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(1)

Wave Propagation Methods for

Compressible Multicomponent Flow with

Moving Interfaces and Boundaries

Keh-Ming Shyue

Department of Mathematics National Taiwan University

(2)

Overview

Illustrative examples Mathematical model

Fluid-mixture type equations of motion for homogeneous two-phase flow

General pressure law for real materials Numerical techniques

Finite volume method based on wave propagation Surface tracking for moving boundaries

Volume tracking for moving interfaces Numerical results

Future work

(3)

Falling Liquid Drop Problem

100 200 300 400 500 600 700 800 900 1000

air t = 0

(4)

Falling Liquid Drop Problem

100 200 300 400 500 600 700 800 900 1000

air

t = 0.2s

(5)

Falling Liquid Drop Problem

100 200 300 400 500 600 700 800 900

air

t = 0.4s

(6)

Falling Liquid Drop Problem

100 200 300 400 500 600 700 800 900

air

t = 0.6s

(7)

Falling Liquid Drop Problem

100 200 300 400 500 600 700 800 900

air

t = 0.8s

(8)

Falling Liquid Drop Problem

100 200 300 400 500 600 700 800 900

air

t = 1s

(9)

Flying Projectile & Ocean Surface

−10 0 10 20 30 40 50

−20

−15

−10

−5 0 5 10 15 20

air

water t = 0

(10)

Flying Projectile & Ocean Surface

−10 0 10 20 30 40 50

−20

−15

−10

−5 0 5 10 15 20

air

water t = 12ms

(11)

Flying Projectile & Ocean Surface

−10 0 10 20 30 40 50

−20

−15

−10

−5 0 5 10 15 20

air

water t = 24ms

(12)

Flying Projectile & Ocean Surface

−10 0 10 20 30 40 50

−20

−15

−10

−5 0 5 10 15 20

air

water t = 36ms

(13)

Flying Projectile & Ocean Surface

−10 0 10 20 30 40 50

−20

−15

−10

−5 0 5 10 15 20

air

water t = 48ms

(14)

Flying Projectile & Ocean Surface

−10 0 10 20 30 40 50

−20

−15

−10

−5 0 5 10 15 20

air

water t = 60ms

(15)

Two Phase Flow Problem

Ignore physical effects such as gravity, viscosity, surface tension, mass diffusion, and so on

Each fluid component satisties

Eulerian form conservation laws

ρt + ∇ · (ρ~u) = 0 (ρ~u)t + ∇ · (ρ~u ⊗ ~u) + ∇p = 0 (ρE)t + ∇ · (ρE~u + p~u) = 0 General pressure law p(ρ, e)

ρ: density, ~u: vector of particle velocity, p: pressure E: specific total energy, e: specific internal energy

(16)

Two-Phase Flow Model

Model derivation based on averaging theory of Drew (Theory of Multicomponent Fluids, D.A. Drew & S. L.

Passman, Springer, 1999)

Namely, introduce indicator function χk as χk(M, t) =

(1 if M belongs to phase k

0 otherwise

Denote < ψ > as volume averaged for flow variable ψ, hψi = 1

V Z

V

ψ dV Gauss & Leibnitz rules

k∇ψi = h∇(χkψ)i−hψ∇χki & hχkψti = h(χkψ)ti−hψ(χk)ti

(17)

Two-Phase Flow Model (cont.)

Take product of each conservation law with χk & perform averaging process. In case of mass conservation equation, for example, we have

kρkit + ∇· < χkρk~uk >= hρkk)t + ρk~uk · ∇χki Since χk is governed by

k)t + ~u0 · ∇χk = 0 (~u0: interface velocity), this leads to mass averaged equation for phase k

kρkit + ∇· < χkρk~uk >= hρk (~uk − ~u0) · ∇χki Analogously, we may derive averaged equation for momentum, energy, & entropy (not shown here)

(18)

Two-Phase Flow Model (cont.)

