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Movies: herringbone flows and flapping waves, 2d inclined inviscid Saint-Venant equations

<p>Supplementary movies for&nbsp;<em><strong>Multidimensional stability and transverse bifurcation of&nbsp;hydraulic shocks and roll waves in open channel flow</strong>.</em></p> <p>See&nbsp;<a href="https://github.com/zyang-pde/Multi-d_inviscid_Saint-Venant_eqs">https://github.com/zyang-pde/Multi-d_inviscid_Saint-Venant_eqs</a>&nbsp;for source codes used to generate these movies.</p> <p><strong>dam_break_F_equals_2_point_25.mp4</strong></p> <p>Dam-break initial data with left fluid height 1, right fluid height 0.7, and Froude number 2.25. The discontinuous hydraulic shock is 2d convectively unstable. Herringbone flows are seen at the end.</p> <p><strong>flat_F_equals_2_point_25.mp4</strong></p> <p>Flat initial data with co-moving speed set to be that of discontinuous hydraulic shock with left limiting fluid height 1, right limiting fluid height 0.7, and Froude number 2.25. Herringbone flows, parabola, and roll waves are seen at the end.</p> <p><strong>dam_break_comparison.mp4</strong></p> <p>Comparison of&nbsp;simulations with dam break initial data and with Froude numbers 2.13,2.14,2.15.</p> <p><strong>flat_comparison.mp4</strong></p> <p>Comparison of flat simulations with flat initial data and&nbsp;with Froude numbers 2.13,2.14,2.15.</p> <p>&nbsp;</p> <p>Movies below are simulated with perturbed roll waves initial data.&nbsp;The Froude number is set to be 6 and the minimum fluid height is set to be 0.28 and the channel width and the y-boundary condition are varied.</p> <p><strong>roll_width_point15.mp4</strong></p> <p>With <strong>wall</strong> boundary condition, the width of the channel is set to be <strong>0.15</strong>. No flapping front is seen&nbsp;at the end. Channel roll waves are stable.</p> <p><strong>roll_width_point16.mp4</strong></p> <p>With <strong>wall</strong> boundary condition, the width of the channel is set to be <strong>0.16</strong>. No flapping front is seen at the end. Channel roll waves are stable.</p> <p><strong>roll_width_point17.mp4</strong></p> <p>With <strong>wall</strong> boundary condition, the width of the channel is set to be <strong>0.17</strong>. No flapping front is seen at the end. Channel roll waves are stable.</p> <p><strong>roll_width_point18.mp4</strong></p> <p>With <strong>wall</strong> boundary condition, the width of the channel is set to be <strong>0.18</strong>. Wave fronts start to flap after a while. The flapping waves are&nbsp;persistent and do not become chaotic.</p> <p><strong>roll_width_point18_refined.mp4</strong></p> <p>With <strong>wall</strong> boundary condition, the width of the channel is set to be <strong>0.18</strong>. Wave fronts start to flap after a while. The flapping waves&nbsp;are&nbsp;persistent and do not become chaotic. Finer mesh grid is used compared with roll_width_point18.py. Raw data files are used to generate figures in the paper.</p> <p><strong>roll_width_point18_periodic.mp4</strong></p> <p>With <strong>periodic</strong> y-boundary condition, the width of the channel is set to be <strong>0.18</strong>. No flapping front is seen at the end. Channel roll waves are stable.</p> <p><strong>roll_width_point36_periodic.mp4</strong></p> <p>With <strong>periodic</strong> y-boundary condition, the width of the channel is set to be <strong>0.36</strong>. Wave fronts start to flap after a while. The flapping waves are persistent and do not become chaotic.</p> <p><strong>roll_width_point19.mp4</strong></p> <p>With <strong>wall</strong> boundary condition, the width of the channel is set to be <strong>0.19</strong>. Wave fronts start to flap after a while. The flapping waves are&nbsp;persistent and do not become chaotic.</p> <p><strong>roll_width_point2.mp4</strong></p> <p>With <strong>wall</strong> boundary condition, the width of the channel is set to be <strong>0.2</strong>. Wave fronts start to flap after a while. The flapping waves are&nbsp;also unstable, transitioning to chaotic flow at the end.</p> <p><strong>roll_width_1.mp4</strong></p> <p>With <strong>wall</strong> boundary condition, the width of the channel is set to be <strong>1</strong>. Wave fronts start to flap after a while. The flapping waves are&nbsp;also unstable, transitioning to chaotic flow at the end.</p> <p>&nbsp;</p>