Bartosiewicz

8/2/2019 Bartosiewicz

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

Ejectors Modeling

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

1615, Boulevard Lionel-Boulet, Varennes (Qc.) J3X 1S6, Canada, Tel: +1 (450) 652 [email protected], [email protected]

P. Desevaux

CREST-UMR 6000, IGE-Parc Technologique, 90000, Belfort, France, [email protected]

Yves Mercadier

THERMAUS, Universit de Sherbrooke, 2500 Boul. Universit, Sherbrooke (Qc.), J1K2R1 Canada,

[email protected]

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 torefrigeration applications. The validation deals particularly with ejectors of zero induced flow. It concentrates on the prediction ofshock 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 numericalsimulation. 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, preliminarytests have also been performed. Laser tomography pictures were used to evaluate the non-mixing length. This parameter has beennumerically 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 highesttotal energy is the motive or primary stream (stream 1 on fig.1), while the other, with the lowest total energy (stream 2) isthe 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 nozzleto reach supersonic velocity. By an entrainment-inducedeffect, the secondary stream is drawn into the flow and

accelerated. Mixing, and recompression of the resulting streamthen occurs in a mixing chamber, where complex interactionstake place between the mixing layer and shocks. In otherwords, there is a mechanical energy transfer from the highestto 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 beenknown for a long time. Indeed, supersonic ejectors aretechnological components that have found many applicationsin engineering. In the aerospace area, they can be used for

altitude testing of a propulsion system by reducing thepressure of a test chamber [1]. The pumping effect is also used

to mix the exhaust gases with fresh air in order to reduce thethermal signature [2]. A mostresearched area is the

application of ejector for thrust augmentation on aircraftpropulsion systems [3,4].

Nevertheless, our primary interest in this paper is the use ofsupersonic ejectors for performing thermal compression in

refrigeration cycles. In view of the numerous publicationsavailable on this subject, it is perhaps one of the most


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important application areas for ejectors. A good overview ofthe different applications in this field may be found in thereview article of Sun and Eames [5]. In this case, ejectors may

either totally replace the mechanical compressors or simply beintroduced as a means of cycle optimization [6]. In thisparticular area, they have become the focus of renewedinterest for many scientists, in an attempt to develop energy

efficient and environment-friendly techniques, in response tothe current practices responsible for environmental damagessuch as ozone depletion or global warming.

Many theoretical and experimental studies have been carried

out in order to understand not only the fundamentalmechanisms in terms of fluid dynamics and heat transfer, butalso ejector operational behaviour. Nevertheless, most of thesestudies still rely on semi-empirical or one-dimensional

models. For theoretical works, Keenan et al. [7,8] set a firststep in the one-dimensional analysis by their model of aconstant area mixing flow for air and the still used concept ofconstant pressure mixing. They were followed by Munday and

Bagster [9], who introduced the fictive throat, which helped toexplain the characteristic ejector maximum capacity

limitations. Later, this kind of models was applied torefrigerants 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 stepforward by developing a one-dimensional model allowing to

track flow properties along the ejector. In their study, fluidproperties were evaluated by using NIST [12] subroutines for

equations of state of refrigerants.

Despite their usefulness and the remarkable progress theyprovided for the general understanding of ejectors, this kind of

studies are still unable to correctly reproduce the flow physicslocally along the ejector. Indeed, it is the understanding oflocal 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 ofgeometrical and operation conditions, for each fluid and eachapplication. In this way, some authors started to be interestedin the CFD approach, which may give at a reasonable cost, agood understanding of local phenomena. Even though

numerous CFD studies [13-19] about supersonic ejectors havebeen achieved since the 90s, some very fundamentalproblems have yet to be overcome, especially concerning themodeling of shock-mixing layer interaction, or ejector

operation through the different conditions. Indeed, some ofthese studies did not consider compressibility [13] orturbulence [14,15]. Moreover, even if some experimentalcomparisons have been performed, they just dealt with global

parameters, and no local validations were achieved. Most ofthem used a relatively poor mesh resolution preventing any

attempts to track shocks [13,16,17]. In cases where turbulencewas taken into account, no experimental validations or any

justifications, except CPU cost, on the use of a particularmodel 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 lasertomography. The authors compared the results with their

simulations. Nevertheless some adjustments for the pressure inthe 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 modelgave acceptable results compared to measurements.

