Applied Thermal Engineering 209 (2022) 118177
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Applied Thermal Engineering
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Research paper
Large Eddy Simulation of a supersonic air ejector
Sergio Croquer a ,∗, Olivier Lamberts b , Sébastien Poncet a , Stéphane Moreau a , Yann Bartosiewicz b
a
b
Department of Mechanical Engineering, Université de Sherbrooke, 2500 boulevard de l’Université, J1K 2R1, Sherbrooke (QC), Canada
Université Catholique de Louvain (UCLouvain), Institute of Mechanics, Materials and Civil Engineering (iMMC), Louvain-la-Neuve, 1348, Belgium
ARTICLE
Keywords:
Supersonic ejectors
Large Eddy Simulation
Turbulence
Refrigeration
INFO
ABSTRACT
This paper presents a study on the flow topology in the mixing chamber of a supersonic ejector using Large
Eddy Simulation (LES). To this end, a supersonic air ejector of squared crossed-section was modelled using a
specialized finite-element code. Comparisons with experimental data showed good agreement, both in terms
of the primary jet shock cell structures and wall pressure measurements (mean deviation of 12%). Results
have been discussed both in terms of time averaged profiles and instantaneous structures in the mixing layer.
The general flow features have been identified by means of instantaneous temperature fields and pressure
profiles through the device. Results show that, under the assessed conditions, the mixing layer is laminar at
first and transitions towards turbulence in the first quarter of the mixing chamber, where 𝛬 vortices have
been identified. These evolve into hairpin vortices and finally break down around half of the mixing chamber.
Time-averaged velocity profiles show self-similarity in this section. In comparison with an unconfined mixing
layer (Fang et al., 2018), the supersonic ejector mixing layer grows slower first but then develops at a similar
rate after the transition region. A shock train occurs towards the end of the mixing chamber, which enhances
mixing. Given its location, it generates a recirculation bubble in the diffuser which narrows the main flow
passage and breaks the flow vertical symmetry. This pioneer study shows the enormous potential that LES
offers for the optimization and detailed analysis of supersonic ejectors.
1. Introduction
The supersonic ejector is a simple device which uses the exergy
of a high pressure flow to compress a secondary stream. It does not
have any moving parts and its design is relatively simple. Hence, it has
attracted research interest during the last decade as a means to reduce
the compressor load in refrigeration systems or to recover the work
normally loss across the throttling stage. Other popular applications
include: novel CO2 capture systems [1], desalination processes [2],
blast furnace gas plants [3] and low-pressure gas reservoirs [4]. The
interested reader is referred to the book of Grazzini et al. [5] and the
reviews of Aidoun et al. [6] for a thorough discussion on its applications
and state of the art.
In a supersonic ejector, the primary (motive) flow is accelerated in
a converging–diverging nozzle. The resulting supersonic jet discharges
into a mixing chamber, dragging along the secondary flow and transferring exergy to it through a mixing layer. Then, the streams go
through a series of shock waves before entering the diffuser, where
they are compressed to the outlet pressure. The performance of supersonic ejectors is commonly assessed in terms of their entrainment
ratio (𝜔𝑟 , the proportion of secondary to primary mass flow rates) and
their compression ratio (𝛱, the proportion of outlet to secondary inlet
pressures). Optimization of these parameters is key, since they have
a direct effect on the efficiency of the whole system [7]. Within this
context, the general device behaviour with regards to the operating
conditions [8], interactions with the rest of the system [9] and the
influence of certain geometrical parameters such as the mixing chamber
length [10] and the area ratios [11] is already well known. From a
global perspective, ejector optimization and design using data driven
models such as Artificial Neural Networks have shown promising results for both single-phase [12] and condensing steam ejectors [13].
Nonetheless, these tools are difficult to extrapolate outside of their
training range and provide no information regarding the underlying
physics. Hence, from a local perspective, recent experimental and
numerical efforts have focused on the relationships between 𝜔𝑟 , 𝛱 and
specific flow characteristics. In this sense, different flow visualization
techniques and wall pressure measurements have been used to study
the influence of the turbulent intensity and spreading rate of the mixing
layer [14], the primary jet non-mixed and core lengths [15,16], the
primary jet condensation fraction [17] and the secondary flow choking type [18]. Furthermore, Reynolds-Averaged Navier–Stokes (RANS)
modelling of supersonic ejectors has become a popular tool to study
the internal flow structure in detail [19]. For instance, in calculating
∗ Corresponding author.
E-mail address: sergio.croquer@usherbrooke.ca (S. Croquer).
https://doi.org/10.1016/j.applthermaleng.2022.118177
Received 27 October 2021; Received in revised form 27 January 2022; Accepted 3 February 2022
Available online 23 February 2022
1359-4311/© 2022 Elsevier Ltd. All rights reserved.
Applied Thermal Engineering 209 (2022) 118177
S. Croquer et al.
Fig. 1. Schematic view of the ejector with relevant notations.
the relationships between shock intensity and exergy losses [20] and
drawing the paths of momentum and exergy through the device [21].
Given its relatively low computational cost, RANS has been adopted
for geometrical optimization, particularly in combination with data
driven algorithms [22]. However, RANS only offers an approximate
description of the mean flow, and is very limited when it comes to
the analysis of turbulent quantities. The next step for the fundamental
study of ejectors and a better quantification of the transfer mechanisms
inside would be the application of more advanced turbulence closures.
Considering the advances in computational power available today, a
feasible approach is Large Eddy Simulation (LES), which has shown
its capacity for the accurate analysis of supersonic mixing layers [23]
and acoustic wave propagation of free supersonic jets [24–26], among
many other applications. To the best of our knowledge, the sole available publication on the analysis of supersonic ejectors using LES is
a brief paper by Bouhanguel et al. [27]. However, its objective was
to propose LES as a better tool to study flow instabilities inside the
device rather than doing a complete flow analysis. Hence, it leaves
out many important details regarding numerical setup and only shows
preliminary results. It also makes a point regarding the complexity of
LES of supersonic ejectors, given that these devices are characterized
by high Mach numbers (𝑀𝑎) and high Reynolds numbers based on the
nozzle diameter or height (𝑅𝑒).
