ROBUST WAF-HLL SCHEME FOR COMPRESSIBLE …procom.kaist.ac.kr/Download/IJP/128.pdf · Engineering...

Engineering Applications of Computational Fluid Mechanics Vol. 6, No. 1, pp. 144–162 (2012)

Received: 6 Sept. 2011; Revised: 24 Oct. 2011; Accepted: Nov. 2011

144

ROBUST WAF-HLL SCHEME FOR COMPRESSIBLE TWO-PRESSURE TWO-VELOCITY MULTIPHASE FLOW MODEL

Geum-Su Yeom *, Keun-Shik Chang ** and Seung Wook Baek **

* School of Mechanical and Automotive Engineering, Kunsan National University, 1170 Daehangno, Gunsan, Jeonbuk 573-701, South Korea

E-Mail: [email protected] (Corresponding Author) ** Division of Aerospace Engineering, Department of Mechanical, Aerospace and Systems Engineering,

Korea Advanced Institute of Science and Technology, 291 Daehak-ro, Yuseong-gu, Daejeon 305-701, South Korea

ABSTRACT: In this paper we have developed a robust WAF-HLL scheme for the compressible two-pressure two-velocity six-equation two-fluid model by improving the earlier WAF-HLL scheme (Yeom and Chang, 2010b) that was also proposed by the authors. The earlier scheme has, however, shown spurious oscillations and instabilities across the shock wave for some severe two-phase flow problems because the WAF (Weighted Average Flux) procedure was applied to the single numerical flux that consists of the gas and the liquid phases. The main idea of the new WAF-HLL scheme is to split the numerical flux of the overall two-phase flow into two phasic components for the gas phase and the liquid phase before we apply the WAF procedure to each of the two phasic numerical fluxes. The present scheme has become numerically very robust as we have removed the oscillations. Furthermore, it makes use of the quadrilateral unstructured grid for more flexibility to deal with complex flow geometry. We demonstrate by solving several two-phase benchmark problems that the new WAF-HLL scheme is significantly more robust than the earlier WAF-HLL scheme, while having accuracy comparable to the earlier one.

Keywords: WAF scheme, HLL scheme, WAF-HLL scheme, two-fluid model, two-phase flow, unstructured grid

1. INTRODUCTION

The weighted average flux (WAF) scheme is a second-order extension of the Godunov-type first-order schemes; its origin can be found from the random flux scheme (Toro, 1986 and 1989). It is a generalization of the Lax-Wendroff scheme and the Warming-Beam method for the nonlinear conservation laws. To improve the WAF scheme, we can further implement a TVD (total variation diminishing) flux limiter to suppress the spurious oscillations near the large gradient of the solution. A merit of the WAF scheme is that we can easily achieve the second-order accuracy for the conventional Godunov-type Riemann solvers without taking the laborious reconstruction and evolution steps of the MUSCL-type schemes (van Leer, 1977, 1979 and 1985; Quirk, 1994). A multi-dimensional extension of the WAF scheme has been made by Toro (1990), and Billett and Toro (1997). Recently, the WAF scheme has been applied to the shallow water equations by Loukili and Soulaïmani (2007) and Fernández-Nieto and Narbona-Reina (2008). Also the WAF scheme has been extended to the compressible two-fluid two-phase model by the present authors (Yeom and Chang, 2010b). Other

type of applications can be found in the papers of Saito et al. (2003), and Titarev and Toro (2005). In this paper, we report on a new robust WAF-HLL scheme that has remarkably improved the numerical stability of the earlier WAF-HLL scheme (Yeom and Chang, 2010b). A compressible two-fluid model, that is an ensemble average of the local instantaneous phasic flow equations, has been widely used to simulate the transient multiphase flows. Although the two-fluid model offers more detailed solution than the mixture model (Wang et al., 1994), it suffers from the numerical instability attributed to the large discrepancy between the flow variables and equations of state of the gas phase and the liquid phase at the boundaries of the two fluids or two phases. One more disadvantage is that the equation system of the two-fluid model is usually given in a non-conservative form: it makes difficult to apply the Godunov-type upwind schemes. Among a variety of two-phase flow models (Ishii and Hibiki, 2006; Toumi, 1996; Tiselj and Petelin, 1997; Städtke et al., 1997; Paillère et al., 2003; Saurel and Abgrall, 1999; Saurel and Lemetayer, 2001; Yeom and Chang, 2006, 2009, 2010a and 2010b; Yoon et al., 2009;