Therefore, as it is the case in transonic compressible flowsinvolving shock reflections, shock-mixing layer interaction,etc, the choice of the turbulence model and grid refinements

are crucial points. In addition, when measurements areperformed, their influence should be evaluated or included inthe model. Consequently, this paper aims at validating thechoice of a turbulence model for the computation ofsupersonic ejectors in refrigeration applications. In order to

limit the complexity of the model and to be able to useavailable experimental data, the working fluid is single-phaseair. 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 numericallyevaluated

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 conservationequations. For variable density flows, the Favre averagedNavier-Stokes equations are more suitable and will be used inthis work. No assumptions are made on the Navier-Stokes

equation (parabolized) except for turbulence modelling. Thetotal energy equation including viscous dissipation is alsoincluded and coupled to the set with the perfect gas law. Thethermodynamics and transport properties for air are hold

constant; their influence was found to be not significant alongthe 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 aneddy viscosity assumption, which makes the Reynolds stresstensor coming from equation averaging, to be proportional tothe mean deformation rate tensor:

2' '

3i juu uji iu u kt t ij x x x j i i

dr m r m d

d

- = + - + (1)


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The advantage of this approach is the relatively lowcomputational cost associated with the determination of theturbulent 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 Launderand 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 kisderived from the exact equations of movement, while theequation for e was obtained using physical reasoning. It is thelatter 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

+ = +

+ - -

(2)

( ) ( )

2

1 2

t

j

uit x x xi j

C G Ckk k

e

e e

m ere r e m

s

e er

+ = +

+ -

(3)

Where kG and MY are respectively the production of

turbulence kinetic energy due to velocity gradients [20] andthe contribution of dilatation-dissipation in compressibleturbulence. 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 turbulentviscosity is computed from k and epsilon:

2

t

kCmm r

e= (4)

The model constants have the default values 1C e = 1.44, 2C e =

1.92, Cm = 0.09, ks = 1 and es = 1.3.

The Realizable k-epsilon model

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

epsilon is

( ) ( )

2

1 2

t

j

uit x x xi j

C S Ck

e

m er e r e m

s

er e r

ne

+ = +

+ -+

(5)

where

1 max 0.43,5

i j i j

C

kS

S S S

h

h

he

=

+

=

=

ij represents the strain rate tensor. The turbulent viscosity isexpressed with the same formulation than the standard model,

but Cm is here no longer constant and taken as a function of

the turbulent fields, strain rate, etc. Its new formulation can befound in [21]. The term Realizable means that the modelsatisfies certain mathematical constraints concerning Reynolds

stresses, in accordance with the turbulence physics, such aspositivity of normal stresses or Schwarz inequality. As aresult, 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 modelsuffers from a lack of extensive validations. Bartosiewicz et al.

[22] have tested this model for underexpanded jets, and nonoticeable improvements have been observed in comparison to

standards models.

The RNG-k-epsilon model (RNG)

The RNG model is derived from the exact Navier-Stokesequations, using a mathematical technique called

ReNormalization Group. The transport equation for epsilondiffers from the standard model by new analyticallydetermined constants and a new term:

( )

2

1 2

effj

uix xi j

C Gk

x

C Rk k

e

e e e

ree

a m

e er

=

+

- -

(6)

with

( )3 20

3

0

1

1

4.38, 0.012

CR

k

m

e

hr h eh

bh

h b

-=

+

= =

(7)

As a result, for weakly strained flow ( 0h h ), this additional

term has no contribution, and RNG tends to give comparableresults 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 thestandard k-epsilon model. In addition, the inverse effectivePrandtl numbers a are computed using analytical formula

derived by the RNG theory [23]. In the same way, the modelconstants are also derived analytically:

1 21.42, 1.68C Ce e= = . The eddy viscosity is computed

using the classical relation (equation (4)) with Cm = 0.0845.


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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

j

uit x x xi j

G Y

w

w w

m wr w r w m

s

+ = +

+ -

(8)

where Gw and Yw 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 performthese 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

( ) ( )

( )11

t

j

uit x x xi j

G Y F D

w

w w w

m wr w r w m

s

+ = +

+ - + -

(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 ofthe 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 ijk

u u u D D P

x

r f e

= + + + +

(11)

where TijD ,LijD , ijP , ijf and ije are, respectively, the

turbulent diffusion, laminar diffusion, stress production,

pressure strain and energy dissipation. TijD , 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 interactionbetween 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

r e=

=(12)

For k-omega models, it is included in Yw 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

= - >

=

(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.


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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 dissipationclose 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 isair 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

Manometer

Pressurecontrol valve

Orificeflowmeter

Orifice flowmeter

Surrounding

atmosphere

Atmospheric

pressure

Ejector

Induced flowratemeasurement

Airfilters

Compressed

air reservoir

Aircompressor

Primary flowratemeasurement

Absolute PressureTransducer

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

.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 isdepicted 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 withone-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


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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, thispressure 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 wellrepresented. At 0.14X = m the maximum difference incomparison 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 beenobserved experimentally.

