Download Navier-Stokes Equations

18.336 spring 2009
lecture 23
05/05/08
Navier-Stokes Equations
�
ut + (u · �)u = −�p +
1
�2 u
Re
[+g] Momentum equation
�·u=0
Incompressibility
Incompressible flow, i.e. density ρ = constant.
Reynolds number:
Re =
U ·L
inertial forces
=
ν
viscous forces
U = Characteristic velocity
L = Characteristic length scale
ν = Kinetic viscosity
� �
u
in 2D: u =
v
1
(1) ut + uux + vuy = −px + Re
(uxx + uyy )
(2) vt + uvx + vvy = −py +
1
(v
Re xx
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+ vyy )
(3) ux + vy = 0
Famous Problems:
Lid driven cavity
Flow around cylinder
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3 unknowns, 3 equations
DAE (Differential Algebraic System), (3) is a constraint
1
Solve by projection approach:
In each time step
1
2
�u
I. Solve ut + (u · �)u =
Re
U∗ − Un
1 2 n
�
U
= −(U n · �)U n +
Δt
Re
Note: � · U ∗ �
= 0
II. Project on divergence-free velocity field
U n+1 − U ∗
= −�p
Δt
!
What is p : 0 = � · U n+1 = � · U ∗ − Δt�2 P
1
⇒ �2 p =
� · U∗
Poisson equation for pressure
Δt
Discretization:
Solution:
u = v = 0,
p = constant
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But: Central differences
on grid allow solution
Uij = Vij = 0,
�
�
P1
i + j even
for
Pij =
P2
i + j odd
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Fix: Staggered grid
× pressure p
• velocity u
◦ velocity v
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2
Boundary Conditions:
U13 = uw
U21 + U22
= us ⇒ U21 + U22 = 2us
2
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Numerical Method:
I. a) Treat Nonlinear Terms
uux + vuy = (u2 )x + (uv)y
uvx + vvy = (uv)x + (v 2 )y
�
�
∂(U 2 )
∂x
�
=
(Ui+ 1 ,j )2 − (Ui− 1 ,j )2
2
�
=
Uij
2
Δx
ij
∂(U V )
∂y
(use ux + vy = 0)
Ui,j+ 1 Vi,j+ 1 − Ui,j− 1 Vi,j− 1
2
2
2
2
Δy
ij
�
�
Ui+ 1 ,j Vi+ 1 ,j − Ui− 1 ,j Vi− 1 ,j
∂(U V )
2
2
2
2
=
∂x
Δx
ij
�
∂(V 2 )
∂y
�
=
Ui,j+ 1
2
(Vi,j+ 1 )2 − (Vi,j− 1 )2
ij
2
2
�
Δy
�
∂(U V )
∂y
ij
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Ui,j + Ui+1,j
Ui,j + Ui,j+1
,
Ui,j+ 1 =
2
2
2
�
�n
�
�n
∗
n
2
Ui,j − Ui,j
∂(U )
∂(U V )
=−
−
Δt
∂x i,j
∂y
i,j
�
�
�
�n
n
Vi,j ∗ − Vi,j n
∂(U V )
∂(V 2 )
=−
−
Δt
∂x
∂y i,j
i,j
where Ui+ 1 ,j =
2
3
I. b) Implicit Diffusion
U ∗∗ − U ∗
1
=
K2D · U ∗∗
Δt
Re
∗∗
∗
V −V
1
=
K2D ·
V ∗∗
Δt
Re
↑
5 point Laplace stencil with Dirichlet boundary conditions
II. Pressure Correction
�
�
1 ∂U ∗∗ ∂V ∗∗
K2D · P =
+
Δt
∂x
∂y
with Neumann boundary conditions
∂p
= 0
∂n
∂U
∂x
U
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n+1
∗∗
−U
∂P
=−
Δt
∂x
n+1
∗∗
V
−V
∂P
=−
Δt
∂y
U
4
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18.336 Numerical Methods for Partial Differential Equations
Spring 2009
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