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Volumes Number 1 1998

Journal of

l ^ E C H N I S C H E XJNIVEElSrrEIT

Archief

Mekelweg 2, 2628 CD D e l f t

T e l s 0 1 5 - 2 7 8 6 8 7 3 / F a s s 2 7 8 1 8 3 6

Marine

Science

and

Teciinoiogy

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J Mar Sci Technol (1998) 3:30-36 JoumB[ of

Marine Science

and Technology

© S N A J 1 9 9 8

Original articles

Distribution of void fraction in bubbly flow through a horizontal

channel: Bubbly boundary layer flow, 2nd report

Y u K i Y O S H I D A \ Y O S H I A K I T A K A H A S H I ' , H I R O H A R U K A T O ^ and M A D A N M O H A N G U I N - '

'Technology Development Department, Ishikawajima-Harima Heavy Industries Co., L t d . , 2-1-1 Toyosu, K o t o - k u , T o k y o 135-0061, Japan -Department of Naval Architecture and Ocean Engineering, University of T o k y o , 7-3-1 Hongo, Bunkyo-ku, T o k y o 113, Japan

- ' I . - E - M Co., L t d . , 2-1-1 Toyosu, K o t o - k u , Tokyo 135, Japan

Abstract: A method of enveloping the huh with a sheet of microbubbles is discussed. I t forms part of a study on means of reducing the skin friction acting on a ship's hull. I n this report, a bubble traveling through a horizontal channel is regarded as a diffusive particle. Based on this assumption, an equation based on flow flux balance is derived f o r determining the void fraction i n approximation. The equation thus derived is used for calculation, and the calculation results are compared with reported experimental data. The equation is further manipu-lated to make it compatible with a mixing length model that takes into account the presence of bubbles in the liquid stream. A m o n g the fart-ors contarTted4n the e q i i ' ^ ^ i ^ ' t l ^ s derived, those affected by the presence of bubbles are the change of mixing length and the difference in the ratio of skin friction between cases with and without bubbles. These fac-tors can be calculated using the mean void fraction i n the boundary layer determined by the rate of air supply into the flow field. It is suggested that the ratio between boundary layer thickness and bubble diameter could constitute a signifi-cant parameter to replace the scale effect i n estimating values apphcable to actual ships f r o m corresponding data obtained i n model experiments.

Key words: bubbly flow, turbulent boundary layer, mixing length, skin friction, void fraction

List of symbols

fl, proportionality constant indicating directionahty of turbulence

B "law-of-the-wall" constant

Cf local skin-friction coefficient in the presence of bubbles Cf„ local skin-friction coefficient i n the absence of bubbles

rfj bubble diameter (m)

g acceleration of gravity (m/s^)

jg flow flux of gas phase accountable to buoyancy (m/s)

ƒ, flow flux of gas phase accountable to turbulence (m/s) /<4 constant relating reduction of hquid shear stress by bubble presence to decrease of force imparted to

Address correspondence to: Y . Yoshida

Received for publication on June 30,1997; accepted on Sept. 30, 1997

bubble by its displacement due to turbulence mixing length of gas phase (m)

4, mixing length of liquid phase (m)

/„,,, diminution of hquid phase mixing length by bubble presence (m)

2g late of air supply to hquid stream (l/min) velocity of bubble rise (m/s)

2R height of horizontal channel (m) T, integral time scale (s)

U,„ mean stream velocity in channel (m/s)

friction velocity in channel (m/s) V' voiume of a bubble (m^)

ll, V time-averaged stream velocities in .r- and y-directions,

respectively (m/s)

ll', v' turbulent velocity components in -v- and y-directions,

respectively (m/s)

v' root mean square of turbulence component in the

y-direction (m/s)

Y root mean square of bubble displacement in y-direction

with reference to turbulent liquid phase velocity (m)

y chsplacement f r o m ceiling (m) a local void fraction

a„, mean void fraction i n boundary layer

?),„ constant relating local void fraction to "law-of-the-wall" constant

AT, reduction of turbulent stress (N/m^)