Engineering Applications of Computational Fluid Mechanics Vol. 6, No. 1 (2012)

145

Tong and Luke, 2010), we consider the eight-equation two-fluid model in two dimensions. There exist many numerical schemes to solve the two-fluid models. The Roe-type schemes (Toumi, 1996; Sainsaulieu, 1995; Mukejord, 2007; Mukejord and Papin, 2007) and VFRoe scheme (Andrianov et al., 2003) are accurate but quite expensive due to repeated costly Jacobian matrix calculations and an additional entropy fix. Moreover, these schemes would not be appropriate because the analytic eigenvalues and eigenvectors are not available for the present two-fluid model. The flux-vector splitting scheme (Städtke et al., 1997) is efficient but rather diffusive. The AUSM schemes (Paillère et al., 2003; Niu, 2001; Niu et al., 2008) have shown accurate and robust results for various two-phase flow problems. They are quite attractive since they can be applied to both the hyperbolic two-fluid models (Saurel et al., 1999 and 2001) and the non-hyperbolic two-fluid models (Yeom and Chang, 2006, 2009, 2010a and 2010b). The HLLC schemes (Tokareva and Toro, 2010; Zein et al., 2010) are very promising because of its accuracy, robustness, and efficiency. However, the HLLC scheme has not been developed for the present two-fluid model. The HLL schemes (Saurel and Abgrall, 1999; Saurel and Lemetayer, 2001; Yeom and Chang, 2006, 2009 and 2010a) are rather diffusive, but simple, robust, and efficient. They can be easily applied to a variety of two-fluid models without much difficulty. Most of these schemes have adopted the MUSCL-type approach to achieve the second-order accuracy. The present authors have recently applied the WAF approach to the HLL scheme in order to investigate the shock wave diffraction in a gas-microdroplet mixture over a wedge (Yeom and Chang, 2010b). This WAF-HLL scheme has produced satisfactory numerical results for the small droplet size in the diameter range, m 5.2 m 5.0 d . Unfortunately, the numerical scheme became unstable as the droplet size became bigger. In addition, the scheme became unstable as the strong cavitation was produced by the expansion waves when we solved the underwater projectile problem. This is mainly because the WAF procedure in the earlier scheme is applied to the single numerical flux that consists of the gas and the liquid phases. In order to overcome the oscillations and instabilities, we split the numerical flux of the overall two-phase flow into two phasic components for the gas phase and the liquid phase before we apply the WAF procedure to each of the two phasic

numerical fluxes. Consequently, the new WAF-HLL scheme has become numerically very robust, while having accuracy comparable to the earlier one. The objective of this paper is to report on a novel WAF-HLL scheme that is more stable and accurate for the compressible two-fluid model; it has also implemented the TVD flux limiters plus the quadrilateral unstructured grid in order to deal with complex geometry. Several one- and two- dimensional benchmark problems are solved and the results are compared with the earlier WAF-HLL scheme and the HLL scheme.

2. COMPRESSIBLE TWO-FLUID MODEL

The eight-equation compressible two-fluid model takes following form in two dimensions are:

SIH

GFU

yxyxtgg

, (1)

with

lll

lll

lll

ll

ggg

ggg

ggg

gg

E

v

u

E

v

u

U

U

U

U

U

U

U

U

8

7

6

5

4

3

2

1

U ,

lllllll

llll

lllll

lll

ggggggg

gggg

ggggg

ggg

puEu

vu

pu

u

puEu

vu

pu

u

2

2

F ,

lllllll

lllll

llll

lll

ggggggg

ggggg

gggg

ggg

pvEv

pv

vu

v

pvEv

pv

vu

v

2

2

G ,

ii

i

ii

i

up

p

up

p

0

0

0

0

H ,

ii

i

ii

i

vp

p

vp

p

0

0

0

0

I ,

y

i

x

i

y

x

y

i

x

i

y

x

DvDuQ

D

D

DvDuQ

D

D

0

0

S .


