Discussion of "Investigation of Flow Upstream of Orifices" by D. B. Bryant, A. A. Khan and N. M. Aziz, Journal of Hydraulic Engineering, January 1, 2008, Vol. 134, No. 1, pp. 98-104

HAL Id: hal-00454440

https://hal.archives-ouvertes.fr/hal-00454440

Submitted on 8 Feb 2010

HAL is a multi-disciplinary open access

archive for the deposit and dissemination of

sci-entific research documents, whether they are

pub-lished or not. The documents may come from

teaching and research institutions in France or

abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est

destinée au dépôt et à la diffusion de documents

scientifiques de niveau recherche, publiés ou non,

émanant des établissements d’enseignement et de

recherche français ou étrangers, des laboratoires

publics ou privés.

Discussion of ”Investigation of Flow Upstream of

Orifices” by D. B. Bryant, A. A. Khan and N. M. Aziz,

Journal of Hydraulic Engineering, January 1, 2008, Vol.

134, No. 1, pp. 98-104

Gilles Belaud, X. Litrico

To cite this version:

Gilles Belaud, X. Litrico. Discussion of ”Investigation of Flow Upstream of Orifices” by D. B. Bryant,

A. A. Khan and N. M. Aziz, Journal of Hydraulic Engineering, January 1, 2008, Vol. 134, No. 1, pp.

98-104. Journal of Hydraulic Engineering, American Society of Civil Engineers, 2009, 135 (2), p. 155

- p. 157. �10.1061/(ASCE)0733-9429(2009)135:2(155.2)�. �hal-00454440�

Khan and N. M. Aziz

Journal of Hydrauli Engineering, January1,2008, Vol. 134, No. 1,pp. 98-104.

DOI: 10.1061/(ASCE)0733-9429(2008)134:1( 98)

GillesBelaud 1

and XavierLitri o 2

Intheirpaper(Bryantetal.,2008)theauthorsusepotentialowtheorytostudyowupstream

of ori es, and ompare theoreti al results to a large set of experimental data. As pointed

out previously by dierent authors among whi h Shammaa et al. (2005) and Belaud and

Litri o (2007,2008), potential ow assumptions give a qui k and e ient method to estimate

thevelo ityeldgeneratedbyanori eandmaybesu ientformanyengineeringappli ations

that do not require to take all the real uid ee ts into a ount, su h asvorti es. Thispaper

thereforeprovidesanother interesting appli ation of the potential ow theoryto real ases.

However, there seems to be some onfusion about the notions used in the paper, espe ially

the radial velo ityandthe velo itymagnitude,and thisdis ussionaimsat bringing some

om-plements to larify these points. We also larify the way one should apply the prin iple of

superposition, and nally provide some theoreti al ba kground for the study of the velo ity

distribution.

1

Resear her, UMR Gestion del'Eau, A teurs, Usages, IRD,Maison des S ien es de l'Eau, 300 av. Emile

Jeanbrau34095MontpellierCedex5,Fran e,belaudmsem.univ-montp2.fr 2

Resear her, UMR Gestion de l'Eau, A teurs, Usages, Cemagref, 361, rue JF Breton, B.P. 5095, 34196

Thepaperusesthepotential methoddes ribed inShammaa etal.(2005). A ording to Bryant

et al. (2008), the appli ation of this method givesa good des ription of the ow far from the

ori e but a rather bad des ription of the owpattern nearthe ori e. Theauthors therefore

propose a so- alled new model that better reprodu es the ow pattern. This dis repan y

between the two modelsis surprising, sin e the method usedbyBryant et al. (2008) to derive

the potential fun tion is exa tly the same as the one originally proposed by Shammaa et al.

(2005). In fa t, the model proposed by the authors is obtained from the original model of

Shammaa et al. (2005) via a simple hange of oordinates, whi h doesnot justifyin our view

the denomination of new model.

Moreover, the dis repan y appears to be due to a misunderstanding of the original model of

Shammaa et al.(2005) whi h orre tly representsthe velo itypattern (seeFig. 3 in Shammaa

et al. (2005)).

First of all, let us re all that the ow potential

Φ

, in the plane

x − z

(the same notation as Bryant etal. isused),isafun tionof

r

and

θ

, ontrarilytothenotation usedbythe authorsin

Eq. (1). The radial and transverse velo ities are then obtained asfollows (see e.g., Bat helor,

1967,pp.100, 600):

V

r

=

∂Φ

∂r

V

θ

=

∂Φ

r∂θ

and

~

V = V

r

e

~

r

+ V

θ

e

~

θ

. Were allinFigure1thedenitionofthe radialandtransversevelo ities

V

r

and

V

θ

and the unitve tors

e

~

r

and

e

~

θ

.

