semi-lagrangian approximation of navier-stokes equations
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Semi-Lagrangian Approximation of Navier-Stokes Equations. Vladimir V. Shaydurov Institute of Computational Modeling of SB RAS Beihang University [email protected]. Contents. Approximation in norm. Modified method of characteristics. Convection-diffusion equation. - PowerPoint PPT PresentationTRANSCRIPT
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Semi-Lagrangian Approximation of Navier-Stokes Equations
Vladimir V. Shaydurov
Institute of Computational Modeling of SB RAS
Beihang University
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Contents
• Approximation in norm.
Modified method of characteristics.
Convection-diffusion equation.
• Approximation in norm.
Conservation law of mass.
• Approximation in norm.
Conservation law of energy.
L
1L
2L
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• The main original feature of semi-Lagrangian approach consists in approximation of all advection members
as one “slant” (substantial or Lagrangian) derivative
in the direction of vector
u vdt y
d
t x
d
d l
.l
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First example: convection-diffusion equation
u v ft x y
d
d lf
1
ˆˆ ˆ( , , ),
.ˆ
ˆ ˆ( , , ),k k
dxu t x y
dt t t tdy
v t x ydt
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Approximation of substantial derivative along trajectory approachL
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( )tf
• The equation at each time level became self-adjoint!
Pironneau O. (1982), …Chen H., Lin Q., Shaidurov V.V., Zhou J. (2011), …
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Computational geometric domain
out
inrigid
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Navier-Stokes equations
(0, )fint , , ,u v eIn the cylinder we write 4 equations in unknowns
( )0,xyxxD u P
Dt x x y
( )0,xy yyD v P
Dt y x y
( ).yx
qqD e u vP
t x y x y
( ) ( ) 0u vt
D
t yD x
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Notation
( , , ) is density;t x y
( , , ), ( , , ) are components of the vector velocity ;u t x y v t x y u
( , , ) is internal energy of mass unit;e t x y
( , , ) is pressure;P t x y
1( , , ) ( , , )
Ret x y t x y
is the dynamic coefficient of viscosity:
21 , 0.76 0.9M e
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Notation
, , are the components of the stress tensor (matrix)
:
xx xy yy
xx xy
xy yy
2 22 , 2 ,
3 3
xx yy
xy
u v v u
x y y x
u v
y x
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Notation
, are components of the vector density of a heat flow:x yq q
( , , ) , ( , , )Pr Prx y
e eq t x y q t x y
x y
( 1) .P e
2 2 222 2
.3 3
u v v u u v
x y x y x y
Re is the Reynolds number; Pr is the Prandtl number;
M is the Mach number; is a gas constant
The equation of state has the following form (perfect gas):
The dissipative function is taken in the form:
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How to avoid the Courant-Friedrichs-Lewy restriction for high Reynolds number
approach
( ) ( ) 0u vt
D
t yD x
Curvilinear hexahedron V:
1
ˆˆ ˆ( , , ),
.ˆ
ˆ ˆ( , , ),k k
dxu t x y
dt t t tdy
v t x ydt
Trajectories:
1L
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0V Q
D
DV d
td dQ
Approximation of curvilinear quadrangle Q:
1 1, ,
4, , 2
1( )
k ki j i j
hk i jQ Q
d dQ O h dQh
Due to Gauss-Ostrogradskii Theorem:
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0V Q R
dV d dQ u dDt
RD
1 1, ,
1 1, ,
4
, , 2 2
( )
1 1
k ki j i j
k ki j i j
Q R
hk i j Q R
d dQ u dR O h
dQ u dRh h
Gauss-Ostrogradskii Theorem in the case of boundary conditions:
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( )0xyxxD u P
Dt x x y
( , ) :x y
( , , ) : ( , , ) ( , ),
0
kt x y t x y x y
u vy
d
xdt t
approach
2L
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( )0xyxxD u P
Dt x x y
( ) ( ) ( )D u D u D u du
Dt Dt Dt dt
( )0xyxxD u P
Dt x x y
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( )0xyxx
V
D u PdV
Dt x x y
1k k
xy yy
t t t tQ
v P vd dQ
y x y
1k k
xyxxt t t t
Q
u P ud dQ
x x y
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( ) 10
2yxqD q u v
PDt x y x y
( ) yxqD e q u v
PDt x y x y
1 2e
2( ) yxqD q u v
PDt x y x y
2 22( ) ( ) ( )
2D D D D
Dt Dt Dt Dt
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( )0
2yx
V
qD q u vP dV
Dt x y x y
1
1
2 k k
yxt t t t
Q
qq u vP d dQ
x y x y
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1k k
xyxxt t t t
Q
u P ud dQ
x x y
1k k
xy yy
t t t tQ
v P vd dQ
y x y
1
1
2 k k
yxt t t t
Q
qq u vP d dQ
x y x y
Finite element formulation at each time level
kt t
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The channel with an obstacle at the inlet
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The component of velocity u
The distribution of density
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Conclusion
• Stability of full energy (kinetic + inner)• Approximation of advection derivatives in the frame of
finite element method without artificial tricks• The absence of Courant-Friedrichs-Lewy restriction on the
relation between temporal and spatial meshsizes• Discretization matrices at each time level have better
properties (positive definite)• The better smooth properties and the better approximation
along trajectories
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• Thanks for your attention!