ISSN 2086-5953 observed. The second case study, however, aims to
show the clearance phenomena when the water enter the clearance of the obstacle.
2 METHOD VALIDATION
2.1 Methods of SPH
The kernel of SPH In the SPH model, the fluid is discretized
with a finite number of macroscopic fluid particles. Each particle a is characterized by mass m
a
, density ρ
a
, pressure p
a
, a velocity vector u
a,
and a position
vector r
a
. The kernel of SPH is the interpolation formula which evaluates the value of any flow
property A at the position r, in relation with all other fluid particles b and is denoted as
2
. ,
1
b b
k b
b
m A
A w h
O h
b
r r r
where w
h
is an interpolating function which plays a central part in SPH. It depends on the distance
between two particles and a parameter h is called the smoothing length and proportional to the initial
particle spacing. In order to reduce the number of particles involved in Equation 1 and thus to
reduce the calculation time, it is convenient to consider kernels characterized by a compact
support of radius h
t
which is proportional to the smoothing length h. Consequently, only particles
located in the disc or sphere, in 3D of radius h
t
and centred on a contribute to the evaluation of the function A relative to the particle a. General
expressions of kernels are given by Morris et al. [9] and Monaghan [7]. In most SPH codes, spline
kernels are used. Herein, we consider the fourth- order spline kernel in the present SPH model [5].
Equations of motion in SPH model
An SPH form of the Lagrangian continuity equation can be written as
d .
. .
2 d
a h
ab b
m w
t
ab ab
u r
2
where u
ab
= u
a
− u
b
and ε
ab
= r
a
− r
b
r
ab
. ˙w
h
corresponds to the spatial partial derivative of the kernel and ddt to a Lagrangian derivative
obtained by the motion of a particle. In a Lagrangian system, the equation of
motion is
d 1
3 d
e
P t
u u F
3 where
ν denotes the kinematic viscosity of the fluid, p
is pressure,
ρ is density and F
e
is external forces such as the gravity or the Lorentz force in
magnetohydrodynamics. In the SPH model, the viscous effect is commonly modelled by an
artificial pressure [9, 11] which is
2
16 .
. .
4
b h
ab b
a b
v v
m w
ab ab
a ab
2 ab
u r u
r r
with η
2
= 0.01h
2
introduced to avoid a zero
denominator and r
ab
= r
a
− r
b
.
The pressure of each particle is determined by the following stiff state equation [7]:
2
. 1
5
o o
o
c P
where ρ
o
represents a reference density, c
o
is a numerical speed of sound and
γ is a constant coefficient equal to 7. In order to simulate a nearly
incompressible flow, c
o
must be at least ten times superior than the maximal velocity of the flow. This
nearly-incompressible assumption thus induces a Mach number M less than 0.1. Consequently, the
relative variation of density, which scales as M
2
, is less than 1 see [9].
2.2 Validation of SPH model