In summary, averaged model system, we have, are

kρkit + ∇· < χkρk~uk >= hρk (~uk ~u0) · ∇χki kρk~ukit + ∇· < χkρk~uk ⊗ ~uk > +∇ hχkpki = hpk∇χki +

k~uk (~uk ~u0) · ∇χki kρkEkit + ∇· < χkρkEk~uk + χkpk~uk >= hpk~uk · ∇χki +

kE (~uk ~u0) · ∇χki kit + h~uk · ∇χki = h(~uk ~u0) · ∇χki

Note: existence of various interfacial source terms

Mathematical as well as numerical modelling of these terms are important (but difficult) for general multiphase flow

problems

(19)

Homogeneous 2-Phase Flow Model

Assume homogeneous (1-pressure & 1-velocity) flow;

i.e., across interfaces: pι = p & ~uι = ~u, ι = 0, 1, 2

Introduce volume fraction for phase k as αk = Vk/V Now, by dropping all interfacial terms, we may obtain a simplified model as

kρk)t + ∇ · (αkρk~u) = 0

kρk~u)t + ∇ · (αkρk~u ⊗ ~u) + ∇ (αkp) = p∇αk kρkEk)t + ∇ · (αkρkEk~u + αkp~u) = p~u · ∇αk

1)t + ~u · ∇α1 = 0

for k = 1, 2, & α1 + α2 = 1. Note this gives 2(2 + Nd) + 1

equations in total for a Nd-dimension 2-phase flow problem

(20)

Homogeneous Flow Model (cont.)

Note that, in practice, rather than using equations αkρk~u &

αkρkEk for each phase, we may write down a system of the form

kρk)t + ∇ · (αkρk~u) = 0 (ρ~u)t + ∇ · (ρ~u ⊗ ~u) + ∇p = 0 (ρE)t + ∇ · (ρE~u + p~u) = 0 1)t + ~u · ∇α1 = 0

ρ~u = P2

k=1 αkρk~u: total momentum ρE = P2

k=1 αkρkEk: total energy

This gives 4 + Nd equations in total, Nd + 1 less than previous model system

(21)

Homogeneous Flow Model (cont.)

Note that it is easy to include, for instance, gravity &

capillary effects to the model

kρk)t + ∇ · (αkρk~u) = 0 (k = 1, 2) (ρ~u)t + ∇ · (ρ~u ⊗ ~u) + ∇p = φ~

(ρE)t + ∇ · (ρE~u + p~u) = φ · ~u~ 1)t + ~u · ∇α1 = 0

1. Gravity case: φ = ~g~

2. Capillary case: φ = σκ∇α~

~g: gravitational constant, σ: surface tension coef.

κ: curvature at interface

(22)

Homogeneous Flow Model (cont.)

Mixture equation of state: p = p(α2, α1ρ1, α2ρ2, ρe) Isobaric closure: p1 = p2 = p

For a class of EOS, explicit formula for p is available (examples are given next)

For some complex EOS, from (α2, ρ1, ρ2, ρe) in model equations we recover p by solving

p11, ρ1e1) = p22, ρ2e2) &

2

X

k=1

αkρkek = ρe This homogeneous two-phase model was called a five-equation model by Allaire, Clerc, & Kokh (JCP

2002) or a volume-fraction model by Shyue (JCP 1998)

(23)

Homogeneous Flow Model (cont.)

Polytropic ideal gas: pk = (γk − 1)ρkek

ρe =

2

X

k=1

αkρkek =

2

X

k=1

αk p

γk − 1 =⇒

p = ρe

 2 X

k=1

αk γk − 1

(24)

Homogeneous Flow Model (cont.)

Polytropic ideal gas: pk = (γk − 1)ρkek

ρe =

2

X

k=1

αkρkek =

2

X

k=1

αk p

γk − 1 =⇒

p = ρe

 2 X

k=1

αk γk − 1

Van der Waals gas: pk = (1−γkb−1kρk )(ρkek + akρ2k) − akρ2k

ρe =

2

X

k=1

αkρkek =

2

X

k=1

αk  1 − bkρk γk − 1



(p + akρ2k) − akρ2k



=⇒

p =

"

ρe −

2

X

k=1

αk  1 − bkρk

γk − 1 − 1



akρ2k

# 2 X

k=1

αk  1 − bkρk

γk − 1



(25)

Homogeneous Flow Model (cont.)

Two-molecular vibrating gas: pk = ρkRkT (ek), T satisfies

e = RT

γ − 1 + RTvib

exp Tvib/T − 1

As before, we now have

ρe =

2

X

k=1

αkρkek =

2

X

k=1

αk

 ρkRkTk γk − 1



+ ρkRkTvib,k exp 

Tvib,k/Tk

− 1

=

2

X

k=1

αk

 p

γk − 1



+ pvib,k exp

pvib,k/p

− 1

(Nonlinear eq.)

(26)

Homogeneous Flow Model (cont.)

