- Email: [email protected]

Study of the oscillatory regime of a turbulent plane jet impinging in a rectangular cavity Amina Mataoui a, Roland Schiestel

b,*

, Abdelaziz Salem

a

a

b

Laboratoire de M ecanique des Fluides, Facult ede Physique, Univ. USTHB, B.P. 32, Bab Ezzouar, 16111 Al Alia, Alger, Algeria Institut de Recherche sur les Ph enom enes Hors Equilibre (IRPHE), UMR 6594 CNRS/Universit es d’Aix-Marseille I & II, 38 rue Fr ed eric Joliot Curie, 13451 Marseille Cedex 20, France Received 12 December 2000; received in revised form 2 May 2002; accepted 30 May 2002

Abstract The present work considers the general case of a turbulent plane jet ﬂowing into a rectangular cavity. The study is relevant to a wide range of practical applications including forced convection, renewal of ﬂuid inside a cavity and ﬂowmeters. The experimental study considers the eﬀect of the conﬁnement on the jet characteristics. All the measurements are made using hot wire anemometry and complemented by visualisation of the various observed ﬂow patterns. Numerical modelling has been carried out using two statistical models of the turbulence: the standard k–e model and a two-scale energy–ﬂux model. The key feature of the multiple scale energy–ﬂux model is the splitting of the turbulence spectrum devised for nonequilibrium turbulence modelling. Three ﬂow regimes are observed depending on the location of the jet exit inside the cavity: oscillatory, transitional and steady. The oscillatory ﬂow characteristics have been analysed from the structural and parametric points of view and the underlying mechanisms are discussed. The onepoint closure models proved to be able to reproduce the measured unsteady behaviour of the ﬂow and some improvements are obtained using the multiple scale concept. Ó 2002 Elsevier Science Inc. All rights reserved. Keywords: Turbulence; Turbulence modelling; Cavity ﬂow; Oscillatory ﬂow

*

Corresponding author. Address: IRPHE, UMR 6594 CNRS/Universite´s dÕAix-Marseille I & II, 49 rue Fre´de´ric Joliot Curie, BP 146, 13384 Marseille Cedex 13, France. Tel.: +33-4-9613-9765; fax: +33-4-9613-9709. E-mail address: [email protected] (R. Schiestel). 0307-904X/02/$ - see front matter Ó 2002 Elsevier Science Inc. All rights reserved. PII: S 0 3 0 7 - 9 0 4 X ( 0 2 ) 0 0 0 5 0 - 1

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1. Introduction In this study we are interested in the interaction of a turbulent plane jet issuing into a rectangular cavity. It is known that when the exit of the plane jet is symmetrically located in the midplane of the cavity, oscillations may occur under certain conditions [11,21,22]. The development of the oscillatory motion is due to the fact that the conﬁguration of a jet ﬂowing between two symmetrical eddies is unstable [21]. When the conﬁguration is slightly asymmetrical, the same phenomenon is observed. Moreover, this small asymmetry can contribute to the initiation of the oscillatory motion. However, for large asymmetries, when the jet nozzle exit is located near the lateral wall of the cavity, the oscillations are damped and a stable regime occurs. The unsteadiness of the hydrodynamic ﬂow ﬁeld is a self-induced phenomenon not enforced by any external force or by the boundary conditions. The unsteadiness of the ﬂow is driven by the ﬂapping motion of the jet between the walls of the cavity, maintained by the Coanda eﬀect. This study is related to various practical problems in industrial applications, such as ﬂuid renewal inside a cavity, ventilation of tunnels, air conditioning inside a closed hall in buildings and cooling or heating by forced convection. Other applications are to the mixing of ﬂuids in an enclosure and enhanced combustion by jets in boilers. The oscillation of a jet issuing into a suddenly enlarged ﬂow passage has been used to devise a new ﬂuidic oscillator [19] and ﬂowmeter without control port. The study of the interaction of the jet with the cavity is closely related to the Coanda eﬀect that has been studied by many researchers. In particular, Bourque and Newman [3], made an analytical study of the jet reattachment on a ﬂat plate that exhibits characteristic similarity groupings of the jet impingement. An analytical and experimental study of the ﬂow produced by a twodimensional jet issuing parallel to a ﬂat plate was performed by Sawyer [15], with detailed measurements of the pressure and velocity ﬁelds along with reattachment distance and reversed ﬂow momentum. A detailed experimental study of a plane jet issuing near a ﬂat plate was made by El-Taher [5], which produced measurements for wall static pressure and mean velocity together with turbulence normal and shear stresses. In this ﬂow, denoted ‘‘ventilated jet’’, an opening exists between the jet and the wall through which part of the ﬂow is entrained preventing the development of a recirculation zone. The author distinguished four regions in this ﬂow: initial merging zone, positive gradient zone, negative gradient zone and fully developed zone. These last three zones are found to exhibit similarity. Ayukawa and Shakouchi [2] and later Shakouchi and Kusuhara [20], studied the oscillation of a jet attaching to an oﬀset parallel plate, considering in particular the frequency of the pressure ﬂuctuations and the frequency of the ﬂapping of the jet that exhibits phase lags. The second paper considers the switching phenomenon of the jet produced by the eﬀect of traversing a second opposite wall. The attachment and detachment of the jet show hysteresis behaviour when varying the oﬀset distance. Shakouchi et al. [21], considered the oscillatory regime of a plane jet issuing between two inclined plates and showed the appearance of two eddies induced at each side of the jet which play an important role in the oscillatory mechanism observed on the periodic ﬂow patterns. The paper also presents the conditions under which the oscillations can occur and the relation between frequency and geometrical parameters and Reynolds number. Ogab [11], developed an experimental study of a turbulent plane jet issuing into a rectangular cavity, the nozzle of the jet being located in the midplane of the cavity. In this situation, oscillatory regimes were observed and the frequency was determined as a function of the