With the main goal of better understanding the supersonic ejector
flow topology and enhance their performance, this study presents an
analysis of the mixing layer characteristics of a supersonic ejector
using LES. For this, a supersonic air ejector operating at 𝑃𝑝0 = 5 bar,
𝑃𝑠0 = 0.974 bar and 𝑃𝑜𝑢𝑡 = 1.2 bar has been modelled using a specialized finite-element code [28]. A further description of the device
and the flow characterization are provided in Section 2. Then, the
numerical setup is described in Section 3. Results are presented and
discussed in Section 4, including a comparison with Schlieren imagery
and wall pressure measurements for experimental validation, as well
as discussions on the main flow features, time averaged profiles, turbulent structures and secondary flow choking. Closing comments are
summarized in Section 6.
directly drawn from the atmosphere. The outflow is discharged out of
the laboratory. The ejector backpressure is tuned with a butterfly valve
placed at the outlet of the diffuser. Pressure, temperature and mass
flow rates are measured at both inlets and the ejector outlet. Endres
Hauser and Kistler transducers are used for pressure measurements
(uncertainty < ±300 Pa) whereas PT100 RTD probes are used for temperature measurement (uncertainty < ±0.5 °C). The mass flow rates are
measured using plate orifices of known geometry. Moreover, the static
pressure along the constant area section and the diffuser is measured
at the locations shown in Table 2. Furthermore, the ejector side panels
are made of Plexiglass to allow for flow visualization. Specifically, the
shock patterns in the mixing chamber are captured using a slightly
modified schlieren Z-type setup with a light source of 150 W and two
mirrors characterized by a 𝜆∕4 surface flatness [29]. Two cameras
were used for capturing the experimental images. A Nikon D5000
with a spatial resolution of about 25 pixels∕mm and a exposure time of
1 × 10−3 s, and LaVision HighSpeedStar 3G with a spatial resolution of
5.5 pixels∕mm and a exposure time of 6.67 × 10−5 s.
The operating conditions summarized in Table 1 were chosen for the
numerical setup. Following isentropic calculations at these conditions,
the motive jet leaves the primary nozzle with velocity 𝑈𝑗 =516.6 m s−1
and temperature 𝑇𝑗 =223 K, which leads to a Reynolds number based on
𝐻𝑗 𝑅𝑒𝑗 =4.14 × 105 and a Mach number 𝑀𝑎𝑗 =1.72. The air properties
at the nozzle exit are: 𝜇𝑗 =1.582 × 10−5 kg m−1 s and 𝜌𝑗 =1.52 kg m−3 .
Based on these values, a characteristic flow through time could be
𝐿
defined as 𝑡𝑐 = 𝑈𝑒𝑗 =0.003 s.
𝑗
3. Numerical methods
Calculations were carried out using the finite element solver AVBP
[28], developed as a joint effort by CERFACS and IFPen. This solver
was originally developed to tackle LES of reacting and non-reacting
flows on unstructured grids as well as accurately solve acoustic wave
propagation problems in supersonic jets [24,26], landing gears [30],
high-lift devices [31] and turbomachines [32] at both high Mach and
Reynolds numbers. Therefore it is considered properly suited for this
study. The solver uses a Two-Step Taylor Galerkin (TTG4 A) high-order
explicit scheme, which is 3rd order in space and 4th order in time [33].
The explicit subgrid scale WALE model [34], is used to model the
subgrid scale effects to yield the proper asymptotic behaviour at the
walls. Air thermodynamic properties are modelled assuming ideal gas
behaviour.
2. Flow configuration
The numerical domain is based on an experimental supersonic
ejector test bench located at the Thermodynamic and Fluid Mechanics
Laboratory of the Université Catholique de Louvain (Belgium). The test
facility is described in detail by Lamberts [29]. Fig. 1 presents a side
view of the ejector profile. The device has a 50 mm wide rectangular cross-section. Other ejector dimensions are given in Table 1. The
working fluid is air, which is stored in a pressurized tank at 16 bar and
served to the primary inlet via a pneumatic valve which regulates the
motive inlet pressure to the desired value. The secondary flow is air
Boundary conditions
A single LES case was performed using the operating conditions
shown in Table 1. Total pressure, total temperature and static pressure
values were imposed at both inlets and outlet using Navier–Stokes
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Applied Thermal Engineering 209 (2022) 118177
S. Croquer et al.
Table 1
Ejector characteristics.
Dimensions
Operating conditions
Parameter
Length [mm]
Parameter
Value
Nozzle throat height, 𝐻𝑡
Nozzle exit height, 𝐻𝑗
Nozzle exit Position, 𝑁𝑋𝑃
Mixing chamber height, 𝐻𝑚
Ejector exit height, 𝐻𝑜
Mixing chamber length, 𝐿𝑚
Ejector length, 𝐿𝑒𝑗
6.0
8.0
18.4
27.0
125.0
280.6
1520.0
Primary inlet total pressure, 𝑃𝑝0
Primary inlet total temperature, 𝑇𝑝0
Secondary inlet total pressure, 𝑃𝑠0
Secondary inlet total temperature, 𝑃𝑝0
Outlet pressure, 𝑃𝑜𝑢𝑡
Entrainment ratio, 𝜔𝑟
Compression ratio, 𝑃 𝑟
5 bar
300 K
0.974 bar
300 K
1.2 bar
0.44
1.23
Table 2
Wall pressure sensor locations through the ejector.
𝑋
𝐿𝑚
Domain initialization and convergence
= 0 is the start of the constant
area section.
