I 7,-AD-Ai39 266 BUBBLES RISING IN A TUBE AND JETS FALLING FROM R NOZZLE 1/iI (U) WISCONSIN UN! V-VIADISON MATHEMATICS RESEARCH CENTER

J M VANDEN-BROECK JAN 84 MRC-TSR-2631 DAAG29-89-C-894iUNCLASSIFIED FIG 12/ NL

.1.

.'.

111.2 5 1111 1.4 "i .

1-11--II I 111

o1_

MICROCOPY RESOLUTION TEST CHARTNATIONAL BUREAU O STANOARDS - 1963 - A

5%%

VA

-.-

A-'...

. .. .

"

4.4k-

"

NRC Technical Surmnary Report #2631

BUBBLES RISING IN A TUBEAND JETS FALLING FROM A NOZZLE

Jean-Marc Vandenw4 roeck

Mathematics Research CenterUniversity of Wisconsin-Madison -

610 Walnut StreetMadison, Wisconsin 53705

DTICELECT

January 1984MA2 SAR2T E8(Received August 29, 1983)

Approved for public releaseDistribution unlimited

Sponsored by

U. S. Army Research Office National Science FoundationP. 0. Box 12211 Washington, DC 20550Research Triangle ParkNorth Carolina 27709

64OTCFILE COPY............

UNIVERSITY OF WISCONSIN - MADISONMATHEMATICS RESEARCH CENTER

BUBBLES RISING IN A TUBE AND JETS FALLING FROM A NOZZLE

Jean-Marc Vanden-Broeck

Technical Summary Report #2631

January 1984

ABSTRACT'/

-The shape of a two-dimensional bubble rising at a constant velocity U

in a tube of width h is computed. The flow is assumed to be inviscid and

incompressible. The problem is solved numerically by collocation. The

results confirm Garabedian's [2] findings. There exists a unique solution for

each value of the Froude number F = U/(gh)2 smaller than a critical value

Here g denotes the acceleration of gravity. It is found that T=

0.36. In addition the problem of a jet emerging from a vertical nozzle is

considered. It is shown that the slope of the free surface at the separation

points is horizontal for F < F and vertical for F > F Graphs and tables

of the results are included.

AMS (MOS) Subject Classification: 76B10

Key Words: Rising bubble, jet

r Work Unit Number 2 - Physical Mathematics

Sponsored by the United States Army under Contract No. DAAG29-80-C-0041.Also, by the Australian Research Grant Committee and the National ScienceFoundation under Grant No. MCS-8001960.

.* - ,. -','.'. . ' . . ,'. -.- , .• % .-. .. .. - -.. *. - .. ,~ .-. . " -..- ,.-..-.*.*. .. -.. , - -... , .-

SIGNIFICANCE AND EXPLANATION

We consider the flow of an incompressible, inviscid, heavy liquid past of

a gas bubble in an infinitely long vertical tube (see Figure la). The flow is

assumed to be two-dimensional. This problem was first considered by Birkhoff

and Carter (1 ) and Garabedian (2 ). Birkhoff and Carter(1) suggested that the

Garabdian(2) Iproblem had a unique solution. On the other hand Garabedian presented

analytical evidence that the problem had a solution for each value of the

Froude number F - U/(gh)/2 smaller than a critical value Fc. Here U is

the velocity at infinity, g the acceleration of gravity and h the width of

the tube.

In the present paper we settle the controversy between the results of

Birkhoff and Carter(1) and those of Garabedian (2 ). We compute accurate

solutions by a collocation method. Our results confirm Garabedian's(2 )

findings. There exists a unique solution for each value of F smaller than a

critical value Fc. However we found Fc - 0.36. This value is about 40

percent higher than that indicated by earlier work on the problem.

In addition we consider the problem of a jet emerging from a vertical

nozzle (see Figures lb and Ic). We show that the slope of the free surface at

the separation points is horizontal for F < Fc and vertical for F > Fc.

odesThe responsibility for the wording and views expressed in this descriptivesuimary lies with MRC, and not with the author of this report.