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impinging distance from the bottom wall and Reynolds number. The inﬂuence of the lateral width of the jet was also considered, and it was shown that in this case a steady regime is recovered. The related topic of the unsteady ﬂow past a rectangular cavity has been investigated both numerically and experimentally by Pereira and Sousa [13]. They showed the occurrence of large amplitude organised oscillations involving coupling between the shear layer and the recirculating ﬂowﬁeld inside the cavity. An experimental and numerical study of self-sustained oscillations of a conﬁned jet in a rectangular cavity, extended downstream by a rectangular channel, was developed by Maurel [10]. In particular, the numerical study provided a description of the phenomenon at the onset of the instability. Also, a linear variation of the frequency with Reynolds number was obtained in the laminar regime. A comprehensive synthesis of the problem of oscillations in impinging free shear layers is given in [14]. Conﬁned jet oscillations have also been observed in thin slab continuous casting problems with liquid metals in the presence of a cross ﬂow [6,7,9]. In this case, pressure diﬀerentials are also a driving force for the oscillating mechanism. The interaction of a jet ﬂowing into a rectangular cavity can lead to diﬀerent ﬂow regimes: steady ﬂow, periodic oscillatory ﬂow or transitional ﬂow in which the unsteadiness is more chaotic. Depending on the location of the jet exit nozzle into the cavity, one of these regimes then develops inside the whole cavity. In the present paper, we are mainly interested in the unsteady periodic regimes. The unsteadiness of the ﬂow develops with no external forcing (it is a selfsustained unsteadiness). From the numerical modelling point of view, one of the questions we try to answer in the present paper is the following: is a one point turbulence closure model able to produce the natural unsteadiness in this kind of ﬂow? In particular, in the case of two-equations models like the k–e model which is based on the turbulent viscosity hypothesis, it is important to ﬁnd out if the viscous character of the model impedes or not the development of self-sustained oscillatory regimes. The unsteadiness of the ﬂow involves important time variations of the turbulence ﬁeld and the turbulence production. Considering that usual one point closures are implicitly based on an underlying equilibrium hypothesis, their applicability may be questionable in strongly varying situations. The approach explored here is the introduction of multiple time scales that mimic the non-equilibrium eﬀects. The model predictions for both single-scale and multiple scale models are compared with experimental data and the results are discussed.

2. Experimental approach 2.1. Flow conﬁguration The experiment is composed of several parts: a wind tunnel which feeds the whole system with air: a nozzle that produces a uniform plane jet with low turbulence level: the rectangular cavity. The small wind tunnel that provides the air supply consists of a variable blower that allows the volume ﬂow rate to be adjusted, followed downstream by a grid and a convergent duct with an air ﬁlter. The air supplied by the wind tunnel is injected into a rectangular duct made of perspex which is composed of several parts (from upstream to downstream): a plane splitter, of expansion ratio

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Fig. 1. Experimental setup.

equal to 5, in which two guiding walls have been disposed in order to obtain an even velocity distribution and to reduce the drag: a settling chamber in which are disposed two grids separated by a suﬃcient gap such that the wake produced by the ﬁrst grid is suﬃciently damped before reaching the second [4]: a honeycomb, with an hexagonal mesh (4 cm depth and side lengths equal to 3 mm), followed by a ﬁne mesh gauze used to further reduce the turbulence and ﬁlter the ﬂow: a small convergent duct joining the settling chamber to a rectangular channel: the nozzle of the plane jet exit. The nozzle was designed to produce two-dimensional ﬂow with relatively low turbulence levels (less than 0.3% at the exit). The cavity shown in Fig. 1 (dimensions given in millimeter) is made of perspex and its shape is a parallelipiped with a square cross section. The nozzle can be moved horizontally and vertically inside the cavity. On the upper wall of the cavity, a longitudinal slot with a sliding ruler allows the introduction of probe support through a small opening located in the centre of the ruler. The vertical displacement of the probe is made by a traversing mechanism across the opening of the ruler. 2.2. Instrumentation and measurements The velocity is measured using constant temperature hot wire anemometry. The measuring device includes a DISA 56C00 multichannel component and signal analysis apparatus. The signal is numerically linearised and ﬁltered through a signal conditioner in order to remove the high

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frequency turbulent part of the signal. For all the measurements, a single hot wire probe (55P11) has been used. This probe is made of a 5 lm diameter platinum-plated tungsten wire of 1 mm length. The probe has been calibrated using the DISA calibration system before and after each session of measurement. A memory oscilloscope is used for sampling and recording the signal during a small time interval. 2.3. Flow visualisation A smoke generator (CFT Taylor 3020) is used to produce white smoke composed of very small droplets of vegetable oil and carbon dioxide. The smoke is injected at the inlet of the channel in order to minimise the possible perturbations of the main ﬂow. A projector located at the bottom end of the cavity produces the lighting of the ﬂow. The light enters through a vertical slot in the centre plane of the end wall. Motion photographs have been taken using a camera with a shutter frequency of 2 pictures per second.

