multi-fluid model rayleigh-taylor mix layer growth with a ... · recent computational simulations...

Los AlamosIntegrated Physics Methods

E.L.Vold

Recent Computational Simulations ofRayleigh-Taylor Mix Layer Growth with a

Multi-fluid Model

Erik VoldLos Alamos National Laboratory

w/ acknowledgements to A.J.Scannapieco, Tim Clark, John Grove, Chuck Cranfill, et.al.

Presented at the 8th International Workshop on The Physics of Compressible Turbulent Mixing

Pasadena, CA, Dec.10-14, 2001

LA-UR-01-6651


E.L.Vold

disclaimer

This is a partial sub-set of the viewgraphs presented at the 8th IWCTM.

The work on ‘resolved scale’ simulationsis currently being prepared for publication, to be submitted to Physics of Fluids.

The work on ‘sub-grid drift flux’ simulationswill be prepared for publication soon, to be submitted to a journal TBD.


E.L.Vold

Recent computational simulations ofRayleigh-Taylor mix layer growth with a multi-fluid model.

Erik VoldLos Alamos National Laboratory

Abstract - LA-UR-01-2562

Recent results of computational simulations of the Rayleigh-Taylor mix layer are presented and discussed. Our previouswork is summarized briefly comparing mix layer growth characteristics observed in different simulation modesincluding single fluid with initial density discontinuity, two-fluids with interface reconstruction and in a full multi-fluiddynamic approach. Recent comparisons under varying compressibility are presented showing negligible influence ofcompressibility on the mix layer growth rate. Using spectral analyses, perturbations intentionally introduced in theinitial conditions are compared to long wave length perturbations introduced inadvertently in these initial conditions.The influence of these initial conditions on late time growth and growth rate are explored. The compressible multi-fluidmodel allows each fluid to have its own ‘drift velocity’ relative to the mass averaged fluid velocity. This can be appliedin several ways within the mix layer to represent a real molecular mixing, a turbulent enhanced diffusive mixing, or anindividual species ‘sub-grid’ convective drift flux. Examples of these in the Rayleigh-Taylor mix layer are discussed.Finally, we consider the combination of these factors which best matches the experimental results for mixing layergrowth rates in incompressible experiments, and how these results may apply to compressible fluids.


E.L.Vold

Introduction• Goals:

– simulate Rayleigh-Taylor mix layers accurately to predict atomic / molecularmixing (e.g., a reactive R-T mixing front) in macroscopic geometries.

– use ‘resolved simulations’ to model mix layer growth and use drift flux(subgrid) simulations to model the mix layers atomically mixing components.

– match experimental ‘alpha’ (α), h = α At g t2

– match refined experimental findings related to mixing front details.

• Central Issues– numerical mixing must be small enough to have a negligible effect on mix

layer growth rates so that ‘sub-grid’ mixing can be represented realistically.

– Hypothesis: the growth rate seen in computations, which have no subgridmixing and small numerical diffusion, should equal or exceed theexperimental value IF the experiment contains small scale dissipation whichreduces the growth rate in the experiment.

focustoday


E.L.Vold

Summary of Relevant Work• Experimental

– wide range of experiments (mostly incompressible) show a mix layergrowth rate which closely approximates the scaling, h = α At g t2

– alpha bubble, αb ~ 0.06-0.07 (earlier work, e.g., Youngs and Read et.al., )- and ~ 0.05 (recent work, e.g., Dimonte, Schneider, et.al.)

• Computational– alpha bubble results range from ~ 0.03 - ~0.1

– many 3-D methods (compressible or incompressible) trending towardslow end, αb ~ 0.03 ~ half experimental mean

– front tracking w/ 2 distinct fluids (‘Frontier code’, Glimm, et.al.) at higherend, ~0.07-0.08

– large variance in alpha just due to random seed in initial perturbation • (~ 0.05 +/- 20-50%, in 2-D compressible isothermal fluids, T. Clark, 2001)

– 2-D results ~ 15% greater than 3-D results (Youngs, 1994).


