Oscillatory phenomena of a turbulent plane jet flowing inside … · 2014. 5. 12. · In this study...

flowing inside a rectangular cavityOscillatory phenomena of a turbulent plane jet

CNRS / Universit& d'Aix Marseille I &I,! France.' h t i t u t de Recherche sur les Phknom2nes Hors Equilibre, U M R.6594U.S.T.H.B., Algirie.'Laboratoire de Mtkanique des Fluides, Facultk des sciences Physique,A. Mataoui ' , R. Schiestel 2,A. Salem'

Abstract

and the Reynolds number.numerically in different situations : the location of the exit of the jet in the cavityof the stable-oscillation flow regime have been detailed experimentally andThe structural properties of the flow and the frequency of the flapping of the jetoscillation regime, stable-oscillation regime and unstable-oscillation regime.have been observed for different positions of the jet inside the cavity : no-(k - E ) and the two-scales energy - flux-models. Three types of flow regimeson two statistical models of the turbulence : the mono-scale energy - dissipationanemometer and supplemented by visualizations. The numerical study is basedin the cavity. The experimental study is made essentially by hot wirecavity is investigated under some conditions of the location of the exit of the jetIn this study , the oscillation of a turbulent plane jet flowing inside a rectangular

1 Introduction

The practical interest of such a phenomenon is related to the various problems in

5 - renewal of fluid inside a cavity4 - air conditioning3 - cooling or heating by forced convection2 - combustion1 - mixing of fluids

industrial applications such as :

© 2002 WIT Press, Ashurst Lodge, Southampton, SO40 7AA, UK. All rights reserved.Web: www.witpress.com Email [email protected] from: Advances in Fluid Mechanics IV, M Rahman, R Verhoeven and CA Brebbia (Editors).ISBN 1-85312-910-0

l.52 A d v m c e s irl Flu id M e c h u n k s W

6 - flowmeters with no moving solid parts.

stable-oscillatory and unstable-oscillation.exit in the cavity. Thee types of flow regimes are observed : no-oscillatory,The jet-cavity interaction is investigated for a great number of location of the jetcavity have been made and its flow characteristics are considerably made clear.Villermaux & Hopfinger [ 5 ] of the self sustained oscillation of a jet inside aSome studies Ogab, [l] Shakouchi &L al. [2], Shakouchi, [3], Maurel [4] and

2 Experimental apparatus and procedure

Ruler

Cavit

suppomAdjustable

\W

I chamber" r r u r r g

Hou~ycom-+++l1 Plane splitterwith guidingwallsI Grids2Figure 1 : Experimental apparatus

4

The velocity measurements are carried out using constant temperature hot wire1) .adapted to produce two- dimensional jet flow with low turbulence level (figurerectangular duct can be displaced horizontally and vertically . The nozzle isThe experimental device consists mainly of a rectangular cavity in which a

one or two signals during a small time interval .experimental session. A memory oscilloscope is used for sampling and recordingprobes have been calibrated using a calibration system before and after eachprobe are made of a 5 pm diameter platinum-plated tungsten wire is used. These(56COO) apparatus including a signal analysis components. A single hot wireanemometry. The measuring device is composed of a Dantec multi channel


A d m c c ~ sill Fluid Mdxzn ics I V 1 5 3

Flow visualisations

per second.photographs series have been shot with a camera using a frequency of 2 picturesof the cavity through a vertical transparent slot in the mid plane. Motionflow at the jet exit. The flow is lightened by a projector located at the bottom endupstream at the inlet of the channel in order to obtain a nice homogeneous whitevegetable oil mixed in compressed carbon dioxide. The smoke is injectedThis generator produces a white smoke composed of very small droplets ofA CFT Taylor smoke generator is used for visualisations of the flows structures.

