Environmental Engineering Reference
In-Depth Information
2.1 Fourier Series Approximation
Since the angular coordinate is 2
-periodic, we can approximate the velocity, the
temperature and the pressure (represented by
ˀ
ˆ
) as a Fourier complex series:
m
ˆ
k
(
e
ik
ʸ
.
ˆ(
r
,ʸ,
z
,
t
)
≈
r
,
z
,
t
)
(2)
k
=
1
So, the first and second derivatives with respect to
ʸ
are:
m
∂ˆ(
r
,ʸ,
z
,
t
)
ik
ˆ
k
(
e
ik
ʸ
,
≈
r
,
z
,
t
)
(3)
∂ʸ
k
=
0
m
2
∂
ˆ(
,ʸ,
,
)
r
z
t
ˆ
k
(
k
2
e
ik
ʸ
,
≈−
,
,
)
r
z
t
(4)
2
∂ʸ
k
=
0
ˆ
k
(
,
,
)
Then, the unknowns are the Fourier coefficients
r
z
t
.
2.2 Projection Method
The method is called “projection” because the velocity field is calculated in two steps.
The first step consists in the calculation of the Navier-Stokes equations by assuming
a constant pressure. In a second step, this velocity field is projected onto a space
of zero divergence and satisfying the appropriate boundary conditions (Fuentes and
Carbajal
2005
).
We assume a uniform pressure field and we approximate the time derivative with
a backward finite difference formula. This leads to the following equation
g
d
3
ʺ
3
u
∗
−
4
u
n
u
n
−
1
+
u
n
,
n
−
1
r
k
2
u
∗
=
+ℵ
(
)
+
2
−
R
a
P
r
(
T
−
T
0
)
+
P
r
∇
0
,
(5)
2
ʔ
t
u
n
,
n
−
1
where
u
∗
is the estimate of the non-
linear term according to the Adams-Bashforth scheme. The term
is the fictitious velocity field and
ℵ
(
)
r
u
n
,
n
−
1
ℵ
(
)
requires
r
knowledge of the velocity field at the two previous times
n
and
n
1 represents
the actual time). On the other hand, the equation for the true velocity field is:
−
1(
n
+
g
d
3
ʺ
3
u
n
+
1
4
u
n
u
n
−
1
−
+
u
n
,
n
−
1
r
k
p
n
+
1
+ℵ
(
)
+
2
−
R
a
P
r
(
T
−
T
0
)
+
∇
ʔ
2
t
2
u
n
+
1
+
P
r
∇
=
0
.
(6)
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