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Numerical prediction of unsteady compressible turbulent coaxial jets
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Computers & Fluids Vol. 27, No. 2, pp. 239±254, 1998 # 1998 Elsevier Science Ltd. All rights reserved Printed in Great Britain 0045-7930/98 $19.00 + 0.00 S0045-7930(97)00023-6
NUMERICAL PREDICTION OF UNSTEADY COMPRESSIBLE TURBULENT COAXIAL JETS PHILIPPE REYNIER1 and HIEU HA MINH1 Institut de MeÂcanique des Fluides de Toulouse, Avenue du Pr. Camille Soula, 31400 Toulouse, France
1
(Received 5 April 1996; revised 4 April 1997) AbstractÐCompressible and turbulent air±air coaxial jets with a strong velocity ratio are simulated by using a ®nite volume scheme and a two-equation turbulence model. For the calculations only the explicit part of the numerical method is used. This allows the simulation of the coherent structures evolving in the ¯ow. The results show a strong instability of the ¯ow near the inlet, this natural unsteadiness decays downstream and vanishes before x/D = 15. In the outer mixing layer the coherent structures evolve as in a simple round jet. Near the inlet, in the inner mixing layer, a recirculation region is found on the jet axis. The presence of this phenomenon in homogeneous coaxial jets is the consequence of the strong velocity ratio between the two streams and explains the lack of the inner potential core. These numerical results agree qualitatively with previous experimental investigations. # 1998 Elsevier Science Ltd. All rights reserved
1. INTRODUCTION
Homogeneous turbulent coaxial jets discharging into stationary air have been intensively investigated in the past (e.g. Ribeiro and Whitelaw [1]; Au and Ko [2]; Dahm et al. [3]). Ko and Au [4] found that the ¯ow was isolated into three separate regions: the initial region, the intermediate region, and the fully merged zone. The initial region is located between the nozzle and the end of the outer potential core, immediately downstream is the intermediate region which ends at the reattachment point and ®nally is found the fully merged zone. The ¯ow is characterized by the presence of two mixing layers: the outer mixing layer, contiguous to the initially stationary air and the inner mixing layer, con®ned between both central and annular jets. The main aspect of coaxial jets is the mixing mechanism of two originally adjacent ¯ows with dierent turbulence levels and anisotropic characters, in fact two dierent turbulence histories. The mixing begins in the initial region but takes place principally in the intermediate region. In an experimental study, Choi et al. [5] showed that the mixing mechanism is promoted by the presence of an adverse pressure gradient. In fact, an adverse pressure gradient increases the shear-layer growth in the initial region. About the mixing increase, So et al. [6] arrived at the same conclusion in an investigation on rotation eects in helium±air coaxial jets. In the same way, Habib and Whitelaw [7] got similar results for air±air con®ned coaxial jets with swirl. The swirling coaxial jets have been more extensively studied by Chen [8] and Dixson et al. [9]. They remarked that the mixing increases with the swirling number. Moreover, they observed a recirculation of the annular jet towards the centerline for swirling numbers superior to 0.5. A consequence of this last phenomenon is the lack of the inner potential core. So far, the presence of a recirculation zone was considered as a characteristic of the annular jets that have been experimentally studied by Ko and Chan [10]. Another feature of coaxial jets, resulting from the ®rst one, is the instability of such a ¯ow pattern. For equal velocities in the two jets, the wake instability prevails, whereas the shearlayer instability dominates for very large or very small velocity ratio (e.g. Wicker and Eaton [11]; Dahm et al. [3]). Recent studies, such as those of Ko and Au [12], Au and Ko [2] and Wicker and Eaton [11] report the presence of instabilities in the inner and outer mixing layers of air±air coaxial jets. On the other hand, the experimental investigation of Gladnick et al. [13] puts in evidence the presence of coherent structures in variable density coaxial jets. Wicker and Eaton [11] and Dahm et al. [3] studied the interaction between the coherent structures evolving in the two shear-layers. Their experimental results show a strong coupling between the instabil239
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ities present in the two mixing layers. The evolution of the inner shear-layer vortices depends on both outer shear-layer evolution and velocity ratio. Tang and Ko [14] studied experimentally the vortice evolution in both initial merging and intermediate regions. They observed azimuthal instabilities in the initial region and vortices as those behind axisymmetric bodies in the intermediate zone. Azimuthal instabilities originate from the bifurcation of initial region structures. Of these studies were carried out for incompressible ¯ows and moderate Reynolds numbers, the results of Moore [15] for a round jet prove that the large scale structures observed at lowReynolds numbers (104) are also observed at Reynolds numbers of about 106. In round jets, the experimental investigation of Morrison and McLaughlin [16] shows the presence of large scale structures for supersonic ¯ows at a Mach number of 2.5. However, the instability process in the initial region is strongly dependent on the initial conditions. DuraÄo [17] had shown already the in¯uence of initial turbulence intensity and velocity pro®le shape on ¯ow evolution. Then Dahm et al. [3] investigated the eects of both lip and initial boundary layer thicknesses. These last authors and Wicker and Eaton [18] observed a strong in¯uence of the initial conditions, especially absolute velocities and velocity ratio, on ¯ow instability. So et al. [6] studied the in¯uence of the turbulence level at the exit and showed a decrease of the inner potential core length and an increase of the mixing when the turbulence level at the inlet is large. This paper attempts to simulate the unsteadiness of coaxial jets with a high velocity ratio. For the calculations a classical turbulence model with the ®nite volume method of MacCormack [19] is used. This numerical investigation is concentrated on the initial region of the ¯ow to study the evolution of coherent structures and the coupling between the two mixing layers. In Section 2 a brief overview of the physical modelling of the problem is exposed. In Section 3 the numerical aspects with the main features of the numerical method used for the calculations are presented. In the same section a validation of the numerical code is shown. The numerical results for compressible coaxial jets are presented in Section 4 with an analysis of the simulated phenomena. In this section, qualitative comparisons with experiments of Ko and Au [12] and Okajima et al. [20] are given. Finally in Section 5 we draw the conclusion. 2. PHYSICAL MODELLING
2.1. Governing equations Two mixing layers are present in coaxial jets, they involve the presence of large scale structures in this ¯ow. Three-dimensional instabilities have been observed in the intermediate region by Tang and Ko [14] and Dahm et al. [3]. They correspond to the loss of organization in the ¯ow and lead to the fully-developed turbulence. In the present paper, the objective is to simulate organized unsteadiness assuming that coherent structures are two-dimensional. So, three-dimensional aspects are not taken account, the loss of organization of the two-dimensional structures is ensured across the diusion of the turbulence model. The problem deals with coaxial jet ¯ows so the equations are solved using axisymmetric coordinates. The equations that describe the physical problem are the unsteady Navier±Stokes equations written in axisymmetric coordinates and the state equation for a perfect gas. The equation system with Favre [21] mass-weighted averaging can be written as: Equation for the conservation of mass density r @ @ ~ 1@ r U r r rV~ 0 @t @x r @r
1
Equation for the conservation of momentum rUÄ @ ~ @ 2 1@ ~ ~ ~ ~ rU rUU rkP ÿ m mt Sxx rf rU~ V~ ÿ m mt S~rx g 0 @t @x 3 r @r
2
Numerical prediction of unsteady compressible turbulent coaxial jets
Equation for the conservation of momentum rVÄ o @ ~ @ n ~ ~ V UV ÿ m mt S~xr r r @t @x 1@ 2 ~ ~ ~ ~ VV P r k ÿ m mt Srr r r r @r 3 1 ~ 2 ~ k ÿ m mt S~yy P r r 3 The terms of the strain tensor are given by the following expressions: ! ~ 2 @U~ @V~ V~ @ V ÿ S~rr 2 r @r 3 @x @r S~xx
! @U~ 2 @U~ @V~ V~ 2 ÿ r @x 3 @x @r @U~ @V~ S~xr S~rx @r @x
S~yy
V~ 2 @U~ @V~ V~ 2 ÿ r 3 @x @r r
241
3
4
5
6 ! 7
Equation for the conservation of total energy EÄ 2 ~ ~ ~ E P r k ÿ m mt Sxx U~ r 3 m mt @ ~ ~ e~i ÿ m mt Sxr V ÿ g Pr Prt @x 1@ m m @ 2r E~ P k~ ÿ m mt S~rr V~ ÿ m mt S~xr U~ ÿ g r r t e~i 0 r @r 3 Pr Prt @r
@ ~ @ E r @t @x
8
State equation for a perfect gas g ÿ 1 P re~i
9
where r is the density, UÄ and VÄ the velocity components with respect to x and r directions (x and r are the longitudinal and radial coordinates, respectively), m the molecular viscosity, mt the turbulent viscosity, Pr the Prandtl number, Prt the turbulent Prandtl number, EÄ the total energy, eÄi the internal energy, P the pressure and g the speci®c heat ratio. 2.2. Turbulence model For the modelling of the turbulence a k ÿ e model is used (where k and e are the turbulent kinetic energy and its dissipation rate). The version of Launder and Sharma [22] which is, according to a review of Patel et al. [23], one of the best performing two-equation models has been chosen. This model is a low-Reynolds number model but the additive terms corresponding to the wall eects have no in¯uence at high Reynolds numbers. As an isotropic model is used in
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this study, the anisotropic character of coaxial jets is not taken into consideration. The equations for turbulent kinetic energy and the dissipation rate are: Equation of turbulent kinetic energy @ ~ @ @ 1@ @ k k~U~ ÿ mk k~ k~V~ ÿ mk k~ P~ k ÿ r e~ Wk r r r r @t @x @x r @r @r
10
with mk m
mt sk
11
p!2 @ k~ Wk ÿ2mmt : @xn
12
The expression of the production term is 8 !2 9 ! !2 !2 < @U~ 2 = ~ ~ ~ @V~ V @ V @ U P~ k 2mt mt : @x r ; @r @r @x 2 @U~ @V~ V~ ÿ r 3 @x @r
!(
!
