Reattached wall jets

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BV N. RAJARATNAM * AND D. MURALIDHAR * *

1niro duetion A number of hydraulic structures like sluices and sluice gates sometimes work under submerged flow conditions. The fluid streams issuing from such outlets could in general, be analysed as turbulent jets. If the confining boundaries are sufIiciently far off from the outlet, the jet can be analysed as a free turbulent jet, about which extensive infor- mation is readily available [lJ. If on the other hand, the jet occupies the full width of the channel and issues tangentially to its bed, then it could be treated as a turbulent wall jet [2, 3, 4, 5, 6

J.

If the full width jet enters the channel at a certain height above the bed, the flow pattern assumes very complex forms. As a first attempt, Rajaratnam and Subramanya [7Jstudied the diffusion of a full width jet issuing horizontally into the tailwater channel at a certain height above the bed with the drop being located at the nozzle outlet itself, as a restricted form of the reaUached wall jet. Haja- ratnam and Subramanya [8J also studied the clif- fusion of a jet produced by a sluice gate over a drop.

In this paper, the drop in the bed is located at a certain distance downstream of the nozzle, (see Fig.

1a) so that the forward flow has the profile of a wall jet before the drop.

.~---_.._ - - -

* Associate Professor, Dept. of Civil Engineering, Univ. of Alberta, Edmonton, Canada.

*" Research Associate, Dept. of Civil Engineering, Univ. of Alberta, Edmonton, Canada.

REATTACHED WALL JETS

In Figure 1a, a jet of depth of !/o and an almost uniform velocity of Do enters the channel. under conditions of deep submergence. Sorne dIstance after the end of the potential core, it grows as a fully developed plane turbulent wall jet. At the sudden drop, the wall jet curves down, towards the lower bed due to the creation of lower pressures (this phenomenon is known as the Coanda. effect [9, 10

J)

and impinges on the bed at a certam section creating a zone of eddying flow of length Le' The forward flow after some distance from this reattachment line (where the separated wall jet reattaches itself to the bed) grows as a wall jet which could be termed the reattached wall jet and an experimental study of the same is presented in this paper.

Experimenis

The experimental arrangement was essentially the same as that used in the earlier studies [7, 8

J.

The velocity distribution in the fOl'ward flow was measured by means of a commercial 3 mm external diameter Prandtl-type Pitot-static tube. The bed shear stress

"0

was measured by means of a Pres- ton tube [11

J

of external diameter of 3 mm and an internaI diameter of about 1.8 mm. The calibration curve used was that of Patel [12J. The reattachment line was found using a tuft probe. The pres- 303

Article published by SHF and available at http://www.shf-lhb.org or http://dx.doi.org/10.1051/lhb/1968023

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Table 1 1 Experimental data

Uo ^1l V 1l L L L R = VoU o

Expt. No. ⁰ ^c _~"

(in) (in) ft/sec Uo ⁽ⁱⁿ⁾ ^h Uo ^Q

~---

FIHST SEHIES

lA 1.00 1.GO ^{~ ')~}

,.- ,

1.69 4.50 2.Glj 12.50 5.27 X 10'1

1 B 1.00 l.G0 0.75 1.69 4.75 2.81 12.50 7.07 X 10'1

2A 1.00 :3.ll8 8.84 :\.08 7.75 2.52 12.50 6.'11 X 10'1

2B 1.00 3.08 7.14 3.08 7.50 2.44 12.50 5.18 X 10'1

3A 1.00 5.14 8.55 5.14 11.50 2.24 12.50 G.19 X 10-1

'lA 1.50 3.29 7.80 2.19 8.25 2.50 12.50 8.4G X 101

5A 1.25 3.25 0.11 2.GO 8.40 2.58 12.50 8.25 X 10 1

SECOND SEHlES

6A 1.00 3.0G 7.94 3.0G 8.50 2.G8 63.50 5.75 X 10-1

GD 1.00 3.04 8.72 3.04 8.50 2.80 50.00 6.32 X 10⁴

Ge l.OO 3.05 8.14 3.05 8.25 2.71 :~9.75 5.90 X 10 1

GD 1.00 3.2:"5 7.97 3.25 8.50 2.61 30.00 :"5.77 X 10'1

GE 1.00 3.24 7.8G 3.24 8.50 2.G2 20.00 5.70 X 101

sure on the bed was observed by means of a large number of piezomoters located on the bed.

