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
(
)
r
u n , n 1
(
)
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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