Consider a fluid inside a square cavity of dimension L bounded by four walls. All the walls except the top wall are fixed in space and time. The top wall is given a constant velocity 𝑢𝑇in the u direction, and we
want to see the development of the u and v velocity profiles of the fluid in the whole domain, as it reaches steady state.
The entire problem statement can be found here.
The incompressible NS equations in primitive variables are given by the equation:
If we apply 2nd order central difference scheme for space discretization and 2nd order Adams Bashforth for time discretization, the above equation can be represented as:
The fractional step, or time-splitting, method solves the unsteady Navier-Stokes equations in a segregated manner. At each time step, an incomplete form of momentum equations is integrated to obtain an approximate velocity field, which is, in general, not divergence-free, then the velocity field is projected into the divergence-free field without changing vorticity. This projection step is achieved by solving the Poisson equation for pressure.
Benchmarking is done against Ghia and Ghia. The project technical report can be found here.