Two series of experiments were conducted. In the first series, the length before the drop L was kept equal to 12.5 Yo and five sets of experiments were conducted with the height of the drop 11 varying from 1.69 to 5.14 times Yo. In the second series, keeping 11/yo almost constant at about 3.0, L/yo was varied from 20 to 63.5. The important experimental results are given in Table 1. In aIl the experiments, the velocity distribution just before the drop was measured and the reattached wall jet was explored in great detai1. Figure 2 is a typical example of the velocity distribution in the forward flow and Figure 3 gives the results of the bed shear stress for the first series. In Figures 2 and 3, x and

.r

are respectively the longitudinal distances from the nozzle and the reattachment line and II

is the turbulent mean velocitv at a normal distance of y from the becl. .

Analysis of Results

Length of the eddying region.

IfLeis the length of the eddying region, for large values of the Reynolds number:

R

=

UoYo v

it has been shown earlier [7,9, 13, 14J that:

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where v is the coefilcient of kinematic viscosity of the fluid. The variation of Le/il ,vith il/yo finalised in referenee [7] is reprodueed in Figure 4 along with the present results for the two series. The present resuIts agree fairly weIl with this curve, thereby showing that Le is not very much affected by the non-uniform veloeity distribution Just before the drop.

Reattached wall jet.

The velocity distribution measurement in the fOl'ward flow region after the reattachment line were tested for similarity by plotting the dimensionless velocity lI/U_m against 'c)= y/01' where lI", is the maximum velocity at that section, and 01 is the value of yat which II = lI,,/2 andOlI/OY is negative (see Fig. 1 b). (lI_m is generaIly called the veloeity seale and 01 the length seale.) Ii was found that for both the series, after a distance of about 18 Yo from the reattaehment line, the velocity distribution is similar and agrees well that of the plane turbulent wall jet issuing into a stagnant surround- ing on a smooth wall with zero pressure gradient, often referred to as the classieal wall jet (CWJ).

Figure 5a shows a few typical plots of the first series and Figure 5b shows a few typieal plots of the second series.

Velocily Scale.

The variation of the climensionless velocity scale (lI₁

n1U

O) is studied in Figure 6a for the fiTSt series and is compared with that of the classical wall jet [15

J.

^It is seen that for any given value of il/yo, after the drop, lI_m is very small, increases slightly due to the favourable pressure gradient after the impingement region and then merges with the curve

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LA HOUILLE BLANCHE/N° 4-1968

10.0 100 110

9.0

ESSAI

~BJ:!.!'L~

• = 12.0,n

=360in

=44.0;n

·S.t.O'n 6C 0

2.0 22 2.4

~'y/8 1 90

8.0 } Present work

Présente étude 80

,380,,.

, 480;n , 580;,.,

·6e.Cm

• 80.0in

7.0

ESSAI .!_lJ..tU~

• "120in

-640;"

'7AOi"

'840,n CWJ " " . [3J .^{900 ,"}

Courbe du jet de paroi classique

70

"39:/5 -660

• 740

• 820 '900

~99,0 : 106,0

- - '::WJ[J] '115.0

jet de parei classique 2

2°

- Ref. 7 1st. Series •

re Série

ESSAI RUN lB . ' 120 : 600 , 800 , 400 , 470,,, ,520,/"1

..

5.0 6.0 h/yo

50 60

X (in)

4.0 40

3.0 30 20

2.0 10

10

NOTE The Or'9ln Ho~Been Vert,eollyDi~plaeed For Eoeh CUfve

o

08

o

06

04

02

~30", 0

" ^495."

'900,,, 750,"

10 ' 10\0,,, ' 840't'I

'10901" '930,n , 1210", • lOBa,,,

'13'JO,c. 1120,,,

0.6

Ibl 0.4 _NOTE.

The Origin Has Been Verlico!1y Displaced

02 fo' Eoch Curve.

00 02 04 06 08 10 12 14 1.6 1.8

10.0~

80

l

\0.18---r

0.16

·

⁰ ^lA^lB

0.14

..

^2A

'" ·

^2B^3A

012

·

⁰ ^4A^5A

.c

st 0.10

~.D

~_?0.08 0.06

0.04

0.02

i

6'0~

Le/h 4.0

, * *"F"'

't'----.--- ^J

2,0

5/

4/

3/

---

(b)

U_m/2

r=~

Free mixing region U Zone de mélonge lIbre

SUI

¹s- ^U^m

^L_

Boundory layer (or wall) region _ J_ ...

Zone de couche linute (ou de porol)

1/

.0

[1 AJ

x = 1.75 in.