146

Here, is the volume fraction, is the density,

u and v are respectively the x- and y-velocities, E is the specific total energy, p is the pressure,

ip is the interfacial pressure, iu and iv are respectively the interfacial velocities in x- and y-direction,

xD and

yD are respectively the viscous

drag force in x- and y-directions, and Q is the

heat transfer between phases. The subscripts g

and l stand for the gas and the liquid phase, respectively. The specific total energy E is defined by

22

2

1vueE , (2)

where e is the specific internal energy. The volume fractions satisfy

1lg

. (3)

Difference of pressure in the gas and the liquid appears as the surface tension force of the droplet,

dpp

gl

4 , (4)

where d is the diameter of the droplet. The interfacial pressure represents the mean pressure between the two phases. A variety of the interfacial pressure models can be found in Stuhmiller (1977), Toumi (1996) and Barre and Bernard (1990). In the present paper, we adopt the following model (Stuhmiller, 1977 and Yeom and Chang, 2009):

22

lglg

gllg

lglg

g

i vvuupp

, (5)

where is the damping coefficient. It is noteworthy that the present two-fluid equation system becomes hyperbolic for 1 though it does not always guarantee a hyperbolic type for the partial differential equations (Paillère et al., 2003).

The interfacial velocities iu and iv are applied to the center of mass:

llgg

lllgggiuu

u

, llgg

lllgggivv

v

. (6)

The drag force terms are given by

lglggDcdx

uuuuCAND 2

1,

lglggDcdx

vvvvCAND 2

1, (7a)

where d

N is the number of droplets per unit

volume, c

A is the cross-sectional area of a droplet,

and 0D

C is the drag coefficient that depends on the local flow regime such as droplet flow, bubbly flow, and so on. In the present paper, we use the following model for DC after Ishii and Hibiki (2006):

,1000for 45.0

,1000for 1.0124 75.0

d

dd

d

D

Re

ReRe

Re

C (7b)

Here the particle Reynolds number is

mdgdggd vvuudRe 22 , where

the mixture viscosity is given as 5.2ggm .

The heat transfer rate per unit volume between gas and liquid phase depends on Nu (Nusselt number) and Pr (Prandtl number), as in the expression

gl

gpgg TTPrd

NuCQ

2

,16

, (8)

where gp

C,

is the specific heat of the gas at

constant pressure.

2.1 Equation of state

We consider each fluid is compressible and use the modified stiffened-gas equation of state (EOS) given by (Saurel et al., 2008)

kkk

kkk

k qe

pp

1,

(with k = g or l) (9)

where k

q is the parameter of the stiffened-gas

EOS, k

p,

is the reference pressure, k

is the ratio

of specific heats. For the gas phase, we take following constants:

4.1g

, Pa 0, g

p , J/kg 0g

q (air),

327.1g

, Pa 0, g

p , J/kg 0g

q (steam).

For the liquid phase, we take following constants:

35.2l , Pa 109

, l

p ,

J/kg 101169 3g

q (liquid water).


147

The speed of sound, k

c , is defined by

k

kkk

k

kkkk

k

pp

pe

epc

)(

/

// ,

2

2

(with k = g or l). (10)

2.2 Transformation of conservative variables to the primitive variables

In order to calculate flux variables, we need to go back to the primitive variables in the two phases ( , p , u , v , , E and e ) from the

conservative variables i

U (i = 1, 2, …, 8). The x-

and y-velocities are given by

1

2

U

Uu

g ,

5

6

U

Uu

l ,

1

3

U

Uv

g ,

5

7

U

Uv

l . (11)

The specific internal energies are given by

22

1

4

2

1ggg

vuU

Ue ,

22

5

8

2

1lll

vuU

Ue . (12)

From Eqs. (3), (4), and (9), we obtain the analytic form of the gas pressure

gp as

2

1,

414

2

1U

dpep

llggg

(13)

where

lllllggpUqeUe

d ,5111

4

.