K "law-of-the-wall" constant in turbulent hquid region

i n absence of bubbles

K^ "law-of-the-wall" constant i n turbulent hquid region

in presence of bubbles

K"2 "law-of-the-wall" constant i n gas phase

A,„ constant indicating representative turbulence scale (m) p viscosity (Pa x s) V kinematic viscosity (m-/s) p density (kg/m^) Suffixes G gas L liquid 0 absence of bubbles

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y . Yoshida et a l : V o i d fraction in bubbly flow 31

Injection of a sheet of microbubbles into the boundary layer along a wall is one means of reducing skin friction between fluid and wall. I n most calculations performed using the bubbly flow model, the distribution of the void fraction is assumed to be known. I n practical machine design, the need occasionally arises to estimate the void fraction f o r a given rate of air feed. Thus, f o r purposes of both theoretical study and practical application, there is a need to identify the factors that govern void fraction distribution in bubbly flow to permit estimations to be made with requisite accuracy.

Precise measurements conducted by Guin et al.' on a stream of bubbly flow through a horizontal channel have indicated that the distribution of the void fraction across the channel is influenced by the quantity of bubbles present in the stream: increasing the rate of air feed without changing the liquid stream velocity tends to draw the bubbles away f r o m the ceiling.

For a vertical channel, studies on upward or down-ward flow have been undertaken by, f o r example, Sato et al.,2 who examined the behavior of bubbles in a stream of low void fraction (less than 5% peak value). They showed that small bubbles similarly tend to be drawn away f r o m the wafl with increasing liquid stream velocfly.

The foregoing tendencies shown by the bubbles can-not be explained solely by consideration of bubble buoyancy, typically represented by Saffman's lifting force.3 The force exerted by the bubbles on the liquid phase still remains to be fully analyzed. Nor has a vahd theoretical mechanism been established to substantiate the empirical formula derived by Kataoka et al* for estimating the void fraction.

I n a preceding report,^ the present authors set f o r t h a simple Lagrangian form.ulation of bubbly flow. A model was proposed to explain the reduction of skin friction that is provided by a given mean void fraction of bubbles in the boundary layer.

As a sequel to the above report, we present a method for calculating an approximation of the distribution of

Um

1

2R i

1

void fraction in a bubbly flow through a horizontal channel. Following the procedure adopted in the pre-ceding report,"" we consider the effect of bubble void fraction in terms of the mixing length. W i t h the bubble considered as a diffusive particle, the master equation is derived on the balance of flow flux in the direction away f r o m the ceiling. This master equation is solved to deter-mine the influence on void fraction of the rate of air feed into the liquid stream and of the stream velocity.

Theoretical consideration

Coordinate system

The object of examination is a two-dimensional, hori-zontal, bubbly water flow that has attained f u l l develop-ment into a steady state. The coordinate system is shown in Fig. 1; the y-axis coincides with the downward direction.

Master equation

The factors governing bubble diffusion are turbulence (mixing length and turbulent velocity), gravity (buoy-ancy), pressure gradient, and l i f t . I n the present in-stance, the bubble flow is approximated by a continuous gaseous stream, and the bubble is assumed to be d i f f u -sive. Once the bubble is considered as a diffusive par-ticle, its velocity distribution no longer needs to be explicitly taken into account.

The possibility of further neglecting the factors of pressure gradient and l i f t will greatly facilitate analysis. As a first step, we formulate the master equation f o r void fraction, taking account solely of the two factors of turbulence and buoyancy.

