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Vortices In A Viscous FluidH. O. Anwar Dr.-Ing. aa Hydraulics Research Station, Ministry of Technology,Wallingford, EnglandVersion of record first published: 01 Feb 2010.

To cite this article: H. O. Anwar Dr.-Ing. (1968): Vortices In A Viscous Fluid, Journal of HydraulicResearch, 6:1, 1-14

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VORTlCES iN A ViSCOUS FLUIDLE FLUIDE VISQUEUX E ECOULEME T TOURBILLO A1RE

by

ORANG. H. O. ANWAR

Hydraulics Research Station, Ministry of Technology, Wallingford, England

Introduction

In many theoretical studies of vortex flow

occurring above an exhaust pipe, the assump­

tion has been made that the motion is laminar

[1,2,3,4]. Analyses based on this assump­

tion have produced useful results for under­

standing the performance of an ideal system,

however. The basis of this assumption is that

the layers in certain zones of the flow are

slowed down by viscosity. It has been shown

in some cases [5] that laminar motion can

occur in a region between the axis of sym­

metry and a radius approximately that of the

outlet.

In many experiments gas or water has been

used for the fluid. These liquids were used

both for convenience and the desirability of

visualizing the flow. It was considered of in­

terest to see whether the experimental results

obtained with these fluids are similar to those

for fluids of high viscosity. Consequently, the

experimental fluids used in the investigation

presented here were glycerol sol utions of var­

ious viscosities.

The experiments were carried out in a spe­

cially designed vortex chamber and tangential

velocities were measured in various horizontal

Paper received 7th December 1967.

Introduction

L'ecoulement tourbillonnaire qui s'etablit au­

dessus d'une conduite d'evacuation a fait I'ob­

jet de nombreuses etudes theoriq ues fondeessur I'hypothese d'un ecoulement laminaire

[1,2,3,4] et les resultats ainsi obtenus ont

perm is de mieux comprendre Ie comportment

d'un systeme ideal. Or, supposer Ie regime

laminaire revient a admettre que, dans cer­

taines regions de I'ecoulement, les couches de

f1uide sont ralenties par I'action de la viscosite

et, pour quelques cas particuliers [5], on a pu

effectivement demontrer qu'un mouvement

laminaire peut s'etablir dans une region cylin­

drique centree sur I'axe et dont Ie diametre est

sensiblement egal a celui de I'orifice d'evacua­

tion.

Lors d'etudes experimentales de ce pheno­

mene, on a generalement utilise soit des gaz,

soit de I'eau- car ces flu ides peu visqueux sont

d'une mise en oeuvre commode tout en per­

mettant une visualisation aisee des ecoule­

ments. Dans Ie travail que nous presentons ici,

nous nous sommes propose d'examiner Ie casde liquides de viscosite elevee afin de deter­

miner si ceux-ci se com portent comme les

fluides moins visqueux. A cette fin, nous avons

L'article re~u Ie 7 dcccmbrc 1967.

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2 Journal of Hydraulic Research / Journal de Recherches Hydrauliques 6 (1968) no. J

planes and at radii which were greater than

the radius of the outlet. The results of these

measurements have been compared with an

analysis based on the assumption of laminar

motion.

Analysis

The following conditions will be assumed to

be satisfied by the vortex flow in a viscous

fluid:

(a) The fluid is incompressible.

(b) The motion is axis-symmetric and steady,

with no body forces.

(c) 0/00 = a;az = 0 i.e. the motion is inde­

pendent of angular and vertical position,

as has been confirmed by experiment for

the region where the axial velocity is zero.

It has been found experimentally that this

region occurs beyond the radius of the

outlet.

In cylindrical polar-coordinates r, z, 0, the

Navier-Stokeseq uation for the tangential com­

ponent of velocity with the above assumptions

is:

choisi comme f1uides experimentaux des solu­

tions de glycerol de viscosites differentes.

Les essais furent effectues dans une cham­

bre tourbillonnaire specialement conc;ue. Les

vitesses tangentielles furent mesurees dans

plusieurs plans horizontaux et a des distances

a I'axe superieures au rayon de I'orifice d'eva­

cuation. Dans cet article, no us comparerons

les resultats obtenus avec les conclusions d'une

etude theorique effectuee en regime laminaire.

Etude theorique

On suppose que I'ecoulement satisfasse auxconditions su ivantes:

(a) Ie fluide est incompressible;

(b) l'ecoulement est axi-symetrique, perma­

nent et ne fait pas intervenir des forces

d'inertie;

(c) 0/00 = a/az = 0: autrement dit, on sup­

pose que Ie mouvement en un point donne

n'est pas fonction des coordonnees angu­

laire ou verticale. Cette condition a ete

confirmee par I'experience pour la zone ou

la vitesse axiale est nulle, cette zone etant

situee au-dela du rayon de I'orifice d'eva­

cuation.

