Finite speed approximations toNavier-Stokes equations
Roberto Natalini
Istituto per le Applicazioni del Calcolo - CNR
INDAM Workshop on Mathematical Paradigms ofClimate Science, Rome, June 2013
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NASA/JPL’s computational model ”Estimating the Circulationand Climate of the Ocean” a.k.a. ECCO2, a high resolution modelof the global ocean and sea-ice
The incompressible Navier-Stokes equations
Find (U,Φ) : IRD × (0,T )→ IRD × IR s.t.{∂tU + div(U⊗U) +∇Φ = ν∆UdivU = 0
Motivations for a finite speed approximation♣ New and more adapted class of estimates♣ Robust and simple numerical approximations♣ Natural treatment: upwinding, pressure term and thedivergence-free constraint♣ Possible coupling with other equations
Plan of the Talk
I Some methods to solve NS eqs.I Relaxation approximationsI A damped wave equation approximationI Boltzmann eq. vs. NS eqsI The Vector BGK approximationI Comparison with Lattice BGK schemesI Some numerical results
Some Numerical Methods
• Finite Element Methods (FEM): variational formulation• + high computational exibility• + rigorous mathematical error analysis → mesh adaptation• - Difficult control of upwinding phenomena and mass conservation
• - A lot of theoretical work for implementation
• Finite volume methods (FVM): conservation equations• + based on physical conservation properties• - problems on unstructured meshes• - difficult stability and convergence analysis
• - heuristic mesh adaptation
• Spectral Methods• + high order approximation
• - special domains
• Finite difference methods (FDM): direct form• + easy implementation,• - problems along curved boundaries• - difficult stability and convergence analysis
• - mesh adaptation difficult
More on Finite Difference Schemes
Projection methods: Chorin,Temam, Kim & Moin, E &Liu, Bell & Collella & Glaz,....
⇒ Instability problems for thePressure
MAC methods: Harlow &Welsh, T. Hou & Wetton....
⇒Staggered grids: differentlocations for pressure andvelocity
High order: Strikwerda,Kreiss...
⇒ Implicit methods
Original projection method (Chorin, Temam)
{∂tu + div(u ⊗ u) +∇φ = ν∆udiv u = 0
Splitting method based on Hodge decomposition. First step
u∗ = un −∆t (un · ∇un − ν∆un) (1)
Second Stepun+1 = u∗ −∆t∇φn+1 (2)
where φn+1 is computed from u∗ to force the incompressibility ofun+1
div∇φn+1 = ∆φn+1 =1
∆tdiv u∗ (3)
The Hyperbolic Relaxation ApproachA one-slide presentation (not this one!)
A simple relaxation model: hyperbolic and diffusive scalings
• Approximation of∂tu + ∂xA(u) = 0
Hyperbolic scaling ( xε ,tε ), for ε→ 0, and λ > |A′(u)| ⇒
uε → u {∂tu
ε + ∂xvε = 0
∂tvε + λ2∂xu
ε = 1ε (A(uε)− v ε)
• Approximation of
∂tu + ∂xA(u) = λ2∂xxu
Diffusive scaling ( xε ,tε2 ), for ε→ 0, uε → u{
∂tuε + ∂xv
ε = 0
∂tvε + λ2
ε2 ∂xuε = 1
ε2 (A(uε)− v ε)
A simple relaxation model: hyperbolic and diffusive scalings
• Approximation of∂tu + ∂xA(u) = 0
Hyperbolic scaling ( xε ,tε ), for ε→ 0, and λ > |A′(u)| ⇒
uε → u {∂tu
ε + ∂xvε = 0
∂tvε + λ2∂xu
ε = 1ε (A(uε)− v ε)
• Approximation of
∂tu + ∂xA(u) = λ2∂xxu
Diffusive scaling ( xε ,tε2 ), for ε→ 0, uε → u{
∂tuε + ∂xv
ε = 0
∂tvε + λ2
ε2 ∂xuε = 1
ε2 (A(uε)− v ε)
A relaxation approximation of Navier Stokes equations
Y. Brenier, R.Natalini, & M. Puel 2004Let u ∈ IR2 and V ∈ IR4
∂tu
ε + divV ε +∇φε = 0∂tV
ε + 1εν∇u
ε = 1ε (uε ⊗ uε − V ε)
∇ · uε = 0
ε→ 0 ⇒{∂tu + div(u ⊗ u) +∇φ = ν∇uε,∇ · u = 0.