Inthe verti alplane

x − z

, the velo itymagnitude

V

an be omputed asfollows:

Figure1: Denitionof velo ity omponents in averti al plane(

x − z

)

Assuming that the potential dependsonly on

r

leads to assume that

V

θ

iszero, whi h in turn

gives

V = |V

r

|

. Notethat

V

r

should benegativein the ase ofa sink.

In fa t, it is not lear whi h quantity is displayed in the gures of the paper. The legend

states

V

r

, i.e. the radial velo ity, but the dis ussers think that the plot orresponding to the

new solution andthe experimental resultsin fa trepresent the velo itymagnitude, whilethe

original solution orrespondsto the magnitudeof the radial velo ity.

Therefore, we think that the apparent dis repan y between the original and new solutions is

due toa onfusion between radial velo ityand velo itymagnitudes. Provided the originof the

polar oordinatesisintheori e,thesetwoquantitiesare losetoea hotherfarfromtheori e

aspointed out bythe authors, butlargely deviate aswe approa h the ori e.

We now usean analyti al modelderived by Belaud and Litri o (2007) to illustrate these

on- epts.

Radial Velo ity and Velo ity Magnitude

Let us onsider a square ori e of side

2c

entered in 0, and use potential ow theory to

express the velo ity omponents in Cartesian oordinates. As shown by Belaud and Litri o

(2007) for any re tangular ori e, the potential ow solution an be expressed in losed-form

omponents in

x − z

plane. Dueto symmetry, the omponent

V

y

is null. We introdu e

M

_{= −}

Q

8πc

,

(2)

r

1

=

pc

2

+ x

2

+ (c − z)

2

,

(3)

r

2

=

pc

2

+ x

2

+ (c + z)

2

,

(4)

λ

1

=

px

2

+ (c − z)

2

+ (c − z)

x

,

(5)

λ

2

=

px

2

+ (c + z)

2

− (c + z)

x

,

(6)

X

1

=

c

r

1

+

px

2

+ (c − z)

2

,

(7)

X

2

=

c

r

2

+

px

2

+ (c + z)

2

,

(8)

in whi h

Q

isthe dis hargethrough the ori e.

Usingtheresultsdevelopedin BelaudandLitri o(2007),

V

x

and

V

z

anbe omputedusingthe

following analyti alexpressions:

V

x

= 2M



arctan(λ

1

X

1

) − arctan

 X

1

λ

1



− arctan(λ

2

X

2

) + arctan

 X

2

λ

2



(9)

V

z

= M log

 (r

1

+ c)(r

2

− c)

(r

1

− c)(r

2

+ c)



.

(10)

From this set of equations, we an ompute the radial velo ity

V

r

and the velo itymagnitude

V

asfollows:

V

r

=

~

V .

−→

OP

|

−→

OP

|

=

xV

√

x

+ zV

z

x

2

_{+ z}

2

(11)

V

=

pV

2

x

+ V

2

z

.

(12)

Thesevelo itiesareplottedatdierentdistan esfromthe ori einFigure2. Thedistan esare

normalized by the equivalent diameter

d = 2p4/πc

whi h gives the same area for the square ori e asfora ir ularori eofdiameter

d

. Theplotsareverysimilarto thoseofBryant etal.

x/d = 0.1

0.2

0.3

0.5

x/d = 0.1

0.2

0.3

0.5

x/d = 0.1

0.2

0.3

0.5

x/d = 0.1

0.2

0.3

0.5

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

−1.5

−1.0

−0.5

0.0

0.5

1.0

1.5

Normalized velocity |Vr|/Uo or V/Uo

z/d

Radial velocity Vr/Uo

Velocity magnitude V/Uo

Radial velocity Vr/Uo

Velocity magnitude V/Uo

Radial velocity Vr/Uo

Velocity magnitude V/Uo

Radial velocity |Vr|/Uo

Velocity magnitude V/Uo

Figure2: Velo ity valuesalong

z

oordinate at dierent distan es

x

fromthe ori e plane.

et al. (2005), while the solution presented as the new solution appears to be the velo ity

magnitude. Far from the ori e, the transverse omponent of the velo ity be omes small and

bothquantitiesbe ome very lose. Figure3shows the iso-velo itylinesin the plane

x − z

and an be ompared to Figure3of Bryant etal. (2008): the dottedlinesdepi tthe radial velo ity

V

r

, while the plain lines depi t the velo itymagnitude. We an also point out that the ori e

shape haslittle inuen ein thedomain of study.

Asa sideremark,we notethat usingEq. (1),wehave:

V

2

r

≤ V

2

.

