12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19–22, 2022

EXPERIMENTAL INVESTIGATION OF COMPRESSIBILITY EFFECTS IN
AXISYMMETRIC SUPERSONIC EJECTORS AND FREE JETS

Kenneth Yi-Nian Hinh1 , Robert J. Martinuzzi and Craig Johansen
Department of Mechanical and Manufacturing Engineering
University of Calgary
2500 University Drive NW, Calgary, Alberta T2N 1N4, Canada
kenneth.hinh@ucalgary.ca, rmartinu@ucalgary.ca and johansen@ucalgary.ca

ABSTRACTdefined by:Particle image velocimetry (PIV) measurements were conducted of supersonic free jets and supersonic entrainment ejectors configured as an axisymmetric compressible jet into a constant area circular duct. Flows from a primary nozzle designed
for a Mach number of 1.25 were investigated at over, perfectand under expanded conditions. Measurements of the velocity
and Reynolds stresses allowed for a characterization of the jetflow not previously reported for a compressible axisymmetricejector. The compressibility effects in the free jet were foundto be in accord with those observed for the compressible mixing layer, suppressing the growth rate and Reynolds stresses
with the convective Mach number, 𝑀𝑐 . Similarity analysis
was performed on the compressible free jet, which indicatedthe conditions for self-preservation in the mean momentum
equation which allowed for the collapse of the Reynolds shear
stresses by the jet spreading rate and centerline velocity. In
the ejector the velocity profiles collapsed with the jet excess
velocity and the jet thickness, but the spreading rate itself function of the both compressibility and confinement. Instead, theReynolds shear stresses were collapsed with Φ(𝑀𝑐 ), which
expresses the effects of compressibility through 𝑀𝑐 .


1
1 𝑑 𝑢(𝑦)¯=𝛿 𝜔 Δ𝑈𝑑𝑦 𝑚𝑎𝑥(1)

where Δ𝑈 is the velocity difference across the layer, 𝑢(𝑦)¯is themean velocity and 𝑦 the transverse distance. The modificationon the vorticity thickness growth rate due to compressibility
through the convective Mach number is expressed the relationship,𝛿 ′𝜔 = 𝛿0′ Φ(𝑀𝑐 )(2)where 𝛿0′ and 𝛿 ′𝜔 is the change of the vorticity thickness inthe streamwise direction for the incompressible and compressible cases, respectively, and Φ(𝑀𝑐 ) is determined empirically.
The single prime represents a spatial derivative. A function
fitted from experimental data to represent Φ(𝑀𝑐 ) is given by
Dimotakis (1991),hi
Φ(𝑀𝑐 ) = 0.8𝑒𝑥 𝑝 −3𝑀𝑐2 + 0.2(3)

Barre et al. (1994) suggests that in the mixing layer, theReynolds shear stress is also related to 𝛿 ′𝜔 by,

INTRODUCTION
The rapid development of low-orbit reusable vehicles motivates interest in high-speed impulse propulsion systems foraerospace applications. With few moving parts, ejector pumps
are an attractive device to provide thrust augmentation for rockets, ramjets and gas turbines. The confined turbulent axisymmetric jet is a laboratory simplification allowing the study ofcompressibility effects on the turbulence. An understandingof these effects is important for characterizing mixing and entrainment ratios determining the performance of the ejector.
Earlier studies mostly focused on the compressible mixing
layer, where compressibility is observed to have a stabilizing
effect on turbulence and is most evident through the suppression of the Reynolds stresses and, as a result, the growth rate.
The convective Mach number, 𝑀𝑐 , is used to scale the compressibility effects and is the Mach number in the frame ofreference of the convective speed of coherent motions between
the inner and outer flows (Papamoschou & Roshko, 1988).
The vorticity thickness, 𝛿 𝜔 , is typically used to characterize the compressible mixing layer (Dimotakis, 1991) and is−𝑢 ′′ 𝑣 ′′ 𝑚𝑎𝑥 1= 𝐾 (𝑠, 𝑞, 𝑀𝑐 )𝛿 ′𝜔
2(Δ𝑈) 2(4)

