Laminar jets of a plane liquid sheet falling vertically in the atmosphere

Fhud Mechamcs, 24 (1987) 11-30 Eilsewer Science Pubhshers B V , Amsterdam - Pnnted m The Netherlands

Journal of Non-Newtoman

11

LAMINAR JETS OF A PLANE LIQUID SHEET FALLING VERTICALLY IN THE ATMOSPHERE

K ADACHI Department of Chemtcal Engmeenng,

Kyoto Unmerwty Kyoto 606 (Japan)

(Recewed December 10,1985, m revised form August 21,1986)

The steady lammar flow of a liqurd sheet falhng vertrcally from a sht &e mto the atmosphere IS studied theoretically under the assumptrons of flat profiles of velocity and pressure over its honzontal cross-se&on. An equatron predmtmg the sheet thickness IS derrved m a way slnular to that whrch has been found useful for a falhng round Jet. The proposed equation IS compared wrth previous results of experiments and numerrcal smulatlons as well as Clarke’s exact soluhon to the ordmary differential equation of a one-&mensronal verbcal Jet, and the results seem to be rather good. The present work IS concerned urlth an mtegral mvanant of the Jet, and lower-boundary conditrons on the vortrcrty are presented for a vlscous-gravity Jet as well as for an inertia-gravity Jet

1. Iutroduction

The study of a plane liqurd Jet is of interest m connection with plane-extension characterrsbcs measurement, curtam coatmg and film castmg. It IS important to mvestrgate the plane Jet in comparison wrth the round Jet because there IS some parallehsm between both of them. As revrewed and drscussed thoroughly by Petne [l], the deformatron and flow of a round hquid filament has been mvestrgated by many researchers. Trouton [2] has shown that, for a one-dunensronal round Jet, the axral velocrty IS governed by the equation

0377-0257/87/$03

50

Q 1987 Elsewer Sauce

Pubhshers B V

12 In tms equation, U IS the mean axial velocity, z 1s the axial distance, v is the kmematm viscosity, g IS the acceleration due to gravity. In the case of a viscous-gravityJet, where the left-hand side of (1) is neghgible, three fannhes of solutions have been found for (l), and the solution for a free-falling Jet was presented first by Trouton [2] and later agam by Cnuckshank [3] as (ii/i&$‘* = (N,*/24)“‘( z/R) + S,,

(2)

where Ii, and R are the mean velocity and Jet radius at the nozzle exit, S, IS a constant, and N,* is defined for a round Jet by NJ* = 12R*pg/i&,q, = 4R*g/P,v.

(3)

In thts equation, p is the fled density and the Trouton vlscosrty qr has been taken as 3 times the shear vrscosity ,u for an umaxral extensional flow. Equation (2) is the only one whtch satisfies the flow condition of free fallmg. l/ii and dPii/dzP + 0 as z + 00,

(4)

where p IS a posrtive defmrte mteger. The equation of the axial velocity for a hqmd sheet, which corresponds to eqn. (1) for a round hqmd filament, was derived by Taylor [4] as

and three farmhes of solutions were given by Cnuckshank [3] m the case of a plane viscous-gravityJet The solution for a slow plane Jet of free falling is gtven by (2) together with (3), if we merely replace m those equations the set of (R, z, 3v) for an axisymmetric flow wrth that of (H, x, 4v) for a plane flow, as follows: (Ei/Eo)1’2= (N,*/24)“‘( x/H) + S,,

(6)

N,* = 12H2pg/UoqT - 3H2g/ii,v.

(7)

Here, H IS the half size of the sht gap, x is the vertical distance, 1)r is taken to be 4~ for a plane extensional flow In the case of a round inertia-gravity Jet, the followmg equations have been presented respectrvely by Scrtven and Prgford [5], Lienhard [6], Kurabayasln [7] and Anno [8]: ’

(9)

13 (10)

where Fr = iii/2 Rg, We = 2 RGi p/a, and u is the surface tension. Equation (8) is exact for an mviscid vertical Jet. Equation (9) was derived by combmmg the Bemomlh equation with the imaginary mean exit velocity of 4U,/3 whtch results from a consideration of the total momentum flux at the nozzle exit. Equation (10) was obtained by mchidmg the surface tension energy m an energy balance of the relevant flow system. Moreover, usmg a perturbation method, Kaye and Vale [9] have obtamed the followmg result for the Jet shape function a(z) a R

-=

+ER+

Wez

...

