Exposed in the flow, the particles will move when the instantaneous fluid force on a particle just outweighs the instantaneous resistant force due to particle gravity and friction[89]. Then particles may roll, slide or saltate along the bed, and even get entrained into the water column above the bed, and move along with the flow. The influence of the fluid phase on the individual particles is represented via the drag force, the pressure gradient force and the buoyant force (see Section 3.3.2). Likewise, the dynamics of the fluid phase will be affected by the presence of the particles.
In the present study, the effects of individual particles on the fluid phase are simulated through a point-source term in the fluid momentum equation. The collective effects from many particles will become apparent on the overall fluid dynamics inside each Eulerian cell. In practice, the realisation of such interactions is usually called two-way coupling. It is four-way coupling when the particle- particle interaction is also implemented. In the present work, the four-way coupling is implemented. In particular, the particle-particle coupling is realised by adopting the inter-particle stress model as described in the preceding sections, and the influence of the flow on particles has been reflected in Section 3.3.2. In this section, the particle-fluid coupling, to be more specific, the particles’ influence on the flow in terms of momentum, viscosity and volume exclusion effect, will be detailed.
3.4.1
Interphase Momentum Transfer
To maintain the computation efficiency, the present study adopts the point-source approach to represent the particle effects. In general, each individual particle is considered as a momentum source/sink in the particular Eulerian cell to the fluid motion. The overall effect from all the particles within the same Eulerian cell can then be integrated as a momentum source/sink in this particular cell. This source term is usually called the interphase momentum transfer in multiphase flow.
Particles are influenced by the flow through the drag force and pressure gradient force, and hence particles gain certain amount of momentum from the fluid phase, which means that the fluid phase lose the same amount of momentum as results of the fluid-particle interaction. Following the work by Snider[73] and Patankar and Joseph[60], the momentum source from the solid phase within a cell, SU, is
integrated as, SU =− Z Z Z φρpVp Dp(Uf −Up)− 1 ρp ∇p dVpdρpdUp, (3.32)
where φ is the particle distribution function, ρp is the particle density, Vp is the
particle volume, and Dp is a parameter derived from the drag coefficient (Eq.
3.21), Uf is the fluid phase velocity and Up is the particle velocity. This is a
simple representation of the particle effects to the fluid phase in general. At the particle scale, the fluid dynamics can be fairly complex and hence there will be
energy losses that cannot be represented exhaustively by Eq. 3.32. However, in scour process, such detailed interactions are not considered important to the overall process. Therefore, the present study will not explore further into these sub-grid scale processes.
3.4.2
Mixture Viscosity
In addition to the momentum transfer, the viscosity of the fluid phase is also in- fluenced by the presence of particles. Past studies show that in dilute suspensions, concentration and viscosity are linearly related[12, 62], and as the concentration approaches the maximum packing status, the viscosity becomes infinite[11, 62].
Several popular viscosity formulas have been reviewed in Section 2.4.4. Consider- ing the Eulerian-Lagrangian framework, as well as the huge amount of particles involved in computation, the Eilers equation and the Krieger-Dougherty equation are more suitable and efficient to implement. Simple sensitivity tests show that there is no significant difference by applying either Eilers equation or Krieger- Dougherty equation in the present model. Therefore Eilers equation is employed hereafter for consistency.
Employing Eilers Equation[11, 62], the bulk viscosity accounting for the presence of the solid particles is modified as
µ0f =µf 1 + 0.5µ0θs 1− θs θcs 2 , (3.33)
where µ0 is the intrinsic viscosity, θs is the volume fraction of the solid particles,
and θcs is the critical value of θs. The intrinsic viscosity accounts for the shape
of particles. For spherical particles, µ0 = 2.5 is recommended, and for irregularly shaped particles, the determination of µ0 stays uncertain[12, 62].
The bulk viscosity is a function of the particle shape, the local sediment volume fraction and critical solid volume fraction. In general, the modified bulk viscosity is no less than the original fluid viscosity. In a cell comprising of water and air only, the solid volume fraction θs is zero, thus the bulk viscosity µ0f converts back into
the original viscosity of the pure fluid µf. When a cell approaches the maximum
packing status, for example, θs = 0.64 and θcs = 0.65, the modified viscosity is
several thousands times the original viscosity, in line with the findings from the past studies.
3.4.3
Volume Exclusion Effect
The presence of particles will influence the volume displacement of the fluid phase. A volume exclusion term (Tve), which accounts for the displacement of the fluid
phase due to particle motion, can be introduced to the l.h.s. of the momentum equation for this purpose. Following the work of Cihonski et al.[9], it reads,
Tve=ρU
∂
∂tlnθf +U· ∇(lnθf). (3.34)
This term was originally derived for the simulation of the volume displacement effects during bubble entrainment in a travelling vortex ring, where gas and liquid phase were involved. By applying it to the present model for scouring, results show that it has very minor effect in the interaction between the fluid phase and solid phase, while it can easily cause numerical instability issues. The examination of this term along with the interphase momentum transfer term will be presented in Chapter 4.
Therefore, the volume exclusion term has been removed from the model prior to model applications. Nevertheless, the work by Cihonski et al.[9] inspires the in- vestigation into the volume displacement effect in terms of liquid-solid interaction, including that during the scour process.
With the modifications described in this section, the final form of the momentum equation for the dispersed fluid phase reads,
∂ρU