[2]Tomoaki Watanabe

1] Department of Aerospace Engineering, Nagoya University, Nagoya, 464-8603,
Aichi, Japan

[2] Education and Research Center for Flight Engineering, Nagoya University,
Nagoya, 464-8603, Aichi, Japan

3] Department of Mechanical Engineering and Science, Kyoto University, Kyoto,
615-8530, Kyoto, Japan

# Nearly homogeneous and isotropic turbulence generated by the interaction of
supersonic jets

Takahiro Mori<EMAIL_ADDRESS>Koji Nagata  [  *  [

###### Abstract

This study reports the development and characterization of a multiple-
supersonic-jet wind tunnel designed to investigate the decay of nearly
homogeneous and isotropic turbulence in a compressible regime. The interaction
of 36 supersonic jets generates turbulence that decays in the streamwise
direction. The velocity field is measured with particle image velocimetry by
seeding tracer particles with ethanol condensation. Various velocity
statistics are evaluated to diagnose decaying turbulence generated by the
supersonic jet interaction. The flow is initially inhomogeneous and
anisotropic and possesses intermittent large-scale velocity fluctuations. The
flow evolves into a statistically homogeneous and isotropic state as the mean
velocity profile becomes uniform. In the nearly homogeneous and isotropic
region, the ratio of root-mean-squared velocity fluctuations in the streamwise
and vertical directions is about 1.08, the longitudinal integral scales are
also similar in these directions, and the large-scale intermittency becomes
insignificant. The turbulent kinetic energy per unit mass decays according to
a power law with an exponent of about 2, larger than those reported for
incompressible grid turbulence. The energy spectra in the inertial subrange
agree well with other turbulent flows when normalized by the dissipation rate
and kinematic viscosity. The non-dimensional dissipation rate is within a
range of 0.51–0.87, which is also consistent with incompressible grid
turbulence. These results demonstrate that the multiple-supersonic-jet wind
tunnel is helpful in the investigation of decaying homogeneous isotropic
turbulence whose generation process is strongly influenced by fluid
compressibility.

###### keywords:

Homogeneous isotropic turbulence, Jet interaction, Compressible turbulence

11footnotetext: Preprint submitted to Experiments in Fluids.

##  1 Introduction

Compressible turbulence plays a crucial role in flows in engineering and
physics fields [3, 14]. For sufficiently large velocity fluctuations,
turbulent motion begins to cause fluid compression and expansion, by which
fluid density varies. The compressibility effects significantly alter the
evolution of turbulence. The growth rate of turbulent shear flows is
suppressed at a high Mach number [12]. When a Mach number defined for velocity
fluctuations is very high, turbulent motion also locally generates shock waves
called shocklets [41]. Velocity, pressure, density, and temperature
drastically vary across shock waves, and the propagation of shocklets also
influences turbulence.

Theoretical studies of turbulence often consider canonical turbulent flows
under relatively simple conditions. These studies also mainly concern
statistical behaviors of turbulence because of the difficulty in constructing
theories and models for an instantaneous flow field based on the Navier–Stokes
equations. One of the simplest problems is homogeneous isotropic turbulence
(HIT) [5], for which various theories and models have been developed in
previous studies, e.g., non-dimensional energy dissipation rate [42],
longitudinal structure functions [10], and decay in absence of external force
[21]. Laboratory experiments are as crucial as numerical simulations in
providing data that support the construction and validation of theories and
models. Previous studies have developed various facilities and methods for
generating HIT in a subsonic regime for which the assumption of
incompressibility is valid. A grid installed in a uniform mean flow in a wind
tunnel can generate nearly homogeneous and isotropic turbulence (e.g., Comte-
Bellot and Corrsin 17, Uberoi and Wallis 69, Valente and Vassilicos 70,
Kitamura et al 36, Djenidi et al 22). Similarly, experiments with water flumes
utilize a grid to generate HIT [65, 63]. Most grids consist of many
intersecting bars, which stir a uniform mean flow. Another type of turbulence-
generating grid is an active grid [43], which utilizes rotating bars with
winglets. The random motion of winglets generates HIT with a high Reynolds
number [48, 40, 83]. A similar method to randomly stir fluid is utilized in a
multi-fan wind tunnel, by which the interaction of fan-induced flows generates
HIT [51, 64]. A mean flow advects the HIT generated in wind tunnels and water
flumes. Because of negligible shear in the mean flow, turbulence decays as it
evolves. The decay of grid turbulence is often compared with the theories of
freely evolving HIT [38].

Previous studies have also developed facilities to generate turbulence without
a mean flow. They often utilize stirring devices in a closed chamber. [8]
reported the generation of HIT with electrical fans placed inside a chamber.
Other studies also developed similar multi-fan facilities [61, 85, 57, 79, 11,
80]. Synthetic jet actuators are also widely used to generate HIT by the jet
interaction [30, 71]. Similarly, the interaction of random jets can generate
HIT: [6] and [15] used two opposing planar arrays of randomly actuated jets
while [52] used a single array. These facilities were used to investigate
various phenomena related to turbulent flows, such as particle disperson [31]
and evaporation of droplets [44].

Fewer experimental studies have been reported for HIT in compressible flows
than in incompressible flows. Experiments of compressible turbulence have been
extensively conducted for turbulent shear flows, such as jets [34], mixing
layers [27], and wakes [9], which have significant application in aerospace
engineering. Compressible grid turbulence has been investigated in shock tube
facilities. Grid turbulence is generated when the mean flow induced by a
planar shock wave passes a grid installed inside a shock tube. Most studies of
grid turbulence in shock tubes focus on the interaction between the shock wave
and grid turbulence [29, 1, 25]. Exceptionally, [13] conducted comprehensive
velocity measurements of grid turbulence in a shock tube. They compared the
statistics of velocity fluctuations for various flow parameters, such as the
Mach number of the mean flow and the Reynolds number based on the mesh size.
The turbulent kinetic energy in grid turbulence is known to decay according to
a power law given by $a_{k}(x-x_{0})^{-n}$, with a coefficient $a_{k}$, a
virtual origin $x_{0}$, and a decay exponent $n$. The experiments by [13]
suggest that compressibility effects cause the decrease of $n$. Later, [25]
also observed the decrease of $n$ in compressible grid turbulence in a shock
tube. [86] reported compressible grid turbulence generated with a perforated
plate installed in a high-speed wind tunnel. They found that $n$ in
compressible grid turbulence exceeds values in incompressible cases when the
Mach number is high. These results imply that the compressibility effects on
grid turbulence are not universal and may depend on various factors, including
the generation process of turbulence, as also discussed in [86]. Indeed, even
in an incompressible fluid, different decay laws of HIT were theoretically
derived for the decay of Saffman turbulence and Batchelor turbulence, which
differ in the shape of an energy spectrum at a low wavenumber range [20]. The
low-wavenumber spectral shape depends on the turbulence generation process.
Therefore, different compressibility effects are possible for the decay of HIT
generated in different facilities.

Recent studies have developed other facilities for compressible turbulence.
[81] developed a compressible-turbulence chamber with opposing arrays of
piston-driven synthetic jet actuators. The piston-driven actuators, initially
proposed by [19], are capable of generating supersonic synthetic jets at a
sufficiently high operation frequency [68, 60, 53]. The interaction of
supersonic synthetic jets generates turbulence strongly influenced by
compressibility effects. Another recently developed facility for compressible
turbulence is the variable density and speed-of-sound vessel. The experiments
use a heavy gas (sulfur hexafluoride SF6) as a working fluid. Even though a
stirring device generates a flow with a velocity lower than 100 m/s, the Mach
number can be high because of the low speed of sound of SF6. These new
facilities will be helpful in the future investigation of compressible
turbulence.

