Commun. Comput. Phys., 2 (2007), pp. 900-932.


The Lid-Driven Square Cavity Flow: From Stationary to Time Periodic and Chaotic

Salvador Garcia 1*

1 Departamento de Ciencias Matematicas y Fisicas, Universidad Catolica de Temuco, Casilla 15-D Temuco, Chile.

Received 12 September 2006; Accepted (in revised version) 14 December 2006
Communicated by Roger Temam
Available online 20 March 2007

Abstract

Ranging from $ \mathrm{Re} = 100 $ to $ \mathrm{Re} = 20,000 $, several computational experiments are conducted, $ \mathrm{Re} $ being the Reynolds number. The primary vortex stays put, and the long-term dynamic behavior of the small vortices determines the nature of the solutions. For low Reynolds numbers, the solution is stationary; for moderate Reynolds numbers, it is time periodic. For high Reynolds numbers, the solution is neither stationary nor time periodic: the solution becomes chaotic. Of the small vortices, the merging and the splitting, the appearance and the disappearance, and, sometime, the dragging away from one corner to another and the impeding of the merging---these mark the route to chaos. For high Reynolds numbers, over weak fundamental frequencies appears a very low frequency dominating the spectra---this very low frequency being weaker than clear-cut fundamental frequencies seems an indication that the global attractor has been attained. The global attractor seems reached for Reynolds numbers up to $\mathrm{Re} = 15,000 $. This is the lid-driven square cavity flow; the motivations for studying this flow are recalled in the Introduction.

AMS subject classifications: 76M20, 76D05, 76F06, 37N10

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Key words: Finite differences, staggered marker-and-cell (MAC) mesh, incremental unknowns, generalized Stokes equations, incompressible Navier-Stokes equations, chaos.

*Corresponding author.
Email: sgarcia@uctemuco.cl (S. Garcia)
 

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