Y. Ito, H. Wakamatsu, T. Nagasaki, “Numerical Simulation of Sub-Cooled Cavitating

Flow by Using Bubble Size Distribution”, Journal of Thermal Science, 12, pp.350-356

(Nov 2003)(Springer)

Numerical Simulation of Sub-cooled

Distribution

Cavitating Flow by Using Bubble Size

Yu ITO Hideki WAKAMATSU Takao NAGASAKI Department of Energy Sciences, Tokyo Institute of Technology, 4259 Nagatsuta-cho, Midori-ku, Yokohama, 226-8502, Japan, E-m叫： ito@es.titech.ac.jp

Anewcav,ta血g model by usmg bubble ,; ね d;stribution based on m邸s of bubbles ;, proposed. Llqwd phase is 如ated with Eu! ロran釦皿ewm-k 邸 a mixture con国=g =•te c,v,tating bubbles. V; 叩o, ph邸e eon,;,i, of vuoous出加sofntino記 vapo, bubbles, wltich ;s <listributed to cl邸s邸 b邸叫 ontherrm邸s. Thee細nge of bubble nm曲o, de面tyl直 e紅hdru;sw邸 solv叫by oo"'idering the eh皿ge of bubble m固 due to phase ch皿ge 邸 well邸 geoo呻on of oow bubbles due to he記rogen切四 nucl四non In this -thod them邸s ofbubbl邸 ;sue叫ed 邸皿

;naepeadent variable, m othec woro, a new cooroin,te, 皿do吋wno,ntv面曲lesaresolv叫mEuleri皿 framewo,kfot spati,I coonlinates 皿d bubblo-mass coo血,te. The pre記nt method ffi ,ppli叫to a c,v; は血g flow ma 四ve屯ent-<livo屯ent no,tle, 皿d the two-ph邸e flow w曲bubbles立e <listribution 皿d ph邸e clumge w邸

successfully predicted.

Key words: cavitation, numerical shnulation, liquid uitrogen, bubble size distribution.

Introduction

Cavitation in turbo-pumps is one of the important problems on developing a liquid fuel rocket engine. In present technology, it is difficult to predict the cavitation and its influence on the ped'ormance of turbo-pumps accorately. Especially叩ogenic cavitation is complex bern,0e cryogenic fluids, such 邸 LH,, LOx and LN,, have sm叫e,latent h四t and lowe, temperature level than onlin町flui曲like H,O, and they 匹 advemly aflected by heat inflnx. The development of reliable and efficient nmnerical code fo, a high-speed叩ogenic cavitating flow is strongly desirable, because it becom蕊 possible to predict the ped'onnance at a design stage and to叫ucenumbcr of prototy pes o, experiments.

The autho,s developed a strict numerical code''1 fo, a high-speed cryogenic cavitating flow. It employed a model that h邸 the following two f四皿es; 1. A comp,essible liquid phase w邸 dealt with Eulerian framewo,k as a m紅ture containing minute cavitating bubbles. 2. Cavitating bubbles were treated with Lagrangian approach, in othe, wo,ds, each bubble was distinguished and traced. The results showed good agreement with NASA experimental data"1 by Simoneau & Hendricks. It needed, howeve,, 1江ge memory and

CPU power in proportion to the volume of calculation donutin, independent of number of numerical grids, b四ause all of the bubbles were distinguished and traced. Such a direct simulation of each bubble is possible for a small flowfield like a Laval no,tle, but it is difficult to handle a lruge flow伽ld such as the turbo-pumps of the rocket.

T皿i et al.131 and Schnerr et aI.1" treated the cavitating flowfield as a mixture and obtained good results, however, they took no thought of bubble sire distribution in a control volume. When we analy瓦 arotating cavitation in the turoo-pumps, for example, it is supposed that analysis for the flowfield, in which both tiny 皿d lruge bubbles exist together, will become important.

