Commun. Comput. Phys., 15 (2014), pp. 596-617.


Chaotic Lid-Driven Square Cavity Flows at Extreme Reynolds Numbers

Salvador Garcia 1*

1 Instituto de Ciencias Fisicas y Matematicas, Universidad Austral de Chile, Casilla 567, Valdivia, Chile.

Received 7 May 2013; Accepted (in revised version) 22 July 2013
Available online 18 October 2013
doi:10.4208/cicp.070513.220713a

Abstract

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 -> \infty, 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 -> \infty (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 = 10^9 the leading eddy - at its qualitative temporal limit's pace - appears to rotate solely counterclockwise.

AMS subject classifications: 76D99, 35Q30, 37N10

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Key words: Navier-Stokes equations, lid-driven square cavity flows, chaos.

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

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