(12)

Takmg mto account that a2ii = R2U,, we find that (12) is the mverse function of (11) as far as the two mam terms m each equation are concerned. In the mvestigation of disintegration of a hqtud sheet, Brown [lo] and Lm and Roberts [ll] have presented the followmg serm-empirical equation for the mean vertical velocity:

&+)‘-1+c(&)“‘,

(13)

where C is an empirical constant which depends upon flow conditions. For an merua-dominant Jet the main terms of free falling due to gravity are the same for both cases of plane and round Jets It must be noted that both (2) and (6) are invahd far downstream because the inertia term U dE/dz becomes a maJor factor balancmg gravity with an mcreasmg vertical &stance, and the equations for an merua-dommant Jet are always useful there. AlI the equations described above are usually valid only either m a low speed region or m a htgh speed region. On the contrary, an equation, apphcable to both flow regions, has been given by Clarke [12,13]. He rewrote (1) and (5) m a nondimensional form as

(14) where U* = (ii/i&)(

Re*Fr/3)1’3,

05)

14 XE (% or s)(

R~?*N,*/72)l’~,

(16)

Re* = 6Rii,p/q,.

(17)

The elongatronal vrscosrty qr should be taken to be 3~ for a round Jet and 4~ for a plane Jet. Clarke [12] derrved the general solution expressed m terms of Any’s fun&on from (14) and fixed rt so that U*(O) = 0. The final result for a free fa.lhngJet is represented by u”(X)=1/21~3[{Ar’(r)/Ar(r)}*-r];

r=(X+k,)/2”3,

(18)

where AZ IS the Aq function, the dash assocrated wrth rt denotes drfferentration wrth respect to r and k, = - 2.94583 . - - . Thrs constant value results from U*(O) = 0. However, rt IS clear from (15) that thrs boundary condrtron is mappropriate except m the case of Re*Fe = 0 Thus, m the present paper, k, wrll be considered as an adjustable parameter. The value of k, should be determmed so that (18) may hold true up to a pomt as far upstream as possible. Equation (18) can be expected to be flexrble enough to cover a wade range of flow condmons. However, the surtable value of k, for an arbitrary flow condrtion IS not presently known. Moreover the numerical table of Az( t) and Az’( r) are not useful, because its range of r IS bted and its mterval Ar IS not always fine, and rt IS also mconvement to have to use numerical tables m calculatmg eqn. (18) The mam purpose of the present work IS to derrve an equation, whrch IS less reasonable but more convement to use than eqn. (18), and to show its htgh apphcabrhty. The equation wrll be produced by a sunple addmon of the two equations expressing the vrscous-gravrty Jet and the inertia-gravrty Jet Unfortunately, however, there are no data presently avarlable wrth which to determme the constant values of the two adjustable parameters contamed in those equatrons. Therefore, those values wrll be chosen so that the two relevant equatrons wrll be consistent with the followmg physical mtumons. The transient flow length of a vertical free Jet between the nozzle and the zone of a flat velocity profile IS O(2H) (or O(2R)) for a low speed Jet, and the Jet thrust, dommatmg the total force exerted on a Jet filament at the extt for a hrgh speed vertical Jet, can be approxunately estunated by the Jet thrust of a gravrty-free Jet computed by Omoder [14] When the Jet swells, the ongm of the vertical coordmate wrll be set at the posrtron of maxrmum Jet thickness. Surface tension effects wrll be occasronally taken mto account although mcompletely The present predrctron of a vertical plane Jet IS compared wrth prevrous related results The present one can be expected to be useful although rt IS not completely Justified by thrs comparison Another support to the present result for a plane vertrcal Jet IS that its counterpart, which IS denved for a