It is easy to show entropies, Sk, k = 1, 2, satisfy

 ∂p1

∂S1



ρ1

DS1

Dt  ∂p2

∂S2



ρ2

DS2

Dt = ρ1c21 − ρ2c22 ∇ · ~u

(27)

Homogeneous Flow Model (cont.)

It is easy to show entropies, Sk, k = 1, 2, satisfy

 ∂p1

∂S1



ρ1

DS1

Dt  ∂p2

∂S2



ρ2

DS2

Dt = ρ1c21 − ρ2c22 ∇ · ~u

Murrone & Guillard (JCP 2005) propsed a reduced two-phase flow model in which

1)t + ~u · ∇α1 = α1α2 ρ2c22 − ρ1c21

P2

k=1 αkρkc2k

!

and now entropy of each phase satisfy

DSk

Dt = ∂Sk

∂t + ~u · ∇Sk = 0, for k = 1, 2

(28)

Some Remarks

1. Model system is hyperbolic under suitable thermodynamic stability condition

(29)

Some Remarks

1. Model system is hyperbolic under suitable thermodynamic stability condition

2. When α2 = 0 (or = 1), ρ2 (or ρ1) can not be recovered from α2 & α2ρ2 (or α1 & α1ρ1), and so take αk ∈ [ε, 1 − ε]

(30)

Some Remarks

1. Model system is hyperbolic under suitable thermodynamic stability condition

2. When α2 = 0 (or = 1), ρ2 (or ρ1) can not be recovered from α2 & α2ρ2 (or α1 & α1ρ1), and so take αk ∈ [ε, 1 − ε]

3. In the model, it is not at all clear on how to compute nonlinear term ρι, ι > 1 from αk & αkρk

(31)

Some Remarks

1. Model system is hyperbolic under suitable thermodynamic stability condition

2. When α2 = 0 (or = 1), ρ2 (or ρ1) can not be recovered from α2 & α2ρ2 (or α1 & α1ρ1), and so take αk ∈ [ε, 1 − ε]

3. In the model, it is not at all clear on how to compute nonlinear term ρι, ι > 1 from αk & αkρk

4. In fact, there are other set of model systems proposed in the literature that are more robust for homogeneous flow & in other more complicated context (examples)

(32)

Some Remarks

1. Model system is hyperbolic under suitable thermodynamic stability condition

2. When α2 = 0 (or = 1), ρ2 (or ρ1) can not be recovered from α2 & α2ρ2 (or α1 & α1ρ1), and so take αk ∈ [ε, 1 − ε]

3. In the model, it is not at all clear on how to compute nonlinear term ρι, ι > 1 from αk & αkρk

4. In fact, there are other set of model systems proposed in the literature that are more robust for homogeneous flow & in other more complicated context (examples) 5. In cases when individual pressure law differs in form

(see below), new mixture pressure law should be

devised first & construct model equations based on that

(33)

Barotropic & Non-Barotropic Flow

Fluid component 1: Tait EOS

p(ρ) = (p0 + B)  ρ ρ0

γ

− B

Fluid component 2: Noble-Abel EOS

p(ρ, e) =  γ − 1 1 − bρ

 ρe

Mixture pressure law (Shyue, Shock Waves 2006)

p =

(p0 + B) ρ ρ0

γ

− B if α = 1

 γ − 1 1 − bρ



(ρe − B) − B if α 6= 1

(34)

Barotropic Two-Phase Flow

Fluid component ι: Tait EOS

p(ρ) = (p0ι + Bι)

 ρ ρ0ι

γι

− Bι, ι = 1, 2

Mixture pressure law (Shyue, JCP 2004)

p =

(p0ι + Bι)

 ρ ρ0ι

γι

− Bι if α = αι (0 or 1) (γ − 1) ρ



e + B ρ0



− γB if α ∈ (0, 1)

(35)

Wave Propagation Method

Finite volume formulation of wave propagation method, QnS gives approximate value of cell average of solution q over cell S at time tn

QnS 1 M(S)

Z

S

q(X, tn) dV

M(S): measure (area in 2D or volume in 3D) of cell S

C E

D F

G H

(36)

Wave Propagation Method (cont.)

First order version: Piecewise constant wave update Godunov-type method: Solve Riemann problem at each cell interface in normal direction & use resulting waves to update cell averages

Qn+1S := Qn+1S −M (Wp ∩ S)

M(S) Rp, Rp being jump from RP

Wp Wp

(37)

Wave Propagation Method (cont.)

