A new adaptative numerical method for kinetic equation St ephane - - PowerPoint PPT Presentation

BGK Model Standard method Presentation of the method (1D) 2D method Conclusions

A new adaptative numerical method for kinetic equation

St´ ephane Brull, Louis Forestier-Coste, Luc Mieussens MNMCFF2014 - Beijing May 21-27, 2014

Forestier-Coste local velocity grid 1/ 49

BGK Model Standard method Presentation of the method (1D) 2D method Conclusions

Introduction

rarefied gas flow Deterministic simulations (DVM) of Boltzmann equation (BGK) A global discrete velocity grid in space and time For practical applications in aerodynamics (atmospheric re-entry problems), the grid is so large that the computational ressources (memory storage and CPU time) require by the simulation are huge Project : construction of local velocity grids (unsteady flows)

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BGK Model Standard method Presentation of the method (1D) 2D method Conclusions

1 BGK Model 2 Standard method 3 Presentation of the method (1D) 4 2D method 5 Conclusions

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BGK Model Standard method Presentation of the method (1D) 2D method Conclusions

1 BGK Model 2 Standard method 3 Presentation of the method (1D) 4 2D method 5 Conclusions

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BGK Model Standard method Presentation of the method (1D) 2D method Conclusions

f (t, x, v) distribution function : mass of particles at time t, at position x ∈ Ω ⊂ RD with a velocity of v ∈ RD.

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BGK Model Standard method Presentation of the method (1D) 2D method Conclusions

f (t, x, v) distribution function : mass of particles at time t, at position x ∈ Ω ⊂ RD with a velocity of v ∈ RD. mass density ρ(t, x) =

• RD f (t, x, v)dv

Forestier-Coste local velocity grid 5/ 49

BGK Model Standard method Presentation of the method (1D) 2D method Conclusions

f (t, x, v) distribution function : mass of particles at time t, at position x ∈ Ω ⊂ RD with a velocity of v ∈ RD. mass density ρ(t, x) =

• RD f (t, x, v)dv

velocity u(t, x) = 1 ρ(t, x)

• RD v f (t, x, v)dv

Forestier-Coste local velocity grid 5/ 49

BGK Model Standard method Presentation of the method (1D) 2D method Conclusions

f (t, x, v) distribution function : mass of particles at time t, at position x ∈ Ω ⊂ RD with a velocity of v ∈ RD. mass density ρ(t, x) =

• RD f (t, x, v)dv

velocity u(t, x) = 1 ρ(t, x)

• RD v f (t, x, v)dv

energy E(t, x) =

• RD

v2 2 f (t, x, v)dv

Forestier-Coste local velocity grid 5/ 49

BGK Model Standard method Presentation of the method (1D) 2D method Conclusions

f (t, x, v) distribution function : mass of particles at time t, at position x ∈ Ω ⊂ RD with a velocity of v ∈ RD. mass density ρ(t, x) =

• RD f (t, x, v)dv

velocity u(t, x) = 1 ρ(t, x)

• RD v f (t, x, v)dv

energy E(t, x) =

• RD

v2 2 f (t, x, v)dv temperature T(t, x) = 1 DR 2E(t, x) ρ(t, x) − u(t, x)2

• Forestier-Coste

local velocity grid 5/ 49

BGK Model Standard method Presentation of the method (1D) 2D method Conclusions

f (t, x, v) distribution function : mass of particles at time t, at position x ∈ Ω ⊂ RD with a velocity of v ∈ RD. mass density ρ(t, x) =

• RD f (t, x, v)dv

velocity u(t, x) = 1 ρ(t, x)

• RD v f (t, x, v)dv

energy E(t, x) =

• RD

v2 2 f (t, x, v)dv temperature T(t, x) = 1 DRρ(t, x)

• RD v − u(t, x)2f (t, x, v)dv

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BGK Model Standard method Presentation of the method (1D) 2D method Conclusions

Notations

collisional invariants : m(v) =

• 1, v, 1

2v2T

moments : ρ = (ρ, ρu, E)T < g >=

• RD gdv

ρ =< mf >

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BGK Model Standard method Presentation of the method (1D) 2D method Conclusions

BGK model

∂tf + v.∇xf = Q(f ) Q(f ) = 1 τ (M[ρ,u,T] − f ) + boundaries conditions

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BGK Model Standard method Presentation of the method (1D) 2D method Conclusions