15 round vertical Jet followmg a snmlar analysis, is vahd over a wide range of flow conditions. However, it is a matter of course that the present equation sould be re-exammed in future concerning its apphcabrhty, and modified by changmg the values of the two adjustable parameters, m comparison with more rehable computational and experimental results. One powerful and reasonable way to obtam a more rehable predictton of a fallmg Jet is through numerical sunulation. This approach was taken for both plane and round Jets by Dutta and Ryan [15], but, regrettably, their results are mcorrect because of the use of mappropnate lower boundary conditions as shown by Ada& and Yoshtoka [16] and discussed by Vrentas et al. [17]. The boundary conditions far downstream from the vertical nozzle are known to be of free fall. In a numerical sunulation of a fallmg Jet, however, it is almost impossible to put the lower boundary far enough downstream for a state of free fall to occur. Thus, the problem of formulating appropriate boundary condtuons at a distance not so far downstream is also important for the analysis of fallmg Jets. Bnef consideration will be given to this problem. Attention should be pard to the fact that rt is rather difficult to determme a pnon the lower boundary conditions m order that the resultant solution may yield a correct constant value to the integral mvaruurt described m the next section. 2. Plane jets 2 1 Macroscoprc balance equatron of momentum We consider a plane hqmd sheet of umt width bounded by the two planes at the sht exit (x = 0) and at an arbitrary distance downstream from it (x = x) and the two free surfaces of the plane Jet ( y = + h( x)), as shown m Ftg 1 For a steadily-falhng plane Jet the macroscoptc balance equation of momentum can be expressed as F0 = P(x)

+ w(x),

(19)

where F(x)

= 2a/(l+

W(x) =

1;2)1’2 + 2/a( rXX-p

2pg ixh(x) JO

dx.

+p, - pu2) dy,

(20)

(21)

Here p, 1s the atmospheric pressure, r is the devlatonc stress and h = dh/dx The term I;b is the force exerted on the upper end of the hquid sheet filament of umt width, F(x) is the force actmg on the filament at the position of interest x, and W(x) is the weight of the suspendmg filament

16

Ag 1 Force balance for a suspendmg plane Jet filament

from its upper end (X = 0) to X. Equation (19) expresses the balance of these three forces, and it indicates that both F(x) and W(x) change urlth X, but that thex sum, F(X) + W(x), 1s an mtegral mvmant whch IS kept constant along the vertical axes of X, as pomted out by Joseph [US]. Tl~s relation 1s useful to test the appropnateness of the observed sheet shape functaon h(x). Equation (20) represents that the total axA force F consists of the four kmds of forces: surface tension, wscous flow resistance, static pressure and inertia force Far downstream from the sht exit it can be assumed that the profiles of u and p are flat over the horizontal cross section of a falhng plane Jet. Then, usmg the followmg boundary con&tion of the force normal to the free surface. P=Pa+r

yy - hTxy - ah/(1 + h2)3’2,

(22)

we can rewnte eqn (20) approximately as F= 20[1/(1

+ P)1’2

+ i&/(1 + l;2)3’z] + 2h [ ( TX.&- TJ

+ /iI&

- pi?], (23)

where ii = i&H/h

(24

It 1s also possible to assume 1 SC=/i2 and hh, and (T,, - Q)

z+ ‘ir_,,

(25)

17 where the last relatton has been derived from the tangential condition on the free surface described by - TYY)= (1 - h2)r

2t;(r-,

For a Newtoman smplified as

u) dy= Jh7,,(0,

2

(*Ix-Tyy)

stress-free (26)

XY’

fhud the terms of the viscous flow resistance -2&Y(O,

h) - u(0, -h)]

= 0,

can be

(27)

=4p dE,dx,

(28)

where the equatron of contmuity has been used m eqn. (27), and u is the y component of the veloctty vector. Thus we have F, = F(0) = 2a/(l+

1;;)1’2 - 2JH( p0 -p,

+ pu;)

dy,

0 F(x)

20 + 2h [4~ di-/dx - pU2].

=

2 2 Bmc

(29) (30)

equatron for ii and Its solution

Subsututmg eqns. (21) and (30) mto (19) and Mferentratmg equation by x, we have

- 2phii2 +2pgh

8ph$j

= 0,

the result

(31)

which 1s eqmvalent to eqn (5) derived by Taylor [4]. Rewntmg this 111terms of the averaged axral velocrty ii and m a dlmensronless form, we obtam

N,* 1 u1+=v=o,

(32)

where 5 = x/H,

U = ii/ii,,

ri = dU/d&

(33)

Equatron (32) indicates that the plane Jet 1s not affected apparently by the surface tension far downstream from the nozzle although the surface tension has not been assumed to be neghgtble. In eqn. (32) the velocrty U 1s considered to be a functron of the vertrcal positron E. However rt appears convement to seek its inverse function t(U). For thts purpose, eqn (32) IS transformed to dT dU+

Re* NJ* 7UT2-ET3+z=

where T = dt/dU.