As also found for the decay exponent of grid turbulence discussed above, flow-
dependent compressibility effects on turbulence have been reported for other
velocity statistics. In fully-developed incompressible turbulence, the
skewness and flatness of a longitudinal velocity gradient, e.g., $\partial
u/\partial x$, have universal relations with a turbulent Reynolds number [62].
Compressible turbulence tends to have larger absolute values of skewness and
flatness than incompressible turbulence with a comparable Reynolds number [23,
77]. One of the critical parameters in compressible turbulence is the
turbulent Mach number, defined as
$M_{T}=\sqrt{u_{rms}^{2}+v_{rms}^{2}+w_{rms}^{2}}/a$, where $u_{rms}$,
$v_{rms}$, and $w_{rms}$ are the root-mean-squared (rms) velocity fluctuations
in three directions, and $a$ is the speed of sound. $M_{T}$ is often treated
as a dominant parameter determining compressibility effects [41, 58]. However,
a comparison of different compressible turbulent flows in experiments and
direct numerical simulations (DNS) has shown that the compressibility effects
on the velocity gradient statistics are not solely determined by local values
of $M_{T}$ [80]: the deviation of the velocity derivative flatness from
incompressible values occurs in supersonic turbulent jets and HIT generated by
supersonic synthetic jets at much lower $M_{T}$ than in compressible HIT
sustained by a solenoidal forcing. The dilatational fluid motion causes
additional kinetic energy dissipation in strongly compressible turbulence. The
dilational and solenoidal dissipation ratio is not also determined solely by
$M_{T}$ [23]. These behaviors are possibly explained by a non-local property
of wave motion [80]: even if a particular location in turbulence has low
$M_{T}$, much higher values of $M_{T}$ can be observed in other regions, where
strong pressure waves, such as shock waves, are generated and propagate into a
flow region of interest. These studies imply that the compressibility effects
on turbulence significantly depend on the generation process of turbulence.
Investigating compressible turbulence in various facilities is crucial in
extending our knowledge of turbulence behavior under compressibility effects.
The further development of compressible turbulence facilities is demanded for
this purpose.

The present study develops a new wind tunnel facility to investigate
compressible decaying HIT. In this wind tunnel, the interaction of many
supersonic jets generates nearly homogeneous and isotropic turbulence, which
continuously decays in a test section without significant influences of mean
shear. Velocity measurements with particle image velocimetry (PIV) and
temperature and pressure measurements characterize the turbulent flow
generated in the multiple-supersonic-jet wind tunnel, which is useful for
investigating decaying HIT in a compressible regime. The paper is organized as
follows. Section 2 describes the multiple-supersonic-jet wind tunnel and
measurement methods. Section 3 presents the measurement results to discuss the
fundamental turbulence statistics compared with other turbulent flows.
Finally, the paper is summarized in Sect. 4.

##  2 Experimental setup and measurement methods

Figure 1: A multiple-supersonic-jet wind tunnel.  Figure 2: A schematic of a
multiple-supersonic-jet wind tunnel. All dimensions are in mm.

###  2.1 Multiple-supersonic-jet wind tunnel

The multiple-supersonic-jet wind tunnel generates nearly homogeneous and
isotropic turbulence by the interaction of supersonic round jets. The
development of the facility is motivated by the experiments of incompressible
HIT generated by the interaction of many subsonic jets [6, 15, 52]. Figure 1
shows a picture of the wind tunnel with three main components: an air supply
system, a plenum chamber, and a test section. Compressed air is supplied to
the cylinder-shaped plenum chamber. The front surface of the plenum chamber is
equipped with 36 Laval nozzles, which generate supersonic round jets. The
interaction of 36 jets generates turbulence which eventually decays in the
streamwise direction in the test section. This section describes the details
of the wind tunnel.

Compressed air is stored in two air tanks with a volume of 220 L (Anest Iwata,
SAT-220C-140). Compressors (Fujiwara Industrial Co., Ltd., SW-L30LPF-01) are
connected to each tank via a moisture separator (SMC, AFM40-02-2A). The outlet
port of each tank is connected to the back surface of the cylinder-shaped
plenum chamber by a flexible hose with an inner diameter of 29.5 mm. Here, the
air is supplied to the chamber through a moisture separator (SMC, AF60-10-A),
a pressure regulator (SMC, AR60-10BG-B), and a pilot kick 2-port solenoid
valve (CKD, ADK11-25A). The pressure regulator is used to adjust the plenum
pressure.

Figure 3: A nozzle plate with 36 Laval nozzles.

Figure 2 shows a schematic of the plenum chamber and test section. The plenum
chamber consists of a 180 mm length pipe and two round plates with a 330 mm
diameter fixed on flanges welded to the pipe. The pipe size is JIS 20A, with
an inner diameter of 206.3 mm and a thickness of 5 mm. The pipe, plates, and
flanges are made of stainless steel. Rubber seals designed for JIS 20A pipes
fill the space between the plates and flanges. The flexible hoses are
connected to two threaded holes on the back plate with a thickness of 6 mm. A
pressure sensor (SMC, ISE30A-C6L-N-M) connected to a port on the front plate
monitors the pressure in the plenum chamber. The plenum pressure is also
measured with another pressure sensor (SMC, PSE-540A R06), whose signal is
recorded by an oscilloscope (Yokokawa, DL850E) at a sampling rate of 1 kHz.

A nozzle plate with $6\times 6$ Laval nozzles is installed on the front plate.
Figure 3 shows a picture of the nozzle plate manufactured with
stereolithography 3D printing with a printing resolution of 10 $\mu$m. All 36
nozzles have the same geometry. The holes along the perimeter of the nozzle
plate are used to fix the plate to the plenum chamber. The inner diameters at
the jet exit and the throat are 4.31 mm and 4.12 mm, respectively. The nozzle
inlet diameter is 6.3 mm. The constant outer diameter is 7.5 mm. The center-
to-center distance between the two nozzles is 12 mm. The inner diameter of the
converging section varies according to a square of a sinusoidal function. The
internal geometry of the diverging section is determined with an analytical
method for nozzle design [24]. Each nozzle with these dimensions generates a
fully expanded supersonic jet with a Mach number of 1.36 in atmospheric air
when the gauge pressure in the chamber is 200 kPaG. The base of the nozzle
plate has a square shape of $110\times 110$ mm2 and a thickness of 5 mm. The
front plate of the chamber has holes with a diameter of 9 mm at the locations
corresponding to the nozzles. The nozzles are inserted and fixed from the
inner side of the chamber plate with bolts and nuts. Air leakage is prevented
with seal washers used with the nuts and a hand-cut PTFE gasket sheet with a 1
mm thickness, which fills the space between the nozzle plate and the chamber
plate.

The test section has $100\times 100$ mm2 square cross-section and 1000 mm
length. The test section is made of an aluminum frame and acrylic plates. For
the side and bottom walls, optical-grade acrylic sheets (Nitto Jushi Kogyo,
CLAREX) with a 1.5 mm thickness are used for optical measurements and
visualizations. The top wall is a conventional acrylic plate with a 3 mm
thickness. The plenum chamber is placed on a urethane pipe-holder designed for
JIS 20A pipes (Nichiei Intec, N-040219). The test section and pipe holder are
fixed on another aluminum frame.

Figure 4: Temporal variation of gauge pressure in the plenum chamber, $P_{C}$.
$t=0$ corresponds to the maximum value of $P_{C}$.

