Discussion
The 2-D lid-driven square cavity flow has been dealt with in countless papers employing a wide variety of methods (such as finite difference, finite volume, finite element, spectral, Lattice Boltzmann, etc), with theoretical orders of accuracy varying from 2 to 20(!) and on a wide range of grid/mesh structures and sizes. Nonetheless, it is only for the special case of Re = 1000 that highly accurate, "benchmark" steady-state results are available. Those were presented by Botella & Peyret, 1998 [1] and later partly confirmed by others like [2] and [5], though I am not aware of any work confirming [1] for all velocity and vorticity values like is done in the present work.
This is not the case for Re > 1000, where the discrepancy between different studies grows with Re. This is partly due to the fact that the complexity of the resulting flow field is higher and consequently it gets increasingly more time-consuming to resolve it with high accuracy. It also becomes increasingly difficult to obtain a steady solution at all. Some don't even believe that it should be possible to obtain steady solutions for Re > 7500-8000 for the time-dependent equations. Another reason might be that some of the more specialized techniques that are frequently applied to this standard problem do not work as well (are less stable) for higher Re. Besides, the problem itself is "problematic", in that it features physically unrealistic discontinuous velocity where the lid meets the side walls. This feature doesn't help neither convergence nor efforts to obtain accurate solutions. Finally as Botella & Peyret note "a large part of the computations concerning this flow are motivated by the validation of a novel method, and do not necessarily pretend to be the most accurate or comprehensive."
Here there is no novel method to validate (for now). In contrast, I sought the best possible result one could get out of simple, entry-level numerical schemes and techniques, which prove just as capable when applied to this problem as any other more sophisticated approach. This way detailed results of steady solutions up to Re = 30000 were presented, whose accuracy should be significantly higher than others currently available (Re = 1000 excluded). This was achieved first by squeezing as much accuracy (and Re range) as possible out of a second order finite difference implementation, and second by using a judiciously graded grid, dense enough near the boundaries and corners while still maintaining smooth gradation and sufficient density in the center as well. This of course meant that the grid point count had to be fairly high. Finishing touches were then applied by means of Richardson extrapolation, (which does work in this case as we saw). Finally, there was no modification to the original problem in order to minimize loss of accuracy, like in [1] and [5] where part of the singularities is subtracted, nor any use of the analytic asymptotics of the flow near the corners like in [2]. Instead, the singularities were resolved by brute force, i.e. simply through local grid refinement.
As mentioned in the front page, a lot of studies suggest that after some moderate Reynolds number the 2-D cavity flow in question becomes unstable. Most agree on the onset of instability being around Re = 8000. Shankar & Deshpande, 2000 [6] note "If the flow does become unsteady, what is the nature of this flow, because it cannot, as a 2-D flow, be turbulent? Are there steady solutions that cannot be computed because they are unstable? Although these are natural questions, they are not of practical relevance, because...2-D flows are almost fictitious." So let's keep in mind that useful as a 2-D cavity analysis is for identifying some features of the flow and testing new numerical schemes, a 2-D cavity flow at Re = 8000 is almost certainly fictitious. So then it would seem more appropriate to talk about numerical or mathematical, rather than hydrodynamic instability. If the flow is unstable, then that has to be with respect to numerical perturbations. The answer then crucially depends on the numerical stability, as well as accuracy characteristics of a particular scheme. So many claim that the flow becomes unstable at Re = 8000 and their solver backs this up by yielding periodic solutions beyond that number. But how can we be sure that this is not due to their scheme introducing numerical oscillations/perturbations that another equally accurate scheme might not? It may also be that their spatial and temporal accuracy is just not sufficient. It is also fair to imagine that some of those periodic solutions and/or failures to find a steady-state, say at Re = 10000, are due to the grid used not being fine enough. Because no matter how high the order of a scheme, when the size of a flow structure (like one of the smaller corner vortices) is near, or smaller than the local grid spacing, it cannot be resolved and that may well cause a periodic solution behavior when the unresolved/ under-resolved structure is large enough to affect the main flow.