@U~
mt
~
~
@x @@rV Vr
) k~ r
13
Equation of turbulent kinetic energy dissipation rate @ @ @ 1 @ @ e~ e~ U~ ÿ me e~ e~ V~ ÿ me e~ r r r r @t @x @x r @x @r Ce1 P~ k ÿ Ce2 with
e~ 2 r We k~ me m
14 mt se
2mmt @2 V~ We r @x2n
15 !2 :
16
where xn is the distance to the wall (here x), so Wk and We are only active in the near-wall regions if a turbulent boundary layer is present. Therefore, these terms have no in¯uence on the present computational results. The set of constants for the turbulence model is given as: Cm 0:09;
Ce1 1:44;
Ce2 1:92;
sk 1;
se 1:3:
17
To simulate unsteadiness without using direct Navier±Stokes simulation or large eddy simulation, an approach close to the semi-deterministic modelling (e.g. Ha Minh and Kourta [24]; Ha Minh [25]) is adopted without modi®cation of the model (the constants are not recalibrated). The classical statistical models have been built in the seventies to get stationary prediction of fully-developed ¯ows. This way has been applied already in the previous investigations of Habib and Whitelaw [7], Nikjooy et al. [26] and Harran [27] to the prediction of homogeneous and heterogeneous mean coaxial jets. Here, a classical model is used with an unsteady numerical scheme and a small time step. Coaxial jet ¯ows are dominated by large scale structures evolving in the two shear-layers. So, this approach leads to the simulation of coherent structures, without any
Numerical prediction of unsteady compressible turbulent coaxial jets
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modi®cation of the turbulence model, if the time steps used for the calculation are less than the time scale of the coherent structures evolving in the ¯ow. 3. NUMERICAL TOOLS
3.1. Numerical method The particular interest of the study on its numerical aspect, is the application of the ®nite volume scheme proposed by MacCormack [19] to the prediction of ¯ows including strong transversal gradients of velocity. Indeed, most of the convective schemes have been performed to capture shock waves, therefore it leads to the prediction of strong longitudinal gradients of velocity, density and pressure. The method uses the prediction±correction step technique and the equations are solved in conservative form. The scheme is accurate to second order in time and space and does not require any additional numerical dissipation for its stability. For the calculation of unsteady problems only the explicit part of the algorithm is executed. In fact, the method of MacCormack [19] is more suitable for the prediction of steady than unsteady problems. This better adaptation for the calculation of steady ¯ows is a result of the approximate treatment of the viscous terms in the implicit part of the scheme. 3.2. Computed cases Two dierent ¯ow patterns have been computed in this study. The ®rst (see Fig. 1) is the experiment of Ribeiro [28] on incompressible coaxial jets. This ¯ow was computed to validate the numerical code. The second con®guration (see Fig. 2) regards compressible coaxial jets. There is no experimental data available for this compressible ¯ow. For the two con®gurations the exit temperature is 300 K, the pressure Pe at the inlet and in the computational ®eld is initially of 0.101 MPa and the initial density in all the ®eld is ro=1.28 kg mÿ3. 3.2.1. Incompressible coaxial jets. The experiment of Ribeiro [28] was executed on air±air coaxial jets with equal velocities. The diameter D1 of the inner jet is 16.1 mm, D2 the exterior diameter of the inner pipe is 21.6 mm, D the interior diameter of the outer pipe is 44.7 mm and the exterior diameter is 50.4 mm. The geometry can be seen in Fig. 1. The velocity Uo in both round and annular jets is equal to 30 m sÿ1. The Reynolds numbers are 34,900 for the round jet and 50,200 for the annular jet. The grid uses 125 75 cells with stretching in the radial direction. It is uniform in the streamwise direction. The computational domain extends 20 diameters D in the streamwise direction and six in the radial direction. 3.2.2. Compressible ¯ow pattern. The compressible ¯ow predicted is an air±air con®guration (see Fig. 2) with a Mach number of 0.96 (ReD'o=181,500, where D'o is the equivalent diameter of the pipe giving the same momentum ¯ux) for the annular jet and a Mach number of 0.26 (ReDi=45,200, Di is the diameter of the central jet) for the central jet. The velocities are equal to 90 m sÿ1 for the round jet and 333 m sÿ1 for the annular jet. The injector diameter D is 11.4 mm and the lip thickness between the two jets is 0.5 mm. The computational grid uses 120 117 points with stretching in the transverse direction outside coaxial jets. The mesh is uniform in the x direction. It extends 16 injector diameters in the streamwise direction and seven injector diameters in the transverse direction.
Fig. 1. Experimental con®guration of Ribeiro [28].
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Fig. 2. Compressible coaxial jets with high velocity ratio.
3.3. Boundary conditions For the con®guration of Ribeiro [8] the inlet conditions are the experimental data measured at x/D = 0.2. The inlet conditions of the compressible coaxial jets, are derived from the experimental data of DuraÄo [17]. In the case of the compressible jets a wall is present (see Fig. 2) perpendicular to the injector. So, on this boundary, Dirichlet boundary conditions (zero) are used for velocity, turbulent kinetic energy and dissipation rate. Neumann conditions (zero) are assumed for the pressure and density. For the two patterns the lower boundary is the central jet axis, so symmetry conditions are assumed. As the upper boundary is far from the ¯ow, Neumann boundary conditions (zero gradients) are applied for the ®rst computed case. For compressible coaxial jets the study takes place in the context of a multi-jet con®guration, so the upper boundary is located between two injectors. In the case of multiple plane jets symmetry conditions are suitable. Here, the coaxial jets are axisymmetric but symmetry conditions are applied. This might not be a perfect condition but it seems to be a good approximation. In fact, every injector is surrounded by several others. Another calculation of the same con®guration with a free boundary led to the same numerical results. As the boundary is far from the ¯ow the conditions applied have a weak in¯uence on the simulation. This is due to the diusion of the numerical scheme. The outlet conditions are deducted from characteristic relationships. They have been recently applied to the direct simulation of compressible viscous ¯ows by Poinsot and Lele [29]. They originated from the characteristic analysis theory and they have been developed for the Euler equations by Thompson [30]. These non-re¯ecting boundary conditions can be written as: @ r 1 @P @ r 1 @P ~ ÿ ÿU ÿ 18 @t c2 @t @x c2 @x ! ~ @P @U~ @ P @ U c c r U~ c r @t @t @x @x ! @V~ @V~ ~ ÿU @t @x @P @U~ @P @U~ c c ÿr ÿ U~ ÿ c ÿr @t @t @x @x
19
20 ! 21
When the ¯ow is subsonic the pressure must be speci®ed at the exit. For the calculation of the experiment of Ribeiro [28] the pressure at the inlet Pe is applied. For the compressible ¯ow, a
Numerical prediction of unsteady compressible turbulent coaxial jets
245
Fig. 3. Axial decrease of the time-averaged streamwise velocity.
zero-gradient-condition is imposed for this quantity. For the turbulent kinetic energy and its dissipation rate the same condition is applied at the end of the domain. 3.4. Validation In Figs 3±5 the results of the prediction of the experiment of Ribeiro [23] are plotted. The comparison leans on time-averaged pro®les. The decrease of the axis velocity is represented in Fig. 3. The prediction shows a very good agreement with the experimental data. The prediction of radial pro®les of the streamwise velocity in the near-®eld (see Fig. 4) puts in evidence a quite good agreement with the measurements. There are some dierences in the transversal pro®les at x = 0.9 D and x = 3 D which should be imputed to the turbulence model. Downstream at x = 6 D and x = 10 D (see Fig. 5) the agreement between the prediction and the experimental data are good. The agreement for the prediction of the streamwise velocity is good for the axis decrease and the transversal pro®les. Then the numerical code is validated for the prediction of coaxial jets.
