Steady Potential Flow through Box with Staggered Inlet and Outlet

​
box aspect ratio
0.25
0.5
1
1.5
2
3
4
Consider the flow through a two-dimensional box with staggered inlet and outlet as show in the figure below:
The flow enters and exits the box with a uniform speed
U=1
. The aspect ratio of the box is
γ=L/W
.
The Demonstration plots the streamlines for this potential flow for a user-set value of
γ
.
These streamlines are the solution of the equation
2
∂
ψ
∂
2
x
+
2
∂
ψ
∂
2
y
=0
subject to the boundary conditions:
BC1: at
x=0
,
ψ=0
for
0⩽y≤W/2
and
ψ=U(y-W/2)
for
W/2⩽y≤W
,
BC2: at
x=L
,
ψ=Uy
for
0⩽y≤W/2
and
ψ=UW/2
for
W/2⩽y≤W
,
BC3: at
y=0
,
ψ=0
for
0⩽x≤L
, and
BC4: at
y=L
,
ψ=UW/2
for
0⩽x≤L
.
The solution methodology is based on Chebyshev orthogonal collocation with
N=31
collocation points.
This problem also admits an analytical solution given by T. Bennett[1]:
ψ=Uy/2+
∞
∑
n=1
2UW
2
λ
n

sin(nπ/2)
sinh(
λ
n
L)
(sinh(
λ
n
x)-sinh(
λ
n
(L-x)))sin(
λ
n
y)
,
where
λ
n
=nπ/W
. We have found excellent agreement between our numerical solution and the analytical solution.

References

[1] T. Bennett, Transport by Advection and Diffusion, New York: Wiley, 2012.

Permanent Citation

Housam Binous, Ahmed Bellagi
​
​"Steady Potential Flow through Box with Staggered Inlet and Outlet"​
​http://demonstrations.wolfram.com/SteadyPotentialFlowThroughBoxWithStaggeredInletAndOutlet/​
​Wolfram Demonstrations Project​
​Published: August 17, 2015