To counter those claiming that the (mathematical/numerical) flow becomes unstable at around Re = 8000, there are other studies that present steady solutions for much higher Re numbers, obtained though time-dependent simulation. Cardoso & Bicudo, 2009 [7] use a scheme that should be second order in space and fourth order in time, on a fine 1024x1024 grid and with a time-step of 0.001s, to obtain steady solutions up to Re = 30000. Hachem et. al., 2010 [8] apply their stabilized finite element variational multiscale (VMS) time-dependent solver on the 2-D cavity, presenting steady solutions up to Re = 20000. Like before, we could possibly argue that those results are made possible precisely because something acts as a stabilizing agent. Cardoso & Bicudo use Thom's vorticity boundary condition which is first order and more diffusive than others and may have a stabilizing effect. They also seem to freeze the advection coefficients (velocities) of the vorticity transport equation at each time step. Hachem et. al. use stabilizing bubble functions whose exact choice could conceivably affect the predicted onset of instability for a flow near the critical Re number. Perhaps more importantly, they indicate that they reached their steady-state solutions using a rather large time-step of 0.1s. That on its own could be making the computation of a steady solution possible whereas that may not be the case with a smaller time-step.
Either way, the fact is that nothing is really stopping us from being able to find an unstable steady-state solution in some cases. The flow past a circular cylinder is a good example of this. While the flow is known to produce the periodic von Kármán vortex sheet wake at about Re = 100, there are steady state computations in the literature for the same and higher Re. There are ways through which this can be achieved. Numerical scheme dissipation is probably one. If the dissipation introduced is strong enough and the flow instability weak, then it seems plausible that the instability could be effectively damped out allowing a steady state to be reached. Using a rather large time step (if the scheme's stability characteristics allow it) is another way as alluded to above. It is again reasonable to imagine that high frequency instabilities could be effectively filtered out when using a sufficiently large time-step. Linearizing the advection coefficients and not strictly enforcing the compatibility between the computed fields (say vorticity and stream function) at each time-step is maybe a third one. With the present simulations I was able to obtain steady-state solutions for up to Re = 30000 starting the time stepping from zero (simulating an impulsively started lid), by carefully relaxing the compatibility between the steam function and vorticity. Though the time-steps used were fairly small (0.002 - 0.004s), numerical experiments showed that solving the stream function equation more accurately at each time-step while keeping the advection coefficients frozen in the vorticity transport equation, led to periodic flow for the higher Reynolds numbers once the main vortex had been established.
This is not the case for Re > 1000, where the discrepancy between different studies grows with Re. This is partly due to the fact that the complexity of the resulting flow field is higher and consequently it gets increasingly more time-consuming to resolve it with high accuracy. It also becomes increasingly difficult to obtain a steady solution at all. Some don't even believe that it should be possible to obtain steady solutions for Re > 7500-8000 for the time-dependent equations. Another reason might be that some of the more specialized techniques that are frequently applied to this standard problem do not work as well (are less stable) for higher Re. Besides, the problem itself is "problematic", in that it features physically unrealistic discontinuous velocity where the lid meets the side walls. This feature doesn't help neither convergence nor efforts to obtain accurate solutions. Finally as Botella & Peyret note "a large part of the computations concerning this flow are motivated by the validation of a novel method, and do not necessarily pretend to be the most accurate or comprehensive."
Here there is no novel method to validate (for now). In contrast, I sought the best possible result one could get out of simple, entry-level numerical schemes and techniques, which prove just as capable when applied to this problem as any other more sophisticated approach. This way detailed results of steady solutions up to Re = 30000 were presented, whose accuracy should be significantly higher than others currently available (Re = 1000 excluded). This was achieved first by squeezing as much accuracy (and Re range) as possible out of a second order finite difference implementation, and second by using a judiciously graded grid, dense enough near the boundaries and corners while still maintaining smooth gradation and sufficient density in the center as well. This of course meant that the grid point count had to be fairly high. Finishing touches were then applied by means of Richardson extrapolation, (which does work in this case as we saw). Finally, there was no modification to the original problem in order to minimize loss of accuracy, like in [1] and [5] where part of the singularities is subtracted, nor any use of the analytic asymptotics of the flow near the corners like in [2]. Instead, the singularities were resolved by brute force, i.e. simply through local grid refinement.