A relaxation approximation of Navier Stokes equations
Y. Brenier, R.Natalini, & M. Puel 2004The same model as a damped Wave equation.
∂tuε + div(uε ⊗ uε) +∇φ = −ε∂ttuε + ν∆uε
∇ · uε = 0
ε→ 0 ⇒{∂tu + div(u ⊗ u) +∇φ = ν∇uε,∇ · u = 0.
A convergence result For all fixed T ≥ 0, let U0 be a smooth divergencefree vector field on T2. Let (uε0,V
ε0 ) be a sequence of smooth initial data
for the relaxation approximation. Assume that there exists C s.t.
||uε0||H1 + ||∂tuε(0, ·)||L2 ≤ C , |uε0|H2 <C0
Ks√ε∫
|uε0(x)− U0(x)|2dx ≤ C√ε
Then, if U is the (smooth) solution of the incompressible Navier Stokesequations with U0 as initial data, we have
supt∈[0,T ]
∫|uε − U|2dx ≤ CT
√ε
Extensions in IR2 and IR3 for less regular initial data in:
• R. Natalini, F. Rousset, Proc. AMS, 2006
• M. Paicu and G. Raugel, ESAIM, Proc.,21:6587, 2007
• I. Hachicha, arXiv:1205.5166v1 May 2013
A convergence result For all fixed T ≥ 0, let U0 be a smooth divergencefree vector field on T2. Let (uε0,V
ε0 ) be a sequence of smooth initial data
for the relaxation approximation. Assume that there exists C s.t.
||uε0||H1 + ||∂tuε(0, ·)||L2 ≤ C , |uε0|H2 <C0
Ks√ε∫
|uε0(x)− U0(x)|2dx ≤ C√ε
Then, if U is the (smooth) solution of the incompressible Navier Stokesequations with U0 as initial data, we have
supt∈[0,T ]
∫|uε − U|2dx ≤ CT
√ε
Extensions in IR2 and IR3 for less regular initial data in:
• R. Natalini, F. Rousset, Proc. AMS, 2006
• M. Paicu and G. Raugel, ESAIM, Proc.,21:6587, 2007
• I. Hachicha, arXiv:1205.5166v1 May 2013
A Kinetic Approach
Goal: a better approximation of the divergence-free constraint
Hydrodynamic limits
The Boltzmann equation in the hyperbolic scaling ( xε ,tε )
∂t fε + ξ · ∇x f
ε =1
εQ(f ε)
ε→ 0
⇓
If f ε → f , then
f (x , t, ξ) =ρ(x , t)
(2πθ(x , t))3/2exp
(−|ξ − u(x , t)|2
2θ(x , t)
)where ρ, u, and θ solve the compressible Euler equations.
Diffusive limits
The Boltzmann equation in the parabolic scaling ( xε ,tε2 )
∂t fε +
1
εξ · ∇x f
ε =1
ε2Q(f ε)
Given the equilibrium configuration M(x , t, ξ) := 1(2π)3/2 exp
(− |ξ|2
2
).
Then
f ε(x , t, ξ) = M(1 + εg) + O(ε2)
where g = ρ+ ξ · u + ( 12 |ξ|
2 − 32 )θ, and
divu = 0, ∇(ρ+ θ) = 0
∂tu + div(u⊗ u) +∇φ = ν∆u
Smooth local solutions: De Masi, Esposito, LebowitzRenormalized solutions: Golse, Saint-Raymond
Diffusive limits
The Boltzmann equation in the parabolic scaling ( xε ,tε2 )
∂t fε +
1
εξ · ∇x f
ε =1
ε2Q(f ε)
Given the equilibrium configuration M(x , t, ξ) := 1(2π)3/2 exp
(− |ξ|2
2
).
