Development of an evaporation boundarycondition for DSMC method with
application to meteoroid entry
F. Bariselli, S. Boccelli, A. Frezzotti, A. Hubin, T. Magin
Annual METRO meeting29th November 2016
Physics of a meteoroid entering the atmosphere
Coupled physico-chemical phenomena occurring during meteoroid ablation
Velocity: from 12 km/s to 72 km/s
Size: from micro size grains to meters
2 / 26
Focus on the gas-surface phenomena
VKI Plasmatron experiment: basaltic sample,
3 MW/m2 heat flux at the stagnation point
Ordinary chondritic sample, 1 MW/m2 heat flux
at the stagnation point
Schematic of gas-surface interactions
3 / 26
Objectives
Goals
Asses the importance in the competition among the differentphenomena
Develop an evaporation boundary condition capable of dealingwith magma compositions (mixture of oxides)
Couple the flow with the material response
Introduce a method to compute chemistry of ablated species
4 / 26
1 Governing equations and numerical methodBoltzmann equationDirect Simulation Monte Carlo methodSPARTA numerical tool
2 Ablative wallKinetic boundary conditionsMelt-vapor equilibriumExtension to mixtures of oxides
3 Material responseMaterial codeApparent heat capacity methodInterface flow-material
4 Trajectory code
5 Gas phase chemistryLagrangian solver
6 Conclusions and future work
4 / 26
1 Governing equations and numerical methodBoltzmann equationDirect Simulation Monte Carlo methodSPARTA numerical tool
2 Ablative wallKinetic boundary conditionsMelt-vapor equilibriumExtension to mixtures of oxides
3 Material responseMaterial codeApparent heat capacity methodInterface flow-material
4 Trajectory code
5 Gas phase chemistryLagrangian solver
6 Conclusions and future work
Governing equations and numerical method 4 / 26
Breakdown of the continuum regime
Knudsen number =λ
D=
mesoscopic scale
macroscopic scale
Governing equations and numerical method Boltzmann equation 5 / 26
Boltzmann equation
f(X, ξ, t) = one-particle velocitydistribution function
∂f
∂t+ ξ · ∂f
∂X+ F · ∂f
∂ξ=
︷ ︸︸ ︷∫ ∞−∞
∫ 4π
0
(f ′f ′
1 − ff1)gσdωdξ1
From microscopic to macroscopic world:
ρ =
∫ ∞−∞
f(X, ξ, t) dξ
ρV =
∫ ∞−∞
ξ f(X, ξ, t) dξ
ρe =
∫ ∞−∞
1
2ξ · ξ f(X, ξ, t) dξ
ϕ =
∫ ∞−∞
f(X, ξ, t) [ξ − vw] · n dξ
q =
∫ ∞−∞
1
2ξ · ξ f(X, ξ, t) [ξ− vw] ·n dξ
Governing equations and numerical method Boltzmann equation 6 / 26
Direct Simulation Monte Carlo method
DSMC algorithm
Each simulated particle represents a large number of real particles
Free motion decoupled from collisions (for ∆t < ν−1coll)
Grid cells used to choose collisions partners and sample averages
DSMC is not Molecular Dynamics
Governing equations and numerical method Direct Simulation Monte Carlo method 7 / 26
SPARTA numerical tool
Developed by Plimpton and Gallis at Sandia National Labs
Open source software (http://sparta.sandia.gov/)
Object-oriented philosophy enables extensions
Parallel implementation through domain decomposition
Flow around MIR space station Apollo re-entering the atmosphere
Governing equations and numerical method SPARTA numerical tool 8 / 26
1 Governing equations and numerical methodBoltzmann equationDirect Simulation Monte Carlo methodSPARTA numerical tool
2 Ablative wallKinetic boundary conditionsMelt-vapor equilibriumExtension to mixtures of oxides
3 Material responseMaterial codeApparent heat capacity methodInterface flow-material
4 Trajectory code
5 Gas phase chemistryLagrangian solver
6 Conclusions and future work
Ablative wall 8 / 26
Kinetic boundary conditions for Boltzmann equation
flux emerging from the wall︷ ︸︸ ︷f(ξ)[ξ − vw] · n =
flux due to evaporation by the wall︷ ︸︸ ︷g(ξ)[ξ − vw] · n +∫
[ξ′−vw]·n<0
KB(ξ′ → ξ)f(ξ′)[ξ′ − vw] · n dξ′︸ ︷︷ ︸flux due to reflection by the wall
[ξ − vw] · n > 0
KB(ξ′ → ξ) = (1− αc)[ξ − vw] · n2π(RTw)2
exp(− |ξ − vw|
2
2RTw
)g(ξ) =
αeρeqw
(2πRTw)3/2exp(− |ξ − vw|
2
2RTw
)αe/αc evaporation/condensationcoefficients (usually αe = αc = 1)
ρeqw equilibrium vapor density
No need to assume equilibrium of thegas with the wall
Ablative wall Kinetic boundary conditions 9 / 26
MAGMA chemical multi-phase equilibrium solver
How do we obtain the equilibrium properties?ρeqwi
= ρeqwi(material composition, Tw)
Developed by Fegley and Cameron, 1987Mass balance, mass action algorithmOnly stoichiometric reactions for change of phaseExtensively validated vs. experimental dataAlready used to model vaporization in silicate lavas (underthermodynamic equilibrium assumption)
Ablative wall Melt-vapor equilibrium 10 / 26
Extension to mixtures of oxides
How do we choose the coefficients?αei , αci i ∈ O, Si, SiO2, Mg, MgO, ...