Then, the liquid pressure l

p is readily obtained

from Eq. (4). Using the EOS, we can obtain the densities as

),(gggg

ep ; ),(llll

ep . (14)

Finally, the volume fractions are given by

g

g

U

1 ; gl

1 . (15)

3. NUMERICAL METHOD

3.1 Finite volume method on unstructured grid

The equation system, Eq. (1), is solved with the fractional-step method by splitting into two subsystems,

0:

H

yxyxtL gg

IH

GFU, (16)

S

U

tL :

S , (17)

where, H

L stands for the hyperbolic operator

solving the homogenous PDE system and S

L

stands for the source operator solved by the conventional fourth-order Runge-Kutta method. The second-order accurate solution at n+1 time step is obtained by the particular sequence,

ntttn LLL UU )2/(

S

)(

H

)2/(

S

1 , (18)

where H

L and S

L have to be, at least, second-

order accurate operators. The homogeneous equation system, Eq. (16), is discretized using the finite volume method (FVM) on the quadrilateral cell-centered unstructured grid. Fig. 1 schematically depicts the computational grid used in the present work. In this figure, the numerical flux *

,snF and the

numerical volume fraction of the gas phase *

,sg

are evaluated at the cell boundary. The resulting FVM formulation is

s

ss

n

i

n

isgs

i

n

i

n

iL

A

t

4

1

*

,

*

,

1 , nIHFUUn

, (19)

where iA is the area of the ith computational cell,

sL is the length of the boundary s of the cell, and

syxs

nn ,n is the unit vector normal to the cell

boundary.

3.2 New WAF-HLL scheme

The WAF scheme is a fully explicit second-order extension of the Godunov first-order upwind scheme. In the WAF scheme, the numerical flux is defined as an integral average of the flux at the half time step:

2/

2/

WAF

2/1 2,

1 x

xi

dxt

xx

UFF , (20)

where U is the solution of the local Riemann problem having the initial data

iU and

1iU .

When two-wave configuration is considered (see Fig. 2), the integral of Eq. (20) becomes a weighted average of the fluxes as in

3

1

)(

2/1

)(WAF

2/1k

k

i

k

iw FF , (21)

where )(

2/1

k

iF is the kth flux in the solution of the

local Riemann problem. The weights )( kw are given by the interval length between two surrounding waves:


148

Fig. 1 Quadrilateral unstructured grid built in the present WAF-HLL scheme.

Fig. 2 Construction of WAF scheme with two basic

waves.

)1(

2/1

)(

2/1

)(

2

1

k

i

k

i

k CCw . (22)

In the above, )(

2/1

k

iC

is the local CFL numbers of

the kth wave as

)(

2/1

)(

2/1

k

i

k

iS

x

tC

, 1)0(

2/1iC , 1)3(

2/1iC , (23)

where )( kS is the speed of the kth wave. Despite that WAF scheme is a second-order accurate in space and time, it suffers from spurious oscillations near a large gradient of the solution. A TVD constraint is enforced as a remedy. The WAF scheme then becomes

3

1

)(

2/1

)(

2/11

WAF

2/1 2

1

2

1

k

k

i

k

iiiiFFFF , (24)

where )(

2/1

)1(

2/1

)(

2/1

k

i

k

i

k

i

FFF and

)(

2/1

)(

2/1

)(

2/1sign k

i

k

i

k

iC . The TVD flux limiter

functions )(

2/1

k

i depend on the local CFL numbers, )(

2/1

k

iC , and a local flow parameter )(

2/1

k

ir . This

parameter is defined by the ratio of the upwind jump to the local jump for some suitable flow variable :

loc

upw)(

2/1

k

ir . (25)

The local jump loc

is

)(

2/1

)1(

2/1loc

k

i

k

i

, (26)

and the upwind jump upw

is

.0 if

,0 if )(

2/1

)(

2/3

)1(

2/3

)(

2/1

)(

2/1

)1(

2/1

upw k

i

k

i

k

i

k

i

k

i

k

i

C

C (27)

The density variable is commonly taken in place of for the single-phase Euler equations. In the previous work (Yeom and Chang, 2010b), the earlier form of the WAF scheme applied to the HLL scheme was investigated by solving shock propagation in the gas-microdroplet two-phase flow. This particular scheme was satisfactory for the smaller droplets, in the diameter range, m 5.2m 5.0 d , but showed undesirable oscillations for the bigger droplets. It was also unstable for the strongly cavitating flows. We hereby introduce a new WAF-HLL scheme that has eliminated all the above instabilities. Generally, flow variables change in a different scale depending on whether the flow is a gas phase or a liquid phase. It would be appropriate henceforth to apply different TVD limiters for the WAF scheme to the gas phase and to the liquid phase. We separate the gas-phase equations and the liquid-phase equations from the original two-fluid equation system before we apply the WAF procedure to each of the two phases. In the formulation, we divide the conservative variables and the flux variables into the phasic components as follows,

l

g

U

UU ,

l

g

F

FF ,

l

g

G

GG , (28)

with

ggg

ggg

ggg

gg

g

E

v

u

U ,

lll

lll

lll

ll

l

E

v

u

U , (29)

ggggggg

gggg

ggggg

ggg

g

puEu

vu

pu

u

2

F ,

lllllll

llll

lllll

lll

l

puEu

vu

pu

u

2

F ,

(30)


