CFD-Experiments Integration in the Evaluation of Six Turbulence Models for Supersonic Ejectors Modeling

Y. Bartosiewicz, Zine Aidoun CETC-Varennes, Natural Ressources Canada, P.O. Box 4800,

1615, Boulevard Lionel-Boulet, Varennes (Qc.) J3X 1S6, Canada, Tel: +1 (450) 652 0352 ybartosi@RNCan.gc.ca, zaidoun@RNCan.gc.ca

P. Desevaux

CREST-UMR 6000, IGE-Parc Technologique, 90000, Belfort, France, philippe.desevaux@univ-fcomte.fr

Yves Mercadier THERMAUS, Université de Sherbrooke, 2500 Boul. Université, Sherbrooke (Qc.), J1K2R1 Canada,

Yves.mercadier@Usherbrooke.ca

Abstract. Supersonic ejectors are widely used in a range of applications such as aerospace, propulsion, and refrigeration… This work evaluates the performance of six well-known turbulence models for the study of supersonic ejector performance and operation. The primary interest of this study is to set up a reliable hydrodynamics model of a supersonic ejector, which may be extended to refrigeration applications. The validation deals particularly with ejectors of zero induced flow. It concentrates on the prediction of shock location, shock strength and the average pressure recovery. In this respect, axial pressure measurements with a capillary probe and performed previously [18,19], have been used in this work for validation and comparison purposes with numerical simulation. A crucial point is that the probe has been included in the numerical model. In these conditions, the RNG and k-omega-sst models have been found to perform satisfactorily for these parameters. For ejectors with an induced flow, preliminary tests have also been performed. Laser tomography pictures were used to evaluate the non-mixing length. This parameter has been numerically evaluated by including an additional transport equation for a passive scalar, which acted as an ideal colorant in the flow. The results have shown significant departures from measurements for secondary pressure with the decrease of the primary pressure. In this condition, the k-omega-sst model has been found to account best for the mixing. Keywords: Shock waves, turbulence modeling, pressure measurements, supersonic ejector, refrigeration. 1. Introduction

Supersonic ejectors are simple mechanical components (Fig. 1), which generally allow to perform the mixing and/or the recompression of two fluid streams. The fluid with the highest total energy is the motive or primary stream (stream 1 on fig. 1), while the other, with the lowest total energy (stream 2) is the secondary or the induced stream. Operation of such systems is also quite simple: the motive stream (high pressure and temperature) flows through a convergent divergent nozzle to reach supersonic velocity. By an entrainment-induced effect, the secondary stream is drawn into the flow and accelerated. Mixing, and recompression of the resulting stream then occurs in a mixing chamber, where complex interactions take place between the mixing layer and shocks. In other words, there is a mechanical energy transfer from the highest to the lowest energy level, with a mixing pressure lying

between the motive or driving pressure and the induction pressure.

Ejectors for compressible fluids are not new and have been known for a long time. Indeed, supersonic ejectors are technological components that have found many applications in engineering. In the aerospace area, they can be used for altitude testing of a propulsion system by reducing the pressure of a test chamber [1]. The pumping effect is also used to mix the exhaust gases with fresh air in order to reduce the thermal signature [2]. A most–researched area is the application of ejector for thrust augmentation on aircraft propulsion systems [3,4].

Nevertheless, our primary interest in this paper is the use of supersonic ejectors for performing thermal compression in refrigeration cycles. In view of the numerous publications available on this subject, it is perhaps one of the most

important application areas for ejectors. A good overview of the different applications in this field may be found in the review article of Sun and Eames [5]. In this case, ejectors may either totally replace the mechanical compressors or simply be introduced as a means of cycle optimization [6]. In this particular area, they have become the focus of renewed interest for many scientists, in an attempt to develop energy efficient and environment-friendly techniques, in response to the current practices responsible for environmental damages such as ozone depletion or global warming.