3 Numerical modelling

3.1 Single scale k - E model

is obtained by an algebraic relation :Prandtl-Kolmogorov’s turbulent viscosity. The turbulent Reynolds stress tensorThe two equations’ model Schiestel [6] and [7] is based on the concept of

where

Vt = E&

The closing is then obtained by calculating the equations of k and E :

d k -- - Ed t (3)

d c- --

dt 2 %

3.2 Two scales energy-flux model

spectrum in three zones (figure 3) Schiestel [6] and [7]:The schema adopted in the present work consists in subdividing the average


154 Advmces irl Fluid Mechmks W

transport equations:The partial kinetic energies and the spectral flows are determined from the next

?-!!E = P - + Diff(k,)d t ( 5 )

k t - E, - + Diff(k,)__ -d t (6)

E = Et

defined by 5 = , knowing that k = k, + k t and vt = cpkx.where C,, , C,, , cl,and C2, may be functions of the spectrum’s formspectrum’s form ratio . The corresponding values in this case are :coefficients . The linking conditions are satisfied for the value {= 3 of theIn first approximation we will attribute constant numerical values to these

kA

CIp=1.65 ; Czp=1.92 ; C,,= 1.75 ; C2t=1.82

P flux out oft heP = product ionK = w a v e n u m b e rE = energy spectrum

h

E -& P product ion rangeenergy in the

kp = part ia l kinet ict ransfer range

= f lux out of the

t ransfer zoneenergy in the

product ion zone

/-

k T = part ia l kinet ic

k P

,’ .+0 K I & = E T K

Figure 2 : Multi-scales model of two levels

of viscous effects and a decrease of the intensity’s levels of the turbulence . ThisThe presence of a rigid wall and the condition of adherence result in an increase


A d m c c . s i l l Fluid M d x z n i c s I V 1 5 5

we have used the method of wall functions Launder & al. [8] .will necessitate a special treatment for the region next to the wall . In this work ,

4 Numerical procedure

variables can be written by the following general form :constant physical properties . The equations for the mean and the turbulentThe considering flow is fully turbulent and instationary. It concerns a fluid with

aCDP d t + P -&( P U i q = -[r, "U] + S,d xj d x j (10)

of the movement. The two-dimensional discrete form of this equation is :where r, and S, are determined for each variable from the global equations

A, CD, = A, CD,+ A, @ W + AN (D,y + As (Ds + b (1 1)

The velocity's components are quantified at shifted nodes and the velocity-

while the A . (i = P , E , W, N and S) and b proceed from the eqn (10) .P ,E , W ,N and S respectively (figure 3) Patankar [9] and Jones &L al. [lo],

where CD p , CDE , CD , CD and CDs are the variable's values of C D at

by m inverse substitution.utilization of the algorithm TDMA which includes a Gaussian process followedthe coefficients of the associated matrix to the eqn (1 1) requires theto be linearized for the stability of the solution . The three-diagonal structure ofpressure coupling is realized with the algorithm SIMPLE . The source term has

1

NJ+l

A X b

J D J.WW

J-l

I-1 I I+1

Figure 3 : cell of integration

then proceeded to the application of the algorithm TDMA along the twovelocity vector especially at the region of the bottom of the cavity . We haveThe examined flow presents a considerable variation of the direction of


1.56 Advmces irl Fluid Mechunks W

increase the stability of the convergence .directions South-North and East-West . This double sweeping is intended to

5 Results

5.1 Geometrical and dynamical parameters

dimensionless coordinates :The location of the point of measurement (M on figure 4) is defined by its

been built on these geometrical parameters :Geometrical parameters are specified on figure 4. Dimensionless quantities have

x=x1/xo H=Hl/Ho (12)

the jet exit and the height of the nozzle and where v is the kinematic viscosity :The dynamical parameter is the Reynolds number based on the mean velocity at

x=(Xl-xl)/X~ and h=hl/Ho

of oscillations :number defined on the same quantities U0 and h0 and where f is the frequencyThe frequency of oscillation in the unsteady case is characterised by a Strouhal

Re=Uoho/v

St=fhO/UO

(13)

(14)

(15)

. .:.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ................................................................................................................................................................................... ~. . .

. . .. . .