LD

• • ~~~~~~~~. .-..-......... . -,.-...... . . .. . -............. . .- • -.- .- - .- . . "--. ~......... ... ..... ........................ _ --.... , -,.... . .. ,- -

BUBBLES RISING IN A TUBE AND JETS FALLING FROM A NOZZLE

Jean-Marc Vanden-Broeck

I. Introduction

Nwe consider the two-dimensional flow of an incompressible fluid past a

gas bubble in an infinitely long tube of width h. We assume that the bubble

extends downwards without limit. We choose a frame of reference moving with

the bubble, so that the flow at large distances from the bubble is

characterized by a constant velocity U (see Fig. (1a)]. We choose the

origin of coordinates at the top of the bubble. We assume that the flow is

symmetric about the x-axis and that gravity g is acting in the negative x-

direction. As we shall see that the shape of the bubble is determined by the

Froude number

UF =" ." (1)

(gh) 2

This problem was first considered by Birkhoff and Carter1 and by Garabedian2.

Birkhoff and Carter attempted to solve the problem numerically. They

assumed that the problem had a unique solution and that the Froude number F

could be obtained as part of the solution. Though they obtained approximate

solutions with F - 0.23, the convergence of their procedure was not really

satisfactory.

Garabedian2 presented analytical evidence that the solution is not

unique. He suggested that a solution exists for each value of F smaller

than a critical value Fc. In addition he showed that Fc > 0.2363 and

guessed the value Fc = 0.24.

Sponsored by the United States Army under Contract No. DAAG29-80-C-0041.Also, by the Australian Research Grant Committee and the National ScienceFoundation under Grant No. MCS-8001960.

.- -.,-- ...... -.. . .... .. ... , ... : . ? .. i. . . ./. . -. ..... ...-. .%..

Figure la

jut"4 LU

h.

'pq

*g S

" Sketch of a bubble rising in a tube.

-- 2

?%o C t

"a. " ' . -' "" " , "2 "-." . .. .". . . - .. . ...

• S .. " ° -+, - " " JgJ'' ' + '. °'' " - """, " " "." -" " " ' , ' -.' ' " ' ' "" "". " " "' - " - .+ '

. -I l4 , I l l ~ l + I 1 " +' I . .-# l +- ++ -J+

In the present paper we compute accurate numerical solutions by

collocation. Our scheme is an improved version of the procedures proposed by

Birkhoff and Carter . Our results confirm Garabedian's2 findings. There

exists a unique solution for each value F smaller than a critical value

Fc. However we found that Fc = 0.36. This value is about 40 percent higher

2than the value guessed by Garabedian

The flow configuration of Fig. 1(a) can also serve to model a jet

emerging from a nozzle [see Fig. 1(b)]. As x tends to infinity, the

velocity is assumed to approach a constant U. It follows from the symmetry

of the flow configurations that the portion SJ of the bubble surface in

Figure Ia is identical, for the same value of F, to the portion SJ of the

jet surface in Figure lb. Therefore our results for the rising bubble imply

that the jet of Figure lb exists for all values of F < Fc. For F > Fc,

this flow configuration fails to exist.

For F > Fc we sought a solution in which the flow separates

tangentically from the nozzle (see Fig. 1(c)]. We found that there exists a

unique solution of the type sketched in Fig. 1(c)] for each value of

F > Fc. For F < Fc these solutions fail to exist.

From our results we conclude that there exists a unique jet for each

value of F. For F > Fc the slope of the free surface is vertical at the

separation points S and S' For F < Fc the slope of the free surface is

horizontal at the separation points and the velocity at these points is equal

to zero.The problem is formulated in Sec. II. The numerical procedure is

described in Sec. III and the results are discussed in Sec. IV.

-3-

16- ~ ** * ** *~~' ~ ' * ' 9 : -

-~ -Nj -4 j w*-.-~~~, ~ * 5 I ~ ' * b .

Figure lb

-4

.- 4

44~~77 77- T -..- ~ *u~ .