Many theoretical and experimental studies have been carried out in order to understand not only the fundamental mechanisms in terms of fluid dynamics and heat transfer, but also ejector operational behaviour. Nevertheless, most of these studies still rely on semi-empirical or one-dimensional models. For theoretical works, Keenan et al. [7,8] set a first step in the one-dimensional analysis by their model of a constant area mixing flow for air and the still used concept of constant pressure mixing. They were followed by Munday and Bagster [9], who introduced the fictive throat, which helped to explain the characteristic ejector maximum capacity limitations. Later, this kind of models was applied to refrigerants and the possibility of the fictive throat being placed in the constant area zone, as assumed by Huang et al. [10]. Very recently, Ouzzane and Aidoun [11] set a step forward by developing a one-dimensional model allowing to track flow properties along the ejector. In their study, fluid properties were evaluated by using NIST [12] subroutines for equations of state of refrigerants.

Despite their usefulness and the remarkable progress they provided for the general understanding of ejectors, this kind of studies are still unable to correctly reproduce the flow physics locally along the ejector. Indeed, it is the understanding of local interactions between shock waves and boundary layers, and their influence on the mixing and recompression rate, that will allow a more reliable and accurate design, in terms of geometrical and operation conditions, for each fluid and each application. In this way, some authors started to be interested in the CFD approach, which may give at a reasonable cost, a good understanding of local phenomena. Even though numerous CFD studies [13-19] about supersonic ejectors have been achieved since the 90’s, some very fundamental problems have yet to be overcome, especially concerning the modeling of shock-mixing layer interaction, or ejector operation through the different conditions. Indeed, some of these studies did not consider compressibility [13] or turbulence [14,15]. Moreover, even if some experimental comparisons have been performed, they just dealt with global parameters, and no local validations were achieved. Most of them used a relatively poor mesh resolution preventing any attempts to track shocks [13,16,17]. In cases where turbulence was taken into account, no experimental validations or any justifications, except CPU cost, on the use of a particular model were carried out. Very recently, Desevaux et al. [18,19] obtained some local pressure measurements along the ejector with a capillary tube, and flow visualizations with laser tomography. The authors compared the results with their simulations. Nevertheless some adjustments for the pressure in the vacuum chamber or mass flow rate at the secondary inlet

were necessary to reproduce reasonably well ejector operation. In this case, they concluded that the standard k-epsilon model gave acceptable results compared to measurements.

Therefore, as it is the case in transonic compressible flows involving shock reflections, shock-mixing layer interaction, etc, the choice of the turbulence model and grid refinements are crucial points. In addition, when measurements are performed, their influence should be evaluated or included in the model. Consequently, this paper aims at validating the choice of a turbulence model for the computation of supersonic ejectors in refrigeration applications. In order to limit the complexity of the model and to be able to use available experimental data, the working fluid is single-phase air. In this respect, six turbulence models, namely k-epsilon, realizable k-epsilon, RNG-k-epsilon, RSM and two k-omega are tested and compared with measurements of Desevaux et al. [18,19]. The effect of the capillary tube is also numerically evaluated

Fig. 1: Ejector geometry

2. Modeling approach

2.1 Governing equations

The flow from a circular nozzle is governed by the steady-state axisymmetric form of the fluid flow conservation equations. For variable density flows, the Favre averaged Navier-Stokes equations are more suitable and will be used in this work. No assumptions are made on the Navier-Stokes equation (parabolized) except for turbulence modelling. The total energy equation including viscous dissipation is also included and coupled to the set with the perfect gas law. The thermodynamics and transport properties for air are hold constant; their influence was found to be not significant along the validation runs.

2.2 Turbulence modeling

Most of the turbulence models used in this paper rely on the Boussinesq hypothesis. It means that they are based on an eddy viscosity assumption, which makes the Reynolds stress tensor coming from equation averaging, to be proportional to the mean deformation rate tensor:

2' '

3i j

uu uji iu u kt t ijx x xj i i

dr m r m d

d

æ ö¶ æ ö¶ ÷ç ÷ç÷ç- = + - + ÷ç÷ç ÷÷÷ ç¶ ¶ ÷ç è øè ø (1)

The advantage of this approach is the relatively low computational cost associated with the determination of the turbulent viscosity. However, the main drawback of this hypothesis is that it assumes the turbulence to be isotropic. The following description of the models k-epsilon, RNG-k-epsilon and k-omega is based on this hypothesis.