E.L.Vold

Summary of Our Methods• Methods

– 2-D multi-fluid Eulerian AMR formulation

– compressible Euler equations in appropriate limit to recover incompressibleapproximation, supplemented with fluid volume fractions

– ideal gas equation of state for each fluid

– advection of fluid volume fractions in mixed cells at the interface• mixed cell treatment (Bowers and Wilson,1991)

• interface reconstruction (D.Youngs, 1984, 1989)

– high-order, monotonic Van Leer advection of fluid quantities

– each fluid has its own density, internal energy and pressure in its fluid volumefraction within the ‘mixed cells’ (containing the interface)

– in ‘drift flux’ representation of sub-grid mixing, each fluid has its own ‘driftmomenta’ relative to the mass average which can be adjusted to representrealistic molecular diffusion or a range of assumed turbulent flux forms


E.L.Vold

Summary of OurPrevious Results

• Previous Results– alpha bubble was found on a 128 x 128 grid with no sub-grid mix

model, to be ~ 0.08 - 0.1, somewhat larger than experiments.

– interface algorithm does not alter the growth rates significantly.

– molecular mixing (by the drift flux) does not influence the mix layergrowth rate but does create a unique distribution of molecularly mixedmaterials controlled primarily by the volume fraction of the lightermaterial.

– drift flux mixing significantly above the molecular diffusion levelreduces the mix layer growth rate (for the set-up in these results, a driftflux ~ 50 times greater matches experimental range (αbub~ 0.055)


E.L.Vold

Issues in Matching ‘alpha’[and what our set-up uses]

• Numerical– grid resolution [1282 or 2562 or 5122]

– Interface treatment [mix cell volume fraction advection w/ Young’s interface reconstruction]

– Differencing schemes [high order monotonic Van Leer like scheme]

– 2-D vs. 3-D [2-D only]

• Initial Conditions– initial perturbation magnitudes, [volume fractions, Vf, set to match interface perturbation]

• perturbation on density [ρ = ρ1Vf1 + ρ2(1.−Vf1) ]

• perturbation on internal energy [ε = ε1Vf1 + ε2(1.−Vf1) ]

– wavelength spectrum, [30 modes, mode numbers 30 - 60, random phase, unit amplitude]

– hydrostatic equilibrium by e(z), or ρ(z) [e(z), w/ ρ = ρο]

• Physics– compressible or incompressible formulation (w/ or w/o internal energy ) [compressible]

• degree of compressibility [varied Ma2 by 2 orders of magnitude]

– fluid equations: Euler, viscid, internal or total energy [Euler using internal energy w/ optionalmulti-fluid drift flux for ‘species momenta’ relative to mass averaged single fluid velocity.]

– Interface physics: surface tension, slip or traction, molecular diffusion, sub-grid mixing


E.L.Vold

rt256bc4 den at t=20,40,60,80


E.L.Vold

RT mix width for 256 case

mix layer width, h mix width growth coef, α

normalized time normalized time0.0 2.5 5.0 7.5 10.0

0.00

0.05

0.10

0.15

0.20

0.25

agt2

alph

abub

alphabub vs agt2

0 2 4 6 8 10

0.0

0.2

0.4

0.6

0.8

agt2

hbub

hbub vs agt2

spikes

spikesbubbles

bubbles


E.L.Vold

RT mix width for varying compressibility

0 1 2 3 4

0.00

0.25

0.50

0.75

1.00

Agt2

h_

b7r

h_b7r vs Agt2

h_b7r h_c11 h_c21

0 1 2 3 4

0.20

0.25

0.30

0.35

0.40

Agt2

a_

b7r

a_b7r vs Agt2

a_b7r a_c11 a_c21

total mix layer width, h mix width growth coef, α

normalized time normalized time

0.30.030.003

0.30.030.003

∼ ∆p/p∆p/p ~


E.L.Vold

BC4- Zint(IC) fixedgrids 1282, 2562 & 5122 - each grid result on 5122


E.L.Vold

BC4- Zint(IC) fixedgrids 1282, 2562 & 5122 - actual grid results

w/ Intrf.Recon. w/o Intrf.Recon.


E.L.VoldAlpha (RT mix growth rate)BC4 -fixed Zint(IC) -

varying grid res. and w/ & w/o Intrf.Recon.

