kine%c’eﬀects’in’aﬂuid’plasmatransport approximaon’and’some ... · 2016-05-20 ·...

Kine%c effects in a fluid plasma transport approxima%on and some recent simula%on results

LLNL Kine%cs Workshop Apr.5-‐7, 2016

presented by E. Vold

with acknowledgement to many: K. Molvig, A. Simakov,

R. Rauenzahn, C. Aldrich, A. Joglekar, M. Ortgea, R. Moll, D. Fenn,

E. Dodd, S. Wilks, L. Yin, B. Albright, B. Haines, N. Hoffman, S. Hsu, P. Hakel, H. Rinderknecht

LA-‐UR-‐16-‐22061

Operated by Los Alamos National Security, LLC for the U.S. Department of Energy's NNSA

Introduction

§  Kinetics includes all collisional effects. –  momentum and energy exchange in:

–  thermal conduction & equilibration, –  viscosity, –  species particle flux relative to Ucm

§  Multi-species fluid transport approximations can represent some kinetic effects. –  assumed ‘nearly’ Maxwellian distributions –  small parameter Kn:

–  MFP < gradient scales –  t >>

§  ICF Shock heating at interfaces: –  fluid approximations may not be formally valid but may perform

reasonably well.

Kn ~ λ / L diffusion⎯ →⎯⎯ λ / LD ~λ

2(D t)1/2= λ2((vthλt)

1/2 =vthτ ij

2((vth2τ ijt)

1/2 =12

τ ijt

⎛⎝⎜

⎞⎠⎟

1/2

τ ij

LD ~ 2[D t]1/2 ~ [Dij ~η ~κ ~ T5/2 ]t1/2

LD ~ [T5/2t]1/2 ~ T 5/4t1/2

Approximated as diffusive processes with characteristic scale length, LD:


Kinetics to plasma fluid transport ** self-consistent plasma particle, heat and momentum flux ** condensed from: Simakov and Molvig, PoP 2016

§  Boltzmann equation:

§  Integrate over velocities: LHS, first order in Kn -> ‘fluid terms’, RHS -> ‘collisions’, consider flux vector.

§  Approximate distribution function w/ Legendre (Sonine) polynomials (for vector piece of solution):

§  Solve coupled equations for particle and heat flux coefficients of each species, i, of N species:

§  Binary system simplifies for light particle flux:

ρi (ui − ucm ) = −µ α11 ∇χ + (χ −Y )∇piapia

+ (z −Y )∇pepia

+ z − Zi2niZ j2nj

j∑

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟αTe

∇TeTi

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟+ 32αTi

∇TiTi

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟

!!εm =mi /mI

∇χ∇Y

= DχDY

= εm(Y + (1−Y )εm )

2 = εm (χ + (1−Y ) / εm )21

pia∇pi −Y∇pia( )

∂ fi∂t

+∇⋅(vi fi )+∇v ⋅(ai fi ) = Cij

fi ~ ( fMax−i + fi1) ~ ( fMax−i + α i

k

k∑ Lk

3/2 ) ~ ( fMax−i +α i0L0

3/2 +α i1L1

3/2 +α i2L2

3/2 )

ν ii + ν ijj∑⎡

⎣⎢

⎤

⎦⎥i

3x3

[α o,α 1,α 2 ]i − ν ji⎡⎣ ⎤⎦i3x3[α o,α 1,α 2 ] j

j∑ = Di ~ [

diχ i

,− 52∇ logTi ,0]i

concentration and ion barodiffusion terms are equivalent in Zimmerman-Schunk model (Hoffman, et.al., PoP, 2015)

concentration given as mass fraction (Kagan, Tang, 2014)

∇χ ⇒ DχDY

∇YFor


Plasma transport in xRage (Eulerian AMR code) compares well to non-linear self-similar diffusion test problem1 under P-T equilib.

[1] D-AL planar mixing at 4 keV, Molvig, Vold, Dodd, Wilks, PRL, 2014

Den

sity

–g/

cc-

Distance -cm-

w/ ion baro-diffusion •

w/o ion baro-diffusion •

• w/ & w/o electron baro-diffusion 8"

10"

12"

14"

16"

18"

20"

22"

(1.00E(03" (5.00E(04" 0.00E+00" 5.00E(04" 1.00E(03"

rho_D41"

rho_F41"

rho_B41"

rho_E41"

rho_C41"

rho_A41"

rho_self(sim"

rho_kiineic"

§  code solutions during development converged on analytic self-sim soln and kinetic soln for D-Al mixing.

Y

V

Χ Y = mass frac X = molar frac v = ‘volume’ frac

tota

l den

sity

–g/

cc-

§  code profiles for X, Y matches analytics in Molvig, et.al., 20141 (for ideal gas or EOS tables )

§  mix profile contributions for low z – high z mixing: DD-Au

§  ion barodiffusion ~ doubles mix width for DD-Au

§  ion barodiffusion contributes less for lower z mixing

-10 x –microns- 10

test case at t = 0.25 ns

-10 x –microns- 10


Viscosity

§  Previous work showed classical binary viscosity can be approximated as a sum over species, and terms simplify considerably in 1D spherical symmetry.