3. Numerical modelling The numerical study has been carried out using two statistical models of turbulence: the singlescale k–e model and the two-scale energy–ﬂux model. The ﬂuid will be supposed to be Newtonian with constant density. 3.1. Single-scale k–e model The two-equation k–e model [8] is used in its standard high Reynolds number form. It is based on the Prandtl–Kolmogorov concept of turbulent viscosity and the Reynolds stress tensor is given by a constitutive relation that assumes a pointwise proportionality between the deviatoric turbulent stresses and the mean strain tensor. Thus, 2 oUi oUj ui uj ¼ k dij mt þ ð1Þ 3 oxj oxi where k 3=2 ð2Þ e The closure of the turbulence ﬁeld is then obtained by solving the following modelled equations for k and e: h i mt m ok o þ rkt oxm rkL dk oUi oUm oUi ¼ þ mt e ð3Þ þ dt oxm oxi oxm oxm h i mt m oe de o reL þ ret oxm e oUi oUm oUi e2 þ þ Ce1 mt Ce2 ¼ ð4Þ dt k oxm oxi oxm k oxm mt ¼ Cl

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where the numerical constants are given by: Cl ¼ 0:09;

Ce1 ¼ 1:44;

Ce2 ¼ 1:92;

rkL ¼ 1:0;

rkt ¼ 1:0;

reL ¼ 1:0;

ret ¼ 1:3

ð5Þ

and the molecular diﬀusion term being indeed negligible in the present application. The value of ret is compatible with the value of the slope of velocity proﬁle in a logarithmic boundary layer with a Karman constant j ¼ 0:41 from the well known relation ret ¼ j2 =Cl1=2 ðCe2 Ce1 Þ. The use of the high Reynolds number form of the model can be questionable when the ﬂow velocities are relatively low. In the present case, the jet Reynolds number (based on the nozzle height and jet exit mean velocity) varies in the range 1300–4000; the air jet is then expands inside the cavity which oﬀers a much larger space. It has been veriﬁed a posteriori that the turbulence Reynolds number Ret ¼ mt =m varies between 50 and 300 inside the cavity for all the cases under consideration. It can be inferred that the molecular viscosity eﬀects are not very inﬂuential and the high Reynolds number model has been retained. This model will be referred to as a single-scale model because a single length scale l ¼ k 3=2 =e is used in all the closure hypotheses of the model. The underlying physical signiﬁcance of such a practice is an assumption of spectral self-similarity that is imbedded in usual one-point closures. 3.2. Multiple scales energy–ﬂux model The multiple scales turbulence model investigated in the present work is based on a simpliﬁed split-spectrum scheme [16,17] obtained by subdividing the turbulence energy spectrum into three zones (Fig. 2). The ﬁrst zone at small and medium wavenumbers corresponds to the spectral range in which production is appreciable, the second zone is the transfer zone in which the energy cascade is the main acting mechanism, and a third zone at large wavenumbers corresponds to the viscous dissipation region. In this latter region the partial kinetic energy will be considered as entirely negligible so that the energy ﬂux eT out of the transfer zone can be considered as equal to the viscous dissipation e. The partial kinetic energies and the spectral ﬂuxes are determined from the following transport equations:

Fig. 2. Sketch of the split-spectrum scheme.

A. Mataoui et al. / Appl. Math. Modelling 27 (2003) 89–114

dkP ¼ dt

h i okP o m þ rmkPt ox m oxm

dkT o ¼ dt

h

mþ

mt rkT

oxm

okT oxm

95

oUi oUm oUi þ þ mt eP oxm oxi oxm

ð6Þ

þ eP eT

ð7Þ

i

e ¼ eT

ð8Þ

The total kinetic energy of turbulence is recovered by k ¼ kP þ kT . h i oeP mt o m þ reP oxm deP eP oUi oUm oUi e2 þ C1P mt C2P P þ ¼ dt kP oxm oxi oxm kP oxm h i oeT mt o m þ reT oxm deT eP eT e2 þ C1T C2T T ¼ dt kT kT oxm

ð9Þ

ð10Þ

where reT , reP , C1P , C2P , C1T and C2T are numerical constants or may be function of the spectral distribution of energy through the shape parameters n ¼ kP =kT and h ¼ eP =eT . In the present study, constant values are given to the numerical coeﬃcients of the model: C1P ¼ 1:65; 2

reP ¼ j

C2P ¼ 1:92;

=Cl1=2 ðC2P

C1P Þ;

C1T ¼ 1:75; reT ¼ j

2

C2T ¼ 1:80;

=Cl1=2 ðC2T

C1T Þ

ð11Þ

The value of C2P gives the correct exponent in the decay law of homogeneous grid turbulence. The diﬀusive coeﬃcients reP and reT are both compatible with the slope of the log law in a turbulent boundary layer. The constitutive relation (1) is now used in conjunction with the turbulent eddy viscosity: mt ¼ Cl k 2 =eP

