Environmental Engineering Reference
In-Depth Information
where the asterisk means dimensionless variables and operators. The dimensionless
parameters are Prandtl
P
r
d
4
=
ʽ/ʺ
=
ʱʲ
/(ʺʽ)
and Rayleigh
R
a
g
numbers. The
unknowns are the velocity field (with components
u
r
,
and
u
z
), the temperature
T
and the pressure
P
. In cylindrical coordinates, the above equations become:
u
ʸ
,
∂
u
r
∂
u
r
r
+
1
r
∂
u
+
∂
u
z
∂
ʸ
∂ʸ
r
+
z
=
0
,
(continuity eq.)
u
2
∂
u
r
∂
u
r
∂
u
r
∂
u
r
∂
u
r
∂ʸ
u
z
∂
u
r
∂
r
=−
∂
p
t
+
r
+
+
z
−
∂
r
P
r
∂
2
u
r
∂
2
u
r
∂ʸ
2
u
r
∂
1
r
∂
u
r
∂
r
2
∂
1
r
2
∂
2
u
u
r
r
2
+
∂
ʸ
∂ʸ
+
+
r
+
−
−
,
(radial eq.)
r
2
2
z
2
∂
u
ʸ
∂
u
ʸ
u
r
r
u
r
∂
u
ʸ
∂
u
r
∂
u
ʸ
∂ʸ
u
z
∂
u
ʸ
∂
1
r
∂
p
∂ʸ
t
+
+
r
+
+
z
=−
P
r
∂
2
u
2
u
2
u
1
r
∂
u
r
2
∂
1
r
2
∂
2
u
r
∂ʸ
u
+
∂
ʸ
ʸ
ʸ
ʸ
r
2
ʸ
+
+
r
+
+
−
,
(angular eq.)
∂
r
2
∂
∂ʸ
2
∂
z
2
g
d
3
ʺ
∂
u
z
∂
u
r
∂
u
z
∂
u
r
∂
u
z
∂ʸ
u
z
∂
u
z
∂
−
∂
p
t
+
r
+
+
z
=−
2
−
R
a
P
r
(
T
−
T
0
)
∂
z
P
r
∂
2
u
z
∂
2
u
z
∂ʸ
2
u
z
∂
1
r
∂
u
z
∂
r
2
∂
1
+
∂
+
+
r
+
,
(vertical eq.)
r
2
2
z
2
2
T
∂
2
T
∂ʸ
2
T
∂
∂
T
∂
u
r
∂
T
∂
u
∂
T
∂ʸ
u
z
∂
T
z
=
∂
1
r
∂
T
r
2
∂
1
+
∂
r
t
+
r
+
+
+
r
+
z
2
.
(Energy eq.)
∂
r
2
∂
2
(1)
In order to solve these equations we use the projection method (Fuentes and
Carbajal
2005
), which consists in the introduction of a fictitious velocity which is
the solution of the Navier-Stokes equation for a constant pressure. For this fictitious
velocity the condition
0 is not fulfilled. In a second step, the pressure is
calculated by solving an equation resulting from taking the divergence of the Navier-
Stokes equations and imposing the condition of zero divergence to the velocity field.
Finally, a real velocity is obtained from the Navier-Stokes equations by including
the pressure calculated in the previous step.
∇·
u
=
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