Thenf ε(x , t, ξ) = M(1 + εg) + O(ε2)
where g = ρ+ ξ · u + ( 12 |ξ|
2 − 32 )θ, and
divu = 0, ∇(ρ+ θ) = 0
∂tu + div(u⊗ u) +∇φ = ν∆u
Smooth local solutions: De Masi, Esposito, LebowitzRenormalized solutions: Golse, Saint-Raymond
Diffusive limits
The Boltzmann equation in the parabolic scaling ( xε ,tε2 )
∂t fε +
1
εξ · ∇x f
ε =1
ε2Q(f ε)
Given the equilibrium configuration M(x , t, ξ) := 1(2π)3/2 exp
(− |ξ|2
2
).
Thenf ε(x , t, ξ) = M(1 + εg) + O(ε2)
where g = ρ+ ξ · u + ( 12 |ξ|
2 − 32 )θ, and
divu = 0, ∇(ρ+ θ) = 0
∂tu + div(u⊗ u) +∇φ = ν∆u
Smooth local solutions: De Masi, Esposito, LebowitzRenormalized solutions: Golse, Saint-Raymond
The Vector BGK Approach
Relaxation + Kinetic
The vector BGK approximationFirst formulation was made in collaboration with F. Bouchut(uncredited)M.F. Carfora & R. Natalini 2008Y. Jobic, R. Natalini & V. Pavan in preparation
Find f εi ∈ IRD+1s.t.
∂t fεi + 1
ελi · ∇x fεi = 1
τε2 (Mi (ρε, ερuε)− f εi )
f εi (x , 0) = Mi (ρ, ερu0), i = 1, . . . ,N
ρε :=∑N
i=1 fi ,0, ερuεl :=
∑Ni=1 f
εi ,l
System of semilinear hyperbolic equationsMain idea: ρε → ρ,uε → U, where U is a solution of theNavier–Stokes eqs.
Compatibility conditions for the Maxwellian functions
N∑i=1
M0i (ρ,q) = ρ (4)
N∑i=1
M li (ρ,q) =
N∑i=1
λilM0i (ρ,q) = ql (5)
N∑i=1
λijMli (ρ,q) =
qjqlρ
+ P(ρ)δjl (P(ρ) = Cργ)
(6)
τ
N∑i=1
λijλik∑r
∂qrMli (ρ, 0)ur = νδjkul (7)
Expansion in the D + 1 Conservation Laws
Set:
ρε :=N∑i=1
fi ,0, ερuεl :=
N∑i=1
f εi ,l
∂tρ+∑j
∂xj
(N∑i=1
λijεf 0i
)= 0
∂t(ερul) +∑j
∂xj
(N∑i=1
λijεf li
)= 0
l = 1, . . . ,D
Velocity equation
To have a the right limit we need
P(ρ)− P(ρ)
ε2→ε→0 ρΦ⇒ ρ = ρ+ O(ε2)
and using two compatibility conditions and the Taylor expansion ofM
M(ρ, ερu) = M(ρ, 0)+∂ρM(ρ, 0)(ρ−ρ)+∇qM(ρ, 0) ·ερu+O(ε2),
⇒ ∂tu + div(u⊗ u) +∇Φ = ν∆u + O(ε)
Incompressibility equation
If, in the first conservation law, we assume
N∑i=1
λilM0i (ρ,q) = ql
0 = ∂tρ+∑j
∂xj
N∑i=1
λijεM0
i − τ∑j ,k
∂2xjxk
N∑i=1
λijλikM0i + O(ε)
=∑j
∂xj (ρuj) + O(ε)
⇒ divu = O(ε)
Hyperbolic compatibility conditions (1)–(3)
As τ → 0 (ε fixed) ; Isentropic Euler Eqs. (A. Sepe in 2011){∂tρ+ div(ρu) = 0
∂t(ρu) + div(ρu⊗ u) +1
ε2∇P(ρ) = 0
Rmk. ε→ 0 in the isentropic Gas-Dynamics yields (formally) the(incompressible) Euler Eqs.{
∂tU + div(U⊗U) +∇Φ = 0divU = 0
The basic Energy (in)equality
H–Theorem
∂tH(f) + Λ · ∇xH(f) ≤ H(M(Uf ))−H(f) ≤ 0
Bouchut’s Theorem (1999): There exist kinetic entropies if eachM ′i has positive real eigenvalues
⇓
∫ [12ρ|u|
2 + C(γ−1)ε2
(ργ − ργ − γργ−1(ρ− ρ)
)]dx
+ Cε4τ
∫∫|f −M|2dxdt ≤
∫1
2ρ|u0|2dx
The basic Energy (in)equality
H–Theorem
∂tH(f) + Λ · ∇xH(f) ≤ H(M(Uf ))−H(f) ≤ 0
Bouchut’s Theorem (1999): There exist kinetic entropies if eachM ′i has positive real eigenvalues
⇓
∫ [12ρ|u|
2 + C(γ−1)ε2
(ργ − ργ − γργ−1(ρ− ρ)
)]dx
+ Cε4τ