where 𝐾 (𝑠, 𝑞, 𝑀𝑐 ) is typically described as an eddy diffusivity
time, which represents the growth rates of the eddies in aconvected coordinate system and is determined empirically.
The ratios, 𝑠 and 𝑞, denote the ratio between the low speed
and high speed sides for density and velocity respectively. The
double prime denotes the temporal fluctuations and the overbar the temporal averaging.
Beside the geometrical differences between the mixing
layer and the axisymmetric jet, the jet convective Mach number
is a function of the streamwise distance, 𝑥, as the centerline
velocity of the jet decays and in the confined jet, the secondary
flow is entrained. In the incompressible confined jet the mean
velocity profiles exhibit similarity when written in the form
(Razinsky & Brighton, 1971),
𝑓 (𝜂) =

1 Corresponding author1𝑈¯
𝑢¯ − 𝑈¯ 2=
𝑢¯ 𝑚 − 𝑈¯ 2 𝑈¯ 𝑚

(5)12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19–22, 2022

Compressed
Air from
Building
BypassStagnation Thermocouple
Pressure Transducer
Light
Sheet
Laser

NozzleCamera POV 1 (50 x 50mm)

Jet core

U2

r,η
x

Laser
Controller
SeederP.C.

Acrylic
MixingTube

Timer
CameraUm

bum

Figure 1: Schematic of experimental setup and nomenclature for velocities for the confined jet scenario. In the free jet, 𝑈2 = 0 and theacrylic mixing tube is omitted.

Whereas the vorticity thickness, 𝛿 𝜔 , is used to describe thethickness in the compressible mixing layers, the radius at half
the maximum excess velocity, 𝑏, is typically used to collapse
jet profiles with 𝜂 = 𝑟/𝑏. By construction, 𝑏 and 𝛿 𝜔 and theirevolution are proportional for self-similar flows. The velocity
difference across the shear layer is considered to be the excess
mean centerline jet velocity 𝑈 𝑚 and the excess mean velocity,
𝑈 are calculated as the difference of the mean flow 𝑢 and thesecondary flow velocity 𝑈 2 . 𝑢 𝑚 is the maximum of the mean
flow occurring at 𝜂 = 0. For the free jet, 𝑈 2 = 0.
Investigations on the local effects of compressibility onjets are limited. Schadow et al. (1990) employed total pressure measurements and Schlieren photography to determine theoverall spreading rate in coaxial jets. The effect on jet swirl on
the entrainment of the jet was investigated by Naughton et al.
(1997) and the growth rate was determined from planar laser
scattering (PLS) measurements. In these cases, the spreading
rates were found to be suppressed by 𝑀𝑐 , but since measurements were taken at a limited number of axial positions, theeffect of 𝑀𝑐 on the local spreading rate and its implications on
the similarity of the jet not been investigated.
Using particle image velocimetry (PIV), a full field measurement is possible to investigate the effects of 𝑀𝑐 on the
spreading rate at a high resolution. In this case, the primary
jet has a convective Mach number is 𝑀𝑐𝑚 = ( 𝑢¯ 𝑚 −𝑈𝑐 )/𝑎 1 andfor the secondary flow, 𝑀𝑐2 = (𝑈𝑐 − 𝑈¯ 2 )/𝑎 2 . The convective
Mach number is taken to be a weighted average of the convective Mach numbers for the high speed√ side and low speed sideof the shear layer, therefore 𝑀𝑐 = 𝑀𝑐𝑚 𝑀𝑐2 . Here 𝑎 𝑚 and𝑎 2 are the speeds of sound of the jet and secondary flow respectively. The convective velocity, 𝑈𝑐 of the large structuresin shear layer is given by,𝑎 𝑢¯ 𝑚 + 𝑎 1𝑈¯ 𝑚
𝑈𝑐 = 2
𝑎1 + 𝑎2

