Computational Fluid Dynamics
Lecture 13
Solving for Pressure
(
)
2
,
,
p
R x y z
∇
=
Elliptic equations such as the Poisson equation are sensitively dependent on Boundary Conditions.
Choices of B.C.’s are:
1. Periodic (if you’re lucky)
2. Dirichlet:
0
B
p
g
=
3. Neumman:
0
B
p
f
z
∂
=
∂
The easiest method (computationally fast) is to use Fast Fourier transforms in any periodic
directions.
For example in 3D if all 3 directions are periodic.
(
)
(
)
(
)
(
)
(
)
2
2
2
2
2
2
2
2
2
2
2
2
Real
Half complex
ˆ
First FFT in ,
,
,
ˆ
ˆ
ˆ
ˆ
,
,
ˆ
ˆ
ˆ
then FFT in direction,
, ,
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
, ,
x
p
p k y z
p
p
p
R k y z
x
y
z
y
p
p k
p
p
p
R k
z
x
y
z
→
→
∂
∂
∂
+
+
=
∂
∂
∂
→
∂
∂
∂
+
+
=
∂
∂
∂
A
A
z
This second transform is a complextocomplex transform.
Finally, FFT in
z
direction (complexto
complex).
(
)
ˆ
ˆ
ˆ
ˆ
ˆ
, ,
p
p k
m
→
A
1
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The PDE has been transformed into
nx
ny
nz
×
×
uncoupled algebraic equations.
(
)
(
)
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
, ,
then using
in wave number space.
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
which is solved for
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
for each , , and
; and then inverse transformi
p
p
p
R k
m
x
y
z
y
k p
p
m p
R
p
R
p
k
m
k
m
∂
∂
∂
+
+
=
∂
∂
∂
∂
= −
∂
−
−
−
=
−
=
+
+
A
A
A
A
A
(
)
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 Spring '07
 Slinn
 Boundary value problem, Fast Fourier transform, pi −1

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