Probe
X [mm]
𝑋
𝐿𝑚
1
2
3
4
5
6
7
8
9
1
9
17
25
33
41
49
57
65
0.004
0.032
0.061
0.089
0.118
0.146
0.175
0.203
0.232
[−]
Probe
X [mm]
𝑋
𝐿𝑚
10
11
12
13
14
15
16
17
18
73
113
153
193
233
273
293
301
309
0.260
0.403
0.545
0.688
0.830
0.973
1.044
1.073
1.101
[−]
Probe
X [mm]
𝑋
𝐿𝑚
19
20
21
22
23
24
25
26
27
317
325
333
350
390
507
624
741
858
1.130
1.158
1.187
1.247
1.390
1.807
2.224
2.641
3.058
The calculations were initialized using a preliminary RANS result
obtained with the 𝑘 − 𝜔 SST turbulence model on the top half of the
geometry (assuming symmetry at 𝑌 = 0). The converged result was
interpolated onto the finer grid used for the LES. The LES was initially
ran until stable values of pressure, temperature, total energy and mass
flow rates were observed through the inlet/outlet boundaries and at
different locations through the domain. From that point onward, the
simulation was ran for about 2𝑡𝑐 (𝐻𝑗 ∕𝑈𝑗 ∼ 380) to collect the flow
statistics. For this, velocity, density, temperature and pressure were
sampled at a frequency of 10 MHz at various locations along the mixing
section and diffuser. A CFL condition 𝐶𝐹 𝐿 ≤ 1.0 was imposed through
the calculations, which lead to a mean time step size under 0.002𝑡𝑐 .
Computations were carried out on the Mammouth Parallele-2B
supercomputer at Université de Sherbrooke, which is managed by
Calcul Québec and Calcul Canada. The special allocation for this project
included 200 AMD Opteron 6172 nodes, each with 31 GB RAM and 24
cores. Using this configuration, the simulation of one flow-through time
(𝑡𝑐 ∼0.003 s) took about 30 days.
[−]
characteristic boundary conditions, which allow compressible waves
to cross the domain boundaries based on the method of characteristics [35]. Additionally, element size was doubled in the regions
adjacent to inflow and outflow boundaries to further ensure numerical
stability [24]. No turbulence is imposed at the inlets at this stage
since no experimental information is available. Walls were modelled as
adiabatic with the adjacent flow velocity imposed using a logarithmic
wall law (wall-modelled LES). Translational periodicity was set in the
transverse direction, with no mass flow or energy penalization.
4. Results
4.1. Comparison with experimental data
Fig. 3 compares a Schlieren image of the primary jet and a corresponding ‖∇𝜌‖ field obtained numerically. The LES captures the main
flow features well. In particular, the double expansion compression fan
arising from the nozzle end, and the location of the shock waves and
their reflections. Nonetheless, the shear layer in the experimental image
appears to be more turbulent already at the nozzle lip. Meanwhile, the
LES, which had no turbulence injected at the primary inlet boundary
condition, shows a quasi-laminar shear layer that becomes unstable by
the appearance of Kelvin–Helmholtz roll ups and vortex pairing further
downstream, which marks the transition towards turbulence. As the jet
leaves the primary nozzle, a series of shock cells are observed, which
indicates that the jet is underexpanded. The first shock cell is well
defined and undisturbed but as the mixing layer widens, the vortices
disturb the jet core flow and the fourth cell becomes very difficult to
distinguish. Observations of the primary jet structure in air supersonic
ejectors using Planar Laser Mie Scattering have captured between 1
and 4 cells in the jet non-mixed length, the number and size of these
structures depending mainly on the inlet pressure ratios [15]. At the
nozzle lip, two rarefaction fans are observed. The first one (slightly
darker) is generated by the geometry change. Whereas the second one
occurs due to the interactions of the incoming secondary flow in the
near wall region with the mixing layer. Observation of subsequent
instantaneous results show that fluctuations in the incoming secondary
flow momentarily disturb the thickness of the secondary fan.
Fig. 4 compares the experimental and numerically obtained static
pressure profiles along the ejector constant area section and diffuser.
The experimental data corresponds to time-averaged values from two
runs taken at two different days with similar ambient conditions. The
time-averaged LES profile was produced by recording the pressure at
several points along the centre line. The shaded area represents the
Grid
The computational domain is based on the as built dimensions of
the experimental ejector, i.e., considering the small deviations from the
nominal values listed in Table 1. For instance, the CAS crossed section
reduces by about 0.3% from inlet to outlet, and the top secondary inlet
crossed section is 0.9% smaller than the bottom one. An unstructured
grid made of tetrahedral elements with 13 prismatic layers adjacent to
wall boundaries was used. The nominal element length scale (𝑙∕𝐻𝑗 )
was set to 0.02 at the nozzle exit and to 0.01 at the nozzle lip. Thus, the
grid has about 6 points across the mixing-layer momentum thickness
at the nozzle lip, which is approximated as 𝛿0 ∼ 0.03𝐻𝑗 [36]. From
these locations, element size increases linearly up to 0.025 at 1∕3 of
the diffuser length. The mesh grid configuration and sizes were chosen
based on the ranges selected in previous simulations of supersonic jets
using the unstructured solver AVBP [26,36]. Furthermore, they lie in
the range ∕𝐻𝑗 ≫ 𝑙∕𝐻𝑗 ≫ 𝜂𝐾 ∕𝐻𝑗 proposed by Bellan [37], where
∕𝐻𝑗 ∼0.1 to 0.5 is the integral length scale to jet height ratio and
𝜂𝐾 ∕𝐻𝑗 ∼3.0 × 10−5 to 5.2 × 10−5 is the Kolmogorov length scale to
jet height ratio for 𝑅𝑒𝑗 =4.14 × 105 . The average dimensionless wall
adjacent element lengths in the streamwise, crosswise and spanwise
directions (𝛥𝑥+ × 𝛥𝑦+ × 𝛥𝑧+ ) were, respectively, 11.4 × 6.6 × 12.4 at the
nozzle tip, 61.1 × 15.7 × 45.6 at the start of the constant area section and
72.7×22.0×63.7 at the start of the diffuser. Details of the resulting mesh,
which has in total 237 × 106 elements and 55 × 106 nodes, are shown in
Fig. 2. Tetrahedral elements adapted to the shear layer spreading have
been used from the nozzle exit. These have similar aspect rations, which
also ensures a proper development of the jet in all directions [26].
