., Tang, T., and Fornberg, B.A compact fourth-order finite difference scheme for the steady incompressible Navier-Stokes equations. Internat. J. Numer. Methods Fluids, 20(10):1137–1151, 1995.CrossRefGoogle Scholar
[36]Mohanty, R. K. and Setia, N.A new fourth-order compact off-step discretization for the system of 2D nonlinear elliptic partial differential equations. East Asian J. Appl. Math., 2:59–82, 2012.Google Scholar
[37]Poullet, P.Staggered incremental unknowns for solving Stokes and generalized Stokes problems. Appl. Numer. Math., 35(1):23–41, 2000.CrossRefGoogle Scholar
[38]Sarin, V. and Sameh, A.An efficient iterative method for the generalized Stokes problem. SIAM J. Sci. Comput., 19(1):206–226, 1998.Google Scholar
[39]Schreiber, R. and Keller, H. B.Driven cavity flows by efficient numerical techniques. J. Comput. Phys., 49(2):310–333, 1983.CrossRefGoogle Scholar
[40]Sekhar, T. V. S., Raju, B. H. S., and Sanyasiraju, Y. V. S. S.Higher-order compact scheme for the incompressible Navier-Stokes equations in spherical geometry. Commun. Comput. Phys., 11:99–113, 2012.CrossRefGoogle Scholar
[41]Shah, A., Guo, H., and Yuan, L.A third-order upwind compact scheme on curvilinear meshes for the incompressible Navier-Stokes equations. Commun. Comput. Phys., 5:712–729, 2009.Google Scholar
[42]Shankar, P. N. and Deshpande, M. D.Fluid mechanics in the driven cavity. Annu. Rev. Fluid Mech., 32:93–136, 2000.CrossRefGoogle Scholar
[43]Singh, S., Krithivasan, S., Karlin, I. V., Succi, S., and Ansumali, S.Energy conserving lattice Boltzmann models for incompressible flow simulations. Commun. Comput. Phys., 13:603–613, 2013.CrossRefGoogle Scholar
[44]Smith, A. and Silvester, D.Implicit algorithms and their linearisation for the transient incompressible Navier-Stokes equations. IMA J. Numer. Anal., 17(4):527–545, 1997.Google Scholar
[45]Song, L. and Wu, Y.Incremental unknowns in three-dimensional stationary problem. Numer. Algorithms, 46(2):153–171, 2007.CrossRefGoogle Scholar
[46]Song, L. and Wu, Y.Incremental unknowns method based on the θ-scheme for time-dependent convection-diffusion equations. Math. Comput. Simulation, 79(7):2001–2012, 2009.CrossRefGoogle Scholar
[47]Swarztrauber, P. N.The methods of cyclic reduction, Fourier analysis and the FACR algorithm for the discrete solution of Poisson’s equation on a rectangle. SIAM Rev., 19(3):490–501, 1977.Google Scholar
[48]Sweet, R. A.Direct methods for the solution of Poisson’s equation on a staggered grid. J. Comput. Phys., 12(3):422–428, 1973.CrossRefGoogle Scholar
[49]Temam, R.Inertial manifolds and multigrid methods. SIAM J. Math. Anal.1, 21(1):154–178, 1990.Google Scholar
[50]Vorst, H. A. van der. Bi-CGSTAB: a fast and smoothly converging variant of Bi-CG for the solution of nonsymmetric linear systems. SIAM J. Sci. Statist. Comput., 13(2):631–644, 1992.Google Scholar
[51]Vanka, S. P.Block-implicit multigrid solution of Navier-Stokes equations in primitive variables. J. Comput. Phys., 65(1):138–158, 1986.CrossRefGoogle Scholar
[52]Wu, Y. L., Shu, C., and Ding, H.Simulation of incompressible viscous flows by local DFD-immersed boundary method. Adv. Appl. Math. Mech., 4:311–324, 2012.CrossRefGoogle Scholar
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