Compte tenu de ces conditions, l'eq uation de

Navier-Stokes exprimant la composante tan­

gentielle de la vitesse s'ecrit, en cOOl'donnees

cylindriq ues:

(1)

where

Vr = the radial velocity

Vo = the tangential velocity

v = the kinematic viscosity of fluid.

Equation (I) will be made dimensionless by

introd ucing the following coefficients [5]:

auVr = la vitesse radiale

Vo = la vitesse tangentielle

= la viscosite cinematique du f1uide.

Pour rendre adimensionnelle I'equation (I),on introduit les coefficients suivants [5]:

rrand/et r = ­

roo

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Anwar / Vortices in a viscous fluid

where ro is a reference radius (defined later)beyond which the circulation roo is constantand rr is the circulation where equation (I) isapplicable.

Thus we can write:

3

ou ro est un rayon de reference (defini ci-des­sous) au-dela duquel Ie vecteur circulation roodemeure constantet ou rr represente Ie vecteur circulation dansIe domaine d'application de I'equation (I).

Cette derniere peut ainsi s'ecrire:

(2)

in the above equation primes denote differen­

tiation with respect to /1. The term vl( Vrr),because of its form, will be called the radialReynolds number and is constant, i.e. vl( Vrr) =

vl( Voro). The term Vrr = constant of the aboveexpression can be found by integrating theequation of continuity subject to assumption

(c). Moreover:

les primes designant des derivees par rapport

a /1. En raison de sa forme, Ie terme vl(Vrr)seraappele nombrede Reynolds radial. Ce terme est

constant, c'est a dire que: vl(Vrr) = vl(Voro).On peut determiner Ie terme Vrr = Cte del'expression precedente en integrant I'equa­

tion de continuite tout en respectant la condi­tion (c) ci-dessus. II convient de noter en outre

que:

v vh- 20m - = constant/Cte

Q

where:

h = the height of the vortex chamber,

Q = the total flux entering the chamber,('J. = a proportionality factor which will be

discussed later.The right hand side of the above expression isnegative because the total flux Q enters in theopposite direction to the radial velocity Vr •

Therefore equation (2) can be written in thefollowing form:

ou:h = hauteur de la chambre tourbillonnaire,Q = flux total penetrant dans la chambre,('J. = coefficient de proportionnalite (dont il

sera question plus loin).Le second membre de I'expression ci-dessusest negatif, car Ie flux total Q et la vitesseradiale Vr sont de sens contraires. On peut

donc ecrire I'eq uation (2) sous la forme:

4A/1r" +r = 0 (3)

where A is a constant according to the aboveexplanation; subject to the following bound­

ary conditions:

A etant une constante, comme nous I'avonsvu, et les conditions aux limites etant:

reO) = 0 and/et reI) = I (4)

The solution of equation (3) by applying theboundary condition (4) can be given as:

La solution de !'equation (3), compte tenu desconditions aux limites (4), peut s'ecrire:

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4 Journal oj Hydraulic Research / Journal de Recherches Hydrauliques 6 (1968) /l0. 1

(5)

where 0 > 1'/ > I.The term:

1

4A

ou 0> 1'/ > I.A mesure que la viscosite du f1uide s'eleve,

Ie terme

4rJ.rr(vh/Q)

of the above equation decreases when the

viscosity of the fluid increases and the expo­

nent of equation (5) becomes closer to unity

showing that the distribution of circulation

approaches that of a solid body rotation.

It is the purpose of the present investigation

to study the effect of various values of the

kinematic viscosity, v, upon the distribution

of circulation. This can be achieved, according

to equation (5) by varying the constant A =

rJ.rr(vh/Q), which, in turn, can be obtained by

varying the kinematic viscosity v when h/Q is

held constant. Moreover, it can be seen that

equation (5), subject to assumption (c), is

valid for a region where the axial velocity is

zero. It has been found experimentally that

this region lies beyond the radius of the outlet.

The motion in a region between the axis of

symmetry and a radius equal to that of the

outlet has been investigated elsewhere [5].

Experiments

It has been shown in many fields of engineer­

ing, and particularly in fluid mechanics, that

dimensionless analysis is a useful tool for

describing natural phenomena provided its

limitations are recognised. An attempt will

therefore be made in this section to apply

dimensional analysis to the problem of in­

vestigating vortices in laminar motion.

