∎

11institutetext: N. Manzano-Miura 22institutetext: Cornell University, Sibley
School of Mechanical and Aerospace Engineering, Ithaca, NY, USA
22email<EMAIL_ADDRESS>33institutetext: D. Gloutak 44institutetext:
University of Colorado Boulder, Ann and HJ Smead Aerospace Engineering
Sciences, Boulder, CO, USA 55institutetext: G.P. Bewley 66institutetext:
Cornell University, Sibley School of Mechanical and Aerospace Engineering,
Ithaca, NY, USA

# Generation of variable compressibility in turbulence at fixed Reynolds
numbers

N. Manzano-Miura  D. Gloutak  G. P. Bewley

(Received: date / Accepted: date)

###### Abstract

The Variable Density and Speed of Sound Vessel (VDSSV) produces subsonic
turbulent flows that are both compressible and observable at all scales with
existing instrumentation including hot wires and particle tracking. We realize
this objective by looking at the flow of a heavy gas (sulfur hexafluoride
SF6), with a speed of sound almost three times lower than for air. By
switching between air and SF6, we isolate the influence of the turbulent Mach
number (up to $M_{t}$ = 0.17) on turbulence statistics from the influences of
changes in the Reynolds number (up to $R_{\lambda}$ = 1600), and boundary
conditions, which we hold constant. A free shear flow is produced by a ducted
fan, and we show that it behaves like a turbulent jet in that the mean
velocity profiles approach self-similarity with increasing distance from the
orifice (up to $x/D_{f}$ = 9). The jet responds like a compressible shear
layer in that it spreads more slowly at higher Mach numbers (up to $M_{j}$ =
0.7) than at low Mach numbers. In contrast, the integral length scales and
Kolmogorov constant of the turbulence are approximately invariant with respect
to changes in either the Reynolds or Mach numbers. We briefly report on
instrumentation under development that will extend the accessible Taylor-scale
Reynolds and turbulent Mach numbers to 4000 and 0.3, respectively.

###### Keywords:

Compressible turbulence Hot-wire anemometry Free-stream turbulence

††journal: Experiments in Fluids

##  1 Introduction

Compressible turbulence plays a determining role in engineered and natural
flows (Lele, 1994; Smits and Dussauge, 2006; Gatski and Bonnet, 2013; Tennekes
and Lumley, 2018). In the incompressible limit, where the fluid density is
approximately constant, a relatively long history of controlled
experimentation combined with theoretical analysis generates a detailed
picture of incompressible turbulence (e.g. Mydlarski and Warhaft, 1996;
Sreenivasan and Antonia, 1997; Pope, 2001; Davidson et al., 2012; Sinhuber et
al., 2015). Though this picture is far from complete, it is much clearer than
the one we have of compressible turbulence, within which turbulent
fluctuations are so strong that they compress and expand the fluid, and the
fluid density is not constant.

Fundamental aspects of subsonic yet compressible turbulence are not well
understood. Even at subsonic mean speeds, turbulent fluctuations can be fast
enough relative to the speed of sound that regions of local compression and
expansion appear in the flow, albeit less frequently than at higher speeds
(Lee et al., 1992; Samtaney et al., 2001; Donzis and Jagannathan, 2013; Wang
et al., 2018). The data needed to generate, validate and improve compressible
turbulence models (Georgiadis et al., 2014; Quadros et al., 2016b) are scarce
due to difficulties in generating fast flows that are also observable. In
laboratory experiments the turbulent scales of motion are too small or too
fast for most instrumentation. In contrast, in nature the conditions cannot be
controlled.

For incompressible turbulence, the main parameter in the problem is the
Reynolds number ($R_{\lambda}$). For compressible turbulence, we have in
addition to this parameter the Mach number ($M_{t}$) and others whose roles
have yet to be established definitively (Federrath, 2013; Danish et al., 2016;
Jagannathan and Donzis, 2016; Wang et al., 2018; Donzis and John, 2019). Due
to the increased mathematical complexity and the larger parameter space that
the turbulence explores, there is less empirical or theoretical information
available about compressible turbulence than about incompressible turbulence.
For example, the energy dissipation rate, the small-scale structure of the
flow, the shape of the energy spectrum, the translation between compressible
forcing in simulations and realistic forcing in experiments, and the
dependence of these features on the parameters in the problem are unknown.

The wide range of applications in which compressible turbulence appears
suggests a broad utility for an understanding of its universal aspects. In
hypersonic boundary layers, for instance, not only the mean flow but also the
fluctuations are compressible (Owen et al., 1975; Spina et al., 1994; Williams
et al., 2018). In scramjets and radial detonation engines combustion occurs at
supersonic speeds, which translates to short residence times for the fuel and
oxidizer in the combustor (Georgiadis et al., 2014; Urzay, 2018; Ladeinde and
Oh, 2021; Sabelnikov et al., 2021). In commercial and military aviation, the
study of turbulence and its radiated sound have guided jet noise control
strategies (e.g. Bodony and Lele, 2008; Jordan and Colonius, 2013). In
astrophysical flows, star formation is slowed by supersonic turbulence (Low
and Klessen, 2004; Federrath, 2013). The early Universe has been modeled as a
turbulent fluid that experiences turbulent mass fluctuations over physical
scales that extend to tens of megaparsecs (Shandarin and Zeldovich, 1989).
These large-scale structures are associated with supersonic flows that
generate cosmological shocks in the intergalactic medium (Ryu et al., 2008).