From the theory of incompressible jets, George (1988) argues that self-preservation of the mean momentum equations is
possible when the Reynolds shear stress scales proportionallyto the jet thickness growth rate, departing from the classicaltheory where 𝑏 must be proportional to 𝑥. In the present
work, similarity arguments are investigated for the free compressible jet and shows that the Reynolds shear stress scaling
can be extended to the compressible scenario when 𝑀𝑐 effects
the spreading rate proportionally to the Reynolds shear stress.
Experimental measurements of the evolution of the spreading
rate, velocities and Reynolds stresses influenced by 𝑀𝑐 as thejet develops shows this is the case in the free compressible
jet. In the confined scenario, when scaled by local variables,
the velocity profiles and shear Reynolds stress profiles collapse
onto a single curve suggesting self-similarity.

EXPERIMENTAL SETUP
The experimental setup is shown schematically in Fig. 1
along with a definition sketch and nomenclature. Planar PIV
was used to measure two-component velocity fields in the planebisecting the center axis of the mixing tube within a supersonicejector. The streamwise position of the measurements begin
where the exit plane of the nozzle and the entrance of the mixing
tube coincide. A Mach 1.25 primary jet with an exit diameterof 𝑑 = 6.35 mm was investigated. Pressure ratios, 𝑃/𝑃𝑠𝑡 𝑎𝑔 , of
0.3378, 0.3890 and 0.4343 resulting in over, perfect and under
expanded conditions, respectively, were investigated.
The mixing tube is cast acrylic (D = 54 mm, l = 914.4mm) and the stagnation temperature upstream of the jet wasmeasured to be 295.15K. Planar measurements in the window
spanned the entire diameter of the mixing tube from a streamwise to nozzle diameter ratio (𝑥/𝑑) of 0 to 25 are patched fromfour contiguous fields-of-view. The nozzle and the mixing tube
were mounted on rails which slide independently of the laser
and the camera in the streamwise direction, ensuring that theposition of the light sheet in respect to the camera is identical between measurements. Contiguous field-of-views overlap
10.8 mm in the streamwise direction.
Both the ambient air and the jet were seeded with 1 𝜇m
Di-Ethyl-Hexyl-Sebacat (DEHS) from a six-jet Laskin nozzle(TSI 9307-6). Ragni et al. (2011) showed that the particle
response time for DEHS of this size is approximately 𝜏𝑝 = 2𝜇s
which was sufficiently low to allow a resolution of the velocity
fluctuations and gradients in this flow.
The PIV consisted of a CMOS camera with a sensor size

(6)

In this investigation, the local relationship between 𝑀𝑐 ,and the suppression of the vorticity thickness growth rate, 𝛿 ′𝜔 ,for the free jet was found to behave similarly to the mixing
layer and is compared to other literature findings. For clarity,
the jet growth rate 𝑏 ′ is confirmed to be proportional to 𝛿 ′𝜔 . In
the mixing layer, by scaling the compressibility effects throughthe vorticity thickness growth of the shear layer, similarity isobserved, and was shown that the compressibility effects areabsorbed into the spreading rate and do not have an effect on
the velocity profiles themselves (Menaa, 2003).

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12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19–22, 2022
of 20 × 20 𝜇m (FastCAM Photron SA-5) and a resolution of1024 × 1024. The camera was equipped with a Nikon AF
Micro-Nikkor 60 mm 𝑓 /2.8𝐷 lens and was mounted perpendicular 20 cm from to the light sheet. A third-order polynomial
spatial calibration was performed before each test to correct
for any distortions from imaging through the acrylic mixing
tube wall. The camera inter-frame time is 0.23 𝜇s and is independent of frame rate and resolution. Rhodamine 6G based
paint was applied on the interior surface of the mixing tube,
combined with a band-pass optical filter, nearly eliminatinginternal reflections.