3
Applied Thermal Engineering 209 (2022) 118177
S. Croquer et al.
Fig. 2. Details of the mesh grid used in the LES computations.
Fig. 3. Details of the flow structure at the exit of the primary nozzle. Comparison between Schlieren imagery (experimental) and a (numerical) pseudo-Schlieren ‖∇𝜌‖.
min/max range of oscillation in the instantaneous LES results. The
RANS profile corresponds to a 2D steady state simulation using the
numerical setup of Lamberts et al. [21] with the 𝑘 − 𝜔 SST turbulence model. A thorough comparison of the LES time-averaged field
versus the RANS results, showing that the LES mean velocity profiles
tends towards the RANS results is provided in [38]. Both numerical
approaches provide similar pressure profiles. At the start of the mixing
chamber (𝑥∕𝐿𝑚 = 0), the LES and RANS time-averaged pressures are,
respectively, 8% and 9% lower than the average experimental values.
Then, between 0 ≤ 𝑥∕𝐿𝑚 ≤ 0.25, the pressure profiles for both LES
and RANS expand further, which corresponds with the overexpanded
jet configuration found in Fig. 3 (LES result), whereas the experimental
data remains almost constant (the maximum variation is +2% relative
to the initial reading). In the mixing chamber until the shock onset
location at 𝑥∕𝐿𝑚 ∼ 1.1, all pressure profiles stabilize. The mean deviation between the time-averaged LES and the experimental data is −21%.
Along this region, the RANS values coincide with the maximum in the
LES instantaneous oscillations (the top of the shaded area). Also, the
difference in between the experimental points grows at the same rate
than the range of the LES fluctuations. It must be noted that, although
the inlet pressure and temperature conditions are similar among the
experimental cases, the secondary inlet air was taken directly from
the atmosphere and thus certain properties such as air humidity were
not controlled. This would help explain the difference between the
experimental results along the region 𝑥∕𝐿𝑚 ≤ 0. Moreover, the experimental test section of the ejector was not isolated. Thus, the adiabatic
assumption in the numerical cases can explain, in part, the differences
with the experimental results. Heat transfer towards the supersonic
flow in the constant area section of the ejector would decelerate the
flow and increase its pressure in comparison with the numerical case.
Moreover, the smaller difference between the RANS and LES (time
averaged) results can be explained by recalling that there was no
induced turbulence in the LES results, leading to a slightly faster flow
and thus, lower average pressure. Further downstream, the shock train
Fig. 4. Comparison of static pressure profiles through the constant area section and
diffuser. The shaded area indicates the variation range of the LES instantaneous results.
is marked by a disturbance in the experimental data around 𝑥∕𝐿𝑚 = 1
yielding first a dip in pressure followed by a sudden pressure increase.
This trend is well captured by the LES time-averaged result. The RANS
model predicts the jump location earlier shortly before 𝑥∕𝐿𝑚 = 1 with
only a consequent pressure increase. Through the diffuser, both LES and
RANS results come closer to the experimental profile. In this section,
the mean deviation between the time-averaged LES values and the
experimental data is −8%. Note that the LES instantaneous pressure
variation increases significantly in this section. As will be shown below,
flow detachment at the beginning of the diffuser leads to the creation
of relatively big turbulent vortices in this area. These lead to important
transverse effects in the diffuser. Thus, the following sections limit the
analysis to the flow topology through the mixing section.
4.2. Main flow features
Fig. 5(a) presents an instantaneous static temperature field over
a streamwise midspan plane. This provides an overview of the flow
4
Applied Thermal Engineering 209 (2022) 118177
S. Croquer et al.
Fig. 5. Instantaneous static temperature field over a streamwise midspan plane. (a) Whole ejector. (b) Close up of the shock train region.
structure through the device, which is typical of supersonic ejectors
according to RANS studies [20] and experimental observations [15].
Namely: a supersonic jet discharging into a mixing section, wherein
both inlet flows interact through the mixing layer, followed by a series
of shocks and the final expansion in the diffuser as shown in the close
up of Fig. 5(b). The flow in the mixing chamber can be considered as
a confined jet. For this type of jet, three stages can be recognized [39]:
close to the nozzle exit the prevailing mixing mechanism is a thin
shear layer, followed by jet mixing (when the shear layer thickness
occupies as much duct crossed section as the potential core) and finally,
turbulent pipe flow. In ejectors working under single-choke regime,
shear mixing followed by pipe flow has been observed using Schlieren
and laser scattering techniques [40]. However, no pipe flow regime is
observed in Fig. 5 since the primary jet potential core extends over
most of the constant area section. This is an indication that the ejector
operates in double-choke regime, which is characterized by a shock
train towards the end of the mixing section when the ratio 𝑃𝑝0 ∕𝑃𝑜𝑢𝑡
is high enough [20]. The choking regime for the assessed ejector
configuration is discussed in detail in Section 4.4. The mixing layer
develops slowly at first and the exchange surface between the streams
becomes important towards the end of the constant area section. At
this point, the shock train enhances mixing and it becomes difficult to
separate the high and low temperature regions.
An interesting feature in Fig. 5(b) is the fact that the shock train
deviates towards the upper wall of the diffuser. Numerical simulations
of supersonic ejectors often assume that the flow is symmetric around
the main axis or the midspan plane. In fact, the LES computation was
initialized using a 3D RANS steady state using this assumption. However, this shock train shifted at the start of the LES run and stabilized as
shown in Fig. 5. A 2D RANS simulation considering the whole domain
in the vertical direction (i.e. no symmetry at 𝑦 = 0) showed the same
trend [38]. The reason for this behaviour might be the recirculating
zone on the lower side of the diffuser, which constraints the flow
passage. In this sense, the preliminary results of Bouhanguel et al.
[27] showed that, in double-choke operation, the shock train enters
the diffuser and flaps between the top and bottom under the effect
of recirculation zones around the supersonic flow region. However,
neither the experimental nor the numerical results in the present study
showed evidence of this flapping phenomenon. A possible explanation
for this behaviour might be the slight geometrical asymmetries in the
domain.