First order version: Transverse-wave included

Use transverse portion of equation, solve Riemann problem in transverse direction, & use resulting

waves to update cell averages as usual

Stability of method is typically improved, while conservation of method is maintained

↓ ↓

Wpq

Wpq

(38)

Wave Propagation Method (cont.)

High resolution version: Piecewise linear wave update wave before propagation after propagation

a) b)

c) d)

αpr

p/2

αpr

p/2

λp∆ t

λp∆ t

(39)

Volume Tracking Algorithm

1. Volume moving procedure (a) Volume fraction update

Take a time step on current grid to update cell averages of volume fractions at next time step (b) Interface reconstruction

Find new interface location based on volume fractions obtained in (a) using an interface reconstruction scheme. Some cells will be subdivided & values in each subcell must be initialized.

2. Physical solution update

Take same time interval as in (a), but use a method to update cell averages of multicomponent model on new grid created in (b)

(40)

Interface Reconstruction Scheme

Given volume fractions on current grid, piecewise linear interface reconstruction (PLIC) method does:

1. Compute interface normal

Gradient method of Parker & Youngs Least squares method of Puckett

2. Determine interface location by iterative bisection

0 0

0 0

0 0

0 0

0 0

0

0 0

0 0

0 0

0 0

0

1 0.29 0.68 0.09

0.51 0.51

0.09 0.68 0.29

↓ interface

interface

Data set Parker & Youngs Puckett

(41)

Volume Moving Procedure

(a) Volume fractions given in previous slide are updated with uniform (u, v) = (1, 1) over ∆t = 0.06

(b) New interface location is reconstructed

0

0 0 0 0

0 0 0 0

0 0

0 0

0 0

0 0

0

1 0.01

0.38 0.11

0.25 0.72

0.06 0.85

0.74 5(−3) 1(−3)

old interface

new interface

(a) (b)

(42)

Surface Moving Procedure

Solve Riemann problem at tracked interfaces & use

resulting wave speed of the tracked wave family over ∆t to find new location of interface at the next time step

o

o

o o

o

o

↑ old front

old front

new front

(43)

Boundary Conditions

For tracked segments representing rigid (solid wall)

boundary (stationary or moving), reflection principle is used to assign states for fictitious subcells in each time step:

zC := zE (z = ρ, p, α)

~uC := ~uE − 2(~uE · ~n)~n + 2(~u0 · ~n)

~u0: moving boundary velocity

C

E F

G H

~n

(44)

Interface Conditions

For tracked segments representing material interfaces, pressure equilibrium as well as velocity continuity

conditions across interfaces are fulfilled by 1. Devise of the wave-propagation method

2. Choice of Riemann solver used in the method

(45)

Stability Issues

Choose time step ∆t based on uniform grid mesh size

∆x, ∆y as

∆t maxp,qp, µq)

min(∆x, ∆y) ≤ 1,

λp, µq: speed of p-wave, q-wave from Riemann

problem solution in normal-, transverse-directions Use large time step method of LeVeque (i.e., wave

interactions are assumed to behave in linear manner) to maintain stability of method even in the presence of

small Cartesian cut cells

Apply smoothing operator (such as, h-box approach of Berger et al. ) locally for cell averages in irregular cells

(46)

Shock-Bubble Interaction Problem

(47)

Shock-Bubble Interaction Problem

(48)

Shock-Bubble Interaction Problem

(49)

Shock-Bubble Interaction Problem

(50)

Shock-Bubble Interaction Problem

(51)

Shock-Bubble Interaction Problem

(52)

Shock-Bubble Interaction Problem

(53)

Shock-Bubble Interaction Problem

(54)

Shock-Bubble Interaction Problem

(55)

Shock-Bubble Interaction Problem

(56)

Shock-Bubble Interaction (cont.)

Approximate locations of interfaces

time=55µs

air R22

time=115µs time=135µs

time=187µs time=247µs time=200µs

time=342µs time=417µs time=1020µs

(57)

Shock-Bubble Interaction (cont.)