T

0,

18 Far downstream from the sht extt the velocity U 1s rather htgh, so we can find the followmg power servessolutron to (34):

where n 1s an integer, p is a defmte posrtive integer, and the constant C,, can be easrly determmed by the conventional procedure so that eqn (34) may be tdenttcally satisfied by (36). Then integration of eqn. (35) yields the general soiutton of E( U) which satrsfres the condrtron of free fall (4): For Re* = 0, x/H = (24/N,*)“*( (ii/U,)“* - 6,))

(37)

and for Re* * 0,

For a creeping Jet the veloctty profile u(y) 1s more umform at the extt plane (X = 0) than the parabohc one and rt becomes almost flat wtthm a distance of the gap size 2h downstream. Therefore the equatron (37) wtth 6, = 1, whtch has been derrved prevtously by Crutckshank [3], 1s not so inaccurate except m a regton sufftctently far downstream from the sht extt. When theJet swells, the ongm of the vertical coordmate 1s set at the posrbon of the maxunum Jet thickness, which extsts withm a distance of 2H for a Newtoman flmd. The creepmg Jet for both cases can be expressed approxrmately by (39)

where in the case of no expansion h, = H, and m the case of Jet swell the computattonal result of swelling ratio, which was presented by Omodet [14] for a gravtty-free Jet, can be used as h,,/H, although this approxrmatton 1s an overestrmatlon. It should be remarked that eqn. (39) is always invahd sufficrently far downstream, because, even for a Jet of a very low Reynolds number, the mertra force wrll eventually grow so that tt overwhelms the viscous force and balances the gravrtattonal force. Equation (38) for a hrgh speed Jet mcludes eqn. (13) presented by Brown [lo] and Lm and Roberts [ll]. For convemence, the following equatron wtll be used as a generahzatton of not only eqn. (13) for a plane vertrcal Jet but also eqns. from (8) to (11) for a round vertical Jet*

19 The terms other than the first two m eqn. (38) are mconvement because of their slow convergence or divergence, when Re*Fr IS not sufficiently large, and so the remaimng terms are ormtted for the present purpose. Far downstream from the nozzle the first term of the right-hand side of eqn. (40) is dommant and 8, is neghgible, but the latter is significant m the neighborhood of the nozzle exit. The assumption of flat profiles of velocity and pressure becomes more mvahd as one approaches the nozzle exit. However it can be expected that a proper choice of the value of 8, veils this mconsistency and makes eqn. (40) useful up to very near the nozzle exit. Thus the mtegration constant 6, is Important, and its value will be deterrnmed m the next section so that the relevant equation may safisfy the requirement of macroscopic momentum balance (19). As Re* decreases, eqn (40) becomes meffective m the neighbourhood of the nozzle exit, but the tail of any let can be represented by eqn (40). In partrcular, the steady lammar Jet of an mvlscid fhud is completely described by eqn. (40) with 6, = 1 Any equation described above is only useful m a hnnted range of flow conditions, and no one equation covers its whole range. So it may be mterestmg to examme the apphcabmty of the followmg equation:

(41) which approaches eqn. (40) as Re* mcreases, and approaches eqn. (39) as Re* decreases. The drawback of eqn. (39) mentioned earlier has also been ehminated here. Moreover, eqn. (41) can be expressed only 111terms of the dimensionless variables (15) and (16) as follows. X=0.5[(U*)2-6,(Re*Fr/3)2’3]

+a

[ (CT*)

(42) If eqn. (41) holds true, the parameter k, m eqn. (18) ought to depend only upon Re *Fr. 2.3 Determnatlon

of the Integral constant 8, for a high-speedjet

For an mvrsctd fhud, the velocity profile mside the nozzle is always flat, and 8, = 1 exactly. For a viscous fluid, however, the velocity profile of a high-speed Jet is parabolic at the nozzle exit, and there is a long transient zone between the nozzle and the zone of flat velocity profile. The proper choice of the constant value 6, is effective for the vahdity of eqn. (40) especially in this transient zone as described before, although 8, depends upon the flow con~tions.

