Do solutions to the 2D driven cavity always exist even for very high values of the Reynolds number, say 10 million or more ? If we use the SIMPLE / SIMPLEC type of algorithms on a simple staggered grid, and solve the problem as a time evolution (unsteady) is there evidence that the velocities begin showing time periodic behaviour after a certain critical Reynolds number ? Can anybody suggest some ref. where such work is reported ?

It is an interesting question. Before you try to get an answer, you probably would like to see a 2D cavity flow problem with a Reynolds number equal or greater than 10 million. At 20 degree C in air, this Reynolds number will translates into ( U * L ) = 150 m2/sec. So, for a cavity of size 10 cm * 10 cm ( L=0.1m ), the velocity (U) will be 1500 m/sec. That is a very large number, about 4 times the speed of sound ( Mach 4 ). You can make the 2D cavity equal to 150m x 150m, and have the velocity (U) equal to 1 m/sec. And the ask the question: " will the flow inside this 150m x 150m cavity remain laminar ? " Based on Schlichting's Boundary-Layer Theory book, the critical Reynolds number in a pipe is 2300, and for the flow over a flat plate, it is 350000. It may be possible to keep the flow more stable using surface suction, surface texture, polymer additive,etc. So the answer is "unknown". But, I guess, most likely, you will get turbulent experimental resluts and laminar numerical solutions ( or un-steady transient flows solutions). If you turn on the turbulence model in the numerical calculation, the results will likely be turbulent. Flow stability is an interesting issue in early days, it also was an important issue for aerospace plane design in 80's. In the 2D cavity flow case, I am not sure whether it is practical to a test at 10 million Reynolds number. If you have access to a CFD code, I am sure that you can check out the results easily by running a few 2D cavity flow cases at 10 million, in either laminar or turbulent mode.

The critical Reynolds number for 2D driven cavity flow is around 11000. If you are not concerned about issues like grid-independence you should be able to get a reasonable solution.

The critical Reynolds number for the 2D lid-driven cavity have been investigated by several authors, and its value depends on the boundary condition on the moving lid:

a) For the standard (non-regularized) lid-driven cavity, which has two singolar corners at the moving lid, the critical Re value (corresponding to a Hopf bifurcation) has been estimated, in Ref. [1], to be in the range 7500-1000. This has been later confirmed - by eigenvalue calculation - in Ref. [2], where the critical Re value has been computed Re_c = 8000.

[1] E. Nobile, Simulation of Time-dependent Flow in Cavities with the Additive-Correction Multigrid Method, Part II: Applications, Numer. Heat Transfer, Part B, vol. 30, 351-370, 1996.

[2] A. Fortin, M. Jardak, J.J. Gervais and R. Pierre, Localization of Hopf Bifurcations in Fluid Flow Problems, Int. J. Numer. Methods Fluids, vol. 24, 1185-1210, 1997.

b) For the regularized lid-driven cavity (which reduce the effect of the singularities since the velocity BC is now continuous), the critical Re value has been estimated by Shen, Ref [3], to be in the range 10000-10500. Later confirmation is also provided in Ref [2], with a direct (eigenvalue) estimation Re_c = 10255.

[3] J. Shen, Hopf Bifurcation of the Unsteady Regularized Driven Cavity Flow, J. Comp. Phys, vol. 95, 228-245, 1991.

Finally, let me add that, although the agreement between these set of calculations is acceptable in terms of critical Re, this is not the case for the fundamental frequency of the oscillation.

Hope it helps,

It is amazing to know that the stability of the flow or the critical Reynolds number somehow can be predicted. Once the flow becomes chaotic, it's hard to predict. Nature does provide us problems with increasing degree of difficulty !

It is interesting that the critical Reynolds number for the driven cavity is this high (11000). Is this value obtained from an experimental investigation? Where is the work published?

I remember reading about some controversy in the hopf-bifurcation reynolds no of the flow; and its between 10000 to 12000 according to that, and the controversy had something to do with the grid indepence of the scheme. I can't place the source though.