4.4.1 Grid 1
Figure 4.3 shows the pressure histories at P4, P6 and P9 for grid 1. Even though grid 1 was the coarsest, the results appear consistent with the physics of the problem. In the transient phase lasting about ten sloshing oscillations, the pressure at P4 is consistent with the water striking the north wall. After 20 sloshing periods the flow has reached a steady periodic state. The peak pressure in figure 4.3(a) remains close to 1200 N/m2, while P6 settles at around 350 N/m2.
4.3(a): Pressure at P4
4.3(b): Pressure at P6 4.3(c): Pressure at P9
The computational data compares rather poorly with the steady-state experimental data, even when considering the discussion in section 3.5.2. The best correlation is observable at P4, the pressure at P6 is underestimated by about 50%. Comparing the computational result with the experimental result at P9, one finds little agreement between the data. In the CFD results, the water does not reach the North wall in the steady state phase, while the experimental results indicate very short-term pressure peaks. The period of oscillation is maintained throughout the CFD simulation, as shown in figure 4.3(a).
4.4(a): Force at East Wall 4.4(b): Force at North Wall
4.4(c): Force at West Wall 4.4(d): Force at South Wall
Figure 4.4: Pressure force on sloshing container [N] using grid 1
As can be seen in figure 4.4, the force plots reach a steady state as well. The forces at the East and West walls, given in figures 4.4(a) and 4.4(c) respectively, are anti-symmetric as expected. The conservation properties of this simulation are quite acceptable, as the mass and momentum imbalances are less than 1%. The maintenance of the correct amount of fluid in the tank is crucial to the success of the simulation, as natural sloshing frequency and hence wall pressure are directly influenced by the filling level. Precise numerical values as well as a summary of the parameters used are given in table 4.2. The imbalance rates are sufficiently low to permit further continuation of the run.
Table 4.2: Summary of grid convergence study, grid 1
Grid Independence Run 1: Summary Run Setup
Grid Nodes 6000
Transient Time [s] 45.0
Required Computational Time [hrs] 41.33
Simulation time per hour CPU∗ [sechr] 1.089
∗2.2 GHz, 64 bit processor with 2 GB RAM
CFX Settings
Fluid Models Air . . .Compressible
Water . . .Incompressible Flow Type Turbulent (k− turbulence model) Spatial Discretisation Second Order Upwind
Transient Scheme First Order Euler
Convergence Criteria Residual RMS 5·10−5
Numerical Precision Double
Conservation of Mass
Imbalance, % of original −0.1603
Imbalance per Simulated Second [seckg] −4.38·10−3 (mean) Conservation of Momentum
X-dir Imbalance, % of original −3.447·10−3 Imbalance per Simulated Second [kg·msec·sec−1] −1.348·10−4 (mean)
Z-dir Imbalance, % of original 2.573·10−3 Imbalance per Simulated Second [kg·msec·sec−1] 6.775·10−5 (mean) 4.4.2 Grid 2
Grid 2 is more than double the total number nodes of grid 1 and it increases the required computational resources by a factor of approximately two. The transient pressure phase at P4, delineated in figure 4.5(a) is similar to the one observed in the coarse grid 1, shown in figure 4.3(a). However, the pressure peaks at P6 and P9, given in figures 4.5(b) and 4.5(c)
4.5(a): Pressure at P4
4.5(b): Pressure at P6 4.5(c): Pressure at P9
Figure 4.5: Pressure [N/m2] at control points using grid 2
respectively, are significantly higher with grid 2. Further, the transition from transient to steady state is more pronounced in the present case as seen in the trough in peak pressure at oscillation 12 and 13. The comparison with experimental data is again rather poor, with qualitative differences similar to the previous grid 1. The pressures in the transient phase (the first 10 oscillations) are higher than in figure 4.3, with an increase at P6 of about 20% of the peak pressure at oscillation six. This makes the grid-dependence of the solution obtained in grid 1 apparent, indicating that grid 1 is unsuitable for further use.
The force plots in figure 4.6 are similar to the previous case, reaching a steady state once the flow has progressed past 20 oscillations. Grid 2 takes longer to progress to a steady state
4.6(a): Force at East Wall 4.6(b): Force at North Wall
4.6(c): Force at West Wall 4.6(d): Force at South Wall
Figure 4.6: Pressure force on sloshing container [N] using grid 2
than the previous grid 1 and the second transient phase is well defined. The transient sloshing force is higher than with grid 1, as might be expected from the previous pressure observations. The conservation properties given in table 4.3 are similar to the previous grid 1, indicating that despite the coarse discretisation the numerical methods used are sufficiently accurate to maintain conservation of mass and momentum.