All experiments are conducted to generate the supersonic jets with a plenum
pressure of $200$ kPaG. In each run of the experiments, the air tanks are
filled with compressed air with a gauge pressure of 900 kPaG. Once the valves
are opened, the compressed air flows into the plenum chamber, increasing the
plenum pressure. The pressure regulators between the air tanks and the plenum
chamber are adjusted to realize the desired value of the plenum pressure.
Figure 4 presents the temporal variation of the gauge pressure in the plenum
chamber, $P_{C}$, during the experiment. In this plot, $t=0$ corresponds to
the maximum value of $P_{C}$. After the valve is opened, the pressure
immediately increases to 200 kPaG. The pressure hardly varies with time for
about 3 s, and then begins to decrease because the pressure in the air tanks
decreases. The plenum pressure decreases from 200 kPaG by 5% at $t=2.9$ s. The
flow in the test section is statistically steady for about 2 s, for which the
measurements are conducted in this study. The streamwise, vertical, and
spanwise directions are denoted by $x$, $y$, and $z$, respectively, and the
origin of the coordinate system is at the center of the jet exits of the
nozzle plate, as shown in Fig. 2.

###  2.2 Particle image velocimetry

Velocity measurements are conducted with two-dimensional and two-component
PIV. The present study uses the DANTEC PIV system, which consists of a double-
pulse Nd:YAG laser (Dantec Dynamics, Dual Power 65-15) and a high-speed camera
(Dantec Dynamics, SpeedSense 9070) with a camera lens with a focal length of
105 mm (Nikkon, AI AF Micro Nikkor 105mm F2.8D). The system is controlled with
a synchronizer (Dantec Dynamics, 80N77) and PIV software (Dantec Dynamics,
Dynamic Studio). Light sheet optics mounted on the laser produce a thin light
sheet with a thickness smaller than 1 mm. The laser unit is placed under the
wind tunnel, and a mirror in front of the laser sheet optics is used to
illuminate an $x$–$y$ plane from the bottom of the test section. The camera
captures particle images from the side of the test section. The time interval
of two laser pulses is adjusted depending on the measurement location, as
discussed below. Each laser pulse has a width of less than 4 ns. The particle-
pair images are captured at 15 Hz. The images are processed with an adaptive
PIV algorithm [66] and universal outlier detection [78] implemented in Dynamic
Studio.

The present PIV uses condensed ethanol droplets generated by the Laval nozzles
as tracer particles. Liquid ethanol is seeded and evaporated in the air tanks
before the experiments. When fluid with evaporated ethanol passes the Laval
nozzles, fluid expansion in the diverging sections causes a temperature drop,
resulting in the condensation of ethanol. This seeding technique has widely
been used in supersonic wind tunnels, which also use divergent nozzles to
produce supersonic flows. Previous studies have confirmed that generated
particles have diameters less than 1 $\mu$m: [54] reported that the diameter
of ethanol droplets was approximately $0.05$–0.2 $\mu$m; [37] used acetone
droplets produced by a Lavar nozzle for PIV measurement, by which the droplet
diameter was estimated as 160 nm. These studies have shown that the
condensation droplets are small enough for PIV measurements of supersonic
flows. The diameter can be estimated from light scattered by droplets. By
visibly observing particles illuminated by white LED (Sigmakoki, SLA-100A),
the droplet diameter is confirmed to be in the Rayleigh-scattering regime and
smaller than visible light wavelength, 360–830 nm. The present study
investigates turbulence generated by the interaction of supersonic jets. As
the jets mix with ambient fluid, the mean velocity in the measurement area is
in a subsonic regime. [37] conducted accurate PIV measurements of supersonic
flows with condensation particles. Therefore, for the present subsonic
velocity range, the ethanol droplets are useful as tracer particles of PIV.

Because how the ethanol is injected in the air tanks affects the number
density of tracer particles in the test section, we describe the detailed
procedure of the experiments. The air tanks have three ports: an inlet port
connected to the compressor, an outlet port connected to the plenum chamber,
and an exhaust port used to release compressed air after experiments. The
exhaust port can be opened or closed by a hand-operated valve, while the inlet
port can be opened by disconnecting a gas hose from the compressor. The
following procedure is repeated for the two air tanks. First, ethanol is
sprayed into the air tank via the inlet port by using an airbrush. Sprayed
ethanol quickly evaporates in the tank. When ethanol is injected, the exhaust
valve is kept partially opened to prevent the reversal flow from the inlet
port. 70 ml of liquid ethanol is injected into each tank. The exact amount of
ethanol in the tank is difficult to estimate because the air partially leaks
from the tank while spraying the ethanol. However, we have confirmed that
increasing the amount from 70 ml does not influence particle images captured
by the PIV system, whereas smaller amounts result in fewer particles. After
the compressor is connected to the inlet port of the tank again and the
exhaust valve is closed, the air is stored in the tank until the gauge
pressure reaches 900 kPaG. Even for a supersaturated case, the mass fraction
of ethanol at this stage is smaller than 1%, for which the ethanol/air mixture
and condensation are expected to have negligible influences on the turbulent
flow in the test section [16, 54]. We have also confirmed that ethanol does
not change the average and standard deviation of dynamic pressure measured
with a Pitot tube. Upon opening the solenoidal valves between the tanks and
the plenum chamber, the camera starts capturing particle images. Velocity
vectors are calculated by analyzing the images captured while the plenum
pressure is constant.

Table 1: Locations and parameters of PIV measurements.  $x_{C}$ | 0.150 m | 0.267 m | 0.360 m | 0.465 m | 0.605 m | 0.746 m
---|---|---|---|---|---|---
$y_{C}$ | 0 m | 0 m | 0 m | 0 m | 0 m | 0 m
$\Delta t$ | 3 $\mu$s | 4 $\mu$s | 4 $\mu$s | 4 $\mu$s | 4 $\mu$s | 4 $\mu$s
$N_{S}$ | 722 | 797 | 700 | 726 | 651 | 595

The PIV measurements are conducted at six streamwise locations. Table 1
summarizes the measurement locations and parameters. The center of the
measurement area is $(x,y)=(x_{C},y_{C})$. These locations are chosen to
investigate turbulence generated by the jet interaction and do not include the
region where each jet evolves before the interaction. This is because our
primary interest is in a nearly homogeneous and isotropic region where
turbulence decays. Although the supersonic jets are issued in the test
section, the mean velocity in the measurement locations is subsonic because
the high-speed jets mix with surrounding fluids with a small mean velocity.
The table also shows the time interval $\Delta t$ between two pulses of the
PIV. At $x_{C}=0.150$ m, the flow is highly inhomogeneous in the cross-
section, and the mean velocity along the centerline is higher than the
surroundings. Therefore, $\Delta t$ is smaller at $x_{C}=0.150$ m than in the
downstream region. These $\Delta t$ values are determined based on the
particle displacement during $\Delta t$. The spatial resolution, defined as
the spacing between points to calculate velocity vectors, is about 1 mm in
both directions. As shown below, the present PIV resolves the scales close to
the small-scale end of the inertial subrange and does not resolve the smallest
scale of turbulent motion. This resolution is sufficient to evaluate most
velocity statistics dominated by large-scale motion. In each wind tunnel
operation, about 35 snapshots of velocity vectors are captured on average. The
experiments are repeated many times for statistical analysis. The total number
of acquired vector images, $N_{S}$, is also shown in Table 1. The snapshots of
velocity vectors are measured at $15$ Hz, for which the time interval between
two snapshots is $0.067$ s. The mean velocity in the fully developed turbulent
region is about 40 m/s. The characteristic length scale of the flow is roughly
estimated as the spacing between the nozzles, 0.012 m. The flow time scale
defined with these velocity and length scales is $0.0003$ s, much shorter than
the sampling interval. Thus, the snapshots of velocity vectors are independent
samples that are not correlated with each other. Statistics are evaluated with
ensemble averages. Two-dimensional velocity vectors are denoted by $(u,v)$.
For a varibale $f$, the $n$th snapshot is denoted by $f^{(n)}(x,y)$. The
average of $f$, $\langle f\rangle$, is evaluated as

$\displaystyle\langle
f\rangle(x,y)=\frac{1}{N_{S}}\sum^{N_{S}}_{n=1}f^{(n)}(x,y).$ (1)

Fluctuations are defined as ${f^{\prime}}^{(n)}(x,y)=f^{(n)}(x,y)-\langle
f\rangle(x,y)$, and the rms fluctuations are evaluated as
$f_{rms}(x,y)=\langle{f^{\prime}}^{2}\rangle^{1/2}=(\langle
f^{2}\rangle-\langle f\rangle^{2})^{1/2}$. In addition, skewness and flatness
of $f$ are defined as $S(f)=\langle f^{\prime 3}\rangle/\langle f^{\prime
2}\rangle^{3/2}$ and $F(f)=\langle f^{\prime 4}\rangle/\langle f^{\prime
2}\rangle^{4}$, respectively.