Maxwellian

Maxwellian M[ρ,u,T](t, x, v) : solution of an entropy minimisation problem (H(f ) =< f ln(f ) >). H(M[ρ,u,T]) = min{H(g), g ≥ 0 s.t. < mg >= ρ} M[ρ,u,T](t, x, v) = ρ(t, x) (2πRT(t, x))

D 2

exp

• − |v − u(t, x)|2

2RT(t, x)

• Forestier-Coste

local velocity grid 8/ 49

BGK Model Standard method Presentation of the method (1D) 2D method Conclusions

1 BGK Model 2 Standard method 3 Presentation of the method (1D) 4 2D method 5 Conclusions

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BGK Model Standard method Presentation of the method (1D) 2D method Conclusions

Discretization

Physical model: ∂tf + v · ∇xf = Q(f ) + BCs Velocity space RD V =

• vk, k ∈ ND

set of discrete velocities Discrete kinetic equation: ∂tfk +vk ·∇xfk = Qk(f ) +BCs

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BGK Model Standard method Presentation of the method (1D) 2D method Conclusions

Velocity grid?

Cartesian grid V:

V = 8 < :vl,m,n = vmin + @ l∆vx m∆vy n∆vz 1 A , @ l m n 1 A = @ 0 : lmax 0 : mmax 0 : nmax 1 A 9 = ;

Constraints: bounds and grid step : the grid must be

large enough fine enough

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BGK Model Standard method Presentation of the method (1D) 2D method Conclusions

How to design a cartesian velocity grid?

Information on the support of f (x, .): assuming f close to its local Maxwellian, then centered on u(x) with standard deviation

• RT(x)

u(x) − c

• RT(x)

u(x) u(x) + c

• RT(x)

2

• RT(x)

The grid contains all the distributions if the bounds are at least vmin = min

x∈Ω{u(x) − c

• RT(x)}

vmax = max

x∈Ω{u(x) + c

• RT(x)}

(c around 4)

At least 3 points into the ”core” of each distribution. The grid step should be: ∆v ≤ min

x∈Ω{

• 2RT(x)}

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BGK Model Standard method Presentation of the method (1D) 2D method Conclusions

Estimation of the bounds and the step of the grid

Temperature and velocity fields are a priori unknown! several solutions :

use the boundaries values: upstream flow (u∞, T∞) and velocity and temperature of the body (uwall, Twall) Flow parameter inside the shock: ushock and Tshock estimated by Rankine-Hugoniot relations. Even better: A compressible Navier-Stokes pre-simulation :

uNS and T NS in Ω let vmin = min

x∈Ω{uNS(x) − c

p RT NS(x)} vmax = max

x∈Ω {uNS(x) + c

p RT NS(x)} ∆v = a min

x∈Ω{

p 2RT NS(x)}

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BGK Model Standard method Presentation of the method (1D) 2D method Conclusions

Drawback of a global grid

For reentry problem : the velocity is very large (≈ 6.000 m/s), the temperature is very large in the shock (∼ 105K), and very small in the upstream and at the boundary (∼ 102K) Consequently: very large grid bounds, and very small step, so very large number of discrete velocities Example: Mach 20, altitude 90 km: V contains 52 × 41 × 41

• points. Around 350 GB memory requirements with a coarse

3D mesh in space!...

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BGK Model Standard method Presentation of the method (1D) 2D method Conclusions

Need of dynamics grids : example

two interacting blastwaves: ` a t = 0 ρ = 1, u = 0, T = [1000, 0.01, 100] max (u + 4 √ RT) = 126 at t = 0.02 (after the interaction): max (u + 4 √ RT) = 236: larger bounds! the optimal velocity grid get 2551 points!

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BGK Model Standard method Presentation of the method (1D) 2D method Conclusions Main Idea

1 BGK Model 2 Standard method 3 Presentation of the method (1D)

Main Idea Illustration Extension of the LVG

4 2D method 5 Conclusions

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BGK Model Standard method Presentation of the method (1D) 2D method Conclusions Main Idea

Local velocity grids

idea: define a velocity grid for each t and x

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BGK Model Standard method Presentation of the method (1D) 2D method Conclusions Main Idea

Local velocity grid: problems

1 how to define the Local Velocity Grid: bounds and step? 2 how to exchange informations between two grids? Example:

∂xf (t, x, v) ≈ f (t, x + ∆x, v) − f (t, x, v) ∆x but f (t, x, .) and f (t, x + ∆x, .) are not defined on the same velocity grid...