The determining features of the problem

are as follows:

(I) Distribution of circulation, r" at radius r.

decrolt et I'exposant de I'equation (5) se rap­

proche de I'unite, ce qui montre que la distri­

bution de la circulation tend vers celie affe­

rente a la rotation d'un corps solide.

Le present travail a ete con9u pour etudier

l'influence de la viscosite cinematique v sur la

distribution de la circulation. Comme Ie mon­

tre I'equation (5), on peut modifier cette dis­

tribution en faisant varier la valeur de A =rJ.rr(vh/Q), c'est-a-dire, par exemple, en jouant

sur la viscosite cinematique v a h/Q constant.

On voit, de surcroi't, que l'equation (5) assu­

jettie a la condition (c) se veri fie pour une

region Oll la vitesse axiale est nulle. On a mon­

tre par I'experience que cette region se situe

au-dela du rayon de I'orifice d'evacuation.

Les caracteristiques de I'ecoulement dans I'es­

pace cyJindrique situe a I'interieur de ce rayon

on t deja ete etud iees [5].

Les experiences

L'analyse dimensionnelle, a condition que I'on

en reconnaisse les limitations, est une techni­

que tres utile pour decrire les phenomenes

naturels. Elle a fait ses preuves dans bien des

domaines scientifiques - et plus particuliere­

ment en Mecanique des f1uides - de sorte que

nous tenterons de I'appliquer maintenant au

probleme de I'ecoulement tourbillonnaire en

regime laminaire.

Les parametres qui interviennent dans ce

probleme sont les suivants:

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Anwar / Vortices in a viscous fluid

(2) Constant circulation, roo.(3) The total flux, Q, entering the vortex

chamber.

(4) Radius fa, beyond which constant circula­

tion occurs.

(5) Physical properties of the experimental

fluid, such as viscosity.

(6) Geometry of experimental apparatus. This

refers to the height, h, and the diameter, D,

of the vortex chamber and the radius f e of

exhaust pipe, although the latter, as men­

tioned before, has not a major effect upon

the present investigation, because we are

concerned with the motion in a region

where I' > f e .

Hence we can write:

5

(I) La distribution de la circulation rr au

rayon f.

(2) Circulation constante roo.(3) Le flux total Q penetrant dans la chambre.

(4) Le rayon fa au-dela duquel la circulation

demeure constante.

(5) Les proprietes physiques du f1uide expe­

rimental, telles que la viscosite.

(6) La geometrie de I'appareil d'essais: hau­

teur h et diametre D de la chambre tour­

billonnaire; rayon f e de la conduite d'eva­

cuation (ce rayon, comme nous I'avons ditplus haut, n'interviendra guere dans la

presente etude, qui s'interesse a I'ecoule­

ment dans la region OU I' > f e).

Nous pouvons donc ecrire:

(r" roo, Q, v, 11, D, ro, r) = 0 (6)

Measurements show that the tangential veloc­

ities depend on the radius only (see experimen­

tal results), and therefore, the vertical coordi­

nate z does not appear in the above expression.

In the experimental set up, described below,

the total flux Q entered the vortex chamber

tangentially and vortices with various initial

circulations, rex" were produced by alteringthe flux Q. Thus we can write the following

expression from the dimensional analysis:

Nos resultats experimentaux montrent que

les vitesses tangentielles ne sont fonction que

d u seul rayon: la coordonnee verticale z ne

figure donc pas dans I'equation (6). Lors des

essais, Ie flux total Q penetrait tangentielle­

ment dans la chambre et des tourbillons posse­

dant des circulations initiales differentes fu­

rent produits en faisant varier Ie flux Q. L'ana­

lyse dimensionnelle conduit donc a I'expres­

sion suivante:

rr (Q t'o D)r = roo = f 11, vl/ h' h (7)

As the experiments were carried out in a fixed

vortex chamber the diameter of the vortex

chamber could only affect the region of con­

stant circulation, because, in the case under

consideration the diameter, D, of the chamber

is taken to be larger than the radius fa. More­

over, it was found experimentally that the

radius fa, at which circulation is equal to roowas the same throughout the experiments.