###  2.3 Pitot tube measurement

Pitot tube measurements are conducted to investigate the flow evolution along
the centerline of the test section. The present study uses an L-shaped
standard Pitot tube with a 7 mm diameter (Testo, 0635 2145) and a pressure
sensor (SMC, PSE543A-R06). The signal from the pressure sensor is recorded
with an oscilloscope (Yokokawa, DL850E) at a sampling rate of 10 kHz. Total
pressure $P_{T}$ is measured at the tip of the tube, while static pressure
$P_{S}$ is measured on the side of the tube, which is separated from the tip
by 30 mm in the streamwise direction. The top wall of the test section is
replaced with many small plates between which the Pitot tube is inserted into
the test section. The measurement location is along the centerline of the test
section. As shown in Sect. 3.1, the streamwise distribution of static pressure
$P_{S}(x)$ is well fitted by an exponential curve
$P_{S,F}(x)=a_{P}(b_{P})^{x}$, where the parameters can be obtained with a
least squares method for measured $P_{S}$. Because the Pitot tube measures
$P_{T}(x)$ and $P_{S,F}(x+\Delta_{Pitot})$ with $\Delta_{Pitot}=30$ mm, the
pressure difference $\Delta P(x)=P_{T}(x)-P_{S}(x)$ is calculated with the
measured $P_{T}(x)$ and the fitting curve $P_{S,F}(x)$. A time average is
calculated for the time series pressure data over 2 s, for which the plenum
pressure is almost constant.

Because the measurements are conducted in turbulence, the velocity
fluctuations also affect $\Delta P$, whose measured values are not related
solely to the dynamic pressure due to the mean flow. Turbulence increases the
total pressure measured by the Pitot tube. The following relation expresses
the pressure difference measured by a Pitot tube in turbulence [4]:

$\displaystyle\Delta P=\frac{1}{2}\rho\left(\langle
u\rangle^{2}+\langle{u^{\prime}}^{2}\rangle+\langle{v^{\prime}}^{2}\rangle+\langle{w^{\prime}}^{2}\rangle\right).$
(2)

The velocity variances are evaluated with the PIV as
$\langle{u^{\prime}}^{2}\rangle+\langle{v^{\prime}}^{2}\rangle+\langle{w^{\prime}}^{2}\rangle\approx\langle{u^{\prime}}^{2}\rangle+2\langle{v^{\prime}}^{2}\rangle$.
Here, the Pitot tube measurement is conducted along the centerline, where
$\langle{v^{\prime}}^{2}\rangle\approx\langle{w^{\prime}}^{2}\rangle$ is
expected to be valid. The density is also estimated from static pressure and
temperature measurements, as described below. Thus, the mean velocity is
calculated from $\Delta P$ measured by the Pitot tube as

$\displaystyle\langle u\rangle=\sqrt{\frac{2\Delta
P}{\rho}-(\langle{u^{\prime}}^{2}\rangle+2\langle{v^{\prime}}^{2}\rangle)}.$
(3)

Another factor that may affect the Pitot tube measurement is the
compressibility effect. However, based on the mean velocity measured with PIV,
we have estimated that this effect results in underestimating the mean
velocity only by less than 2%. Therefore, the compressibility correction for
mean velocity is not applied to the Pitot tube measurement.

###  2.4 Temperature measurement

Temperature is measured with a fine sheathed K-type thermocouple (J Thermo,
TJK-LS1501GP), whose sheath has a 0.15 mm diameter and a 100 mm length. The
response time of this thermocouple is less than 1 s. The output voltage of the
thermocouple is processed and recorded by a data logger (Hioki, LR8431) at a
sampling rate of 100 Hz. Because of the response time, the temperature
recorded over 1 s after 1 s from the start of the wind tunnel is averaged to
estimate the mean temperature. The thermocouple is inserted from the top of
the test section in the same manner as the Pitot tube. Temperature
measurements are conducted at different streamwise locations.

###  2.5 Estimation of the dissipation rate of turbulent kinetic energy

The decay of turbulent kinetic energy along the centerline is investigated for
a nearly homogeneous and isotropic region corresponding to the downstream
region. The turbulent kinetic energy per unit mass,
$k_{T}=(\langle{u^{\prime}}^{2}\rangle+\langle{v^{\prime}}^{2}\rangle+\langle{w^{\prime}}^{2}\rangle)/2$
is evaluated as
$(\langle{u^{\prime}}^{2}\rangle+2\langle{v^{\prime}}^{2}\rangle)/2$ with the
PIV data. The governing equation for $k_{T}$ can be derived from the
conservation equation of momentum for a compressible fluid, which is rewritten
in the form of

$\displaystyle\frac{\partial u_{i}}{\partial t}+u_{j}\frac{\partial
u_{i}}{\partial x_{j}}=-\frac{1}{\rho}\frac{\partial p}{\partial
x_{i}}+\frac{1}{\rho}\frac{\partial\tau_{ij}}{\partial x_{j}}.$ (4)

Here, $p(x,y,z,t)$ and $\rho(x,y,z,t)$ are local values of pressure and
density, respectively, and $\tau_{ij}$ is the viscous stress tensor.
Multiplying Eq. (4) by $u^{\prime}_{i}=u_{i}-\langle u_{i}\rangle$ and
rearranging terms yield the following governing equation for $k_{T}$:

$\displaystyle\frac{\partial k_{T}}{\partial t}+\langle
u_{j}\rangle\frac{\partial k_{T}}{\partial x_{j}}=\underbrace{-\langle
u^{\prime}_{i}u^{\prime}_{j}\rangle\frac{\partial\langle
u_{i}\rangle}{\partial
x_{j}}}_{P_{k}}\underbrace{-\frac{1}{2}\frac{\partial\langle
u^{\prime}_{i}u^{\prime}_{i}u^{\prime}_{j}\rangle}{\partial
x_{j}}}_{D_{T}}+\underbrace{\left\langle\frac{u^{\prime}_{i}u^{\prime}_{i}}{2}\frac{\partial
u^{\prime}_{j}}{\partial x_{j}}\right\rangle}_{\Theta_{k}}$
$\displaystyle\underbrace{-\left\langle\frac{1}{\rho}\frac{\partial
u^{\prime}_{j}p}{\partial
x_{j}}\right\rangle}_{D_{P}}+\underbrace{\left\langle\frac{p}{\rho}\frac{\partial
u^{\prime}_{j}}{\partial
x_{j}}\right\rangle}_{\Theta_{P}}+\underbrace{\left\langle\frac{1}{\rho}\frac{\partial
u^{\prime}_{i}\tau_{ij}}{\partial
x_{j}}\right\rangle}_{D_{V}}-\underbrace{\left\langle\frac{\tau_{ij}}{\rho}\frac{\partial
u^{\prime}_{i}}{\partial x_{j}}\right\rangle}_{\varepsilon}.$ (5)