. .

of entering flowMean velocity profile

X I -+2

~ ~

............. P A. . . .. . .& : g X I

................ W X I , h )

out flowprofile goingMean velocity

ha HIh

................................ ............... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

0 X

Figure 4 : Geometrical parameters

5.2 Identification of the flow regime’s zones

the jet exit being successively located on each node of a rectangular mesh. Wepoints are obtained by sweeping in the two directions x and h the whole cavity,the time analysis of the mean velocity for every point of the cavity . TheseThe nature of the interaction is determined numerically and experimentally by


A d m c c . s i l l Fluid M d x z n i c s I V 1 5 7

because gravity effect is negligible.This diagram is symmetrical toward the mid transverse plane of the cavity5. This diagram is valid for Reynolds number varying from 1000 to 5000.11 (unstable oscillation) and 111(stable oscillation) are summarized one the figuredetected thee types of flow regimes. This distinct zones named I (no oscillation),

H

0 0.2 0.4 0.6 0.8 x

Figure 5: - Diagram of the several flow regime zone’s

5.3 Time analysis of oscillations

determined by the first peak in the Fourier modes A/Uo distribution. The twosame locations by the two-scales model. The fundamental frequency is clearlyThe figure 7 shows the calculated Fourier transforms of these velocity at theturbulent kinetic energy (k) etc .....velocity modulus (U), transverse velocity component (V), mean pressure (P) ,periodic variations is observed for the mean and the turbulent variables such asmodel and T=1.81s for the two-scales model. For the numerical prediction thethe period for the same case are very close, with T=l.87s for the single scaleexperimental period in time is found to be T=1.84 S and the calculated values ofat the same point in the cavity x=O.3 , X=02 , H=OS , Re=4000 ,h=0.300. Theand several secondary peaks that are also clearly visible on the calculated signalThe measured filtered velocity signal shows a periodic pattern with a main peakusing the k-E model and the two-scales model both predict an oscillatory regime .experience velocity signals are practically similar. The numerical calculationsthe periodicity of the mean velocity (figure 6). The two-scales model and theThe oscillation of the jet have been evidenced experimentally and numerically by

harmonics that are noticeable on the time signals of the velocity modulus.models give similar results, but the two-scales model produces more energetic


1 5 8 Advmces irl Fluid Mechmks W

a) Photograph of the signal of the velocity modulus as function of time

0.6

0.5

0.4

f o.33

0.2

0.1

0.00 2 4 6 8

t 6)

b) Calculated based on single scale model

0.71 10.6

0.5

0 0.43.3 o.3

0.2

0.1

0.00 2 4 6 8 10

c) Calculated signal based on two - scales modelt 6)

methods ( x=O.3 , X=O.8 , H=0.5 , Re=4000 ,h=0.300)Figure 6 : Variation of the velocity modulus as function of time by different


A d m c c . s ill Fluid Mdxzn ics I V 1 5 9

0.006 ~>

3.me,

Q 0 004 --

2al.*5 0.002La

0.0 0.5 1.0 1 . 5 2.0 2.5 3.0 3.5 4.0 4.5 5.0

f W )

for ( x=O.3 , X=O.8 , H=0.5 , Re=4000 , h=0.300)Figure 7 : Fourier modes of the mean velocity time signal, two scales prediction

obtained by many authors in similar flow geometries [l] to [ 5 ] .frequency varies linearly with the Reynolds number. This result has been alsobetter agreement for the two-scales model. This result also means that theexperimental results are finely reproduced by the two models, with a slightlypractically constant over the domain of Reynolds numbers considered. Theis represented versus the Reynolds number. The Strouhal number is foundcavity ( H and X ). On figure 9 the Strouhal number based on jet exit conditionscavity varies linearly as the Reynolds number and the location of the jet in theThe frequency of the flapping of the jet between the two laterals wall of theVillermaux & Hopfinger [5].of impinging jets pressure effects are important in the feedback mechanismsthe recirculating flow and amplified by the instability of the jet, while in the casejet flowing in an enlarged channel the perturbations are convected upstream byseveral origins depending on the particular geometry considered. In the case of aconfined jets are linked to feedback effects of perturbations but seem to haveadvance phase before the velocity signal. The mechanisms of oscillation inbased on the two-scales model. The pressure oscillation is somewhat ahead inof mean velocity and the mean pressure is evidenced by the numerical predictioncalculated point. For this case, the phase shift between the transverse componentAyukawa & Shakouchi [l l]. We can see this phenomenon on figure 8 for oneshift proves the existence of a feedback mechanism with time lag effectsshift between the pressure and the mean velocity in the oscillating jet. This phaseAnother phenomenon have been shown in this study is the existence of a phase