Figure lcNOP

Sktc of ae aln rmanzze hresraei

e tS

h ;-

S' Y S

"I"

im

Sketch of a Jet falling from a nozzle. The free surface is "

~assumed to be vertical at the separation points S and '.

'%" ~~-5 -

""1.

II. Formulationi.

Let us consider a two-dimensional jet emerging from a nozzle of width

h [see Figs. 1(b) and 1(c)]. As x + -, the velocity approaches a

constant U. The fluid is assumed to be incompressible and inviscid.

We define dimensionless variables by choosing h as the unit length

and U as the unit velocity. We introduce the potential function # and the

stream function 4. Without loss of generality we choose * - 0 at

x - y - 0 and * = 0 on the streamline IJ. It follows from the choice of

the dimensionless variables that 4 = -1 on IJ'. The complex potential

plane is sketched in Fig. 2.

We denote the complex velocity by C = u - iv and we define the function

T - i by

=u - iv e • (2)

We shall seek T - i0 as an analytic function of f = + i4 in the strip

- < <0.

On the surface of the jet, the Bernoulli equation yields

12 1 2.q + gx M qc " (3)

Here q is the flow speed, g the acceleration of gravity and qc the

' velocity at the separation points. In dimensionless variables (3) becomes

22T 2 qce +- x - on SJ and S'J' • (4)

F2 U2

Here F is the Froude number defined by (1).

It is convenient to eliminate y and qc from (4) by differentiating

(4) with respect to *. Using the relation

ax + = 1 -T+i8+ ea. +i u-iv

we obtain

e2T3r+ e cos 0 on SJ and S'J' (6)C a, 2

F

-6-

.4*

o " , - , t , q , , -. , . , % . , , %. .

% " % . " ......% % , , % " .' ... , . .- . . " . . ." ." . .. % . .... .. .. .... , . - ,. . . . . ..

- 04

*1%I

0%0

0 I4

-4)

4430)

0

41.4

('e Cl)

% 0

% 0

The kinematic conditions on IS and IS' yield

e=o * -1 < o (7)

- oo *<o . (8)

The flow configuration of Fig. 1(b) is characterized by stagnation points

at S and S'. This yields the additional conditions

T -, 8 =H at 0=O, -02 (9)

= -, 8 " -H at * - 0, 4 = -12

The flow configuration of Figure Ic is characterized by finit .rl non-

zero velocities at S and S'. This yields the additional condil .'s

T ±, 8 -0 at *0, =0(10)

T Y , e 0 at 0, * =-1

This completes the formulation of the problem of determining T - i.

This function must be analytic in the strip -1 < * < 0 and satisfy the

conditions (6) - (8) and (9) for the flow configuration of Fig. 1(b) and the

conditions (6) - (8) and (10) for the flow configuration of Fig. 1(c).

-8-

4%

.... .... .. . . .. . .. *%. .

II'.-."III. Numerical procedure

Following Birkhoff and Carter we define the new variable t by the

relation

-7rf 1t 1e C t +- ) . (11)t

" This transformation maps the flow domain onto the unit circle in the complex

t-plane so that the walls of the nozzle go onto the real diameter and the free

surfaces onto the circumference (see Fig. 3).

In order to obtain solutions for the flow configuration of Fig. 1(b), we

note that

(In(1 it)]11 as t + ±i (2

t as t + 1 (13)

(see Birkhoff and Carter1 for details). Therefore we define the function

Al(t) by the relation

e -i [-In C(I + t2 )]1/ 3 (-tn C)-1 /3 (0 - t2)efl)W (14)

Here C is an arbitrary constant between 0 and 0.5. We choose C - 0.2.

The function 11(t) is bounded and continuous on the unit circle and analytic

in the interior. The conditions (7) and (8) show that n(t) can be expanded

in the form of a Taylor expansion in even powers of t. Hence

5 e - Iftn C(I + t2)]3(-n C)- 1 3 (1 - t2)exp( a (15)s". n-1

The functions T and 9 defined by (15) satisfy (7) - (9). The

coefficients an have to be determined to satisfy (6) on SJ. The condition

(6) on S'J' will then be autotomically satisfied by symmetry.