A close up of the shock train is shown in Fig. 6, which depicts
the magnitude of the static pressure gradient along with streamlines
from the primary and secondary inlets. This train is characterized by
an important first shock followed by smaller ones. The analysis of
subsequent images showed that the first shock location moves around
a centre position, taking ∼ 1.75𝑡𝑐 to return to its starting location [38].
These oscillations have been linked to pressure disturbances generated
as the boundary layer becomes turbulent [41]. As seen in Fig. 6, the
first shock cell imposes oblique pressure jumps which accelerate the
flow in the vertical direction and draw the streamlines closer together.
Moreover, as the main flow passage is narrowed across each shock,
detachment is observed near the walls. This leads to the development
of the large recirculating zone pointed out in Fig. 5 and the flow
asymmetry in the diffuser. Despite this large recirculation zone the
primary and secondary flows remain parallel along the diffuser.
Time averaged profiles
A discussion on the mean flow characteristics through the mixing
section will be presented in the following. These profiles were obtained by processing the velocity statistics collected during 2𝑡𝑐 (roughly
380𝐻𝑗 ∕𝑈𝑗 ) once the simulation stabilized. First, Fig. 7 shows the mean
streamwise velocity ⟨𝑈 ⟩ profiles at subsequent 𝑥∕𝐿𝑚 locations (made
dimensionless by the centreline mean streamwise velocity ⟨𝑈𝑐 ⟩). In
Fig. 7(a), the velocity profile just after the nozzle exit (𝑥∕𝐿𝑚 = −0.04)
has a prominent maximum at the centre. This is explained by the fact
that the rarefaction fans stemming from the nozzle lip meet at this
point. After that, the profiles between −0.02 ≤ 𝑥∕𝐿𝑚 ≤ 0.18 (Fig. 7(b))
alternate between a double peaked and a smooth shape as they cross
the mixing layer. The velocity dips close to the centre correspond to
the shock diamond expansion waves. Furthermore, as the mixing layer
grows, the profiles change from top hat to bell shaped. The mean
streamwise velocity profiles show self-similarity further down the tube
(𝑥∕𝐿𝑚 ≥ 0.46), particularly around −0.1 ≤ 𝑦∕𝐻𝑚 ≤ 0.1. Furthermore,
as the main flow diverges towards the top wall in the diffuser, the
peaks of the ⟨𝑈 ⟩ profiles also move along 𝑦 ≥ 0. This is clear when
comparing the superposition of the profiles around 𝑦 ∼ 0.2 in contrast
with their separation around 𝑦 ∼ −0.2 in Fig. 7(d). Regarding the
⟨𝑈 ⟩
tails of the profiles, the mean velocity tends to 𝑈 ∼ 0.2 for the
𝑐
most part of the mixing chamber. This is in contrast with free jets
profiles, where the velocity tends towards zero. This reflects the effect
of the jet confinement and its acceleration as the flow expands through
the mixing section. Such profile characteristics, namely the tail ends
⟨𝑈 ⟩
separation and the limiting 𝑈 end values have been reported in the
𝑐
self-similar region of confined jets using hot-wire measurements [42].
The development of the mixing layer is shown in Fig. 8, which
depicts the root-mean-square (rms) of the velocity fluctuations in the
vertical direction at five positions along the constant area section.
Slightly after the nozzle lip, at 𝑥∕𝐿𝑚 = 0, fluctuations are very weak.
Energy transfer between the primary and secondary flow at this location is weak and dominated by molecular diffusion. Further down
the constant area section, at position 𝑥∕𝐿𝑚 = 0.25, the maxima in
the profile mark the vertical location of both shear layers. The peak
𝑣′rms ∕⟨𝑈 𝑐⟩ in this position is ∼ 0.005. Then, at positions 𝑥∕𝐿𝑚 = 0.5
and 0.75, the intensity of the fluctuations does not augment but they
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S. Croquer et al.
Fig. 6. Pressure gradient magnitude (∇𝑝) field showing the instantaneous shock structure at the end of the constant area section.
Fig. 7. Vertical profiles of the mean streamwise velocity ⟨𝑈 ⟩∕⟨𝑈𝑐 ⟩ along the ejector constant area section. ⟨𝑈𝑐 ⟩ is the mean streamwise velocity at the centreline.
Fig. 8. Profiles of the normalized crosswise rms velocity 𝑣′rms ∕⟨𝑈 𝑐⟩ at different locations through the constant area section.
where 𝑈1 =467 m s−1 and 𝑈2 =156 m s−1 are, respectively, the average
velocities of the primary and secondary flows at the nozzle exit location. In Fig. 9, 𝛿𝑤0 is the vorticity thickness at the NXP. The top of
Fig. 9 shows an isocontour of the Q-criterion coloured by the rotational
direction (the rotation vector would be normal to the paper), green
indicates clockwise rotation and pink counterclockwise rotation. The
experimental data of Fang et al. [23] for an unconfined supersonic mixing layer has been added as a reference with the objective of assessing
the effects of jet confinement. Regarding the supersonic ejector results,
both mixing layers evolve similarly. Mild differences are observed at
the end of the constant area section, where the shock train and the main
flow shift towards the upper wall affect each mixing layer differently.
The growth rate of the vorticity thickness is characterized by the slope
cover a wider section of the ejector which indicates the extent of the
mixing layer. Shortly after 𝑥∕𝐿𝑚 = 0.5 the boundary layers developing
on both walls start interacting with the mixing layer. Towards, 𝑥∕𝐿𝑚 =
1 the symmetry in the profile starts to break down and the whole
ejector crossed section is turbulent. The vertical fluctuations intensity
is increased, specially because of the oblique shock waves which, as
mentioned above,
√ create a pressure gradient in the vertical direction.
The maximum 𝑣′2 ∕⟨𝑈𝑐 ⟩ intensity has increased to 0.01 with the
fluctuations on the bottom (𝑦∕𝐻𝑚 < 0) being slightly stronger.