4. RESULTS
The calculations led to the prediction of the dynamic and thermodynamic characteristics for coaxial jets. Unsteady ®elds of velocity, pressure and density and energetic balances of turbulent kinetic energy and dissipation rate are predicted. Section 4.1 focuses on the unsteadiness in coaxial jets and the interaction between the instabilities of the two mixing layers. The evolution of ¯ow quantities is analysed in Section 4.2.
Fig. 4. Transversal pro®les of the streamwise velocity for x = 0.9 D and x = 3 D.
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Fig. 5. Transversal pro®les of the streamwise velocity for x = 6 D and x = 10 D.
4.1. Natural ¯ow unsteadiness The calculations lead to the simulation of unsteadiness in the near-®eld of coaxial jets. In the Fig. 6, the time-dependent variations of the streamwise velocity for two points of the computational ®eld are plotted. The ®rst point is located in the inner mixing layer at x = 3.2 D and r = 0.3 D, the second is situated in the outer mixing layer at x = 3.2 D and r = 0.6 D. The ¯ow unsteadiness in the initial region is a quasi-periodic phenomenon (see Fig. 6). The corresponding spectra (see Fig. 7) obtained by Fourier analysis for the two points over 100 periods are very similar. The dominant frequency of 5400 Hz is the same for the two spectra and the same ®rst harmonic is found. The corresponding Strouhal number based on the injector diameter and the annular jet velocity is equal to 0.18. In Figs 8±13, the unsteady variations of streamwise and radial velocities, pressure, density and turbulent quantities are represented for four grid sections situated at x = 0.34 D, x = 2 D, x = 5 D and x = 15 D, for four times of a period (T/4, T/2, 3 T/4 and T). For all quantities a high unsteadiness is observed near the inlet at x = 0.34 D. By the in¯uence of the diusion mechanisms the natural unsteadiness is damped downstream and has vanished at x = 15 D. The instabilities which are present near the inlet, are at the origin of the coherent structures evolving in the ¯ow. The presence of vortices is due to the entrainment of the outer ¯uid by the annular jet on the one hand and to the mixing mechanism between the annular ¯ow and the central jet on the other hand. They originate from the Kelvin±Helmholtz instability. These vortices are toroidal coherent structures. As the numerical code works with an axisymmetric coordinate system, azimuthal structures as those observed in the initial zone of coaxial jets by Tang and Ko [14] cannot be predicted. The presented results are consistent with the fact, corroborated by previous experiments (Gladnick et al. [13]; Wicker and Eaton [11]), that in coaxial jets the near-®eld is dominated by unsteady large scale structures which evolve as instabilities in the (a)
(b)
Fig. 6. Time-variations of the streamwise velocity in the near-velocity in the near-®eld: (a) at x = 3.2 D and r = 0.3 D; (b) at x = 3.2 D and r = 0.6 D.
Numerical prediction of unsteady compressible turbulent coaxial jets (a)
247
(b)
Fig. 7. Spectra of the unsteady streamwise velocity in the near-®eld: (a) at x = 3.2 D and r = 0.3 D; (b) at x = 3.2 D and r = 0.6 D.
mixing layers between the two jets and at the border with the outer ¯uid. The streamwise velocity pro®les (see Fig. 8) and the visualization of the velocity ®eld in Fig. 14 show a central region of periodically negative velocity in the initial region at x = 0.34 D. This phenomenon is the result of the growth of the large scale structures which penetrate the central jet. The structure genesis is located in the inner shear-layer but their evolution depends on both velocity ratio and initial conditions (Dahm et al. [3]; Wicker and Eaton [11]). Owing to the recirculation of the annular jet, the central potential core practically vanished in this simulation, as in the experimental study of Dixson et al. [9] on swirling coaxial jets. In the present study on homogeneous coaxial jets, the recirculation is not due to swirl but to velocity ratio. The velocity ratio lÿ1 is equal to 0.27 (lÿ1=Ui/Uo where Ui is the exit velocity of the central jet and Uo this of the annular jet). Okajima et al. [20] observed experimentally the same phenomenon in air±air coaxial jets for velocity ratio inferior to 0.35. For this range of the velocity ratio, Au and Ko [2] did not observe the presence of a recirculation region. This discrepancy between these two experiments is a consequence of dierent initial conditions. In the study of Au and Ko [2] the turbulence intensities at the exit are 0.5% for the inner and 1.2% for the outer nozzles. In the experiment of Okajima et al. [20] the turbulence intensity at the outer nozzle is 0.4% and the turbulence in the
Fig. 8. Pro®les of the unsteady streamwise velocity.
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P. Reynier and H. H. Minh
Fig. 9. Pro®les of the unsteady radial velocity.
central jet is fully-developed. In the present numerical study, the turbulence level at the exit is stronger than in the experiment of Au and Ko [2]. It is concluded that in coaxial jets a high initial level of turbulence favours the presence of a recirculation in the central region for high velocity ratio. This agrees with previous results on swirling coaxial jets obtained by Dixson et al. [9] in which the presence of a swirling eect results in a strong level of turbulence at the inlet. The presence of a recirculation region is not a characteristic of homogeneous coaxial jets. In an
Fig. 10. Pro®les of the unsteady pressure.
Numerical prediction of unsteady compressible turbulent coaxial jets
249
Fig. 11. Pro®les of the unsteady density.
experimental investigation, Camano and Favre-Marinet [31] report the same phenomenon in heterogeneous coaxial jets for high momentum ratio. In the present study the simulation of a recirculation zone and the same frequencies for the dominant large scale structures in the two shear-layers prove the strong interaction between the inner and the outer shear-layers. This coupling phenomenon between the large scale structures of the inner and outer mixing layers, predicted in this simulation, has been already observed experimentally by Wicker and Eaton [11,18].
Fig. 12. Pro®les of the unsteady turbulent kinetic energy.
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Fig. 13. Pro®les of the unsteady dissipation rate.
4.2. Evolution of dynamic and turbulent quantities In the Figs 15±20 the steady pro®les of streamwise and radial velocities, pressure, density and turbulent quantities are plotted for four grid sections located at x = 0.34 D, x = 2 D, x = 5 D, and x = 15 D. The mean ¯ow has been obtained by averaging the results over 100 periods. The streamwise velocity decreases in the initial region, the minimum is reached at x = 0.5 D where it is periodically negative, a fact that agrees with the predicted recirculation of the annular jet towards the axis. At x = 0.34 D (see Fig. 8) the streamwise velocity varies in the annular jet.
Fig. 14. Streamwise velocity ®eld at T/4, T/2, 3 T/4 and T (from the head of the page, respectively). The high velocities are represented in red and yellow, low velocities in blue and negative velocities in purple.
Numerical prediction of unsteady compressible turbulent coaxial jets
251
Fig. 15. Pro®les of the time-averaged streamwise velocity.
Fig. 16. Pro®les of the time-averaged radial velocity.
Fig. 17. Pro®les of the time-averaged pressure.
This is a consequence of the Biot and Savart induction eect, due to the entrainment, which sets up in the outer mixing layer. The time-averaged pro®le (Fig. 15) shows near the axis, at x = 0.34 D, a lower level than the exit velocity, it is a consequence of the presence of the recirculation zone. Then the centerline velocity grows up to the reattachment point at x = 4.3 D. Downstream, the pro®le evolution of the longitudinal velocity (see Fig. 8) is similar to a fullydeveloped round jet. For the same velocity ratio, Ko and Au [12] found the reattachment point to be at x = 5.2 D. The dierence in the reattachment point location can be explained by the lack of recirculation in their experiment due to dierent inlet conditions. The unsteadiness of the radial velocity (see Fig. 9) decays rapidly downstream and has vanished at x = 15 D. So, it is in the near-®eld that the entrainment of the outer ¯uid by the ¯ow is strong. This is con®rmed
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Fig. 18. Pro®les of the time-averaged density.
Fig. 19. Pro®les of the time-averaged turbulent kinetic energy.