ð12Þ

It has also been veriﬁed that this choice of numerical coeﬃcients permits a numerical prediction of turbulent steady channel ﬂow which is close to that from a standard k–e calculation. As this is a near equilibrium ﬂow, it is indeed desirable that the two approaches give similar results. In more general situations, however, the use of multiple scales enables partial account to be taken of non-equilibrium turbulence, as the time lags involved in the cascade process are introduced in the model equations. Non-equilibrium turbulence is understood here as a situation in which the production is strongly varying in time or in space. The shape parameter n ¼ kP =kT allows the model to distinguish a spectral energy distribution having a bump in the production range from an energy distribution having a bump in the smaller scales range. The ability of the multiple scale models to mimic the cascade process with non-standard spectral distributions is achieved by solving the split-spectrum turbulence kinetic transport equations for each spectral zone. The splitting between the production and the transfer regions in the energy spectrum allows the energy ﬂux eP out of the production zone to be directly inﬂuenced by mean strain while the energy ﬂux eT out of the transfer zone (which equals the true dissipation) is no longer linked to the mean

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strain and can evolve by itself. The dissipation is then only weakly coupled with the mean ﬂow. In the case of an equilibrium ﬂow eP ¼ eT ¼ e and the behaviour of a single-scale model is recovered. Eq. (12) can be written in an equivalent form mt ¼ Cl ðeT =eP Þk 2 =e which shows that compared with the standard formula in the single-scale k–e model, this assumption take into account some non-equilibrium eﬀects via the ratio h1 ¼ eT =eP . The turbulent viscosity is then increased in the case of weak production with a low shape parameter n and is decreased under the opposite conditions. The high Reynolds number turbulence equations cannot be used to describe the near wall region of turbulent ﬂows since the no-slip condition involves an increase of viscous eﬀects and a damping of turbulence ﬂuctuations. In the present case, the ﬂow is not strongly inﬂuenced by the near-wall turbulence structure, so that an approximate treatment for the wall region will be suﬃcient. Indeed pressure eﬀects are found to be the driving force maintaining the unsteadiness of the conﬁned jet. So, in this study economical logarithmic wall functions are used, and the turbulence model applied to the fully turbulent internal region. The generalised form of the wall functions given by Spalding have been used, based on the extended deﬁnition of pseudo friction 1=2 velocity U ¼ sw =qc1=4 l kN , where kN is the value of turbulent kinetic energy at the ﬁrst grid point 1=2 near the wall. Accordingly, the dimensionless wall distance is given by y þ ¼ yc1=4 l kN =m and the þ þ þ logarithmic law of the wall reads U ¼ U =U ¼ ð1=jÞ log EyN for yN > 11:0 with j ¼ 0:41 and E ¼ 9:0. If yNþ < 11:0 at exceptional points then the relation U þ ¼ y þ is used. 3.3. Numerical implementation and boundary conditions The governing equations for mean values and turbulent quantities are transport equations with an elliptic spatial operator. They are solved using the ﬁnite volume discretization method and staggered meshes for velocity components in order to prevent spurious pressure modes developing. The well-known SIMPLE algorithm is used for pressure–velocity coupling [12]. The convection–diﬀusion scheme is an hybrid ﬁrst order scheme [12] behaving like central diﬀerencing when diﬀusion is dominant and like an upwind scheme when convection is dominant. As usual, the source terms in the turbulence equations are linearised to guarantee the stability of the procedure and the solution algorithm is based on internal iterations within each time step. The numerical method was tested ﬁrst on a steady (plane channel) ﬂow in order to obtain grid independence to a good approximation. The grid size required was (NX ¼ 100, NY ¼ 80). Grid tests conﬁrmed that, for example, a grid size of (NX ¼ 120, NY ¼ 100) produced almost identical results, with diﬀerences in computed values appearing in the third signiﬁcant digit. In these tests however, the location of the ﬁrst grid point near the wall was not modiﬁed when changing the total number of grid points. The inﬂuence of the size of the time step was then tested for an unsteady case (the present jet in a cavity conﬁguration) in order to determine step values giving time variations and frequencies results that are independent of the time step. Considering that the frequencies of oscillation of the unsteady ﬂow are relatively low, a ﬁrst order time scheme has been retained, with small time steps used in order to obtain suﬃcient accuracy. For studying periodic regimes, 500 time steps are used within one period of time. For the boundary conditions at inlet, constant values are imposed for mean velocity and turbulence quantities:

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U ¼ U0 ;

V ¼ 0;

k ¼ 0:03U02

and e ¼ k 3=2 =kh0 ðk ¼ 0:1Þ

97

ð13Þ

At the exit plane, two cases have to be distinguished for a conserved quantity / (being V, k, or e). First, if the ﬂow is entering the cavity (U < 0), then / ¼ /ext where /ext is the value prevailing in the ambient environment. Second, if the ﬂow is leaving the cavity (U > 0), then o/=ox ¼ 0 is enforced. In each case, the pressure is imposed and the U component of velocity is recovered from the continuity equation. Near the wall, the extended logarithmic law of the wall is used as boundary condition as explained in the previous paragraph. It appeared, in the analysis of this ﬂow that the wall region do not seem to play a decisive role in the triggering of the oscillations, so the wall function method was used in spite of its limitations. Probably, low Reynolds number turbulence models would have been useful for reﬁning the description of the wall region but we do not expect important inﬂuences on the oscillation mechanism which is rather linked to pressure eﬀects (as explained in next paragraphs).