0.9 = 12.0

" ^{= 17.0}

v = 21.0

0.8 x =26.0

• ^{= 30.0}

'"

₊ ₌₄₄₀^{= 38.0}

6 =50.0

•

^{= 56.0}

0 = 68.0

( ft) 0.5

0.4

0.3

0.2

0.1

0 2 3 4 7

(ft/sec)

2/

1/

Definition sketch / Schéma de définition.

t---Le-""1

(a)

2/ Typical velocity distributions, (Note the change of origin for the first two sections.)

Répartitions-types des vitesses. (Noter le clll/n(Jement d'ori(Jine pour les deux premières sections.)

3/

Bed shear stress distributions.

Répartitions des efl'orts de cisaillement ail j'ond.

4/Lenght of eddying region.

Lon(Jl1eur de la zone tourbillonnaire.

5/Velocity distribution. Heattached wall jet.

Répartition des vitesses, .Jet de paroi rattaché.

305

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of the classical wall jet. The cune for l'uns lA and 1 B is shown by a dotted cune in Figure (j a.

The section at vvhich un/U" variation merges with that of the dassical wall jet cu ne is farther l'rom the section at which the velocity distribution beco- mes similar.

The results for the second series are shown in Figure () band they also show that after some distance, the velocity scale varies in the same manner as the elassical wall jet.

Length scale.

The results of the dimensionless length sc~de

8dUo for the first series are shown plotted in Fi- gure 7a. Il is seen that just before the drop, the length scale points fall on the C\VJ cu ne [15] and in the region influenced by the drop the values are much higher. After some distance, the experimental curve merges with that of the C\V.J cuneo The curve for l'uns lA and lB is shown as a dotled line. The section at which merging with the cor- responding C\V.J curve takes place is farther away th an the section at which merging takes place for the veloci ty scale. The magnitude of these distances is roughly of the same orcIer as obtained by Hajaratnam and Subramanya [7]. The length scale results for the second series are shown in Figure 7b. The same trend is noticable, but ,vith more scatler.

Bed shear stress.

Bed shear stress variations have been given earlier in Figure :3. Following the method used earlier by Hajaratnam and Subrmnanya [7, 8J, the se profiles were plotted with

"O/"Om

against X/el' where

"Om

is the maximum value of "" in any given

l'un, el is the value of x' al which "0

=

"om/2 and d'to/dx is negative. Figure 8a shows the results of the first séries and Figure 8 b shows the results of the second series along with the similarity curve of Haj aratnam and Subramanya [7]. Il is seen that the present results agree very well with the cune of reference [7] thereby showing that the shear stress distrihution is not afTœtcd hy thl~ nature of the velocity distrihution existing before the drop.

For the bed shear plot, the results of the dilnen- sionless shear stress and length sc~des are shown in Figures Ç)a and b. The present results for the shear stress scale are lower and higher for the length scale, when compared with the curves of Hajaratnam and Subramanya [7J.

Conclusions

An experimental study has been made of the diffusion of submerged full width reetangular out1ets, when there is a drop in the bed after some distance l'rom the outlet. This could also be considered as a study of plane turbulent reattached wall jets. Il has been found that the length of the eddying region below the drop is not afl'ected appreciably hy the nature of the velocity distribution existing before the drop. The velocity distribution in the reattached jet has heen found to be si milaI', which agrees dosely with that of the classical wall jet.

The velocity and length scales merge with the variation of classical wall jet after certain distances l'rom the reattachment line. A study has been made of the bed shear stress in the reatlached fiow.

4.0 À

de porOi D..

( b)

100 120 140

60 X/Yo80 40

V

o 20

12 ₀₆₈• 6A Llyo =₌63.5_50.0 •

À 6C =^39.75

10 ^l:. 6D =30.0

V 6E =20.0

2.0

10.0 , - - - , - - - - , - - - , - - - , - - - - , - - - , - - - - , - - - , - - - _ , - _ - - , 8.0

"-::2

~6.0

o ( bl

o • '"

oGA

• G B

'" Ge _ _ eWJ [8]

.. 6 0 ..Jet de poroiclaSSique o GE

10

0.2

20 30 ·<iO

6/VclocH\' scale.

Echelle- des vitesses.

90 100 no 120

8.0

6.0 ,.!?

"- t.O4.0

2.0

Legend sarne as fig. 6 a Même légende que fig. 6a

CWJ[8J

Jet de paroi classique

(a)

7/Lcngth scale.

Echelle des fOllfluellrs. o 10 20 30 40 50 60 70 80 90 100 x/Yo

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LA HOUILLE BLANCHE/N° 4-1968

1.4 1.2

2

•

1,0

Series Série

• °

(a)

• •

0,4

• •

o 0.2

1.0 3.0 4.0 50,0

40,0

Ref,7_

30,0

-'= ° _~ (b)

"

⁸

cO ⁸

20.0 °CI

Series

~ ⁸ _Série

° ²

10,0

::;:;-'

o

X E 2.0 8/Bed shear stress distribution.