The vorticity thickness of the top and bottom mixing layers is shown
in Fig. 9. The vorticity thickness 𝛿𝑤 is calculated as [43]:
𝛿𝑤 =
𝑈1 − 𝑈2
𝜕⟨𝑈 ⟩ |
|
𝜕𝑦 |𝑚𝑎𝑥
(1)
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Applied Thermal Engineering 209 (2022) 118177
S. Croquer et al.
similar shape but opposing rotating sense are generated at the top and
bottom nozzle lips. Small vortices are generated just at the nozzle lips,
but these are not advected into the main flow. Instead, they are engulfed by the recirculation zones created at the blunt ends of the nozzle.
The close up of the top nozzle lip structures in Fig. 10(a) shows the
appearance of small hairpin vortices which are rapidly dissipated by the
incoming flow. As pointed out in Fig. 3, the secondary expansion fans
are a result of these recirculation pockets disturbing the jet boundary.
As the vortices are generated and destroyed in this region, the thickness
of the secondary expansion fans changes. Moreover, these fans cross
the jet and interact with the opposite jet boundary. At the point of
this interaction, the shear layer is destabilized and roller type vortices
are observed. These are clearly seen at the start of the second shock
diamond. These roller vortices lead to Kelvin–Helmholtz structures with
counter rotating cores typical of shear mixing layers [44,45].
Further downstream, 𝛬-type coherent structures are formed close to
the third shock cell. A close up of such structures is shown in Fig. 10(b).
Their formation (roll-up) leads to the first slope change point (𝑥∕𝐿𝑚 ∼
0.14) in the vorticity thickness profile shown in Fig. 9. LES of supersonic
mixing layers show that these vortices transform into hairpin vortices
further downstream [43]. Fig. 10(c) presents a close up of a hairpin
vortex in the bottom mixing layer. In comparison with the LES data
of Fang et al. [23], these appear elongated in the streamwise direction,
probably under the effect of shear and flow confinement. These hairpin
are eventually broken into smaller vortices through the second half of
the ejector.
Fig. 9. Vorticity thickness (𝛿𝑤 , Eq. (1)) of the top and bottom mixing layers.
Comparison with the LES results of Fang et al. [23] for a free supersonic mixing layer.
𝛿𝑤0 is the vorticity thickness at the NXP.
changes marked with vertical lines in Fig. 9. The latter reveal changes
𝜕⟨𝑈 ⟩
in the main shear rate ( 𝜕𝑦 ) which, in turn, are related to vortex
formation and roll-up [43]. Shortly after the nozzle exit the mixing
layer grows at a slow pace, confirming that both flows are laminar at
the nozzle exit position. Then, for 0.14 ≥ 𝑥∕𝐿𝑚 ≥ 0.32, the mixing
layer becomes unstable and starts transitioning towards turbulence.
Structures of increasing size are observed in this region, in particular
𝛬 type vortices. In contrast with the first section where the shock
cells in the jet core are clearly distinguishable, the shear layers in this
section are wide enough that the shock cells break down and disappear.
The structures observed along the mixing layer will be discussed in
detail in the next section. The vortex pairing of the 𝛬 type structures
triggers the transition to turbulence and leads to the rapid increase in
vorticity thickness along 0.14 ≤ 𝑥∕𝐿𝑚 ≤ 0.32 [23], and the narrowing
of the jet potential core as observed in free jets. Further downstream,
between 𝑥∕𝐿𝑚 ∼ 0.32 and 𝑥∕𝐿𝑚 ∼ 0.67, the mixing layer is fully
turbulent. Recall that the velocity profiles in Fig. 7 become self-similar
in this region. The vorticity thickness grows at a slower pace than
in the previous region because the contribution by vortex pairing is
balanced by the confinement within the mixing chamber. Moreover,
the boundary layers appearing close to 𝑥∕𝐿𝑚 ∼ 0.5 reach the mixing
layer. As these two regions interact, 𝛿𝑤 on its own cannot be used
to interpret the behaviour of the mixing layer. Nonetheless, it can be
observed that 𝛿𝑤 decreases in the region 𝑥∕𝐿𝑚 ≥ 0.67, meaning that the
vertical streamwise gradient augments.
In comparison with the free mixing layer of Fang et al. [23], both
profiles augment rapidly at the start of the mixing section. Then, the
free mixing layer slightly decays and plateaus at 𝑥∕𝐿𝑚 ∼ 0.10, which
was related with the formation and sustaining of hairpin vortices over a
short period before breaking up into smaller vortices. After the hairpin
vortices break up, the free-mixing layer becomes self-preserved and
its growth rate is almost constant. In contrast, the supersonic ejector
mixing layer has a higher growth rate in the region 0.1 ≤ 𝑥∕𝐿𝑚 ≤ 0.32,
which indicates that the flow is still transitioning towards turbulence.
Nonetheless, in the following section (𝑥∕𝐿𝑚 ∼ 0.32 and 𝑥∕𝐿𝑚 ∼ 0.67)
the 𝛿𝜔 slope reduces and approximates the corresponding growth rate of
the free mixing layer in the self-preserving region (these are indicated
by the red dashed lines). Similarly, the mean velocity profiles of Fig. 7
show self-similarity in this region.
4.4. Compound choking
Finally, as was noted previously, the present ejector flow condition
does not lead to a fully turbulent pipe flow and the primary jet potential
core extends all the way to the diffuser at the end of the mixing
chamber, yielding two separate streams. Indeed, Fig. 11(a) shows a
Mach number contour over the ejector midspan plane for the averaged
flow along with the dividing streamlines and the sonic line (the isoline
𝑀𝑎 = 1). The dividing streamline is defined as the streamline passing
through a point between the primary and secondary flow at the nozzle
exit plane [21]. By definition, there is no mass flow across this line.
