Viscous Compaction Waves Joseph M. Powers and Michael T. Cochran - - PowerPoint PPT Presentation

Viscous Compaction Waves Joseph M. Powers and Michael T. Cochran University of Notre Dame, Notre Dame, Indiana SIAM Annual Meeting New Orleans, Louisiana 12 July 2005

Compaction Wave Schematic

u ~ 100 m/s

p

φ ~ 0.73 φ ~ 0.98

D ~ 400 m/s

s s 100 µm particles

Introduction

• Heterogeneous energetic solids composed of

100 µm crystals in plastic binder.

• Engineering length scales on the order of many cm.
• Macrobehavior (ignition, etc.) strongly linked to

microstructure.

• Continuum mixture models with non-traditional

constitutive theories needed to capture grain scale physics.

Review

• Gokhale and Krier, Prog. Energy Combust. Sci.,

1982.

• Baer and Nunziato, Int. J. Multiphase Flow, 1986.
• Powers, Stewart, Krier, Combust. Flame, 1990.
• Saurel and Abgrall, J. Comput. Phys., 1999.
• Bdzil, et al. Phys. Fluids, 1999, 2001.
• Gonthier and Powers, J. Comput. Phys., 2000.
• Powers, Phys. Fluids, 2004.

Issues with Continuum Mixture Theories

• Well-posedness not always straightforward.
• Second law complicated.
• Shock jumps not clearly defined for non-conservative

equations.

• Consequent numerical difficulties.

Inviscid Theory of Bdzil, et al.

• First theory to unambiguously satisfy the second law.
• Hyperbolic and well-posed for initial value problems.
• Fundamentally non-conservative.
• Some regularization needed for discontinuities.
• No viscous cutoff mechanism for multidimensional

instabilities.

• Grid-dependent numerical viscosity problematic.

Viscous Extension

∂ ∂t (ρsφs) + ∇ · (ρsφsus) = C, ∂ ∂t (ρgφg) + ∇ · (ρgφgug) = −C, ∂ ∂t (ρsφsus) + ∇ ·

• ρsφsusuT

s + φs (psI − τ s)

• = M,

∂ ∂t (ρgφgug) + ∇ ·

• ρgφguguT

g + φg (pgI − τ g)

• = −M,

Viscous Extension (cont.)

∂ ∂t

• ρsφs
• es + 1

2us · us

• +∇ ·
• ρsφsus
• es + 1

2us · us

• + φsus · (psI − τ s) + φsqs
• = E,

∂ ∂t

• ρgφg
• eg + 1

2ug · ug

• +∇ ·
• ρgφgug
• eg + 1

2ug · ug

• + φgug · (pgI − τ g) + φgqg
• = −E,

∂ρs ∂t + ∇ · (ρsus) = −ρsF φs , ∂ ∂t (ρsφsηs + ρgφgηg) +∇ · (ρsφsusηs + ρgφgugηg) ≥ −∇ · φsqs Ts + φgqg Tg

• .

Constitutive Equations

φg + φs = 1, ψs = ˆ ψs(ρs, Ts) + B(φs), ψg = ψg(ρg, Tg), ps = ρ2

s

∂ψs ∂ρs

• Ts,φs

, pg = ρ2

g

∂ψg ∂ρg

• Tg

, ηs = − ∂ψs ∂Ts

• ρs,φs

, ηg = − ∂ψg ∂Tg

• ρg

, βs = ρsφs ∂ψs ∂φs

• ρs,Ts

, es = ψs + Tsηs, eg = ψg + Tgηg,

Constitutive Equations (cont.)

τ s = 2µs (∇us)T + ∇us 2 − 1 3(∇ · us)I

• ,

τ g = 2µs (∇ug)T + ∇ug 2 − 1 3(∇ · ug)I

• ,

qs = −ks∇Ts, qg = −kg∇Tg, C = C(ρs, ρg, Ts, Tg, φs), M = pg∇φs − δ(us − ug) + 1 2(us + ug)C, E = H(Tg − Ts) − pgF + us · M +

• es − us · us

2

• C,

F = φsφg µc (ps − βs − pg).