Plasma viscosity due to electrons is important in high z material. ( X = 0 here)

1.0E‐04

1.0E‐03

1.0E‐02

1.0E‐01

1.0E+00

1.0E+01

1.0E+02

1.0E+03

1.0E+04

0.00 0.20 0.40 0.60 0.80 1.00

plasma viscosity ‐ kg m

‐1 s‐1 ‐

X ‐ molar frac4on of light ion‐

Viscpls1

Viscpls2

Viscplse

Viscpls

light ions heavy ions electrons total

§  Ion viscosity in classical binary transport (Molvig, Simikov, Vold, PoP 2014):

§  Ion viscosity in approx as species summation

§  Viscous terms simplify in 1D spherical geometry –  viscous tensor:

–  viscous energy dissipation

ηM = mini43

vTi2

ν i[n ]αη[Δ I ]xIZI

2 = 43mivTi

2

ν ii[ni ]ninαη[Δ I ]xIZI

2 ∝xiαη[Δ I ]xIZI

2 ~αη[Δ I ]Δ I

ηM ≈ kT niν i

⎡

⎣⎢

⎤

⎦⎥

i∑ = kT ni

ν ijj∑

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥i

∑

Δ I =xIZI

2

xiZi≈ (1− xi ) ZI

2

xi= εm (1− yi ) ZI

2

yi

ν ij = Cνα ijK zi

2zj2Lijnj

mi1/2kT 3/2


Temperature profiles from 1D Lagrange simulations are sensitive to viscous terms and at center show a peak in Ti for the inviscid case.

Position (µ m)0 20 40 60 80 100 120

Tem

pera

ture

(eV

)

102

103

104T

e - Viscous

Ti

Tr

Te - Inviscid

Ti

Tr

§  at shock convergence –  (1.42 ns inviscid) –  (1.52 ns viscous)

160 170 180 190 200 210 220 230 240 25010

−2

10−1

100

101

102

103

Position (µ m)

Tem

pera

ture

(eV

)

Te − Viscous

Ti

Tr

Te − Only Viscous M

Ti

Tr

Te − Inviscid

Ti

Tr

§  after shock convergence –  (1.56 ns )

§  during incoming shock convergence

0 20 40 60 80 100 12010

1

102

103

104

Position (µ m)

Te

mp

era

ture

(e

V)

Te − Viscous

Ti

Tr

Te − Inviscid

Ti

Tr

0 r -microns- 120 0 r -microns- 120 160 r -microns- 250

tem

pera

ture

s –e

V-

103 104 104 Ti Ti Ti

Te Te Te

Tr Tr

inviscid

plasma viscosity

plasma viscosity only in momentum equation Inviscid case shows Ti peak on center at shock convergence

and rapid relaxation.


Density [r,t] surface in a 1D Lagange code: ICF implosion with and without plasma viscosity

rad

ius

– µ

m-

0

500

0 1 2

time – ns-

0.0#

50.0#

100.0#

150.0#

200.0#

250.0#

300.0#

350.0#

400.0#

450.0#

0.00# 0.50# 1.00# 1.50# 2.00# 2.50#

radius'(m

icrons('

-me'(ns('

VISC_ALL#

INVISC#

§  ‘Standard’ Lagrange w/ artificial viscosity

rad

ius

– µ

m-

0

500

0 1 2

time -ns-

§  Lagrange w/ plasma and artificial viscosity


Density [r,t] surface in ICF implosion with and without plasma viscosity AND w/ ‘late time’ artificial viscosity zeroed

§  q

radi

us –

µm

-

0

500

0 1 2

time -ns- ra

diu

s –

µm

-

0

500

0 1 2 time – ns-

46.0%

48.0%

50.0%

52.0%

54.0%

56.0%

58.0%

60.0%

1.7% 1.9% 2.1% 2.3%

radius'(m

icrons'

-me'(ns('

VISC_ALL%

INVISC%

VISCwQavOFF_150607%

§  ‘Standard’ Lagrange w/ artificial viscosity

§  Lagrange w/ plasma viscosity §  NO artificial viscosity (after

shock convergence)


-3 -2 -1 0 1 2 3

0.01

0.1

1

10

100

Kinetic coefficients**, for ion temperature gradient drive in binary species mixing, are of order unity when using molar concentrations but appear large using mass concentrations.