Hence, it establishes a boundary between the primary and secondary
flows through the ejector. Given than the ejector in this study has a
rectangular crossed section, two separate dividing streamlines are constructed. Although this configuration implies a greater shear perimeter,
the difference in entrainment ratio in comparison with a circular nozzle
ejector should be small [46].
These two streamlines define three regions: the top secondary flow,
the primary flow and the bottom secondary flow. Following their paths
in Fig. 11(a), the three flows remain practically horizontal through
the constant area section. At the diffuser, the primary flow deviates
slightly and remains parallel to the top wall, in agreement with the
instantaneous structure shown in Fig. 5. With regards to the 𝑀𝑎
contour, as the primary jet interacts with the secondary flows, the area
with 𝑀𝑎 > 1 increases but it does not reach the ejector walls. According
to the sonic line criterion (also known as Fabri-choking criterion), this
would mean that the secondary flows do not reach the sonic condition
and thus, the ejector works in single-choke regime. However, if this
were the case, no shock train should be observed at the start of the
diffuser. This shock train indicates the ejector works in double-choke
regime. From an operational point of view, the double-choke regime
is preferred since it maximizes the ejector entrainment ratio. Thus,
being able to accurately determine the working condition for a given
operating point is important in the analysis of supersonic ejectors.
For compound flows (i.e. multiple parallel streams with different
inlet conditions), Bernstein et al. [47] shows that the overall flow can
4.3. Turbulent structures in the mixing layer
Fig. 10 provides a qualitative visualization of the turbulent structures generated along the mixing layers using isocontours of the Qcriterion coloured by the vorticity rotation direction, i.e.: considering
a rotation vector parallel with the spanwise direction, green indicates
clockwise rotation and pink counterclockwise rotation. The density
gradient magnitude field has also been added to indicate the position
of the primary jet shock cells. As expected, two mixing layers with very
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Applied Thermal Engineering 209 (2022) 118177
S. Croquer et al.
Fig. 10. Instantaneous Q-criterion contour at the start of the constant area section along with details of (a) the recirculation zone at the nozzle lip, (b) 𝛬-vortices and (c) hairpin
vortices in the mixing layer. The structures are coloured by vorticity rotation direction. i.e.: considering a rotation vector parallel with the span wise direction, green indicates
clockwise rotation and pink counterclockwise rotation.
Fig. 11. (a) Average Mach number field through the ejector along with the sonic line (𝑀𝑎 = 1) in thin red and the dividing streamlines in black. (b) Individual 𝑀𝑎 and compound
choke criterion (𝛽) profiles through the constant area section.
behave as a single sonic or supersonic flow even if some of the substreams remain subsonic. This can be verified by using the compound
choke indicator 𝛽 which, for perfect gases, is [47]:
(
)
𝑛
∑
𝐴𝑖
1
𝛽=
−1 ,
(2)
𝛾
𝑀𝑎2𝑖
𝑖=1 𝑖
as in double-choke regime even though the secondary flow is not
choked [48,49]. If the flow behaves as compound-sonic or compoundsupersonic, the ejector behaves as in double-choke and the entrainment
ratio remains independent of changes in the back-pressure.
The 𝛽 parameter for the compound flow, along with the individual
𝑀𝑎 profiles for the three flows along the constant area section are
shown in Fig. 11(b). The individual 𝑀𝑎 profiles were calculated by
considering that each flow is confined within the corresponding dividing streamlines and/or the ejector walls and averaging the velocity at
subsequent crossed sections. Concerning the primary flow, it remains
supersonic through the constant area section as expected. It becomes
subsonic after the shock trains in the diffuser. On the other hand, the
secondary flows enter the mixing constant area section in subsonic
where 𝐴𝑖 , 𝛾𝑖 and 𝑀𝑎𝑖 are the cross-sectional area, the heat capacity
ratio, and the Mach number of each substream (in this case: topsecondary, primary and bottom-secondary) respectively. The subscript
‘‘i’’ represents the individual sub-streams. Based on this parameter, the
total flow can be regarded as compound-subsonic (𝛽 > 0), compoundsonic (𝛽 = 1) or compound-supersonic (𝛽 < 0). Using this theory, it has
been shown that close to the limiting pressure the ejector might behave
8
Applied Thermal Engineering 209 (2022) 118177
S. Croquer et al.
regime but are accelerated at different rates. The secondary top flow
barely surpasses 𝑀𝑎 > 1 at 𝑥∕𝐿𝑚 = 0.2 whereas the secondary bottom
flow becomes supersonic right at the entrance of the mixing chamber
reaching 𝑀𝑎 = 1.3 at 𝑥∕𝐿𝑚 = 0.2. At this location the three flows
are closed to their top speeds and the 𝛽 parameter has a minimum.
The higher velocity for the secondary bottom flow might be related to
the deviation of the main jet towards the upper wall of the diffuser.
Both secondary flows begin their deceleration and become subsonic at,
respectively, 𝑥∕𝐿𝑚 = 0.8 and 𝑥∕𝐿𝑚 = 0.9. The shock train, which can
be located based on the bumps in the primary flow 𝑀𝑎 profile begins
at about this location.
With regards to the 𝛽 parameter, this is an indicator of the sonic
regime of the (overall) compound flow. It rapidly decreases and becomes negative just before the constant area section, indicating the
location where the flow becomes compound-sonic, even though at this
point the secondary top flow is still subsonic. 𝛽 remains negative for
most of this region and increases rapidly along with the deceleration
of the secondary flows. It becomes positive again after the first shock
(𝑥∕𝐿𝑚 ∼ 0.9). From that point onward, the flow is compound-subsonic,
event though 𝑀𝑎 = 1.5 for the primary flow. Thus, based on the 𝛽
indicator, the flow configuration shown here corresponds to an ejector
in double-choke regime. The fact that the primary flow shows a slower
deceleration and becomes subsonic later in the diffuser is in agreement
with the 𝑀𝑎 contour of Fig. 11(a).
this regard is the use of tabulated equations of state, which have shown
to greatly reduce the time dedicated to property calculation at each
time step in RANS models of 𝐶𝑂2 ejectors [52].
