Numerical solution of the Navier–Stokes equations in the modeling of two-dimensional viscous incompressible fluid flows

In this paper, the effectiveness of the implicit iterative line-by-line recurrence method for solving difference elliptical equations arising in numerical simulations of two-dimensional viscous incompressible fluid flows is analyzed. The research is carried out by an example of the problem about a steady two-dimensional lid-driven cavity flow formulated in primitive variables $(u,v,p)$. It is shown that applying the line-by-line recurrence method allows one to reduce the total time for solving the problem in comparison with the use of the present-day effective bi-conjugate gradients method with stabilization. As an illustration of the achieved results, the structure of the flow at $\mathrm{Re}=15000$ is shown. Here, in terms of the use of a non-uniform grid, it became possible to obtain a sequence of bottom-corner vortices up to the fourth level. As a validation of the received solution, the comparison of basic parameters of all vortices with results of other authors was carried out at $\mathrm{Re}=1000$. In addition, the mass imbalance was estimated; it did not exceed $10^{-8}\div10^{-6}$ depending on the location of the cross section in the cavity, and a comparison of the relative size and ‘intensity’ of bottom-corner vortices of the third and fourth levels with the Moffatt analytical solution of the problem of a viscous fluid flow near a sharp corner was carried out.