The first and second terms of the left-hand side represent the temporal
variation of $k_{T}$ and the advection due to mean flow, respectively. On the
right-hand side, $P_{k}$ is the production term, $D_{T}$ is the turbulent
diffusion term, $D_{P}$ is the pressure diffusion term, $D_{V}$ is the viscous
diffusion term, and $\varepsilon$ is the dissipation term. These terms also
appear in the governing equation for an incompressible fluid. The remaining
terms represent direct influences of fluid compressibility: $\Theta_{k}$ is
the change in the turbulent kinetic energy per unit mass due to fluid
expansion or compression, and $\Theta_{P}$ is the energy conversion between
the kinetic and internal energies by pressure work. Even though the
interaction of the supersonic jets generates turbulence near the nozzles, the
velocity fluctuations in the decay region are subsonic, implying that
$\Theta_{k}$ and $\Theta_{P}$ are not dominant in the variation of $k_{T}$
once the nearly homogeneous and isotropic turbulence has fully developed. As
long as the Reynolds number is not too small, the viscous contribution to the
spatial transport of kinetic energy is negligible, and $D_{V}$ is much smaller
than other terms in turbulence [75, 73]. In addition, the flow is assumed to
be statistically steady and homogeneous in the $y$ and $z$ directions, and the
diffusion terms in these directions are negligible. The turbulent and pressure
diffusions in the streamwise direction are also negligible compared with the
transport due to the mean flow in decaying HIT [73]. Then, Eq. (5) is
simplified as

$\displaystyle\langle u\rangle\frac{\partial k_{T}}{\partial x}=-\varepsilon.$
(6)

This relation is often used to evaluate the dissipation rate in grid
turbulence because $\langle u\rangle$ and $k_{T}$ are much easier to measure
than $\varepsilon$ [35, 67]. The decay of HIT depends on the spectral shape at
large scales, which can differ depending on how turbulence is generated [20].
The compressibility effect on the decay properties may also emerge through the
generation process of turbulence, although this influence does not appear in
the governing equation of $k_{T}$. Such influences may appear in the virtual
origin and decay exponent in the power law decay of $k_{T}$ [13, 86].

Equation (6) is used to estimate the turbulent kinetic energy dissipation rate
from the PIV data. The decay of $k_{T}$ in HIT is well represented by a power
law given by

$\displaystyle k_{T}=a_{k}(x-x_{0})^{-n},$ (7)

where $a_{k}$ is a coefficient related to the level of velocity fluctuations,
$x_{0}$ is the virtual origin, and $n$ is the decay exponent [45, 20]. A least
squares method is applied to the streamwise distribution of $k_{T}$ obtained
by the PIV data. Then, $\varepsilon$ is evaluated as

$\displaystyle\varepsilon=\langle u\rangle an(x-x_{0})^{-n-1}.$ (8)

Figure 5: Streamwise variations of (a) mean static pressure $P_{S}$ and (b)
mean temperature $T$. The error bars represent the standard deviation.

##  3 Results and discussion

###  3.1 Pressure and temperature profiles

Figure 5(a) presents the streamwise variation of the mean static pressure
(gauge pressure), $P_{S}$. The error bars represent the standard deviations of
time series pressure histories, which indicate the degree of temporal pressure
fluctuations. The pressure increases in the downstream direction and
approaches the atmospheric pressure. The pressure distribution is well
approximated by an exponential curve $P_{S}=a_{P}(b_{P})^{x}$, where
$a_{P}=-15.0$ kPaG and $b_{P}=6.61\times 10^{-5}$ are obtained with a least
squares method. As the results are shown as gauge pressure, this pressure
variation is slight compared to the atmospheric pressure and hardly affects
the mean fluid density in the test section. Figure 5(b) presents the mean
temperature $T$ along the centerline. The error bars also represent the
standard deviation. The streamwise variation of $T$ is slight, and the average
of all data points yields 303 K.

These measurement results enable us to estimate fluid density $\rho$ with the
equation of state for an ideal gas as $\rho=P/RT$ with the gas constant
$R=287.05$ J/kg K. Here, the exponential function $P_{S}=a_{P}(b_{P})^{x}$ and
mean temperature of 303 K are used to evaluate $\rho(x)$. The mean temperature
yields the viscosity coefficient $\mu$, calculated with Sutherland’s law as

$\displaystyle\mu=\mu_{0}\left(\frac{T}{T_{0}}\right)\left(\frac{T_{0}+S}{T+S}\right),$
(9)

with $T_{0}=273$ K, $S=110.4$ K, and $\mu_{0}=1.724\times 10^{-5}$ Pa s. As
the mean temperature does not vary with $x$, $\mu$ is also treated as a
constant. These fluid properties are used in the analysis of velocity
statistics obtained with the PIV.

###  3.2 Velocity statistics

Figure 6:  Instantaneous fluctuating velocity vectors
${\bf{u}^{\prime}}=(u^{\prime},v^{\prime})$ measured at (a) $x_{C}=0.150$ m
and (b) $x_{C}=0.465$ m. The color represents the vector magnitude
$|{\bf{u}^{\prime}}|$.

Figure 6 shows instantaneous fluctuating velocity vectors
${\bf{u}^{\prime}}=(u^{\prime},v^{\prime})$ measured at $x_{C}=0.150$ m and
$0.465$ m. The color represents the vector magnitude. For both measurement
locations, the turbulent jets from the nozzles have fully developed and have
interacted with each other, and the clear imprints of each jet are not
observed in the velocity field. At $x_{C}=0.150$ m, the velocity fluctuations
reach about 150 m/s. As turbulence decays, velocity fluctuations become small
from $x_{C}=0.150$ m to $0.465$ m. At both locations, the vector fields
exhibit typical rotating patterns associated with vortices, e.g.,
$(x,y)=(0.11$ m, $-0.01$ m$)$ in Fig. 6(a), indicating that the present PIV
well captures these turbulent structures.

Figure 7:  Statistical convergence tests for mean velocity and rms velocity
fluctuations with the PIV data measured at $(x_{C},y_{C})=(0.360$ m, $0$ m$)$.
Totally 700 vector snapshots are divided into two datasets, each containing
350 snapshots. The mean velocity and rms velocity fluctuations calculated
separately for the two datasets are plotted as functions of $x$ at $y=0$.

The degree of statistical convergence is examined before the velocity
statistics are discussed. For the measurement location of
$(x_{C},y_{C})=(0.360~{}$m, $0$ m$)$, 700 snapshots of velocity vectors are
acquired. These snapshots are divided into two datasets, each containing 350
snapshots. The velocity statistics are separately calculated for the two
datasets whose differences arise from statistical errors. Figure 7 compares
the streamwise distributions of the average and rms fluctuations of streamwise
velocity, $\langle u\rangle$ and $u_{rms}$, at $y=0$. The two datasets yield
slightly different distributions. A statistical quantity $f$ calculated with
datasets 1 and 2 is denoted by $f_{1}$ and $f_{2}$, respectively. For the
distribution of $f$ along the $x$ axis at $y=0$, the statistical error arising
from the finite number of samples is evaluated with $f_{1}(x,0)$ and
$f_{2}(x,0)$ as an rms error:

$\displaystyle
E(f)=\sqrt{\frac{1}{x_{S}-x_{E}}\int_{x_{S}}^{x_{E}}(f_{1}-f_{2})^{2}dx},$
(10)

where $x_{S}\leq x\leq x_{E}$ is the measurement area. For Fig. 7, the
relative errors, $E(\langle u\rangle)/\langle u\rangle$ and
$E(u_{rms})/u_{rms}$, are about 5%, which is a typical statistical error when
the number of the snapshots is about 350. Thus, statistical errors for the
present PIV measurement with $N_{S}>500$ are expected to be smaller than 5%.
The statistical errors of other velocity statistics are evaluated with this
method for each PIV measurement location, and $E(f)$ is shown in figures
below.