20 The only relatron, which IS useful in the relevant zone for the present purpose, IS the macroscopic balance equatron of momentum (19). For a h&-speed Jet the effect of surface tension IS negligrble in comparrson wtth that of mertra. The right--handsrde of eqn. (19) IS roughly estnnated from eqns. (21) and (30) at a drstance very far downstream from the nozzle in order to obtain a constant of the integral mvariant. Substrtutmg eqn. (40) mto the relevant equations, we have

+

-@,Y, w+ 4

(44

It must be noted here that the unknown velocrty profile in the transrent zone has not been used m the above estunatron of the constant, and that only the mean velocity U(x) has been utrhzed. The left-hand side, the axral exrt force FO, m the case of no surface tensron, IS equal to theJet thrust T, wrth a mmus sign: Fo= -To,

W

where T,=2

*(~,,-~a+~a:)dy. /0

If To IS known for any gtven flow condrtron, then the mtegral constant 6, can be determined by combmmg eqns. (44) and (45) as 8, = ( T0/2HpS;)2

(47)

The Jet thrust for a plane vertrcal Jet IS not been known as yet. For a hrgh-speed flow, however, the Jet thrust IS dommated by the mertra term whrch can be estrmated well wrth the parabolic velocity profile. Thus we can assume that the Jet thrust for a htgh-speed vertrcal Jet IS little affected by the gravrtatronal force and by the surface tension, as shown m the expernnental work of Ada&r and Yoshtoka [16] for a round vertrcal let. Then the Jet thrust for a htgh-speed vertrcal Jet IS almost equal to that for a high-speed horrzontal Jet. In thrs case, the weight of a plane Jet filament IS zero, and its tlnckness 2h approaches a constant 2h, far downstream from the nozzle. By usmg the macroscoprc balance equauon of momentum for a plane gravrty-free Jet, we have To/2 Hpii; = H/h,. Here h,

(48)

has been computed numerrcally by Omoder [14], and the final

21 I

12

I

I

I

6/S

I

11 10 Q9 CI8

loo

lo’

102

i?

Ret-1 Fig 2 Thud of a phe jet on the batasof Omalds grawty and no surface tension

eomputatior& results in

the case of no

result of TO 1s given in Fig. 2. Usmg eqn. (46) and the fully-developed velocity profile U(Y)/& = 1511 - (Y/W2],

(49)

we find that the curve m Fig. 2 should approach to 6/5 as Re* mcreases, for ( pO -pa) becomes neghgible in comparison with pui m eqn. (46). As Re* decreases, the effects of gravitational force and surface tension become significant. Then the value of S, obtamed m the above way is inaccurate. In such a case, however, the inaccuracy of 8, m eqn. (41) 1s veiled to some extent by the extstence of the last term for a low-speedjet. 2.4 Compamon

witi p~vma

works

As described in the mtroducuon, Clarke’s equation (18) is an exact solution to eqn. (32), so that eqn. (41) dertved m the present work must almost accord with Clarke’s equation by a stutable choice of the value of the adjustable parameter k,. The two relevant equatrons are compared m Fig. 3 for three different cases. Case 1, Case 2 and Case 3 correspond, respectively, to an inertia-gravity Jet, a viscous vertical Jet assocrated wrth die swell, and a viscous-gravityjet, as shown m Table 1, where Re = 2Hii,p/p = 3Re*/4,

6* = 8, + (24/Re*Fr)“‘S,.

N, = 4H2pg/iiop

= 3N,*/4,

(50) (51)

= l/&X for Case 2. The values Here, 8, = lforCaselandCase3,and6, of k, have been determined so that two correspondmg curves for each case may intersect at the pomt of an open circle gtven in Fig. 3.

---

Clarke

( modlf led )

XlHIFr

L-1

Fig 3 Agreement between the present and the Clarke predxtioon

Thus figure mdicates that eqn. (41) IS a very good substitute to Clarke’s equation (18) with an adJustable parameter k, over a mde range of flow condltlons. Attention may be piud to the fact that eqn. (41) can be expressed only m terms of Clarke’s dimensionless vanables U* and X as eqn. (42) accordmg to a necessary condition for agreement between the two relevant equations On the contrary, the broken hne for Clarke’s ongmal solution (18) with k, = - 2.94583 1s mvahd m the neighborhood of the nozzle exit even for a viscous-gravity Jet of Re Fr = 10m4, because Clarke’s boundary con&tlon of U*(O) = 0 yields an mfintte value of h(O), which IS derived from the law of mass balance, although the relevant condition U*(O) = 0 IS almost satisfied for a very low value of Re Fr as seen from the deflmtion of U* gven by eqn. (15). The appropnate value of the adJustable parameter k, mcreases with an increasing value of Re Fr as seen m Table 1. Thus it can be concluded at least that Clarke’s ongmal solution IS not adequate for the present purpose, and that the usefulness of Clarke’s equation (18) wth an adJustable parameter has been hrmted at present because the smtable value

