I have one doubt as
what is the meaning of 2d driven cavity flow?
Classic Driven-Cavity Flow: Impermeable square box wherein the motion of an incompressible fluid is driven by the uniform motion of one of its sides, the other three sides being fixed.
Albensoeder, S.1 ; Kuhlmann, H.C.1
Institute of Fluid Mechanics and Heat Transfer, Vienna University of Technology, Resselgasse 3/1/2, A-1040 Vienna, Austria
The lid-driven-cavity problem is one of the most important benchmarks for numerical Navier–Stokes
solvers. Its importance results from the fundamental rectangular or square geometry and the simple driving
of the flow by means of the tangential motion with constant velocity of a single lid, representing Dirichlet
boundary conditions. Moreover, the driven-cavity flow exhibits a number of interesting physical features.
The Prandtl–Batchelor theorem is easily demonstrated as the primary vortex, driven by the wall motion,
develops a core of constant vorticity as the Reynolds number is increased (see, e.g. ). In addition, flow
separation from the stationary wall and the existence of an infinite sequence of viscous corner eddies in the
rigid 90-corners can be observed . The system also exhibits a particular singularity in the boundary
conditions where moving and stationary walls meet [47,24,27]. Last but not least, the flow in an infinitely extended system undergoes a sequence of instabilities and transitions when the speed of the lid is increased
before becoming turbulent. The particular scenario depends very much on the aspect ratio of the cavity. A
recent review on the flow physics in the driven cavity was given by Shankar and Deshpande .
Ghia et al.  and Schreiber and Keller  were among the first to publish benchmark data on the liddriven
cavity flow. These classical papers are frequently referenced even today. At about the same time, the
interest in the flow physics of interior recirculating flows was revived, after the pioneering works of Burggraf
 and Pan and Acrivos . The renewed interest led to a series of papers by Koeseff et al. , Freitas
et al. , and coworkers, focussing on the three-dimensional vortical structures and on end-wall effects.
During the course of time the grid resolution and the numerical accuracy were significantly improved
and even more accurate solutions of the two-dimensional problem have been calculated. Highly accurate
two-dimensional solutions have been obtained by Botella and Peyret  using a spectral method in which
the effect of the corner discontinuity on the numerical solution was reduced by the incorporation of an
asymptotic solution for the flow in the direct vicinity of the singular corners. For an efficient elimination
of the singularity, they made use of the leading-order Stokes flow  plus the first-order nonlinear correction
(see, e.g. ). A similar procedure, merely taking into account the leading-order Stokes-flow solution,
was previously introduced by Schultz et al. .
Since two-dimensional, time-dependent calculations are significantly less costly than full three-dimensional
simulations, the Hopf bifurcation at high Reynolds numbers of the pure two-dimensional flow
was investigated with high precision by Goodrich et al. , Shen , Abouhamza and Pierre , and Auteri
et al. [3,4]. While these calculation are of fundamental interest for benchmarking, they are less relevant
for real flows. The oscillatory two-dimensional flows exist only for such high Reynolds numbers
Re = O(104) that they are very unlikely to be observed experimentally. Albensoeder et al.  have shown
that the two-dimensional steady flow becomes unstable to genuinely three-dimensional flows at Reynolds
numbers one order of magnitude smaller than those at which the two-dimensional flow oscillations have
Some of the first three-dimensional cavity-flow calculations were carried out by De Vahl Davis and
Mallinson  and Goda . The relevance of three-dimensional flows in general was demonstrated by
Freitas et al. . In particular, three-dimensional effects near the end-walls of a finite-size system which
can be realized in the laboratory were pointed out by Koseff and Street . In order to clarify the observed
three-dimensional flow structures numerically, a series of benchmark tests for the lid-driven square cavity
was undertaken for a Reynolds number of 3200. The results, however, published in Deville et al. ,
remained inconclusive, because the numerical solutions obtained by different methods and resolutions scattered
significantly. An important point in this regard is the fact that end-wall effects in finite-length systems
can, to a certain degree, suppress the intrinsic three-dimensional flow instabilities in the bulk of the cavity
. During the last years, further three-dimensional calculations have become available, see e.g. Ku et al.
, Cortes and Miller , Babu and Korpela , and Wang and Sheu . Further contributions of Iwatsu
et al. , Chiang et al. [11,12], and Sheu and Tsai  primarily focussed on the flow structure and
topology, and not on benchmarking.
Until recently, numerical calculations have predominantly be performed for two-dimensional flows. We
have, however, carried out the first correct three-dimensional linear stability analysis of the two-dimensional
cavity flow with periodic boundary conditions in the spanwise direction . The uncertainty of the critical
Reynolds number was of the order of magnitude of 1%. Despite the sparseness of reliable stability data,
our results were validated by comparison with neutral-stability data of Ding and Kawahara [16,17] and
by critical Reynolds numbers of Kuhlmann et al. . Additional confidence was gained from our own
experimental results  and from the numerical work of Spasov et al.  and Shatrov et al. . After
having become three-dimensional, the cavity flow develops into a turbulent flow upon a further increase
of the Reynolds number. Apart from the work of Leriche and Gavrilakis , the turbulent flow regime
has not yet been seriously tackled numerically for Reynolds numbers of the order of Re = O(104) and above.
S. Albensoeder, H.C. Kuhlmann / Journal of Computational Physics 206 (2005) 536–558 537
During our investigation of the transition to three-dimensional flow the lack of numerical data for
nonlinear three-dimensional flows became obvious. Owing to their physical significance and the fact that
laminar three-dimensional flows have become accessible by numerical methods, we have carried out a number
of systematic calculations in order to provide high-accuracy numerical data for the three-dimensional
cavity flow. In the next section, the problem will be formulated. Thereafter, the numerical methods will be
described and a validation of the code is performed. Our benchmark results are presented and discussed in
Section 5. Finally, a summary of the results is provided in the conclusion.
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