integral.
1 𝜕 ( 𝜌¯ 𝑣¯ 𝑟) 𝜕 ( 𝜌¯ 𝑢)
¯
+
=0
𝑟 𝜕𝑟
𝜕𝑥
𝜕 𝑢¯
𝜕 𝑢¯
1 𝜕 ( 𝜌𝑢¯ ′′ 𝑣 ′′ 𝑟)𝜌¯ 𝑢¯+ 𝜌¯ 𝑣¯
=−
𝜕𝑟
𝜕𝑥
𝑟
𝜕𝑟
∫ ∞
2
𝑀0 = 2𝜋𝜌¯ 𝑢¯ 𝑟d𝑟

(7)
(8)
(9)

0

Note that the velocity-density correlations are assumed to besmall and thus omitted from the momentum equation. This isessentially Morkovin’s hypothesis, which is typically valid in
free shear flows with 𝑀𝑐 less than 1.5 (Bradshaw, 1977). The
shape function for the density profile, 𝑔(𝜂) is not universal, andis a function of 𝑠 and 𝑞 through the Crocco-Busemann relations.
The Reynolds shear stress is represented by its maximum value,
𝑅𝑠 , and a shape function, ℎ12 . Performing a similarity analysis
following George (1988), the functional dependencies of 𝑈¯ 𝑚 ,𝜌¯ 𝑠 , 𝑅𝑠 and 𝑏 are left remain determined. For the compressible
free jet:

The illumination source was a dual cavity Nd:YLF laser(Photonics DM20-527-DM) with a repetition rate up to 10 khz
with a maximum laser energy of 20 mJ per pulse. A 1 mm
thick light sheet was formed from two spherical lenses and acylindrical lens. Measurements were conducted in dual-pulse
mode with a separation interval of 1 to 1.5 𝜇s to ensure that theparticles moved about 7 pixels at the center of the jet between
image pairs. A silicon photodetector (Thorlabs DET02AFC)
was used to ensure that the pulse separation time matches thetime sent by the trigger.
Image pairs were collected and processed using LaVision
DaVis 10.2.0 PIV software. Each test consisted of 1000 image pairs, with the Reynolds stresses converging after approximately 750 pairs. Processing was done using a multi-passstrategy resulting in a vector field with 32 × 32 pixel window
size with 75% overlap. This provided about 20 vectors acrossthe jet exit diameter.

𝑈¯ = 𝑈¯ 𝑚 𝑓 (𝜂)(10)𝜌¯ = 𝜌¯ 𝑠 𝑔(𝜂 : 𝑠, 𝑞)(11)−𝑢 ′′ 𝑣 ′′ = 𝑅𝑠 ℎ12 (𝜂)(12)𝜂 = 𝑟/𝑏(13)

In the free jet, the jet momentum, 𝑀0 , is assumed tobe constant. From (9), the relationship between the density,
centerline velocity and jet spreading rate is,

RESULTS AND DISCUSSION
𝑀0 1/2 −1𝑏𝑈¯ 𝑚 = 𝐵𝜌¯ 𝑠

In this study, the local convective Mach number 𝑀𝑐 was
calculated at each streamwise position with the jet centerline
velocity as 𝑢¯ 𝑚 and the secondary flow velocity as 𝑈¯ 2 , with𝑎 2 = 342 m/s, calculated from a temperature of 291.15 K. As
jet temperature measurements were not available, 𝑎 1 , was approximated by averaging the temperature of the ambient flow
and jet exit static temperature calculated from isentropic relations.