Equations of State Modified Tait equation for solid (correction courtesy

• D. W. Schwendeman)

ψs(ρs, Ts, φs) = cvsTs

• 1 − ln

Ts Ts0

• + (γs − 1) ln

ρs ρs0

• + 1

γs ρs0 ρs εs + q +(ps0 − pg0) (2 − φs0)2 ρs0φs0 ln

• 1

1−φs0

• ln

  2 − φs0 2 − φs (1 − φs)

1−φs 2−φs

(1 − φs0)

1−φs0 2−φs0

 

Virial equation for gas

ψg(ρg, Tg) = cvgTg

• 1 − ln

Tg Tg0

• + (γg − 1)
• ln

ρg ρg0

• + bg(ρg − ρg0)

Viscous Dissipation Function

Φs = 2µs     ∇us + (∇us)T 2

• strain rate

− 1 3(∇ · us)

• mean strain rate

I    

• deviatoric strain rate

:     ∇us + (∇us)T 2

• strain rate

− 1 3(∇ · us)

• mean strain rate

I    

• deviatoric strain rate

.

• similar expression for Φg.

Dissipation: Clausius-Duhem Equation

I ≡ (−C) βs ρsTs + es − eg − pg(1/ρg − 1/ρs) Tg + ηg − ηs

• +δ (us − ug) · (us − ug)

Tg +H(Tg − Ts)2 TgTs +φsφg µc (ps − βs − pg)2 Ts +φsΦs Ts + φgΦg Tg +ksφs∇Ts · ∇Ts T 2

s

+ kgφg∇Tg · ∇Tg T 2

g

≥ 0.

Characteristics

• Three real characteristics us, us, ug,
• Three associated eigenvectors,
• Not enough eigenvectors for eleven equations:

parabolic,

• Eight additional conditions from boundary conditions
• n Ts, Tg, us, ug.

Numerical Method

• One-Dimensional: Fortran 90 code

– Second order central spatial discretization – High order implicit integration in time with DLSODE

• Two-Dimensional: FEMLAB software tool

– Finite element method for the form

∂q ∂t + ∇ · f(q) = s(q).

– Unstructured mesh

1D Verification: Shock Tube

0.1 0.2 0.3 0.4 0.5 x (m) 300 305 T (K) 290 310 295

Tg Ts

0.001 0.01 0.1 ∆x (m) 0.01 0.1 1 L (K)

1

0.001 0.0001

slope=1.95 slope=0.75 A1 A2 inviscid analytical viscous numerical

1D Verification: Piston-Driven Shock

0.1 0.2 0.3 0.4 0.5 x (m) 310 320 330 340 T (K) 300

solid gas B1

0.18 0.22 0.24 0.26 0.28 x (m) 305 310 315 320 325 T (K)

g

steady solution time-dependent solution

0.20

viscous shock in gas B3

0.37 0.38 0.39 x (m) 310 320 330 340 T (K)

steady solution time-dependent solution

0.36 300

s

viscous shock in solid B2

1D Subsonic Piston-Driven Compaction

0.1 0.2 0.3 0.4 0.5 x (m) 0.5 1 5 10 50 p (MPa)

p

s

p

g

βs E1

0.1 0.2 0.3 0.4 0.5 x (m) 20 40 60 80 100 u (m/s)

u , u

s g

E3

0.1 0.2 0.3 0.4 0.5 x (m) 301 302 303 304 T (K)

T , T

s g

0.1 0.2 0.3 0.4 0.5 x (m) 0.2 0.4 0.6 0.8 1 φs

E2

1D Dissipation: Subsonic Case

0.1 0.2 0.3 0.4 0.5 x (m) 50 100 150 200

I (MW/m /K)

3

Total Compaction Solid Momentum Diffusion

E

FEMLAB vs. F90 Verification: 1D Shock Tube

x [m]

310

305 295

290

300

x [m]

1.4 x 10-3 0 0.1 0.2 0.3 0.4 0.5 0 0.1 0.2 0.3 0.4 0.5 R e l a t i v e E r r

• r

T [K] g 0.5 1.0

Narrow 2D Shock Tube vs. 1D Shock Tube

300 K 292 K 310 K 300 K x (m) 0 0.1 0.2 0.3 0.4 0.5

290 300 310 x (m) t = 50 µs

T (K)

g 0.0 0.1 0.2 0.3 0.4 0.5

2D 1D

Small Energy Pulse: 2D Response

. . .

t = 18 μs t = 0 μs Solid Pressure small temperature perturbation radial pressure wave at ~ 2000 m/s reflection at wall

Large Energy Pulse: 2D Response

φs

Max = 0.867 Min=0.730 (x,y) 0.040 0.000

• 0.015

0 0.035 0.05 x (m) y (m)

Conclusions

• Diffusion enables use of simple numerical techniques.
• Diffusion suppresses short wavelength instabilities,

e.g. Kelvin-Helmholtz.

• Diffusion suppresses subgranular length scales.
• Compaction dominates the dissipation.
• Rigorous subscale physical justification for diffusion

models presently lacking.

• Such justification necessary for a validated model.