Lid-driven Cavity Flow.pdf

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).


‣ 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

Section C

(Δt ∂2

∂x2)(Δt ∂

2

∂y2)ψ n = (Δt ∂

2

∂x2)gn (10)

gi, jn =

ΔtΔy2

ψi, j−1n −2ΔtΔy2

ψi, jn +

ΔtΔy2

ψi, j+1n

= −a×ψi, j−1n − (b−1)×ψi, j

n − c×ψi, j+1n

(11)

(12)

(Δt ∂2

∂x2)gi, j

n =ΔtΔx2

gi−1,1n −2ΔtΔx2

gi, jn +

ΔtΔx2

gi+1, jn (13)

= −a× gi−1, jn − (b−1)× gi, j

n − c× gi+1, jn (14)

SectionA = SectionC +ψi, jn +Δtωi, j

n

SectionB = fi, jn+1(afterTDMA)


When i = 0 , j = 1 ⇒ ⇒ ⇒

When i = 1 , j = 1 ⇒ ⇒

When i = 2 , j = 1 ⇒ ⇒

When i = 3 , j = 1 ⇒ ⇒

When i = 4 , j = 1 ⇒ ⇒ ⇒

a0ψg +b0ψL=0,1 + c0ψ1,1 = B.C a0 = 0,b0 =1,c0 = 0 b0ψL=0,1 = B.C

a1ψL=0,1 +b1ψ1,1 + c1ψ2,1 = S1,1 a1ψL=0,1 +b1ψ1,1 + c1ψ2,1 = S1,1a2ψ1,1 +b2ψ2,1 + c2ψ3,1 = S2,1 a2ψ1,1 +b2ψ2,1 + c2ψ3,1 = S2,1a3ψ2,1 +b3ψ3,1 + c3ψ4,1 = S3,1 a3ψ2,1 +b3ψ3,1 + c3ψ4,1 = S3,1a4ψ3,1 +b4ψR=4,1 + c4ψg = S4,1 a4 = 0,b4 =1,c4 = 0 b4ψR=4,1 = B.C

b0 c0 0 0 0a1 b1 c1 0 00 a2 b2 c2 00 0 a3 b3 c30 0 0 a4 b4

!

"

#######

$

%

&&&&&&&

ψL=0,1

ψ1,1ψ2,1

ψ3,1

ψR=4,1

!

"

#######

$

%

&&&&&&&

=

S0,1 = B.CS1,1S2,1S3,1

S4,1 = B.C

!

"

#######

$

%

&&&&&&&

b0 c0 0 0 0a1 b1 c1 0 00 a2 b2 c2 00 0 a3 b3 c30 0 0 a4 b4

!

"

#######

$

%

&&&&&&&

ψL=0,1

ψ1,1ψ2,1

ψ3,1

ψR=4,1

!

"

#######

$

%

&&&&&&&

=

S0,1 = B.CS1,1S2,1S3,1

S4,1 = B.C

!

"

#######

$

%

&&&&&&&

1 c0 /b0 =C[0] 0 0 0a1 b1 c1 0 00 a2 b2 c2 00 0 a3 b3 c30 0 0 a4 b4

!

"

#######

$

%

&&&&&&&

ψL=0,1

ψ1,1ψ2,1

ψ3,1

ψR=4,1

!

"

#######

$

%

&&&&&&&

=

S0,1 /b0 = D[0]S1,1S2,1S3,1S4,1

!

"

#######

$

%

&&&&&&&

1 C[0] 0 0 00 b1 −a1C[0] c1 0 00 a2 b2 c2 00 0 a3 b3 c30 0 0 a4 b4

"

#

$$$$$$$

%

&

'''''''

ψL=0,1

ψ1,1ψ2,1

ψ3,1

ψR=4,1

"

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$$$$$$$

%

&

'''''''

=

S0,1 /b0 = D[0]S1,1 −a1D[0]

S2,1S3,1S4,1

"

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$$$$$$$

%

&

'''''''

1 C[0] 0 0 00 1 c1 / (b1 +a1C[0] )=C[1] 0 00 a2 b2 c2 00 0 a3 b3 c30 0 0 a4 b4

!

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#######

$

%

&&&&&&&

ψL=0,1

ψ1,1ψ2,1

ψ3,1

ψR=4,1

!

"

#######

$

%

&&&&&&&

=

S0,1 /b0 = D[0](S1,1 −a1D[0] / (b1 −a1C[0] )= D[1]

S2,1S3,1S4,1

!