149

ggggggg

ggggg

gggg

ggg

g

pvEv

pv

vu

v

2G ,

lllllll

lllll

llll

lll

l

pvEv

pv

vu

v

2G .

(31)

The numerical fluxes of gas phase and liquid phase and the numerical volume fraction of the gas phase are individually computed by the HLL solver using the initial data,

LU and

RU , as

,0 if

,0 if

,0 if

HLL

,

RgR

RL

LR

LRRLgRLgLR

LgL

sg

S

SSSS

SSSS

S

n

nn

n

n

F

UUFF

F

F

(32)

,0 if

,0 if

,0 if

HLL

,

RlR

RL

LR

LRRLlRLlLR

LlL

sl

S

SSSS

SSSS

S

n

nn

n

n

F

UUFF

F

F

(33)

,0 if

,0 if

,0 if

HLL

,

RgR

RL

LR

gRLgLR

LgL

sg

S

SSSS

SS

S

(34)

where sKgggK

nGFFn

, and sKlllK

nGFFn

, .

The intermediate states , *g

U and *l

U are given by

LR

gRgLgLLgRR

g SS

SS

nnFFUU

UHLL

* , (35)

LR

lRlLlLLlRR

l SS

SS

nnFFUU

UHLL

* . (36)

The fastest wave speeds, L

S and R

S , can be

estimated by the effective sound speeds, g

c and

lc , as follows:

RllRggLllLggL

cucucucuS ˆ,ˆ,ˆ,ˆmin,,,,

nnnn , (37)

RllRggLllLggR

cucucucuS ˆ,ˆ,ˆ,ˆmax,,,,

nnnn , (38)

Fig. 3 Construction of new WAF-HLL scheme for compressible two-fluid model.


150

where g

c and l

c are given by the analytic

eigenvalues of Nguyen et al. (1981) as

22

2

ˆllgggl

ll

gg cc

ccc

and

.ˆ

22

2

llgggl

gg

ll cc

ccc

(39)

From the phasic numerical fluxes and the volume fraction of the gas phase given in Eqs. (32)-(34), we can construct the new WAF-HLL formulation as described in Fig. 3. The TVD flux limiters of the WAF scheme are then applied individually to the phasic numerical fluxes and the numerical volume fraction of the gas phase in consideration of different scales of flow-variable change.

)(

,

2

1

)(

,

)(HLL WAF

,sign

2

1

2

1 k

sgk

k

sg

k

sgRgLsgC

nnnnFFFF

,

(40)

)(

,

2

1

)(

,

)(HLLWAF

,sign

2

1

2

1 k

slk

k

sl

k

slRlLslC

nnnnFFFF

,

(41)

)(

,

2

1

)(

,

)(HLLWAF

,sign

2

1

2

1 k

sgk

k

s

k

sgRgLsgC

,

(42)

where )()1()( kkk , xtSC k

s

k

s )()( ,

)()(

,

)(

,, k

s

k

sm

k

smCr .

The parameter )(

,

k

smr are defined by

loc,

upw,)(

,

g

gk

sgr

,

loc,

upw,)(

,

l

lk

slr

,

loc,

upw,)(

,

k

sr . (43)

The local jumps are defined by

)(

,

)1(

,loc,

k

LRg

k

LRgg

, )(

,

)1(

,loc,

k

LRl

k

LRll ,

)(

,

)1(

,loc,

k

LR

k

LR , (44)

and the upwind jumps are by

,0 if

,0 if )()(

,

)1(

,

)()(

,

)1(

,

upw, k

LR

k

RUPg

k

RUPg

k

LR

k

LUPg

k

LUPg

g C

C (45)

,0 if

,0 if )()(

,

)1(

,

)()(

,

)1(

,

upw, k

LR

k

RUPl

k

RUPl

k

LR

k

LUPl

k

LUPl

l C

C (46)

.0 if

,0 if )()(

,

)1(

,

)()(

,

)1(

,

upw, k

LR

k

RUP

k

RUP

k

LR

k

LUP

k

LUP

C

C

(47)

The flow variables m

are chosen as

ggg

U 1 , (48)

lll

U 5 , (49)

gg

U 1 or ll

U 5 . (50)

Among several limiter functions (for example, see Toro (1999)), we investigate the following four limiter functions.