1 6 0 A d v m c e s irl Flu id M e c h u n k s W

0.2

. .~ v / u o- 0 . 5 , a I , , , I , I , , , I , , ,

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 I

t (S)

l

Figure 8 : Phase shifts between the pressure and the transverse velocity signals atx=O.6, h=0.50 , X=5 , H=0.5 , Re=4000

1 O 3 %+l.5 at X = O . 5 Two-scales m o d 1

n

- I O 3 S t + l . O a t X = O . 6

- IO3 S t + 0 3 at X = O . 75r

. - ' - - - - - - ~ - - - . . . . ~ . _ ~ ~ ~ ~ ~ ~ ~............................. m

I I t I2 3 4

103ReFigure 9 : Variation of Strouhal number as Reynolds number at H=0.5

5.4 Mean structure of the oscillatory flow

moving; the largest eddy fi-om the unattached side go upstream and produces atime. When the jet is deflected fi-omits mean position, the two lateral eddies startcounter-rotating eddies are clear on each side of the jet inside the cavity at eachprediction lead to the description of the mechanism of one oscillation. Twoduring one period of time ( T). The analysis of these photos and numericalThe figure 10 illustrate photographs of the flapping motion of the jet oscillating


Ad\wzcc.s i l l Fluid M d x z n i c s I V 1 6 1

and the mechanism is periodically reproduced [2].to be deflected to the opposite side where a pressure maximum is then createdcavity increases, the corresponding eddy goes downstreamand then forces the jetin the inside of the cavity at the opposite side; this new flow rate entering thepressure defect; when it approaches the lateral exit, the fluid is sucked (figure 4)

t l T=O.O t l T d . 4 4

oscillation (Re=4000,X=023 and H=0.425).Figure 10: Experimental flow visualizationof six stages of one period of time of

6 Conclusion

to improve the numerical predictions compared to the standard k-E model.jet is due to pressure effects.The use of multiple scale turbulence model allowedsatisfactoryby two-equationsmodels because the mechanism of oscillationof themodifications [12,13,14]. In this present work, the oscillations are predictedunsteadiness, to capture of unsteadiness behind cylinder necessitateflow behind the square cylinders (Jones and Launder model [lo]) do not produceOscillatory flows studied in the literature using two-equations models like thevisualization.moving eddies on both sides of the jet are evidenced experimentally bythe Reynolds number when the Strouhal numberis constant. Two distinguishablethe cavity. Dimensional analysis predicts that the frequency is in proportion tounstable oscillation, and no oscillation flows depending the location of the jet inThe flow patterns are classified into three classes, namely stable oscillation,oscillationscharacterizedby well-defined frequencyof thejet.sufficiently large Reynolds numbers, this system exhibits self-sustainedconfined in a rectangular cavity. For certain geometrical configurations and forWe have presented an experimental and numerical study of a turbulent plane jet


162 Advmces irl Fluid Mechmks W

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[2] Shakouchi, T., Suematsu, Y. & Ito, T., A study on oscillatory jet in a cavity,

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[3] Shakouchi, T., A new fluidic oscillator, flowmeter, without control port and

Paris VI, Juillet 1994.[4] Maurel, A., InstabilitC d'un jet confinC, Thkse de Doctorat de I'Universite

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[5] Villermaux, E. & Hopfinger, E.J., Self-sustained oscillations of a confined jet

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closures, Phys. Fluids,vol. 30, N"3, p. 722-731, March 1987.[6] Schiestel, R., Multiple-time-scale modelling of turbulent flows in one point

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[S] Launder, B. E. Lk Spalding, D. B. , the numerical computation of turbulent

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equations model of turbulence, Int. J. Heat Mass Transfer, vol. 15,[lo] Jones, W.P 2% Launder, B. E., The prediction of laminarization with a two-

Kyoto, Vol. 9, p. 10.4.1, 1993.stationary and vibrating square cylinder, gfhSymp. on Turbulent Shear Flows,

[13] Kato, M. & Launder, B. E., The modelling of turbulent flow around a

engineering applications, A I A A J., Vol. 32, no8,pp. 1598-1605,1994.[14] Menter, F. R., Two-equation eddy-viscosity turbulence models for

Bull. ofJ.S.ME., Vol. 25, N"206, pp. 1258-1265 ,August 1982.

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