"

#######

$

%

&&&&&&&


1 C[0] 0 0 00 1 C[1] 0 00 0 b2 −a2C[1] c2 00 0 a3 b3 c30 0 0 a4 b4

"

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$$$$$$$

%

&

'''''''

ψL=0,1

ψ1,1ψ2,1

ψ3,1

ψR=4,1

"

#

$$$$$$$

%

&

'''''''

=

S0,1 /b0 = D[0](S1,1 −a1D[0] / (b1 −a1C[0] )= D[1]

S2,1 −a2D[1]S3,1S4,1

"

#

$$$$$$$

%

&

'''''''

1 C[0] 0 0 00 1 C[1] 0 00 0 1 c2 / (b2 −a2C[1] )=C[2] 00 0 a3 b3 c30 0 0 a4 b4

"

#

$$$$$$$

%

&

'''''''

ψL=0,1

ψ1,1ψ2,1

ψ3,1

ψR=4,1

"

#

$$$$$$$

%

&

'''''''

=

S0,1 /b0 = D[0](S1,1 −a1D[0] / (b1 −a1C[0] )= D[1](S2,1 −a2D[1] ) / (b2 −a2C[1] )= D[2]

S3,1S4,1

"

#

$$$$$$$

%

&

'''''''

1 C[0] 0 0 00 1 C[1] 0 00 0 1 C[2] 00 0 0 b3 −a3C[2] c30 0 0 a4 b4

"

#

$$$$$$$

%

&

'''''''

ψL=0,1

ψ1,1ψ2,1

ψ3,1

ψR=4,1

"

#

$$$$$$$

%

&

'''''''

=

S0,1 /b0 = D[0](S1,1 −a1D[0] / (b1 −a1C[0] )= D[1](S2,1 −a2D[1] ) / (b2 −a2C[1] )= D[2]

S3,1 −a[3]D[2]S4,1

"

#

$$$$$$$

%

&

'''''''

1 C[0] 0 0 00 1 C[1] 0 00 0 1 C[2] 00 0 0 1 c3 / (b3 +a3C[2] )=C[3]0 0 0 a4 b4

!

"

#######

$

%

&&&&&&&

ψL=0,1

ψ1,1ψ2,1

ψ3,1

ψR=4,1

!

"

#######

$

%

&&&&&&&

=

S0,1 /b0 = D[0](S1,1 −a1D[0] / (b1 −a1C[0] )= D[1](S2,1 −a2D[1] ) / (b2 −a2C[1] )= D[2](S3,1 −a3D[2] ) / (b3 −a3C[2] )= D[3]

S4,1

!

"

#######

$

%

&&&&&&&


1 C[0] 0 0 00 1 C[1] 0 00 0 1 C[2] 00 0 0 1 C[3]0 0 0 0 b4 −a4C[3]

"

#

$$$$$$$

%

&

'''''''

ψL=0,1

ψ1,1ψ2,1

ψ3,1

ψR=4,1

"

#

$$$$$$$

%

&

'''''''

=

S0,1 /b0 = D[0](S1,1 −a1D[0] / (b1 −a1C[0] )= D[1](S2,1 −a2D[1] ) / (b2 −a2C[1] )= D[2](S3,1 −a3D[2] ) / (b3 −a3C[2] )= D[3]

S4,1 −a4D[3]

"

#

$$$$$$$

%

&

'''''''

1 C[0] 0 0 00 1 C[1] 0 00 0 1 C[2] 00 0 0 1 C[3]0 0 0 0 1

!

"

#######

$

%

&&&&&&&

ψL=0,1

ψ1,1ψ2,1

ψ3,1

ψR=4,1

!

"

#######

$

%

&&&&&&&

=

S0,1 /b0 = D[0](S1,1 −a1D[0] / (b1 −a1C[0] )= D[1](S2,1 −a2D[1] ) / (b2 −a2C[1] )= D[2](S3,1 −a3D[2] ) / (b3 −a3C[2] )= D[3](S4,1 −a4D[3] ) / (b4 −a4C[4 ] )= D[4 ]

!

"

#######

$

%

&&&&&&&

1 C[0] =c0b0

0 0 0

0 1 C[1] =c1

b1 −a1C[0]0 0

0 0 1 C[2] =c2

b2 −a2C[2]0

0 0 0 1 C[3] =c3

b3 −a3C[3]0 0 0 0 1

"

#

$$$$$$$$$$$$

%

&

''''''''''''

ψL=0,1

ψ1,1ψ2,1

ψ3,1

ψR=4,1

"

#

$$$$$$$

%

&

'''''''

=

S0,1b0

= D[0]

S1,1 −a1D[0]b1 −a1C[0]

= D[1]

S2,1 −a2D[1]b2 −a2C[1]

= D[2]

S3,1 −a3D[2]b3 −a3C[2]

= D[3]

S4,1 −a4D[3]b4 −a4C[3]

= D[4 ]

"

#

$$$$$$$$$$$$$$$

%

&

'''''''''''''''

Lid-driven Cavity Flow.pdf

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