Page 273 - The Combined Finite-Discrete Element Method

256 FLUID COUPLING

The fluid moves over the grid. Inertia terms are taken into account through convective
terms of the type

∂v x ∂x ∂v x ∂y ∂v x ∂z
dv x ∂v x
= + + + (8.1)
dt ∂t ∂x ∂t ∂y ∂t ∂z ∂t
∂v x ∂v x ∂v x ∂v x
= + v x + v y + v z
∂t ∂x ∂y ∂z
where v x ,v y and v z are velocity components in the x, y and z directions of the inertial
reference frame fixed to the Eulerian grid. These velocity components do not correspond
to fluid particles, but to points fixed in space. The acceleration field

dv x
a x (x,y,z,t) = (8.2)
dt
is an instantaneous acceleration field, which for any spatial point (x, y, z) represents
acceleration of the fluid particle that at a given time instance, occupies a spatial position
defined by specific spatial coordinates x, y and z. At every time instance, t it is a different
particle. This is achieved by defining velocity at grid points which are not fixed to fluid
particles and do not move in space, i.e. the coordinates of these grid points do not change
in time. If the grid is regular, there is even no need to remember the coordinates of the
grid points. However, the formulation is nonlinear in terms of velocity.
The combined finite-discrete element method is based on Lagrangian grid, where nodes
of the finite element are fixed to the moving solid particles and move together with solid
particles in space. Thus, inertia terms are simply expressed as

dv x ∂v x
= (8.3)
dt ∂t
where, for a specific solid particle defined by its fixed initial coordinates at time t = 0,
i.e. coordinates x i ,y i and z i
v x = v x (x i ,y i ,z i ,t) (8.4)

is the velocity component in the x-direction of the inertial reference frame. This velocity
component represents the velocity of the same specific solid particle at all time instances.
This solid particle can be recognised by the fact that at time t = 0 (initial configuration)
it occupied the spatial point
(x i ,y i ,z i ) (8.5)

where x i ,y i and z i are clearly not functions of time. In practical terms, this is achieved
through defining velocity at the nodes of the finite element mesh. These nodes are fixed to
solid particles and move together with solid particles. Thus, the coordinates of the nodes
change in time. The formulation is linear in terms of velocity. However, the coordinates
of the nodal points have to be updated at every time instance considered (usually every
time step).
In short, the CFD grid comprises grid points fixed in space and fluid particles moving
relative to these grid points, while the combined finite-discrete element method comprises
grid points moving together with solid particles.