Examples of simulations of complex collisions

Examples of simulations of complex collisions ´ Celine Baranger, Laurent Desvillettes [email protected] [email protected]

CEA-DAM & CMLA, ENS de Cachan

` Cargese, September 19-23 2005 Trends in Physical and Numerical Modeling for Industrial Multiphase Flows

MIMF - Cargèse

September 19-23 2005

1

– p. 1/2

p Outline

Complex collision: context and equations Numerical approach Particle method for the kernel part of the kinetic equation

Examples of simulations Homogeneous case: comparison with an explicit solution Application in a code of simulation (CEA) 1-D computations of polytropic gases and comparison with Euler equations.

MIMF - Cargèse


2

– p. 2/2

p Context: Kinetic theory Kinetic theory is used to described many situations: study of clouds, mixture of fluid and air in diesel engine, medical sprays, reacting mixture of polytropic gases. In our study, two kind of problems Example 1: description of sprays = suspension of particles of fluid in a gas. Kinetic equation (description of the dispersed phase) coupled with hydrodynamic system (the surrounding gas). Example 2: description of a reacting mixture of polytropic gases : 4 kinetic equations (one per gas). In the hydrodynamic limit, it corresponds to the reactive Euler equations.

MIMF - Cargèse


3

– p. 3/2

p Context: complex collision Description (kinetic level): distribution function f on time t, position x, velocity v and other variables (radius, energy) : f ≡ f (t, x, v, · · · ), solution of the Vlasov-type equation : ∂f + v · ∇x f = 0 ∂t Complex phenomena like collisions are taking into account in the kinetic description by a collision term Qcollision (f, f ). ∂f + v · ∇x f =Qcollision (f, f ) ∂t

MIMF - Cargèse


4

– p. 4/2

p Example 1 : sprays Sprays = Suspension of particles of fluid in a gas. Gas: classical fluid description with macroscopic quantities α(t, x) , ρ(t, x) , u(t, x) , I(t, x) Particles: kinetic description with density of probability f (t, x, v, r, T ) Physical Effects: exchange between gas and particles (Drag force, momentum and energy exchanges (J. Mathiaud )) collision and coalescence. breakup of the particles (T.A.B model). compressibility of droplets turbulence (in progress ).

MIMF - Cargèse


5

– p. 5/2

p Example 1: sprays (2) Equations of the model : 8 > > > > > < > > > > > :

∂t (α ρ) + ∇x · (α ρ u) ∂t (α ρ u) + ∇x · (α ρ u ⊗ u) + ∇x p ` ´ ∂t (α ρ I) + ∇x · (α ρ uI) + p ∂t α + ∇x · (αu)

= = =

0, F s, Q˙ s ,

∂t f + v · ∇x f + ∇v · (F f ) + ∂T (T f )

=

Qcollision (f, f ),

with the closure p = p(ρ, I), the volume fraction of gas R α = 1 − 43 πr3 f dvdT dr and the interaction term (momentum and energy exchanges) « Z „ 4 4 Fs = − πr3 ρd F f dv dT dr = π r3 ∇x p + D(v − u) f dv dT dr 3 3 Z ` ´ D(v − u)2 − mcl T f dv dT dr. Q˙ s = Z

MIMF - Cargèse


6

– p. 6/2

p Example 2: reacting mixture of polytropic gases Mixture of four gases undergoing a chemical reversible reaction A1 + A 2 A 3 + A 4 For each specie, fi ≡ fi (t, x, v, I), with I internal energy. Four kinetic equations ∂ t fi + v · ∇ x fi =

Qm i (fi , fi )

+

X

QbS ij (fi , fj ) + Qreact , i

i = 1, . . . , 4

j6=i

Left hand side: classical transport part Right hand side: kernels correspond to the following collision’s types: - Qm i elastic mechanical collisions between molecules of the same specie i - QbS i,j elastic mechanical collisions between molecules of different species i and j - Qreact reactive collisions between different species i and j i

MIMF - Cargèse


7

– p. 7/2

p Example 2: reacting mixture of polytropic gases(2) In the hydrodynamic limit (w.r.t. mechanical collisions), f i has the following form : ni mi 3/2 − fi (t, x, v, I) = ( ) e T 2πT

mi |v−u|2 +I 2 T

with n(t, x) =

P4

i=1

ni the total number of molecules,

u(t, x) the mean velocity of the mixture, T (t, x) the temperature of the mixture. Theses quantities are solutions of Euler system of reactive gases (in a 1D case) 8 > > > >
> > > : ∂t 1 2

∂t ni + ∂x (ni u)

=

σi S

∂t ( mi ni u) + ∂x ( mi ni u2 + nκT ) i=1 « –« „ » 4 i=1 4 P P mi ni u2 + 52 nκT + ∂x u 12 mi ni u2 + 72 nκT

=

0

=

ES


8

4 P

i=1

MIMF - Cargèse

4 P

i=1

– p. 8/2

p Collision kernel: an ’simple’ example In the 2 previous examples, the collision term is of the form Qcollision (f, g)

=

Z

···

Z Z

v ∗ ,r ∗ ,T ∗

π θ=0

Z

2π φ=0



0

0

f (t, x, v , r, T )g(t, x, v ∗ , r∗ , T ∗ )

−f (t, x, v, r, T )g(t, x, v ∗ , r∗ , T ∗ )

ff

B

dv ∗ dr∗ dT ∗ dθ dφ.