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BGK Model Standard method Presentation of the method (1D) 2D method Conclusions Main Idea

Definition of a LVG: conservation laws

at tn: space cell [xi− 1

2 , xi+ 1 2 ]

f n

i,k ≈ f (tn, xi, vn i,k), where

vn

i,k ∈ Vn i =

• vn

i,1, vn i,2, . . . , vn i,K n

i

• local velocity grid (with K n

i points)

Un

i ≈ U(tn, xi) with a quadrature on the LVG:

U(tn, xi) = (ρ, ρu, E)(tn, xi) =

• R

m(v)f (tn, xi, v) dv = mf (tn, xi, .) ↓ Un

i = (ρn i , ρn i un i , E n i ) = K n

i

• k=1

m(v n

i,k)f n i,kωn i,k = mf n i Vn

i

where m(v) = (1, v, 1

2|v|2). Forestier-Coste local velocity grid 19/ 49

BGK Model Standard method Presentation of the method (1D) 2D method Conclusions Main Idea

Definition of a LVG: conservation laws

approximation of U(tn+1, xi) by discrete conservation laws: conservation laws ∂tU + ∂xvmf = 0 finite volume upwind scheme Un+1

i

− Un

i

∆t + Φn

i+ 1

2 − Φn

i− 1

2

∆x = 0 where the numerical fluxes are defined by Φn

i+ 1

2 =

• v+mf n

i

• Vn

i +

• v−mf n

i+1

• Vn

i+1 Forestier-Coste local velocity grid 20/ 49

BGK Model Standard method Presentation of the method (1D) 2D method Conclusions Main Idea

Definition of a LVG

approximations of u(tn+1, xi) and T(tn+1, xi): un+1

i

and T n+1

i

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BGK Model Standard method Presentation of the method (1D) 2D method Conclusions Main Idea

Definition of a LVG

approximations of u(tn+1, xi) and T(tn+1, xi): un+1

i

and T n+1

i

bounds of Vn+1

i

set to vn+1

min,i = un+1 i

− 4

• RT n+1

i

and vn+1

max,i = un+1 i

+ 4

• RT n+1

i

remark: corrects bounds if f close of its local Maxwellian

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BGK Model Standard method Presentation of the method (1D) 2D method Conclusions Main Idea

Definition of a LVG

approximations of u(tn+1, xi) and T(tn+1, xi): un+1

i

and T n+1

i

bounds of Vn+1

i

set to vn+1

min,i = un+1 i

− 4

• RT n+1

i

and vn+1

max,i = un+1 i

+ 4

• RT n+1

i

remark: corrects bounds if f close of its local Maxwellian new discrete velocity grid Vn+1

i

: uniform with a constant number of points (K n+1

i

= 10 ` a 30) remark: variables step and number of points could be necessary (discontinuous f ).

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BGK Model Standard method Presentation of the method (1D) 2D method Conclusions Main Idea

Computation of f n+1

i

kinetic equation: ∂tf + v∂xf = Q(f ) approximation of f (tn, xi, v): f n

i (v)

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BGK Model Standard method Presentation of the method (1D) 2D method Conclusions Main Idea

Computation of f n+1

i

kinetic equation: ∂tf + v∂xf = Q(f ) approximation of f (tn, xi, v): f n

i (v)

finite volume - upwind scheme (continuous v):

f n+1

i

(v) − f n

i (v)

∆t +v + f n

i (v) − f n i−1(v)

∆x +v − f n

i+1(v) − f n i (v)

∆x = Q(f n

i )(v)

problem: f n+1

i

, f n

i , f n i−1 and f n i+1 not defined on the same

grids.

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BGK Model Standard method Presentation of the method (1D) 2D method Conclusions Main Idea

Computation of f n+1

i

reconstruction procedure: for each i, reconstruction of f n

i

• n its LVG Vn

i

¯ f n

i (v) =

( R(f n

i )(v)

if v n

min,i ≤ v ≤ v n max,i

else,

¯ f n

f n

vn

vn

vn

numerical scheme: f n+1

i

(v) − f n

i (v)

∆t + v + f n

i (v) − f n i−1(v)

∆x + v − f n

i+1(v) − f n i (v)

∆x = Q(f n

i )(v) Forestier-Coste local velocity grid 23/ 49

BGK Model Standard method Presentation of the method (1D) 2D method Conclusions Main Idea