Fig. 20. Pro®les of the time-averaged dissipation rate.
by the time-averaged pro®les of the radial velocity in Fig. 16. The mean pro®les of the radial velocity show two maxima and minima in the near-®eld at x = 0.34 D. They correspond to the two mixing layers and they agree with the presence of two dierent trains of structures, one in the outer and the other in the inner shear-layer. After the reattachment point, at x = 5 D, there is only one train of structures which disappear progressively by the in¯uence of the diusion. Pressure and density pro®les (see Figs 10 and 11) present the same evolution, due to weak temperature variations in this ¯ow. In the initial region at x = 0.34 D strong unsteadiness of these two quantities occur. The pressure minima correspond to the coherent structure passage (the pressure reaches a minimum in the center of a vortex), the pressure maxima correspond to the interval between two coherent structures (e.g. Hussain [32]). The pressure and density variations decay downstream and have disappeared at x = 15 D. Time-averaged pressure and den-
Numerical prediction of unsteady compressible turbulent coaxial jets
253
sity pro®les are plotted in Figs 17 and 18. The pressure pro®les show two minima and maxima at x = 0.34 D. The maxima are located on the axis and in the potential core of the annular jet, the minima are located in the shear-layers. After x = 2 D, the pressure becomes progressively uniform in the ¯ow. Downstream the reattachment point, one maximum and one minimum are predicted for the density and the pressure as in a round jet. At x = 0.34 D (Figs 12 and 13), the turbulent kinetic energy and the dissipation rate pro®les (the dissipation rate is scaled by e0=U30/D) are strongly unsteady with two peaks corresponding to the two mixing layers. In the initial region, high turbulence levels occur as a consequence of the high velocity gradients which involve a strong production of turbulence. At x = 2 D, strong turbulence levels on the jet axis caused by the recirculation phenomenon are predicted. At x = 15 D, the pro®les for turbulent quantities are steady, in fact the instabilities are progressively blunted by the diusion mechanism. Time-averaged pro®les of turbulent quantities (Figs 19 and 20) show strong levels in the near-®eld because of the high stress in this zone. As in the unsteady pro®les, the turbulent kinetic energy is high on the centerline at x = 2 D in the mean pro®les (Fig. 19), due to the recirculation in this region. The absolute maximum of turbulent quantities is located in the outer shear-layer as generally in the coaxial jet ¯ows (e.g. Ko and Au [12]; Okajima et al. [20]). Downstream, the turbulent kinetic energy and the dissipation rate pro®les become ¯at due to the diusion eects but at x = 15 D the maxima of these quantities are not located on the axis. The fully-developed turbulent ¯ow is not yet reached. 5. CONCLUSION
In this paper a two-equation turbulence model has been applied in order to predict the behaviour of compressible coaxial jets. This allows the numerical study of an unsteady air±air compressible and turbulent ¯ow dominated by large scale structures. As small time steps are used for the calculations, toroidal structures are simulated in the near-®eld. The unsteadiness decays downstream and has vanished before x = 15 D by the in¯uence of the diusion mechanism. It was predicted, for a velocity ratio of 0.27 between the two jets, a recirculation of the annular jet towards the central jet. This result agrees qualitatively with previous experimental studies. The presence of such a phenomenon in homogeneous coaxial jets depends on both inlet conditions (turbulence level) and velocity ratio. Three zones of high turbulence levels are predicted, two of them in the initial region of the shear-layers and the third one, weaker than the others, in the recirculating region. The absolute maximum is located in the outer mixing layer as generally in coaxial jet ¯ows. The presence of a recirculation zone and the same frequency for the dominant instability in the two mixing layers prove the strong coupling between the two shear-layers. This result agrees with previous experimental investigations. The purpose of this numerical study was the simulation of coherent structures in coaxial jets with a methodology close to the semi-deterministic modelling. The approach is able to simulate unsteadiness, so good perspectives are opened in the prediction of unsteady ¯ows at high-Reynolds numbers. AcknowledgementsÐThe authors are grateful to the SocieÂte EuropeÂenne de Propulsion for its support under Contract PRC/CNRS No. 90.0018 C and to the CNES which supports this work under the post-doctorate grant of P. Reynier.
REFERENCES 1. Ribeiro, M. M. and Whitelaw, J. H., Coaxial jets with and without swirl. Journal of Fluid Mechanics, 1980, 96, 769± 795. 2. Au, H. and Ko, N. W. M., Coaxial jets of dierent mean velocity ratios. Part 2. Journal of Sound and Vibration, 1987, 116, 427±443. 3. Dahm, W. J. A., Frieler, C. E. and Tryggvason, G., Vortex structure and dynamics in the near ®eld of a coaxial jet. Journal of Fluid Mechanics, 1992, 241, 371±402. 4. Ko, N. W. M. and Au, H., Initial region of subsonic coaxial jets of high mean velocity ratio. Journal of Fluids Engineering, 1981, 103, 335±338. 5. Choi, D. W., Gessner, F. B. and Oates, G. C., Measurements of con®ned coaxial jet mixing with pressure gradient. Journal of Fluids Engineering, 1986, 108, 39±46. 6. So, R. M. C., Yu, M. H., Otugen, M. V. and Zhu, J. Y., Rotation eects on inhomogeneous mixing in axisymmetric sudden-expansion ¯ows. International Journal of Heat Mass Transfer, 1987, 30, 2411±2421. 7. Habib, M. A. and Whitelaw, J. H., Velocity characteristics of con®ned coaxial jets with and without swirl. Journal of Fluids Engineering, 1980, 102, 47±53. 8. Chen, C. P., Calculation of con®ned swirling jets. Communications in Applied Numerical Methods, 1986, 2, 333±338.
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9. Dixson, T. F., Truelove, J. S. and Wall, T. F., Aerodynamics studies on swirled coaxial jets from nozzles with divergent quarls. Journal of Fluids Engineering, 1983, 105, 197±203. 10. Ko, N. W. M. and Chan, W. T., The inner regions of annular jets. Journal of Fluid Mechanics, 1979, 93, 549±584. 11. Wicker, R. B. and Eaton, J. K., Near-®eld of a coaxial jet with and without axial excitation. AIAA Journal, 1994, 32, 542±546. 12. Ko, N. W. M. and Au, H., Coaxial jets of mean velocity ratios. Journal of Sound and Vibration, 1985, 100, 211±232. 13. Gladnick, P. G., Enotiadis, A. C., Larue, J. C. and Samuelsen, G. S., Near-®eld characteristics of a turbulent co¯owing jet. AIAA Journal, 1990, 28, 1405±1414. 14. Tang, S. K. and Ko, N. W. M., Coherent structures interactions in an unexcited coaxial jet. Experiments in Fluids, 1994, 17, 147±157. 15. Moore, C. J., The role of shear-layer instability waves in jet exhaust noise. Journal of Fluid Mechanics, 1977, 80, 321±367. 16. Morrison, G. L. and McLaughlin, D. K., Instability process in low Reynolds supersonic jets. AIAA Journal, 1980, 18, 793±800. 17. DuraÄo, D., Turbulent mixing of coaxial jets. M.Sc. Thesis. University of London, London, 1971. 18. Wicker R. B. and Eaton J. K., Flow visualization study of the structure of a co-¯owing jet with and without controlled excitation. AIAA Paper, 93-0651, 1993. 19. MacCormack, R. B., A numerical method for solving the equations of compressible viscous ¯ow. AIAA Paper, 810110, 1981. 20. Okajima, A., Tiwata, T. and Ueno, H., Experimental and numerical studies on coaxial jets. In Proceedings of FLUCOME Conference. Toulouse, France, 1994. 21. Favre, A., Equations des gaz turbulents compressibles. Journes de MeÂcanique, 1965, 4, 361±390, 391±421. 22. Launder, B. E. and Sharma, B. I., Application of the energy dissipation model of turbulence to the calculation of ¯ow near a spinning disc. Letters Heat Mass Transfer, 1974, 1, 131±138. 23. Patel, V. C., Rodi, W. and Scheuerer, G., Turbulence models for near wall and low Reynolds number ¯ows: a review. AIAA Journal, 1985, 23, 1308±1319. 24. Ha Minh, H. and Kourta, A., Semi-deterministic turbulence modelling for ¯ows dominated by strong organized structures. In Proceedings of 9th International Symposium on Turbulent Shear Flows. Kyoto, 1993. 25. Ha Minh, H., The use of experiment for improving turbulence models. In Proceedings of FLUCOME Conference. Toulouse, France, 1994. 26. Nikjooy, M., Karki, K. C., Mongia, H. C., McDonell, V. G. and Samuelsen, G. S., A numerical and experimental study of coaxial jets. International Journal of Heat and Fluid Flow, 1989, 10, 253±261. 27. Harran, G., Analyse physique, modeÂlisation et simulation numeÂrique des meÂcanismes de meÂlange dans les jets simples et coaxiaux turbulents. Ph.D. Thesis. INPT No. 841, Toulouse, France, 1994. 28. Ribeiro, M. M., Turbulent mixing of coaxial jets. M.Sc. Thesis. University of London, London, 1972. 29. Poinsot, T. J. and Lele, S. K., Boundary conditions for direct simulations of compressible viscous ¯ows. Journal of Computational Physics, 1992, 101, 104±129. 30. Thompson, K. W., Time dependent boundary conditions for hyperbolic systems. Journal of Computational Physics, 1987, 68, 1±24. 31. Camano, E. B. and Favre-Marinet, M., On the initial region of inhomogeneous coaxial jets. In Advances in Turbulence V, ed. R. Benzi. Kluwer Academic, Amsterdam, 1994, pp. 58±62. 32. Hussain, A. K. M. F., Coherent structures and turbulence. Journal of Fluid Mechanics, 1986, 173, 303±356.