Thus expression (7) becomes:

Comme les essais furent executes dans une

chambre de dimensions constantes et comme

Ie diametre D adopte pour celle-ci fut supe­

rieur a la distance radiale fa, Ie parametre D

ne put influer que sur la region a circulation

constante. De surcroi't, les experiences ont

montre que Ie rayon fa pour lequella circula­

tion est egale a roo demeure constant d'un

essai a I'autre. Ainsi, I'expression (7) devient­

elle:

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6 Journal of Hydraulic Research / Journal de Recherches Hydrauliques 6 (1968) no. 1

(8)

which agrees with the theoretical approach

given before. Further, the theoretical ap­

proach showed that the non-dimensional cir­

culation, r, varies as a simple power of non­

dimensional radius 1'/, i.e.:

ce qui est en accord avec I'etude theorique

ci-dessus. L'analyse theorique indiquait, de

plus, que la circulation adimensionnelle est

fonction du rayon adimensionnel eleve a une

puissance simple:

r = 1'/" (9)

/1

where: ou:

4cm(vh/Q)

The exponent /1 is less than unity and is a

function of h, Q and v, so that by varying one

of these parameters, say v the kinematic vis­

cosity, we obtain a set of curves on I'/-r plot.

H is thus necessary to plot the measured cir­

culation against radius 1'/ to determine whether

such a relation between rand 1'/ exists. There­after, in order to proceed with expression (8)

empirically, it is necessary to determine the

variation of the exponent of radius 1'/ with the

non-dimensional parameter given in expres­

sion (8).

Experimental apparatus

Test vortices were formed in a transparent

cylindrical tank of 8 in. internal diameter and

16 in. high. Glycerol solutions of various vis­

cosities l11.2, 19.6 and 40 CTS) were fed tan­

gentially into the tank. The tank had two

inlet nozzles of -} in. bore at the circumference,

positioned at opposite ends of a diameter. Thetank was provided with a central exhaust pipe

at its base of internal diameter 0.875 in. and

length 2 ft. The top end of the tank was closed

in order that vortices with various initial cir-

L'exposant /1, fonction de h, Q et v, est infe­

rieur a l'unite, de sorte qu'en faisant varier

l'un des trois parametres precites, mettons v,

viscosite cinematique, il est possible d'obtenir

un abaque de courbes de r en fonction de 1'/.

Pour determiner s'il existe une telle relation

entre r et '1, il convient done de porter la cir­culation mesuree en fonction du rayon 1'/. A

partir de la, I'elaboration empirique de I'ex­

pression (8) necessite la determination de la

fa~on dont I'exposant du rayon 1'/ varie en

fonction du parametre sans dimensions figu­

rant dans l'expression (8).

L'installation d'essais

Les tourbillons a etudier sont engendres dans

une cuve cylindrique transparente d'une hau­

teur de 40 em et d'un diametre interne de

20 em. Des solutions de glycerol de viscosites

differentes (11,2, 19,6 et 40cst.) penetrent

dans la cuve par deux entrees tangentielles de

12,5 mm, placees aux deux extremites d'un

diametre. La cuve est munie d'une conduite

d'evacuation situee dans I'axe. Cette conduite,

d'un diametre de 22 mm, est longue de 60 em.

La cuve est fermee a sa partie superieure pour

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Anwar / Vortices in II viscous fluid 7

window --l--t-==

fenctrc

lightbeam~falsccau lumineux

ai r outletpurge d'air'

~­

I

vortex chamberchambrc tourbillonnairc

container

cooling coilserpcntin de

rcfroidlsscment

insulationl~olation therm,que

nozzlebusc d'rnjcctton

copper cooling tankcuve de rcfroldlsscmcntdraphragme

pompe

to nozzles

pressure tappings-+l-~

prlscs de pression

tapwate r fo r coo ling ----&::=~r=====~~;;;;)cau de rdroidissemcnt

volumetric discharge measuring tankcuve de mesure de debit

Layout of experimental apparatlls Fig. I. L'installation d'essais

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8 Journal of Hydraulic Research / Journal de Recherches Hydrauliques 6 (1968) no. 1

Fig. 2. Circulation at different levelsVariation de la circulation it differents niveaux

qu'il soit possible de creer des tourbillons

ayant des circulations initiales differentes. Le

debit entrant dans la chambre est mesure aI'aide d'un diaphragme situe entre la pompe

et Je reservoir de glycerol. Au surplus, Ie debit

total sortant de la conduite d'evacuation est

contrale au moyen d'une cuve de mesure gra­

duee (fig. I). Les buses d'entree, alimentees

par pompe sont munies de prises de pression

qui permettent d'assurer un debit constant.

11 est apparu tres tat que la temperature du

glycerol s'elevait rapidement au cours d'une

mesure. Pour pallier cet inconvenient, Ie reser­

voir de glycerol fut place dans une cuve de

refroidissement, I'espace entre celle-ci et Ie

reservoir etant traverse par Ie debit d'une eau

de robinet a 14 °e. De surcrolt, un serpentin

a eau de robinet fut place a I'interieur du re­

servoir de glycerol. Grace a ces precautions,

la temperature des solutions de glycerol a pu

etre maintenue a environ 16°C.