TABLE 1 The flow conditions for the sheet tiuckness curves presented 111Fig 3

CaSe 1 2 3

Re 100 001 001

Fr 1 1 001

% 100 001 1

** Modhed values (Clarke’s ong,mal one IS - 2 94583)

ko

s*

-14 8329 l * -2 4349 ** - 2 6999 **

1888 52 56 566 4

23

/

.I---

_____..-

Nj=O

<--mm-.-_-

.8 Nj .4

t

0

2

I

--

-

---

_

o-o-

O-d*,

52

-

,

l$

\

Ryan

\

I

\\

I

loo x/H

-

I

A

10’

lo*

1-I

Fig 4 Comparison of present prechctionsof sheet thwknessW&I its computational results of others at Re = 0

of k, has not been known yet and, moreover, the use of the numerical table of Ai@s function is mconvement. In the case of Re = 0, eqn. (41) reduces to eqn (39), whose predction for h,,/H = 1 is compared with that of numerical simulation reported by Dutta and Ryan [15] for Re = 0 and N, = 1 and 4 m Fig. 4, where the computational result of Omodel [14] for Re = N, = 0 IS also shown It appears that the Jet length computed by Dutta and Ryan is too short to expect a reasonable result and that the almost constant sheet thickness at Its lower end is caused by the mappropnate lower-boundary condition of zero vorticlty. An mconsistency whtch results from tlus boundary condition has also been discussed for a round Jet by Vrentas et al [17] The use of the mappropnate boundary condition descrtbed above is one reason for the large disagreement betweeen the prediction of eqn (41) and that of Dutta and Ryan [15] A sun&u large discrepancy has also been found for round Jets between the Jet shape curves computed by Dutta and Ryan m an analogous way and emprrical ones, as shown m a previous work of Ada& and Yoshloka [16] In the case of a vertical planeJet with dte swell eqn. (41) can be expressed as ==($(&~+(&i”2((;,“‘-(&)1’2), Fr H

which 1s compared with the experimental

(52) data of Lm and Roberts [ll] m

FIN 5 Comparison of present pred~ctmns of sheet thickness wth the expenmental data of Lm and Roberts

Fig. 5, where the dotted hnes are then empuxal curves obtamed from eqn. (13) by substituting eqn. (24), and the correspondmg predrctrons of eqn. (52) are the smgle-dotted hnes The value of h,,/H has been assumed to be 1 19 for the two relevant cases accordmg to the computational result of Omoder [14] for a hoIlz0nta.l plane Jet of no surface tensron and Re = 1. The curve fittmg of the two empnical curves of Lm and Roberts wrth eqn. (41) is shown m Fig. 6, where the value of the adJustable parameter 6* defined by eqn. (51) has been determmed so that the two correspondmg curves may

cuse

1 c=13.5

$=0.444

case

2 C= 4

6*=1.12

-

0.8

_____ exp

I 10

I 5 x/H/Fr

L-3

Fig 6 Curve flttmg of empnxxl data of Lm and Roberts WA the present equation

25 coincide at the pomt of each open circle. The agreement IS reasonable; however, those values of S* presented in the figure mdicate that the adJustable parameter h,JH III eqn. (52) is larger than 2. ‘I&s is only possible when the fhud adheres to the downward edge faces of two sht plates at its exit, for the maxunum swelhng ratio for a vertmal plane Jet is less than 1.19 for a horizontal plane Jet. Thus the experimental data of Lm and Roberts seem to be unrehable, and the disagreement between the present theoretical and then empirical curves shown in Fig. 5 is not unreasonable. In the present section, it has been shown that eqn. (41) 1s a good substitute to the Clarke exact solution to the one dlmens1ona.lbasic equation of a verticalJet. However aJustification of the flow model (41) for a vertical Jet has neither been given by previous experimental data nor by exlstmg computatronal results because of their Imperfection. In the next section, the validity of eqn. (41) \nll be exammed by considermg a parallel correspondence between plane and round vertical Jets 3. Interrelation between plane and round jets A treatment parallel to the present one of a falhng plane Jet will be presented for a falhng round Jet m a separate paper. However, the mam results are summarized below for ease of reference The basic equation for a round Jet, i.e., the counterpart of eqn. (34) is T2 T 1 6cau’/2+~=0’ where T=dt/dU Ca = i&p/a.