4. Results and discussion 4.1. Diﬀerent ﬂow regimes and parametric study The jet–cavity interaction has been studied for many diﬀerent locations of the jet exit inside the rectangular cavity in order to achieve a complete sweep of the cavity. The jet exit locations were chosen based on a regular rectangular mesh of 1 cm size both in the horizontal and the vertical directions. Depending on the location of the jet exit, diﬀerent ﬂow regimes are observed in the whole cavity: steady ﬂow, periodic oscillatory ﬂow or transitional ﬂow in which the unsteadiness is more chaotic. The kind of interaction is determined practically by the temporal analysis of the mean velocity signal at diﬀerent ﬁxed locations. The experimental study allowed the outline of these three zones to be deﬁned. Each zone corresponds to a diﬀerent regime of interaction (Fig. 3). Such zones have been also found in a conﬁned jet ﬂow with a diﬀerent geometry [19]. The contours of theses zones have also been veriﬁed to be in entire agreement with the numerical predictions. These zones remain valid when varying the Reynolds number within the range considered (1000 < Re < 5000 where Re is the jet Reynolds number at the jet exit). Here, we are mainly interested in the unsteady periodic regime.

Fig. 3. Outline of the three regimes of interaction.

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Fig. 4. Geometrical parameters in the jet–cavity interaction, Aðx1 ; h1 Þ is the probe location.

In this study, dimensionless quantities have been used for the geometrical and dynamical parameters (Fig. 4): X ¼ X1 =X0 ;

H ¼ H1 =H0 ;

h ¼ h1 =H0 ;

St ¼ fh0 =U0

x ¼ ðX1 x1 Þ=X1 ;

Re ¼ U0 h0 =m; ð14Þ

where U0 is the jet exit velocity, f the frequency of oscillations, m the kinematic viscosity and X0 , X1 , H0 , H1 , x1 , h0 , h1 are geometric characteristics given in Fig. 4. 4.2. Time analysis of the periodic regime In the oscillatory regime, the analysis of measured velocity signals shows a purely periodic phenomenon. The numerical calculations using the k–e model and the two-scales model both predict an oscillatory regime (Fig. 5). The measured ﬁltered velocity signal shows a periodic pattern with a main peak and several secondary peaks that are also clearly visible on the calculated signal at the same point in the cavity (X ¼ 0:8, H ¼ 0:5, Re ¼ 4500 at the location x ¼ 0:3, h ¼ 0:7). The experimental period in time is found to be T ¼ 1:83 s and the calculated values of the period for the same case are very similar, these being T ¼ 1:785 s for the single-scale model and T ¼ 1:875 s for the multiscale model. Also, it is apparent from Fig. 5, that the standard k–e model underestimates the amplitude of oscillation whereas the multiscale model produces the correct value. For the two models considered here, the periodic variations of various quantities such as velocity modulus (UM ), transverse velocity component (V), mean pressure (P) and tur-

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Fig. 5. Time signal of the measured and calculated mean velocity modulus for X ¼ 0:8, H ¼ 0:5 and Re ¼ 4500, at the location x ¼ 0:3, h ¼ 0:7: (a) experiment, (b) single-scale model and (c) two-scales model.

bulent kinetic energy (k) are shown in Figs. 6 and 7. For a point located near the lateral wall (Figs. 6a and 7a) the oscillation curve, for all the quantities, exhibits a distinct major peak for each period. However, for a point located on the centreline in front of the jet exit (Figs. 6b and 7b), the shape of the instantaneous signals is totally diﬀerent, in particular for UM , P and k, due to the nearly symmetrical ﬂapping motion of the jet in the transverse plane, the ﬁrst harmonic becomes

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Fig. 6. Computed time signal of velocity modulus, transverse component of velocity, Pressure and kinetic energy of turbulence for the single-scale model at: (a) x ¼ 0:3, h ¼ 0:086, X ¼ 0:8, H ¼ 0:425, Re ¼ 4000 and (b) x ¼ 0:3, h ¼ 0:490, X ¼ 0:8, H ¼ 0:425, Re ¼ 4000.

dominant and the frequency of the signal looks to be around double the true frequency of the phenomenon. This is indeed visible on the transverse velocity signal, which changes sign at each

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Fig. 7. Computed time signal of velocity modulus, transverse component of velocity, pressure and kinetic energy of turbulence for the two-scales model at: (a) x ¼ 0:3, h ¼ 0:086, X ¼ 0:8, H ¼ 0:425, Re ¼ 4000 and (b) x ¼ 0:3, h ¼ 0:490, X ¼ 0:8, H ¼ 0:425, Re ¼ 4000.

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Fig. 8. Fourier modes of the mean velocity time signal: (a) single-scale prediction for X ¼ 0:8, H ¼ 0:8, Re ¼ 4000 and (b) two-scales prediction for X ¼ 0:8, H ¼ 0:425, Re ¼ 4000.