(13)

Thisinequalityexplains why, in Figures3and 4ofBryantet al.(2008), the originalsolutionis

always lower than the so- alled newsolution, in whi h the velo itymagnitude

V

is al ulated

fromEq. (12)of thisdis ussion andEqs. (5)to (7)of Bryant et al. (2008).

Prin iple of Superposition

A ordingto theprin ipleofsuperposition, thepotential fun tion,the streamfun tionandthe

0.2

0.5

0.4

0.3

0.2

0.5

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

−1.0

−0.8

−0.6

−0.4

−0.2

0.0

0.2

0.4

0.6

0.8

1.0

x/d

z/d

Radial velocity |Vr|/Uo

Velocity magnitude V/Uo

Figure3: Iso-velo itylinesin

x − z

plane

an ertainlynot beadded.

Let us illustrate this point in the ase of the superposition of two velo ityelds. We want to

nd the resulting velo ityelds bysuperposition ofthe elds

~

V

1

and

~

V

2

. The resulting eldis

givenby:

~

V = ~

V

1

+ ~

V

2

.

This anbeexpressedinany oordinatesystembyaddingthe omponentsofthevelo ityve tor

alongthe oordinate basis. InCartesian oordinates, we get:

V

x

= V

1

x

+ V

2

x

V

z

= V

1

z

+ V

2

z

.

Inpolar oordinates, weget:

V

r

= V

1

r

+ V

2

r

the transverse omponent

V

θ

is needed in order to ompute the velo ity ve tor. This is valid onlyifthesame oordinatesystemisusedfor allthesuperposedsinks. Inthepolar oordinates

ase, itis important to he kthatthe two systems aredes ribed usingthe same origin.

Theexpressionsprovided byShammaaet al.(2005)inEqs. (1011)are orre tandleadtothe

same results as in Cartesian oordinates. The dis repan y between the original solution and

the newsolutionin Figures10,12,13and 14ofBryantet al.(2008)isagainprobablydue toa

onfusionbetweenradialvelo itiesandvelo itymagnitudes. A ording tothedis ussers'

al u-lations, theseplots (originalsolution)mayresultfromtheaddition ofthe velo itymagnitudes

or fromaddition ofradial velo ities al ulated with a dierent origin for both sour es.

Ee t of the Velo ity Distribution

The authors used an empiri al method to determine the limit when the velo ity distribution

should be onsidered. Thepresent dis ussion brings some theoreti alelements onsistent with

theexperimentalresults. Tosimplify, we onsiderasquareori eofheight

2c

. Theee tofthe

ori e size an be analyzed by onsidering that an ori e is omposedof two parts, an upper

partof height

c

, entered in

+c/2

, and alower part ofheight

c

, entered in

−c/2

.

Themeanheadabove theupperori e is

h

0

− c/2

, whilethe meanhead above thelower ori e is

h

0

+ c/2

. Sin etheori estrengthisproportionaltoitsdis hargeandthereforeto thesquare

rootofthe head,themeanerror

ǫ

onthe strengthbetween thelowerand theupperori es an

beestimatedby:

ǫ =

ph

0

+ c/2 −

_√

ph

0

− c/2

h

0

(14) whi h gives

ǫ ≃

c

2h

0

(15)

If

d = 2c

is the height of the ori e and ifwe take

ǫ = 0.02

asthe authors do in their paper, orresponding to an error of 2%, we nd a limit of

h

0

/d = 12.5

, whi h is exa tly the result

obtained experimentally bythe authors.

Thesameanalysis ondu tedwitha ir ularori e wouldleadto aslightlydierent result,but

the analysisgivesa rough andrapid estimation ofthe ee t of thevelo ity distributionwithin

the ori e. Thismay be helpful to de ide whether, or not, the velo ity distribution should be

usedin the potential owsolution. Ifnot, we mayuse losed-form expressions for the velo ity

eld(see equations (9)and (10)ofthis dis ussionfor the squareori e).

Referen es

Bat helor,G.K. (1967). An introdu tion touid me hani s. Cambridge University Press.

BelaudG.andLitri oX.(2007). 3Dvelo ityeldgeneratedbyasideori einanopen- hannel.

32th CongressIAHR, Veni e, 1-4 July 2007.

Belaud G. and Litri o X. (2008). Closed-form solution of the potential ow in a ontra ted

ume J. of Fluids Me hani s,599:299307.

Bryant D.B.,Khan,A.A.andAziz,N.M. (2008). Investigationof FlowUpstreamof Ori es.

J. of Hydrauli Engineering,134(1):98104.

Shammaa, Y., Zhu, D. Z. and Rajaratnam, N. (2005). Flow upstream of ori es and slui e