Figure 8:  Streamwise distributions of (a) mean streamwise velocity $\langle
u\rangle$ and (b) rms velocity fluctuations of streamwise and vertical
velocities, $u_{rms}$ and $v_{rms}$, at $y=0$. The gray shaded area represents
the statistical errors estimated by Eq. (10).

Figure 8(a) shows the streamwise distribution of mean streamwise velocity
$\langle u\rangle$. The statistical errors estimated with Eq. (10) are shown
in gray for PIV data. The mean velocity obtained with the Pitot tube is also
plotted for comparison. For the Pitot tube measurement, the error bars
represent the mean velocities which correspond to the pressure differences of
$\Delta P\pm(\Delta P)_{rms}$, where $(\Delta P)_{rms}$ is the rms
fluctuations of $\Delta P$. For the Pitot tube measurement at $x=0.09$ m, the
PIV data for turbulence correction, Eq. (3), is unavailable. Therefore, an
extrapolation of the PIV data provides the velocity variance used for the
correction. The mean velocity distribution hardly differs for the PIV and the
Pitot tube, ensuring the measurement accuracy of both methods. The mean
velocity rapidly decreases in the $x$ direction up to $x\approx 0.3$ m and
then stays at an almost constant value of 37 m/s.

Figure 8(b) shows the distribution of rms velocity fluctuations of streamwise
and vertical velocities, $u_{rms}$ and $v_{rms}$. The rms velocity
fluctuations increase with $x$ for $x\lesssim 0.2$ m, where turbulence is
generated. They continuously decay in the $x$ direction after the peaks at
$x\approx 0.2$ m. Although $u_{rms}$ is much larger than $v_{rms}$ in the
upstream region, the difference between $u_{rms}$ and $v_{rms}$ becomes small
with $x$. Their ratio $u_{rms}/v_{rms}$ is often used to evaluate the isotropy
at large scales. The average of $u_{rms}/v_{rms}$ for $x>0.4~{}$m is 1.08.
This value is close to those in grid turbulence, for which
$u_{rms}/v_{rms}\approx 1.1$ has been reported [39, 32, 36]. Thus, grid
turbulence and turbulence generated by the supersonic jet interaction have a
similar degree of isotropy.

Figure 9:  Vertical distributions of (a) mean velocity and (b) rms velocity
fluctuations for streamwise and vertical velocities.

Figure 9 shows the vertical distributions of mean velocities, $\langle
u\rangle$ and $\langle v\rangle$, and rms velocity fluctuations, $u_{rms}$ and
$v_{rms}$. At $x=0.16$ m, the mean velocity varies significantly with $y$. The
$y$ dependence of the mean velocity becomes weak as $x$ increases. For $x\geq
0.36$ m, $\langle u\rangle$ and $\langle v\rangle$ are almost uniform in both
$x$ and $y$ directions, and the mean velocity gradient is negligible in the
downstream region. The rms velocity fluctuations are also inhomogeneous in the
$y$ direction at $x=0.16$ m, where $u_{rms}$ and $v_{rms}$ are large away from
the center ($y=0$). Large $u_{rms}$ and $v_{rms}$ at $x=0.16$ m appear in the
region of $y$ with the large mean velocity gradient $\partial\langle
u\rangle/\partial y$, and the production of turbulent kinetic energy due to
the mean shear still influences the flow at $x=0.16$ m. The rms velocity
fluctuations become homogeneous in the $y$ direction for $x\geq 0.27$ m. These
results suggest the nearly homogeneous and isotropic turbulence with a uniform
mean flow forms for $x\gtrsim 0.3$ m. With the nozzle spacing $L_{J}=12$ mm
and the jet exit diameter $D_{E}=4.31$ mm, $x\gtrsim 0.3$ m corresponds to
$x/L_{J}\gtrsim 25$ and $x/D_{E}\gtrsim 70$.

Figure 10:  Streamwise distributions of skewness and flatness of streamwise
and vertical velocities at $y=0$.

Figure 10 shows the streamwise distribution of skewness $S$ and flatness $F$
of velocity components. A variable with a probability distribution following a
Gaussian function has $S=0$ and $F=3$, shown as horizontal lines in Fig. 10.
These Gaussian values of $S$ and $F$ have been obtained for various types of
fully-developed isotropic turbulence [30, 38]. For the present facility, $S$
and $F$ greatly differ from the Gaussian values in the upstream region of
$x\lesssim 0.15$ m. The flow is highly inhomogeneous and anisotropic in this
region. Large $F$ is related to the intermittency of large-scale fluctuations.
The intermittent behavior of the jet interaction may cause large values of
$F(u)$ and $F(v)$, as also reported for the interaction of two incompressible
jets [84]. As turbulence develops, $S$ and $F$ approach the Gaussian values,
and $S\approx 0$ and $F\approx 3$ are achieved for $x\gtrsim 0.2$. Therefore,
the intermittent velocity fluctuations become insignificant once the flow
becomes statistically homogeneous in the cross-section.

The large-scale turbulent motion is characterized by integral scales, which
can be evaluated with velocity auto-correlation functions. The longitudinal
auto-correlation functions for streamwise and vertical velocities are
respectively defined as

$\displaystyle f_{u}(x,y,r_{x})=\frac{\langle
u^{\prime}(x,y)u^{\prime}(x+r_{x},y)\rangle}{\langle u^{\prime
2}(x,y)\rangle},$ (11) $\displaystyle f_{v}(x,y,r_{y})=\frac{\langle
v^{\prime}(x,y)v^{\prime}(x,y+r_{y})\rangle}{\langle v^{\prime
2}(x,y)\rangle},$ (12)

where $r_{x}$ and $r_{y}$ are separation distances. Because the PIV measures
the velocity profiles in a finite area, the available ranges of $r_{x}$ and
$r_{y}$ are limited. For evaluations of the correlation function with large
distances, $f_{u}(r_{x})$ is calculated for $(x,y)$ corresponding to a
location near the upstream end of the measurement area at $y=0$. Similarly,
$f_{v}(r_{y})$ is calculated for $(x,y)$ corresponding to the upstream bottom
corner of the measurement area. In this way, $r_{x}$ and $r_{y}$ range up to
the size of the measurement area.

The longitudinal integral scales are defined as the integrals of $f_{u}$ and
$f_{v}$:

$\displaystyle
L_{u}=\int_{0}^{r_{x0}}f_{u}dr_{x},~{}~{}~{}L_{v}=\int_{0}^{r_{y0}}f_{v}dr_{y}.$
(13)

Here, $r_{x0}$ and $r_{y0}$ are often taken as the first zero-crossing point
[13]. For the present PIV, the measurement area is insufficient to calculate
the correlation functions up to the zero-crossing point. In addition, as also
observed for hot-wire measurements in wind tunnels, an accurate evaluation of
correlation functions for large $r_{x}$ and $r_{y}$ is problematic because of
a limited number of large-scale samples. For these reasons, the integral scale
is often evaluated by approximating the correlation function at large $r$ with
an exponential function [47, 7, 28]. The present study also adapts this
methodology for the calculation of integral scales. An exponential function
$a_{f}\mathrm{exp}(-b_{f}r_{x})$ is obtained with a least squares method
applied for measured $f_{u}$ in the range of $r_{x}$ with $f_{u}<0.5$. The
numerical integration of $f_{u}$ is calculated with the measured $f_{u}$ and
the exponential function as

$\displaystyle
L_{u}=\int_{0}^{r_{1/2}}f_{u}dr_{x}+\int_{r_{1/2}}^{r_{end}}a_{f}\mathrm{e}^{-b_{f}r_{x}}dr_{x},$
(14)

where $r_{1/2}$ is defined as $f_{u}(r_{1/2})=0.5$ and $r_{end}$ corresponds
to $r_{x}$ where $a_{f}\mathrm{exp}(-b_{f}r_{x})$ is equal to $10^{-5}$.
Because of the exponential decay of the correlation, larger values of
$r_{end}$ do not affect the evaluation of the integral scale. The integral
scale in the vertical direction $L_{v}$ is also calculated by the same method
applied for $f_{v}$.