The k-epsilon and Realizable k-epsilon (Rk-epsilon) models

The standard k-epsilon model was first proposed by Launder and Spalding [20] and is a semi-empirical model based on transport equations for the turbulence kinetic energy ( k ) and its dissipation rate ( e ). The model transport equation for k is derived from the exact equations of movement, while the equation for e was obtained using physical reasoning. It is the latter that provides weakness or strength of the considered model. The equations for k and epsilon are the following

( ) ( )

ti

i j k j

k M

kk ku

t x x x

G Y

mr r m

s

r e

é ù¶ ¶ ¶ æ ö ¶÷çê ú+ = + ÷ç ÷çê úè ø¶ ¶ ¶ ¶ë û+ - -

(2)

( ) ( )

2

1 2

t

juit x x xi j

C G Ckk k

e

e e

m ere r e m

s

e er

é ù¶ ¶ ¶ æ ö ¶÷çê ú+ = + ÷ç ÷çê úè ø¶ ¶ ¶ ¶ë û

+ -

(3)

Where kG and MY are respectively the production of turbulence kinetic energy due to velocity gradients [20] and the contribution of dilatation-dissipation in compressible turbulence. The formulation of the latter will be further detailed. 1C e , 2C e are calibration constants and ks , es , the turbulent Prandtl number for k and epsilon. The turbulent viscosity is computed from k and epsilon:

2

tk

C mm re

= (4)

The model constants have the default values 1C e = 1.44, 2C e = 1.92, C m = 0.09, ks = 1 and es = 1.3.

The Realizable k-epsilon model

This model is relatively recent [21]. The transport equation for epsilon is deduced from an exact equation for the mean vorticity fluctuations. In this respect, the new formulation for epsilon is

( ) ( )

2

1 2

t

juit x x xi j

C S Ck

e

m ere r e m

s

er e r

ne

é ù¶ ¶ ¶ æ ö ¶÷çê ú+ = + ÷ç ÷çê úè ø¶ ¶ ¶ ¶ë û

+ -+

(5)

where

1 max 0.43,5

ij ij

C

kS

S S S

hh

he

é ù= ê ú

+ê úë û

=

=

ijS represents the strain rate tensor. The turbulent viscosity is expressed with the same formulation than the standard model, but C m is here no longer constant and taken as a function of

the turbulent fields, strain rate, etc. Its new formulation can be found in [21]. The term “Realizable” means that the model satisfies certain mathematical constraints concerning Reynolds stresses, in accordance with the turbulence physics, such as positivity of normal stresses or Schwarz inequality. As a result, the model is supposed to better resolve the round jet anomaly, or boundary layers with adverse pressure gradients. However, contrary to older models, the R-k-epsilon model suffers from a lack of extensive validations. Bartosiewicz et al. [22] have tested this model for underexpanded jets, and no noticeable improvements have been observed in comparison to standards models.

The RNG-k-epsilon model (RNG)

The RNG model is derived from the exact Navier-Stokes equations, using a mathematical technique called “ReNormalization Group”. The transport equation for epsilon differs from the standard model by new analytically determined constants and a new term:

( )

2

1 2

effj

uix xi j

C Gk

x

C Rk k

e

e e e

r ee

a m

e er

¶ ¶=

¶ ¶

+

é ù¶ê úê ú¶ë û

- -

(6)

with

( )3 2

03

0

1

14.38, 0.012

CR

km

e

hr h ehbh

h b

-=

+= =

(7)

As a result, for weakly strained flow ( 0h h» ), this additional term has no contribution, and RNG tends to give comparable results than the k-epsilon model. But in regions of large strain rate ( 0h h> ), the Re term may have a significant contribution, which yields a lower turbulent viscosity than the standard k-epsilon model. In addition, the inverse effective Prandtl numbers a are computed using analytical formula derived by the RNG theory [23]. In the same way, the model constants are also derived analytically:

1 21.42, 1.68C Ce e= = . The eddy viscosity is computed using the classical relation (equation (4)) with C m = 0.0845.

The standard and SST-k-omega models

This approach was first proposed by Wilcox [24] and consists in replacing the equation for epsilon by a transport equation for kw e= . This equation, in the case of the standard

formulation is

( ) ( )

t

juit x x xi j

G Y

w

w w

m wrw rw m

sé ù¶ ¶ ¶ æ ö ¶÷çê ú+ = + ÷ç ÷çê úè ø¶ ¶ ¶ ¶ë û

+ -

(8)

where Gw and Y w are the production and the dissipation of omega respectively. The former has the same form than the k-epsilon model and is then also evaluated in a manner consistent with the Boussinesq hypothesis [24]. The latter includes the compressibility effects and will be further explained. The turbulent viscosity is evaluated by combining k and w :

*t

krm a

w= (9)

where the coefficient *a is a function of the turbulence Reynolds number, and provides a damping of the turbulent viscosity in low Reynolds regions; In high Reynolds regions,

* 1a = . This model works very well inside the boundary layer, but it has to be abandoned in the wake region and outside. The reason is that the standard k-omega model has a very strong sensitivity to the freestream conditions [25].

