Session B.pdf - Clarkson University

TRANSFORMATION AND COMPUTATION OF BASIC EQUATIONS
Let ξx
= ξx
⋅ J , ξ
y
= ξ
y
⋅ J , η
x
= ηx
⋅ J , η
y
= ηy
⋅ J , uξ = uξ
x
+ vξ
y
, vη = uηx
+ vηy
,
qξ
= q x
ξ
x
+ qvξ
y
, qη = q x
ηx
+ qy
ηy
, in which uξ
and are velocities in ξ and η
directions respectively. The governing equations under BFC can be deduced through
coordinate transformation of equations (1) − (3):
∂
∂
h
t
1 ∂ qξ
+
J ∂ ξ
+
1
J
∂ q
η
∂ η
= 0 , (16)
∂q
∂t
x
+
1
J
∂
∂ξ
qxq
(
h
ξ
+ ξ
x
gh
2
2
) +
1
J
∂
∂η
qxq
(
h
η
+ η
x
2
gh
) = bx
, (17)
2
∂q
y
∂t
+
1
J
∂
∂ξ
qxq
(
h
ξ
+ ξ
y
gh
2
2
) +
1
J
∂
∂η
qyq
(
h
η
+ η
y
gh
2
2
) = b . (18)
y
The boundary conditions can be expressed as A ϕ + B∂ϕ
/ ∂n
= C with A, B, and C
given, ∂ ϕ / ∂n is normal derivative at boundary, ϕ is any variable desired, gradient of
ρ
function f is ∇f
= ( fξξx
+ fηηx
) i + ( fξξ
y
+ fηηy
)j . The transformation equations of
boundary conditions can be obtained as:
∂ϕ
∂n
( ξ)
=
J
1
q
11
( q
11
∂ϕ
+ q
∂ξ
12
∂ϕ ∂ϕ 1 ∂ϕ ∂ϕ
) = ( q12
+ q22
)
( η)
∂η ∂n
J q ∂ξ ∂η
22
, (19)
in which: q
2 2 2 2
ξx + ξy
= xη
+ = α q 12
= ξxηx
+ ξyηy
= −β
q
22
= ηx
+ ηy
= γ .
11
= yη
As illustrated in Fig.4, ABCD is control volume, (i, j) is control node,
= ( h,
q , q ) T
2
2
, F ( q , q / h + gh / 2, q q / h) T
area. Let U
x y
=
x x
x y
,
2
2 T
G = ( q , q q / h,
/ h + gh / 2)
, then equations (16) − (18) give:
where
y
x y
q y
U
= U
∆ t
− ( ) ( H
∆ξ ∆η
+ H
+ H
+ H
n+
1 n
i, j i,
j
AB BC CD DA
)
2
∆ξ∆η
2
is control
, (20)
H = F AB
∆η
, H = G BC
∆ξ
, H −F
∆η, H = H DA
∆ξ
. The two-step
AB
BC
CD
=
CD
MacCormack scheme is used with stagger difference scheme. Boundary conditions
include incoming flow, outflow and non-penetrable wall condition u ⋅ n = 0 .
¦З
DA
j+1
j
j-1
C
D
(i,j)
B
A
i-1 i i+1
¦О
Fig. 4. Discrete scheme
188