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Fig. A.1


Effect of three different methods of treating the pressure at the solid top and bottom boundaries. In all three panels, the vertical velocity component, w, is plotted on a split logarithmic colour scale. We use Θ = 10-3, L = 101, constant kinematic viscosity and set the resolution to only 16 × 64 to make the spurious oscillations visible. We obtain the result plotted in the left panel using the simple symmetry conditions for pressure (Eqs. (A.19) and (A.20)). Preceding the pressure-gradient computation by third-order pressure extrapolation to the ghost cells reduces the oscillations’ amplitude by a factor of ~100 (middle panel). Increasing the extrapolation order to six brings about another decrease by a factor of ~30 in the oscillations’ amplitude (right panel). The pressure gradient is in all three cases computed by the second-order central differences in the whole computational domain.

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