Computation of f n+1

i

reconstruction procedure: for each i, reconstruction of f n

i

• n its LVG Vn

i

¯ f n

i (v) =

( R(f n

i )(v)

if v n

min,i ≤ v ≤ v n max,i

else,

¯ f n

f n

vn

vn

vn

numerical scheme: f n+1

i

(v) − f n

i (v)

∆t + v + f n

i (v) − f n i−1(v)

∆x + v − f n

i+1(v) − f n i (v)

∆x = Q(f n

i )(v)

for every v n+1

i,k

• f Vn+1

i

: f n+1

i,k

− ¯ f n

i (v n+1 i,k )

∆t + v n+1

i,k + ¯

f n

i (v n+1 i,k ) − ¯

f n

i−1(v n+1 i,k )

∆x + v n+1

i,k − ¯

f n

i+1(v n+1 i,k ) − ¯

f n

i (v n+1 i,k )

∆x = ¯ Q(f n

i )(v n+1 i,k ) Forestier-Coste local velocity grid 23/ 49

BGK Model Standard method Presentation of the method (1D) 2D method Conclusions Main Idea

Advantages of this scheme

1 advantage 1: the local velocity grid adapt in time and space

to the local temperature T and velocity u

2 advantage 2: only initial values for u and T are required Forestier-Coste local velocity grid 24/ 49

BGK Model Standard method Presentation of the method (1D) 2D method Conclusions Main Idea

Properties of the numerical scheme

1 if R positive, then the scheme is positive, provided that

∆t satisfies a standard CFL condition T n+1

i

positive

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BGK Model Standard method Presentation of the method (1D) 2D method Conclusions Main Idea

Properties of the numerical scheme

1 if R positive, then the scheme is positive, provided that

∆t satisfies a standard CFL condition T n+1

i

positive

2 under this slight modification on the quadrature:

f n

i Vn

i := ¯

f n

i

T n+1

i

is positive

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BGK Model Standard method Presentation of the method (1D) 2D method Conclusions Main Idea

Properties of the numerical scheme

1 if R positive, then the scheme is positive, provided that

∆t satisfies a standard CFL condition T n+1

i

positive

2 under this slight modification on the quadrature:

f n

i Vn

i := ¯

f n

i

T n+1

i

is positive

3 in practice: even with non positive R, the scheme generally

preserves the positivity

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BGK Model Standard method Presentation of the method (1D) 2D method Conclusions Illustration

1 BGK Model 2 Standard method 3 Presentation of the method (1D)

Main Idea Illustration Extension of the LVG

4 2D method 5 Conclusions

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BGK Model Standard method Presentation of the method (1D) 2D method Conclusions Illustration

Illustration 1: Sod test case

velocity profile:

0,1 0,2 0,3 0,4 0,5 0,1 0,2 0,3 0,4 0,5 0,6

... : global velocity grid (wrong bounds, with initial data)

• : global velocity grid (correct bounds, 100 points: bounds and step

estimated by an Euler computation, then a grid convergence study) .- : local velocity grid (10 points)

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BGK Model Standard method Presentation of the method (1D) 2D method Conclusions Illustration

Illustration 2: two interracting blast waves

before the interaction (top), after (bottom)

T u p

0,2 0,4 0,6 0,8 1 0,5 1 1,5 2 2,5 3 0,2 0,4 0,6 0,8 1

• 5

5 10 15 20 0,2 0,4 0,6 0,8 1 100 200 300 400 0,2 0,4 0,6 0,8 1 1 2 3 4 0,2 0,4 0,6 0,8 1 8 10 12 14 16 18 20 0,2 0,4 0,6 0,8 1 300 350 400 450 500 550 600

• global velocity grid: 2531 points (bounds and step estimated by an

Euler computation , then grid convergence study)

• local velocity grid: 30 points

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BGK Model Standard method Presentation of the method (1D) 2D method Conclusions Illustration

Reconstruction procedure

piecewise polynomial interpolation high order interpolation is necessary discontinuous distributions ( large Knudsen number), a non

• scillatory method is necessary (ENO interpolation used)

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BGK Model Standard method Presentation of the method (1D) 2D method Conclusions Illustration

