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WOLFRAM NOTEBOOK
Steady Potential Flow through Box with Staggered Inlet and Outlet
Steady Potential Flow through Box with Staggered Inlet and Outlet
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 . The aspect ratio of the box is .
U=1
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 ψ+ψ=0 subject to the boundary conditions:
2
∂
∂
2
x
2
∂
∂
2
y
BC1: at , for and for ,
x=0
ψ=0
0⩽y≤W/2
ψ=U(y-W/2)
W/2⩽y≤W
BC2: at , for and for ,
x=L
ψ=Uy
0⩽y≤W/2
ψ=UW/2
W/2⩽y≤W
BC3: at , for , and
y=0
ψ=0
0⩽x≤L
BC4: at , for .
y=L
ψ=UW/2
0⩽x≤L
The solution methodology is based on Chebyshev orthogonal collocation with collocation points.
N=31
This problem also admits an analytical solution given by T. Bennett [1]:
ψ=Uy/2+2UW(sinh(x)-sinh((L-x)))sin(y)
∞
∑
n=1
2
λ
n
sin(nπ/2)
sinh(L)
λ
n
λ
n
λ
n
λ
n
where =nπ/W. We have found excellent agreement between our numerical solution and the analytical solution.
λ
n