In this study, consequently, a new model is proposed to deal with the bubble si,e distribution. The liquid model is the same as the previous one. Vapor phase consists of v江ious s1'es of minute vapor bubbles, which is distributed to classes based on their mass. The change of bubble number density for each class was solved by considering the chnnge of bubble m邸s due to phase change as well as generation of new bubbles due to heterogeneous nucleation. In this method the 皿ss of bubbles is treated 邸 an independent v虹1曲le, in other

woro, a new coo咄nate, and dependant variables are solved in Eulerian framewmk fo, spatial coonlinates and bubble-m邸s coordinate. The present model can deal with the bubble size distribution with much less computational effort than the Lagrangian simulation of each bubble.

Mixture Model

Foe the Euleriao fi:amewock of the liquid phase, the flowfield is divided into control volumes, each filled up with the mixture containing minute vapor bubbles. Geoerally it is well known that pure liquid behaves like incompressible fluid whose average sound speed is Ia,ge, however, liquid cootaining bubbles behaves as com­pressible fluid whose average sound speed is sm叫Based on void fraction in the control volume, the average sound speed is, therefore, calculated, so the effect of conはining bubbles is taken into accouot. Besides, since compressibility of the pure liquid is coosidered, sub-cooled conditioo can be rigornusly calculated. By using this model, basic equations for the liquid phase 四

given 邸 follows:M邸s consemtioo四uation"

一(a,p,)+V(a必V,)a-'1,-l(N,r.)"'

Momentum equatioo a

-(aLPL VL)+V(aLPL V[) =aL▽ P+ a,

aLI仕AVL+知VL ー L{Nf'o,CVrn -Vd)­,

LN瓦+aLpL{roxVL 辺x(roxr))

Ene屯y equation a

a, ー(a,p心+V(a,p凶V,)=—A,,h,, 一L(N凡，％）

(I)

(2)

(3)

Here, aL is the volume fraction occupied with liquid in a control volume, i.e. aL = 1-<知(aadenotes void fraction). p, V, A, r, P, µ, N, F, w, ,,, 皿d h are density, velocity vectot, mass of nucleation pet unit volume匹, unit time, mass of ph邸e change四one existing bubble 匹, unit time, pressure, viscosity, numbcr-density of bubble pet unit volume, friction vectot pet one existing bubble, angul紅 velocity of the system,<li叩n•e from the cente, of the totational system, total ene屯y and enthalpy, resp虹tively. Subscript L, G and i denote liquid, bubble and level of the bubble m邸Sclass.

Bubble Size Distribution Model

The vapor ph邸e in intended flow伽Ids consists of

minute bubbles m the order of sub-micrometer to sub-mHHmeter. Then the bubbles are assumed to be a cloud of spherical bubbles, and all bubbles are filled with saturated vapor. In the control volume the bubbles are distributed to the classes baぬd on their mass. Each class has bubble number-density N, m邸s of bubbles ma per urut volume, mass of phase change for ooe bubble r 0 per unit time , mass of bubble nucleation Aa per unit time aod unit volume in the smallest class only, and bubble velocity知. A coordinate of bubble mass (w) is defined m addition to the spatial coordinates (x , y , z). N and ma are treated with Eulenan framework with respect to x, y, z and w. By using tlus model, basic equations for the vapor phase are written as follows:

Num如density for each mass class of bubbles a il

-N+V(N,a)+ー(N氏）＝”a, aw

(4)

where ,i : Number of nucleation per unit time and unit volume based on the heterogeneous nucleation theory Mass conservation equation for each mass class of bubbl邸

o o ー（灰） +V(哄,ua)+ ー（叱応） =Aa+NI'a., ow

"'a=a⑲G

Volume fraction

aa, =(4函N,)/3

aL 十 �a0,�1ヽ•W

(5)

(6)

(7)

(8)

- The exp皿,ion/contract,on rate of the vapot bubble,R=dR/dt, is dominste<l by two mびhsni<m<; a momeotum that is needed to push away/pull up ambient liquid, and a heat transfcr that is needed to dnve ph邸e change at the intedace between the bubble and ambient liquid. They are called the inertia (momentum) control and the heat transfet control, tespectively. Genetally speaking, in the expansion ptocess the bubble radial movcrnent is dominated hy the inertia only at first and then changes to the heat transfet control. Ovcr the whole contraction process it is controlled by the inertia1".

