A test case initially presented by Almgren et al [2]. Three elliptical bodies are placed in the unit square. Constant unity inflow and outflow are specified on the left and right boundaries. Projection is then performed to obtain a potential flow solution around the bodies.
Tables 5 and 6 illustrate the errors and convergence orders obtained for both components of the velocity when the resolution varies. Richardson extrapolation is used. The errors are computed either on the whole domain (All cells) or on the cells whose parents at level 7 are entirely contained in the fluid (Full 128 cells).
Close to second-order convergence is obtained in the bulk of the fluid, reducing to first-order close to the boundaries. The errors are small in all cases (with a maximum of 6%) and comparable to that obtained by Almgren et al using a different discretisation.
All cells Full 128 cells 128-256 Rate 256-512 128-256 Rate 256-512 L_{1} 3.72e-04 1.87 1.02e-04 2.25e-04 2.00 5.63e-05 3.72e-04 1.87 1.02e-04 2.25e-04 2.00 5.63e-05 L_{2} 2.13e-03 1.27 8.85e-04 4.77e-04 1.96 1.23e-04 2.13e-03 1.27 8.85e-04 4.77e-04 1.96 1.23e-04 L_{∞} 6.68e-02 0.60 4.41e-02 6.74e-03 1.27 2.79e-03 6.68e-02 0.60 4.41e-02 6.74e-03 1.27 2.79e-03
All cells Full 128 cells 128-256 Rate 256-512 128-256 Rate 256-512 L_{1} 4.00e-04 1.91 1.07e-04 2.82e-04 2.00 7.03e-05 4.00e-04 1.91 1.07e-04 2.82e-04 2.00 7.03e-05 L_{2} 1.75e-03 1.41 6.59e-04 7.56e-04 1.70 2.33e-04 1.75e-03 1.41 6.59e-04 7.56e-04 1.70 2.33e-04 L_{∞} 5.35e-02 0.70 3.29e-02 9.28e-03 1.33 3.69e-03 5.35e-02 0.70 3.29e-02 9.28e-03 1.33 3.69e-03