We use the notation t - Itle ic so that points on SJ are given by

t e 0 < 2 <I. Using (11) we rewrite (6) in the form

2T dT 1 -TW cotg e d -2e cose= 0• (16)dO F

-9-

. L%.._......... ..... .......... ...... ...... .-.. ,......' - ' . ' ' ' ' ,- L ", ..", "- ' ,'." :"' ,,' '.,, ". o. f .t : - " .% "" , " . ', - ." ' -"." '

r~rr ~ -~ *

.4

.4

1%*

Cl).c.

0A

'U4--.! I-I

04-''I - I* .~.. 41

0 1.-Il

0*1'4r14

0

.1

in

144.4.

'U

-10-4-4 *~ .4

~ I:- ~4 4

-, %.~ -- - .- 4-~-~2%.*4-4-Th. I

Here T a) and e(o) denote the values of T and 6 on the free surface

SJ.

We solve the problem approximately by truncating the infinite series in

(15) after N terms. We find the N coefficients an by collocation. Thus

we introduce the N mesh points

1 N= 2"-gI"17 --

11 drUsing (15) and (17) we obtain [T(a)] [(0)1 anid [-1. in terms

of the coefficients an. Substituting these expressions into (16) we obtain;AA

N nonlinear equations for the N unknowns an , n = 1,...,N. We solve this

system by Newton's method. Once this system is solved for a given value of

F, we calculate the functions T and 8 by setting Iti - 1 in (14). The

shape of the jet is then found by numerically integrating the relation C5).

We shall refer to this numerical procedure as scheme I.

Solutions for the flow configuration of Fig. 1(c) can be obtained by

omitting the factor I - t2 in (14). Thus (15) becomes

T-ie e2p 1/3 b / n

e [-In C(I + t2 )]/ 3 (-In C) - I / 3 t 2 n) . (18)

n1 1

The functions T and 9 defined by (18) satisfy (7), (8) and (9). The

coefficients bn have to be found to satisfy (6) on SJ. We truncate the

infinite series in (18) after N terms and determine bn, n - 1,...,N by the

collocation method described in scheme I. We shall refer to this numerical

procedure as scheme II.

Solutions for the flow configuration of Figs. 1(b) and 1(c) can also be

obtained by assuming the expansion

-[-In C(1 + t2 (-In C1/3 1/3 2n (19)

n = 1 n-

-11-

• - -. '..... ,. , ,'. ..... '. . ,. ... ,...... . . . . .. . . . . . . . ........ . . . . . . .-.... . ... . . . . . . . . . ..:.;. ..". . ..,.., - . ., ,, . , ._ , %

-* -Yk -, % T. - k _W Wk.

It can easily be verified that (19) satisfies (7) and (8). In addition (19)a dn

satisfies (9) when d -- and (10) when dn ' -1. Therefore (19)n-1 n-0

is appropriate to describe the flow configuration of Fig. 1(b) as well as the

flow configuration of Fig. 1(c). The infinite series in (19) is truncated4.

after N terms and the coefficients dn, n - 1,...,N are determined by the

collocation method described in scheme I. We shall refer to this scheme as

scheme III.

1°-2

,...

'V."

4NW..

N

-12-

IV. Discussion of the results

We used scheme I to compute solutions for the flow configuration of Fig.

1(b). We obtained accurate solutions for F < 0.3. However we were unable to

obtain reliable solutions for F > 0.3. Similarly we obtained accurate

solutions for the flow configuration of Fig. 1(c) for F > 0.4 by using

scheme II. However we could not calculate reliable solutions for

F < 0.4.

In order to obtain solutions for 0.3 < F < 0.4, we used scheme III. We

found that scheme III gives accurate solutions for all values of F. As n

increases the coefficients dn decrease rapidly. For example

10 ^310 30 6.10, d6 0 2.10 , d90 - 2.10 - for F 0.374. In

addition the solutions of scheme III agree with those of scheme I for F < 0.3

and with those of scheme II for F > 0.4.q c

In Fig. 4 we present values of the velocity F- at the separation

point 8 versus F. The velocity qc is equal to zero for F < 0.36 andV.