∫∫|f −M|2dxdt ≤
∫1
2ρ|u0|2dx
A 5 velocities scheme in 2DOrthogonal Velocities Model (D. Aregba-Driollet & R. Natalini2003). Setting W = (ρ,q) and
A1(W ) =
(q1,
q21
ρ+ P(ρ),
q1q2
ρ
), A2(W ) =
(q2,
q1q2
ρ,q2
2
ρ+ P(ρ)
)Maxwellian functions in the form
Mi (W ) = aiW +2∑
j=1
bijAj(W )
The velocities are λi = λci ,, for some λ > 0, with
c1 = (1, 0), c2 = (0, 1), c3 = (−1, 0), c4 = (0,−1), c5 = (0, 0)
a1 = · · · = a4 = a,a5 = 1− 4a;
b11 = b22 = −b31 = −b42 = 12λ ,
bij = 0 otherwise.
Consistency, Stability, and Global Existence
♣ The continuous model is consistent if τ = ν2λ2a
♣ The Maxwellian functions are positive and the model has akinetic entropy if the following conditions are verified:
1
4> a >
1
2λ
(√P ′(ρ) + εum
)So it is enough to take λ > 2
(√P ′(ρ) + εum
)♣ Under these conditions, for fixed ε and τ and small initial data,the smooth solution is global in time (Kawashima conditions →Hanouzet-Natalini ARMA 2003)
The fully discrete schemeSolve (using the upwind scheme) a discrete version of
∂t fi + 1ελi · ∇x fi = 0 tn ≤ t < tn+1
fi (x , tn) = f ni (x),(8)
(ρn+1
ερn+1un+1
)=
N∑i=1
fi (tn+1−) and fn+1 = M(ρn+1, ερn+1un+1
)
Consistent with the Navier-Stokes equations (order 2 in space) if
ε = aλ∆xν , ∆t ≤ a(∆x)2
ν
Main idea: the artificial viscosity is used to reconstruct theNavier-Stokes viscosity
The fully discrete schemeSolve (using the upwind scheme) a discrete version of
∂t fi + 1ελi · ∇x fi = 0 tn ≤ t < tn+1
fi (x , tn) = f ni (x),(8)
(ρn+1
ερn+1un+1
)=
N∑i=1
fi (tn+1−) and fn+1 = M(ρn+1, ερn+1un+1
)
Consistent with the Navier-Stokes equations (order 2 in space) if
ε = aλ∆xν , ∆t ≤ a(∆x)2
ν
Main idea: the artificial viscosity is used to reconstruct theNavier-Stokes viscosity
Comparison with the Lattice BGK models:
(McNamara & Zanetti, Higuera & Jimenez, Succi, Benzi, H. Chen,S. Chen, Doolen,.....)Give a set of velocities ci , and a grid such that ∆x = ∆tci
fi (x + ∆tci , t + ∆t) = fi (x , t) +1
τ(Mi − fi )
The D2Q9 lattice
ρ :=∑N
i=1 fi ,q :=∑N
i=1 ci fi
Mi (ρ,q) = Wiρ
{ρ+ 3ci · q− 3
2 |q|2 + 9
2 (ci · q)2}
ConsistencyTo reach consistency with the Navier-Stokes equations, fix ∆x andω = ∆t
τ ∈ (0, 2)⇓
|c | =3ν
∆x
(2ω
2− ω
)∆t = ωτ =
∆x2
3ν
(2− ω
2ω
)
BGK Lattice Boltzmann models vs. Kinetic schemes
• Lattice grids (µ = 1);
• Scalar distribution function (for fixed i);
• No nonlinear stability criteria;
• boundary conditions
; Junk & Klar (2000): finite difference version
ConsistencyTo reach consistency with the Navier-Stokes equations, fix ∆x andω = ∆t
τ ∈ (0, 2)⇓
|c | =3ν
∆x
(2ω
2− ω
)∆t = ωτ =
∆x2
3ν
(2− ω
2ω
)
BGK Lattice Boltzmann models vs. Kinetic schemes
• Lattice grids (µ = 1);
• Scalar distribution function (for fixed i);
• No nonlinear stability criteria;
• boundary conditions
; Junk & Klar (2000): finite difference version