In a steady stream of fully developed turbulence, the sum of flow flux in the 3'-direction is nil:

(1) A control surface perpendicular to the y-axis is con-sidered, as iUustrated in Fig. 2. I n steady state, a bubble oscülates across this control surface within a range equal to the mean free path. Assuming diffusion exhib-its directionality independent of the y-axis, the turbu-lence created by the bubble crossing the control surface generates a flow flux

Fig. 1. Basic coordinate system

dy 1^ dy

d^ÏÏ, da dn, ^ dn,

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32 Y. Yoshida et al.: V o i d fraction in bubbly flow

Bubble

O

Conti'ol surface

Fig. 2. Control surface

I n Eq. 2, if ttie flow field is isotropic, A, = 1. Moreover, buoyancy generates a flow flux

(3) Substitution of Eqs. 2 and 3 into Eq. 1 formulates the master equation for the void fraction:

a-da dll^^

dy dy + aq = 0 (4)

where the mixing length 4 of the gas phase is sought as described below, using the value of mixing length dimi-nution /„,j of the liquid phase defined in our earher report.^

As described previously,^ the presence of bubbles causes a reduction of shear stress equal to

The law of action and reaction applying between the gas and liquid phases calls for balance of the reduction of shear stress in the liquid phase with the increase of the corresponding stress in the gas phase, i.e., the turbulence stress. Hence, f r o m Eq. 5, the shear stress reduction

dy dy

Assuming that 0^ = ïiu X /„

Substituting E q . 7 into Eq. 4,

" l 'mb PG a dhif^ da düj^ ^ dy- d}' dy + CCq, (7) (8)

I n Eq. 8, the first term on the left hand side corre-sponds to the gradient of turbulence, the second term to that of void fraction, and the third term to buoyancy, so that akogether the equation indicates balancing of the three factors.

Solution of tiie master equation

We will here limit our aim to estimating the

approxi-mate mixing length by considering solely the upper half of the channel, in which half alone the bubbles are inferred to be gathered upon development of the lence to a stabilized steady state. Assuming the turbu-lence stress to become nil along the channel axis, and the velocity distribution to be logarithmic, the mixing length^'

R (9)

I n seeking to determine the diminution /,„,, of mixing length, the constant Km Eq. 9, which becomes K2 for the gas phase, is defined by Eq. 29 of our previous report,' in the f o r m Ki =

K-K2 = n,„c(T (10)

where ;]„, = XJd,,, as also defined in the same report. Here, by approximation, 77;,, 0= Sd,, (see Appendix).

I f the boundary layer in the flow field is small com-pared with that in the experimental tank in which 7]„, was determined empirically, the value of ?]„, is reduced in proportion to the boundary layer thickness; it is mag-nified similarly in the opposite case.

I n the liquid phase, the mixing length diminution

/„ '^23',! R dy d%^ (6) dy^ R

I n a field of logarithmic velocity distribution,

Kl y

dü^ _ J_lf

Kl y'

Substituting Eqs. 11, 12, and 13 into 4,

3 1 -where da dy R a aipLlPoU.nfnC^f (11) (12) (13) (14) (15)

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Y. Yoshida et al.: V o i d fraction in bubbly flow 33 I n Eq. 14, i f a is considered to be a function solely of

y, the partial differentiation can be replaced by f u l l differentiation:

3

da

(16)

which, being a differential equation of separated vari-able f o r m , is easily solved analytically in the f o r m

R (17)

where the constant of integration JC, is determined so as to let the mean void fraction obtained by integrating Eq. 17 along the y-axis become equal to the mean void fraction a„, determined experimentally at the prescribed air supply rate.

Calculation

Conditions applied

Adopting as the subject of calculation the channel used by Guin et al.' in their experiment, of height 2R = 10mm, we simulate the stream through this channel. The examination w i l l cover the five comhinatinns of experimental conditions f o r which the bubble size was measured by Guin et al.'

I n the lower half of the channel, below the centerline, both the turbulence gradient and buoyancy impart an upward velocity to the bubbles, moving them to the upper side. This would justify the earlier assumption that, in a stabilized steady condition and beyond a certain distance downstream along the channel, the bubbles are gathered in the upper half of the channel, and more specifically in the upper boundary layer of 0 < y < 0.005m.