Computers & Fluids Vol. 27, No. 2, pp. 239±254, 1998 # 1998 Elsevier Science Ltd. All rights reserved Printed in Great Britain 0045-7930/98 $19.00 + 0.00 S0045-7930(97)00023-6
NUMERICAL PREDICTION OF UNSTEADY COMPRESSIBLE TURBULENT COAXIAL JETS PHILIPPE REYNIER1 and HIEU HA MINH1 Institut de MeÂcanique des Fluides de Toulouse, Avenue du Pr. Camille Soula, 31400 Toulouse, France
1
(Received 5 April 1996; revised 4 April 1997) AbstractÐCompressible and turbulent air±air coaxial jets with a strong velocity ratio are simulated by using a ®nite volume scheme and a two-equation turbulence model. For the calculations only the explicit part of the numerical method is used. This allows the simulation of the coherent structures evolving in the ¯ow. The results show a strong instability of the ¯ow near the inlet, this natural unsteadiness decays downstream and vanishes before x/D = 15. In the outer mixing layer the coherent structures evolve as in a simple round jet. Near the inlet, in the inner mixing layer, a recirculation region is found on the jet axis. The presence of this phenomenon in homogeneous coaxial jets is the consequence of the strong velocity ratio between the two streams and explains the lack of the inner potential core. These numerical results agree qualitatively with previous experimental investigations. # 1998 Elsevier Science Ltd. All rights reserved
1. INTRODUCTION
Homogeneous turbulent coaxial jets discharging into stationary air have been intensively investigated in the past (e.g. Ribeiro and Whitelaw [1]; Au and Ko [2]; Dahm et al. [3]). Ko and Au [4] found that the ¯ow was isolated into three separate regions: the initial region, the intermediate region, and the fully merged zone. The initial region is located between the nozzle and the end of the outer potential core, immediately downstream is the intermediate region which ends at the reattachment point and ®nally is found the fully merged zone. The ¯ow is characterized by the presence of two mixing layers: the outer mixing layer, contiguous to the initially stationary air and the inner mixing layer, con®ned between both central and annular jets. The main aspect of coaxial jets is the mixing mechanism of two originally adjacent ¯ows with dierent turbulence levels and anisotropic characters, in fact two dierent turbulence histories. The mixing begins in the initial region but takes place principally in the intermediate region. In an experimental study, Choi et al. [5] showed that the mixing mechanism is promoted by the presence of an adverse pressure gradient. In fact, an adverse pressure gradient increases the shear-layer growth in the initial region. About the mixing increase, So et al. [6] arrived at the same conclusion in an investigation on rotation eects in helium±air coaxial jets. In the same way, Habib and Whitelaw [7] got similar results for air±air con®ned coaxial jets with swirl. The swirling coaxial jets have been more extensively studied by Chen [8] and Dixson et al. [9]. They remarked that the mixing increases with the swirling number. Moreover, they observed a recirculation of the annular jet towards the centerline for swirling numbers superior to 0.5. A consequence of this last phenomenon is the lack of the inner potential core. So far, the presence of a recirculation zone was considered as a characteristic of the annular jets that have been experimentally studied by Ko and Chan [10]. Another feature of coaxial jets, resulting from the ®rst one, is the instability of such a ¯ow pattern. For equal velocities in the two jets, the wake instability prevails, whereas the shearlayer instability dominates for very large or very small velocity ratio (e.g. Wicker and Eaton [11]; Dahm et al. [3]). Recent studies, such as those of Ko and Au [12], Au and Ko [2] and Wicker and Eaton [11] report the presence of instabilities in the inner and outer mixing layers of air±air coaxial jets. On the other hand, the experimental investigation of Gladnick et al. [13] puts in evidence the presence of coherent structures in variable density coaxial jets. Wicker and Eaton [11] and Dahm et al. [3] studied the interaction between the coherent structures evolving in the two shear-layers. Their experimental results show a strong coupling between the instabil239
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ities present in the two mixing layers. The evolution of the inner shear-layer vortices depends on both outer shear-layer evolution and velocity ratio. Tang and Ko [14] studied experimentally the vortice evolution in both initial merging and intermediate regions. They observed azimuthal instabilities in the initial region and vortices as those behind axisymmetric bodies in the intermediate zone. Azimuthal instabilities originate from the bifurcation of initial region structures. Of these studies were carried out for incompressible ¯ows and moderate Reynolds numbers, the results of Moore [15] for a round jet prove that the large scale structures observed at lowReynolds numbers (104) are also observed at Reynolds numbers of about 106. In round jets, the experimental investigation of Morrison and McLaughlin [16] shows the presence of large scale structures for supersonic ¯ows at a Mach number of 2.5. However, the instability process in the initial region is strongly dependent on the initial conditions. DuraÄo [17] had shown already the in¯uence of initial turbulence intensity and velocity pro®le shape on ¯ow evolution. Then Dahm et al. [3] investigated the eects of both lip and initial boundary layer thicknesses. These last authors and Wicker and Eaton [18] observed a strong in¯uence of the initial conditions, especially absolute velocities and velocity ratio, on ¯ow instability. So et al. [6] studied the in¯uence of the turbulence level at the exit and showed a decrease of the inner potential core length and an increase of the mixing when the turbulence level at the inlet is large. This paper attempts to simulate the unsteadiness of coaxial jets with a high velocity ratio. For the calculations a classical turbulence model with the ®nite volume method of MacCormack [19] is used. This numerical investigation is concentrated on the initial region of the ¯ow to study the evolution of coherent structures and the coupling between the two mixing layers. In Section 2 a brief overview of the physical modelling of the problem is exposed. In Section 3 the numerical aspects with the main features of the numerical method used for the calculations are presented. In the same section a validation of the numerical code is shown. The numerical results for compressible coaxial jets are presented in Section 4 with an analysis of the simulated phenomena. In this section, qualitative comparisons with experiments of Ko and Au [12] and Okajima et al. [20] are given. Finally in Section 5 we draw the conclusion. 2. PHYSICAL MODELLING
2.1. Governing equations Two mixing layers are present in coaxial jets, they involve the presence of large scale structures in this ¯ow. Three-dimensional instabilities have been observed in the intermediate region by Tang and Ko [14] and Dahm et al. [3]. They correspond to the loss of organization in the ¯ow and lead to the fully-developed turbulence. In the present paper, the objective is to simulate organized unsteadiness assuming that coherent structures are two-dimensional. So, three-dimensional aspects are not taken account, the loss of organization of the two-dimensional structures is ensured across the diusion of the turbulence model. The problem deals with coaxial jet ¯ows so the equations are solved using axisymmetric coordinates. The equations that describe the physical problem are the unsteady Navier±Stokes equations written in axisymmetric coordinates and the state equation for a perfect gas. The equation system with Favre [21] mass-weighted averaging can be written as: Equation for the conservation of mass density r @ @ ~ 1@ r U r r rV~ 0 @t @x r @r
1
Equation for the conservation of momentum rUÄ @ ~ @ 2 1@ ~ ~ ~ ~ rU rUU rkP ÿ m mt Sxx rf rU~ V~ ÿ m mt S~rx g 0 @t @x 3 r @r
2
Numerical prediction of unsteady compressible turbulent coaxial jets
Equation for the conservation of momentum rVÄ o @ ~ @ n ~ ~ V UV ÿ m mt S~xr r r @t @x 1@ 2 ~ ~ ~ ~ VV P r k ÿ m mt Srr r r r @r 3 1 ~ 2 ~ k ÿ m mt S~yy P r r 3 The terms of the strain tensor are given by the following expressions: ! ~ 2 @U~ @V~ V~ @ V ÿ S~rr 2 r @r 3 @x @r S~xx
! @U~ 2 @U~ @V~ V~ 2 ÿ r @x 3 @x @r @U~ @V~ S~xr S~rx @r @x
S~yy
V~ 2 @U~ @V~ V~ 2 ÿ r 3 @x @r r
241
3
4
5
6 ! 7
Equation for the conservation of total energy EÄ 2 ~ ~ ~ E P r k ÿ m mt Sxx U~ r 3 m mt @ ~ ~ e~i ÿ m mt Sxr V ÿ g Pr Prt @x 1@ m m @ 2r E~ P k~ ÿ m mt S~rr V~ ÿ m mt S~xr U~ ÿ g r r t e~i 0 r @r 3 Pr Prt @r
@ ~ @ E r @t @x
8
State equation for a perfect gas g ÿ 1 P re~i
9
where r is the density, UÄ and VÄ the velocity components with respect to x and r directions (x and r are the longitudinal and radial coordinates, respectively), m the molecular viscosity, mt the turbulent viscosity, Pr the Prandtl number, Prt the turbulent Prandtl number, EÄ the total energy, eÄi the internal energy, P the pressure and g the speci®c heat ratio. 2.2. Turbulence model For the modelling of the turbulence a k ÿ e model is used (where k and e are the turbulent kinetic energy and its dissipation rate). The version of Launder and Sharma [22] which is, according to a review of Patel et al. [23], one of the best performing two-equation models has been chosen. This model is a low-Reynolds number model but the additive terms corresponding to the wall eects have no in¯uence at high Reynolds numbers. As an isotropic model is used in
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P. Reynier and H. H. Minh
this study, the anisotropic character of coaxial jets is not taken into consideration. The equations for turbulent kinetic energy and the dissipation rate are: Equation of turbulent kinetic energy @ ~ @ @ 1@ @ k k~U~ ÿ mk k~ k~V~ ÿ mk k~ P~ k ÿ r e~ Wk r r r r @t @x @x r @r @r
10
with mk m
mt sk
11
p!2 @ k~ Wk ÿ2mmt : @xn
12
The expression of the production term is 8 !2 9 ! !2 !2 < @U~ 2 = ~ ~ ~ @V~ V @ V @ U P~ k 2mt mt : @x r ; @r @r @x 2 @U~ @V~ V~ ÿ r 3 @x @r
!(
!