Quantitative assessment of prominent flow velocities:

Velocity (m/s) Vs VR VT Vui Vuf Vdi Vdf Haas & Sturtevant 415 240 540 73 90 78 78 Quirk & Karni 420 254 560 74 90 116 82 Our result (tracking) 411 243 538 64 87 82 60 Our result (capturing) 411 244 534 65 86 98 76

Vs (VR, VT) Incident (refracted, transmitted) shock speed t ∈ [0, 250]µs (t ∈ [0, 202]µs, t ∈ [202, 250]µs ) Vui (Vuf) Initial (final) upstream bubble wall speed t ∈ [0, 400]µs (t ∈ [400, 1000]µs)

Vdi (Vdf) Initial (final) downstream bubble wall speed t ∈ [200, 400]µs (t ∈ [400, 1000]µs)

(58)

Aluminum-Plate Impact Problem

0 1 2 3

−2

−1 0 1 2

Density

Al target Al flyer

vacuum

0 1 2 3

−2

−1 0 1 2

Pressure

0 1 2 3

−2

−1 0 1 2

Volume fraction

t = 0µs

(59)

Aluminum-Plate Impact Problem

0 1 2 3

−2

−1 0 1 2

Density

Al target Al flyer

vacuum

0 1 2 3

−2

−1 0 1 2

Pressure

0 1 2 3

−2

−1 0 1 2

Volume fraction

t = 0.5µs

(60)

Aluminum-Plate Impact Problem

0 1 2 3

−2

−1 0 1 2

Density

Al target Al flyer

vacuum

0 1 2 3

−2

−1 0 1 2

Pressure

0 1 2 3

−2

−1 0 1 2

Volume fraction

t = 1µs

(61)

Aluminum-Plate Impact Problem

0 1 2 3

−2

−1 0 1 2

Density

Al target Al flyer

vacuum

0 1 2 3

−2

−1 0 1 2

Pressure

0 1 2 3

−2

−1 0 1 2

Volume fraction

t = 2µs

(62)

Aluminum-Plate Impact Problem

0 1 2 3

−2

−1 0 1 2

Density

Al target Al flyer

vacuum

0 1 2 3

−2

−1 0 1 2

Pressure

0 1 2 3

−2

−1 0 1 2

Volume fraction

t = 4µs

(63)

Cylinder lift-off Problem

Moving speed of cylinder is governed by Newton’s law Pressure contours are shown with a 1000 × 200 grid

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0 0.05 0.1 0.15 0.2

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0 0.05 0.1 0.15 0.2

0 0.05 0.1 0.15 0.2

t = 0

t = 0.1641s

t = 0.30085s

(64)

Cylinder lift-off Problem

A convergence study of center of cylinder & relative mass loss for at final stopping time t = 0.30085s

Mesh size Center of cylinder Relative mass loss 250 × 50 (0.618181, 0.134456) −0.257528

500 × 100 (0.620266, 0.136807) −0.131474 1000 × 200 (0.623075, 0.138929) −0.066984

Results are comparable with numerical appeared in literature

(65)

Falling Rigid Object in Water Tank

−2 0 2

−3

−2

−1 0 1 2 3

Density air

water

−2 0 2

−3

−2

−1 0 1 2 3

Pressure

−2 0 2

−3

−2

−1 0 1 2 3

Volume fraction

t = 0ms

(66)

Falling Rigid Object in Water Tank

−2 0 2

−3

−2

−1 0 1 2 3

Density air

water

−2 0 2

−3

−2

−1 0 1 2 3

Pressure

−2 0 2

−3

−2

−1 0 1 2 3

Volume fraction

t = 1ms

(67)

Falling Rigid Object in Water Tank

−2 0 2

−3

−2

−1 0 1 2 3

Density air

water

−2 0 2

−3

−2

−1 0 1 2 3

Pressure

−2 0 2

−3

−2

−1 0 1 2 3

Volume fraction

t = 2ms

(68)

Falling Rigid Object in Water Tank

−2 0 2

−3

−2

−1 0 1 2 3

Density air

water

−2 0 2

−3

−2

−1 0 1 2 3

Pressure

−2 0 2

−3

−2

−1 0 1 2 3

Volume fraction

t = 3ms

(69)

Future Work

3D volume tracking method

General curvilinear grid system

Body-fitted grid for complicated geometries Low Mach number flow

Remove sound-speed stiffness by preconditioning techniques or pressure-based method

Include more physics towards real applications

Diffusion, phase transition, or elastic-plastic effect Hybrid surface-volume tracking algorithm for balance laws with interfaces & boundaries

(70)

Liquid Drop Problem (Revisit)

Tracking Capturing

air air

t = 0 t = 0

(71)