(t=z/R),

(54) (55)

It can be shown from the comparison between eqns. (34) and (53) that differences between two correspondmg one-dmensional Jets of plane and axlsymmetry appear m the terms of surface tension and an integral constant. The solution to eqns. (53) and (54), the counterparts of eqns (37) and (38) are, respectively, as follows For Re* = 0 (56)

26 and for Re* z+ 0,

(57) where Bo = 4R2pg/a.

(58)

When Bo = 0, eqn (56) reduces to eqn. (2) By the same discussions as m the case of a plane vertrcal Jet, the followmg equatron can be denved as the apparent counterpart of eqn. (41) for a round vertical Jet. “=(+?,+[~+((+.J+&J2](~-*“), Fr R

(59)

where S, = (T,/~R*pii;)~ = ( R/cz,)~,

(60)

and S, = 1 for no dre swell, and 8, = R/a,, for dre swell. Here, the swelling ratro of a round horrzontal Jet, am/R, has been computed numerrcally by Omoder [19]. By putting a/R = ( iidu)“2

and z/R = x/H,

(61)

we fmd that eqn. (59) IS exactly eqtuvalent to eqn. (41) m the case of no surf ace tension

L/K/rr

1-j

Fig 7 Wide apphcabhty of the equation predxtmg the shape of a falhg

round Jet

27

Re=0.553 ----

Fr=31

8

eXp of Lm & Roberts ( 1981)

-

Eq 59 (6*=2 558 1

2--__ 0

0 16’

I

I

I

loo

10’

lo2

x/H/Fr , zIRIFr

C-1

Fig 8 Agreement between the computational result of Shuav equation for the shape of a falhng round Jet with &e swell

and the predxtion

of the

Experimental data of Ada& and Yoshtoka [16] on the Jet shape of a round vertical let are compared with the predictron of eqn. (59) m Fig 7, which mdlcates that eqn. (59) is vahd m a wide range of flow comhtions. The computational result of Shtraw [ZO] m the case of die swell and no surface tension 1s also predicted well by eqn (59) as shown m Ag. 8. In these figures, Re = Re * and N, = N,* for a round vertical Jet, and 6, 1s given by eqn. (51) Thus a wide apphcabrhty of eqn (59) has been guaranteed to some extent by the good predictions described above. Thus there seems to be no reason why eqn (41) for a plane vertical Jet can not be so vahd as its counterpart, eqn. (59) for a round vertical Jet, although the valnbty of eqn (41) must be tested by experrments m future. 4. Lower boundary conditions for a plane vertical jet As mentioned 111the mtroduction, the problem of formulatmg appropriate boundary condnlons at a distance not too far downstream is important for the present flow problem. For instance, the numerical simulatron based on more appropriate boundary conditions enables the constant value of the integral mvarrant -FOto be determmed more correctly, and the latter gives a more proper value of 6* m eqn (41). In tins sectron, the lower boundary condrtrons on the stream function 1c/and the vorticity w is considered.

28 We begm with the exact equatmns of motion and contmurty m henslonless forms,

where u and u are the x- and Y-velocrty components, respectrvely, the vortrcrty u IS defined by au

au

“=x-V and Q,= p - N,x/4.

(66)

The lower boundary IS assumed to be located at a posrtion where u and p are uniform over the horrxontal cross-se&on of the plane Jet. Then eqn. (64) produces dii v= -ydx-

(67)

Combmmg eqn. (24) m the dimensronlessf&m urlth the relatron (68)

u = a+/aY,

we obtam the followmg lower boundary condition on the stream function a/~ (6%

JI =Y/h

The lower boundary condition of w can be denved from eqns. (62) and (63) For brevity we restrict our analysrs to the case of We= l/0 = 0 for a whde. Then the relevant equatrons can be snnphfied respectively as

au -= ax a+ -E= ax

m

0,

au -5

(

orw=-Yz.

d+ 1

(71)

Substrtutlon of G(X) and eqn. (67) mto the defmmg equation of o yrelds *=-Y-

d2ii dx2 -

(72)

29 Equatmg equation (71) to (72) and mtegratmg obtam

the resultant equation, we

cp= dZi/dx + const

(73)

The left-hand side of the above equation has been defined by eqn. (66), m which the pressure p can be grven approxrmately by P “Pa+

2Wi3Y,

(74)

wluch has been derrved from one of the free surface comhtrons (22) by using the present assumptrons, descrrbed before, together with the relatron (25) for a Newtoman fhud. Thus, using eqns. (66), (74) and (67), we can derive from eqn. (73) dc/dx

= - N,x/12 + const.

