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Ljd-drjven Cavjty Flow
‣ Domain is a “uniform” grid n×n
j = 4
0,4 1,4 2,4 3,4 4,4
j = 3
0,3 1,3 2,3 3,3 4,3
j = 2
0,2 1,2 2,2 3,2 4,2
j = 1
0,1 1,1 2,1 3,1 4,1
j = 0
0,0 1,0 2,0 3,0 4,0
i = 0 i = 1 i = 2 i = 3 i = 4
�ψ[0][1]
�ψ[2][2]
�ψ[4 ][3]
�ψ[3][0]
�ψ[1][3]
�ψ[3][4 ]
�ψ[0][0]
�ψ[2][1]
�ψ[4 ][2]
�ψ[0][4 ]
�ψ[1][2]
�ψ[3][3]
�ψ[2][0]
�ψ[4 ][1]
�ψ[0][3]
�ψ[2][4 ]
�ψ[1][1]
�ψ[3][2]�ψ[0][2]
�ψ[2][3]
�ψ[4 ][4 ]
�ψ[4 ][0]�ψ[1][0]
�ψ[3][1]
�ψ[1][4 ]
Update boundary condition to zero again when solving the RHS in section C, term (13)(14).
Ljd-drjven Cavjty Flow
‣ Stream function
(1−Δt ∂2
∂x2)(1−Δt ∂
2
∂y2)ψ n+1 =ψ n +Δtω n + (Δt ∂
2
∂x2)(Δt ∂
2
∂y2)ψ n
(1−Δt ∂2
∂x2)× f =ψ n +Δtω n + (Δt ∂
2
∂x2)(Δt ∂
2
∂y2)ψ n
f = (1−Δt ∂2
∂y2)ψ n+1
(1)
(2)
(3)
// Expand RHS with (△t×∂2/∂x2) (△t×∂2/∂y2) first
// Recall (△t×∂2/∂x2) gn
// Solve equation with TDMA
(1−Δt ∂2
∂x2)× f = fi, j
n+1 −ΔtΔx2
( fi−1, jn+1 −2 fi, j
n+1 + fi+1, jn+1)
= −ΔtΔx2
fi−1, jn+1 + (1+ 2Δt
Δx2) fi, j
n+1 −ΔtΔx2
fi+1, jn+1
(4)
(5)
= a× fi−1, jn+1 +b× fi, j
n+1 + c× ΔtΔx2
fi+1, jn+1 (6)
Section A // Expand (1-△t×∂2/∂x2) term on LHS
(1−Δt ∂2
∂y2)×ψi, j
n+1 =ψi, jn+1 −
ΔtΔy2
(ψi, j−1n+1 −2ψi, j
n+1 +ψi, j+1n+1) (7)
= −ΔtΔy2
ψi, j−1n+1 + (1+ 2Δt
Δy2)ψi, j
n+1 −ΔtΔy2
ψi, j+1n+1 (8)
= a×ψi, j−1n+1 +b×ψi, j
n+1 + c× ΔtΔy2
ψi, j+1n+1 (9)
Section B // Expand (1-△t×∂2/∂y2) term of “f ” on LHS