Hp. 0: MxOy (l) x M (g) + y O (g) ⇒ αci = 0 ∀ i 6= M, O
Hp. 1: αcM= αeM
= αM = α
Hp. 2: αcO= αeO
= αO
⇒ Thermodynamic equilibrium has to be retrieved by kineticapproach
ϕeqi = ϕeqi (αi, ρeqwi, Tw)
ϕeqeM
= ϕeqcM∀ M
ϕeqeO= ϕeqcO
⇒∑k∈R ϕ
eqeOk
= ϕeqcO
ykϕeqeMk
= xkϕeqeOk∀ k ∈ R
Ablative wall Extension to mixtures of oxides 11 / 26
Electron concentration around a non-ablating meteoroid
Kn = 0.1
D = 1 cm
H = 80 km
V∞ = 72 km/s
non-ablating meteoroid1011 ÷ 1020 m−3 typical e− density for meteorsAmbipolar diffusion assumption
Gas phase chemistry frozen above 90 km ⇒ electrons from metal speciesAblative wall Extension to mixtures of oxides 12 / 26
Ablating meteoroid at 95 km altitude
Pure magnesium:Twall = 925 K (melting temperature)vwall = 0 m/sαc = αe = 1
D = 1 cm
H = 95 km
V∞ = 72 km/s
Translational temperature
Cooling effect of evaporation at the wallAdiabatic T ≈ 2.5M K >> TwallGeometric temperature: particles notcharacterized by this temperature in thethermal sense
Ablative wall Extension to mixtures of oxides 13 / 26
Ablating meteoroid at 95 km altitude
Delay in the excitation of the internal dofsStrong shielding effects of Mg vapor
Rotational temperature Ablation products molar fraction
Ablative wall Extension to mixtures of oxides 14 / 26
1 Governing equations and numerical methodBoltzmann equationDirect Simulation Monte Carlo methodSPARTA numerical tool
2 Ablative wallKinetic boundary conditionsMelt-vapor equilibriumExtension to mixtures of oxides
3 Material responseMaterial codeApparent heat capacity methodInterface flow-material
4 Trajectory code
5 Gas phase chemistryLagrangian solver
6 Conclusions and future work
Material response 14 / 26
Material response
Randomly and rapidly rotating sphere: ∂T∂t = k
ρcp∂2T∂r2 + 2k
ρcp1r∂T∂r
Ablating wall: moving mesh (fixed reference frame)
Re-mapping procedure at each time step
Finite differences, explicit time integration
0.00 0.05 0.10 0.15 0.20Distance from the surface of the sphere [m]
0
100
200
300
400
500
600
700
800
Deri
ved
vari
ab
le [
K m
]
Time [s]:
1
20
100
200
Numerical solution
Verification of the spherical coordinates:
unsteady solution
0.00 0.05 0.10 0.15 0.20Distance from the initial surface position [m]
0
1000
2000
3000
4000
Tem
pera
ture
[K
]
Time [s]:
20
100
120
Numerical solution
Verification of the moving wall (re-mapping):
steady solution
Material response Material code 15 / 26
Tracking of the melting front
Liquid
Solid
2 T
T
CP
CPS
CPL
(CPS + CP
L)/2 + L/(2 T)
Heat capacity
0 100 200 300 400 500Time [s]
0.00
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
Posi
tion
of
the l
iqu
id-s
oli
d i
nte
rface
[m
]
Latent heat of melting [J/Kg]:
4e+05
4e+06
2e+07
Numerical solution
Verification of the apparent heat capacity method:
position of the melting front
Apparent heat capacity method
No need to deform the mesh to track the solid-liquid interface
Position of the melting front obtained a posteriori
Material response Apparent heat capacity method 16 / 26
Interface flow-material
Surface mass balance
ϕe − ϕc = ρwvw
ϕe computed theoretically
ϕc directly from DSMC simulation
No contribution of reflection (no net flux)
qc + qr − qe + εσT 4∞ =
ρwhwvw + k∂T
∂r
∣∣∣∣w
+ εσT 4w
ϕe computed theoretically
qc, qr directly from DSMC simulationSurface energy balance
Material response Interface flow-material 17 / 26
1 Governing equations and numerical methodBoltzmann equationDirect Simulation Monte Carlo methodSPARTA numerical tool
2 Ablative wallKinetic boundary conditionsMelt-vapor equilibriumExtension to mixtures of oxides
3 Material responseMaterial codeApparent heat capacity methodInterface flow-material
4 Trajectory code
5 Gas phase chemistryLagrangian solver
6 Conclusions and future work
Trajectory code 17 / 26
Trajectory code
Python implementation (cython to improve performances)
Interface with MAGMA for wall equilibrium properties
Interface with NASA atmospheric model for free-stream properties
Sub time stepping for material response
Flow update resolution can be fixed through input file