The SST (Shear Stress Transport) version [26]of the k-omega model overcomes this deficiency because it is based on a blending approach. Indeed, to achieve the different desired features, the standard k-epsilon, turned into a k-omega formulation, is used from the wake region and outside; while the original model is used in the near wall region. To perform these features, a blending function 1F is designed to be one inside the boundary layer and to change gradually to zero in the wake region. The resulting equation for w is

( ) ( )

( )1 1

t

juit x x xi j

G Y F D

w

w w w

m wrw r w m

sé ù¶ ¶ ¶ æ ö ¶÷çê ú+ = + ÷ç ÷çê úè ø¶ ¶ ¶ ¶ë û

+ - + -

(10)

The term Dw comes from the transformation of the k-epsilon model into a k-omega model [26]. It can be view as an additional cross diffusion term. Details on the expressions of

1F and Dw are provided in [26]. In addition, the eddy viscosity is redefined so as to take into account the transport of the principal turbulent shear stress [26].

The Reynolds Stress Model (RSM)

For this model, Reynolds stresses are directly computed by their transport equations, and the isotropic eddy viscosity assumption (Boussinesq hypothesis) is no longer used:

( )' ' T Lk i j ij ij ij ij ij

ku u u D D P

xr f e

¶= + + + +

¶ (11)

where TijD , L

ijD , ijP , ijf and ije are, respectively, the turbulent diffusion, laminar diffusion, stress production, pressure strain and energy dissipation. T

ijD , ijf and ije require to be modeled, contrary to the other terms. Further details about the RSM model and evaluation of various terms may be found in Bartosiewicz et al. [27]. Moreover, it has been found [27,22] that the RSM model gives better agreement with measurements for underexpanded jets than the other k-epsilon based models. Based on the results of these investigators, it is reasonable to assume that the use of that model in the case of supersonic ejector, where the flow downstream the primary nozzle is actually an underexpanded (or over) jet may be successful.

Wall functions

When appropriate, standards logarithmic wall functions are considered. Special care is given to the first cell location, by a local adaptation following y + , such as 30y + £ . If

11y + < , the classical linear law is taken for the sublayer. This treatment is consistent with the use of standard wall functions.

2.3 Compressibility corrections

An additional term should be used to model the interaction between turbulence and compressibility. Indeed, it is well known that the spreading rate of a mixing layer is decreased with the Mach number [28]. In the k-epsilon based models, and RSM, this term is included in MY (equation (2)):

2

2

2M t

t

Y M

kM

c

re=

= (12)

For k-omega models, it is included in Y w and it consists in replacing tM by a function ( )tF M :

( )

t 0

2 20 t 0

0

0 if M

if M

0.25

t

tt t t

t

MF M

M M M

M

£ìïïï= íï - >ïïî=

(13)

The advantage of this latter is that it does not take effect everywhere in the flowfield, but just in some places where compressibility becomes important.

2.4 Algorithm

The governing equations are solved using the commercial CFD package FLUENT. In this respect, they are discretized using a control volume technique. For all equations, convective terms are discretized using a second-order upwind scheme. Inviscid fluxes are derived using a second order flux splitting, achieving the necessary upwinding and dissipation close to shocks. The resulting system is time-preconditioned, in order to overcome the numerical stiffness encountered at low Mach number. The discretized system is solved in a coupled way with a Block Gauss Seidel algorithm. The time marching procedure uses a first order implicit Euler scheme, where the time step is set up by a Courant-Friedrichs-Lewy (CFL) number. In a second step, other scalar equations such as turbulence are solved in a segregated fashion. In addition, an adaptative unstructured mesh is used to better track shock waves and gradients. For each case, a grid convergence study is performed to get minor differences between the final and the previous adaptation step. The computation is stopped when mass and energy residuals are small enough and global conservations reached.