The main liquid stream is propelled at velocities U,„ = 4.5, 6.3, and 8.1 m/s and air is fed at rates 2 ^ = 23, 40, and 50 l/min.

Representative values of bubble diameter, cited f r o m G u i n et al.,' are rf,, = 700,um f o r the case of U,„ = 4.5m/ s, and dl, = 500/.mi f o r U,„ = 6.3 m/s and 8.1 m/s.

The master equation f o r given in the preceding report' is

1

In UR + B _1_ (18)

where / f = 0 . 4 1 , S = 4.9.

Between cases with and without bubbles present in the liquid stream, the skin-friction ratio

uju,,)' (19)

As regards the constant A, of Eqs. 2, 4, 8, and 15, indicating the directionality of turbulence, the well-known study by KlebanofP^ indicates the turbulence in the direction normal to that of the main stream (i.e., in the y-direction in this instance) to be 0.5 to 1.0 times that of the main stream. Based on this fact, we adopt in this instance «, = 0.75 as a representative valtie.

For 7]„, of Eq. 10, a value of 0.85 was given in the preceding report' for a boundary layer thickness that was about 10 times what is prescribed f o r the present study. Since the main stream velocity was of the same order of magnitude as in this case, only the difference in boundary layer thickness need be taken into account i n this instance, and we let 7j,„ = 0.085.

For the velocity of bubble rise, applying the bubble friction coefficient given by Stokes, balance is achieved between buoyancy and friction when

6Kp^(dj2)q^ = p,gV Hence, PLSV 6KPi^[dj2] (20) (21) Results of calculation

Presented in Fig. 3 is the skin-friction ratio CflCf^ as function of air supply rate a,„ calculated using Eq. 19. The corresponding measured data given by Guin et al.' are also drawn in for comparison. The calculation and the measurements are roughly similar. What differs be-tween the two is the trend shown by the measured plots, but not by the calculated plots, of first lowering rapidly with increasing air supply, then flattening out, and then rising again. The reason for this discrepancy might lie in the fact that the calculation did not account for minute localized differences in void fraction, while in the actual flow the bubbles may have been very un-evenly distributed.

The distribution of void fraction a calculated using Eq. 17 is presented in Fig. 4, normalized to the mean vahie «„, across the boundary layer. The inference adopted earlier that the bubbles were gathered in the upper half of the channel is substantiated by the curves that do not cross the y-axis.

W i t h the air supply rate QQ held flxed at 23 l/min, reduction of the stream velocity f r o m 8.1 m/s to 4.5 m/s is seen to have tended to draw the peak of the void frac-tion away f r o m the ceiling. A similar tendency is seen when the air supply rate is increased with the stream velocity held fixed at 4.5 m/s.

Corresponding to the calculated data of Fig. 4, Fig. 5 presents the measured plots given by Guin et al.'

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Re-34 E O ; é 0.8 -

I

-OT 0.75 -

i 1

0.7 r I I I I i . I I I i , , , , -O 0.05 0.1 0.15 0.2 Mean void fraction a

m

Fig. 3. Skin-friction ratio

u ,Q ,d

m' a' b

0 0.002 0.004 0.006 0.008 0.01 y[m]

Fig. 4. Calculated void fraction ratio

.01 y[m]

Fig. 5. Measured void fraction ratio

Y. Yoshida et al.: Void fraction in bubbly flow

garded qualitatively, the curves in the two figures can be considered to agree well.

Di.sciissioii

Distribution o f void fraction

The void fraction expressed by Eq. 17 separates, bro-adly speaking, into the two cases of ^ 1 and /CQ < 1-The former case appears when U,„ is large and when a,„ is small. In such case, the effects of the turbulence gra-dient /, (Eq. 2) and of the buoyancy (Eq. 3) do not balance off between the terms of Eq. 4 at any value of y, and the bubble void concentrates close to the ceiling. When /Co < 1, a balance between the two terms takes place at a certain value of y, to produce a peak of the void fraction at the relevant distance f r o m the ceiling.