Figure 11:  (a) Longitudinal auto-correlation function of streamwise velocity
at $x=570$ mm and $y=0$ mm. The measured correlation is compared with an
exponential function $f_{u}=a_{f}\mathrm{exp}(-b_{f}r_{x})$ and the
correlation functions in incompressible turbulent jets (Namer and Ötügen 49,
Watanabe et al 76). A least squares method determines $a_{f}=0.843$ and
$b_{f}=-32.8$ m-1 from the measured $f_{u}$. The separation distance $r_{x}$
is normalized by the integral scale $L_{u}$. (b) The streamwise variation of
integral scales $L_{u}$ and $L_{v}$, which are calculated from the
longitudinal auto-correlation functions of streamwise and vertical velocities,
respectively. The inset shows the ratio between $L_{u}$ and $L_{v}$.

Figure 11(a) shows the correlaton function $f_{u}$ and its approximation based
on the exponential function at $(x,y)=(570$ mm$,0$ mm$)$. The separation
distance $r_{x}$ is normalized with the integral scale for comparisons with
other turbulent flows. The exponential function accurately approximates the
decay of $f_{u}$ for large $r_{x}$. In addition, the profile of $f_{u}$ agrees
well with the correlation functions measured in other turbulent flows.

Figure 11(b) presents the streamwise variation of the integral scales. In the
inhomogeneous and anisotropic region, $x<0.3$ m, the streamwise integral scale
$L_{u}$ is larger than the vertical scale $L_{v}$, and large-scale velocity
fluctuations are also anisotropic. On the other hand, the downstream region
satisfies $L_{u}\approx L_{v}$. This is also confirmed by the inset presenting
$L_{u}/L_{v}$. The average of $L_{u}/L_{v}$ for $x>0.3$ m is 1.03, and the
integral scales hardly differ in both directions, suggesting that turbulent
motion at large scales is statistically isotropic. For $0.3\lesssim x\lesssim
0.6$, the integral scales increase as turbulence decays. However, in the most
downstream location of the measurement, $L_{u}$ and $L_{v}$ decrease with $x$.
In DNS of decaying isotropic turbulence in a triply periodic domain, the decay
is influenced by the boundary conditions when the domain size divided by the
integral scale is smaller than about 3 [2]. The side length of the cross-
section, 0.1 m, is about $3.7L_{u}$ at $x=0.57$ m. The decrease of $L_{u}$ and
$L_{v}$ in the further downstream region may be caused by the confinement
effects.

Figure 12:  (a) The decay of turbulent kinetic energy $k_{T}$ along the
centerline. The red broken line represents a power law
$k_{T}=a_{k}(x-x_{0})^{-n}$ with $a_{k}=21.0$ m2/s2, $x_{0}=0.12$ m, and
$n=2.1$, which are obtained with a least squares method applied for $x>0.32$
m. (b) The streamwise variations of Taylor microscale $\lambda$, Kolmogorov
scale $\eta$, and turbulent Reynolds number $Re_{\lambda}$, estimated with the
power law decay of $k_{T}$.

###  3.3 The dissipation rate of turbulent kinetic energy

Figure 12(a) shows the streamwise variation of turbulent kinetic energy per
unit mass
$k_{T}\approx(\langle{u^{\prime}}^{2}\rangle+2\langle{v^{\prime}}^{2}\rangle)/2$
along the centerline. The decay of $k_{T}$ is investigated for the nearly
homogeneous and isotropic region, $x>0.32$ m. A nonlinear least squares method
(Levenberg–Marquardt method) applied for $x>0.32$ m yields the parameters in
the power law, Eq. (7), as $a_{k}=21.0$ m2/s2, $x_{0}=0.12$ m, and $n=2.1$.
Equation (7) with these parameters is also shown in the figure. The decay of
$k_{T}$ is well approximated by the power law. The exponent $n=2.1$ is larger
than typical values reported for incompressible grid turbulence at high
Reynolds numbers, $n\approx 1.0$–1.5 [35, 33, 67, 83]. As discussed in Sect.
1, [86] also reported that $n$ in compressible grid turbulence tends to be
larger than incompressible cases. This tendency is qualitatively consistent
with the large exponent in this study. However, these results contradict the
decrease of $n$ observed in shock-tube experiments [13, 25], and the
compressibility effects on the decay exponent should be addressed in future
studies.

The small-scale turbulence properties are often related to the kinetic energy
dissipation rate. The Taylor microscale $\lambda$ and Kolmogorov scale $\eta$
are defined as

$\displaystyle\lambda$ $\displaystyle=\sqrt{\frac{10\mu
k_{T}}{\rho\varepsilon}},$ (15) $\displaystyle\eta$
$\displaystyle=(\mu/\rho)^{3/4}\varepsilon^{-1/4}.$ (16)

The turbulent Reynolds number $Re_{\lambda}$ is define with $\lambda$ as

$\displaystyle Re_{\lambda}=\frac{\rho\sqrt{2k_{T}/3}\lambda}{\mu}.$ (17)

The dissipation rate estimated from the decay of $k_{T}$ with Eq. (8) is used
to evaluate $\lambda$, $\eta$, and $Re_{\lambda}$. Figure 12(b) shows their
streamwise variations in the nearly homogeneous and isotropic region.
$\lambda$ and $\eta$ increase as turbulence decays in the $x$ direction. The
orders of the length scales are $\lambda=O(10^{-3}$ m$)$ and $\eta=O(10^{-5}$
m$)$. The spatial resolution of PIV is about $1.0\times 10^{-3}$ m, and the
scales greater than the Taylor microscale are resolved by the PIV. This
resolution is sufficient to investigate the large-scale properties, such as
mean velocity and rms velocity fluctuations. The turbulent Reynolds number
decreases with the decay of turbulence. It ranges between about 800 and 400.
These values of $Re_{\lambda}$ are larger than those of most subsonic grid
turbulence in large wind tunnel facilities because of the large velocity
fluctuations of supersonic jets.

Figure 13:  Normalized energy spectra of (a) streamwise velocity,
$E_{u}(k_{x})$, and (b) vertical velocity, $E_{v}(k_{y})$, where $k_{x}$ and
$k_{y}$ are the wavenumbers in the streamwise and vertical directions,
respectively. The spectra and wavenumbers are non-dimensionalized with the
kinematic viscosity $\nu=\mu/\rho$, the energy dissipation rate per unit mass
$\varepsilon$, and the Kolmogorov scale $\eta$. The present results are
compared with direct numerical simulation of forced HIT ($Re_{\lambda}=202$)
[74] and experiments of a boundary layer ($Re_{\lambda}=202$) [59, 50] and
grid turbulence ($Re_{\lambda}=520$) [35].