Les vitesses tangentielles ont ete mesurees

par une methode optiq ue a I'aide de petites

bulles d'air injectees au travers des bu es. Les

bulles se trouvant a une distance radiale don­

nee etaient illuminees par un mince faisceau

lumineux: elles apparaissaient comme des

eclairs brillants lors qu'on les observait au

telescope et que I'on les maintenait momen­

tanement immobile au moyen d'un prisme

tournant [6].La figure 2 indique les resultats des mesures

de vitesses tangentielles faites a plusieurs ni-

--° level above outlet 3 in.

niveau au-dessus de la sortie

• level above outlet 7 in.~ level above outlet 10 in. ~

- --~'Q

11

--~.~ '" ~~

~g ~ 0

r-- --0.05

0.10

0.15

0.20

0.00.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0_ radius (in.)

rayon (pouces)

g 0.30'w'"~ 0.25'u

~ 0.35~

culation would be produced. The flux entering

the tank was measured by an orifice placed

between the glycerol container and the pump.

In addition, the total flow was checked vol u­

metrically after final discharge from the out­

let pipe (Fig. I). The nozzles were fed from

the pump and each of them was provided with

a pressure tapping to ensure a constant dis­

charge through the nozzles. Preliminary tests

showed that the temperature of the glycerol

solution rose during the test very rapidly. ]n

order to keep the temperature of the solution

constant, the glycerol container was therefore

placed in a water-cooled tank. Tap water at a

temperature of 57 of was run into the space

between the glycerol container and the tank,

and in addition a coil was placed inside the

glycerol container, fed with the tap water (see

Fig. J). With this arrangement the tempera­

ture of the glycerol sol utions were kept con­stant at about 60 oF.

Tangential velocities were measured optical­

ly by means of minute air bubbles injected

into the nozzJes. These were illuminated by a

narrow light beam at a known radius and they

were observed as bright flashes in a telescope

when momentarily held stationary by a ro­

tating prism [6].

Fig. 2 shows the results of tangential

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Anwar / Vortices in a viscous fluid

velocity measurements obtained at various

levels and radii, indicating as mentioned be­

fore, that the tangential velocities were inde­

pendent of elevation, which confirms assump­

tion (c) made in the theoretical approach.

Results of measurements

Equation (5) shows that the distribution of

circulation should vary as a simple algebraic

function of the non-dimensional radius 11.When the measured non-dimensional circula­

tions are plotted against the radius on double

logarithmic graph it is found that all profiles,

within the limits of experimental scatter, fall

on straight lines, which are shown on Fig. 3,

for kinematic viscosity v = 11.2, 19.6 and

40 CST. The relation between non-dimen­

sional circulation and radius is thus confirmed

as being:

9

veaux et a pi usieurs distances radiales. Comme

nous I'avons dit plus haut, ces resultats de­

montrent que la vitesse tangentielle est inde­

pendante de I'elevation, ce qui confirme I'hy­

pothese (c) de I'analyse theorique.

Les resultats d'essais

Selon I'equation (5) de I'analyse theorique, la

distribution de la circulation est une fonction

simple du rayon sans dimensions '1. Lorsqu'on porte en coordonnees logarithmiques

les circulations adimensionnelles en fonction

du rayon, on obtient des droites, a la disper­

sion experimentale pres. Celles-ci sont tracees

sur la figure 3 pour les trois valeurs de la vis­

cosite cinematique v (11,2, j 9,6 et 40 cst.).

La relation (9) qui exprime la circulation adi­

mensionnelle en fonction d u rayon se trouve

ainsi confirmee:

r = 11"

