bubbles a tube and jets falling from r nozzle i un! v ... · i 7,-ad-ai39 266 bubbles rising in a...
TRANSCRIPT
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
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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
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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
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Approved for public release; distribution unlimited.
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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)
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