Numerical Validation
in collaboration with V. Pavan and Y. Jobic
(IUSTI, Aix-Marseille Universite)
Lid-driven cavity : Computational domain
u = U, v = 0
wa
ll wa
ll
wall
primary
vortex
top left
vortex
(T)
bottom
left vortex
(BL1)
bottom
right vortex
(BR1)
BL2 BR2
Figure: Setting of the problem
Lid-driven cavity : results 1
Figure: streamlines at Re 400, Nx = Ny= 400
Figure: streamlines at Re 7500, Nx =Ny = 7500
Lid-driven cavity : results 2
y/N
−0.2
0
0.2
0.4
0.6
0.8
1
1.2
−0.2
0
0.2
0.4
0.6
0.8
1
1.2
u/U
−0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 1.2
−0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 1.2
a Re = 100 present work
b Re = 400 present work
c Re = 1000 present work
d Re = 3200 present work
e Re = 5000 present work
f Re = 7500 present work
Re 100 Ghia&al
Re 400 Ghia&al
Re 1000 Ghia&al
Re 3200 Ghia&al
Re 5000 Ghia&al
Re 7500 Ghia&al
a
bcde
f
Figure: u values at the centerline
v/U
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
x/N
−0.2 0 0.2 0.4 0.6 0.8 1 1.2
−0.2 0 0.2 0.4 0.6 0.8 1 1.2
a Re = 100 present work
b Re = 400 present work
c Re = 1000 present work
d Re = 3200 present work
e Re = 5000 present work
f Re = 7500 present work
Re 100 Ghia&al
Re 400 Ghia&al
Re 1000 Ghia&al
Re 3200 Ghia&al
Re 5000 Ghia&al
Re 7500 Ghia&ala
b
c
de
f
Figure: v values at the centerline
Transient couette flow : Computational domain
Wall
UmPeriodic
Periodic
H
Figure: boundary conditions
analytical solution
ux(y , t) = Um
(1− y
H− 2
π
∞∑k=1
1
kexp
(−k2π2
H2νt
)sin
(kπ
Hy
))(9)
Transient couette flow : results 1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 0.2 0.4 0.6 0.8 1
Flu
id v
eloc
ity U
x
y / H
t = 0.125t = 0.5
t = 1t = 2t = 3
Figure: Different time solutions at Re 10
0
5
10
15
20
25
30
35
40
0 0.2 0.4 0.6 0.8 1
Flu
id v
eloc
ity U
x
y / H
t = 0.1t = 1t = 2t = 3t = 4t = 6
Figure: Different time solutions at Re800
Transient couette flow : results 2
L1 relative error
1e−07
1e−06
1e−05
0.0001
1e−07
1e−06
1e−05
0.0001
Number of points
100 200 300 400 500 600 700
100 200 300 400 500 600 700
slope : 2
Figure: Order 2 in space
Backward-Facing Step : Computational domain
H
Um
h
hi
L
x1
x2
x3
Figure: Conditions limites
Backward-Facing Step : results 1
0 5 10 150
1
2
Figure: streamlines at Re 100, Nx = 1000, Ny = 250, ε = 0.016
0 5 10 15 200
1
2
Figure: streamlines at Re 800, Nx = 20000, Ny = 1000, ε = 0.25
Backward-Facing Step : results 1
0 5 10 150
1
2
Figure: streamlines at Re 100, Nx = 1000, Ny = 250, ε = 0.016
0 5 10 15 200
1
2
Figure: streamlines at Re 800, Nx = 20000, Ny = 1000, ε = 0.25
Backward-Facing Step : results 2
2
4
6
8
10
12
14
100 200 300 400 500 600 700 800
x1/h
Re
present workArmalyErturk
Biswas
Figure: attachment point for the firstvortex
6
8
10
12
14
16
18
20
22
24
450 500 550 600 650 700 750 800 850