@U~
mt
~
~
@x @@rV Vr
) k~ r
13
Equation of turbulent kinetic energy dissipation rate @ @ @ 1 @ @ e~ e~ U~ ÿ me e~ e~ V~ ÿ me e~ r r r r @t @x @x r @x @r Ce1 P~ k ÿ Ce2 with
e~ 2 r We k~ me m
14 mt se
2mmt @2 V~ We r @x2n
15 !2 :
16
where xn is the distance to the wall (here x), so Wk and We are only active in the near-wall regions if a turbulent boundary layer is present. Therefore, these terms have no in¯uence on the present computational results. The set of constants for the turbulence model is given as: Cm 0:09;
Ce1 1:44;
Ce2 1:92;
sk 1;
se 1:3:
17
To simulate unsteadiness without using direct Navier±Stokes simulation or large eddy simulation, an approach close to the semi-deterministic modelling (e.g. Ha Minh and Kourta [24]; Ha Minh [25]) is adopted without modi®cation of the model (the constants are not recalibrated). The classical statistical models have been built in the seventies to get stationary prediction of fully-developed ¯ows. This way has been applied already in the previous investigations of Habib and Whitelaw [7], Nikjooy et al. [26] and Harran [27] to the prediction of homogeneous and heterogeneous mean coaxial jets. Here, a classical model is used with an unsteady numerical scheme and a small time step. Coaxial jet ¯ows are dominated by large scale structures evolving in the two shear-layers. So, this approach leads to the simulation of coherent structures, without any
Numerical prediction of unsteady compressible turbulent coaxial jets
243
modi®cation of the turbulence model, if the time steps used for the calculation are less than the time scale of the coherent structures evolving in the ¯ow. 3. NUMERICAL TOOLS
3.1. Numerical method The particular interest of the study on its numerical aspect, is the application of the ®nite volume scheme proposed by MacCormack [19] to the prediction of ¯ows including strong transversal gradients of velocity. Indeed, most of the convective schemes have been performed to capture shock waves, therefore it leads to the prediction of strong longitudinal gradients of velocity, density and pressure. The method uses the prediction±correction step technique and the equations are solved in conservative form. The scheme is accurate to second order in time and space and does not require any additional numerical dissipation for its stability. For the calculation of unsteady problems only the explicit part of the algorithm is executed. In fact, the method of MacCormack [19] is more suitable for the prediction of steady than unsteady problems. This better adaptation for the calculation of steady ¯ows is a result of the approximate treatment of the viscous terms in the implicit part of the scheme. 3.2. Computed cases Two dierent ¯ow patterns have been computed in this study. The ®rst (see Fig. 1) is the experiment of Ribeiro [28] on incompressible coaxial jets. This ¯ow was computed to validate the numerical code. The second con®guration (see Fig. 2) regards compressible coaxial jets. There is no experimental data available for this compressible ¯ow. For the two con®gurations the exit temperature is 300 K, the pressure Pe at the inlet and in the computational ®eld is initially of 0.101 MPa and the initial density in all the ®eld is ro=1.28 kg mÿ3. 3.2.1. Incompressible coaxial jets. The experiment of Ribeiro [28] was executed on air±air coaxial jets with equal velocities. The diameter D1 of the inner jet is 16.1 mm, D2 the exterior diameter of the inner pipe is 21.6 mm, D the interior diameter of the outer pipe is 44.7 mm and the exterior diameter is 50.4 mm. The geometry can be seen in Fig. 1. The velocity Uo in both round and annular jets is equal to 30 m sÿ1. The Reynolds numbers are 34,900 for the round jet and 50,200 for the annular jet. The grid uses 125 75 cells with stretching in the radial direction. It is uniform in the streamwise direction. The computational domain extends 20 diameters D in the streamwise direction and six in the radial direction. 3.2.2. Compressible ¯ow pattern. The compressible ¯ow predicted is an air±air con®guration (see Fig. 2) with a Mach number of 0.96 (ReD'o=181,500, where D'o is the equivalent diameter of the pipe giving the same momentum ¯ux) for the annular jet and a Mach number of 0.26 (ReDi=45,200, Di is the diameter of the central jet) for the central jet. The velocities are equal to 90 m sÿ1 for the round jet and 333 m sÿ1 for the annular jet. The injector diameter D is 11.4 mm and the lip thickness between the two jets is 0.5 mm. The computational grid uses 120 117 points with stretching in the transverse direction outside coaxial jets. The mesh is uniform in the x direction. It extends 16 injector diameters in the streamwise direction and seven injector diameters in the transverse direction.
Fig. 1. Experimental con®guration of Ribeiro [28].
244
P. Reynier and H. H. Minh
Fig. 2. Compressible coaxial jets with high velocity ratio.
3.3. Boundary conditions For the con®guration of Ribeiro [8] the inlet conditions are the experimental data measured at x/D = 0.2. The inlet conditions of the compressible coaxial jets, are derived from the experimental data of DuraÄo [17]. In the case of the compressible jets a wall is present (see Fig. 2) perpendicular to the injector. So, on this boundary, Dirichlet boundary conditions (zero) are used for velocity, turbulent kinetic energy and dissipation rate. Neumann conditions (zero) are assumed for the pressure and density. For the two patterns the lower boundary is the central jet axis, so symmetry conditions are assumed. As the upper boundary is far from the ¯ow, Neumann boundary conditions (zero gradients) are applied for the ®rst computed case. For compressible coaxial jets the study takes place in the context of a multi-jet con®guration, so the upper boundary is located between two injectors. In the case of multiple plane jets symmetry conditions are suitable. Here, the coaxial jets are axisymmetric but symmetry conditions are applied. This might not be a perfect condition but it seems to be a good approximation. In fact, every injector is surrounded by several others. Another calculation of the same con®guration with a free boundary led to the same numerical results. As the boundary is far from the ¯ow the conditions applied have a weak in¯uence on the simulation. This is due to the diusion of the numerical scheme. The outlet conditions are deducted from characteristic relationships. They have been recently applied to the direct simulation of compressible viscous ¯ows by Poinsot and Lele [29]. They originated from the characteristic analysis theory and they have been developed for the Euler equations by Thompson [30]. These non-re¯ecting boundary conditions can be written as: @ r 1 @P @ r 1 @P ~ ÿ ÿU ÿ 18 @t c2 @t @x c2 @x ! ~ @P @U~ @ P @ U c c r U~ c r @t @t @x @x ! @V~ @V~ ~ ÿU @t @x @P @U~ @P @U~ c c ÿr ÿ U~ ÿ c ÿr @t @t @x @x
19
20 ! 21
When the ¯ow is subsonic the pressure must be speci®ed at the exit. For the calculation of the experiment of Ribeiro [28] the pressure at the inlet Pe is applied. For the compressible ¯ow, a
Numerical prediction of unsteady compressible turbulent coaxial jets
245
Fig. 3. Axial decrease of the time-averaged streamwise velocity.
zero-gradient-condition is imposed for this quantity. For the turbulent kinetic energy and its dissipation rate the same condition is applied at the end of the domain. 3.4. Validation In Figs 3±5 the results of the prediction of the experiment of Ribeiro [23] are plotted. The comparison leans on time-averaged pro®les. The decrease of the axis velocity is represented in Fig. 3. The prediction shows a very good agreement with the experimental data. The prediction of radial pro®les of the streamwise velocity in the near-®eld (see Fig. 4) puts in evidence a quite good agreement with the measurements. There are some dierences in the transversal pro®les at x = 0.9 D and x = 3 D which should be imputed to the turbulence model. Downstream at x = 6 D and x = 10 D (see Fig. 5) the agreement between the prediction and the experimental data are good. The agreement for the prediction of the streamwise velocity is good for the axis decrease and the transversal pro®les. Then the numerical code is validated for the prediction of coaxial jets.
