A pressure-stabilized characteristic-curve FE scheme
for the Navier-Stokes equations
HN
Nov. 8, 2009
The characteristic-curve method
Nov. 8, 2009
2
The Navier-Stokes eqns.
Find (u, p) :   (0, T )  R d  R s.t.
R
u
 (u  ) u  2 D(u )   p  f
t
Material derivative
 u  0
d
(d  2, 3)
u 0
u(0)  u 0


1
f , u : given, Dij (u )  ui , j  u j ,i .
2
0
Nov. 8, 2009
3
A pressure-stabilized characteristic-curve FEM for the NS eqns.
 uhn  uhn 1  X 1 (uhn1 , t ) 

, vh 

t


Scheme (P1/P1)
The matrix is symmetric
Find (u hn , phn ) n 1  Vh  Qh s.t. for n  1, 2,  , N T ,
 A BT 


n
n 1
n
n
n

Mh (u h , u h ; t ), vh  ah (uh , vh )  bh (vh , ph )  ( f h , vh ), vh  Vh ,  B C 
NT
u

 (u  )u
t
bh (uhn , qh ) Ch ( phn , qh )  0,
qh  Qh ,
Pressure stabilization
Vh and Qh : P1- finite element spaces,
uh0 : an approximat ion of u 0 , X 1 ( w, t )( x)  x  w( x)t , f hn   h f (, nt ),
 u  w  X 1 ( w, t ) 
M h (u , w; t ), vh  
, vh ,
t


ah (u , v)  2 D(u ), D(v) , bh (v, q )    v, q , Ch ( p, q )  
  0, hK  diam( K ),   ,  K : L2 ( K ) d inner product.
2
h
 K p, q K ,
K T h
K
Nov. 8, 2009
4
Numerical results
The Navier - Stokes eqns. :
(u  )u  2D(u )   p  f

 u  0


ug

(1,1,1)
in ,
in ,
on .
(0,0,0)
g1 ( x1, x2 , 1)
A regularize d cavity flow problem (Re  100, 400, 1,000)
0.5, 0.5,1
  (0, 1) 3 ,   10  2 , 2.5 10 3 , 10 3 , f  0,
x2
Dirichlet BC :
16 x1 (1  x1 ) x2 (1  x2 ) ( x3  1)
g1 ( x1 , x2 , x3 )  
,
0
(
otherwise
)

1
O
1
x1
g 2  g 3  0.
P1/P1
Nov. 8, 2009
Solver: CR
5
A regularized cavity flow, Re=1,000
N  64
DOF : 691,860
(1,1,1)
t 
1
32
Ne : 972,288
(0,0,0)
Velocity (evolution)
Nov. 8, 2009
Pressure at stationary state
6
Re=1,000
N  64
DOF : 691,860
Ne : 972,288
t 
1
32
Nov. 8, 2009
7
The graphs of u1 (0.5,0.5, ) and u3 (  ,0.5,0.5)
(1,1,1)
(0,0,0)
x2  0.5
Nov. 8, 2009
8