As mentioned in the front page, a lot of studies suggest that after some moderate Reynolds number the 2-D cavity flow in question becomes unstable. Most agree on the onset of instability being around Re = 8000. Shankar & Deshpande, 2000 [6] note "If the flow does become unsteady, what is the nature of this flow, because it cannot, as a 2-D flow, be turbulent? Are there steady solutions that cannot be computed because they are unstable? Although these are natural questions, they are not of practical relevance, because...2-D flows are almost fictitious." So let's keep in mind that useful as a 2-D cavity analysis is for identifying some features of the flow and testing new numerical schemes, a 2-D cavity flow at Re = 8000 is almost certainly fictitious. So then it would seem more appropriate to talk about numerical or mathematical, rather than hydrodynamic instability. If the flow is unstable, then that has to be with respect to numerical perturbations. The answer then crucially depends on the numerical stability, as well as accuracy characteristics of a particular scheme. So many claim that the flow becomes unstable at Re = 8000 and their solver backs this up by yielding periodic solutions beyond that number. But how can we be sure that this is not due to their scheme introducing numerical oscillations/perturbations that another equally accurate scheme might not? It may also be that their spatial and temporal accuracy is just not sufficient. It is also fair to imagine that some of those periodic solutions and/or failures to find a steady-state, say at Re = 10000, are due to the grid used not being fine enough. Because no matter how high the order of a scheme, when the size of a flow structure (like one of the smaller corner vortices) is near, or smaller than the local grid spacing, it cannot be resolved and that may well cause a periodic solution behavior when the unresolved/ under-resolved structure is large enough to affect the main flow.
To counter those claiming that the (mathematical/numerical) flow becomes unstable at around Re = 8000, there are other studies that present steady solutions for much higher Re numbers, obtained though time-dependent simulation. Cardoso & Bicudo, 2009 [7] use a scheme that should be second order in space and fourth order in time, on a fine 1024x1024 grid and with a time-step of 0.001s, to obtain steady solutions up to Re = 30000. Hachem et. al., 2010 [8] apply their stabilized finite element variational multiscale (VMS) time-dependent solver on the 2-D cavity, presenting steady solutions up to Re = 20000. Like before, we could possibly argue that those results are made possible precisely because something acts as a stabilizing agent. Cardoso & Bicudo use Thom's vorticity boundary condition which is first order and more diffusive than others and may have a stabilizing effect. They also seem to freeze the advection coefficients (velocities) of the vorticity transport equation at each time step. Hachem et. al. use stabilizing bubble functions whose exact choice could conceivably affect the predicted onset of instability for a flow near the critical Re number. Perhaps more importantly, they indicate that they reached their steady-state solutions using a rather large time-step of 0.1s. That on its own could be making the computation of a steady solution possible whereas that may not be the case with a smaller time-step.
Either way, the fact is that nothing is really stopping us from being able to find an unstable steady-state solution in some cases. The flow past a circular cylinder is a good example of this. While the flow is known to produce the periodic von Kármán vortex sheet wake at about Re = 100, there are steady state computations in the literature for the same and higher Re. There are ways through which this can be achieved. Numerical scheme dissipation is probably one. If the dissipation introduced is strong enough and the flow instability weak, then it seems plausible that the instability could be effectively damped out allowing a steady state to be reached. Using a rather large time step (if the scheme's stability characteristics allow it) is another way as alluded to above. It is again reasonable to imagine that high frequency instabilities could be effectively filtered out when using a sufficiently large time-step. Linearizing the advection coefficients and not strictly enforcing the compatibility between the computed fields (say vorticity and stream function) at each time-step is maybe a third one. With the present simulations I was able to obtain steady-state solutions for up to Re = 30000 starting the time stepping from zero (simulating an impulsively started lid), by carefully relaxing the compatibility between the steam function and vorticity. Though the time-steps used were fairly small (0.002 - 0.004s), numerical experiments showed that solving the stream function equation more accurately at each time-step while keeping the advection coefficients frozen in the vorticity transport equation, led to periodic flow for the higher Reynolds numbers once the main vortex had been established.