Inertia control When a radial speed of bubble mterface is higher

thao a developing speed of a thermal boundary layer around the bubble in the expaosion process, the thermal boundary layer is thin enough for the phase chaoge at the interface. Therefore it seems that equihbrium condition " kept al the interface. So does ii in the contraction proc岱s.

T,•T,

P0 •P,(T,)

P,•p,(T,)

(9)

(10)

(11)

4 dR1

I'a=-;r:pa-3 dt

Heat transfer control

(12)

(13)

When the radial speed of the bubble interface islower than the developing speed of the thermal boundarylayer around the bubble in the expansion process, thethermal boundary layer is well developed and itsthickness is nearly the same as the bubble size. It seemsthat dRldt is much smaller than the other terms in theequation of interfacial momentum because the interfcialmovement becomes instantaneously quasi- steady st.ate.

I'a =4,rRJ(T,. -Ta)/L (14)

in .!!..(PaR')=I'a 3 dt (15)

Pa =Pi, +'lu/R (16)

Ta =Ta(Pa) (17)

Pa= Pa(Po) (18)

Numerical Method

Fig. l shows a flowchart on a calculation procedure.In the babble nucleation stage, initial radius Ro, thenumber n and the mass Ao of the bubble nucleation arecalculated by the heterogeneous nucleation theory ofbubbles.

Ro= 2aj(P,a1 -PL) (19)

(20)

(21)

Here, CT, P ... ,, N0 , m, A. and rp are coefficient of surfacetension, saturated pressure, number of "seeding nuclei"per unit volume, mass of one molecule, thermalconductivity and parameter of contact angle on the"seeding nuclei". Ao takes a value in the smallest class ofthe mass only. In the bubble interior stage, it is necessaryto judge the appropriate expansion model of each class.Figs.2(a) and (b) show a temporal change of the bubbleradius. As shown in Fig.2(a), the bubble radialmovement is controlled by the inertia only at first and bythe heat transfer in almost all period. Therefore the heattransfer control is employed in the expansion process,and the inertia control is employed in the contraction

process. In the advection stage, SHUS (SimpleHigh-resolution Upwind Scheme) approach161 with 3rd

· order MUSCL interpolation is applied to the spatial integration. Each physical property is function of temperature and pressure. In the pressure stage, the pressure is implicitly calculated by using the balancebetween the liquid density and the void fraction. In thebubble movement stage, a slip velocity between the liquid and the bubbles of each class is calculated. Afriction force between them is also taken into account. Inthe temporal advance stage for unsteady analysis,four-stage 2nd order Runge-Kutta temporal integrationscheme is applied so as to maintain the time accuracy.

10,,10•

[E] "'"'""""""

'°'

• loilialrad」ss100X10 m Ambieomいµs,atsra90K�"しio,,IPra�s「e02MPaLN,

102 --- ------

100 ニ―---—―�

98゜

--·H翌"""·「c1oool『ALーloertia必““

2 3 time t[seo[

4 5,10 a

(,)fapfil四onprr心ss

1'0110 ふ

120

[Elと'"""』"""'

.o

0

0

0

0 0

0

0

8

6

4

2

ー

• lm<,al rad'"'100 X10 m Ambeo<T,m伝,.しl「'90KAmb,eo<Pre,'"re0.2MPa

LN,

lo,rtiaw'"'"'

' 3 time t[sec]