cross over. Fig. 8a and b show the Fourier transforms of these velocity time signals at the same locations. The fundamental frequency is clearly determined by the ﬁrst peak in the Fourier modes A=U0 distribution. Accordingly, for the near centreline point (h ¼ 0:49) the fundamental mode becomes very weak and the ﬁrst harmonic seems to be dominant due to the up and down ﬂapping of the jet. The two models give similar results, but the multiscale model produces more energetic harmonics. The presence of these harmonics is also noticeable on the physical signals (Fig. 7), which exhibit sharper peaks. The global amplitude of the mean velocity signal in physical space is shown in Fig. 9 for several cross sections in the cavity. The amplitude decreases downstream and decays smoothly to zero at the bottom wall. The amplitude has a local maximum on the centreline of the jet and secondary maxima near the lateral walls due to the alternative occurrence of forward and backward ﬂow near the wall. The multiscale model gives higher values of the amplitude than the single-scale model and thus reaches a better agreement with the experiment. In a diﬀerent geometrical conﬁguration, Ayukawa and Shakouchi [2] have shown the existence of a phase shift between the pressure and the mean velocity in the oscillating jet attaching to an oﬀset parallel plate. Although the driving mechanism seems to be somewhat diﬀerent in this experiment, there are also some similar features involving the pressure–velocity coupling. This phase shift reveals the existence of a feedback mechanism with time lag eﬀects and suggests also that pressure eﬀects are important in the observed phenomenon. In the present case, the calculated phase shifts between the transverse component of mean velocity and the mean pressure are plotted

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103

Fig. 9. Amplitude distribution of the oscillating motion along several cross sections of the cavity (X, H and Re, see Fig. 8).

in Fig. 10 for the two prediction models. Comparing the time evolution of the two signals suggests that the pressure oscillation is in advance phase compared to the velocity signal. Generally speaking, the mechanisms of oscillation in conﬁned jets are linked to feedback eﬀects of perturbations but seem to have several origins depending on the particular geometry considered. In the case of a jet ﬂowing in an enlarged channel the perturbations are convected upstream by the recirculating ﬂow and ampliﬁed by the instability of the jet, while in the case of impinging jets pressure eﬀects are more important in the feedback mechanisms. 4.3. Inﬂuence of geometrical and physical parameters on the frequency of oscillation The main parameters of the problem, the Reynolds number and the location of the jet in the cavity have been varied in order to estimate their inﬂuence on the oscillating ﬂow and in particular on the frequency of oscillation. In Fig. 11a–c, the frequency is plotted against the distance of the jet exit to the bottom wall for several values of Reynolds number and several height locations (H) of the jet exit. In all cases the frequency is found to decrease moderately when the impingement distance (X) increases. The predictions from the two models are in good overall agreement with the experiment, the standard k–e model slightly overpredicts the measured values and the multiscale model gives lower values that are in better agreement with the experiment. Fig. 12a–c show the Strouhal number based on jet exit conditions (St ¼ fh0 =U0 ) plotted against the Reynolds number based on the same scaling (Re ¼ U0 h0 =m). The Strouhal number is found to remain practically constant over the range of Reynolds numbers investigated. The two models closely reproduce the experimental ﬁnding, with slightly better agreement for the two-scales model. This result also means that the frequency varies linearly with the Reynolds number. This kind of behaviour has been obtained by many other authors in similar ﬂow geometries [10,11,19,21]. Another geometrical parameter is the ﬁnite length of the cavity, and additional experiments and calculations were made for a longer cavity (twice as long as in

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Fig. 10. Phase shifts between the pressure and the transverse velocity signals at x ¼ 0:3, h ¼ 0:490, X ¼ 0:8, H ¼ 0:425, Re ¼ 4000: (a) single-scale model and (b) two-scales model.

the case above). It was found that the changes in the ﬂow structure are not important. The eddies remain comparable in size and a low velocity cushion then ﬁlls the closed end of the cavity. The frequencies also remain almost unchanged. These numerical results are not included in the present paper. 4.4. Model analysis of the oscillatory motion The NLDS model proposed by Villermaux and Hopﬁnger [23] for the study of self-sustained oscillations in conﬁned jets is based on memory eﬀects associated with a recirculation loop. But, as the authors point out in their paper, the mechanism they invoke diﬀers from the self-sustained oscillations that arise from a global pressure feedback loop which is characteristic of impinging jets. It appears that in the present case, the driving mechanism is indeed a pressure feedback. According to this point of view, a very crude physical model of the mechanisms involved is

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105

Fig. 11. Frequency of oscillation f versus impingement distance X at: (a) H ¼ 0:325, (b) H ¼ 0:425 and (c) H ¼ 0:500 ((d) experiment, ( ) single-scale model, (.) two-scales model).

proposed. If the transverse velocity is supposed to be mainly governed by the pressure eﬀects, then the y-component of the momentum equation reduces to:

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Fig. 12. Strouhal number St versus Reynolds number Re at: (a) H ¼ 0:325, (b) H ¼ 0:425 and (c) H ¼ 0:500 ((d) experiment, ( ) single-scale model, (.) two-scales model).

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dV oðP þ qv2 Þ dt oy

107

ð15Þ

If g denotes the shift of the jet axis from the symmetry line due to the deﬂection of the jet at the mean location of the impingement L (Fig. 13), then: dg ¼V dt

and

d2 g oðP þ qv2 Þ dt2 oy

ð16Þ

Supposing now that the pressure is created by kinetic energy of the normal component at the jet impact: P¼

ðua sin aÞ2 2

and

oðP þ qv2 Þ u2a jgjg ¼ 2 oy 2L H0

ð17Þ

If jgj is approximated by its mean value H0 =4, one gets: oðP þ qv2 Þ u2 ¼ a2 g oy 8L and ﬁnally the equation of motion for the ﬂapping of the jet d2 g u2a ¼ g dt2 8L2 allows the frequency of oscillation to be determined: f ¼

x 1 ua pﬃﬃﬃ ¼ 2p 2p L 8

ð18Þ

Fig. 13. Sketch of the self-sustained oscillation mechanism.