Table 4.3: Summary of grid convergence study, Grid 2
Grid Independence Run 2: Summary Run Setup
Grid Nodes 12000
Transient Time [sec] 45.0
Required Computational Time [hrs] 99.26
Simulation time per hour CPU∗ [sechr] 0.456
∗2.2 GHz, 64 bit processor with 2 GB RAM
CFX Settings
Fluid Models Air . . .Compressible
Water . . . Incompressible Flow Type Turbulent (k− turbulence model) Spatial Discretisation Second Order Upwind
Transient Scheme First Order Euler
Convergence Criteria Residual RMS 5·10−5
Numerical Precision Double
Conservation of Mass
Imbalance, % of original −0.1619
Imbalance per Simulated Second [seckg] −3.633·10−3 (mean) Conservation of Momentum
X-dir Imbalance, % of original −1.801·10−2 Imbalance per Simulated Second [kg·msec·sec−1] −4.778·10−4 (mean)
Z-dir Imbalance, % of original −3.031·10−2 Imbalance per Simulated Second [kg·msec·sec−1] 7.111·10−4 (mean) 4.4.3 Grid 3
Grid 3, encompassing 28000 nodes, provides additional spatial resolution. However, the in- crease in required time steps due to reduction in grid spacing reduces the conservation prop- erties of mass and momentum, given in table 4.4 considerably. The steady state pressure at P4 depicted in figure 4.7(a) shows the split peak observed in experimental studies. The
4.7(a): Pressure at P4
4.7(b): Pressure at P6 4.7(c): Pressure at P9
Figure 4.7: Pressure [N/m2] at control points using grid 3
transient peaks at P4 and P6, shown in figure 4.7(b) are about 10% higher than with grid 2. Note the large spike in the pressure history of P9 between oscillations 5 to 7 displayed in figure 4.7(c). Further, one may note the absence of the secondary peak at oscillation 15, the pressure obtained with grid 3 progresses directly to the steady state. Considering the experimental data in figure 4.7(a), there is better agreement than with previous grids, as the pressure disagreement at P4 is in the regions of oscillating experimental data. Again, the pressures at locations P6 and P9 do not match the experimental data any better. The pressure peak in the transient region has increased again compared to grid 2, the steady-state solution is similar in grids 2 and 3. The pressure and force histories obtained with grid 3
match those from grid 2 better than those from grid 1, indicating the result is now less grid dependent.
The force plots in figure 4.8 are slightly higher than in the previous grid 2 (figure 4.6). Also note the typical sloshing peaks shown at the anti-symmetric East and West walls illustrated in figures 4.8(a) and 4.8(c) respectively. Mass and momentum conservation are satisfactory for this run. However, the mass residual given in table 4.4 appears slightly larger than expected. The computational requirements have increased significantly compared to grids 1 and 2, with the simulated second per CPU hour decreasing by about 90% to nearly 0.12 sec/hr. Shown later in figure 4.11 is the free surface position throughout the simulation. The free surface elevation correlates well to the pressure graphs, illustrating the significance of the static pressure component.
4.8(a): Force at East Wall 4.8(b): Force at North Wall
4.8(c): Force at West Wall 4.8(d): Force at South Wall
Table 4.4: Summary grid convergence study, grid 3
Grid Independence Run 3: Summary Run Setup
Grid Cells 28000
Transient Time [sec] 32.0
Required Computational Time [hrs] 260.53
Simulation time per hour CPU∗ [sechr] 0.123
∗2.2 GHz, 64 bit processor with 2 GB RAM
CFX Settings
Fluid Models Air . . .Compressible
Water . . . Incompressible Flow Type Turbulent (k− turbulence model) Spatial Discretisation Second Order Upwind
Transient Scheme First Order Euler
Convergence Criteria Residual RMS 5·10−5
Numerical Precision Double
Conservation of Mass
Imbalance, % of original 5.234
Imbalance per Simulated Second [seckg] 1.636·10−1 (mean) Conservation of Momentum
X-dir Imbalance, % of original −8.987·10−3 Imbalance per Simulated Second [kg·msec·sec−1] −2.808·10−4 (mean)
Z-dir Imbalance, % of original 4.083·10−3 Imbalance per Simulated Second [kg·msec·sec−1] 1.276·10−4 (mean)
4.4.4 Grid 4
4.9(a): Pressure at P4
4.9(b): Pressure at P6 4.9(c): Pressure at P9
Figure 4.9: Pressure [N/m2] at control points using grid 4
Grid 4 is the finest grid used for the grid independence study, and it is evident that it is not suitable for practical applications due to the significant computational requirements. Thus only a limited time history is available. Nonetheless, the transient region can still be used as a basis for assessment. It compares well, both when considering the pressure peaks given in figure 4.9 as well as the local negative pressures observable in the transient regions between oscillations 4 and 6, to the previous grid 3. Note that the transient peaks at P6, shown in figure 4.9(b), are lower than those observed using grid 3.
4.10(a): Force at East Wall 4.10(b): Force at North Wall
4.10(c): Force at West Wall 4.10(d): Force at South Wall
Figure 4.10: Pressure force on sloshing container [N] using grid 4
Table 4.5: Settings for grid convergence study, grid 4. Residual data not available due to short run
Grid Independence Run 4: Summary Run Setup
Grid Cells 56000
Transient Time [sec] 9
Required Computational Time [hrs] 140.15
Simulation time per hour CPU∗ [sechr] 0.064
∗2.2 GHz, 64 bit processor with 2 GB RAM
CFX Settings
Fluid Models Air . . .Compressible
Water . . . Incompressible Flow Type Turbulent (k− turbulence model) Spatial Discretisation Second Order Upwind
Transient Scheme First Order Euler
Convergence Criteria Residual RMS 5·10−5
Numerical Precision Double
Comparing the computational results to the experimental data, it is apparent that they do not match in the steady state. The static pressure appears to be predicted correctly when comparing the water surface elevation to the discussion in section 3.5.1. However, the exper- imental pressure readings indicate water impacting the North wall but this is not observed in the computational study. In summary, the results do not match the experimental data as well as one would expect, but the results appear to be independent of the computational grid. Therefore, the current combination of parameters is not sufficient for simulating violent sloshing. This is considered in greater detail in section 8. The next section gives an analysis of the results using the theory outlined in section 4.3 to formally establish grid independence