G … · improved is a CIP(Constrained Interpolation Profile) and CSL2(Conservative Semi-Lagrangian) based Cartesian Grid Method. .H\ZRUGV: CIPa , CSL2a , CCUPa , surface tracking,

$Page 1: G … · improved is a CIP(Constrained Interpolation Profile) and CSL2(Conservative Semi-Lagrangian) based Cartesian Grid Method. .H\ZRUGV: CIPa , CSL2a , CCUPa , surface tracking,$
(26일 목) 제 1 발 표 장 25 다상유동 Ⅰ

��

��

��

��.��,�� 3�� CIP��.CIP�� 2�� Hermit �� 2��. CIP�� .CIP�� CIP-CSL��.��CIP�� CIP-CSL4, CIP-CSL2, CIP-CSL3 �� . � �� CSL2� �� (��)�� CIP��. CSL2��.�� CIP-CSL2��

�� CIP-CSL2��.

��2.1 CIP�(Cubic Interpolated Pseudo Particle Method/Constrained

Interpolation Profile Method)

�� ( , )f x t ��(1)�� 1�� CIP��.

0f fut x� �

� ��

(1)

�(1)��(2)��.

( , ) ( ,0)f x t f x ut� � (2)

��(�)�� u �� x ,�� t �� dt �� ix �� iu ��.��(2)��(3)��.

( , ) ( , )f x t dt f x udt t� � � (3)

� Fig. 1�� t �� ix �� dt ��(3)��.

Fig. 1 Advection of physical property

��

Interface-tracking simulation of multi-phase flow using CIP-CSL2 scheme

H.N Im

In this study, it is performed numerical simulation on multi-phase flow by means of CIP-CSL2 scheme. It is applied to a two-phase free surface flow problem at a high density ratio equivalent to that of an air-water system, for examining the computational capability. The method that is being developed and improved is a CIP(Constrained Interpolation Profile) and CSL2(Conservative Semi-Lagrangian) based Cartesian Grid Method.

: CIP , CSL2 , CCUP , surface tracking, multi-phase

1��,��* Corresponding author E-mail: [email protected]

2011 전산유체 학술대회-2차 수정.indd 25 2011-05-23 오전 9:55:40

26 제 1 발 표 장 (26일 목) 다상유동 Ⅰ

CIP f /g f x Hermit . (1) (4) .

g g uu gt x x

(4)

g f . 2 (5)~ (6) 3(7)~ (8) .

3 2( ) ( ) ( ) ( )i i i i i i i if x a x x b x x c x x d (5)( )i i if x f , ( ) /i i idf x dx g

1 1( )i i if x f , 1 1( ) /i i idf x dx g (6)1 3 2n n n

i i i i if a X b X g X f1 23 2n n

i i i ig a X b X g (7)

2 3

2( )i iup i iupi

g g f fA

D D

2

3( ) (2 )iup i i iupi

f f g gB

D D (8)

X udt iup( )iiup i sign u , ( )iD dx sign u .

( )isign u iu +1 –1(9) Fortran code .

If ( iu >=0 ) then sign( iu ) = 1

else sign( iu ) = -1

end if (9)

(7) t

Navier-Stokes.

CIP

. CIP CSL4, CSL2

. CSL2 Multi-phase CIP

CSL2. CSL2 . 1

(10) (1).

0D Dt x

(10)

CIP Hermit (10)

(11) .

0D Dt x

(11)

(11) ( )D f , (10) D= fdx

CIP. CIP f

/f x 3 CSL2f fdx 2 .

. 2 3 .

,

( ) ( )i

x

i xD x f x dx (12)

3 2( ) 1 2 ni i i iD x A X A X f X (13)

3(14) .

2( )( ) 3 1 2 2 ni i i i

D xF x A X A X fx

(14)

.

( )iD x CIP(15) .

( ) 0i iD x , ( ) ( ) ni iup i icellD x sign u

( ) nx i i iD x f , ( ) n

x i iup iupD x f (15)

, nicell

( ( ) / 2iicell i sign u ), ( )iiup i sign u . (16) .

( ) 2 ( )1n n n

i iup i icellf f sign uAx x x x x

2( ) 2 ( )2n n n

i iup i icellf f sign uAx x x

(16)



�� i iup ix x x� � � ��.

2.3��CIP�� f � �� x f� � ��

CIP-CSL2�� f � �� .�� (17)� �� CIP-CSL2�� sub-step 1~3��.

/ ( ) / 0f t uf x� � � � � � (17)

sub-step 1. �(17)� advection phase� non-advection phase��(18), (19)��.

Advection phase

* 2( ) 3 1 2 2 ni i if F x udt A A f� �� (18)

Non-advection phase

1 *ni i if f G dt� � � , * /i iG f u x� � � � (19)

sub-step 2.�� Fig. 2��(20)��.

1/ 2 1/ 2 1

1 ( )i i i i

n n n n� � � ��

� � � � � � (20)

i

n�� (21)� ��.

( ) ( )i

i

xni i ix

F x dx D x�

� ��

� �� 3 2( 1 2 )n

i i iA A f� � �� (21)

sub-step 3. sub-step1), 2)��.

��, iu t� � � � ��. �(20), �(21)� � �� (��)��.

Fig. 2 ��

2.4��

CIP-CSL2�� CIP�� TYPE-M�� Directional splitting��. Directional splitting

�� 1��. ��(22)�� 2��advection equation ��.

( ) ( ) 0f uf vft x y� � �

� � ��

(22)

2�� Fig. 3��.