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Table 1 f (Hz) H ¼ 0:5 Experiment Single-scale model Two-scales model Analytical method

X ¼ 0:5

X ¼ 0:6

X ¼ 0:7

X ¼ 0:8

0.740 0.760 0.720 0.748

0.700 0.720 0.700 0.682

0.670 0.700 0.650 0.622

0.540 0.680 0.620 0.599

An estimate of the axial velocity decay due to the jet expansion is necessary to approximate the impact velocity ua . To do this, a formula for free jets [1] is used: ,sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 0:11L ð19Þ ua ¼ 1:2U0 0:41 þ h0 =2 It is clear that the free jet theory is not applicable to conﬁned jets but in the present case, considering that the rate of spread of the jet is not dramatically diﬀerent from free jet expansion, Eq. (19) limited to the calculation of axial velocity gives a rough estimate. The values of frequency calculated using Eqs. (18) and (19) are given in Table 1 compared with the experimental and the predicted values (for the single-scale and multiple scale models) for Re ¼ 4000. In Eqs. (18) and (19) the value of L was estimated from visualisations. There is good agreement between the measured, computed and theoretical values given in Table 1, considering the very crude assumptions underlying the derivation of the model. The fact that the nozzle protrudes into the cavity with an open end behind it allows the development of oscillations by pressure eﬀects as explained in the previous simpliﬁed model. The open end of the cavity is in fact divided into two openings above and below the plane channel feeding the jet. The intermediate channel walls practically prevent any interaction of pressure waves between them. 4.5. Mean structure of the unsteady ﬂow Fig. 14 shows photographs of the jet using white smoke seeds. Eight stages during one period of time are presented. The Reynolds number chosen is Re ¼ 4000 and the location of the jet exit given is X ¼ 0:8 and H ¼ 0:425. The pictures illustrate the periodic up and down ﬂapping of the oscillating jet. These eight instants of time have been selected uniformly within a period (at intervals dt=T ¼ 0:125). Reverse ﬂow occurs on the opposite side of the impingement of the jet, which is strongly diverted. The streamlines contours computed using the two-scale model are plotted in Fig. 15 and the corresponding pressure contours in Fig. 16 for the same instants of time. The contour plots show that the interaction of the jet into the cavity produces two deterministic eddies on each side of the jet with opposite swirl. Once the jet axis deﬂects from its horizontal position, the eddies begin to move while changing in size (Fig. 15). The largest eddy from the unattached side drifts upstream and then produces a pressure defect (Fig. 16) as it approaches the

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109

Fig. 14. Visualisation of a period of oscillationof the jet seeded with an inert white smoke Re ¼ 4000, X ¼ 0:8, H ¼ 0:425.

lateral exit; simultaneously the ﬂuid is sucked inside the cavity at the opposite side. This new ﬂuid supply entering the cavity increases and the corresponding eddy drifts downstream and forces the jet to be deﬂected and to attach to the opposite side, where a pressure maximum is created. The pressure within the two lateral eddies periodically increases and decreases and the attachment of the jet switches from one sidewall to the other. The same phenomenon is repeated periodically. These mechanisms of switching motion are in agreement with the observations of Shakouchi [18] for similar ﬂow conditions. For illustration purposes, the contour lines of turbulent kinetic energy are given in Fig. 17. The ﬂapping phenomenon is also clearly visible, with the two lateral eddies on either sides of the jet. High values of turbulent kinetic energy are found on the impinging side of the jet whereas low values of turbulent kinetic energy appear on the opposite side. The bottom region of the cavity is characterised by a somewhat weaker turbulence activity with relatively lower values of mean velocity and turbulence intensity. The contours of the spectral shape parameter n shown in Fig. 18 indicate the non-equilibrium state of turbulence represented by the two-scale model. High values

110

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Fig. 15. Calculated instantaneous streamline contours within one period of time Re ¼ 4000, X ¼ 0:8, H ¼ 0:425, coutours intervals: DW ¼ 0:07 m2 /s.

of the n parameter appear in the vicinity of the jet exit where maximum shear production occurs, whereas low values prevail in the bottom part of the cavity. The order of magnitude of the ratio of the characteristic frequencies of the turbulence fT and of the oscillatory motion fJ has been found to be of order fT =fJ 10. showing that the jet oscillation is relatively low compared to the characteristic time scales of turbulence, therefore non-equilibrium eﬀects are expected to have

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111

Fig. 16. Calculated instantaneous pressure contours within one period of time Re ¼ 4000, X ¼ 0:8, H ¼ 0:425, contours intervals: DðP =qÞ ¼ 0:2 m2 /s2 , (––) positive values, (– – –) negative values.

relatively mild eﬀects. These eﬀects are however perceptible through the phase lag between the diﬀerent quantities, as mentioned previously.