- different from zero for F > 0.36. Therefore

Fc - 0.36 . (20)

This value is about 40% higher than the value guessed by Garabedian2 .

Profiles of the jet for various values of F are presented in Fig. 5.

For F - ., the free surface reduces to the vertical line y - 0. As F

decreases from infinity, the jet becomes thinner. As F approaches zero, the

thickness of the jet tends to zero and the free surface approaches the

horizontal line x - 0. For F > Fc the slope of the free surface at

x - y - 0 is vertical. For F < Fc the slope at x - y - 0 is horizontal.

For all values of F, the free surface approaches the vertical line y -

1as x * -. Therefore the jet is described far downstream by the slender

Jet theory of Keller and Geer3 . This theory shows that

-13-

,.. ... . .. ., . ,. .... .. -.... -. - .' % .' '' . "" "

h,.

L

ii 0

to)

0 000.14

.41

o

lo:1.71 0"44

- 41

0

,i °

,* h4

,-'*1

-14 - -":::: .. , , . . . ..0o .' ..., ....... ...... . .. . , . -.. . ... , . .. ... ... .- , . , - . .. -. , , , , ., . , , , , , -, -, 2

Figure 5

~x

y -.

.1*I

.1/.-- 0.2

bc.d --0.4

Computed free surface profiles for various values of F. Thecurves (a), (b), (c) and (d) correspond respectively to F = 0.2,F F F - 0.36, F =0.45 and F = 1.

.- 15-

" _I . l-~ l. -. j ._k -. Oo Oo- -- e .mD o-o- 'o. o . .'- .o ,-.. ' .'. '. " ,' ". ", ".. %b " ".o'.' '-" ,° ° 4,o

u -iv ~-V(x) as x.-e . (21)

Here V(x) is real and positive. Conservation of mass and equation (4) yield

the relations

(1-2y)V(x) - 1 (22)2lx 2V(x) - - x as x+. . (23)

Eliminating V(x) between (22) and (23) gives

2F2(1-2y)2

Formula (24) is the asymptotic shape of the jet far downstream. Our numerical

results were found to agree with (24) for x large and negative. This

constitutes an important check on our numerical scheme.

The solutions of Fig. 5 for F < FC are also solutions of the flow

configuration of Fig. 1(a). Therefore there exists a unique "rising bubble"

for each value of F smaller than Fc . This confirms Garabedian's findings.

Garabedian2 used a criterion of stability to suggest that the unique

44physically significant "rising bubble" is the one for which F - Fc. Collins4

reported the experimental values F - 0.25. The discrepancy between this

experimental data and our theoretical value Fc - 0.36 is presumably due to

the three-dimensionality of the real flow and to the effect of surface

tension. The bubble profile corresponding to F - Fc is shown in Fig. 6.

JC

JV

%% N.-N

-16-,

,

"..%" . . . ... •. *." .' .".. 4"- ". . ""4. " ." " . . .. 4. ...... ".'. ,. ".-.- '.'•'' . . .. '' . ,i4. * 4 * * ,44 - '. •' , 4 4 4" • - r r , .i. :' 1 ft".. " " :

Figure 6

V.-7

%'V

'.e

-d

."-0.5 -0.25 0 0.25 0.5

Z '- Bub:ble pr ofile for F .. F i.36

: C

#-,.em....- "-

i-F ... , ... .."5.%% '". ": " "''' . - ."-" - ." ." " . ".. .i . J '" ""-• .' " " ." ".' , ." ''''" -'' .".'." ". '. '"'." " .. -17- ," '. ,", . '. - . . " ' . " " . . .

, , ,;,= -,.. ,]'], ',' , ,- , .,-,,... ' ._,' , . ,..'. ." ,,-_, ,.. . . -. , . .- .... . .. .. . . . .•".