3. Experimental setup

3.1 Flow facility

The experimental installation is schematized in Figure 2. A screw compressor of sufficient capacity is used to ensure the continuous operation of the ejector. Compressed air (at a maximum pressure of 8 bars) is filtered to remove large particles such as dust, and compressor oil droplets. The compressed air is then directed towards an air reservoir, which is connected to the entrance of the primary nozzle of the ejector just after passing through a pressure control valve to adjust the primary stagnation pressure P1. The induced fluid is air taken from the surrounding atmosphere. The induced mass flow rate m2 can be regulated by means of a valve located at the entrance of the aspiration duct. Apparatus installed on the primary and secondary air circuits for measurement of stagnation pressures and flow rates is also shown in figure 2.

Manometer

ManometerPressure

control valve

Orificeflowmeter

Orifice flowmeter

Surrounding

atmosphere

Atmospheric

pressure

Ejector

Induced flowratemeasurementAir

filters

Compressed

air reservoir

Aircompressor

Primary flowrate measurement

Absolute Pressure Transducer

Fig. 2: Experimental apparatus.

3.2 Centerline pressure measurement system

Static pressure measurements along the ejector centerline were performed by means of a sliding measuring system [29]. The probe, as shown in figure 3, consists of a capillary tube (external diameter of 1 mm, internal diameter of 0.66 mm) located on the ejector axis. Static pressure is measured through a hole of 0.3 mm diameter, which crosses the tube radially, and the value is then transmitted to an absolute pressure transducer. The translation of the tube provides static pressure measurements along the pseudo-shock region (between the inlet section of the primary nozzle to the exit section of the diffuser (section 4).

Capillary tube

Static pressure hole

.

Outlet

P1

P2

Fig. 3: Capillary probe to measure the centerline pressure.

Measurements were taken for three different pressures at the primary inlet: 4atm, 5atm, and 6atm respectively. Concerning the comparison with the numerical results presented in this paper, some tests were performed with no induced flow but with the probe insertion; and some with an induced flow without the probe insertion, for each pressure cited above.

4. Results and discussion

4.1 Flow Physics and boundary conditions

The ejector geometry with its characteristics dimension is depicted Fig.1. At some appropriate level of the exit pressure ( 3P ), the primary stream (1) is accelerated through a convergent-divergent nozzle to a supersonic velocity. The secondary stream (2) is then drawn by an entrainment effect. Depending on the geometrical design, the resulting stream may reach sonic speed again inside the constant area duct, although it is not the case in this study. Along this duct, the flow follows a succession of normal and/or oblique shock waves, also called a shock train region (Fig.4), and involving a pressure rise [30]. Farther downstream, the pressure inside the motive stream is adjusted with that of the secondary stream to attenuate the shocks until they disappear. Following this zone, the primary stream may be still supersonic as observed by [27]. In addition, it is noted that the pressure still increases until a maximum value is reached. This fact is consistent with one-dimensional theory for a supersonic flow inside an adiabatic duct with friction. The region of the total pressure rise, from the outlet of the primary nozzle, is referred as “pseudo-shock” [30]. Downstream of this point, the motive jet becomes subsonic and friction makes the pressure decrease. Depending on this pressure recovery, the flow should have sufficient energy in terms of total pressure to reach the final

pressure ( 3P ) at the diffuser. Further details on the different flow regions can be found in [30].

At inlets (labels 1 and 2 on Fig. 1), total pressures, total temperatures, and flow direction consistent with flow characteristics (subsonic) are prescribed. At exit (label 3), the pressure is fixed when the flow is subsonic, and extrapolated from interior when the flow is supersonic. In refrigeration, this pressure is that of the condenser, while 1P is the generator pressure and 2P is the evaporator pressure. In this study all the walls are considered to be adiabatic with no slip. For turbulence, it is considered that inlet flows have a turbulence intensity of 5% that matches with the value in a fully developed pipe.

Constant-area duct

Edge ofboundarylayer

Separation region

Pseudo-shock region

Mixing regionShock train region

.

..

3

2

1

Staticpressure

Distance x

Pressure riseby shock train

Pressure peaks

Pressure riseby pseudo-shock

Fig. 4: Description of the different flow regions.