Trajectory code 18 / 26
Trajectory-material response coupling
No DSMC coupling: evaporation into vacuum (Knudsen-Langmuir)
qflow =Λρ∞(H)V 3
∞8
6080100120140160180200
Altitude [km]
200
400
600
800
1000
1200
1400
1600
Te
mp
era
ture
[K
]
V = 15 km/s, D = 0.002 m
Core temperature
Surface temperature
9095100105110
Altitude [km]
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Po
sitio
n f
rom
th
e c
ore
[m
]
×10-3 V = 15 km/s, D = 0.002 m
Evaporating front
Melting front
Molten thickness
D = 2 mm
V∞ = 15 km/s
Trajectory code 19 / 26
1 Governing equations and numerical methodBoltzmann equationDirect Simulation Monte Carlo methodSPARTA numerical tool
2 Ablative wallKinetic boundary conditionsMelt-vapor equilibriumExtension to mixtures of oxides
3 Material responseMaterial codeApparent heat capacity methodInterface flow-material
4 Trajectory code
5 Gas phase chemistryLagrangian solver
6 Conclusions and future work
Gas phase chemistry 19 / 26
Lagrangian solver
Atmospheric
entry ows
many chemical products
(air, ablated species)
thermal non-equilibrium
(T, Tr, Tv)
2D / 3D geometries
TO
O
EXPEN
SIV
E!
Result: only simple models currently employed
METEOROID TRAIL: detailled ionization mechanisms required...
IDEA: introduce chemistry a posteriori!
HOWEVER. . .
LAgrangianReactor forStrEams inNonequilibrium
LARSEN
Gas phase chemistry Lagrangian solver 20 / 26
Chemistry of ablated species computed a posteriori
1) Simple simulation 2) Extract streamlines
3) Lagrangian solver4) Refined resultsYi
s
u(s) ρ(s) + ICs
Velocity and density fields from baseline simulationassumed good enough
Gas phase chemistry Lagrangian solver 21 / 26
LARSEN formulation
From baseline simulation: u, ρ are given
We still need:
Species mass eq. → ∂syi =ωi −∇ · J i
ρu
Total enthalpy eq. → ∂sH = Qext
Species internal energy → ∂seini =
∇ ·Dini + Ωin
i − hini ωiρyiu
Q
Jiuωi
=⇒ System of ODEs
Gas phase chemistry Lagrangian solver 22 / 26
Hypersonic inert flow around a cylinder
Axisymmetric DSMCsimulation
Pure argon
M = 10
Kn = 0.05
(a)
(b)
(a)
(b)
(a)
(b)
0
1000
2000
3000
4000
5000
6000
7000
0 0.5 1 1.5 2curvilinear abscissa [m]
SPARTLARSEN
3
DSMC
LARSEN
(a)
(b)
Tem
pera
ture
[K
]
T correctly reproduced
Gas phase chemistry Lagrangian solver 23 / 26
Relaxation behind a shock wave - refining chemistry
Inviscid flow
Fire-II capsulefree-stream conditions
air5: N2 O2 N O NO
air11: air5 +N+ O+ NO+ N+
2 O+2 e−
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 0.002 0.004 0.006 0.008 0.01 0.012 0.014
Mass f
racti
ons
LARSEN - air5 to air11SHOCKING - air11
NO
O
O+
N2
N
N+
O2
0
5000
10000
15000
20000
0 0.001 0.002 0.003 0.004 0.005
distance from shock [m]
Tem
pera
tu
SHOCKING - air5
1distance from shock [m]
1
0
5000
10000
15000
20000
25000
30000
35000
40000
0 0.001 0.002 0.003 0.004 0.005
distance from shock [m]
Tem
pera
ture
[K
]
T well predictedions concentrations well predictedneutral species concentrations improved
Gas phase chemistry Lagrangian solver 24 / 26
1 Governing equations and numerical methodBoltzmann equationDirect Simulation Monte Carlo methodSPARTA numerical tool
2 Ablative wallKinetic boundary conditionsMelt-vapor equilibriumExtension to mixtures of oxides
3 Material responseMaterial codeApparent heat capacity methodInterface flow-material
4 Trajectory code
5 Gas phase chemistryLagrangian solver
6 Conclusions and future work
Conclusions and future work 24 / 26
Conclusions and future work
Conclusions
Develop a DSMC evaporation boundary condition for silicates
Couple flow-material-trajectory
Develop a method to implement detailed chemistry a posteriori
Future work
Include dynamic of molten layer to gas surface interactions
Add ionization of metallic species
Assess recombination time of free electrons in the trail
Conclusions and future work 25 / 26
Thanks for your attention
Thanks to:
B. Dias for the useful discussion
G. Bellas for the help with LARSEN
L. Zavalan, B. Helber, P. Collins for the experiments
Conclusions and future work 26 / 26