The magnitude of both the interphase momentum transfer term and the volume exclusion term are examined in the preceding section for a single particle falling in the water column. However, the influence of a single particle on the fluid
phase is not significant. In order to test the impacts of the particles at high volume fraction on the fluid phase and hence examine the implementation of these two terms, large number of particles are needed. In this section, a large number of particles in nearly fully packed conditions will be used for this purpose. In particular, an isolated block of particles is placed in the flow. Tests with the isolated block located in the middle of the flow, on the bottom wall and at a trench corner are performed respectively. Details of the flow passing this block can reveal whether the implementation of the two terms are correct in terms of the fluid hydrodynamics. The detailed set-up of the tests are listed in Table 4.5. To minimise other effects, particles are fixed, and the focus is placed on the reaction of the flow field to the presence of particles. The modelling results with both terms on, only the interphase momentum transfer term (SU) on, and only the volume
exclusion term (Tve) on are presented.
Test number Block location in the flow Interphase momentum transfer Volume exclusion term MDL-1 middle X X MDL-2 X ◦ MDL-3 ◦ X BTM-1 bottom X X BTM-2 X ◦ BTM-3 ◦ X CRN-1 corner X X CRN-2 X ◦ CRN-3 ◦ X
Table 4.5: Model set-up in the isolated block tests.
The mesh resolution in these tests is 2.5 mm, and it is one cell wide in the trans- verse z direction. Slip boundary conditions are applied to the front and back
xy−planes. A steady current boundary condition is imposed at the inlet bound- ary. The mean flow velocity is 0.35ms−1. The diameter of the parcel is 0.72mm,
and particle d50 is 0.36 mm. The maximum solid volume fraction in the block is 0.58. The time step is 0.0005s.
The modelled flow fields and velocity profiles at selected sections in MDL tests are shown in Figure 4.3. The flow velocity fields produced with both the source terms on (MDL-1), and those produced with only the interphase momentum transfer term on resemble with each other, and the physical effects of the source terms are well represented. As shown in Figure 4.3a, 4.3b, 4.3c, and 4.3d, the fluid phase is affected by the presence of the particle block and flows around the block as expected, and vortices in the lee-wake side and flow reversal are captured. As the block of particles is a porous medium, water can flow into the block, and the flow inside it is very minor as expected. The magnitude and direction of the velocity vectors in MDL-1 and MDL-2 agree with each other very well. The only difference between these two tests is that in MDL-2, the volume exclusion term is turned off. It implies that, the volume exclusion term has much less effect than the interphase momentum transfer term, which is consistent with the results in the particle falling tests. When the interphase momentum transfer term is turned off and the volume exclusion term is on, instead of getting blocked by the particles, the flow squeezes into the block and accelerates there due to the spatial gradient of the fluid volume fraction in and around the block. Inside the block, flow accelerates to two to three times the mean flow velocity (see Figure 4.3e and 4.3f). This effect is not desirable for scour studies.
To testify whether the presence of the particles at boundaries can cause prob- lems, the block of particles is placed on the bottom boundary (BTM tests). The modelled flow velocity field and velocity profile at selected sections are shown in Figure 4.4. The flow velocity around the block decelerates and the vortices in the
lee-wake side are fully resolved when the interphase momentum transfer term is on. The velocity profiles are well developed and captured as well. When only the volume exclusion term is on, the flow accelerates inside the block and no vortex is observed. In short, the interphase momentum transfer term achieves the physical impact of the solid phase back on the fluid phase reasonably while the volume exclusion term results in unrealistic flow acceleration inside the block, and the effect of the latter is much less than the former. In the CRN tests, the block of particles is placed at a trench corner. Again, the same phenomena are observed as shown in Figure 4.5.
To further confirm the difference in the tests with both source terms on and with only the interphase momentum transfer term on, the velocity field in the former tests is deducted by that in the latter tests. The velocity difference in the three series of tests is shown in Figure 4.6 to visualise the contribution of the volume exclusion term when both source terms are on. It is clear that the volume exclusion term has little effect in the upstream and the far field. It only has impact on the circulation zone immediately downstream of the block. Water in the downstream side tends to flow upstream towards the block, which is not realistic physically. It again gives a preliminary suggestion that this term is not desirable for scour studies and needs further investigation.
The magnitudes of the interphase momentum transfer term and the volume ex- clusion term are shown in Figure 4.7. The maximum values of both terms are observed at the two corners of the block at the upstream side in the MDL tests, and at the upper left corner in BTM and CRN tests where the block rests on the bottom boundary. The maximum value of the interphase momentum transfer term is one order larger than that of the volume exclusion term. In the majority
(a)MDL-1 (b)MDL-1
(c)MDL-2 (d) MDL-2
(e) MDL-3 (f) MDL-3
Figure 4.3: Velocity vector field (left column) and velocity profiles at selected
(a) BTM-1 (b) BTM-1
(c) BTM-2 (d) BTM-2
(e) BTM-3 (f)BTM-3
Figure 4.4: Velocity vector field (left column) and velocity profiles at selected
(a) CRN-1
(b)CRN-2
(c) CRN-3
Figure 4.5: Velocity vector field in Tests CRN. Red line: the boundary of the block.
(a) MDL (b)BTM
(c)CRN
Figure 4.6: Velocity difference field. Red line: the boundary of the block.
area, the magnitude of the interphase momentum transfer term is three orders larger than that of the volume exclusion term.
In conclusion, the interphase momentum transfer term decelerates the flow around and inside the particle block as expected. As the fluid volume fraction inside the block is smaller, the effect of the volume exclusion term alone results in water squeezing into the block and getting accelerated there. This is correct mathe- matically, not realistic physically on its own though. Because the magnitude of the interphase momentum transfer term is several orders larger than that of the volume exclusion term, the overall effect of this two source terms is dominated by the interphase momentum transfer term.
The volume exclusion term is indispensable to compensate for the non-divergence- free flow field caused by the presence of particles. For the two-fluid methodology
(a)MDL-1 (b)MDL-1
(c) BTM-1 (d) BTM-1
(e) CRN-1 (f)CRN-1
Figure 4.7: Magnitude of interphase momentum transfer term (left column)
employed in the hydrodynamic module, zero velocity divergence is satisfied. With the MP-PIC approach coupled with the hydrodynamic module, the density and velocity field of the fluid phase does not implicate any additional elements due to the introduction of particles. Therefore, the fluid phase is still divergence free. For this reason, the volume exclusion term is not a core element to the current model. In addition, considering the unrealistic physical effects and its potential hazard to numerical stability, the volume exclusion term is removed from the model hereafter.