3.5.1
Boundary Conditions
The two typical numerical boundary conditions, namely, the Dirichlet boundary condition and the von Neumann boundary condition, are available in the hydro- dynamic module[30, 71]. Dirichlet boundary condition prescribes the value of dependent variables on the boundary directly, and the von Neumann boundary condition prescribes the gradient of the variables normal to the boundary. In the former case, a fixed value φB can be specified on the boundary and therefore, the
values on the cell faces along this boundary are all assigned as φf = φB. In the
latter case, the face gradient ∇φ is specified, and the boundary face value can be computed by
S· ∇φ=|S|φf −φP dn
, (3.36)
where φf and φP are the value of the variable on the boundary face and at the
cell centre of this boundary cell, respectively, and dn is the distance from the cell
Therefore, fixed value or fixed gradient of the dependent variables for the bound- aries can be implemented directly in the model. For complex boundary conditions such as wave boundary conditions, new boundary condition types can be developed on top of the existing options.
A typical model setting involves the flow over a sandy bed. The sands are placed on the bottom boundary of the domain, and the water and air are above the bed. There are basically four types of boundaries: inlet, outlet, atmosphere and walls (see Figure 3.2). The inlet is on the l.h.s. and the outlet is downstream at the r.h.s. of the domain. For the inlet boundary, the velocity can be specified directly. Zero normal gradient of the velocity is usually applied at the outlet boundary. The slip/no-slip condition can be used on the front and back wall as needed. Zero velocity is imposed on the bottom wall and the surface of structures. The pressure gradient is set such that it provides the specified flux on each boundary according to the velocity.
Figure 3.2: Sketch of a computational domain. Red: water; blue: air.
The boundary conditions in the hydrodynamic module are also applicable to the solid-phase-related Eulerian variables such as solid volume fraction θs. For the
Lagrangian variables such as particle position and particle velocity, as they are determined by Newton’s Law of Motion, once the initial values are assigned, those values will be calculated accordingly. When a particle reaches the downstream
boundary, it will no longer remain in the solution domain. When periodic bound- ary condition is assigned, particles will re-enter from the corresponding boundary into the domain again once exit a periodic boundary.
The time step ∆tused in a simulation is determined by both the Courant-Friedrichs- Lewy (CFL) condition and the particle time scale τt =
ρpd2p
18µ. ∆t must be smaller
than the time step required by the CFL condition and the particle time scale. The time step determined this way will be used as the Eulerian time step in the sim- ulation for both the hydrodynamic module and particle module. The Lagrangian sub-time-step used to evolve particles will then be determined automatically dur- ing simulation.
3.5.2
Initial Conditions
The initial conditions in the hydrodynamic module can be specified for each Eule- rian variable according to test configurations. The initialisation of the Lagrangian variables needs special treatment as they are not based on the Eulerian grid. A particle initialiser was devised for this purpose.
Initialisation of the Parcel Positions
As parcels are the actual computational unit in the particle module, the particle input actually refers to that of the parcels. The parcels’ diameter, initial velocity and positions are the necessary input for the particle module. Parcels’ initial velocity can be set as required. Parcels’ diameters can be determined by other factors, as long as it is smaller than the grid size and larger than particle size. It will be determined along with the initialisation of the particle positions.
In scour process, it usually starts with a sand bed where the cells within the bed are fully packed or close to fully packed. In MP-PIC method, the parcel diameter should be smaller than the grid size. It is worth mentioning that parcel diameter is different from the median particle size d50. Particle d50 is assigned according to the experimental set-up or in-situ observations, and will be used to calculate particle Reynolds number and particle motion. The parcel diameter is only used to calculate the parcel volume during the simulation where necessary. Sensitivity tests show that cases with the ratio of the grid spacing to parcel diameter within 3−4 produce reasonable results (See Chapter 4). Bigger ratios than 4 should be suitable as well, but the amount of parcels involved will increase remarkably, which increases the computational expenses dramatically. With these rules in mind, an initialisation method is proposed here.
In the particle initialiser, the particle region is specified first. If it is fully packed and the critical solid volume fraction θcs = 0.65, each cell in this region will
length scale. In this way, the solid volume fraction will be approximately 0.6488, close to θcs. In each cell, the parcels will be distributed randomly. A parcel’s
position, xp, is determined by the cell centre c, cell length L and the random
number vector R. Each component of this random number vector Ris between 0 and 1. The parcel’s position is derived by
xp =c+
1
2(2R−1)·L. (3.37)
In this way, parcels are all located within the specified particle region and the distribution results satisfy the solid volume fraction as needed. The numbers in the example above, such as number of parcels per cell, can be easily modified to suit the experimental conditions.