It will be noted that by definition of 1'0 and

/00 the equation must pass through the point

/ = I, 11 = I. Fig. 3 also shows that the ex­

ponent 11 of equation (9) is less than unity and

increases with increase of viscosity of the test

fluids. These results are in good agreement

with those predicted by the theoretical ap­

proach (see equation (5». The non-dimen­

sional parameter Q/( vh), as suggested from

dimensional analysis, has been evaluated and

the variations of the exponent with this para­

meter is shown in Fig. 4. This figure shows

that the exponent increases with decrease of

the non-dimensional parameter and the dis­

tribution of circulation approaches that of

solid body rotation (i.e. J, a, 11) when the

parameterQ/(vh)approaches zero. Thisasymp­

totic condition can be achieved either by an

increase of kinematic viscosity, v, or by an

increase of the height, h, without altering the

On notera que, d'apres les definitions de 1'0

et / OC' cette equation doit passer par Ie point

/ = I, 11 = I. La figure 3 montre egalement

que I'exposant 11 de I'equation (9) est inferieur

a I'unite et qu'il augmente quand la viscosite

des fluides s'accrolt. Ces conclusions sont en

bon accord avec les resultats de I'etude theo­

rique (voir I'equation (5». On a evalue Ie

parametre sans dimensions Q/(vh) ressortant

de I'analyse dimensionnelle; la fayon dont I'ex­

posant 11 varie en fonction de ce para metre est

indiquee figure 4. On voit que la valeur de

I'exposant augmente quand celie du para­

metre adimensionnel diminue et que la distri­

bution de la circulation tend vel's celie d'un

corps solide (c'est a dire: r, a, /1) lors que Ieparametre Q/(vh) tend vel's zero. On peut s'ap­

procher de ces conditions limites en augmen­

tant soit la viscosite cinematique v, soit la

hauteur II sans modifier Ie flux total Q. L'exa-

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- -kinematic viscosity v = 40 cstViSCoslte cincrnatiquc

Joumal of Hydraulic Research / Journal de Recherches Hydrauliques 6 (/968) no. /

f--------!-----!---- -----+ -r-~J-I--+-

i L 1- 1

-, - - .-l---t-1--t-­I

o Q/vh=47.1f) Q/ vh = 41• Q/vh=32.4

0.1 L- ---L__L-~_::_:_--'-~--,-L-::_l_:_..L.!:___:_---~--'-----____:_'___,__--'-~__'_::_'_:_~

0.01 0.02 0.04 0.06 0.08 0.1 0.2 0.4 0.6 0.8 1.0-'YJ

}O

~ 1.0

I0.90.80.70.6

0.5

0.4

0.3

0.2

kinematic viscosity v = 19.6 cstviscositc cincmatiquc

---- ---+----+---l----+---+-I-Ol.-:+~

r='YJ°~5 0 Qlvh=71.6r='YJ° 56 I • Q/vh=59

0.2 "'=:--:----."-L;:=----L..---;"'=..,..-l~~~'_::;_;!;_+.____--__,;~-~-____,,_'_.,_____--'--.,,.....,~~..._...,.J0.01 0.02 0.04 0.06 0.0801 0.2 0.4 0.6 0.8 1.0

-'YJ

0.3

~16:~0.80.70.6

0.51------ -+-

0.4

L• I

L

0.3

kinematic viscosity v = 11.2 cstr ~ 'YJ 0.33 _ vrSCOSllC clI1crnatlquc

r= 'YJ 037 0 Q/vh=134f) Qlvh =110

r= 'YJ0t.! • Q/vh = 770.2 ....",.."---~~---L..---;~..,..---1~--,,,...,~ld+,-----------.,,"""~~-~-,-----~~~""""'~

0.01 0.02 0.2 0.4 0.6 0.8 1.0-'YJ

~r b:~-0.80.7

0.6

0.5

0.4

Measured distribution of circulation Fig. 3. Distribution de circulation mesuree

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Anwar / Vortices in a viscolls fluid 11

Fig. 4. Variation of non-dimensional parameter Q/(vh)with exponent 11

Variation du parametre sans dimensions Q/(Iih)en fonction de !'exposant 11

total flux Q. The above conclusion can also be

seen from equation (5), where the constant Aincreases by reason of an increase in viscosity

of the test fluid or the height, h, and, con­

sequently, the exponent of equation (5) ap­

proaches unity. Furthermore, Fig. 4 shows

that the exponent t1 approaches zero with in­

crease of the parameter Q/(vh). In this case

equation (9) shows that the distribution of

circulation approaches that of an inviscid

fluid, i.e. r = const. This result can be ob­

tained from equation (2) in which the first

term on the left hand side of the equation

approaches zero for a fluid of low viscosity.

In the theoretical approach we have intro­

duced the proportionality factor, CI., which is

the ratio between the total flux Q entering the

vortex chamber and the sum of the radial flow

within the body of the vortex (i.e. CI. = Q/q).This means that the radial flux q flows through

the body of vortex and the rest (i.e. Q-q)flows along the rigid boundaries. This in fact

men de I'equation (5) permet d'aboutir a la

meme conclusion: en augmentant ou la visco­

site ou la hauteur h, la valeur de la constante As'eleve I'exposant de I'equation (5) tend en

consequence vers I'unite. De plus, la figure 4

montre que I'exposant t1 tend vers zero quand

Q/(vh) s'accroi't. Dans ce cas, I'equation (9)

indique que la distribution de la circulation

tend vers celie d'un fluide non-visqueux, c'est

a dire: r = Cte. Ce meme resultat ressort

egalement de I'equation (2), dont Ie premier

terme du premier membre tend a s'annuler

pour les f1uides de faible viscosite.