4 '''° <

(b) Contraction process

Fig,2 Inertia/He,t transfe, control

Results

Analys;,; on static bubble growing Ficstly, bubble nucleation and gmwth in a

supe<heated liquid: Fig.3(a), and bubble collapse in a sub-cooled liquid: Fig.3(b) were simulated. Figs.3(a) and (b) show the tempo,al change of bubble numbe,- density N,(1厨kg) and void fraction aG, (I/kg) of each class (.As shown in Fig.3(a), numbe,-density is increased due to the nucleation in the smallest class, i=l, the new bubblegrnws with time, and the bubble shi応tol四ge,cl邸s,i>l.On the othe, hand, in the case of bubble collapse in a sub-cooled liquid as shown in Fig.3(b), the existmgbubble is contrac<ed, it shifts to smalle, class, 皿dfinrulyit becomes extinct at the smallest one. Figs.3(c) and (d) show the comparison of results by using the bubble si,edistribution model with various dw. The dw is the gridinte,val with respect to w. In叫dition results of single bubble growth and collapse without using bubble si,edistribution model (non- distribution model) 紅e plotted Almost the same results are obtained fo, various dw. At 炉IO

―", all bubbles are cl邸sified in the smallest class,

therefo,e dw=W-" is叫opted. They c皿 be cal四la<edwithin the 5% erro, as compared to the non-distribution

model.

Analysis on cavitating flow in 1D Laval nozzle el·

Experiment,! results of Simoneau & Hendrie訟 mNASA 肛e reliable and useful because s皿pies are quite enough.Hence we use Lav,! nozzle whose profile is the same as them. Fig.4 shows the nozzle profile, which has the 7deg conve屯ent section, the 9 mm straight throat, the 3 deg dive,gent section and the width is constant at 10 mm. The beginning of the tlrroat is set as xecO and positive v,Iue in the flow direction. Fig.5 shows pressure profiles c,lculated with vurious grid sires, dx. Th函紅eoscillations around the throat, howevec, positions of shock waves 紅e obtained steadily. We employed d司mm.

Fig.6(a) shows pressure and void fraction proftles for various N0, which corresponds to number-density of impurities existing in the liquid as the "seeding nuclei". Representative values of M。 in tap watec is roughly about JO'to !Ow (1/mり，which is Japanese st皿d叫of water­purity for tap watec, however, various v,lues have beeo used, for ex血pie JO" (llm') in tap water of Schue炉and JO" (1/mりin liquid血rogen of Taoi1". In practic,I step, it is necess紅y to get optimum value of Na and¢for the workin g fluid. As shown in Fig.6(a) the pc蕊suretends to rise with inc, ⑫ sing Na in the divergent 記ction.The void fractions in ,II cases 紅e ,!most the same Flg.6(b) shows pressure and void fraction profiles for various ¢, which is the p紅ameter how easily the nucleation occurs. The smallcr¢is, the more easily the nucleation occurs. In the tlrroat section, the pressure profile for Sin¢= 1炉is diff血nt from the othern. In the divergent section the pressure tends to rise with decrea­sing¢. In the latter p紅t of the divergent section, the void fractionfo,Sin¢=10

―'varies slightly from the others.

Table I shows a bound紅y and initl,I condition to produce a transouic flowpattern. Fig.7 shows results by using the bubble size distribution model in detail. Fig.7(a) shows pressure, tempecature and void fraction proftles. There is a shock wave around た70 mm. Through the shock wave, the pcessure uses and the void fraction decreases immediately. The temperature increases as void fraction decreases, aad vice ven;a. It seems that this phenomena is caused by the effect of the latent heat, i.e. thermodynamic effect. Fig.7(b) shows the void fraction profile on the classes. The liquid in a sub-cooled condition flows into the tlrroat (!), the bubble nucleation occun; around the throat (II) \III), the bubbles grow up in the divergent section (IV), and they contract due to the increase of the pressure through the shock wave (V). Fig.7(c) shows numbげdensity and average 呻us profiles of each class. It shows that the bubbles contract due to the increase of the pressure through the shock wave as well.