One-dimensional energy spectra of streamwise and vertical velocities ($E_{u}$
and $E_{v}$) are calculated with Fourier transform in the $x$ and $y$
directions, respectively. The wavenumbers in these directions are denoted by
$k_{x}$ and $k_{y}$. The energy spectra in turbulence with a high Reynolds
number are known to have a universal shape in an inertial subrange in various
turbulent flows [55]. Figure 13 presents $E_{u}$ and $E_{v}$ measured at
several streamwise locations. The spectra and wavenumbers are normalized by
the dissipation rate $\varepsilon$ and kinematic viscosity $\nu=\mu/\rho$. The
present results are compared with incompressible turbulent flows. The spectra
are evaluated for normalized wavenumbers between $10^{-3}$ and $10^{-1}$,
determined by the measurement area and spatial resolution. The present results
quantitatively agree with other turbulent flows. They are also consistent with
the power laws expected for the inertial subrange, $E_{u}\sim k_{x}^{-5/3}$
and $E_{v}\sim k_{y}^{-5/3}$. Thus, the present PIV with condensation
particles is capable of measuring the scale dependence of velocity
fluctuations in the inertial subrange. In addition, the flow at a wide range
of scales is in a statistically isotropic state, as attested by $E_{u}\approx
E_{v}$.

Figure 14:  Dependence of non-dimensional energy dissipation rate
$C_{\varepsilon}$ on the turbulent Reynolds number $Re_{\lambda}$. The present
results are compared with experimental data of subsonic grid turbulence: [35],
[18], and [82] for passive-grid generated turbulence; [26], [33], [67], [56],
[46], and [83] for active-grid generated turbulence.

The dissipation rate is often related to characteristic velocity ${\cal U}$
and lenght ${\cal L}$ of large-scale turbulent motion as

$\displaystyle\varepsilon=C_{\varepsilon}\frac{{\cal U}^{3}}{\cal L},$ (18)

where $C_{\varepsilon}$ is a non-dimensional dissipation rate.
$C_{\varepsilon}$ plays a central role in theories and models of turbulent
flows [72]. The velocity and length scales in the present experiment are
evaluated as ${\cal U}=\sqrt{(\langle u^{\prime 2}\rangle+\langle v^{\prime
2}\rangle)/2}$ and ${\cal L}=(L_{u}+L_{v})/2$, and $C_{\varepsilon}$ is
evaluated with $\varepsilon$ in the nearly homogeneous and isotropic region.
Figure 14 plots $C_{\varepsilon}$ as a function of $Re_{\lambda}$ for the
present experiment and other studies of incompressible grid turbulence. In
fully-developed grid turbulence with high $Re_{\lambda}$ in incompressible
flows, $C_{\varepsilon}$ takes values approximately between 0.4 and 1.1 while
$C_{\varepsilon}$ increases rapidly as $Re_{\lambda}$ becomes smaller than
about 100. The present values of $C_{\varepsilon}=0.51$–0.87 are within a
range reported for grid turbulence with high $Re_{\lambda}$. The decay
exponent of the turbulent kinetic energy is known to be influenced by the
spatial variation of $C_{\varepsilon}$. For the present experiment,
$C_{\varepsilon}$ slightly increases in the streamwise direction. The increase
of $C_{\varepsilon}$ causes a larger decay exponent $n$ [38]. The spatial
variation of $C_{\varepsilon}$ is possibly relevant to the large values of $n$
in the present experiment and in compressible grid turbulence [86].

##  4 Conclusion

The present study has reported the development and characterization of the
multiple-supersonic-jet wind tunnel, which is designed to investigate decaying
nearly homogeneous and isotropic turbulence in a compressible flow regime. The
PIV, pitot tube, and thermocouple measurements characterize the generated
turbulence, where velocity vectors, static and dynamic pressures, and
temperature are measured. The present wind tunnel can sustain a steady mean
flow over 2–3 s. It can also seed tracer particles in the test section by
condensation. The accuracy of the PIV is confirmed by comparing the velocity
statistics with other turbulent flows. In addition, the mean velocity measured
by the PIV agrees well with the Pitot tube measurements, for which turbulence
correction is applied with the PIV data of velocity variances.

The mean velocity in the upstream region is large along the centerline of the
test section. However, as the flow evolves, the mean velocity profile becomes
uniform in the test section. Then, the mean velocity gradient becomes
negligibly small in the downstream region. Once the mean flow profile becomes
uniform, nearly homogeneous and isotropic turbulence begins to decay. At this
decay regime, rms velocity fluctuations are also uniform in the cross-section.
The ratio of rms values of streamwise- and vertical-velocity fluctuations is
about 1.08, close to the values reported for incompressible grid turbulence.
The longitudinal integral length scales are also similar in the streamwise and
vertical directions. These statistics have confirmed that the supersonic jet
interaction generates nearly homogeneous and isotropic turbulence. In the
inhomogeneous and anisotropic region near the jet nozzles, the flatness of
velocity fluctuations is larger than 3, indicating that the large-scale
velocity fluctuations are highly intermittent. The skewness and flatness of
velocity fluctuations are close to the Gaussian values in the decay region,
where the large-scale intermittency is insignificant.

The decay of turbulent kinetic energy per unit mass is investigated in the
nearly homogeneous and isotropic region. The turbulent kinetic energy decays
according to a power law, as reported for subsonic grid turbulence. The decay
exponent $n$ for the supersonic jet interaction is about 2, which is larger
than the values reported for incompressible grid turbulence. This large value
of $n$ is consistent with experiments of compressible grid turbulence in a
high-speed wind tunnel by [86]. In addition, the dissipation rate of turbulent
kinetic energy per unit mass has been evaluated with the decay of the
turbulent kinetic energy. The energy spectra of velocity fluctuations
normalized by the dissipation rate and kinematic viscosity follow the
universal power law of the inertial subrange, observed for other
incompressible turbulent flows. The non-dimensional dissipation rate
$C_{\varepsilon}$ for the supersonic jet interaction is within a range of
0.51–0.87, which is also consistent with grid turbulence at a high Reynolds
number.

The nozzle component in the present wind tunnel can be replaced with other
components with different nozzle geometries, generating turbulent jets with
different Mach and Reynolds numbers. This feature is helpful in varying the
degree of the compressibility effects, which will be considered in future
studies. These results have demonstrated that the multiple-supersonic-jet wind
tunnel is helpful in the investigation of decaying homogeneous isotropic
turbulence in a compressible flow regime.

## Statements and Declarations

### Funding

This work was supported by Japan Society for the Promotion of Science (KAKENHI
Grant Numbers JP22K03903 and JP22H01398).

### Competing interests

The authors have no competing interests to declare that are relevant to the
content of this article.

### Availability of data and materials

The data that support the findings of this study are available from the
corresponding author upon reasonable request.

### Authors’ contributions

Takahiro Mori: Data curation (equal); Formal Analysis (equal); Investigation
(lead); Methodology (equal); Resources (equal); Software (supporting);
Validation (equal); Visualization (equal); Writing - original draft
(supporting); Writing - review & editing (equal). Tomoaki Watanabe:
Conceptualization (lead); Data curation (equal); Formal Analysis (equal);
Funding acquisition (equal); Investigation (supporting); Methodology (equal);
Project administration (lead); Resources (equal); Software (lead); Supervision
(lead); Validation (equal); Visualization (equal); Writing - original draft
(lead); Writing - review & editing (equal). Koji Nagata: Conceptualization
(supporting); Data curation (supporting); Funding acquisition (equal);
Investigation (supporting); Methodology (supporting); Project administration
(supporting); Resources (equal); Software (supporting); Supervision
(supporting); Validation (equal); Visualization (supporting); Writing -
original draft (supporting); Writing - review & editing (equal).

Acknowledgments The authors acknowledge Prof. K. Mori (Osaka Prefecture
University), Mr. N. Iwakura (Nagoya University), and Ms. R. Nakayama (Nagoya
University) for their help and valuable comments in developing the multiple-
supersonic-jet wind tunnel. The authors also acknowledge Mr. K. Ishizawa
(Nagoya University) for assistance in experiments.

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