Fig. 5: Axial pressure vs Measurements or several turbulencemodels.

In the following computations, a capillary probe is accounted

for the numerical model. This has been implemented byincluding 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 shockreflections become annular. Figure 6 shows the adapted mesharound 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 pressureprofiles in the case of the k-epsilon model (Fig.7 a) and theRNG model (Fig.7 b). In both cases, the correlation with

measurements has significantly improved, that showing theprobe has a noticeable effect on the flow as long as itrelatively small. Indeed the area ratio between the probe andthe mixing duct is about 1.7.10-2.

Fig. 6: adapted mesh around shocks and near the probesurface.

For the RNG model, the improvement is even better, becausethe reflections phase is in excellent agreement withmeasurements (Fig.7 b). The effect of adaptation is alsodemonstrated where sharper shock can be observed. In

addition, it is observed that errors are more important in


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expansions than in compressions. For the former thedifference is about 10%, while for the latter it is 35-50%. Onceagain, the presence of condensation in the flow may inhibit

compressibility effects by smoothing out the shock andexpansion waves. However, further investigations are thoughtto be necessary and will be performed in order to check thishypothesis. Nevertheless, shock locations and the average

pressure recovery seem to be correctly represented by theRNG model.

a)

b)

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

For all the other turbulence models, the modeling of thepressure probe provided clear improvement over the

computations without the probe insertion, but theirperformance was lower than that of the RNG.

Figure 8 compares the performance of the two k-omegamodels with that of the RNG. The standard k-omega modelover-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 pressurerecovery after the fifth compression. Farther downstream (notshown), both models give the same pressure. Howeveradditional tests are required to check which model performsbetter for ejector modeling.

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

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 firstcompression 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 samediscrepancy is observed for expansion amplitudes.Measurements are still a little more dissipative compared to

numerical results. However, pressure recovery is very wellpredicted, although the sst-k-omega model tends to under-predict the average line of pressure before eventually reachingthe same value (Fig.9 a).

For 1 6P = atm (Fig.9 b), shock locations are also well

represented, even though the dissipation effect ofmeasurements is more pronounced downstream of

0.1X = m. The two first shocks are under-predicted, and themodel fails to predict the location of the second shock asaccurately 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 aboutoptimal ejector operation and design.


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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 comparedwith experimental data. In these tests, the capillary pressureprobe is removed from the ejector, and a laser tomography

facility is used to visualize the flow in the mixing duct. Detailsof this technique can be found in [18,19]. From thesevisualizations, it is possible to establish the non-mixing lengthbetween the core and the secondary flows: The core of the

flow is darker than the fringes and the outer flow. As anumerical tracer, a passive scalar is used for comparison with

the laser tomography pictures. For a scalar Gthe steady statetransport equation can be written:

i

i i i

u

x x x

rmG G

G G = (14)

where r rG = and l t eff m m m mG = + = . This tool gives

qualitative information about the mixing regions (length ofnon-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 thesecondary 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 toincrease 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 designof 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)

RNG0.16 0.18 0.21

ml (computed) (m)

k-omega-sst0.14 0.17 0.21

23 6Error/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 zerosecondary flux so as to be concentrated on the problem ofshocks position, shocks strength and shock-boundary layerinteraction.

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


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fail in predicting expansions strength. This under-predictionmay be attributed to the occurrence of condensation, asobserved in the flow during the experiments. However,

computation-experiments iterations are necessary to clarifythis hypothesis. For these validations, the pressure probe hasalso been modeled, and its effect has been found to besignificant 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 asecondary flow, a numerical colorant has been used. Theresults have been compared with those of laser tomography

pictures. The secondary pressure has also been accounted forcomparisons. For the highest pressure, numerical results werein excellent agreement with experimental data. But, as theprimary pressure was decreased, some divergences between

the models and measurements were observed for thesecondary pressure. At this point, there is no satisfactoryexplanation 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 thecontrol valve (at the secondary inlet); in addition the tracers

(water particles due to condensation, oil droplets) used toevaluate the non-mixing length are probably non-ideal, andtheir diffusion coefficient is perhaps a function of local

conditions. At this point, the k-omega-sst model provides thebest overall results in so far as it is able to better account ofthe mixing between the two streams.

Nevertheless, further tests are required for an induced flowejector over a wide range of operation conditions until theejector chocking, and with more realistic boundary conditionsat the secondary inlet: indeed, in refrigeration, the mass flow

rate is generally not known a priori. At this inlet, theevaporator pressure should be prescribed, and the entrainmentratio should be computed along iterations.

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