4.2 Results: operation with no secondary flow

For these tests, there is no secondary inlet. The motive total pressure is successively set to 4, 5, 6 atm, and the total temperature is kept constant at 300 K. In order to validate the choice of the turbulence model, the case for 1 5P = atm is taken as a reference. The selected model will then be applied for other operating pressures. In all cases, the exit pressure is kept constant equal to 3 1P = atm.

Figure 5 illustrates the results for the axial pressure based on different turbulence models, plotted from the primary nozzle outlet. It is clear that neither k-epsilon based models (k-epsilon, R-k-epsilon or RNG) nor RSM model are able to represent correctly the shock reflection pattern. Numerical results almost show phase opposition for the shock reflections. However, the qualitative evolution of pressure recovery is well represented. At 0.14X = m the maximum difference in comparison with measurements is about 6%. In addition, a difference of 20% is observed at the outlet of the primary nozzle. For this case, the one-dimensional theory gives a ratio

10.075P

P = at the nozzle exit. The discrepancy between this

value and the predicted numerical value is approximately

6.5 %. The difference may due to condensation that has been observed experimentally.

Fig. 5: Axial pressure vs Measurements or several turbulence models.

In the following computations, a capillary probe is accounted for the numerical model. This has been implemented by including a solid core around the axis. The probe diameter is 1mm, and coupled heat transfer conditions are applied on it. The probe changes the flow topology, making the shock reflections become annular. Figure 6 shows the adapted mesh around the probe, where it is possible to guess the location of incident and reflected waves.

Figure 7 presents the effect of the probe on the axial pressure profiles in the case of the k-epsilon model (Fig.7 a) and the RNG model (Fig.7 b). In both cases, the correlation with measurements has significantly improved, that showing the probe has a noticeable effect on the flow as long as it relatively small. Indeed the area ratio between the probe and the mixing duct is about 1.7.10-2.

Fig. 6: adapted mesh around shocks and near the probe surface.

For the RNG model, the improvement is even better, because the reflections phase is in excellent agreement with measurements (Fig.7 b). The effect of adaptation is also demonstrated where sharper shock can be observed. In addition, it is observed that errors are more important in

expansions than in compressions. For the former the difference is about 10%, while for the latter it is 35-50%. Once again, the presence of condensation in the flow may inhibit compressibility effects by smoothing out the shock and expansion waves. However, further investigations are thought to be necessary and will be performed in order to check this hypothesis. Nevertheless, shock locations and the average pressure recovery seem to be correctly represented by the RNG model.

a)

b)

Fig. 7: Effect of the probe on computational results.

For all the other turbulence models, the modeling of the pressure probe provided clear improvement over the computations without the probe insertion, but their performance was lower than that of the RNG.

Figure 8 compares the performance of the two k-omega models with that of the RNG. The standard k-omega model over-predicts shock amplitude, and fails in predicting shock location after the fourth peak. This confirms observations of [25] on the use of k-omega model outside boundary layers. The k-omega-sst model gives comparable results to those of RNG, but it tends to under-predict the average line of pressure recovery after the fifth compression. Farther downstream (not shown), both models give the same pressure. However additional tests are required to check which model performs better for ejector modeling.

Fig. 8: Comparison between the RNG and the two k-omega models.

Two other motive pressures have been tested for these two turbulence models. Figure 9 shows results for 1 4P = (a) and

1 6P = atm. (b). For 1 4P = atm, the jet issuing from the primary nozzle is strongly overexpanded, the first compression is strong. For 1 6P = atm, it becomes underexpanded and the flow firstly proceeds to an expansion. The first compression is smoothed because it occurs in the more confined mixing duct. For 1 4P = atm, shock locations are in very good agreement with experimental data. The same discrepancy is observed for expansion amplitudes. Measurements are still a little more dissipative compared to numerical results. However, pressure recovery is very well predicted, although the sst-k-omega model tends to under-predict the average line of pressure before eventually reaching the same value (Fig.9 a).

For 1 6P = atm (Fig.9 b), shock locations are also well represented, even though the dissipation effect of measurements is more pronounced downstream of

0.1X = m. The two first shocks are under-predicted, and the model fails to predict the location of the second shock as accurately as for the remaining shocks. In addition the total pressure available at 0.14X = m is larger for the strongly overexpanded case of 1 4P = atm, positively impacting on ejector operation in case of refrigeration applications. However, additional studies are needed to investigate about optimal ejector operation and design.

a)

b)

Fig. 9: Centerline pressure for 1 4P = atm and 1 6P = atm.

