# Lid-driven Cavity in 2D

## Problem Description

A square cavity with a side length of L°= 1e-3° m is filled with water (density: 1000 kg/m3, dynamic viscosity: 1e-3 kg/m/s). The upper wall is moved with a constant velocity U0 imposing a rotational movement of the fluid in the cavity. Here, the configurations with Re°=°100, Re°=°1000 and Re°=°10000 are simulated by adjusting the velocity of the lid correspondingly. The highly-resolved Finite-Difference calculations by Ghia et al. 2 with a grid resolution of 257x257 are taken as reference.

## Numerical Setup

Three different resolutions of 50x50, 100x100 and 200x200 fluid particles are used for each Reynolds-Number. Figure 1 shows the initial particle distribution for the coarsest resolution where wall boundary conditions are imposed by three rows of particles. The simulation is run until a fully developed flow is achieved.

## Results

The velocity profiles in the horizontal and vertical centerlines are shown in Figure 2 to Figure 4 for each Reynolds-Number. For the Re = 100 case, all the profiles show very good agreement with the reference profiles by Ghia et al. Velocity profiles at Re = 1000 and Re = 10000 match those of 2 at finer resolutions of 100x100 and 200x200, respectively. This is expected as higher Reynolds numbers results in finer structures and requires more particles to resolve.

1 S. Adami, H. Hu and N. Adams, "A generalized wall boundary condition for smoothed particle hydrodynamics," Journal of Computational Physics, vol. 231, pp. 7057-7075, 2012.
2 U. Ghia and K. S. C. Ghia, "High-Re solutions for incompressible flow using the Naview-Stokes equations and a multigrid method," Journal of Computational Physics, vol. 48, pp. 387-411, 1982.