When the calculated curves of Fig. 4 and the mea-sured curves of Fig. 5 are compared, the separation into two distinct groups is seen in both. The noticeable de-viation of the calculated curves f o r the 23 l/min air sup-ply rate can be ascribed to the large void volume causijig the void fraction peak to descend close to the channel axis; as a result, the calculated data reveal the conse-quence of neglecting the conditions beyond the channel axis.

Skin friction and scale effect

The skin friction in the presence of bubbles, Cj-, can be determined f r o m the skin friction i n the absence of bubbles, Cf^: the local void fraction a i s derived f r o m the void fraction a,„ averaged over the boundary layer thickness through calculations using Eqs. 10, 18, 15, and 17. I n the course of this calculation, substituting determined by Eq. 18 into E q . 19 will give the value of

Cfl CfQ.

W i t h reference to the scale effect, the agreement of the trend seen between the calculated and measured data of Fig. 3 would point toward the possibility, subject to further verification studies, of the ratio between boundary layer thickness and bubble diameter coming to constitute a significant parameter that could serve i n place of the scale effect. The calculated data indicated i n Fig. 3 were derived by adopting a value of parameter

7],,, = 0.85, and the measurement data a corresponding value of 7]„, = 0.085, f o r boundary layer of thickness S differing by a factor 10. Through the boundary layer the fluid flowed at a similar velocity, which led us to infer that the bubble diameter df, was roughly similar in both cases. The approximate agreement seen in Fig. 3 be-tween calculated and measured data would thus indi-cate that the relation 77,,, Sid,, adopted in Eq. 10 would hold true. This would indicate that the ratio between

(7)

Y. Yoshida et ai.: V o i d fraction in bubbly flow 35

boundary layer thickness ö and bubble diameter cl,, could serve as coefficient to relate systems of different sizes for parameters such as i],„ governing the local void fraction. Such an expedient is permissible since /],„ is a parameter that considers the conditions within the boundary layer in macroscopic scale.

Observations on testing Inill models in a laboratory basin

We here consider resistance measurements in a labora-tory basin on a hull model equipped with bubble-ejecting slit.

I f a value applicable to the actual hull is to be ob-tained f r o m experiments on a model in a macroscopic approach, this would caU f o r matching the values of /c,, i.e., [(boundary layer thickness)/(bubble diameter)] x [2/3 power of mean void fraction], and of the Reynolds number to those of the actual hull.

Matching either of the above values would, however, be difficult in practice. Now, the measured data on bubble diameter given by Guin et al.' would indicate that, i n a f u l l y developed stream, the bubble diameter is strongly influenced by the stream velocity. Thus, in the model experiment, which is performed at a stream ve-locity lower than that for the actual hull, it would be logical to consitler tiie btrbbies to be of diameter ferger than in the case of the actual hull. Also, the thickness of the boundary layer would be smaller in the model than in the actual hull, on account of the shorter total model length.

The foregoing circumstances would tend to diminish /]„, of the model compared with the actual hull, which would mean that, even if a,„ could be matched between the model and the actual hull, the reduction of skin friction derived f r o m the model would be smaller than that f o r the actual hull. Moreover, increasing the air supply rate in order to raise the value of a,„ would be limited by the necessity of maintaining the fluidity of bubbly flow.

A l l in all, it can be considered difficult to correctly determine the skin-friction ratio Cf/Cjo of the actual hull f r o m model basin experiment. Nevertheless, i f theoreti-cal analysis of the phenomena occurring in the model basin prove to substantiate the experimentally obtained data, the theoretical considerations adopted in the model experiment might validly be applied to the actual hull. This is a matter that needs to be further examined.