MINBEE limiter:

.1 if

,10 if 11

,0 if 1

,

rC

rrC

r

Cr (51)

SUPERBEE limiter:

.2 if 12

,21 if 11

,10.5 if

,5.00 if 121

,0 if 1

,

rC

rrC

rC

rrC

r

Cr (52)

van Leer limiter:

.0 if 1

211

,0 if 1

,r

r

rC

r

Cr (53)

van Albada limiter:

.0 if 1

1 11

,0 if 1

,2

rr

rrC

r

Cr (54)

3.3 Boundary and CFL condition

The boundary conditions are implemented by using the ghost cells which are the reflected image of the boundary cells. The time step t is determined by

max

smin

L

CFLt s , (55)

where 1,0CFL is the Courant-Friedrichs-Lewy (CFL) number. The maximum wave speed at the current time level,

max , is computed by

llgg

cVcV ,maxmax

, (56)

where 22 vuV . The actual time step is determined by taking the minimum as

ii

tt min for all cells i. (57)


151

4. NUMERICAL TESTS

4.1 One-dimensional Toumi’s two-phase shock tube

We consider the two-phase shock tube problem studied by many researchers (Toumi , 1996; Tiselj and Petelin, 1997; Paillère et al., 2003; Yeom and Chang, 2006 and 2010a; Niu et al., 2008). Fig. 4 shows a schematic of the problem and the complex wave structure involved: a left rarefaction wave, a left shock, a right contact discontinuity, and two right shocks. The diaphragm is at the midsection of a 10-m long horizontal pipe and divides the left and the right fluids with the following initial states:

3

3

7

kg/m 0.622

kg/m 2.225

m/s 0

m/s 0

Pa 102

25.0

Ll

g

l

g

g

u

u

p

,

3

3

7

kg/m 0.612

kg/m 6.112

m/s 0

m/s 0

Pa 101

1.0

Rl

g

l

g

g

u

u

p

.

It should be noted that this problem is very sensitive to the damping coefficient of the interfacial pressure term, as indicated by the earlier researchers (Toumi, 1996; Paillère et al., 2003; Yeom and Chang, 2006 and 2010a). We take 5 following the earlier practice. In Fig. 5, the numerical results of the new WAF-HLL scheme (with a van Leer limiter), obtained at t = 5 ms with 500 cells, are compared with the HLL and the earlier WAF-HLL (with a van Leer limiter) by Yeom and Chang (2010b). The reference solution was calculated with a grid as fine as 20,000 cells by the HLL scheme (Yeom and Chang, 2006). The results indicate that the present new WAF-HLL scheme presents a more accurate solution than the HLL scheme for the field variables like volume fraction, pressure and velocities; the present scheme shows almost similar resolution to the earlier WAF-HLL scheme. Fig. 6 compares the four limiters tested with the new WAF-HLL scheme for the same problem. Among the flux limiters showing similar results, the SUPERBEE limiter seems to present slightly better resolution than others.

4.2 Two-dimensional shock wave diffraction about a wedge in a gas-droplet mixture

Next, we consider the two-dimensional shock wave diffraction about a wedge in a gas-droplet mixture, which was earlier investigated by Yeom

and Chang (2010b). Fig. 7 shows a schematic of this problem: the pure gas medium is joined by the gas-droplet mixture at the vertical material interface located where the wedge is started. The wedge is 1 m long with a half angle of 25 degree. The incident shock in the pure gas medium has shock Mach number 7.1

