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

Similarly the nodal velocities are corrected using
( ′ ′)
u = u + d p − p
(42)
* u
P P P w e
( ′ ′)
v = v + d p − p
(43)
* v
P P P s n
where
d
u
P
αuAe v αvAn
= and d = (44)
P
( a ) ( a )
P
P
P
P
The pressure corrections at the cell faces in Eqs. (42) and (43) are calculated by linear
interpolation from the nodal values as
+ +
p′ = f p′ + (1 − f ) p′
(45)
w w P w W
+ +
p′ = f p′ + (1 − f ) p′
(46)
e e E e P
+ +
p′ = f p′ + (1 − f ) p′
(47)
s s P s S
+ +
p′ = f p′ + (1 − f ) p′
(48)
n n N n P
Boundary Conditions for Pressure
Since there is no equation for the pressure, no boundary conditions are needed for the
pressure at the boundary points. The pressure values at the boundary points can be
calculated by linear extrapolation using the two near-boundary node pressures.
Boundary Conditions for Pressure Correction Equation
When the velocities at the boundaries are known, there is no need to correct the velocities
at the boundaries in the derivation of the pressure correction equation. For example if the
velocity at the west boundary is known then for a control volume near the west boundary:
( ′ ′ )
u = u + d p − p
(49)
u
* u
e e e P E
w
= u
(50)
wall
( ′ ′ )
v = v + d p − p
(51)
* v
n n n P N
( ′ ′ )
v = v + d p − p
(52)
* v
s s s S P
Substituting equations (49)-(52) into the discretized continuity equation (7) we obtain the
following pressure correction equation for a control volume near the west boundary
a p′ = a p′ + a p′ + a p′ + a p′
+ b
(53)
P P W W E E S S N N
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