Hard sphere cross section B(θ, |v − v ∗ |, r, r ∗ ) = π|v − v ∗ | (r + r ∗ )2

sin θ . 4π

Post- collisional velocities 8 r3 v + r ∗3 v ∗ r∗3 0 > ∗ > v = + |v − v | σ, > > < r3 + r∗3 r3 + r∗3 > > > 0 > : v∗

=

r3 v + r ∗3 v ∗ r3 ∗ − |v − v | σ. r3 + r∗3 r3 + r∗3

( σ ∈ S 2 parametrized by θ, φ)

MIMF - Cargèse


9

– p. 9/2

p Numerical approach Aim: compute the kinetic equation and specially the collision term. ∂f + v · ∇x f =Qcollision (f, f ) ∂t Tool: particular method f ≈ sum of Dirac masses in x, v, r, T .

f =

N X

ωp δxp ,vp ,rp ,Tp

p=1

where ωp is the numerical representativity of the numerical particle xp , v p , r p , T p . Splitting in time : transport part: collision part:

MIMF - Cargèse

∂f ∂t ∂f ∂t

+ v · ∇x f = 0 (leads to ODEs on xp , vp , rp , Tp ). R =Qcollision (f, f ) = Q+ − ( f B) f


10

– p. 10/2

p Numerical computation of ∂t f

= Qcollision (f )

Assumption: in each cell Ci of an Eulerian grid, f is a constant w.r.t x, that is: N n (i)

X

n

f (t , x, v, r, T ) =

p=1

ωpn |Ci |

δvpn ,rpn ,Tpn (x, v, T ) 1Ci (x)

Spurious collision method: Total number of spurious collisions during ∆t: N (i)2 ∆t where Bmax =

picking

max

p1 ,p2 ∈Ci

π|vp1 − vp2 |(rp21 + rp22 ) =

ω B N (i)2 ∆t |C | max i

2

with the probability

ω Bmax |Ci | max

p1 ,p2 ∈Ci

B(p1 , p2 ).

pairs of particles (p1 , p2 )

Bp1 ,p2 it’s a real collision Bmax

then, picking σ in S 2 and computing the new velocities of p1 et p2 .

MIMF - Cargèse


11

– p. 11/2

p Example 0: Comparison with an explicit solution 0-D case i.e. no position dependency, with B(θ, |v − v∗ |, r, r ∗ ) = 1 Property: if f0 ≡ f0 (v) =

2√ 2 e−|v|2 , |v| 3π π

the solution of ∂t f = Qcoll (f ) is f (t, v) = (a(t) + b(t)|v|2 )e−c(t)|v|

2

Comparison between analytic solution and simulation result (arbitrary units): Collision simulation Theorical and numerical momentum of order 4 11 numerical theorical (125/12-5/3*exp(-t/3))

momentum of order 4

10,5

10

9,5

9

0

MIMF - Cargèse

5

10

15 time t

20

25

30


12

– p. 12/2

p Example 1: Simulation of collisions in a CEA code

without collision with collision, Bird’s method with collision, Nanbu’s method

MIMF - Cargèse


13

– p. 13/2

p Example 2: reactive mixture work in collaboration with Sandra Pieraccini, Politecnico di Torino We used adimensional equations. Boundary conditions: free flow BC on a bounded slab (x ∈ [−1, 1]). The reaction considered is H2 O + H OH + H2 |{z} |{z} | {z } |{z} A1

A2

A3

A4

corresponding to m = (0.018, 0.001, 0.017, 0.002) and E = 6.33 · 104 Kj/mol. Initial data: special interest in the Riemann Problem. Data used ( L-values correspond to x < 0, R-values correspond to x ≥ 0) 2 6 6 6 6 6 6 6 6 4 MIMF - Cargèse

n1 n2 n3 n4 u p

3

2

3

0.06 6 7 7 6 0.05 7 7 6 7 7 6 0.15 7 7 7 =6 7 6 0.1 7 , 7 6 7 7 6 7 7 4 0 5 5 2690 L

2 6 6 6 6 6 6 6 6 4

n1 n2 n3 n4 u p

2

3 7 7 7 7 7 7 7 7 5

R

3

0.03 6 7 6 0.02 7 6 7 6 0.1 7 7 =6 6 0.05 7 6 7 6 7 4 0 5 500


14

– p. 14/2

p Example 2: reactive mixture, density

ρ

Comparison of the numerical simulations of the 2 models: kinetic and euler The case without chemical reactions

MIMF - Cargèse


15

– p. 15/2

p Example 2: reactive mixture, velocity

MIMF - Cargèse

u


16

– p. 16/2

p Example 2: reactive mixture, Temperature

MIMF - Cargèse


T

17

– p. 17/2


ρ

The case with chemical reactions

MIMF - Cargèse


18

– p. 18/2

p Example 2: reactive mixture, velocity

MIMF - Cargèse

u


19

– p. 19/2


MIMF - Cargèse


T

20

– p. 20/2


ρ

Comparison far from the collisional regime (small C)

MIMF - Cargèse


21

– p. 21/2


MIMF - Cargèse


T

22

– p. 22/2