Smoothed Particle Hydrodynamics for Free-Surface Flows - Selected Topics of Computational and Experimental Fluid Mechanics - page 121

n + 1

dt is calculated using the updated values of v n + 1

ˁ /

At the end of the time step d

a a

and r n + 1 (Monaghan 2005 ).

a

2.8 Variable Time Step

The time step depends on the flow properties. For example, the time step decreases
when the fluid collides with fixed boundaries since the forces increase. In general, the
time step depends on the force per unit mass and the Courant condition. A variable
time step

ʔ t is calculated according to (Monaghan and Kos 1999 ):

t = C · min ( t F , t CV ),

min a h f a t F = / , (18)

h

max b

t CV = min a ,

h v ab x ab
r ab

c s +

here t CV depends on the Courant

condition and the viscosity of the system. C is a constant that can vary between 0.1
and 0.3.

t F is based on the force per unit mass | f a | , and

2.9 Boundary Conditions

The boundary conditions implemented in DualSPHysics are the Dynamic Boundary
Conditions (DBCs). The boundary particles (BPs) must satisfy the same equations as
the fluid particles (FPs), but they are not free to move in any direction. The movement
of these particles is equal to zero or externally imposed such as a flap or a piston in
a wave maker or any other kind of moving object (gates, elevators, etc.).
The behaviour of these boundary particles can be summarized as follows. The
FPs are free to move and interact with each other but when a FP approaches a
boundary and the distance between a FP and the BPs becomes smaller than 2 h ,the
BPs increase their density, leading to a pressure increase through the equation of state
( 9 ). This process modifies the force exerted on the FP due to the pressure term on
the momentum Eq. ( 6 ). For a more detailed description of the boundaries the reader
is referred to the work of Crespo et al. ( 2007 ).

2.9.1 Periodic Boundary Conditions

Periodic boundary conditions are also implemented in DualSPHysics. In this case,
the particles near an open lateral boundary interact with the particles on the other
side of the domain since the area of influence of a particle extends beyond the lateral
boundary.