(14)

where 𝐵 is a constant. Upon substitution of (11)-(13) into(8)-(9), the mean-momentum equation is:

∫ 𝜂
′ 
1 𝜌¯ 𝑚(𝑔 𝑓 ) ′𝑔 𝑓 𝜂d¯ 𝜂¯ −
𝑔𝑓2−
2 𝜌¯ 𝑚𝑔𝜂 0
 ′ 
 

∫ 𝜂
′
𝑏
(𝑔 𝑓 )𝑅𝑠 (𝜂𝑔ℎ12 ) ′𝑔𝑓2+
𝑔 𝑓 𝜂d¯ 𝜂¯ =
2
𝑏
𝑔𝜂 0
𝜂
𝑏𝑈¯ 𝑚


The governing equations in this region of flow are thoseof conservation of mass and momentum and the momentum

(15)

0.25

1.00

0.15
0.75
0.05
0.50
-0.05
0.25
-0.15
0.00

-0.25
-2

0

2

-2

0

2

2

Figure 2: The velocity profile (left) with overlay of Gaussian shape function, 𝑓 (𝜂) = 𝑒 −𝑙𝑛(2) 𝜂 (– –) and Schlichting’s jet equation withconstant eddy viscosity 𝑓 (𝜂) = (1 + 𝜂2 /2) −2 (—-). The Reynolds shear stress (right) for perfectly expanded free jet. Axial positions
𝑥/𝑑 = 12.5 (◦); 15.0 (□); 17.5 (△); 20.0 (◁); 22.5 (▷).

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12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19–22, 2022
0.70

0.40

0.60
0.30

0.50
0.40

0.20
0.30
0.20

0.10

0.10
0.00

0.00
-2

0

2

-2

0

2

Figure 3: Reynolds normal streamwise (left) and transverse (right) stresses for the perfectly expanded free jet. Axial positions 𝑥/𝑑 =12.5 (◦); 15.0 (□); 17.5 (△); 20.0 (◁); 22.5 (▷).

The first term in (15) differs from the incompressible free
jet and, if large, needs to be considered if similarity conditions
are to be satisfied. However, in the present flow, this term may
be assumed negligible. Examining the shape functions, thefirst term consists of a difference between two even functions,in contrast to a sum in the second term. In most cases, theshape functions are the same order of magnitude, thus the firstterm is small when compared to the second.
In jets, the centerline temperature, and through the ideal
gas law the density, diffuses to ambient conditions and approaches unity as the jet develops. If the density ratio can beconsidered nearly constant, the shape function 𝑔(𝜂) becomes aconstant function. For these conditions, the equation of George(1988) is recovered.

−𝑓2−



∫ 𝜂
(𝜂ℎ12 ) ′
𝑓′
𝑅𝑠
𝑓 𝜂d
¯ 𝜂¯ =
2 𝑏′
𝜂 0
𝜂
𝑈¯ 𝑚

the vorticity thickness is scaled 0.142, which is the value for a
free incompressible jet from experiments by Moore (1977).
From (16), self-preservation of the mean momentum equation of the free jet requires that the streamwise variation of 𝑏 ′ be
2 . As the evolution of 𝛿 and 𝑏 ′ is proproportional to 𝑅𝑠 /𝑈¯ 𝑚𝜔
′2
portional, 𝑏 is similarly modified by Φ(𝑀𝑐 ) and thus 𝑅𝑠 /𝑈¯ 𝑚must also be altered by 𝑀𝑐 in the same way. In Fig. 5, the2 , is plotted
normalized maximum Reynolds shear stress, 𝑅𝑠 /𝑈¯ 𝑚against 𝑀𝑐 and appears to be similarly modified by Φ(𝑀𝑐 ) forcomparison, the relation for Φ(𝑀𝑐 ) from Dimotakis (1991),
multiplied by 0.02 is also shown. The modification is similar
to that for the compressible mixing layer where the Reynolds
shear stress is modified with Φ(𝑀𝑐 ) (Goebel & Dutton, 1991).
The relationship between 𝑈¯ 𝑚 , 𝑅𝑠 and 𝑏 ′ in (16) in the jet
literature is referred to as an amplification factor, but perhaps
the analogue from the compressible mixing layer, the eddy
diffusivity time, 𝐾 (𝑠, 𝑞, 𝑀𝑐 ), introduced in (4) which has a(16)

