Environmental Engineering Reference
In-Depth Information
Fig. 1
Integration domain to solve Navier-Stokes and continuity equations. This domain is
decomposed in four parallelepipeds.
a
Upper
view
b
view in
perspective
D
u
Dt
=−
1
ˁ
∇
1
Re
∇
2
u
+
P
(1)
∇
·
u
=
0
.
(2)
Here,
D
is the material derivative.
The forcing is introduced through a sinusoidal flow rate:
/
Dt
≡
∂/∂
t
+
(
u
·
∇
)
Q
(
t
)
=
Q
0
sin
(
2
ˀ
St
)
(3)
The dimensionless parameters appearing in Eqs.
1
and
3
are the Reynolds (
Re
)
and the Strouhal (
S
) numbers. The first one is defined ad
Re
=
UH
/ʽ
while the
last one is defined as
S
=
H
/
UT
, where
H
is the channel width,
U
the maximal
velocity in the channel,
is the kinematic viscosity and
T
is the forcing period.
The Strouhal number can be see like a dimensionless frequency (López-Sánchez and
Ruiz-Chavarría
2013
; López-Sánchez
2013
).
Equations
1
-
2
are supplementedwith boundary conditions. The bottom, the lateral
wall in the channel and some lateral walls in the open domain are solid, then velocity
must vanish there. In the free surface we assume that both vertical velocity and
tangential stress vanish. Finally, along the channel input (wavy line in Fig.
1
a) and
along the far boundary in front of the channel (dark side in Fig.
1
b) the flow rate
given by Eq.
3
must be fulfilled.
Open Field Operation And Manipulation (OpenFOAM
2008
) is a collection of
solvers for partial differential equations. This program is structured in modules writ-
ten in
C++
. It incorporates pre and post-processing utilities to generate meshes and
present the results of simulations (The open source
2004
).
In order to test the spatial convergence, we carried simulations with different
numbers of points in the grid. We start with 108,000 points and we finish with
2,100,000. The criterion to stop a further increase is that the difference of velocity
values in two subsequent meshes becomes less than a threshold
ʽ
10
−
6
.Onthe
other hand, to guarantee time convergence we introduce the Courant number (Co),
defined as follow:
ʵ
=
U
q
ʔ
t
Co
=
q
,
(4)
ʔ
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