x/h
Re
X2
X3
present workArmalyErturk
Biswaspresent work
ArmalyErturk
Biswas
Figure: detachment/attachment pointfor the second vortex
Backward-Facing Step : results 2
2
4
6
8
10
12
14
100 200 300 400 500 600 700 800
x1/h
Re
present workArmalyErturk
Biswas
Figure: attachment point for the firstvortex
6
8
10
12
14
16
18
20
22
24
450 500 550 600 650 700 750 800 850x/
hRe
X2
X3
present workArmalyErturk
Biswaspresent work
ArmalyErturk
Biswas
Figure: detachment/attachment pointfor the second vortex
Figure: Landsat 7 image of clouds off the Chilean coast near the JuanFernandez Islands (also known as the Robinson Crusoe Islands)
Figure: Von Karman vortices off the coast of Rishiri Island in Japan
Von Karman streets
click here to open the movie in your browserThe instabilities of the steady flow make the Von karman streets
appear
Conclusions (?) and possible directions
• Hyperbolic approximations furnish a nice framework to studyNSE
• Some analytical problems are still open.
• It is possible to derive simple and effective schemes for NSE.
• 3D dimensions, complex geometries (coming soon...)
• coupling with other equations, multidomains...
Conclusions (?) and possible directions
• Hyperbolic approximations furnish a nice framework to studyNSE
• Some analytical problems are still open.
• It is possible to derive simple and effective schemes for NSE.
• 3D dimensions, complex geometries (coming soon...)
• coupling with other equations, multidomains...
Conclusions (?) and possible directions
• Hyperbolic approximations furnish a nice framework to studyNSE
• Some analytical problems are still open.
• It is possible to derive simple and effective schemes for NSE.
• 3D dimensions, complex geometries (coming soon...)
• coupling with other equations, multidomains...
Conclusions (?) and possible directions
• Hyperbolic approximations furnish a nice framework to studyNSE
• Some analytical problems are still open.
• It is possible to derive simple and effective schemes for NSE.
• 3D dimensions, complex geometries (coming soon...)
• coupling with other equations, multidomains...
Conclusions (?) and possible directions
• Hyperbolic approximations furnish a nice framework to studyNSE
• Some analytical problems are still open.
• It is possible to derive simple and effective schemes for NSE.
• 3D dimensions, complex geometries (coming soon...)
• coupling with other equations, multidomains...
Conclusions (?) and possible directions
• Hyperbolic approximations furnish a nice framework to studyNSE
• Some analytical problems are still open.
• It is possible to derive simple and effective schemes for NSE.
• 3D dimensions, complex geometries (coming soon...)
• coupling with other equations, multidomains...
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