0.0 2.5 5.0 7.5 10.0

0.00

0.05

0.10

0.15

0.20

agt2 _r14

ab_

r14

ab_r14 vs agt2 _r14

0.00 1.25 2.50 3.75 5.00 6.25

0.00

0.05

0.10

0.15

0.20

agt2 _r24ab

_r2

4

ab_r24 vs agt2 _r24

0.00 1.25 2.50 3.75 5.00 6.25

0.00

0.05

0.10

0.15

0.20

agt2 _r54

ab_

r54

ab_r54 vs agt2 _r54

0.00 1.25 2.50 3.75 5.00 6.25

0.00

0.05

0.10

0.15

0.20

agt2 _r4i

ab_

r4i

ab_r4i vs agt2 _r4i

1282 2562

2562 w/o Intrf5122


E.L.VoldBC2- Vf(IC) fixedgrids 1282, 2562, 5122 actual grid dimensions


E.L.VoldAlpha (RT mix growth rate)BC2 -fixed Vf(IC) -

varying grid res. and w/ & w/o Intrf.Recon.

0.0 2.5 5.0 7.5 10.0

0.00

0.05

0.10

0.15

0.20

0.25

agt2 _r12

ab_

r12

ab_r12 vs agt2 _r12

0.00 1.25 2.50 3.75 5.00 6.25

0.00

0.05

0.10

0.15

0.20

0.25

agt2 _r22ab

_r2

2

ab_r22 vs agt2 _r22

0.00 1.25 2.50 3.75 5.00 6.25

0.00

0.05

0.10

0.15

0.20

0.25

agt2 _r52

ab_

r52

ab_r52 vs agt2 _r52

0.00 1.25 2.50 3.75 5.00

0.00

0.05

0.10

0.15

0.20

0.25

agt2 _r2i

ab_

r2i

ab_r2i vs agt2 _r2i

1282 2562

2562 w/o Intrf5122


E.L.VoldBC2 (fixed VF(IC)) vs. BC4 (fixed Zint(IC))are the same structures seen across grid res. in either case?

128 256 512

BC2

BC4


E.L.Vold

RT resolved simulation casesalpha summary for At=0.8

Base cases - bubbles

Base cases - spikes

delta(IC)=delta(base case)/2 - bubble or spike

w/af- atomic mixing bydrift flux momenta

x-ave contours eval.

IC: e = e0, ρ = ρ(z).

100 200 300 400 500 600

0.00

0.05

0.10

0.15

0.20

N grid points

alph

a

alpha-spike and alpha-bubble vs. number of grid points

4 pts inbase case

Possible base case point

Alphaspikes

Alphabubbles


E.L.Vold

0 2 4 6 8 10

0.0

0.1

0.2

0.3

0.4

0.5

Agt2lt10

hmoa

gt2l

t10d

_5a

1

hmoagt2lt10d _5a1 vs Agt2lt10

alpha0 = 1

alpha0 = 0.5

Analytic model compared to fluid equations

alpha(tot)for Dc=0.5alpha0 = 1alpha0=0.5

h/Agt2(alpha0=1,Agt2>>100) -> 0.25

h/Agt2(alpha0=0.5,Agt2>>100) -> 0.17

∂∂h

tu=

∂∂ δ α

βu

tC

u

hAgi

Dii

o o ii+

+=

2

( )

Agt2

alph

a

alpha result (h/Agt2)Analytic model:

Fluid model:

∂∂h

tum

g~

∂

∂ ρ

u

tu u a Ag

p+ ⋅∇ = −

∇~

∂

∂ω

u

t

uu as+

∇− × =

2

2∂

∂ω

u

t

u

L

u

Lu ax

x

y

x

+ + − ×2 2

2 2~

OR:

OR:

d < 1.e-4


E.L.Vold

h/Agt2 as f[delta0]

1e-05 1e-04 1e-03 1e-02 1e-01 1e+00

0.00

0.05

0.10

0.15

delta0

h/ A

gt2

h / Agt2

hoh0 _hvt hoh0 _hintvt hoh0 _vot

hoh0_hintvt:v determined as:∂∂ δ α

v

t

v

v dta

o c

++

=∫

2

( ) αc = 0.12

δο or tο

Normalize δo to L = Agt2

h/Agt2 is not f[delta]

h/Agt2 isf[delta]

for comparison to comput. setupδo/L=1.e-2/5=0.002

comput on 128 grid:δo/L~var(Vf-IC)/L~0.7dx/70dxor δo/L(128)~1.e-2

δo ~ variance of Vf(IC)