Dans I'analyse theorique, nous avons intro­

duit Ie facteur de proportionnalite CI., qui est

defini comme etant Ie rapport entre Ie flux

total Q entrant dans la chambre tourbillon­

naire et la somme des flux radiaux au sein du

tourbillon: CI. = Q/q. Cela signifie que Ie flux

radial q passe au sein du tourbillon et que Ie

flux residuel (= Q-q) longe les parois rigides.

Les experiences ont pu mettre en evidence ce

phenomene, grace a un faisceau lumineux

d'une epaisseur de 2 mm qui servait a eclairer

une suspension de bulles d'air de 50 a 100

microns que I'on avait introduit dans Ie tour­

billon. En plus de I'ecoulement radial au sein

du tourbillon, on a egalement observe un

ecoulement radial dans deux couches, cha­

cune epaisse de 5 mm, au sommet et sur Ie

fond de la chambre. La vitesse radiale au sein

du tourbillon etait trop faible pour etre aise­

ment mesuree, mais la vitesse radiale mesuree

dans les couches au voisinage des parois rigi­

des etait nettement superieure ala vitesse dans

Ie coeur du tourbillon. Figure 5 nous donne

les resultats de ces mesures, lesquels ont servi

acalculer Ie facteur de proportionnalite CI. pour

diverses valeurs de Q/(vh). On voit que CI. aug­

mente avec Q/(vh), ce qui signifie que Ie flux

radial au sein du tourbillon s'annule lorsque

la viscosite cinematique v s'identifie a zero et

que Ie mouvement du tourbillon tend vers

10 15--_. 1O-1 Q/:vh

---- experimentalexperience

------ theoreticaltheorie

c 1.0

10.9

0.8

0.7 -

0.6 -

0.5--

0.4 -

0.3-

0.20 5

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12 Journal of Hydraulic Research / Journal de Recherches Hydrauliqlles 6 (/968) no. J

celui d'une tube tourbillonnaire classique [6].L'exposant

••10

15 -

50~----5"""-----1,.L;0-------,.J15

--_,10-1 Qlvh

Fig. 5. Variation of u-value with non-dimensionalparameter Q/(1ih)Variation de a en fonction du parametre sansdimensions Q/(I!h)

has been observed experimentally by means

of a beam of light 0.08 in. thick projected into

the vortex chamber to illuminate a suspension

of air bubbles of size range 50-100 ~ intro­

duced into the flow. In addition to the radial

flow within the body of vortex, radial flow

was also observed in two layers each about

0.2 in. thick at the top and the bottom of the

chamber. The radial velocity in the body of

vortex was too low to be measured satisfac­

torily, but the radial velocity measured in the

layers close to the rigid boundaries was much

higher than the velocity in the body of the

vortex. The results of these measurements

from which the proportionality factor a. has

been evaluated for various non-dimensional

parameters Q/(vh) are given in Fig. 5. This

figure shows that the value of a. increases with

increase of Q/(vh). This implies that there is no

radial flow within the body of vortex when

the kinematic viscosity v becomes zero and the

motion in the vortex approaches that of a

classical vortex tube [6].The exponent

n=14a.n: (vh /Q)

has been evaluated from the a.-curve in Fig. 5

and the results of this calculation is shown in

Fig. 4 for comparison, which shows that the

agreement between calculation and measure­

ment is satisfactory. Fig. 4 further shows that

the theoretical and the experimental curve of

the exponent n increases with increase in vis­

cosity of the test fluid.

A good agreement between theory and ex­

periment can be obtained also when the initial

circulation, r "" is not too high. Unfortunately

the experimental set up was not suitable for

vortices with a very low initial circulation.

a ete determine a partir de la figure 5. Les

resultats de ce calcul sont indiques sur la

figure 4, ce qui nous permet de constater que

calculs et mesures s'accordent de fayon satis­

faisante. La figure 4 montre egalement que ns'accrolt quand la viscosite du fluide aug­

mente.

Lors que la circulation initiale roo n'est pas

trop importante, il y a egalement un bon

accord entre theorie et experience. Malheu­

reusement les conditions d'essai ne permirent

pas I'obtention de tourbillons ayant une tres

faible circulation initiale.

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Anwar / Vortices in a viscolls fluid

Summary

[n order to check the theoretical conclusionsbased on the assumption that the motion in avortex flow is laminar, vortices were producedusing glycerol as a test ftuid with three dif­ferent kinematic viscosities.

The distribution of circulation was mea­sured across the vortex chamber at variouslevels. It was found that the circulation is afunction of the radius only.

