Development of an evaporation boundary condition for DSMC ...

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




4 Trajectory code



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




4 Trajectory code



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


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





4 Trajectory code



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




4 Trajectory code



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-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





4 Trajectory code



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


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


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


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





4 Trajectory code



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


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


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Development of an evaporation boundary condition for DSMC ...

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