Fig. 3 CIP-CSL2 ( x� , y� � )

��,��(23)~(25)��.1

, ( , , )i

i

xnxi j ix

f x y t dx� �� (23)

1

, ( , , )j

j

ynyi j jy

f x y t dy� �� (24)

1 1

, ( , , )i i

i i

x yni j ix y

f x y t dxdy� � �� (25)

�(16)~(20)� ��(sub-step 1.~ sub-step 3.)�1 11 ( , , , , , )n n n nCIPCSL D u f f x� ��

� �� 2�� CIP-CSL2

�� STEP1, STEP2� �� .

STEP1 : ( ) 0f uft x� �

� ��

1 11 ( , , , , , )n step n stepx xCIPCSL D u f f x� �

1 11 ( , , , , , )n step n stepy yCIPCSL D u x� � � � (26)

1

2ij iju u

u �� (27)

STEP2 : ( ) 0f vft y� �

� ��

1 1 1 11 ( , , , , , )step n step ny yCIPCSL D v f f y� ��

1 11 1 11 ( , , , , , )step n step nx xCIPCSL D v y� � � �� (28)

(x)F 1i�

(x)Fi

1ix�

ixx

1i��

i��

tui�tui ��1

, 1i jf �

1,i jf �,i jf

1, 1i jf � �

1, 1i j� � �,yi j� 1,yi j� �

,xi j�

, 1xi j� �



1

2ij i jv v

v

(29)

3.1 ne-dimensional CIP, CSL2 mass, density error, profile

(17) (30) CIPCSL2 .

1(1 0.5sin(2 ))i

i

ux (30)

Fig. 4

( a ) T=0.8 sec. profile, CSL2

( b )T=0.8 sec. Mass Error, CSL2

( c ) T=0.8 sec. f Mass Error, CSL2

(d ) T=0.8 sec. f Mass Error, CIP

Fig. 4 T=0.8sec Profile, CSL2 , CIP

Fig. 4 (b) ~ (d) Original CIP CSL2 mass error Density Mass Error . Mass (

CSL2 mass error 0%.

2-dimensional water drop down 2 (31)~(34)

. (33) CSL2(31), (32)

CIP (Type M-directional split).

, Navier-Stokes ,

.

( ) 0t

u (31)

21( ) p ftu u u u (32)

X

F

1 1.1 1.2 1.3 1.4

0

0.2

0.4

0.6

0.8

1

1.2

1.4

TIME

Err

orof

RH

O(%

)

0 0.2 0.4 0.6

-2E-13

-1E-13

0

1E-13

TIME

Err

orof

MA

SS

(%)

0.2 0.4 0.60

5E-06

1E-05

1.5E-05

2E-05

TIME

Err

orof

mas

s(%

)

0 0.2 0.4 0.6

0

0.005

0.01

0.015

0.02

0.025



2( ) ( )sp p Ct

u u (33)

( ) 0iit

u (34)

u , , , p, i i , sC , f , ,

.,

Cell . (fractionalstep ) ,

(35) . Navier-Stokes C-CUP []. C-CUP

.

homogeneous phase.

**

2 2

1 1( )n

n

s

p ppC t t

u (35)

Poisson 1 Bi-CGSTAB .

CIP-CSL2(Color function, Density function) .

310

(36)~(37)F

.

tan[(1 ) ( 0.5)]F (36) tan

0.5(1 )

arc F (37)

1 0, (38)

.

(1 ) Air Water

(1 ) Air Water (38)

Water ( ) , Air

. 2 water drop domain Fig. 5

. slip 0.001x y , 0.00001t .

Fig. 5 Schematic of water droplet

Water air table 1 .

Table 1 Property of Water & Air Water Air

Density [kg/m3] 1000 1.25 Viscosity [kg/m-s] 1.004e-6 15.12e-6Color function [-] 1 0 Pressure [Pa] 101325 Ratio of specific heat [-] 20000 1.4

Water drop Fig. 6~Fig. 9 .

Fig. 6 T=0.0sec, Density function


0 0.02 0.04 0.06 0.08 0.1

0 0.02 0.04 0.06 0.08 0.1





��

CIP-CSL2�� 1�� 2��. �� CIP-CSL2�� .�� CIP-CSL2�� 3�� Boiling,

Defrosting��.

��

[1] 2001, T.Yabe, “An Exactly Conservative Semi-Lagrangian Scheme (CIP-CSL) in One Dimension”, ASM(American Meteorogical Society) journals, Vol.129, pp.332-344.

[2] 2001, T. Kawai, “Numerical Simulation of the Ricochet Phenomenon using CIP Scheme” �15��(JSCFD), pp.

[2] 1985, T. Yabe, “Cubic Interpolation Pseudo-particle Method(CIP) for Solving Hyperbolic-Type Equations”, in Journal of Computational Physics, Academic Press, Vol.61, pp.261-268.

[3] 2001, T. Yabe, “The Constrained Interpolation Profile Method for Multiphase Analysis”, in journal of Computational Physics, Academic Press, Vol.169, pp. 556-593.

[4] 2004, T. Yabe, “Higher-order schemes with CIP method and adaptive Soroban grid towards mesh-free scheme”, in Journal of Computational Physics, Elsevier B.V, Vol.194, pp. 57-77.

[5] 2006, K. Takizawa, “Computation of Free-surface flow and Fluid-Object Interactions with the CIP Method Base on Adaptive Meshless Soroban grid”, World Congress on Computational Mechanics, Los Angeles, USA.

[6] 2000, R Tanaka, T Nakamura, T Yabe, “Constructions an exactly conservative scheme in a non-conservative form”, Comput. Phys. Commun., vol 126, pp.230

0 0.02 0.04 0.06 0.08 0.1

0 0.02 0.04 0.06 0.08 0.1


Documents

G … · improved is a CIP(Constrained Interpolation Profile) and CSL2(Conservative Semi-Lagrangian) based Cartesian Grid Method. .H\ZRUGV: CIPa , CSL2a , CCUPa , surface tracking,