sM and is diffracted in

the gas-droplet mixture. In Fig. 7(b), when the incident shock hits the material interface, a reflected shock moves back to the gas-only medium and a transmitted shock passes through the gas-droplet mixture. The transmitted shock is diffracted over the wedge which consists of a wedge-reflected shock and a Mach stem, with a slip line emanated from the triple point. A relaxation zone is developed behind the transmitted shock where the gas and the droplet phase are in kinematic and thermodynamic non-equilibrium. In the earlier work (Yeom and Chang, 2010b), calculation was safely performed for small droplet sizes, in the diameter range m 5.2m 5.0 d but no bigger, because of numerical instability. In the present work, we have made successful calculations for the much bigger droplet cases, for d = 1, 10 and 100 m. Figs. 8 and 9 respectively show the gas density contours in the flow field and the wall values of the gas density, all dependent on the droplet sizes: d = 1, 10, and 100 m. We used the initial volume fraction of the gas phase, %02.0

g . We tested

two different grid sizes, the coarse grid with 20,000 cells and the fine grid with 156,800 cells. We compared the numerical results obtained by the present new WAF-HLL scheme and by the earlier WAF-HLL scheme (Yeom and Chang, 2010b). For d = 1 m, both schemes gave stable solution; for d = 10 and 100 m, the earlier WAF-HLL scheme showed spurious oscillations in the region where there is shock and at the compression corner, in contrast to the stable new WAF-HLL scheme.

Fig. 4 Toumi’s two-phase shock tube problem.


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Fig. 5 Toumi’s shock tube problem at t = 5 ms, solved with 500 cells by new WAF-HLL scheme (dash-dotted line).

Comparison is made with the solution by HLL scheme (dotted line), by the earlier WAF-HLL scheme (Yeom and Chang, 2010b) (dashed line), and the reference fine-grid solution solved with 20,000 grid cells (solid line)


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Fig. 6 Solution with four flux limiters (MINBEE, SUPERBEE, van Leer, and van Albada) obtained with 500 grid

cells for the Toumi’s shock tube problem.

Fig. 7 Shock diffraction in a gas-droplet two-phase mixture over a wedge: (a) initial time, (b) later time with the shock diffracted in the gas-droplet mixture.


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(a) Coarse grid (20,000 cells)

(b) Fine grid (156,800 cells)

Fig. 8 Gas density contours dependent on the droplet sizes: d = 1, 10, and 100 m. Calculation is made with the initial droplet volume fraction 0.02%. The first six plots (a) are by a coarse grid of 20,000 cells, and the next six plots (b) are by a fine grid of 156,800 cells. The left three plots in each (a) and (b) are by the earlier WAF-HLL scheme (Yeom and Chang, 2010b), and the right three plots in each (a) and (b) are by the present new WAF-HLL scheme.


155

(a) Coarse grid (20,000 cells)

Fig. 9 Gas density distribution on wall is dependent on droplet size: d = 1, 10, and 100 m. The 12 plots are all corresponding to the cases of Fig. 8. The first six plots (a) are by a coarse grid of 20,000 cells, the next six plots (b) are by a fine grid of 156,800 cells. The left three plots in each (a) and (b) are by the earlier WAF-HLL scheme (Yeom and Chang, 2010b), the right three plots in each (a) and (b) are by the present new WAF-HLL scheme.


156

(b) Fine grid (156,800 cells)

Fig. 9 continued


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(a) Wall velocity

Fig. 10 Wall distribution of the velocity (a) and the temperature (b), for both phases, is dependent on the droplet size: d = 1, 10, and 100 m. The results are obtained with a fine grid of 156,800 cells and initial droplet volume fraction 0.02 %. For each figure, left column is by the earlier WAF-HLL scheme (Yeom and Chang, 2010b), right column is by the present new WAF-HLL scheme.


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(b) Wall temperature

Fig. 10 continued


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Fig. 10 shows the velocity and the temperature distribution on the wall for both the gas phase and the droplet phase for the same wedge problem. The computational results are obtained with a fine grid of 156,800 cells. The initial droplet volume fraction is fixed with %02.0

g : it means that

the number density of the droplet is decreased as the droplet size becomes bigger and vice versa. The velocity distribution and the temperature distribution on the wall are almost the same for the gas and the liquid phases when the droplet size is d = 1 m. These curves begin to disagree more and more as the droplet size is increased, as observed with the cases d = 10 m and 100 m. This phenomenon is explained as follows: since the number density of the droplet is less for the bigger droplet size, the momentum and the thermal interactions between the two phases are reduced, resulting in the escalated discrepancy between the gas and the droplet phases in the curves of the velocity and temperature distributions on the wall. Fig. 10 also shows that, in contrast to the new WAF-HLL scheme, the earlier WAF-HLL scheme is locally unstable where there is a shock for the larger droplet sizes.

