5.2 Particles with non-circular shape
5.2.1 Elliptic disk falling with various initial angle
Table 5.9: Points in time where collision effects occur for the calculation of the refined mesh in Figure 5.15.
Effect Cut Cell DG Wan and Turek (2006)
Kiss 1.13 1.13
Tumble 1.74 1.53
Separate 1.93 1.73
P1 hits bottom 5.94 6.23
P2 hits bottom 7.37 7.57
(a)θ0=0◦ (b)θ0=45◦ (c)θ0=90◦
Figure 5.16: Different starting angles of an elliptic disk falling in incompressible fluid.
Table 5.10: Setting for falling elliptic disk.
k DoF att0 Adaptive Mesh
1 33 600 yes
2 72 000 yes
3 124 800 yes
The values for x-position, y-position, x-velocity and y-velocity are plotted over time and compared for different polynomial degreesk, leading to increasing spatial resolution.
All calculations will be also carried out after the ellipse collided with the bottom wall until a termination timetend =2.0 is reached. For all collisions the model based on the conservation of momentum is being used.
Elliptic disk with 0◦
At first, the disk is placed with a starting angle ofθp = 0◦in the channel. Of course, this starting angle is expected to result in the slowest possible falling velocity, not depending on spatial resolution. In Figure 5.17 the results can be seen. It is clearly notable that all quantities coincide perfectly for the free-fall part. In all calculations the ellipse is falling with the same speed and no distraction in x-direction. In detail, this means that the ellipse is not tilted during its falling process. Right before reaching the wall, the ellipse is slowing down due to the presence of the bottom-wall. The ellipse is slowed down due to hydrodynamical forces such that the collision model is first active attcol =1.77.
The behavior of the elliptic disk after collision differs slightly. For a polynomial degree ofk =3, a small velocity in x-direction occurs after the collision model is triggered.
However, reaching and staying in a position of rest is also the most challenging part for a collision model. This is due to the equilibrium state between gravitational forces and collision forces acting from the wall unto the elliptic disk. As the sign of the
0 0.5 1 1.5 2
−0.3
−0.2
−0.1 0 0.1 0.2 0.3
t
x-position
k=1
k=2
k=3
(a) x-position over time.
0 0.5 1 1.5 2
−2
−1.5
−1
−0.5 0 0.5 1 1.5
t
y-position
(b) y-position over time.
0 0.5 1 1.5 2
−0.6
−0.4
−0.2 0 0.2 0.4 0.6
t
x-velocity
(c) x-velocity over time.
0 0.5 1 1.5 2
−5
−4
−3
−2
−1 0 1
t
y-velocity
(d) y-velocity over time.
Figure 5.17: Results of an ellipse falling withθ0 =0◦.
momentum is changed in every timestep, an ”oscillatory” equilibrium state has to be kept by the collision model.
The comparison of all calculations leads to the conclusion, that those instabilities can be damped by choosing a low polynomial degree, leading to ’stable’ position of rest.
Therefore, for high-order calculations the presented collision model has to be improved.
Nonetheless, a general prediction of the point of collision and the falling trajectories during the free fall process is not sensitive to spatial resolution.
Elliptic disk with 45◦
In the next case, the elliptic disk is tilted 45◦. The simulations are carried out with the same spatial resolutions. It can be predicted that a tilted disk will gain an acceleration in x-direction due to hydrodynamical forces. In Figure 5.18, it can be seen that for all
resolutions the acceleration in horizontal direction is present. In addition, the time of collision is identical for all polynomial degrees and can be denoted withtcol =1.14.
0 0.5 1 1.5 2
−1
−0.8
−0.6
−0.4
−0.2 0 0.2
t
x-position
k=1
k=2
k=3
(a) x-position over time.
0 0.5 1 1.5 2
−2
−1.5
−1
−0.5 0 0.5 1 1.5
t
y-position
(b) y-position over time.
0 0.5 1 1.5 2
−1.2
−1
−0.8
−0.6
−0.4
−0.2 0 0.2
t
x-velocity
(c) x-velocity over time.
0 0.5 1 1.5 2
−6
−4
−2 0 2 4
t
y-velocity
(d) y-velocity over time.
Figure 5.18: Results of an elliptic disk falling withθ0=45◦.
However, in contrast to the first setting withθ0 =0◦, some differences between resolu-tions occur. First, it will be focused on the free falling part of the calculation fort<tcol. The vertical positions and velocities coincide quite well between all simulations. For the x-Position it can be seen, that for the coarsest spatial resolution an overprediction in hydrodynamical force takes place. Therefore, the ellipse is accelerated more in x-direction than fork =2 andk =3.
Lastly, the x-position and velocity differs much depending on resolution. Fork=1 the x-Position att =2.0 renders to be close tox =−0.6 whereas for both other calculations the position of rest is atx=−0.71. The aforementioned overprediction of x-Velocity leads to a different behavior after collision as well. Here, the convergence ofk =2 and
k=3 to a position of rest can be seen, whereas the calculation fork =1 still moves in positive x-direction.
Elliptic disk with 90◦
In the last variation of the current problem, the elliptic disk is tilted by 90◦. This leads to the fastest falling velocity. Therefore, a collision time which is smaller than in the calculations with angles of 0◦and 45◦can be expected. Results of this calculation can be seen in Figure 5.19. Here, the collision time can be denoted withtcol =1.06 for a first collision and witht2ndcol =1.55 for a second collision. Nonetheless, until the time of the first collision large agreement for the free-falling part can be yield. Only the velocity in vertical direction differs slightly atk =1 but has almost no impact on the collision time.
0 0.5 1 1.5 2
−1
−0.8
−0.6
−0.4
−0.2 0 0.2 0.4 0.6 0.8 1
t
x-position
k=1
k=2
k=3
(a) x-position over time.
0 0.5 1 1.5 2
−2
−1.5
−1
−0.5 0 0.5 1 1.5 2
t
y-position
(b) y-position over time.
0 0.5 1 1.5 2
−0.6
−0.4
−0.2 0 0.2 0.4 0.6
t
x-velocity
(c) x-velocity over time.
0 0.5 1 1.5 2
−4
−2 0 2 4
t
y-velocity
(d) y-velocity over time.
Figure 5.19: Results of an elliptic disk withθ0=90◦.
After colliding the elliptic disk increase velocity in x-direction for all three cases. This results from a normal collision vector which is not exactly~n ={0, 1}. Resulting out
of this, the acceleration in the opposite x-direction fork =3 occurs. However, if the x-position and x-velocity is mirrored at x = 0, resp. Ux = 0, almost no difference betweenk =2 andk=3 can be seen. Therefore, a convergence can be observed.
Fort>t2ndcol, the most significant difference is the already mentioned velocity increase after the second collision. Fork =1 andk=3 the velocity att=2.0 is far from rest.
This means, that the ellipse is still moving in x-direction whereas in y-direction it is almost at rest. This happens due to different turns in orientation leading to increasing hydrodynamical forces in horizontal direction. However, convergence in y-position and y-direction can be seen for all investigated cases.
All in all, the current test shows the ability of the method to predict the falling behavior even for coarse sparse resolutions. However, as soon as strong antisymmetric hydro-dynamical effects and collisions come into play, a fine resolution at the interface for integration and collision determination is required. It also shows the big challenge of reaching convergence for a numerical scheme of collisions which is very sensitive to fluctuations in hydrodynamical forces and collision direction calculation.