Influence of the reconstruction step: Sod test

ρ u T

• 1
• 0,5

0,5 1 0,2 0,4 0,6 0,8 1

• 1
• 0,5

0,5 1 0,2 0,4 0,6 0,8

• 1
• 0,5

0,5 1 0,6 0,8 1 1,2 1,4 1,6 1,8 2

• 1
• 0,5

0,5 1 0,2 0,4 0,6 0,8 1

• 1
• 0,5

0,5 1 0,2 0,4 0,6 0,8

• 1
• 0,5

0,5 1 0,5 1 1,5 2 2,5 3

Free transport regime (no collisions): top: global grid (30 points) bottom: local velocity grid (30 points): affine interpolation, ENO3, ENO4

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BGK Model Standard method Presentation of the method (1D) 2D method Conclusions Illustration

Boundaries conditions

diffuse reflection: f (t, x = 0, v > 0) = − v−f

v+MwMw

ghost cell approach Mw on its own local velocity grid (based on the wall temperature and velocity) however, such reflection can generate very discontinuous and non-symmetric distributions!

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BGK Model Standard method Presentation of the method (1D) 2D method Conclusions Illustration

Boundaries conditions

example: Heat transfert problem

TR TL t = 0   ρ0 u0 = 0 T0 = TL   x = 0 x = 1

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BGK Model Standard method Presentation of the method (1D) 2D method Conclusions Illustration

Boundaries conditions

example: Heat transfert problem for small t, the support of f (t, x ≈ 1, .) is non symmetric:

TR TL t = 0   ρ0 u0 = 0 T0 = TL   x = 0 x = 1

• RTL
• RTR

Forestier-Coste local velocity grid 32/ 49

BGK Model Standard method Presentation of the method (1D) 2D method Conclusions Illustration

Boundaries conditions

example: Heat transfert problem for small t, the support of f (t, x ≈ 1, .) is non symmetric:

TR TL t = 0   ρ0 u0 = 0 T0 = TL   x = 0 x = 1

• RTL
• RTR

local velocity grid (symmetric by construction) is not large enough for v < 0 (−1900 instead of −2200).

0,1 0,2 0,3 0,4 0,5 0,6

• 50
• 40
• 30
• 20
• 10

(Kn = 0.01, global velocity grid=100 points, LVG=30 points)

Forestier-Coste local velocity grid 32/ 49

BGK Model Standard method Presentation of the method (1D) 2D method Conclusions Extension of the LVG

1 BGK Model 2 Standard method 3 Presentation of the method (1D)

Main Idea Illustration Extension of the LVG

4 2D method 5 Conclusions

Forestier-Coste local velocity grid 33/ 49

BGK Model Standard method Presentation of the method (1D) 2D method Conclusions Extension of the LVG

Extension of the LVG: algorithm

compute f n+1

i,k

for every vn+1

i,k

• f Vn+1

i

w = vn+1

i,1

(leftmost point) loop

w = w − ∆v n+1

i

compute f n+1

i

(w) by the numerical scheme:

f n+1

i

(w) − ¯ f n

i (w)

∆t +w + ¯ f n

i (w) − ¯

f n

i−1(w)

∆x +w − ¯ f n

i+1(w) − ¯

f n

i (w)

∆x = ¯ Q(f n

i )(w)

if f n+1

i

(w) is too large then add w to the grid and continue the loop else stop

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BGK Model Standard method Presentation of the method (1D) 2D method Conclusions Extension of the LVG

Extension of the LVG: illustration

Heat transfert problem: T u p

0,1 0,2 0,3 0,4 0,5 0,6 300 400 500 600 700 800 900 1000 0,1 0,2 0,3 0,4 0,5 0,6

• 50
• 40
• 30
• 20
• 10

0,1 0,2 0,3 0,4 0,5 0,6 1 1,5 2 2,5 3 0,1 0,2 0,3 0,4 0,5 0,6 300 400 500 600 700 800 900 0,1 0,2 0,3 0,4 0,5 0,6

• 50
• 40
• 30
• 20
• 10

0,1 0,2 0,3 0,4 0,5 1 1,5 2 2,5 3

no extension, symmetric LVG (top) with extension (bottom) Kn = 0.01, global velocity grid=100 points, LVG=30 points

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BGK Model Standard method Presentation of the method (1D) 2D method Conclusions 2D preliminary results

1 BGK Model 2 Standard method 3 Presentation of the method (1D) 4 2D method

2D preliminary results Embedded local velocity grid Illustration Mesh Refinment

5 Conclusions

Forestier-Coste local velocity grid 36/ 49

BGK Model Standard method Presentation of the method (1D) 2D method Conclusions 2D preliminary results

