Numerical Validation

Numerical and experimental displacement histories with a structural frequency of 1.8 Hz for uncontrolled, 0 (controlled), 8, 13, 20 and 26 degrees inclination are displayed in Figures

105 4.3(a) to 4.3(f). Almost identical displacement histories are achieved between the numerical and experimental cases for the uncontrolled case in Figure 4.3(a).

Increase in numerical damping is predicted for all controlled cases, displayed in Figures 4.3(b), 4.3(c), 4.3(d), 4.3(e) and 4.3(f). For numerical displacement histories of the cases with inclinations of 13, 20 and 26 degrees, in Figures 4.3(d), 4.3(e) and 4.3(f), enough excess energy dissipation is predicted to affect the structure’s damped frequency, ωd. As indicated in

Equation (4.2) below, as ζeq increases, ωd decreases.

2 1 _eq n

d ω ζ

ω = − (4.2)

Where ωn is the undamped natural frequency of the structure, and ζeq is the equivalent

viscous damping ratio caused by energy dissipation within the fluid. This results in the developing phase difference between the predicted and experimentally observed structural displacement histories in Figures 4.3(d), 4.3(e) and 4.3(f).

As inclination increases from 0 degrees, the liquid settles to one side of the container, increasing the liquid height. Energy dissipation characteristics vary with the increased liquid height producing standing waves, instead of travelling waves, which occur at inclinations of 0 and 8 degrees at a structural frequency of 1.8 Hz. At this structural frequency, increases in liquid velocity are observed in the travelling waves with larger free surface deformation when compared to cases with lower structural frequencies.

106 Standing waves, which occur at inclinations above 8 degrees, possess lower energy dissipation characteristics than travelling waves. This is due to a reduction in liquid velocities and the liquid being less energetic with a calmer free surface. At these instances, the increase in numerical damping results in a phase difference between numerical and experimental displacement histories.

At inclinations of 0 and 8 degrees, although higher numerical damping is also predicted than observed experimentally, similar structural frequencies are produced between numerical and experimental cases. This is a result of SPH being able to accurately predict the timing and motion of the structure and the travelling waves, with increased liquid velocities and larger free surface deformation. The increase in artificial numerical damping that occurs at a structural frequency of 1.8 Hz is too high to validate the SPH model. As a result, a method to improve the accuracy of the numerical predictions is investigated next.

Increased numerical energy dissipation is related to artificial numerical damping associated with SPH, which relies on integration, as part of its smoothing procedure. This integration takes place from the centre of each fluid particle and has to do with its interaction with its neighbouring particles. As the particle size gets smaller, along with the corresponding smoothing length, the level of artificial damping is expected to diminish, at the expense of required computational effort. A particle size of 0.8 mm x 0.8 mm was initially chosen due to it achieving a successful validation analysis with experimental observations, at a similar shallow liquid height, for the case in Appendix 2 Resolution Study. However, due to this case

107 having a larger free surface length than compared to the case analysed in Appendix 2 Resolution Study unphysical results are numerically predicted, at this particle size. Therefore, a reduction in the particle size from 0.8 mm x 0.8 mm and 0.4 mm x 0.4 mm is analysed.

Numerical displacement histories of the case with no inclination, at a structural frequency of 1.8 Hz, are displayed in Figure 4.4 with particle sizes 0.8 mm x 0.8 mm and 0.4 mm x 0.4 mm. The artificial damping reduces slightly as particle size increases. Therefore, this gives confidence that SPH is able to achieve similar damping to the experiments by increasing the particle size further. A particle size of 0.4 mm x 0.4 mm for the cases with a free surface length of 150 mm has been determined to be at the edge of the practical limit with the currently available computational facility. Therefore, the particle size required to achieve acceptable damping, at a structural frequency of 1.8 Hz, is unable to be achieved. Halving the particles size increases the run time by 4 times.

Chapter 2 gave evidence that SPH was capable of achieving acceptable comparisons with a similar experimental (inverted pendulum) setup up to a structural frequency of 0.92 Hz. Therefore, a lower structural frequency of 0.85 Hz is now investigated, in order to be able to present information with acceptable accuracy. Numerical and experimental displacement histories with a structural frequency of 0.85 Hz for uncontrolled, 0 (controlled), 8, 13, 20 and 26 degrees inclination are displayed in Figure 4.5(a), (b), (c), (d), (e) and (f). Again almost identical displacement histories are achieved between the numerical and experimental cases for the uncontrolled case in Figure 4.5(a).

108 The container length of 150 mm was chosen so that the liquid frequency of 0.85 Hz, be tuned at an inclination of 0 degrees, as per Equation 4.1. As a result, the most effective case is at this inclination, dissipating the majority of the energy within the first three cycles from release in Figure 4.5(b). With effective tuning, the fluid travels out-of-phase with the structure, eliminating structural oscillations quickly, therefore controlling the structure.

As inclination increases, free surface length is reduced and tuning is lost, resulting in prolonged structure oscillations, displayed in Figures 4.5(c), (d), (e) and (f). Smaller amounts of liquid producing increased velocities are observed in these un-tuned cases, compared to the tuned case. The reduced amounts of high velocities in the liquid sloshing waves result in decreased energy dissipation at the wave-to-wall interactions. Consequently, the fluid is unable to dissipate the energy from the structure, transferring it back as a result, increasing the amount of time to cease oscillating. These observations are presented later, in Section 4.3.2, in the form of liquid velocity flow field snapshots.

Increasing the inclination of the container also varies the dissipation characteristics of one side of each container. The inclined cases’ free surface stretches and shrinks as the structure oscillates, travelling up and down the sloped base of the container. Consequently, the wave travelling up and down the sloped base does not reach the wall of the opposing side of the container. Therefore, dissipation on this side of the container is limited to shearing between the travelling wave and container base. This differs from the 0 degree inclination case where wave-to-wall interactions occur at both walls of the container.

109 A small increase in numerical damping is also predicted for inclination cases above 0 degrees, at a structural frequency of 0.85 Hz, displayed in Figures 4.5(c), (d), (e) and (f). Also, minute phase differences occur between numerical and experimental displacement histories. This is again due to the increase in artificial numerical damping. However, even with these small variations, at a structural frequency of 0.85 Hz, SPH predictions show acceptable similarities with experimental observations.

Efforts to validate SPH at a structural frequency of 1.8 Hz were unsuccessful, due to the numerical damping being too high, due to limitations of the numerical tool at a high frequency. However, the numerical model is capable of producing quite acceptable comparisons to experiments at a structural frequency of 0.85 Hz. This critical observation gives confidence to explore numerically at lower structural frequencies, around 0.85 Hz, which are relevant frequencies in large structures (Wu et al. 2009).