Natalini nse slide_giu2013

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

click here to open the movie in your browser

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

http://www.youtube.com/watch?v=CCmTY0PKGDs

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


∂t fε +

1

εξ · ∇x f

ε =1

ε2Q(f ε)


(− |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



Diffusive limits


∂t fε +

1

εξ · ∇x f

ε =1

ε2Q(f ε)


(− |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



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



0 5 10 150

1

2


0 5 10 15 200

1

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


Biswaspresent work

ArmalyErturk

Biswas

Figure: detachment/attachment pointfor the second vortex


2

4

6

8

10

12

14

100 200 300 400 500 600 700 800

x1/h

Re


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


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

http://www.youtube.com/watch?v=GO9n2yY7wFY&feature=share&list=UU3gnDD9awON0EpIVtfXTH7g

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...































Technology

Natalini nse slide_giu2013