The foregoing analysis has implications for the characteristics in compressible jets. If the effects of the shape function𝑔(𝜂) can be neglected, the velocity profiles are expected to resemble the incompressible scenario as long as the first term in(15) remains small and that the compressibility effects are notexpressed in the scaled velocity profiles themselves but throughthe growth rate of the jet. In Fig. 2, the velocity profile agrees
with two common functions used to represent free incompressible jets, indicating that at the current jet conditions, 𝑔(𝜂), hasa small effect.
George (1988) hypothesizes for self-preservation of themean momentum equation, the quantities 𝑈¯ 𝑚 , 𝑅𝑠 and 𝑏 ′ must
maintain the same relative balance at the same relative locations. In the incompressible jet, this explained the effect of
initial conditions on the self-preservation. In the current case,this can be extended to determine the effect of compressibility
on self-preservation for the compressible jet.
The incompressible free jet is expected to grow linearly
within the similarity region. Thus, any change in the growthrate in the compressible scenario is due to the effect of 𝑀𝑐 .
The relationship between the convective Mach number and thejet thickness growth rate, recalling that 𝑏 ′ ∼ 𝛿 ′𝜔 and shown in
Fig. 4, agrees well with the function from (3). This shows that
locally, the vorticity thickness growth rate in compressible freejets behaves like the mixing layer counterpart. For comparison,

1

0.5

0
0

0.5

1

Figure 4: Vorticity thickness growth rate suppression. Experimental results in markers: Papamoschou & Roshko (1988)(△); Samimy & Elliott (1990) (◦); Goebel & Dutton (1991)(□); Barre et al. (1994) (◁); Naughton et al. (1997) (▷). Mixing
Layer DNS in black lines: Freund et al. (2000) (– –); Matsuno& Lele (2020) (—). Relation from Dimotakis (1991) (—);
Current investigation of Mach 1.25 free jets: Under expanded(—), Perfectly expanded (—), Over expanded (—)

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12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19–22, 2022

0.025

0.30

0.020

0.25

0.015

0.20

0.010

0.15

0.005

0.10

0.000

0.05
0

0.2

0.4

0.6

5

Figure 5: Suppression of maximum Reynolds shear stresses
with increasing convective Mach number. Only includes region
where the velocity profile is self-similar. Free: Under expanded(∗); Perfectly expanded (◁); Over expanded (▷). Confined:
Under expanded (△); Perfectly expanded (□); Over expanded(◦). Relation from Dimotakis (1991) now multiplied by 0.02
(—)

𝑅𝑠
2 𝑏′
¯
𝑈𝑚

(17)

1.00

0.02

0.75

0.01

0.50

0.00

0.25

-0.01

0.00

-0.02
0

20

25

The behavior of 𝐾jet due to compressibility can be explained by analogy to the mixing layer eddy diffusivity time,
𝐾 (𝑠, 𝑞, 𝑀𝑐 ). Menaa (2003) found that 𝐾 (𝑠, 𝑞, 𝑀𝑐 ) is relatively
constant, but especially when the density ratio is near unity.
In the confined jet, the confinement introduces a secondary
flow, momentum loss and pressure gradient and as a consequence, the flow is not self-preserving in the mean momentum
equation and 𝐾jet is not constant. The velocity profiles collapses when scaled by the excess velocity, and is nearly identical to the free jet profile with some deviation at the edges of
the jet. In Fig. 5, the suppression of the Reynolds shear stressdue to compressibility is still captured by Φ(𝑀𝑐 ), suggestingthat the Reynolds shear stress can be collapsed by normalizing
2 Φ(𝑀 ), as shown in Fig. 7. This indicates that theby 𝑈¯ 𝑚𝑐effects of confinement and compressibility can be quantified
through the local variables, 𝑈¯ 𝑚 , 𝑀𝑐 , and 𝑏, making this flow
self-similar.
Due to the confinement, the incompressible spreading rate
is generally no longer constant and is unknown, and without amethod of normalizing 𝑏 ′ , Φ(𝑀𝑐 ) cannot be determined fromthe confined experimental results. Although this is the case,as the Reynolds shear stress is suppressed through Φ(𝑀𝑐 ),Fig. 6 shows 𝐾jet as a function of 𝑥/𝑑. The observationthat 𝐾jet is nearly constant is then consistent with the assumption that the suppression of 𝑏 ′ and 𝑅𝑠 due to compressibility can be expressed as 𝑏 ′ = Φ(𝑀𝑐 )𝑏 0′ and 𝑅𝑠 = Φ(𝑀𝑐 )𝑅𝑠,0 ,where the subscript, 0, indicates the value in the incompressible case. A constant 𝐾jet as the jet develops suggests that theflow is self-preserving to at least the mean momentum equation. Satisfying this condition results in the Reynolds shear
2 𝑏 ′ asstress collapsing onto a single curve when scaled by 𝑈¯ 𝑚shown in Fig. 2.

