Falling Liquid Drop Problem

(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

ρ_t + ∇ · (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 χ_k as χ_k(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

hχ_k∇ψi = h∇(χ_kψ)i−hψ∇χ_ki & hχ_kψ_ti = h(χ_kψ)_ti−hψ(χ_k)_ti

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Two-Phase Flow Model (cont.)

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

hχ_kρ_ki_t + ∇· < χ_kρ_k~u_k >= hρ_k(χ_k)_t + ρ_k~u_k · ∇χ_ki Since χ_k is governed by

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

hχ_kρ_ki_t + ∇· < χ_kρ_k~u_k >= hρ_k (~u_k − ~u0) · ∇χ_ki Analogously, we may derive averaged equation for momentum, energy, & entropy (not shown here)

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Two-Phase Flow Model (cont.)

In summary, averaged model system, we have, are

hχ_kρ_ki_t + ∇· < χ_kρ_k~u_k >= hρ_k (~u_k − ~u⁰) · ∇χ_ki hχkρk~uki_t + ∇· < χkρk~uk ⊗ ~uk > +∇ hχkpki = hpk∇χki +

hρk~uk (~uk − ~u⁰) · ∇χki hχ_kρ_kE_ki_t + ∇· < χ_kρ_kE_k~u_k + χ_kp_k~u_k >= hp_k~u_k · ∇χ_ki +

hρ_kE (~u_k − ~u⁰) · ∇χ_ki hχ_ki_t + h~u_k · ∇χ_ki = h(~u_k − ~u⁰) · ∇χ_ki

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 α_k = V_k/V Now, by dropping all interfacial terms, we may obtain a simplified model as

(α_kρ_k)_t + ∇ · (α_kρ_k~u) = 0

(α_kρ_k~u)_t + ∇ · (α_kρ_k~u ⊗ ~u) + ∇ (α_kp) = p∇α_k (α_kρ_kE_k)_t + ∇ · (α_kρ_kE_k~u + α_kp~u) = p~u · ∇α_k

(α¹)_t + ~u · ∇α¹ = 0

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

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

(20)

Homogeneous Flow Model (cont.)

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

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

(αkρk)_t + ∇ · (αkρk~u) = 0 (ρ~u)_t + ∇ · (ρ~u ⊗ ~u) + ∇p = 0 (ρE)_t + ∇ · (ρE~u + p~u) = 0 (α¹)_t + ~u · ∇α¹ = 0

ρ~u = P²

k=1 α_kρ_k~u: total momentum ρE = P²

k=1 α_kρ_kE_k: total energy

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

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Homogeneous Flow Model (cont.)

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

capillary effects to the model

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

(ρE)_t + ∇ · (ρE~u + p~u) = φ · ~u~ (α¹)_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

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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

p1(ρ1, ρ1e1) = p2(ρ2, ρ2e2) &

2

X

k=1

α_kρ_ke_k = 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)

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Homogeneous Flow Model (cont.)

Polytropic ideal gas: p_k = (γ_k − 1)ρ_ke_k

ρe =

2

X

k=1

α_kρ_ke_k =

2

X

k=1

α_k p

γ_k − 1 =⇒

p = ρe

² X

k=1

α_k γk − 1

(24)

Homogeneous Flow Model (cont.)

Polytropic ideal gas: p_k = (γ_k − 1)ρ_ke_k

ρe =

2

X

k=1

α_kρ_ke_k =

2

X

k=1

α_k p

γ_k − 1 =⇒

p = ρe

² X

k=1

α_k γk − 1

Van der Waals gas: pk = (₁₋^γ^k_b⁻¹_k_ρ_k )(ρkek + akρ²_k) − akρ²_k

ρe =

2

X

k=1

α_kρ_ke_k =

2

X

k=1

α_k 1 − b_kρ_k γ_k − 1

(p + a_kρ²_k) − a_kρ²_k

=⇒

p =

"

ρe −

2

X

k=1

α_k 1 − bkρk

γk − 1 − 1

a_kρ²_k

# ² X

k=1

α_k 1 − bkρk

γk − 1

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Homogeneous Flow Model (cont.)

Two-molecular vibrating gas: p_k = ρ_kR_kT (e_k), 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





ρ_kR_kT_k γ_k − 1

+ ρ_kR_kTvib^,k exp

Tvib^,k/T_k

− 1





=

2

X

k=1

α_k





p

γ_k − 1

+ pvib^,k exp

pvib^,k/p

− 1



 (Nonlinear eq.)

(26)

Homogeneous Flow Model (cont.)

It is easy to show entropies, S_k, 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, S_k, 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

(α¹)_t + ~u · ∇α¹ = α¹α² ρ²c²2 − ρ¹c²1

P²

k=1 αkρkc²_k

!

and now entropy of each phase satisfy

DS_k

Dt = ∂S_k

∂t + ~u · ∇S_k = 0, for k = 1, 2

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Some Remarks

1. Model system is hyperbolic under suitable thermodynamic stability condition

(29)

Some Remarks

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

(30)

Some Remarks

3. In the model, it is not at all clear on how to compute nonlinear term ρ^ι, ι > 1 from α_k & α_kρ_k

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Some Remarks

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

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) = (p⁰_ι + B_ι)

ρ ρ⁰_ι

γ^ι

− B_ι, ι = 1, 2

Mixture pressure law (Shyue, JCP 2004)

p =











(p⁰ι + Bι)

ρ ρ⁰_ι

γ^ι

− 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, Qⁿ_S gives approximate value of cell average of solution q over cell S at time t_n

Qⁿ_S ≈ 1 M(S)

Z

S

q(X, t_n) 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⁺¹_S := Qⁿ⁺¹_S −M (W_p ∩ S)

M(S) R_p, R_p being jump from RP

↓

W_p W_p

(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

W_pq

(38)

Wave Propagation Method (cont.)

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

a) b)

c) d)

α_pr

p/2

α_pr

p/2

λ_p∆ 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

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

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

↑

↑ 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:

z_C := z_E (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 max_p,q (λ_p, µ_q)

min(∆x, ∆y) ≤ 1,

λ_p, µ_q: 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 V_R V_T Vui V_uf V_di V_df 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 (V_R, V_T) Incident (refracted, transmitted) shock speed t ∈ [0, 250]µs (t ∈ [0, 202]µs, t ∈ [202, 250]µs ) V_ui (V_uf) Initial (final) upstream bubble wall speed t ∈ [0, 400]µs (t ∈ [400, 1000]µs)

V_di (V_df) 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

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 = pref(ρk) + ρkΓ(ρk)[ek − eref(ρk)]

ρe =

2

X

k=1

α_kρ_ke_k =

2

X

k=1

α_k

p − pref(ρk)

Γ(ρ_k) + ρ_keref(ρ_k)

=⇒

p =

"

ρe −

2

X

k=1

αk

−pref(ρ_k)

Γ(ρk) + ρkeref(ρk)

#

(80)

Mie-Grüneisen Equations of State

(pref^{, e}ref⁾ 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^{, e}ref⁾ lies along a Hugoniot locus

Assume linear shock speed us & particle velocity up

u_s = c0 + s u_p 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 )

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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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