SIMPLE Method on Non-staggered Grids - Department of ...

⎧ + +
⎡fe ( aP) uE+ ( 1−
fe )( aP)
u ⎤
P
⎪⎣
E
P ⎦
⎪ ⎡ + + +
( fe ( sc) ( 1 fe )( sc)
) δ xe y fe ( sc)
x
E P E E
y⎤
⎪
+ − Δ − Δ Δ
α ⎢
⎥
⎪+ u ⎢ +
( 1 fe )( sc)
x
P P
y
⎥
⎪ ⎣
− − Δ Δ
⎦
1
⎪
u = ⎨+ α ⎡ y p p f y p p f y p p
E
( aP
) ⎣
−Δ − + Δ − + − Δ −
e ⎪
n− 1 + n− 1 + n−1
⎪+ ( 1−αu) ⎡( aP) ue − fe ( aP) uE −( 1−
fe )( aP)
u ⎤
e E P
P
⎪ ⎣
⎦
⎪ o o + o o + o
+ α ⎡
o
u
au
e e
− fe ( aP) uE −( 1−
fe )( aP)
u ⎤
⎪ ⎣
E
P
P⎦
⎪
⎩ ⎪
+ +
( ) ( ) ( 1 ) ( )
e u E P e e w e e w
where ( a )
P
e
is found from equation (26). It should be noted that the term in the
parenthesis, which is multiplied by (1–α u ) is incorrect in the paper of Yu et al.’s (2002).
It should also be noted that the cell face velocities found from the momentum
interpolation method are used to determine the mass fluxes across the cell faces.
They should not be used for the cell face value of the independent variable φ in the
deferred correction term b dc in eq. (9) in the case φ stands for u or v in the x- or y-
momentum equations. The face values of the independent variable φ are calculated
using a suitable convection scheme, such as UPWIND or QUICK.
P
⎫
⎪
⎪
⎪
⎪
⎪
⎤
⎪
⎦⎬
⎪
⎪
⎪
⎪
⎪
⎪
⎪ ⎭
(27)
The <strong>SIMPLE</strong> Algorithm
For a guessed pressure field p * the corresponding face velocity
Eq. (22) as
( )
( )
* * *
o
* u i i i p
e u E P n−1
u e o
e ( 1 αu)
e e
( aP)
( aP)
( aP)
e e e
*
u
e
can be written using
au b y p p a
u α ∑ + α Δ −
= − + − u + α u
(28)
*
A similar equation can be written for the face velocity v
n
. Now we define the correction
p' as the difference between the correct pressure field p and the guessed pressure field p *
so that
*
p = p + p′
(29)
Similarly we define velocity corrections u' and v' as
u = u + u′
(30)
*
e e e
9