comput on 256 grid:δo/L~var(Vf-IC)/L~0.7dx/140dxor δo/L(256)~0.5e-2


E.L.Vold

0 50 100 150 200 250 300

0.0

0.2

0.4

0.6

0.8

waveNo _or _freq _512

FF

T_

PS

Dam

p_

512

FFT_PSDamp_512 vs waveNo _or _freq _512

0 50 100 150 200 250 300

0.0

0.5

1.0

1.5

2.0

waveNo _256

FF

T_

PS

DA

mp

_25

6

FFT_PSDAmp_256 vs waveNo _256

0 50 100 150 200 250 300

0.0

0.5

1.0

1.5

2.0

2.5

waveNo _128

FF

T_

PS

DA

mp

_12

8


0 50 100 150 200 250 300

1e-03

1e-02

1e-01

1e+00

1e+01

1e-03

1e-02

1e-01

1e+00

1e+01

waveNo _256

FF

T_

PS

DA

mp

_25

6

FF

T_

PS

DA

mp

_128


512 pts 256 pts 128 pts

0 50 100 150 200 250 300

1e-07

1e-06

1e-05

1e-04

1e-03

1e-02

1e-01

1e+00

1e+01

1e-07

1e-06

1e-05

1e-04

1e-03

1e-02

1e-01

1e+00

1e+01

waveNo _or _freq _512

FF

T_

PS

DA

mp

_51

2

FF

T_

PS

DA

mp

_256

FFT_PSDAmp_512 vs waveNo _or _freq _512

256 & 512 pts 256 & 128 pts

lin-lin plot lin-lin plot lin-lin plot

log-lin plot log-lin plot

source modes

mode spill-out amplitudes

mode spill-out amplitude- 256 pts

mode spill-out amplitude- 512 pts

RT spectral density for IC perturbationw/ modes k=30-60: zint(t=0) ~ Vf


E.L.Vold

0.00 0.25 0.50 0.75 1.00 1.25

1001.25

1001.00

1000.75

1000.50

1000.25

1000.00

col

ro

w

1.125e-061.250e-061.375e-061.500e-06r24t80_4_md

locations for FFTs at t80

0.00 0.25 0.50 0.75 1.00 1.25

1001.25

1001.00

1000.75

1000.50

1000.25

1000.00

col

ro

w

0.00025 0.00050 0.00075 0.00100r24t80_3_md

z=0.5675,0.6374,0.7775

sing

le d

ata

line

inpu

ts to

FFT

rang

e of

dat

a lin

es in

put t

o av

erag

e FF

T


E.L.Vold

FFTs p-t80 128 & 512

1e+00 1e+01 1e+02 1e+03 1e+04

1e-10

1e-09

1e-08

1e-07

1e-06

1e-10

1e-09

1e-08

1e-07

1e-06

k_512

FF

Tco

_pt

80g5

12Z

0reg

avx1

0

FF

Tco

_pt80g128z0regav

FFTco_pt80g512Z0regavx10 vs k _512

FFTco_pt80g512Z0regavx10 FFTco_pt80g128z0regav

1e+00 1e+01 1e+02 1e+03 1e+04

1e-11

1e-10

1e-09

1e-08

1e-07

1e-11

1e-10

1e-09

1e-08

1e-07

k_128

FF

Tco

_pt

80g1

28z0

rega

v

FF

Tco

_pt80g512Z

0regav

FFTco_pt80g128z0regav vs k _128

FFTco_pt80g128z0regav FFTco_pt80g512Z0regav

p-t80 128 & 512 p-t80 128 & 10x p-t80 512

Therefore, unresolved components on 128 grid appear to beirrelevant to small k, long wavelength mode growth**.

p spectra (left) w/ 10x shift on 512 grid (right) to compare

= modecut-offon coarsergrid

= modecut-offon coarsergrid

** alternative explanation: numerical errors on 128 grid and resolved highmode numbers on 512 grid both have the same effect on the small k, longwavelength mode growth***.*** alternative alternate: the alternate is true and the effect in either case is ~ 0.

