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

Ranging from Re=100 to Re=20,000,several computational experiments are conducted,Re being the Reynolds number.The primary vortex stays put,and the longterm 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 Re=15,000.This is the lid-driven square cavity flow;the motivations for studying this flow are recalled in the Introduction.

Hopf Bifurcations, Drops in the Lid–Driven Square Cavity Flow

The lid-driven square cavity flow is investigated by numerical experiments.It is found that from Re=5,000 to Re=7,307.75 the solution is stationary,but at Re=7,308 the solution is time periodic.So the critical Reynolds number for the first Hopf bifurcation localizes between Re=7,307.75 and Re=7,308.Time periodical behavior begins smoothly,imperceptibly at the bottom left corner at a tiny tertiary vortex;all other vortices stay still,and then it spreads to the three relevant corners of the square cavity so that all small vortices at all levels move periodically.The primary vortex stays still.At Re=13,393.5 the solution is time periodic;the long-term integration carried out past t_(∞)=126,562.5 and the fluctuations of the kinetic energy look periodic except slight defects.However at Re=13,393.75 the solution is not time periodic anymore:losing unambiguously,abruptly time periodicity,it becomes chaotic.So the critical Reynolds number for the second Hopf bifurcation localizes between Re=13,393.5 and Re=13,393.75.At high Reynolds numbers Re=20,000 until Re=30,000 the solution becomes chaotic.The long-term integration is carried out past the long time t_(∞)=150,000,expecting the time asymptotic regime of the flow has been reached.The distinctive feature of the flow is then the appearance of drops:tiny portions of fluid produced by splitting of a secondary vortex,becoming loose and then fading away or being absorbed by another secondary vortex promptly.At Re=30,000 another phenomenon arises—the abrupt appearance at the bottom left corner of a tiny secondary vortex,not produced by splitting of a secondary vortex.

Chaotic Lid-Driven Square Cavity Flows at Extreme Reynolds Numbers

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.