Two-Fluid Effective-Field Equations

and are void fraction function, with 1g l g l

2

2

( ) ( )0

( ) ( )( )

( ) ( )

( ) ( )0

( ) ( )( )

( ) ( )

g g g g g

g g g g g gg

g g g g g g

g

g

l

l

g

l l l l l

l l l l l ll

l l l l l l l

u

t x

u up

t x

E u H

t x

u

t x

u up

t

p

pt

px

Ep

H

t t

u

x

Mathematical Issues

• Non-conservative: – Uniqueness of Discontinuous solution?– Pressure oscillations

( ) ( )i ik kp p

• Non-hyperbolic system: Ill-posedness?– Stability– Uniqueness

• How to sort it out?

2

2

( )

( )

( ) ( )0

( ) ( )( )

( ) ( )

( ) ( )0

((

) ( )) )(

( )

g g g g g

g g g g g gg

g g g g g g g

l l l l l

l l l l l ll

l l l

vmg

i vmg

vm

g

l g

gp

p

u

t x

u up

t x

E u H

t x

u

t x

u

f

u f

fu

pt x

t

p

p p

p

E

t

( )

( ) i vmg

l l ll lufp u

xp

t

H

Remedy for hyperbolicity: Interfacial pressure correction term and

virtual mass term

Modeling – Interfacial Pressure (IP)

* 2g l g l

g l l g

p w

gl uuw Stuhmiller (1977):

2** wCp gp

Here, we have

1* pC

Faucet Problem: Ransom (1992)

• Hyperbolicity insures non-increase of overshoot, but suffering from smearing

• Location and strength of void discontinuity is converged, not affected by non-conservative form

Effect of hyperbolicity Solution convergence

0 2 4 6 8 100.0

0.1

0.2

0.3

0.4

0.5

0.6

C*

P=0

Vo

id F

ract

ion

X

0 2 4 6 8 10

0

20

40

60

80

100 C*

P=0

Ga

s V

elo

city

( m

/s)

X

0 2 4 6 8 100.0

0.1

0.2

0.3

0.4

0.5

0.6

C*

P=0

C*

P=1.0

Vo

id F

ract

ion

X

0 2 4 6 8 10

0

20

40

60

80

100 C*

P=0

C*

P=1.0

Ga

s V

elo

city

( m

/s)

X

Results for Toumi's shock tube problem. Interfacial correction terms only.

7

7

Initial condition: 1.0 10 pa, 0.25

2.0 10 pa, 0.10

L L

R R

p

p

Modeling – Virtual Mass (VM)

Drew et al (1979)

( )cvm vm vmd c dc cd dvmd

u u uuf u ut x

Cx

VM is necessary if IP is not present, the coefficients are

unreasonably

high for droplet flows.

Requirement of VM can be reduced with IP.

0 2 4 6 8 100.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

C*

P=0

Vo

id F

ract

ion

X

0 2 4 6 8 10

0

20

40

60

80

100 C*

P=0

Ga

s V

elo

city

( m

/s)

X

0 2 4 6 8 100.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

C*

P=0

C*

P=2.0

Vo

id F

ract

ion

X

0 2 4 6 8 10

0

20

40

60

80

100 C*

P=0

C*

P=2.0

Ga

s V

elo

city

( m

/s)

X

Results for Toumi's shock tube problem. With virtual mass model (Type II, = 0.4).vmC

7

7

Initial condition: 1.0 10 pa, 0.25

2.0 10 pa, 0.10

L L

R R

p

p

Numerical Method

• Extended from single-phase AUSM+-up (2003).

• Implemented in the All Regime Multiphase Simulator

(ARMS).

- Cartesian.

- Structured adaptive mesh refinement.

- Parallelization.

A case with 40% liquid fraction

Ugas=1km/s

L=0.4, liquid mass =400kgVL=150m/s(in radial)Liquid area: l=2m, r=0.4m

L=60m

R=12m

Axis

( Grid size 10cm, calculation time :0-150ms Calculation domain:,L=60m,R=12m )

Liquid fraction, pressure and velocity contours of particle cloud for time 0-150 ms.

Lquid fraction (Min:10-8 -Max:10-3)

Pressure (Min:1bar-Max:7bar)

Gas Velocity (Min:0m/s -Max:1,000m/s)

Droplet radius R = 3.2mm, incoming shock speed M = 1.509

Current and future works

• Complete the hyperbolicity work on the multi-fluid system.

• Complete the adaptive mesh refinement into our solver –

ARMS

• Expand Music-ARMS to solve 3D problems.

• Introduce physical models:

•Surface tension model

•Turbulence model

• Verification and validation.

• Real world applications.