7. Primitive variable formulation
i) A mixed elliptic-parabolic equn. ( two unknowns : pressure, velocity )
ii) no direct link for the pressure between continuity & momentum equn.
Poisson equn. for pressure
Vorticity-Stream function formulation
i) vorticity-stream function formulation does not include the pressure term.
the velocity field is determined initially
For the pressure field, Poisson equn.
ii) The lack of al simple stream function in three diminsions
8. Poisson Equation for Pressure
PrimitiveVariables
∂u ∂ 2 ∂p ∂ 1
+ (u ) + + ( uv ) = ( ∇2 u )
∂t ∂x ∂x ∂y Re
∂v ∂ ∂ ∂p 1
+ ( uv ) + ( v 2 ) + = ( ∇2 v )
∂t ∂x ∂y ∂y Re
∂u ∂u ∂ 2 ∂2 p ∂2 1 ∂
( )
+ 2 u2 + 2 +
∂t ∂x ∂x ∂x ∂x∂y
( uv ) =
Re ∂x
∇2 u ( )
addition
∂v ∂ ∂ ∂2 2 ∂2 p 1 ∂
+
∂t ∂y ∂x∂y ∂y
( )
( uv ) + 2 v + 2 =
∂y Re ∂y
∇2 v ( )
∂ ∂u ∂v ∂ 2 ∂2 ∂2 2 ∂2 p ∂2 p
+ ÷+ 2 u + 2
∂t ∂x ∂y ∂x
2
( )
∂x∂y
( uv ) + 2 v + 2 + 2
∂y ∂x ∂y
( )
1 ∂ ∂v 2
= ( ) (
∂x ∇ u + ∂y ∇ v
Re
2
) 0 continuity
equn.
9. ∂2 p ∂2 p ∂D ∂ 2 ∂2 ∂2 2
∂x 2
+ 2 =−
∂y
− 2 u2 − 2
∂t ∂x
( )
∂x∂y
( uv ) − 2 v
∂y
( )
1 ∂2 ∂2
+ 2 ( D) + 2 ( D)
Re ∂x ∂y
∂u ∂v
D= +
∂x ∂y
For an incompressible flow, D=0
However, due to numerical consideration, this term will not be
set to zero
It must be evaluated to prevent error accumulation, as well as
to prevent nonlinear instability