C: 0

"0.::; ·- u

��

4X 10.g

• •x•:• 1j

ii>� 2X10 � ·� 2X106:::, Q)

t=tx10-esec t=0.36X10 sec 61 _______ _

zo O 4 .............................. :·X-10-.9-=1.=::.:c .... :c:c ..... ::::: ..... ::::: .... ::::: ..... ::c: ..... c:::: ..... :::::: ..... ::::c_. __ - -- - - --- ----- ---�- 4 X 1 :• 1 :---····.

--. 0

I t=200X10-esec t=0.72X10 sec 2X106

L 2X 106 j .............................. ........................... ............ . ..... ...! ........................ .............. ........ ............. o.g ......................... ........ ........ -·--................. .• x,g.,1 • I . •x•:

a

0

I t=400X10-e sec t= 1.08X10-e sec

2X 1061 � --��---

2X106

�- -A�-- o A -·-·--�:;-g�r--·-·· ·---- --- - -- - --- -- --- •x ·�r_------

. 0 I

A t=600X10.a sec t=1.44X1cr'sec2X 106

1

2X10e A . 1 ........... ........................ --..=..==�====------....... .................. , _____ .. ,0 ............... ,

4X1g.9

�'--�•�-

2X1�

1

�--�A __o Bubble mass class

25

20

15

10

5

(a) Nucleation and growth

Ambient Temperature 90K Ambient Pressure 0.2MPa Heat Ira nsfer control LN2

o-,,.. ..... _._. _ _,.-...-.-...... ..--1

0.0 0.2 0.4 0.6 0.8 1.0x10-$

lime t(secJ

•x•:• I t= 1.aox 1cre sec 6 _L_________ _

2X10

0

.,::,

Bubble mass class (b) Contraction and extinction30x10.a ...---------------,Ambient Temperature 90K Ambient Pressure O.SMPa

25 ,---- Inertia control

20

LN2

'a · 15 I!

10 ---- dW=10•14 - ctw.:10·'3 5 • • • dw=10·12- Non-<listribution Model

Q i--T""T"T""TTTT'T-r,--,-,rr-.....--l-1-,1 0.0 0.5 1.0 1.5 2.0x10.a

lime I [sec] (c) Temporal change of bubble radius in expansile process (d) Temporal change of bubble radius in contractile processFig.3 Calculation by using the bubble size distribution model

Table 1 Boundary 皿d initial conditions

Total inlet temperature

95.6(K)

Total inlet pressure

2.48 (MPa)

Inlet vapor pressure

0.561 (MPa)

Inlet void fraction

。

Outlet pressure

l.02(MPa)

Initial number density

Parameter of contact angel Grid interval

出

1013 (1/mり Sin¢= 10-4

Grid interval

d ー五10·3 (m) 10―13 (kg)

+l、 竺子るFig.4 Laval nozzle profile

(width is constant at 10 mm)

。6ーx6-edl

Ce』nsse』d

4 2

今，•O

rJw>1o·"kg Sin r-10"'

rt.,•2.510やa P,..•1.0 rn'Pa ゜ ., .... , .... , .... ,.

-200 -150 -100 -50 。

"· • ·N =10 . "— N=10

ー一叱10"

50

N UO!P1?11 P!O >

10D

1.0 0.8 0.6 0.4 0.2 0.0 . , .... , .... , .... ,.

-200 -150 -100 -50

...,..o

血=10·1\g Sin F10" Pt.,=2.s 10'P• P,..=1.0 10°Pa

。 50 100

(a) Effect of number- density of seeding nuclei N,。

6 。ーx

3

-ed}de nSS31d

tsuoB3e

I P!O,\

2

゜ ., .... , . . , .... , .... , .. .. ---,---200 -150 -100 -50 0 50 100

Distance from tho, bo,ginnlng of me throat x (mm]

Fig.5 Effect of grid interval d, €゜ーx6

[01』den切ecl

4 2

a,.�a ゃ10" 11m• 年10.10kg

. 'PtJn"2.510 Pa I'; ダ1.010Pa

•- - Slr,;-10 — Sin♦:10

｀• • ·Sin♦-10

゜ ., .... , ......... ,.