-2

15

Figure 6: Jet eddy diffusivity time, 𝐾jet , for the free jet. Overexpanded (△); Perfectly expanded (□); Under expanded (◦).

similar form, is more appropriate. To differentiate from themixing layer, the eddy diffusivity time for the jet is:

𝐾jet =

10

2

-2

0

2

2

Figure 7: The velocity profile (left) with overlay of Gaussian shape function, 𝑓 (𝜂) = 𝑒 −𝑙𝑛(2) 𝜂 (– –) and Schlichting’s jet equation
with constant eddy viscosity 𝑓 (𝜂) = (1 + 𝜂2 /2) −2 (—-). The Reynolds shear stress (right) for supersonic ejector with primary jet Mach
number = 1.25. Axial positions 𝑥/𝑑 = 12.5 (◦); 14.125 (□); 15.75 (△); 17.375 (◁); 19.0 (▷).

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12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19–22, 2022
ACKNOWLEDGEMENTS
Funding for this work was provided by the Natural Sciences Engineering Research Council (NSERC) of Canada andAtlantis Research Labs Inc. through a Collaborative Researchand Development (CRD) grant.

4.0
3.0
2.0

REFERENCES

1.0

Barre, S., Quine, C. & Dussauge, J. P. 1994 Compressibility
Effects on the Structure of Supersonic Mixing Layers: Experimental Results. Journal of Fluid Mechanics 259, 47–78.
Bradshaw, P. 1977 Compressible Turbulent Shear Layers. Annual Review of Fluid Mechanics 9 (1), 33–52.
Dimotakis, P. E. 1991 Turbulent Free Shear Layer Mixing
and Combustion. Progress in Astronautics and Aeronautics
135 (Chapter 5), 76.
Elliott, G. S. & Samimy, M. 1990 Compressibility Effects inFree Shear Layers. Physics of Fluids A: Fluid Dynamics
2 (7), 1231–1240.
Freund, J. B., Lele, S. K. & Moin, P. 2000 Compressibility
Effects in a Turbulent Annular Mixing Layer. Part 1. Turbulence and Growth Rate. Journal of Fluid Mechanics 421,
229–267.
George, W. K. 1988 The Self-Preservation of Turbulent Flowsand Its Relation to Initial Conditions and Coherent Structures. Advances in Turbulence pp. 39–73.
Goebel, S. G. & Dutton, J. C. 1991 Experimental Study of Compressible Turbulent Mixing Layers. AIAA Journal 29 (4),538–546.
Matsuno, K. & Lele, S. K. 2020 Compressibility Effects inHigh Speed Turbulent Shear Layers – Revisited. In AIAA
Scitech 2020 Forum. Orlando, FL: American Institute of
Aeronautics and Astronautics.
Menaa, M. 2003 Density and Compressibility Effects on the
Structure of Supersonic Mixing Layer. In 12th AIAA International Space Planes and Hypersonic Systems and Technologies. Norfolk, Virginia: American Institute of Aeronautics
and Astronautics.
Moore, C. J. 1977 The Role of Shear-Layer Instability Waves
in Jet Exhaust Noise. Journal of Fluid Mechanics 80 (2),321–367.
Naughton, J. W., Cattafesta III, L. N. & Settles, G. S. 1997
An Experimental Study of Compressible Turbulent Mixing
Enhancement in Swirling Jets. Journal of Fluid Mechanics
330, 271–305.
Papamoschou, D. & Roshko, A. 1988 The Compressible Turbulent Shear Layer: An Experimental Study. Journal of Fluid
Mechanics 197, 453–477.
Ragni, D., Schrijer, F., van Oudheusden, B. W. & Scarano, F.
2011 Particle Tracer Response Across Shocks Measured by
PIV. Experiments in Fluids 50 (1), 53–64.
Razinsky, E. & Brighton, J. A. 1971 Confined Jet Mixing forNonseparating Conditions. Journal of Basic Engineering
93 (3), 333–347.
Samimy, M. & Elliott, G. S. 1990 Effects of Compressibilityon the Characteristics of Free Shear Layers. AIAA Journal
28 (3), 439–445.
Schadow, K. C., Gutmark, E. & Wilson, K. J. 1990 Compressible Spreading Rates of Supersonic Coaxial Jets. Experiments in Fluids 10-10 (2-3), 161–167.