To the 2D

reduct model : F(t, x, y, vx, vy, vz) f (t, x, y, vx, vy) = Fz g(t, x, y, vx, vy) = F |vz|2 2 z cartesian velocity grid reconstruction : 2D ENO interpolation

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BGK Model Standard method Presentation of the method (1D) 2D method Conclusions 2D preliminary results

Heat transfert problem on a cylinder:

• utside wall temperature : 1000K

inside wall and domain temperature : 300K gas at equilibrium

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BGK Model Standard method Presentation of the method (1D) 2D method Conclusions 2D preliminary results

Heat transfert problem on a cylinder:

• utside wall temperature : 1000K

inside wall and domain temperature : 300K gas at equilibrium

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BGK Model Standard method Presentation of the method (1D) 2D method Conclusions 2D preliminary results

computational time too long Each velocity of each cell interpolated at least 8 times.

Forestier-Coste local velocity grid 39/ 49

BGK Model Standard method Presentation of the method (1D) 2D method Conclusions 2D preliminary results

computational time too long Each velocity of each cell interpolated at least 8 times. ⇒ new definition of local velocity grids

Forestier-Coste local velocity grid 39/ 49

BGK Model Standard method Presentation of the method (1D) 2D method Conclusions Embedded local velocity grid

1 BGK Model 2 Standard method 3 Presentation of the method (1D) 4 2D method

2D preliminary results Embedded local velocity grid Illustration Mesh Refinment

5 Conclusions

Forestier-Coste local velocity grid 40/ 49

BGK Model Standard method Presentation of the method (1D) 2D method Conclusions Embedded local velocity grid

Embedded LVG : definiton

u ± 4 √ RT reference velocity step.

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BGK Model Standard method Presentation of the method (1D) 2D method Conclusions Embedded local velocity grid

Embedded LVG : definiton

u ± 4 √ RT reference velocity step. reference point for the grids : (0, 0).

Forestier-Coste local velocity grid 41/ 49

BGK Model Standard method Presentation of the method (1D) 2D method Conclusions Embedded local velocity grid

Embedded LVG : definiton

u ± 4 √ RT reference velocity step. reference point for the grids : (0, 0). take into account the accuracy wished.

Forestier-Coste local velocity grid 41/ 49

BGK Model Standard method Presentation of the method (1D) 2D method Conclusions Embedded local velocity grid

Need of auto extension

v_y v_x 2482.1

• 2482.1

3427.7

• 2009.3
• 43.4373
• 12.412

cellule 1130 v_y v_x 2954.9

• 2954.9

3427.7

• 2363.9
• 41.4344
• 12.4799

cellule 1130

without with

Forestier-Coste local velocity grid 42/ 49

BGK Model Standard method Presentation of the method (1D) 2D method Conclusions Illustration

1 BGK Model 2 Standard method 3 Presentation of the method (1D) 4 2D method

2D preliminary results Embedded local velocity grid Illustration Mesh Refinment

5 Conclusions

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BGK Model Standard method Presentation of the method (1D) 2D method Conclusions Illustration

Shock wave on a cylinder

high velocity wave coming from left. Knudsen number around 1.

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BGK Model Standard method Presentation of the method (1D) 2D method Conclusions Illustration

1000 2000 T 236 2567

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BGK Model Standard method Presentation of the method (1D) 2D method Conclusions Illustration

0.02 0.04 Error 0.057

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BGK Model Standard method Presentation of the method (1D) 2D method Conclusions Illustration

20 30 40 50 60 N 17 63

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BGK Model Standard method Presentation of the method (1D) 2D method Conclusions Mesh Refinment

1 BGK Model 2 Standard method 3 Presentation of the method (1D) 4 2D method

2D preliminary results Embedded local velocity grid Illustration Mesh Refinment

5 Conclusions

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BGK Model Standard method Presentation of the method (1D) 2D method Conclusions Mesh Refinment

If extrem rarefied then far from equilibrium. Distributions can be multimodal. last method : good boundaries but lot of points to see the thinest mode AMR idea :

take the boundaries from last method. refine only where the mode are thin.

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BGK Model Standard method Presentation of the method (1D) 2D method Conclusions

1 BGK Model 2 Standard method 3 Presentation of the method (1D) 4 2D method 5 Conclusions

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BGK Model Standard method Presentation of the method (1D) 2D method Conclusions

Conclusions

smaller velocity grid time-consuming high order ENO interpolation Embedded LVG

Perspectives

better extension of 2D LVG 3D

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