It was shown theoretically and confirmedexperimentally that the distribution ofcircula­tion varies as a simple power, 11, of the radiusand that the power is less than unity. It wasfound that for kinematic viscosities between11.2 and 40 CST the power varied between0.334 and 0.725. This was in good agreementwith the theoretical approach within the lim­itation of the experiments.

The power 11 thus increased with increase inkinematic viscosity of the test fluid and thedistribution of circulation in a viscous vortexmust therefore approach that of a solid bodyrotation for a fluid of high viscosity.

Acknowledgements

The work described herein was conducted aspart of a research programme of the Hydrau­lics Research Station of the Ministry of Tech­nology, and the paper is published by permis­sion of the Director of Hydraulic Research.The writer wishes to express his thanks toMessrs. J. A. WELLER and R. J. C. BONN fortheir careful experimental work.

/3

Resume

Pour verifier les conclusions d'une analysetheorique fondee sur I'hypothese d'un ecoule­ment tourbillonnaire laminaire, des tourbil­Ions ont ete etudies experimentalement dansune chambre contenant des solutions de gly­cerol de viscosites cinematiques differentes.

On a mesure la distribution de la circula­tion a plusieurs niveaux dans la chambre et adifferentes distances radiales. On a constateque la circulation n'est fonction que de laseule distance radiale.

La theorie prevoit - et I'experience I'a con­firme - que la distribution de la circulation estegale au rayon eleve a une puissance 11 qui estinferieure a I'unite. Pour des viscosites cine­matiques comprises entre 11,2 et 40 cst., I'ex­posant 11 varie entre 0,334 et 0,725. Ces valeurssont en bon accord avec I'analyse theorique,compte tenu des limites de ['experimentation.

Comme I'exposant 11 augmente avec la vis­cosite cinematique, la distribution de la cir­culation dans un tourbillon visqueux doit ten­dre vers celie d'un corps solide en rotation amesure que la viscosite du fluide s'eleve.

Remerciements

Ce travail fait partie d'un programme de re­cherches actuellement en COllfS a la Hydrau­lics Research Station du Ministere de la Tech­nologie de Grande-Bretagne. L'auteur remer­cie M. Ie Directeur des Recherches d'en avoirbien voulu autoriser la publication, ainsi queMM. J. A. WELLER et R. J. C. BONN pourla far;on soignee dont ils ont mene les travauxexperimentaux.

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14 Journal of Hydraulic Research / Journal de Recherches Hydrauliques 6 (1968) no. 1

otations _ --,

Radial Reynolds number A

Diameter of the vortex chamber D

Height of the vortex chamber hTotal flux entering the vortex chamber QSum of radial flow within the body of q

vortex

Radius measured from the axis of sym- ,

metry

Radius beyond which circulation IS '0

constant

Cylindrical polar coordinates " e, Z

Radius of the exhaust pipe 'eRadial velocity V,

Tangential veloctty VoRatio of the total flux to the sum of r:t

radial flux within the body of vortex

Non-dimensional circulation 1

Circulation at any radius, 1,

Constant circulation where, > '0 1 00

Non-dimensional radius YJ

Kinematic viscosity of the test fluid v

References

Nombre de Reynolds radial

Diametre de la chambre tourbillonnaire

Hauteur de la chambre tourbillonnaire

Flux total penetrant dans la chambre

Somme des flux radiaux au sein du tour­

billon

Rayon (mesure a partir de l'axe de

symetrie)

Rayon au-dela duquel la circulation

demeure constante

Coordonnees cylindriques

Rayon de la conduite d'evacuation

Vitesse radiale

Vitesse tangentielle

Rapport du flux total a la somme des

flux radiaux au sein du tourbillon

Vecteur circulation, sans dimensions

Vecteur circulation au rayon,

Vecteur circulation pour, > "0

Rayon sans dimensions

Viscosite cinematique du fluide d'essai

Bibliographies

I. LONG, R. R., Vortex motion in a viscous fluid. Journal of Meteorology, Vol. 15, 1958.2. Lo G, R. R., A vortex in an infinite viscous fluid. Journal of Fluid Mech. II. 1961,611.3. DERGARABEDIAN, P., The behaviour of vortex motion in an emptying container. Proc., Heat Transfer and

Fluid Mech. Institute, 1960.4. ROTI, N., On the viscous core of a line vortex. Z.A.M.P. Vol. IXb, 1958.5. ANWAR, H. 0., Formation of a weak vortex. Journal of Hydraulic Research, Vol. 4, No. I, 1966.6. ANWAR, H. 0., Flow in a free vortex. Water Power April 1965.

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