0.0
0

0.2

0.4

0.6

Figure 8: Reynolds stress anisotropy with 𝑀𝑐 . Ratios of maximum Reynolds stresses. 𝑢 ′′ 𝑢 ′′ /𝑢 ′′ 𝑣 ′′ , Free: Under expanded(□); Perfectly expanded (□); Over expanded (□), Confined: Under expanded (□); Perfectly expanded (□); Over expanded (□).
𝑣 ′′ 𝑣 ′′ /𝑢 ′′ 𝑣 ′′ , (◦); Perfectly expanded, (◦); Over expanded (◦),
Confined: Under expanded (◦); Perfectly expanded (◦); Overexpanded (◦). Relation from Dimotakis (1991) now multipliedby 1.8 (—)

the spreading rate is also expected to remain modified through𝑏 ′ = Φ(𝑀𝑐 )𝑏 0′ .
The anisotropy of the Reynolds stresses is illustrated inFig. 8. Although anisotropy is expected, there remains disagreement as to how the stresses are affected by 𝑀𝑐 . The
streamwise Reynolds stress, 𝑢 ′′ 𝑢 ′′ , in some studies was foundto be unaffected by 𝑀𝑐 (Goebel & Dutton, 1991; Freund et al.,2000) but was found to be suppressed with 𝑀𝑐 in others (Elliott& Samimy, 1990; Samimy & Elliott, 1990; Matsuno & Lele,
2020). In the compressible free and confined jet, the streamwise Reynolds stress was found to agree with the latter case.
As indicated in squares in Fig. 8, the constant ratio implies that
the Reynolds transverse and shear stress are suppressed similarly by 𝑀𝑐 . Unlike studies in the mixing layer, the transverse
stress ratio 𝑣 ′′ 𝑣 ′′ /𝑢 ′′ 𝑣 ′′ shows stronger suppression with 𝑀𝑐
than the streamwise and shear stresses.

CONCLUSION
The compressible shear layer within the supersonic freejets and axisymmetric supersonic ejector and of Mach number
1.25 was investigated with particle image velocimetry (PIV),
providing a detailed measurement of the turbulent stresses andjet thickness growth rate as the jet developments, whichedallowed for an investigation relating the convective Mach number, 𝑀𝑐 , to the Reynolds stresses and spreading rate at eachstreamwise position of the jet. Similarity analysis introduced
a condition for self-preservation in the mean momentum equation for the free compressible jet, where the eddy diffusivity
time for the jet, 𝐾jet , must be constant. In the ejector, the velocity and Reynolds shear stress collapsed with local variables
as 𝑏 ′ is a function of the confinement and the compressibility.
Through the modification of 𝑅𝑠 with Φ(𝑀𝑐 ), the spreading
rate is expected to also be suppressed with 𝑀𝑐 .

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