The document discusses various formulations of the incompressible Navier-Stokes equations. It presents the equations in dimensional and non-dimensional forms using primitive variables, vorticity-stream function formulations, and the Poisson equation for pressure. The primitive variable formulation results in a coupled system for pressure and velocity while the vorticity-stream function formulation avoids including pressure directly but requires solving a Poisson equation to determine pressure from the velocity field.

1. Incompressible Navier-Stokes
Equation
2004-4-18

2. Non-dimensionalization
∂ui ∂ui 1 ∂p ∂2ui
+u j =− +υ (Navier-Stokes Eq.)
∂t ∂x j ρ ∂xi ∂x j 2
∗ xi tu0
∗ ∗ tυ ∗ ui ∗p u0 L
x = t = (t = 2 ) u = p = Re =
i
L L L
i
u0 ρu02
υ
(
∂ ui∗u0 ) + u ∗u0
( ) = − 1 ∂ ( p ρu ) + ∂ ( u u )
∂ ui∗u0 ∗ 2
0
2 *
i 0
(
∂ t ∗ L / u0 ) j
∂( x L)
∗
j
ρ ∂( x L) ∗
i
(∂x L) *
i
2
∂ui* ∂ui* ∂p* 1 ∂2ui*
+u * * = − * +
j
∂t * ∂x j ∂xi Re ∂x*
j
∂ui ∂ui ∂p 1 ∂2ui
∴ +u j =− + (non-dimensionalization)
∂t ∂x j ∂xi Re ∂x j 2

3. Primitive Variable Formulations
 Two dimensional (CONSERVATIVVE FORM)
∂u ∂v
+ =0
∂x ∂y
∂u ∂  2 P ∂  ∂2u ∂2u 
+ u + ÷ + ( uv ) =υ 2 + 2 ÷
∂t ∂x  ρ  ∂y  ∂x ∂y 
∂v ∂ ∂  2 P  ∂2 v ∂2 v 
+ ( uv ) + v + ÷=υ 2 + 2 ÷
∂t ∂x ∂y  ρ  ∂x ∂y 
 Non-dimensional (CONSERVATIVVE FORM)
∂u ∗ ∂v∗
∗
+ ∗ =0
∂x ∂y
∂u ∗ ∂ ∂ 1  ∂2u ∗ ∂2u ∗ 
∗
+ ∗ ( u ∗2 + P ∗ ) + ∗ ( u ∗v∗ ) =  ∗2 + ∗2 ÷
∂t ∂x ∂y Re  ∂x ∂y 
∂v∗ ∂ ∂ 1  ∂2 v∗ ∂2 v∗ 
∗
+ ∗ ( u v ) + ∗ ( v +P ) =
∗ ∗ ∗2 ∗
 ∗2 + ∗2 ÷
∂t ∂x ∂y Re  ∂x ∂y 

4. Vorticity-Stream Function Formulation
• Vorticity ; Ω= 2ϖ =∇×v
∂v ∂u
→For a two dimensional flow Ωz = −
∂x ∂y
∂ψ ∂ψ
•Stream function; u = , v =−
∂y ∂x
∂u ∂u ∂u 1 ∂p  ∂2 u ∂2 u 
+u +v =− +υ  2 + 2 ÷
∂t ∂x ∂y ρ ∂x  ∂x ∂y 
∂v ∂v ∂u 1 ∂p  ∂2 v ∂ 2 v 
+u +v =− +υ  2 + 2 ÷
∂t ∂x ∂y ρ ∂y  ∂x ∂y 
∂2u ∂u ∂u ∂2u ∂v ∂u ∂2 u 1 ∂2 p  ∂3u ∂3u 
+ +u + +v 2 = − +υ  2 + 3 ÷
∂t ∂y ∂y ∂x ∂x∂y ∂y ∂y ∂y ρ ∂x∂y  ∂x ∂y ∂y 
∂2 v ∂u ∂v ∂2 v ∂v ∂v ∂v 1 ∂2 p  ∂3u ∂3u 
+ +u 2 + +v =− +υ  3 + ÷
∂t ∂x ∂x ∂x ∂x ∂x ∂y ∂x∂y ρ ∂x∂y  ∂x ∂x∂y 2 

5. O
∂  ∂u ∂v  ∂  ∂u ∂v  ∂  ∂u ∂v   ∂u ∂v  ∂u ∂v 
 − ÷+ u  − ÷+ v  − ÷+  + ÷ − ÷
∂t  ∂y ∂x  ∂x  ∂y ∂x  ∂y  ∂y ∂x   ∂x ∂y  ∂y ∂x 
 ∂2  ∂u ∂v  ∂2  ∂u ∂v  Continuity eqn.
=υ  2  − ÷+ 2  − ÷ 
 ∂x  ∂y ∂x  ∂y  ∂y ∂x 
∂Ω ∂Ω ∂Ω  ∂ 2 Ω ∂2 Ω 
+u +v =υ  2 + 2  Vorticity transport eqn.
∂t ∂x ∂y  ∂x ∂y 
∂2 Ψ ∂2 Ψ
−Ω = + Stream function eqn.
∂x 2
∂y 2
L Ψ
nondimensional form Ω∗ = Ω Ψ∗ =
u∞ u∞ L

6. Vorticity-Stream Function Formulations
Dimensional CONSERVATIVVE FORM
∂Ω ∂ ∂  ∂2 Ω ∂2 Ω 
+ (uΩ) + (vΩ) = υ  2 + 2 
∂t ∂x ∂y  ∂x ∂y 
∂2 Ψ ∂2 Ψ
−Ω = +
∂x 2
∂y 2
 Non-dimensional (CONSERVATIVVE FORM)
∂Ω∗ ∂ ∗ ∗ ∂ ∗ ∗ 1  ∂2Ω∗ ∂2 Ω∗ 
∗
+ ∗ (u Ω ) + ∗ (v Ω ) =  ∗2 + ∗2 
∂t ∂x ∂y Re  ∂x ∂y 
∗ ∂2 Ψ∗ ∂2 Ψ∗
−Ω = ∗2
+
∂x ∂y ∗2

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

10. Poisson Equation for Pressure
Vorticity-Stream function formulation
∂2 p ∂2 p  ∂ 2ψ  ∂ 2ψ   ∂ψ 
+ 2 = 2  2 ÷ 2 ÷−  ÷
∂x 2
∂y  ∂x  ∂y   ∂x∂y 