4. RESULTS
The calculations led to the prediction of the dynamic and thermodynamic characteristics for coaxial jets. Unsteady ®elds of velocity, pressure and density and energetic balances of turbulent kinetic energy and dissipation rate are predicted. Section 4.1 focuses on the unsteadiness in coaxial jets and the interaction between the instabilities of the two mixing layers. The evolution of ¯ow quantities is analysed in Section 4.2.
Fig. 4. Transversal pro®les of the streamwise velocity for x = 0.9 D and x = 3 D.
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P. Reynier and H. H. Minh
Fig. 5. Transversal pro®les of the streamwise velocity for x = 6 D and x = 10 D.
4.1. Natural ¯ow unsteadiness The calculations lead to the simulation of unsteadiness in the near-®eld of coaxial jets. In the Fig. 6, the time-dependent variations of the streamwise velocity for two points of the computational ®eld are plotted. The ®rst point is located in the inner mixing layer at x = 3.2 D and r = 0.3 D, the second is situated in the outer mixing layer at x = 3.2 D and r = 0.6 D. The ¯ow unsteadiness in the initial region is a quasi-periodic phenomenon (see Fig. 6). The corresponding spectra (see Fig. 7) obtained by Fourier analysis for the two points over 100 periods are very similar. The dominant frequency of 5400 Hz is the same for the two spectra and the same ®rst harmonic is found. The corresponding Strouhal number based on the injector diameter and the annular jet velocity is equal to 0.18. In Figs 8±13, the unsteady variations of streamwise and radial velocities, pressure, density and turbulent quantities are represented for four grid sections situated at x = 0.34 D, x = 2 D, x = 5 D and x = 15 D, for four times of a period (T/4, T/2, 3 T/4 and T). For all quantities a high unsteadiness is observed near the inlet at x = 0.34 D. By the in¯uence of the diusion mechanisms the natural unsteadiness is damped downstream and has vanished at x = 15 D. The instabilities which are present near the inlet, are at the origin of the coherent structures evolving in the ¯ow. The presence of vortices is due to the entrainment of the outer ¯uid by the annular jet on the one hand and to the mixing mechanism between the annular ¯ow and the central jet on the other hand. They originate from the Kelvin±Helmholtz instability. These vortices are toroidal coherent structures. As the numerical code works with an axisymmetric coordinate system, azimuthal structures as those observed in the initial zone of coaxial jets by Tang and Ko [14] cannot be predicted. The presented results are consistent with the fact, corroborated by previous experiments (Gladnick et al. [13]; Wicker and Eaton [11]), that in coaxial jets the near-®eld is dominated by unsteady large scale structures which evolve as instabilities in the (a)
(b)
Fig. 6. Time-variations of the streamwise velocity in the near-velocity in the near-®eld: (a) at x = 3.2 D and r = 0.3 D; (b) at x = 3.2 D and r = 0.6 D.
Numerical prediction of unsteady compressible turbulent coaxial jets (a)
247
(b)
Fig. 7. Spectra of the unsteady streamwise velocity in the near-®eld: (a) at x = 3.2 D and r = 0.3 D; (b) at x = 3.2 D and r = 0.6 D.
mixing layers between the two jets and at the border with the outer ¯uid. The streamwise velocity pro®les (see Fig. 8) and the visualization of the velocity ®eld in Fig. 14 show a central region of periodically negative velocity in the initial region at x = 0.34 D. This phenomenon is the result of the growth of the large scale structures which penetrate the central jet. The structure genesis is located in the inner shear-layer but their evolution depends on both velocity ratio and initial conditions (Dahm et al. [3]; Wicker and Eaton [11]). Owing to the recirculation of the annular jet, the central potential core practically vanished in this simulation, as in the experimental study of Dixson et al. [9] on swirling coaxial jets. In the present study on homogeneous coaxial jets, the recirculation is not due to swirl but to velocity ratio. The velocity ratio lÿ1 is equal to 0.27 (lÿ1=Ui/Uo where Ui is the exit velocity of the central jet and Uo this of the annular jet). Okajima et al. [20] observed experimentally the same phenomenon in air±air coaxial jets for velocity ratio inferior to 0.35. For this range of the velocity ratio, Au and Ko [2] did not observe the presence of a recirculation region. This discrepancy between these two experiments is a consequence of dierent initial conditions. In the study of Au and Ko [2] the turbulence intensities at the exit are 0.5% for the inner and 1.2% for the outer nozzles. In the experiment of Okajima et al. [20] the turbulence intensity at the outer nozzle is 0.4% and the turbulence in the
Fig. 8. Pro®les of the unsteady streamwise velocity.
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Fig. 9. Pro®les of the unsteady radial velocity.
central jet is fully-developed. In the present numerical study, the turbulence level at the exit is stronger than in the experiment of Au and Ko [2]. It is concluded that in coaxial jets a high initial level of turbulence favours the presence of a recirculation in the central region for high velocity ratio. This agrees with previous results on swirling coaxial jets obtained by Dixson et al. [9] in which the presence of a swirling eect results in a strong level of turbulence at the inlet. The presence of a recirculation region is not a characteristic of homogeneous coaxial jets. In an
Fig. 10. Pro®les of the unsteady pressure.
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249
Fig. 11. Pro®les of the unsteady density.
experimental investigation, Camano and Favre-Marinet [31] report the same phenomenon in heterogeneous coaxial jets for high momentum ratio. In the present study the simulation of a recirculation zone and the same frequencies for the dominant large scale structures in the two shear-layers prove the strong interaction between the inner and the outer shear-layers. This coupling phenomenon between the large scale structures of the inner and outer mixing layers, predicted in this simulation, has been already observed experimentally by Wicker and Eaton [11,18].
Fig. 12. Pro®les of the unsteady turbulent kinetic energy.
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Fig. 13. Pro®les of the unsteady dissipation rate.
4.2. Evolution of dynamic and turbulent quantities In the Figs 15±20 the steady pro®les of streamwise and radial velocities, pressure, density and turbulent quantities are plotted for four grid sections located at x = 0.34 D, x = 2 D, x = 5 D, and x = 15 D. The mean ¯ow has been obtained by averaging the results over 100 periods. The streamwise velocity decreases in the initial region, the minimum is reached at x = 0.5 D where it is periodically negative, a fact that agrees with the predicted recirculation of the annular jet towards the axis. At x = 0.34 D (see Fig. 8) the streamwise velocity varies in the annular jet.
Fig. 14. Streamwise velocity ®eld at T/4, T/2, 3 T/4 and T (from the head of the page, respectively). The high velocities are represented in red and yellow, low velocities in blue and negative velocities in purple.
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Fig. 15. Pro®les of the time-averaged streamwise velocity.
Fig. 16. Pro®les of the time-averaged radial velocity.
Fig. 17. Pro®les of the time-averaged pressure.
This is a consequence of the Biot and Savart induction eect, due to the entrainment, which sets up in the outer mixing layer. The time-averaged pro®le (Fig. 15) shows near the axis, at x = 0.34 D, a lower level than the exit velocity, it is a consequence of the presence of the recirculation zone. Then the centerline velocity grows up to the reattachment point at x = 4.3 D. Downstream, the pro®le evolution of the longitudinal velocity (see Fig. 8) is similar to a fullydeveloped round jet. For the same velocity ratio, Ko and Au [12] found the reattachment point to be at x = 5.2 D. The dierence in the reattachment point location can be explained by the lack of recirculation in their experiment due to dierent inlet conditions. The unsteadiness of the radial velocity (see Fig. 9) decays rapidly downstream and has vanished at x = 15 D. So, it is in the near-®eld that the entrainment of the outer ¯uid by the ¯ow is strong. This is con®rmed
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Fig. 18. Pro®les of the time-averaged density.
Fig. 19. Pro®les of the time-averaged turbulent kinetic energy.