4.3 Two-dimensional high-speed underwater projectile

Finally, we consider an underwater high-speed cavitating projectile in two dimensions. This problem has been considered by earlier investigators, either experimentally or numerically using the mixture model (Hrubes, 2001; Vlasenko, 2003; Neaves and Edwards, 2006; Li et al., 2008). Fig. 11 shows a schematic of this problem. The projectile moves at a velocity of 700 m/s in the liquid water having a pressure of 100 kPa and a density of 1150 kg/m3. A small amount of gas (1%) is mixed numerically in the liquid water because of the two-fluid requirement of the formulation. As the projectile moves with a high velocity in the water, a compression region is followed by the expansion region along the body. The strong symmetric rarefaction waves, which are initiated by the leading edge, significantly reduce the local pressure. As a result, strong cavitation is created along the body and in the wake region behind the projectile. We used a quadrilateral unstructured grid of 308,447 cells and CFL = 0.4 for this computation. We assume that the velocities of the gas and the liquid phases are locally identical by enforcing the interface condition between the two phases. This condition can be achieved by the

Fig. 11 High-speed underwater projectile problem.

(a)

(b)

Fig. 12 Steady-state gas volume fraction and mixture density for the high-speed underwater projectile: Cavitation predicted by HLL scheme (a), and by present new WAF-HLL scheme (b).


160

instantaneous velocity relaxation procedure proposed by Saurel et al. (1999 and 2001). Fig. 12 shows the steady-state results for the gas volume fraction and the mixture density obtained by both the HLL and the new WAF-HLL schemes. In this figure, two strong rarefaction waves are created from the leading edge and a wide cavitation zone are developed along the body. In this figure, the new WAF-HLL scheme resolves the rarefaction waves and the interface of the cavitation zone much more clearly than the HLL scheme does. For this particular problem, a converged solution was not produced by the earlier WAF-HLL scheme. Fig. 13 shows the transient solutions of gas volume fraction for four different time instants, solved by the present new WAF-HLL scheme. In this figure, three gas cavities, which are initiated from the leading edge and the back side of the projectile, are developed with time to be merged into a large cavity region covering the projectile.

5. CONCLUSIONS

In this paper, we have developed a novel WAF-HLL scheme for the compressible two-fluid two-phase model. The new WAF-HLL scheme is robust without suffering from the undesirable instabilities in contrast to the earlier WAF-HLL scheme by the present authors. The main idea of the new WAF-HLL scheme is to separate the numerical flux of the scheme into the gas-phase and the liquid-phase components and we apply the WAF procedure individually to each of the phasic components. In addition, we enhanced the grid flexibility by adopting the quadrilateral unstructured grid for two-phase problems having complex geometry. We have compared the new WAF-HLL scheme with the earlier WAF-HLL scheme and the HLL scheme by solving several 1-D and 2-D benchmark problems. It has been proved that the new WAF-HLL scheme is significantly more robust than the other schemes, while having accuracy comparable to the earlier WAF-HLL scheme. Further, the WAF method is easier to implement than the MUSCL-type schemes. The present study has been limited to the 2-D gas-liquid particle system. It is noteworthy that the present WAF-HLL scheme can be applied to the gas-solid two-phase system by using the equation of state for the solid particle and the corresponding drag coefficient model. In order to extend to the 3-D system, one needs to change 2-D quadrilateral computational cells and 1-D interfaces into 3-D hexagonal cells and 2-D

Fig. 13 Transient solutions of gas cavity evolution for

high-speed underwater projectile solved by present new WAF-HLL scheme.

rectangular interfaces; and then the same FVM method and WAF-HLL procedure are applied to the 3-D system. In conclusion, the new WAF-HLL scheme is expected to be applied to a wide range of two-phase flow problems in the future.

ACKNOWLEDGEMENTS

This work has been supported by the Low Observable Technology Research Center program of Defense Acquisition Administration and Agency for Defense Development.

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