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A pressure-stabilized characteristic-curve FE scheme for the Navier-Stokes equations. HN. The characteristic-curve method. The Navier-Stokes eqns. Material derivative. A pressure-stabilized characteristic-curve FEM for the NS eqns. Scheme (P1/P1). The matrix is symmetric. - PowerPoint PPT Presentation
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Nov. 8, 2009
A pressure-stabilized characteristic-curve FE schemefor the Navier-Stokes equations
HN
Nov. 8, 2009 2
The characteristic-curve method
Nov. 8, 2009 3
The Navier-Stokes eqns.
0
)(2)(
u
fpuDuut
u 0u
dR )3,2( d
Material derivative
s.t.Find RR dTpu ),0(:),(
0)0( uu
.2
1)(:, ,,
0ijjiij uuuDuf given,
Nov. 8, 2009 4
A pressure-stabilized characteristic-curve FEM for the NS eqns.
product. innerdiam
of ionapproximat an
spaces element finiteP1- and
for s.t.Find
dKK
KKKhhh
hhh
hnhh
hh
hhhnhhh
nhh
hhhnh
nhhhh
nhhh
nh
nhh
Thh
N
nnh
nh
KLKh
qphqpqvqvbvDuDvua
vt
twXwuvtwu
tnfftxwxxtwXuu
QV
Qqqpqub
Vvvfpvbvuavtuu
NnQVpu
h
T
)(:,),(,0
,,),(,),(,)(),(2),(
,,),(
),;,(
),,(,)())(,(,:
,:
,,0),(),(
,),,(),(),(),;,(
,,,2,1),(
2
2
1
100
1
1
T
C
M
C
M
,
Scheme (P1/P1)
K
uut
u)(
Pressure stabilization
h
nh
nh
nh v
t
tuXuu,
),( 11
1
The matrix is symmetric
CB
BA T
Nov. 8, 2009 5
.
,0
,)(2)(
on
in
in
:eqns. Stokes-Navier The
gu
u
fpuDuu
.0
,)(0
)1()1()1(16),,(
,0,10,105.2,10,)1,0(
)000,1,400,100(Re
32
322113211
3323
gg
xxxxxxxxg
f
otherwise
:BC Dirichlet
problemflowcavity dregularizeA
Numerical results
P1/P1
)0,0,0(
)1,1,1(
Solver: CR
1x
)1,,( 211 xxg
O
1,5.0,5.0
1
1
2x
1x
)1,,( 211 xxg
O
1,5.0,5.0
1
1
2x
Nov. 8, 2009 6
A regularized cavity flow, Re=1,000
Velocity (evolution) Pressure at stationary state
)0,0,0(
)1,1,1(64N
288,972
860,691
:Ne
:DOF 32
1t
Nov. 8, 2009 7
Re=1,000
64N
288,972
860,691
:Ne
:DOF
32
1t
Nov. 8, 2009 8
)5.0,5.0,(),5.0,5.0( 31 uu and of graphs The
5.02 x
)0,0,0(
)1,1,1(
