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

v = v + v′
(31)
*
n n n
Subtracting Eq. (28) from Eq. (22) gives
e
( ′ )
( a )
P
e
( − )
( a )
αu ∑
i
au
i i
+ bp α
e uΔy p′ E
p′
P
u′ = − (32)
P
e
As an approximation, in <strong>SIMPLE</strong> method the first term in the above equation is neglected
giving
( )
u′ = d p′ − p′
(33)
u
e e P E
where
d
α A
= (34)
u u e
e
( aP
)
e
where A e = Δy is the area of the control volume at the east face. Similarly,
( )
v′ = d p′ − p′
(35)
v
n n P N
where
d
α A
= (36)
v v n
n
( aP
)
n
Then the corrected velocities become
( ′ ′ )
u = u + d p − p
(37)
* u
e e e P E
( ′ ′ )
v = v + d p − p
(38)
* v
n n n P N
Substituting the corrected face velocities such as that given by Eq.s (37)and (38) into the
discretized continuity equation (7) gives
a p′ = a p′ + a p′ + a p′ + a p′
+ b
(39)
P P W W E E S S N N
where
a = ( ρ Ad) a = ( ρAd) a = ( ρAd) a = ( ρAd)
E e W w N n S s
* * * *
( ρ ) ( ρ ) ( ρ ) ( ρ )
b= u A − u A + v A − u A
w e s n
After solving the p' field from Eq. (39) the face velocities are corrected using Eq.'s (37)
and (38) and the pressure field is corrected by using
= + α ′
(41)
*
p p
p
p
where α p is the pressure under-relaxation factor which is chosen to be between 0 and 1.
(40)
10