2-D laminar/turbulent driven square cavity flow
A classical test problem is numerical simulation of a laminar/turbulent flow of an incompressible viscous fluid in a square cavity with the upper moving boundary. It has long since been a "testing area" for approbation of approximation schemes for terms in initial equations as well as of computational models and methods. A large number of calculation data concerning this problem have been accumulated. Therefore, it makes sense to return periodically to the solution of the indicated problem at different levels of development of computer systems and software for verification and analysis of the computational codes developed.
The interest shown by researchers in the problem on a circulation flow in a cavity was stimulated in the past by the relatively low level of computational resources required for solving it because, in this case, the computational region is limited and, therefore, a small number of grid nodes are selected and simple boundary conditions are set. For this problem, the first solutions of Navier–Stokes equations written, for economy of computational resources, in transformed vorticity–stream function variables were obtained. The calculations were performed not even on a computer but by brigades of specialists.
When computers appeared, the indicated problem remained, as before, the focus of attention of specialists developing computational methods. A detailed analysis of methods concerning the problem considered that appeared in the period from the 1960s to the late 1980s is given in a set of monographs (for example 1, 2, 3). One of the first works in the area of numerical investigation of a viscous-fluid flow in a square cavity began thirty years ago  on a BESM-4 computer containing 21 × 21 grid nodes in one memory cube. At that time, in the mid-1970s, high-accuracy Arakava schemes of the second and fourth order of approximation and nonuniform grids were used for the first time for solving Navier–Stokes equations.
One of the most important scientific achievements of the CFD in the 1980s was a numerical solution of the diffusion problem, since this problem is associated with errors in approximation of convective terms of equations. In simulation of separation flows, a necessary condition for obtaining an exact result is the use of schemes with a low numerical viscosity (upwind schemes of the second and higher orders of approximation of the type of the Leonard scheme with quadratic interpolation, the Agarval scheme, and others) for representation of convective terms in transfer equations. At the same time, it has been established that the first-order schemes give erroneous solutions at high Reynolds numbers even in the case where multigrid methods are used.
Progress in computational engineering and, especially, widespread use of personal computers in the 1980s and subsequent years have made it possible, first of all, to develop universal first-wave codes of applied programs, such as the PHOENIX, FLOW3D, and FIDAP, and then more modern information products/commercial CFD codes such as FLUENT, StarCD, and CFX. When computers with a large memory and a high CPU speed appeared, instead of the Navier–Stokes equations in transformed variables, Navier–Stokes equations written in physical variables — Cartesian velocity or pressure components — began to be predominantly used. Moreover, this made it possible to substantially (by an order of magnitude and more) increase the number of computational cells and increase the resolution of nearwall region for this problem.
 I. A. Belov, I. P. Ginzburg, and S. A. Isaev, Motion and heat transfer in a closed region with moving boundaries, Vestn. LGU, No. 13, 41–50 (1976).
 I. A. Belov and N. A. Kudryavtsev, Heat Transfer and Resistance of Tube Bundles [in Russian], Energoatomizdat, Leningrad (1987).
 I. A. Belov, S. A. Isaev, and V. A. Korobkov, Problems and Methods of Calculation of Separation Incompressible-Fluid Flows [in Russian], Sudostroenie, Leningrad (1989).
 Isaev S. A., Baranov P. A., Kudryavtsev N. A., Lysenko D. A., and Usachov A. E. SIMULATION OF A CIRCULATION LAMINAR FLOW AROUND A SQUARE CAVITY WITH A MOBILE BOUNDARY AT HIGH REYNOLDS NUMBERS WITH THE USE OF VP2/3 AND THE FLUENT PACKAGE. Journal of Engineering Physics and Thermophysics, Vol. 78, No. 4, 2005
 Isaev S. A., Baranov P. A., Kudryavtsev N. A., Lysenko D. A., and Usachov A. E. Complex Analysis of turbulence models, algorithms, and grid structures at the computation of recirculating flow in a cavity by means of VP2/3 and FLUENT packages. Part 1. Scheme factors influence. Thermophysics and Aeromechanics, 2005, Vol: 12, No 4.
 Isaev S. A., Baranov P. A., Kudryavtsev N. A., Lysenko D. A., and Usachov A. E. Complex Analysis of turbulence models, algorithms, and grid structures at the computation of recirculating flow in a cavity by means of VP2/3 and FLUENT packages. Part 2. Estimation of models adequacy. Thermophysics and Aeromechanics, 2006 , Vol 13, No 1.