(75)

The fmal result 1s produced by substitutmg eqn. (75) mto (72). w = NJy/12.

(76)

Thrs boundary condrtron on w wrll be useful for numerrcal srmulatron m the case where Re IS very small and the lower boundary 1s not located so far downstream that the Jet can be expressed approxrmately by eqn (40). When the Jet at the lower boundary is described by eqn. (40), an alternative lower boundary comhtron on w can be obtamed from substrtutron of eqn (40) m the dunensronless form mto eqn. (65) as w=z?’

1

Y

(77)

which indicates that o approaches to zero as x + co. In contrast, Dutta and Ryan [15] used the zero vortrcrty condition at a drstance of 0(2H) downstream from the sht nozzle. Thrs 1s the reason why then computatronal results have been considered to be inaccurate m Sectron 2. Attentron must be paid to the fact that the condrtron (77) is only vahd at a high Reynolds number or at a posruon sufficiently far downstream from the sht nozzle 5. Conclusions A new equation for pre&ctmg the thrckness of a falhng hqmd sheet has been produced by a smple ad&tron of the two terms of a high-speedJet, and a low-speed Jet, m which their adjustable parameters have been determmed from both an mtegral mvarrant and the maxrmurn swelhng ratio The resultant equatron IS a good substitute for the Clarke exact solutron to the or&nary differential equation of a one-dimensronal vertrcal Jet, and moreover the former IS more convement m use than the latter. It has not

30

been justified either by previous experimental data or by existmg computational results because of their defects. However, there seems to be no reason why the relevant equation of a plane vertical jet is not so valid as its counterpart of a round vertrcal jet, since the latter was derived by an analysis completely parallel to that in the case of the former and found to be valid by its comparrson with experimental data and computational results for round vertical jets, although the final confirmation of the vahdtty of the relevant equatron by means of experiment IS still needed Possible lower boundary conditions on the stream fun&on and the vortrcity have also been presented for a vrscous-gravity jet as well as for a inertia-gravrty jet because they are important for a further investigation of the relevant flow problem. Acknowledgement

The author is mdebted very much to one of the revrewers for many helpful suggestrons. References 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

D J S Petne, Elongational Flows, pltman Pubhshmg, London, 1979, pp 191-198 F T Trouton, Proc Roy Sac, A77 (A519) (1906) 426-440 J 0 Cnuckshank, Trans ASME J Fhud Eng ,106 (1984) 52-53 G I Taylor (m an appenti to [lo]) L E Scnven and R L P@ord, AIChE J ,5 (1959) 397-402 J H Lienhard, Trans ASME J Baw Eng ,90 (1%8) 262-268 T Kurabayashl, Trans Jpn Sot Mech Eng ,25 (1959) 185-198 J N Anno, The Mechamcs of Llqmd Jets, Heath, Lexmgton, 1977, pp 47-55 A Kaye and D G Vale, Rheol Acta, 8 (1%9) l-5 D R Brown, J Fhud Mech , 10 (1961) 297-305 S P Lm and G Roberts, J Flmd Mech ,112 (1981) 443-458 N S Clarke, Mathematia, 12 (1966) 51-53 N S Clarke, J Fhud Mech, 31 (1%8) 481-500 B J Omoda, Comput Fhuds, 7 (1979) 79-% A Dutta and M E Ryan, AIChE J ,28 (1982) 220-232 K Ada& and N Yoshloka, m Mena et al (ed ), Advances 111Rheology, UNAM, Mexico, Vol 2,1984, pp 329-337 J S Vrentas, C M Vrentas and A F Shua~, AIChE J ,31 (1985) 1044-1046 D D Joseph, Arch Ration Mech Anal, 74 (1980) 389-393 B J Omoda, Comput Fhuds, 8 (1980) 275-289 A F !&ran, Ph D Thess, Ilhnols Inst Technology, 1978