-200 -150 -100 -50

1.0-' .. 0.8 0 、6〇.40.2 0.0

。50 100

13 8 ·13 a.,--o �10 1/m 心.,10 kg ． ．l\,•2510 Pa P.,,,,1.0 10 Pa

., - - Si叫=10.. - Si�=10 ..• •• S1�=10

-200 -150 -100 -50 0 50 100 Dis1ance from Illa beginning of Ille throat x[mmJ

(b) Effect of parameter of contact angle on nuclei¢

Fig.6 Variation of Pressure and void fraction

紅101

[ed]daヒmmgaJi

ヽ{

•• ．．．

••• |i

:＇ .. , ．．

．．．．

．

＼

I ••.••

． ．

ヽ

．． ．

-

[

-

i

i

i

＿

i

一

ぃ

一

(

e1 r

.e

u-

．

t

.ru

a r-

＂ •Sser

.

p

5-

T-

．

Void fraction ug peraturo T[K]

96

95

94

93

-＿_ー-_-――-―――――ー―――-―――――――――-―

Void fraction a

8

6

4

2

鼻u

n5

n5

nぃ

1.0

゜ .'.

-200 0.0

-150 ← 100 -50 0 50 100 D ista11Ce fr伽 the beginning of the throat , [叫

(a) Pressure, temperature and void fraction x=O

30xto94tng

M ...... 25 82ne E� 20 130ng ........ �· llt: 15

-- 184ng ... 248ne

10 32·2ng

5

-30 -20 -10 0 10 20 30 40 50 60 100 110

250x10-4 ----------------------,

322 ·200411.li ------::::::::\

�!:, ······ .. •: ... :�:�:.�=.::: ... : .. : .. :: .. :,\�::::::_� � ... ..

a:

150

100 184ng 41 � ••. :: •• :: .. :::.:::-.. ;::-......

so 248ns 322ng

0 ........................................ .,....,_,.. ....... r"""l'"'T'".,...,--.--r-r-r--r-,.--,--r...--1'""""'-r ��� � ro 20 � � so � ro � � 1001m

Diat•nc• from the beginning of the thro•t x [mu

(c) Number density and average radius of each class

Fig.7 �ulta l>y using bubble si7.e di&tribntion model

Cond .. on

A numerical code of cavitating flow with bubble size distribution model based on the mass of the bubbles was developed. The present method was applied to 1D convergent-divergent nozzle flow, a.nd a reasonable behavior of bubble distribution was obtained which indicates the potenti$1 of the present method fm: the simulati� of complex cavitating flows.

References

[1) Ito, Y, Nagashima, T. Numerical Simulation of Sub-cooled LN1 NO'LZ!e Flow with Cavitation. Proc. of the 5111 ISAIF. Poland, 2000. 715-722

{2] Simoneau, R J, Hendricks , R C. Two-phase Choked Flow of

Cryogenic Flnid8 in Cooverging,.diveqillg Nozzle&. NASA TP 1484, 1979

[3) Tani, N, Nagasbima, T, Yamaguchi, K. Numerical Analysis of Q:yogeni<: TM>-p� Flow in 2D Laval Nozzle. Proc. of the Sdi ISAIP.Poland. 2000. 705-713

[41 'Tuan, W, Sauer, J, Sclmerr, G H. Modeling and Computation of Unsteady Cavitating Flows in Jajection Nozzle. Mee, Ind., 2001,2:333-394

[5] Hao, Y, Prosperetti, A. The Dynamics of Vapor Bubbles inAcoustic PressUl'e Flelds. Physics of Fluid, 1999, 11(8):2008-2019

[6) Shima, B, Jonouchi, T. Role of CPD in Aeronautical Engineering (No.12) AUSM Type Upwind Schemes. 12111

NAL Symposium on Ain:raft Computational Aerodynamics. 1994.255-260