Liquid Drop Problem (Revisit)

Tracking Capturing

air air

t = 0.2s t = 0.2s

(72)

Liquid Drop Problem (Revisit)

Tracking Capturing

air air

t = 0.4s t = 0.4s

(73)

Liquid Drop Problem (Revisit)

Tracking Capturing

air air

t = 0.6s t = 0.6s

(74)

Liquid Drop Problem (Revisit)

Tracking Capturing

air air

t = 0.8s t = 0.8s

(75)

Liquid Drop Problem (Revisit)

Tracking Capturing

air air

t = 1s t = 1s

(76)

Thank You

(77)

References

(JCP 1998) An efficient shock-capturing algorithm for compressible multicomponent problems

(JCP 1999, 2001) A fluid-mixture type algorithm for

compressible multicomponent flow with van der Waals (Mie-Grüneisen) equation of state

(JCP 2004) A fluid-mixture type algorithm for barotropic two-fluid flow Problems

(JCP 2006) A wave-propagation based volume tracking method for compressible multicomponent flow in two space dimensions

(Shock Waves 2006) A volume-fraction based algorithm for hybrid barotropic & non-barotropic two-fluid flow

problems

(78)

Thermodynamic Stability

Fundamental derivative of gas dynamics

G = −V 2

(∂2p/∂V 2)S

(∂p/∂V )S , S : specific entropy Assume fluid state satisfy G > 0 for thermodynamic stability, i.e.,

(∂2p/∂V 2)S > 0 & (∂p/∂V )S < 0 (∂2p/∂V 2)S > 0 means convex EOS

(∂p/∂V )S < 0 means real speed of sound, for c2 =  ∂p

∂ρ



S

= −V 2  ∂p

∂V



S

> 0

(79)

Homogeneous Flow Model (cont.)

Mie-Grüneisen EOS: pk = prefk) + ρkΓ(ρk)[ek − erefk)]

ρe =

2

X

k=1

αkρkek =

2

X

k=1

αk

p − prefk)

Γ(ρk) + ρkerefk)



=⇒

p =

"

ρe −

2

X

k=1

αk

−prefk)

Γ(ρk) + ρkerefk)

#

(80)

Mie-Grüneisen Equations of State

(pref, eref) lies along an isentrope

1. Jones-Wilkins-Lee EOS for gaseous explosives

Γ(V ) = γ − 1, V = 1/ρ eref(V ) = e0 + A V0

R1 exp −R1V V0



+ B V0

R2 exp −R2V V0



pref(V ) = p0 + A exp −R1V V0



+ B exp −R2V V0



2. Cochran-Chan EOS for solid explosives

Γ(V ) = γ − 1

eref(V ) = e0 + −A V0 1 − E1

"

 V V0

1−E1

− 1

#

+ B V0 1 − E2

"

 V V0

1−E2

− 1

#

pref(V ) = p0 + A V −E1

− B  V −E2

(81)

Mie-Grüneisen EOS (cont.)

(pref, eref) lies along a Hugoniot locus

Assume linear shock speed us & particle velocity up

us = c0 + s up We may derive the relations

Γ(V ) = Γ0  V V0

α

, Γ0 = γ − 1 pref(V ) = p0 + c02(V0 − V )

[V0 s(V0 − V )]2 eref(V ) = e0 + 1

2 pref(V ) + p0 (V0 − V )

(82)

Material Quantities for Model EOS

JWL EOS ρ0(kg/m3) A(GPa) B(GPa) R1 R2 Γ

TNT1 1630 371.2 3.23 4.15 0.95 0.30

TNT2 1630 548.4 9.375 4.94 1.21 1.28

Water 1004 1582 −4.67 8.94 1.45 1.17

CC EOS ρ0(kg/m3) A(GPa) B(GPa) E1 E2 Γ

TNT 1840 12.87 13.42 4.1 3.1 0.93

Copper 8900 145.67 147.75 2.99 1.99 2

Shock EOS ρ0(kg/m3) c0(m/s) s Γ0 α

Aluminum 2785 5328 1.338 2.0 1

Copper 8924 3910 1.51 1.96 1

Molybdenum 9961 4770 1.43 2.56 1

MORB 2660 2100 1.68 1.18 1

Water 1000 1483 2.0 2.0 10−4

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Falling Liquid Drop Problem

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Falling Liquid Drop Problem

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Falling Liquid Drop Problem

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Falling Liquid Drop Problem

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