REFERENCES

11G. Birkhoff and D. Carter, J. of Mathematics and Physics 6,1 769 (1957).

(21 P. R. Garabedian, Proc. R. Soc. London Ser A241, 423 (1957).

4. (31 J. B. Keller and J. Geer, J7. Fluid Mech. 59, 417 (1973).

(4] R. Collins, J. Fluid Mech., 22, 763 (1965).

4-

%

.%9

SECURITY CLASSIFICATION OF THIS PAGE (M-,n Dae Fntered)

REPORT DOCUMENTATION PAGE [READ INSTRUCTIONSBEFORE COMPLETING FORM

1. REPORT NUMBER 2. GOTCS N .3. RECJ6IENT'S CATALOG NUMBER

#2631 cO cs 2 (Q4. TITLE (and Subtitle) S. TYPE OF REPORT A PERIOO COVERED

Summary Report - no specificBubbles Rising in a Tube and Jets Falling d.,.'.reporting periodfrom a Nozzle S. PERFORMING ORG. REPORT NUMBER

. AUTHOR(e) S. CONTRACT OR GRANT NUMBER(e)

MCS-8001960.Jean-Marc Vanden-Broeck DAAGZ9-80-C-0041

9. PERFORMING ORGANIZATION NAME AND ADDRESS 10. PROGRAM ELEMENT. PROJECT, TASK

% AREA A WORK UNIT NUMBERSMathematics Research Center, University of Wr Unit NMer2 SWork Unit Number 2-

610 Walnut Street Wisconsin Physical MathematicsMadison, Wisconsin 53706I . CONTROLLING OFFICE NAME AND ADDRESS 12. REPORT DATE

January 1984See Item 18 below 1,. NUMBER OF PAGES

1814. MONITORING AGENCY NAME & ADDRESS(if different from Controlling Office) IS. SECURITY CLASS. (of this report)

UNCLASSIFIEDISa. DECL ASSIFICATION/ DOWNGRADING

SCHEDULE

,e. I. DISTRIBUTION STATEMENT (of thle Report)

Approved for public release; distribution unlimited.

17. DISTRIBUTION STATEMENT (of the abetract entered In Block 30. I different from Report)

IS. SUPPLEMENTARY NOTESU. S. Army Research Office National Science FoundationP. O. Box 12211 Washington, DC 20550Research Triangle ParkNorth Carolina 27709

It. KEY WORDS (Continue on reveree side if neceeeery and identify by block number)

Rising bubble, jet

20. AOSTRACT (Continue an ,everee side It neceeeary and identify by block number)

--'. The shape of a two-dimensional bubble rising at a constant velocity U in a--". tube of width h is computed. The flow is assumed to be inviscid and incom-

pressible. The problem is solved numerically by collocation. The results confirmGarabedian's 2) findings. There exists a unique solution for each value of theFroude number F = U/(gh) I/ 2 smaller than a critical value Fc. Here g denotesthe acceleration of gravity. It is found that Fc = 0.36. In addition the problem

77of a jet emerging from a vertical nozzle is considered. It is shown that the%.slope of re sr0% . sope ofthe free surface at the separation points is horizontal for F < P

and vertical for F > Fc . Graphs and tables of the results are included.

DO M 1473 EOIT103R OF I NOV GS IS OBSOLETEDD , A. ,*UNCLASSIFIED

SECURITY CLASSIFICATION OF THIS PAGE (ohen Veto Entered)

:~~~~~.......... .. . . "'...". '-.:..........- , ,'..' ' . . . ,..........,"-'..- - . ..-....... .. " .". -, .-

'2. .', -, ;,*e. ;.,~ • . ....~~

~~44

Itr:;~

!~.L~~ I ~ ~ t%

0,W 7~

~ .~ r '0 ~j7KM *'ao-4 0 a~ wVi

£ ~ - *I >.~AkI ~ *oI'.V

VT.

I4 t . .,

". ft . . .

41 T

i~E It

t

ki .

* .