Yiannis Papadopoulos, June 2015
References
[1] O. Botella & R. Peyret, Benchmark spectral results on the lid-driven cavity flow, Computers and Fluids, 27 (1998), pp. 421-433.
[2] A. Shapeev & P. Lin, An asymptotic fitting finite element method with exponential mesh refinement for accurate computation of corner eddies in viscous flows, SIAM Journal of Scientific Computing, 31 (2009), pp. 1874–1900.
[3] U. Ghia, K.N. Ghia & C.T.Shin, High-Re solutions for incompressible flow using the Navier-Stokes equations and a multigrid method, Journal of Computational Physics, 48 (1982), pp. 387-411.
[4] E. Erturk & C. Gokcol, Fourth-order compact formulation of Navier-Stokes equations and driven cavity flow at high Reynolds numbers, International journal for Numerical Methods in Fluids, 50 (2006), pp. 421-436.
[5] R.K. Shukla, M. Tatineni, X. Zhong, Very high-order compact finite difference schemes on non-uniform grids for incompressible Navier-Stokes equations, Journal of Computational Physics, 224 (2007), pp. 1064-1094.
[6] P.N. Shankar & M.D. Deshpande, Fluid mechanics in the driven cavity, Annual Review of Fluid Mechanics, 32 (2000), pp.93-136.
[7] N. Cardoso & P. Bicudo, Time dependent simulation of the driven lid cavity at high Reynolds number, arXiv:0809.3098v2.
[8] E. Hachem, B. Rivaux, T. Kloczko, H. Digonnet, T. Coupez, Stabilized finite element method for incompressible flows with high Reynolds number, Journal of Computational Physics, 229 (2010), pp. 8643-8665.
[9] T. Coupez & E. Hachem, Solution of high-Reynolds incompressible flow with stabilized finite element and adaptive anisotropic meshing, Comput. Methods Appl. Mech. Engrg. , 267 (2013), pp. 65-85.
[10] K. Yapici, Y. Uludag, Finite volume simulation of 2-D steady square lid driven cavity flow at high Reynolds numbers, Brazilian Journal of Chemical Engineering, Vol. 30, No. 4 (2013), pp. 923-937.
[2] A. Shapeev & P. Lin, An asymptotic fitting finite element method with exponential mesh refinement for accurate computation of corner eddies in viscous flows, SIAM Journal of Scientific Computing, 31 (2009), pp. 1874–1900.
[3] U. Ghia, K.N. Ghia & C.T.Shin, High-Re solutions for incompressible flow using the Navier-Stokes equations and a multigrid method, Journal of Computational Physics, 48 (1982), pp. 387-411.
[4] E. Erturk & C. Gokcol, Fourth-order compact formulation of Navier-Stokes equations and driven cavity flow at high Reynolds numbers, International journal for Numerical Methods in Fluids, 50 (2006), pp. 421-436.
[5] R.K. Shukla, M. Tatineni, X. Zhong, Very high-order compact finite difference schemes on non-uniform grids for incompressible Navier-Stokes equations, Journal of Computational Physics, 224 (2007), pp. 1064-1094.
[6] P.N. Shankar & M.D. Deshpande, Fluid mechanics in the driven cavity, Annual Review of Fluid Mechanics, 32 (2000), pp.93-136.
[7] N. Cardoso & P. Bicudo, Time dependent simulation of the driven lid cavity at high Reynolds number, arXiv:0809.3098v2.
[8] E. Hachem, B. Rivaux, T. Kloczko, H. Digonnet, T. Coupez, Stabilized finite element method for incompressible flows with high Reynolds number, Journal of Computational Physics, 229 (2010), pp. 8643-8665.
[9] T. Coupez & E. Hachem, Solution of high-Reynolds incompressible flow with stabilized finite element and adaptive anisotropic meshing, Comput. Methods Appl. Mech. Engrg. , 267 (2013), pp. 65-85.
[10] K. Yapici, Y. Uludag, Finite volume simulation of 2-D steady square lid driven cavity flow at high Reynolds numbers, Brazilian Journal of Chemical Engineering, Vol. 30, No. 4 (2013), pp. 923-937.
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