4.3 Preliminary results for an induced flow ejector

In this part, the secondary mass flow rate is experimentally kept constant ( 2 0.028m =& kg/s) by means of a control valve. In the model, the mass flow rate is also fixed, the secondary pressure being computed to match this flow rate is compared with experimental data. In these tests, the capillary pressure probe is removed from the ejector, and a laser tomography facility is used to visualize the flow in the mixing duct. Details of this technique can be found in [18,19]. From these visualizations, it is possible to establish the non-mixing length between the core and the secondary flows: The core of the flow is darker than the fringes and the outer flow. As a numerical tracer, a passive scalar is used for comparison with the laser tomography pictures. For a scalar G the steady state transport equation can be written:

i

i i i

ux x x

rmG

G¶ G ¶ æ ¶ G ö÷ç= ÷ç ÷çè ø¶ ¶ ¶

(14)

where r rG = and l t effm m m mG = + = . This tool gives qualitative information about the mixing regions (length of non-mixing), and even about the quality (under certain conditions) of mixing by checking G values. At the primary inlet 0G = is fixed, while 1G = is prescribed at the secondary inlet. Finally, walls are considered non-permeable to this numerical colorant.

Table 1 illustrates results for the computed secondary pressure 2P and for the non-mixing length. The non-mixing length is

numerically evaluated by the location at which Gstarts to increase from zero. Both models give comparable results concerning the secondary pressure 2P . It is shown that for

1 6P = atm the agreement with experimental data is excellent for both the secondary pressure and the non-mixing length. For 1 5P = and 1 4P = atm the models under-predict 2P of 23% and 24% respectively. It is also shown that the k-omega-sst gives better performance for the prediction of the non-mixing length. This parameter is very important in the design of ejectors in refrigeration.

1P (atm) 4 5 6

2P (measured) (atm) 0.78 0.68 0.4

2P (computed) (atm) 0.61 0.52 0.4

ml (measured) (m) 0.13 0.17 0.21

Measurement error (+/- %) 15 12 9.5

ml (computed) (m) RNG

0.16 0.18 0.21

ml (computed) (m) k-omega-sst

0.14 0.17 0.21

23 6 Error/measurements

(%) 8 0

0

Tab. 1: Non-mixing length and secondary pressure.

5. Conclusion

This paper deals primarily with the validation of six turbulence models for the computation of supersonic ejector. This validation has first been applied to the case with zero secondary flux so as to be concentrated on the problem of shocks position, shocks strength and shock-boundary layer interaction.

It has been shown that the RNG and k-omega-sst models are the best suited to predict the shock phase, strength, and the mean line of pressure recovery. However, all models seem to

fail in predicting expansions strength. This under-prediction may be attributed to the occurrence of condensation, as observed in the flow during the experiments. However, computation-experiments iterations are necessary to clarify this hypothesis. For these validations, the pressure probe has also been modeled, and its effect has been found to be significant in spite of the fact that much care was taken by reducing the probe to not disturb the flow.

In order to study the mixing, for an ejector operating with a secondary flow, a numerical colorant has been used. The results have been compared with those of laser tomography pictures. The secondary pressure has also been accounted for comparisons. For the highest pressure, numerical results were in excellent agreement with experimental data. But, as the primary pressure was decreased, some divergences between the models and measurements were observed for the secondary pressure. At this point, there is no satisfactory explanation to this departure from experimental data. However, there are some aspects that cannot be modeled such as pressure losses induced by the upstream fluid flow and the control valve (at the secondary inlet); in addition the tracers (water particles due to condensation, oil droplets) used to evaluate the non-mixing length are probably non-ideal, and their diffusion coefficient is perhaps a function of local conditions. At this point, the k-omega-sst model provides the best overall results in so far as it is able to better account of the mixing between the two streams.

Nevertheless, further tests are required for an induced flow ejector over a wide range of operation conditions until the ejector chocking, and with more realistic boundary conditions at the secondary inlet: indeed, in refrigeration, the mass flow rate is generally not known a priori. At this inlet, the evaporator pressure should be prescribed, and the entrainment ratio should be computed along iterations.

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