.s

Répartition des efTorts de cisaillement all fond.

'O~-t;;:: , "" ^l

JOi

1'~~

^{--R,r 7}

i

Dar'

~.~ ~

^.,.

~

'/'orro ~.~. ~ :;~~ ¹

06

·,.~~:E ~

~ -:. SA i

04 .. ..,1

0211 '

^{IA,lB,2A 2} l

oo

~..

02^", 0.4 06 08 10

~~--,-,

i2 14^--

-"---,-~

1.6;:/8

11.8 20 22 2J

91

Seale~ for the bed shear distribution.

Echelles de la répartition des ef-

forts de cisaillement all fond. o ^1.0 ^2.0 ^3,0 ^4,0

h/yo

5,0 6.0 7,0

Acknowledgment

This work was done Ül the Civil Engineering vepartment of the University of Alberta at Edmon- ton, Canada. The authors are thankful ta Ml'. J.

Bater and Ml'. A. Dwight for their help in the experimental worle They are also grateful ta the Na- tional Research Council of Canada for the financial assistance provided.

[1J

[2J [3J

[4J

[5J

References

ABH.HlOVICH (G. N.). - The Theorv of Turbulent .Jets.

English translation published by'M.!.T. Press (1!Jfi:l).

GLAUEHT (l\!.B.). - The 'Val! .Jet. J. of Flllid Meehanies, vol. 1, pp. 625-fi43 0!J56).

HA.JAHATNAM (N.). - Flo'w Below a Submerged Sluiee Gate as \Vall .Tet Problem. l'roc. 2nd Australasian Conference and Fluid Meehanies, New Zealand 0!Jfi5).

IlA.IAHATNAM (N.). - Submerged Hydraulic .Tump. J. of the Hydrallfies Division, ASCE, vol. !JI, No. HY 4 (.Tuly 1%5).

HA.JAHATNA~1 (N.). - Hydraulie .lump as a \Vall .Tet.

J. of the Hydralllies Division, .4SCE, vol. !JI, No. HY;, (September 1%5).

[6J I1A.JAHATNA~1 (N.). and SUBHAMAYA (K.). - Annotated Bibliographie on 'Vall .Jets. Tech. Dept. of the Civil Engrg., [j'niv. of Alberta, Edmonton, Canada 0%7).

[7] HA.JAHATNBI (N.) and SIJBHA~IANYA (K.). Plane Turbulent Heattaehed 'Vall .Jets. Tech. Hept. of the Dept. of Civil Engrg., Univ of A lberta, Edmonton, Canada (May 1!Jfi7).

[8] HA.L\HA'l'NA~1(N.) and SUBH.HL\NYA (K.). - Diffusion of Submerged Sluice Gate Flow Over a Drop. Froc. of the 12th I.i1.l1.R. Congress, Fort Collins, Colorado (September 1!Jfi7).

[!J] BOUHQUE (C.) and NEW~L\N (B.G.). - Heattachment of a Two-Dimensional Incompressible .Jet to an Adjacent Flat Plate. Aero. Qllarterly, vol. II (August 1%0).

[10] NEWMAN (B.G.). - The Dellection of Plane .Tets by Adjacent Boundaries Coanda E1Tects, in Boundary Layer and Flow Control, vol. 1 Ed. by G. V. Lachmon, Perga mon Press 0 !)(il).

[11] pHESTON (.LIl.). - The Determination of Turbulent Sldn Friction bv Means of Pitot Tubes. J. of tlle Royal Aeronal/tical Society, vol. 54 0%4).

[12] pATEL (V. C.). - Calibration of the Preston Tube and Limitations on its llse in Pressure Gradients. J. 01

the Fll/ids Mec1wnies, vol. 2:l (l!Jfi;').

[1:)] SAWYEH (IL A.). - The Flow due to a Two-Dimensional .Jet Issuing Parallel to a Flat Plate. J. of Fll/id Mec1wnics, vol. !J (1%0).

[141 SAWYEH (H. A.). - Two-Dimensional Heattaching .Tet Flows Including the Effects of Curvature on Entrain- ment. J. of Flllid Mec1wnics, vol. 17 (l!J63).

[15] HA.IA1UTNAM (N.) and SUBH.DIANYA OZ.). - Diffusion of Hectangular 'Vall .Jets in \Vider Channels. J. of H!Jdrmllic Rescarch, Delft, 1%8.

307

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