Upward flow tiirough a vertical ciiannel

We w i f l here add a discussion on the void fraction distri-bution across a vertical channel with flow in the upward direction. I n this case, gravity acts not in the y-direction, but in the upstream direction, and as a result the

bubbles would tend to i ise faster than the liquid phase. It is also known that, particularly with slow stream ve-locity, the void fraction peaks in the vicinity of v/all.

These phenomena can be explained in the following manner. Buoyancy generates a fairly high bubble veloc-ity relative to liquid. I f under this condition a bubble enters the boundary layer, which is a domain of shear flow, buoyancy is generated in the direction that impels the bubble toward the wall. As a resuh, the frequency of bubbles traversing a channel cross section will concen-trate toward the wall.

In the preceding sections dealing with a horizontal channel, the treatment covered cases where the pres-sure gradient and buoyancy could be neglected f o r the purpose of obtaining a better overall perspective of the situation. I n dealing with a low-velocity upward flow through a vertical channel, the above two factors would have to be taken into account in the master equation. However, whether or not such a treatment will in itself provide a valid solution cahs for further scrutiny and f o r experimentation with a flow of void fraction that is in a range similar to that adopted in the present study.

Conclusions

— A peak appears i n thfr void fraction ratio plotted against distance f r o m the wall in the boundary layer of a flow containing bubbles. This peak moves away f r o m the ceiling with increasing flow rate. The above behavior is explained by the balance of the effects of turbulence and of buoyancy. A simple master equa-tion governing the distribuequa-tion of void fracequa-tion has been derived f r o m theoretical consideration, which should prove conveniently applicable in practice. — The analytical solution of the above master equation

proved to agree qualitatively with experimental data. The accuracy of estimation could not be con-sidered good toward the outer zone of the boundary layer. This calls f o r further examination of such fac-tors as the boundary conditions.

— I t is suggested that the ratio between boundary layer thickness and bubble diameter could constitute a signiflcant parameter to replace the scale effect in estimating values applicable to actual ships f r o m corresponding data obtained in model experiments.

A p p e n d i x

Derivation ofthe expression ?],„ °^ &d,,

The reduction of mixing length

(8)

36 Y. Yoshida et al.: V o i d fraction in bubbly flow where turbulent domain, i.e., y+ = 1200 and yl5 = 0.35.

Substi-^ tuting these values into Eq. 24,

K,-5 (25)

1 , \ , , Substituting Eq. 25 into 23,

[K,y] (23)

1^ ^ R e f e r e n c e s L

I 3/2 1- G u i n M M , ICato H , Yamaguchi f t et al (1996) Reduction of V ^ — ^ / c ^ ' / ^ skin friction by microbubbles and its relation with near-wall

£ ^ bubble concentration in a channel. J M a r Sci Technol 1:241-_ i 254 ) 2. Sato Y , Honda T , Saruwatavi S et al (1977) Two-phase bubble flow

(2nd report, influence of liquid stream velocity and channel size on f / j j y air bubble motion) (in Japanese), Trans Jpn Soc Mech E n g

— 4 3 : 2 2 8 8 - 2 2 9 6

'^y J 3. Saffman PG (1965) The lift on a small sphere in a slow shear flow. J Fluid Mech 22:385-400

ICataoka I , Serizawa A (1995) Modeling and prediction of turbu-t / j . lence in bubbly turbu-two-phase flow. 2nd Inturbu-ternaturbu-tional Conference on

X Muhiphase Flow '95. Kyoto, Japan, vol 2. A p r i l 37,1995, pp M 0 2

-y+ I ^ y'^'^) 5. Yoshida Y , Takahashi Y , Kato H , Masuko A et al (1997)

Simple Lagrangian formulation of bubbly flow in a turbulent boundary layer (bubbly boundary layer flow). J M a r Sci Technol

While any given value may be- s e t ler y/y% we h«r^- g. t k l f l T, Inoue M (1981) Dynamics o f viscous flm-d (fn Japanese).

1',

y

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