Column collapse

Martin and Moyce [40] presented an experiment of column collapse and many numerical models are built for this example [4, 9, 30, 41]. Figure 5.35 shows the results of experiments by Koshizuka [41]. In the experimental setup, the tank is made of glass, with a base length of 0.584m. The water column, with a base length of 0.146m and a height of 0.292m, is initially supported on the right by a vertical plate drawn up at time t=0.0s. For the numerical calculation non-slip boundary conditions are applied to the bottom and sides of the tank. The top boundary is modeled as open boundary which pressure is fixed. The density of fluid 1 is 1000 and the viscosity is 0.01. For the fluid 2 the density is 1 and the viscosity is 0.001.

In this study, there are four types of gird configurations adopted: 48*28, 120*70, 240*104 and 120*165. The first three types of configurations have the same scale of computational domain (0.584m*0.340m). The last one increases the height of the tank and built for testing the open boundary condition.

The time evolutions are shown in Figure 5.37 to 5.40. The contour is 0.4≤α ≤0.6 with 11 levels. At first, water column starts to collapse and the covered area at bottom increase until the leading edge touch the right wall. Then, water liquid splashes and starts to leave the domain at the top right corner. To continue, the water against the right wall starts to fall back under the influence of gravity. The backward moving wave has folded over and a small amount of air is trapped at t=0.8s. The water spray is produced. Finally, the spray touches the left wall again and traps a large air bubble.

Compare each configurations, the coarse grid introduces oscillation obviously. The flow field becomes unstable and numerical diffusion is produces when the water column returns to the left wall, especially in the coarse grid. Due to the region of air trapped is complex, the manner of reconstructing grid configuration can improve this problem.

In addition, surface tension force term is considered in the calculation and Figure 5.41 to 5.44 show the numerical results. It is not surprising that the results are not different from the former without surface tension. The shape of air bubbles in the returning water column is smoother.

Figure 5.45 shows the leading edge versus time. The calculated results show that the leading edge moves faster when the resolution of the grids increases. The height of the collapsing water column versus time is shown in Figure 5.46. For each calculations the predicted height is the same and it correspond very well with the experiment, except for the finer grids (240*140).

Figure 5.36 Experimental results of a collapsing water column by Koshizuka [40].

Figure 5.37 The numerical results of column collapse which the time step size is 0.00025s and the spatial size is 4.86e-3m (grids: 120*70). The maximum Courant number during the computation is 0.699.

Figure 5.38 The numerical results of column collapse which the time step size is 0.001s and the spatial size is 0.0121m (grids: 48*28). The maximum Courant number during the computation is 0.621.

Figure 5.39 The numerical results of column collapse which the time step size is 0.000125s and the spatial size is 2.43e-03m (grids: 240*140). The maximum Courant number during the computation is 0.563.

Figure 5.40 The numerical results of column collapse which the time step size is 0.00025s and the spatial size is 4.86e-03m (grids: 120*165). The maximum Courant number during the computation is 0.615.(continue)

Figure 5.40 The numerical results of column collapse which the time step size is 0.00025s and the spatial size is 4.86e-03m (grids: 120*165). The maximum Courant number during the computation is 0.615.

Figure 5.41 The numerical results of column collapse with surface tension which the time step size is 0.00025s and the spatial size is 4.86e-3m (grids:

120*70). The maximum Courant number during the computation is 0.699.

Figure 5.42 The numerical results of column collapse with surface tension which the time step size is 0.001s and the spatial size is 0.0121m (grids: 48*28). The maximum Courant number during the computation is 0.611.

Figure 5.43 The numerical results of column collapse with surface tension which the time step size is 0.000125s and the spatial size is 2.43e-03m (grids:

240*140). The maximum Courant number during the computation is 0.485.

Figure 5.44 The numerical results of column collapse with surface tension which the time step size is 0.00025s and the spatial size is 4.86e-03m (grids: 120*165). The maximum Courant number during the computation is 0.597.(continue)

Figure 5.44 The numerical results of column collapse with surface tension which the time step size is 0.00025s and the spatial size is 4.86e-03m (grids: 120*165). The maximum Courant number during the computation is 0.597.

t*sqrt(2g/a)

240*140 with surface tension 120*165 with surface tension 120*70 with surface tension 48*28 with surface tesnion Exp. 1.125"

Exp. 2.25"

CICSAM of Ubbink

Figure 5.45 The position of the leading edge versus time.

t*sqrt(g/a)

240*140 with surface tension 120*165 with surface tension 120*70 with surface tension 48*28 with surface tension Experiment

CICSAM of Ubbink

Figure 5.46 The height of the collapsing water column versus time.

b z

Collapse with an obstacle

The example of column collapse with an obstacle is more complicated than the previoes one. The dimensions of tank are the same as the previous experiment but a small obstacle is placed in the way of the wave front. The obstacle (0.024m*0.048m) is placed on the bottom of the tank, in the way of the moving front with its lower left corner in the center of the tank.

For the numerical calculation, the computational domain increases in height. In this study, open boundary and close boundary (wall) at top are tested, respectively. Figure 5.47 shows the experimental result by Koshizuka. Figure 5.48 and 5.49 show the calculated results. The solid line stands for wall and the dashed line presents open boundary. The water column collapse and is the same with previous example until touch a small obstacle. Then, the movement of the leading edge has been obstructed by the small obstacle and a tongue of water splash up.

As time goes by, the tongue continues its movement toward the opposite wall until impinges the wall and trapping air beneath it. Finally, water falls down under the action of gravity.

Although the velocity flow fields are not totally similar, the distribution of fluids is almost the same.

Surface tension term is considered and Figure 5.50 and 5.51 show the results. As well as the previous case, the influence of surface tension is not important.

Figure 5.47 Experimental results of a collapsing water column hitting an obstacle (Koshizuka [40])

Figure 5.48 The numerical results of column collapse with obstacle without surface tension. The upper boundary at the computational domain is wall.

Figure 5.49 The numerical results of column collapse with obstacle without surface tension. The upper boundary at the computational domain is open boundary which pressure is fixed.

Figure 5.50 The numerical results of column collapse with obstacle with surface tension. The upper boundary at the computational domain is wall.

Figure 5.51 The numerical results of column collapse with obstacle with surface tension. The upper boundary at the computational domain is open boundary which pressure is fixed.