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
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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.
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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.
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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
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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 ).
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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
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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
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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
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4 P
i=1
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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)
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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:
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∂f ∂t ∂f ∂t
+ v · ∇x f = 0 (leads to ODEs on xp , vp , rp , Tp ). R =Qcollision (f, f ) = Q+ − ( f B) f
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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 .
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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
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p Example 1: Simulation of collisions in a CEA code
without collision with collision, Bird’s method with collision, Nanbu’s method
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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
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p Example 2: reactive mixture, density
ρ
Comparison of the numerical simulations of the 2 models: kinetic and euler The case without chemical reactions
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p Example 2: reactive mixture, velocity
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p Example 2: reactive mixture, Temperature
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p Example 2: reactive mixture, density
ρ
The case with chemical reactions
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p Example 2: reactive mixture, velocity
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p Example 2: reactive mixture, Temperature
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p Example 2: reactive mixture, density
ρ
Comparison far from the collisional regime (small C)
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p Example 2: reactive mixture, Temperature
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T
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