Another point of improvement is the full resolution of the near
wall flow. In this study, a 2.5D wall-modelled approach was chosen.
However, the imposed near wall velocity profile is based on a standard
logarithmic law of the wall, which does not consider complex phenomena such as high compressibility and adverse pressure gradients.
Adopting a full 3D wall-resolved LES approach would be ideal to
remove these simplifications, however this would greatly increase the
mesh element count in the order of 𝑅𝑒13∕7 [53], which would be
impossible to handle with the existing computational resources.
Finally, in terms of applicability, the higher computational cost of
LES makes it difficult to use this approach for geometrical optimization,
specially when comparing several designs at once. An alternative to
circumvent this would be to shortlist a few design alternatives using a
less expensive approach (such as RANS or thermodynamic modelling)
and then apply LES for the final comparison. Similarly, all in all, it is
evident that LES offers greater insight and a more detailed description
of the flow through the ejector than RANS. Its true limitation being the
currently available computational power.
6. Conclusions
A pioneer study on the flow topology in the mixing chamber of
a supersonic ejector using LES has been presented. To this end, a
supersonic air ejector of squared crossed-section was modelled using
a specialized finite-element code. Comparisons with experimental data
showed good agreement, both in terms of the primary jet shock cell
structures and wall pressure measurements (mean deviation of 12%).
Results were discussed both in terms of time averaged profiles and
instantaneous structures in the mixing layer. The general flow features
have been identified by means of instantaneous temperature fields and
pressure profiles through the device. The following conclusions can be
drawn:
5. Discussion
The results presented above show the main value of LES over
RANS for the study of supersonic ejectors. In essence, RANS provides
the mean flow characteristics. Thus, it is not surprising that the time
averaged LES results are very similar to the RANS solution. However,
the statistical nature of RANS prevents a fine description of any kind of
transfer mechanism [50]. More so when turbulence plays an important
role in these mechanisms. On the other hand, LES is an unsteady
approach which only models the structures smaller than the filter
threshold (i.e., the local grid size) while fully resolves the larger ones.
This results in a stricter description of the flow unsteady behaviour
and topology. For instance, this approach offers a more accurate quantification of the crossed terms 𝑢′ 𝑣′ , which play a central role in the
energy and exergy accounting of supersonic ejectors. This would lead
to a better understanding of the link between the flow topology and the
loss coefficients, similar to the work carried out in [51], as well as to
the improvement of detailed analysis tools such as the exergy transport
tubes proposed by [21]. Moreover, LES results allow for a thorough
visualization of the larger turbulent structures that develop across the
ejector, namely those in the shear mixing layer and the diffuser. It also
allows studying the nature of the shock waves at the end of the constant
area section, which are a major source of exergy losses through the
device. This visualization has been difficult to achieve experimentally
given the limited size of typical test benches and the handling of
hazardous fluids. The ejector used in this study is a clear example
of these limitations, as its design deviates from typical ejectors for
industrial applications to allow for flow visualization. Another added
value of the LES presented in this study is the influence of small
geometrical imperfections in the domain. These are normally averaged
out in RANS studies, given the coarser mesh size and treatment of the
governing equations. However, its influence can be captured in the LES
study, as reflected by the flow asymmetry observed in this study.
Nonetheless, LES still presents some limitations for the generalized
study of supersonic ejectors. For instance, the working fluid in this
study is air, whose thermodynamic behaviour has been modelled using
the simple perfect gas model. However, most ejectors for industrial
applications work with refrigerants such as 𝑅134𝑎, 𝑅1234𝑦𝑓 , 𝑁𝐻3 or
𝐶𝑂2 . These often require the use of more complex equations of state.
Moreover, modelling transcritical and two-phase ejectors would require
different formulations at extra computational costs. An alternative in
• Instantaneous pressure fluctuations suggest that turbulent structures spawn at the mixing layer and increase in size as they travel
down the ejector. A noticeable enlargement is observed in the
diffuser.
• At the assessed conditions, the shock train onsets towards the end
of the mixing chamber. This interacts with a recirculation zone in
the diffuser, which narrows the main flow passage.
• Primary jet velocity profiles show self-similarity slightly before
the first half of the mixing chamber. Moreover, their tail end velocity is ⟨𝑈 ⟩∕𝑈𝑐 ∼ 0.2, in agreement with hot-wire measurements
in confined jets.
• Vorticity thickness and instantaneous images showcase the development of different turbulent structures in the mixing layer,
namely rollers close to the NXP and then 𝛬 vortices which evolve
into hairpin vortices and eventually break down before the shock
train.
• Following the compound-choke indicator 𝛽, under the assessed
conditions the ejector works in double-choke regime.
This study shows the great potential that LES offers for the detailed analysis of supersonic ejectors. LES offers a different kind of
information than RANS, most notably in the unsteady description of
the flow, visualization and quantification of the fluctuation terms.
Although its higher cost prevents the direct application in geometrical
optimization problems, LES fully resolves the energy transport in the
larger turbulent structures, which in turn leads to a more accurate
energy and exergy accounting. This information can be proven useful
for a better estimation of loss coefficients and detailed exergy studies.
The next stage would be the generalization of this approach to ejectors
for industrial applications. Further steps in this direction would be:
9
Applied Thermal Engineering 209 (2022) 118177
S. Croquer et al.
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Declaration of competing interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to
influence the work reported in this paper.
Data availability statement
The data that support the findings of this study are available from
the corresponding author upon reasonable request.
Acknowledgements
This project is part of the NSERC chair on industrial energy efficiency financially supported by Hydro-Québec (Laboratoire des technologies de l’énergie), Natural Resources Canada (CanmetEnergy in
Varennes) and Emerson Commercial and Residential Solutions. The
authors thank and recognize the Centre Européen de Recherche et de
Formation Avancée en Calcul Scientifique (Cerfacs), which provided
us with AVBP for academic research, as well as support all along the
project. All calculations have been performed using the HPC facilities
of the Calcul Québec and Compute Canada networks. They are all here
gratefully acknowledged.
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