5. Concluding remarks The oscillatory phenomenon of a turbulent plane jet issuing into a rectangular cavity has been studied experimentally and by numerical modelling. The conditions that cause the oscillating

112

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Fig. 17. Calculated instantaneous turbulent kinetic energy contours within one period of time Re ¼ 4000, X ¼ 0:8, H ¼ 0:425, contours intervals: Dk ¼ 0:02 m2 /s2 .

behaviour and the mechanisms involved have been discussed. The jet–cavity interaction is characterised by the existence of diﬀerent ﬂow regimes (steady regime, oscillatory regime, transitional regime) depending on the location of the jet exit inside the cavity. The result is practically independent of Reynolds number (provided that the Reynolds number is large). In the periodic oscillatory regime, which is studied in the present paper, the fundamental frequency has been found to increase linearly with Reynolds number and also to decrease moderately when the distance of the jet to the bottom plate increases. The underlying mechanisms of oscillation have been suggested by a simple model, which is based on a coupling between velocity and pressure with feedback. This crude scheme helps elucidate the source of the periodic phenomenon and the frequency derived from the model is in good agreement with both the experimental ﬁndings and

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113

Fig. 18. Calculated contours of spectral shape parameter X ¼ 0:8, H ¼ 0:425, contours intervals Dn ¼ 0:4.

the turbulence modelling predictions. Relatively simple one point turbulence closures proved to be able to reproduce the unsteady behaviour of the ﬂow without any artiﬁcial triggering of the oscillations. The use of a multiple scale turbulence model was shown to improve the numerical predictions compared to the standard k–e model, as the multiple scale model embodies lag eﬀect mechanisms that occur in the non-equilibrium turbulence produced in unsteady ﬂows.

References [1] G.N. Abramovich, The Theory of Turbulent Jets, The MIT Press, Cambridge, MA, 1963. [2] K. Ayukawa, T. Shakouchi, Analysis of a jet attaching to an oﬀset parallel plate (oscillation of a jet), Bull. JSME 19 (130) (1976) 395–401.

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[3] C. Bourque, B.G. Newman, Reattachment of a two dimensional incompressible jet to an adjacent ﬂat plate, Aeronaut. Quart. XI (1960) 201–232. [4] P. Bradshaw, An Introduction to Turbulence and its Measurements, Pergamon Press, Oxford, NY, 1971. [5] R.M. El-Taher, Similarity in ventilated wall jets, AIAA J. 20 (2) (1982) 161–166. [6] B.M. Gebert, M.R. Davidson, M.J. Rudman, Computed oscillations of a conﬁned submerged liquid jet, Appl. Math. Modell. 22 (11) (1998) 843–850. [7] T.A., Honeyands, N.A. Molloy, Oscillations of submerged jets conﬁned in a narrow deep rectangular cavity, in: R.W. Bilger (Ed.), Proceedings of 12th Australasian Fluid Mechanics Conference, Sydney, 10–15 December 1995, pp. 493–496. [8] W.P. Jones, B.E. Launder, The prediction of laminarization with a two-equations model of turbulence, Int. J. Heat Mass Transfer 15 (1972) 301–313. [9] N.J. Lawson, M.R. Davidson, Crossﬂow characteristics of an oscillating jet in a thin slab casting mould, J. Fluids Eng. 121 (1999) 588–595. [10] A. Maurel, Instabilite dÕun jet conﬁne, These de Doctorat de lÕUniversite Paris VI, Juillet 1994. [11] A. Ogab, Contribution a lÕetude de lÕevolution dÕun jet plan turbulent dans une cavite de section rectangulaire, These de Magistere, Mecanique des Fluides, USTHB Alger, September 1985. [12] S.V. Patankar, Numerical heat transfer and ﬂuid ﬂow, Series in Computational Methods in Mechanics and Thermal Sciences, Hemisphere Publishing Corp. & Mc Graw Hill, 1980. [13] J.C.F. Pereira, J.M.M. Sousa, Experimental and numerical investigation of ﬂow oscillations in a rectangular cavity, Trans. ASME J. Fluids Eng. 117 (1995) 68–74. [14] D. Rockwell, E. Naudascher, Self-sustained oscillations of impinging free shear layers, Ann. Rev. Fluid Mech. 11 (1979) 67–94. [15] R.A. Sawyer, The ﬂow due to two-dimensional jet issuing parallel to a ﬂat plate, J. Fluid Mech. 9 (4) (1960) 543– 560. [16] R. Schiestel, Multiple-time-scale modeling of turbulent ﬂows in one point closures, Phys. Fluids 30 (3) (1987) 722– 731. [17] R. Schiestel, Les Ecoulements Turbulents, Modelisation et Simulation, 2nd ed., Hermes, Paris, 1998. [18] T. Shakouchi, An experimental study on the cavity type ﬂuidic oscillator, Research Reports of the Faculty of Engineering, vol. 6, Mie University, Japan, December 1981, pp. 1–13. [19] T. Shakouchi, A new ﬂuidic oscillator, ﬂowmeter, without control port and feedback loop, J. Dyn. Syst. Measur. Control 111 (1989) 535–539. [20] T. Shakouchi, S. Kusuhara, Analysis of a jet attaching to an oﬀset parallel plate (inﬂuence of an opposite wall), Bull. JSME 25 (203) (1982) 766–773. [21] T. Shakouchi, S. Kusuhara, J. Yamaguchi, Oscillatory phenomena of an attached jet, Bull. JSME 29 (250) (1986) 1117–1123. [22] T. Shakouchi, Y. Suematsu, T. Ito, A study on oscillatory jet in a cavity, Bull. JSME 25 (206) (1982) 1258–1265. [23] E. Villermaux, E.J. Hopﬁnger, Self-sustained oscillations of a conﬁned jet: a case study for the non-linear delayed saturation model, Physica D 72 (1994) 230–243.

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