§  Assume (to simplify): §  Y (mass frac) = 0.5 Erfc[ξ] §  ξ = z / LD = z / (2(D t)1/2)

ρi (ui − ucm ) = −µ α11 ∇χ + (χ −Y )∇pipi

+ (z −Y )∇pepi

+ z − Zi2niZ j2nj

j∑

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟αTe

∇TeTi

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟+ 32αTi

∇TiTi

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟

!!εm =mi /mI∇χ∇y

= DχDy

= εm(y + (1− y)εm )

2 = εm (χ + (1− χ ) / εm )2

-3 -2 -1 1 2 3

0.2

0.4

0.6

0.8

1.0

D - Au

ξ = z / LD ξ = z / LD -3 -2 -1 0 1 2 3

0.5

1.0

1.5

2.0

ξ = z / LD ξ = z / LD

D - Au D - Au

-3 -2 -1 0 1 2 3

0.01

0.1

1

10

Y

χ

α11 Y χ α11

αTi

χ i =ni

ni + nI= ρi /mi

ρi /mi + ρI /mI

= yiyi + (1− yi )εm

** Molvig, Simikov, Vold, PoP, 2014

Dχ /DY αTi

Dχ /DY⎛⎝⎜

⎞⎠⎟

αTi

Dχ /DY⎛⎝⎜

⎞⎠⎟

αTi

Dχ /DY⎛⎝⎜

⎞⎠⎟

= grad[Ti] coef. relative to mass fraction gradient (e.g., Kagan-Tang form)

D - Al

∇χ = DχDY( )∇Y


Preliminary simulations (xRage) of Rinderknect planar shock experiment using MSV** ion transport model show a depletion of Ne at the ion shock front followed by a small enhancement of Ne concentration behind the shock front.

§  Depletion/Enhancement is a result of ion thermodiffusion.

t1 = 2.5 ns

• Depletion and enhancement are persistant w/ ion shock front.

Ti[t1]

Ne

mas

s fra

ctio

n

Ne

mas

s fra

ctio

n –

log

scal

e-

0.1

0.2

0.3 t2 = 5 ns

x distance –cm- (spanning 0.35 to 0.75 cm)

x distance –cm- (spanning 0.35 to 0.75 cm)

Ti[t2]

Te[t1]

Te[t2]

Ne[t1] Ne[t2] w

/ Ti a

nd T

e -e

V-

103

100

shock[t1]

** MSV- Molvig, Simakov, Vold, PoP, 2014

H w/ Ne = 2 % atm

shock[t2]

shock[t1] shock[t2]


Simulations (xRage) of OMEGA shot #78199 using MSV multispecies ion-transport model show a depletion of Ar at the incoming ion shock front followed by a persistent enhancement of argon concentration behind the shock and following shock reflection at center.

§  Enhancement is a result of ion thermodiffusion –  Simulations with ion thermodiffusion turned off do not show this effect.

shock

t = 1.2 ns

radius (out to100 microons)

• Depletion and Enhancement persist for many 100s of ps.

• Bang time ~ 1.35 ns

• times (ns) are:

1., 1.1, 1.15, 1.2, 1.3, 1.4

t1=1.

t = 1.1

t = 1.15

t = 1.2

t = 1.3

t = 1.4

radius (out to100 microons)

At t = 1.2 ns, Ti shock converges on center

Ar m

ass

fract

ion

Ar m

ass

fract

ion

– lo

g sc

ale-

0.1

0.2

0.1

0.01

0.5

0.0


Plasma transport simulations (xRage) showing mix layer profile evolution at early times (< ~ psec, Kn ~ mfp/LD large) and into fluid regime ( ~ ns, Kn ~ mfp/LD small) for DD-Al mixing at 4 keV).

3 distinct phases: 1 - << 1 psec: u grows starting ‘tri-model’ profile 2 - ~ 1- 2 psec: u tri-model profile relaxes in magnitude, spreading in space 3- < 0.1-0.2 ns: dP relaxed -> ~ 0, while div*u ≠ 0, and relaxes on diffusion time scale

2.5 psec 1. psec

0.2 psec 0.025 psec 0.25 nsec

0.1 nsec

§  pressure §  velocity

••• Results are being compared to kinetic simulations in VPIC (L.Yin) and iFP (Chacon, Taitano) to understand early time (large Kn number) mix behavior.

0.2 psec

1. psec

2.5 psec

0.25 nsec

0.025 psec

- 4 x -microns- 4 - 4 x -microns- 4

3 distinct phases: 1 - << 1 psec: pressure discontinuity, dP, grows at interface. 2 - ~ 1- 2 psec: dP propagates into each fluid at its sonic speed. 3- < 0.1-0.2 ns: dP has relaxed -> 0, in the mix region

D D Al Al


Small scale structures: CAUTION: numerics can mask the plasma transport mixing

Slide 13

§  diffusion mix front grows in time LD ~ 2(D t)1/2

‘side’ view ‘top’ view

-4 -2 2 4

0.2

0.4

0.6

0.8

1.0

-4

-2

24

-1.0

-0.5

0.5

1.0

mix volumes: early time: Vm[t] ~ Am[t] LD[t] …..and finally… Vm[t] ~ f[ Am[t],LD[t],λ[t] ]

§  Numerical solutions for ICF ‘sym-cap’ mixing** show detailed structure w/ more realistic IC.

§  When structure sizes are comparable to numerical diffusion scales, the plasma diffusion may not greatly modify the solution.

λ

** Haines, Grimm, et.al. in preparation, 2016

time

simple IC geometry ‘detailed’ IC geometry

§  For RM, RT, KH instabilities, mix volume related to interfacial area, A, in more complex manner

kine%c’eﬀects’in’aﬂuid’plasmatransport approximaon’and’some ... · 2016-05-20 ·...

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