Fig. 20. Pro®les of the time-averaged dissipation rate.
by the time-averaged pro®les of the radial velocity in Fig. 16. The mean pro®les of the radial velocity show two maxima and minima in the near-®eld at x = 0.34 D. They correspond to the two mixing layers and they agree with the presence of two dierent trains of structures, one in the outer and the other in the inner shear-layer. After the reattachment point, at x = 5 D, there is only one train of structures which disappear progressively by the in¯uence of the diusion. Pressure and density pro®les (see Figs 10 and 11) present the same evolution, due to weak temperature variations in this ¯ow. In the initial region at x = 0.34 D strong unsteadiness of these two quantities occur. The pressure minima correspond to the coherent structure passage (the pressure reaches a minimum in the center of a vortex), the pressure maxima correspond to the interval between two coherent structures (e.g. Hussain [32]). The pressure and density variations decay downstream and have disappeared at x = 15 D. Time-averaged pressure and den-
Numerical prediction of unsteady compressible turbulent coaxial jets
253
sity pro®les are plotted in Figs 17 and 18. The pressure pro®les show two minima and maxima at x = 0.34 D. The maxima are located on the axis and in the potential core of the annular jet, the minima are located in the shear-layers. After x = 2 D, the pressure becomes progressively uniform in the ¯ow. Downstream the reattachment point, one maximum and one minimum are predicted for the density and the pressure as in a round jet. At x = 0.34 D (Figs 12 and 13), the turbulent kinetic energy and the dissipation rate pro®les (the dissipation rate is scaled by e0=U30/D) are strongly unsteady with two peaks corresponding to the two mixing layers. In the initial region, high turbulence levels occur as a consequence of the high velocity gradients which involve a strong production of turbulence. At x = 2 D, strong turbulence levels on the jet axis caused by the recirculation phenomenon are predicted. At x = 15 D, the pro®les for turbulent quantities are steady, in fact the instabilities are progressively blunted by the diusion mechanism. Time-averaged pro®les of turbulent quantities (Figs 19 and 20) show strong levels in the near-®eld because of the high stress in this zone. As in the unsteady pro®les, the turbulent kinetic energy is high on the centerline at x = 2 D in the mean pro®les (Fig. 19), due to the recirculation in this region. The absolute maximum of turbulent quantities is located in the outer shear-layer as generally in the coaxial jet ¯ows (e.g. Ko and Au [12]; Okajima et al. [20]). Downstream, the turbulent kinetic energy and the dissipation rate pro®les become ¯at due to the diusion eects but at x = 15 D the maxima of these quantities are not located on the axis. The fully-developed turbulent ¯ow is not yet reached. 5. CONCLUSION
In this paper a two-equation turbulence model has been applied in order to predict the behaviour of compressible coaxial jets. This allows the numerical study of an unsteady air±air compressible and turbulent ¯ow dominated by large scale structures. As small time steps are used for the calculations, toroidal structures are simulated in the near-®eld. The unsteadiness decays downstream and has vanished before x = 15 D by the in¯uence of the diusion mechanism. It was predicted, for a velocity ratio of 0.27 between the two jets, a recirculation of the annular jet towards the central jet. This result agrees qualitatively with previous experimental studies. The presence of such a phenomenon in homogeneous coaxial jets depends on both inlet conditions (turbulence level) and velocity ratio. Three zones of high turbulence levels are predicted, two of them in the initial region of the shear-layers and the third one, weaker than the others, in the recirculating region. The absolute maximum is located in the outer mixing layer as generally in coaxial jet ¯ows. The presence of a recirculation zone and the same frequency for the dominant instability in the two mixing layers prove the strong coupling between the two shear-layers. This result agrees with previous experimental investigations. The purpose of this numerical study was the simulation of coherent structures in coaxial jets with a methodology close to the semi-deterministic modelling. The approach is able to simulate unsteadiness, so good perspectives are opened in the prediction of unsteady ¯ows at high-Reynolds numbers. AcknowledgementsÐThe authors are grateful to the SocieÂte EuropeÂenne de Propulsion for its support under Contract PRC/CNRS No. 90.0018 C and to the CNES which supports this work under the post-doctorate grant of P. Reynier.
REFERENCES 1. Ribeiro, M. M. and Whitelaw, J. H., Coaxial jets with and without swirl. Journal of Fluid Mechanics, 1980, 96, 769± 795. 2. Au, H. and Ko, N. W. M., Coaxial jets of dierent mean velocity ratios. Part 2. Journal of Sound and Vibration, 1987, 116, 427±443. 3. Dahm, W. J. A., Frieler, C. E. and Tryggvason, G., Vortex structure and dynamics in the near ®eld of a coaxial jet. Journal of Fluid Mechanics, 1992, 241, 371±402. 4. Ko, N. W. M. and Au, H., Initial region of subsonic coaxial jets of high mean velocity ratio. Journal of Fluids Engineering, 1981, 103, 335±338. 5. Choi, D. W., Gessner, F. B. and Oates, G. C., Measurements of con®ned coaxial jet mixing with pressure gradient. Journal of Fluids Engineering, 1986, 108, 39±46. 6. So, R. M. C., Yu, M. H., Otugen, M. V. and Zhu, J. Y., Rotation eects on inhomogeneous mixing in axisymmetric sudden-expansion ¯ows. International Journal of Heat Mass Transfer, 1987, 30, 2411±2421. 7. Habib, M. A. and Whitelaw, J. H., Velocity characteristics of con®ned coaxial jets with and without swirl. Journal of Fluids Engineering, 1980, 102, 47±53. 8. Chen, C. P., Calculation of con®ned swirling jets. Communications in Applied Numerical Methods, 1986, 2, 333±338.
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9. Dixson, T. F., Truelove, J. S. and Wall, T. F., Aerodynamics studies on swirled coaxial jets from nozzles with divergent quarls. Journal of Fluids Engineering, 1983, 105, 197±203. 10. Ko, N. W. M. and Chan, W. T., The inner regions of annular jets. Journal of Fluid Mechanics, 1979, 93, 549±584. 11. Wicker, R. B. and Eaton, J. K., Near-®eld of a coaxial jet with and without axial excitation. AIAA Journal, 1994, 32, 542±546. 12. Ko, N. W. M. and Au, H., Coaxial jets of mean velocity ratios. Journal of Sound and Vibration, 1985, 100, 211±232. 13. Gladnick, P. G., Enotiadis, A. C., Larue, J. C. and Samuelsen, G. S., Near-®eld characteristics of a turbulent co¯owing jet. AIAA Journal, 1990, 28, 1405±1414. 14. Tang, S. K. and Ko, N. W. M., Coherent structures interactions in an unexcited coaxial jet. Experiments in Fluids, 1994, 17, 147±157. 15. Moore, C. J., The role of shear-layer instability waves in jet exhaust noise. Journal of Fluid Mechanics, 1977, 80, 321±367. 16. Morrison, G. L. and McLaughlin, D. K., Instability process in low Reynolds supersonic jets. AIAA Journal, 1980, 18, 793±800. 17. DuraÄo, D., Turbulent mixing of coaxial jets. M.Sc. Thesis. University of London, London, 1971. 18. Wicker R. B. and Eaton J. K., Flow visualization study of the structure of a co-¯owing jet with and without controlled excitation. AIAA Paper, 93-0651, 1993. 19. MacCormack, R. B., A numerical method for solving the equations of compressible viscous ¯ow. AIAA Paper, 810110, 1981. 20. Okajima, A., Tiwata, T. and Ueno, H., Experimental and numerical studies on coaxial jets. In Proceedings of FLUCOME Conference. Toulouse, France, 1994. 21. Favre, A., Equations des gaz turbulents compressibles. Journes de MeÂcanique, 1965, 4, 361±390, 391±421. 22. Launder, B. E. and Sharma, B. I., Application of the energy dissipation model of turbulence to the calculation of ¯ow near a spinning disc. Letters Heat Mass Transfer, 1974, 1, 131±138. 23. Patel, V. C., Rodi, W. and Scheuerer, G., Turbulence models for near wall and low Reynolds number ¯ows: a review. AIAA Journal, 1985, 23, 1308±1319. 24. Ha Minh, H. and Kourta, A., Semi-deterministic turbulence modelling for ¯ows dominated by strong organized structures. In Proceedings of 9th International Symposium on Turbulent Shear Flows. Kyoto, 1993. 25. Ha Minh, H., The use of experiment for improving turbulence models. In Proceedings of FLUCOME Conference. Toulouse, France, 1994. 26. Nikjooy, M., Karki, K. C., Mongia, H. C., McDonell, V. G. and Samuelsen, G. S., A numerical and experimental study of coaxial jets. International Journal of Heat and Fluid Flow, 1989, 10, 253±261. 27. Harran, G., Analyse physique, modeÂlisation et simulation numeÂrique des meÂcanismes de meÂlange dans les jets simples et coaxiaux turbulents. Ph.D. Thesis. INPT No. 841, Toulouse, France, 1994. 28. Ribeiro, M. M., Turbulent mixing of coaxial jets. M.Sc. Thesis. University of London, London, 1972. 29. Poinsot, T. J. and Lele, S. K., Boundary conditions for direct simulations of compressible viscous ¯ows. Journal of Computational Physics, 1992, 101, 104±129. 30. Thompson, K. W., Time dependent boundary conditions for hyperbolic systems. Journal of Computational Physics, 1987, 68, 1±24. 31. Camano, E. B. and Favre-Marinet, M., On the initial region of inhomogeneous coaxial jets. In Advances in Turbulence V, ed. R. Benzi. Kluwer Academic, Amsterdam, 1994, pp. 58±62. 32. Hussain, A. K. M. F., Coherent structures and turbulence. Journal of Fluid Mechanics, 1986, 173, 303±356.