wave number wave number


E.L.Vold

FFTs p-t80, grids compared128 & 512, 256 & 512, 128 & 256

1e+00 1e+01 1e+02 1e+03 1e+04

1e-11

1e-10

1e-09

1e-08

1e-07

1e-11

1e-10

1e-09

1e-08

1e-07

k_128

FF

Tco

_pt

80g1

28z0

rega

v

FF

Tco

_pt80g512Z

0regav



1e+00 1e+01 1e+02 1e+03 1e+04

1e-11

1e-10

1e-09

1e-08

1e-07

1e-11

1e-10

1e-09

1e-08

1e-07

k_256

FF

Tco

_pt

80g2

56z0

rega

v

FF

Tco

_pt80g512Z

0regav



1e+00 1e+01 1e+02 1e+03

1e-11

1e-10

1e-09

1e-08

1e-07

1e-11

1e-10

1e-09

1e-08

1e-07

k_128

FF

Tco

_pt

80g1

28z0

rega

vF

FT

co_

pt80g256z0regav


FFTco_pt80g128z0regav FFTco_pt80g256z0regav

= range of IC modes

128 & 512 256 & 512

128 & 256

= mode cut-offon coarser grid


E.L.Vold

RT mix at late time -varying AtwAt = 0.96ρ1/ρ2 = 50t = 60 (z = 2.6)

At = 0.8ρ1/ρ2 = 9t = 80 (z = 3.84)

At = 0.33ρ1/ρ2 = 2t = 120 (z = 3.6)

At = 0.048ρ1/ρ2 = 1.1t = 320 (z = 3.66)


E.L.Vold

Alpha (RT growth rate)for varying Atwood number and w/ & w/o Interface Recon.

0.00 1.25 2.50 3.75 5.00 6.25

0.00

0.05

0.10

0.15

0.20

agt2 _r24

ab_

r24

ab_r24 vs agt2 _r24

0.00 1.25 2.50 3.75 5.00 6.25

0.00

0.05

0.10

0.15

0.20

agt2 _r4i

ab_

r4i

ab_r4i vs agt2 _r4i

0.0 2.5 5.0 7.5 10.0

0.0

0.1

0.2

0.3

0.4

agt2 _a3i

ab_

a3i

ab_a3i vs agt2 _a3i

0.0 2.5 5.0 7.5 10.0

0.0

0.1

0.2

0.3

0.4

agt2 _a33

ab_

a33

ab_a33 vs agt2 _a33

A=0.8, w/IR A=0.8, w/o IR

A=0.33, w/IR A=0.33, w/o IR

BC4


E.L.Vold

0 2 4 6 8

-0.1

0.0

0.1

0.2

0.3

0.4

agt2 _a4i

ab_

a4i

ab_a4i vs agt2 _a4i

0 2 4 6 8

-0.1

0.0

0.1

0.2

0.3

0.4

agt2 _a04

ab_

a04

ab_a04 vs agt2 _a04

0.00 1.25 2.50 3.75 5.00 6.25

0.00

0.05

0.10

0.15

0.20

0.25

agt2 _a96

ab_

a96

ab_a96 vs agt2 _a96

Alpha (RT growth rate)for varying Atwood number and w/ & w/o Interface Recon.

A=0.96, w/IR

A=0.96, w/o IR

A=0.048, w/IR A=0.048, w/o IR

0.0 0.5 1.0 1.5 2.0 2.5 3.0

0.00

0.05

0.10

0.15

0.20

0.25

agt2 _a9iab

_a9

i

ab_a9i vs agt2 _a9i


E.L.Vold

Conclusions: R-T Mix Layer Growth• Results apply to 2-D multi-mode-IC simulations for At=0.8

• Grid convergence is good for alpha bubbles - less certain for alpha spikes.

• Alpha bubble computed here, ~ 0.05-0.065, agrees with experimental data.

• Discrepancy with other computations predicting lower alpha (~0.03)

– may be mostly due to treatment of internal energy discontinuity at interface and/or the internalenergy in the long wavelength IC which contributes to growth rate through energy fluctuations.

– a smaller difference (~15%) is expected between 2-D and 3-D.

– It is shown to be unlikely that discrepancy is related to compressibility, hydrostatic equilibriumform, IC mode amplitudes or IC mode spectra details, or front evaluation methods..

• Internal energy fluctuations dominate over density fluctuations where (eo/ρρρρo) is sufficientlysmall in the mix layer -

– this occurs in the heavier fluid even in limit as compressibility becomes 'negligible'.

– eo is irrelevant in ‘ideal’ incompressible fluid, so only density fluctuations matter.

• Transition from early time IC dominated regime to later time self-similar solutions is evidentand agrees with analytic results.

• The resolved simulations appear to be adequately represented in the multi-fluid model so thatwe can now proceed to use the multi-fluid drift-flux model to represent the molecular mixingand/or sub-grid scale turbulent mixing within the Rayleigh-Taylor unstable mix layer.

multi-fluid model rayleigh-taylor mix layer growth with a ... · recent computational simulations...

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