One of the few universal findings about compressible turbulent flows is that
free shear layers grow more slowly at high Mach numbers than at low Mach
numbers (Sarkar and Lakshmanan, 1991; Gatski and Bonnet, 2013). We briefly
review these findings with a focus on turbulent jet experiments, since they
most closely resemble the experiments we performed. Schadow et al. (1990)
reports Schlieren photography and total-pressure measurements showing
decreased spreading rates in coaxial jets at transonic Mach numbers
($0.25<M_{c}<2.25$, where $M_{c}=(U_{1}-U_{2})/(c_{1}-c_{2})$ is the
convective Mach number and the subscripts $1$ and $2$ denote the faster and
slower streams), and a strong dependence on the practical definitions of the
jet width and spreading rate. Samimy et al. (1993) finds that vortex
generators at the jet orifice ($0.3<M_{j}<1.81$, where $M_{j}=U_{j}/c$ is the
jet Mach number and $U_{j}$ is the jet speed at the orifice) increase jet
spreading rates in a way that depends on geometry but not on Mach number.
Zaman (1998, 1999) also finds different behaviors for different nozzle
geometries as well as decreased spreading and centerline velocity decay rates
for round jets ($0.29<M_{j}<1.97$) above Mach one. Fleury et al. (2008) uses
Particle Image Velocimetry (PIV) to find self-similar space-time correlations
both off-axis and on-axis in subsonic isothermal round jets ($0.6<M_{j}<0.9$),
as well as smaller integral scales at higher Mach numbers. Feng and McGuirk
(2016) studies subsonic but compressible annular shear layers
($0.2<M_{c}<0.6$) and states that their slower growth at higher Mach numbers
starts at lower $M_{c}$ than for planar mixing layers and is stronger at
comparable $M_{c}$.

Numerical simulations add detail to the experimental observation that
compressible shear layers grow more slowly than incompressible ones. For
instance, Sarkar and Lakshmanan (1991) implement a closure model and find
reduced compressible shear layer growth rates consistent with experiments.
Blaisdell et al. (1993) performs simulations of decaying isotropic,
homogeneously sheared turbulence, and also finds a reduced growth rate of
turbulence attributed to an increase in the dissipation rate accompanied by an
energy transfer to internal energy by the pressure-dilatation correlation.
Freund et al. (2000) reports direct numerical simulations (DNS) of annular
mixing layers ($0.1<M_{c}<1.8$, $0.2<M_{j}<3.5$, $M_{t}$ up to 0.8), which
resemble the early development of a round jet, and explains suppressed growth
rates by the budget of the streamwise component of Reynolds stresses – whereas
shear stresses and radial and azimuthal normal stresses are suppressed at high
Mach numbers, the axial normal stresses stay the same, causing a shear stress
anisotropy which decreases with Mach number. This latter study also identifies
a decrease in transverse length scales due to an acoustic timescale
limitation, which eventually causes large flow structures to deform faster
than sound can propagate through them, disabling the formation of coherent
eddies. Sandberg et al. (2012) reports DNS of a compressible jet (0.46 ¡
$M_{j}$ ¡ 0.84) generated by a fully-developed pipe flow exiting into a low
Mach number coflow, and finds that the self-similarity of coflowing jets
breaks down for coflow values larger than about 40$\%$ of the jet speed. Arun
et al. (2019) reports a DNS of high-speed mixing layers ($0.2<M_{c}<1.2$) and
finds that with increasing $M_{c}$, an increase in vortex-dominated regions
accompanies a reduction in shear layer growth rates.

These findings are generally consistent with the observation that mixing
layers spread more slowly at higher Mach numbers, an observation embodied in
the Langley curve (Slessor et al., 2000). Taken together, the results also
suggest an important dependence on the conditions of the flow at any given
Mach number, and that different geometries produce different behaviors. The
dependence on the Mach number needs to be explained and the mechanisms at work
need to be separated from those associated with changes in the Reynolds number
and geometry.

Incompressible turbulence within jets exhibits an approximate -5/3rds power-
law in energy spectra even in the near-field and at distances from the orifice
smaller than those over which the jet is self-similar (Fellouah et al., 2009).
This rapid development of the characteristic turbulence spectrum has also been
observed in wakes and behind grids (Braza et al., 2006; Valente and
Vassilicos, 2012). The spectrum may be steeper for compressible flows (Donzis
and Jagannathan, 2013; Jagannathan and Donzis, 2016), when shocks are dominant
for instance, and even for $M_{t}$ as low as 0.1 (Bertoglio et al., 2001;
Donzis and Jagannathan, 2013; Federrath, 2013; Wang et al., 2017, 2018). Such
a steeper spectrum was reported in Biagioni and D’Agostino (1999) in a
hypersonic wind tunnel flow albeit at an axial distance less than one nozzle
diameter, where a -11/3rds power law was in agreement with the compressible
spectrum predicted by an Eddy Damped Quasi-Normal Markovian (EDQNM) model but
inconsistent with a cascading picture characteristic of fully developed
turbulence.

This paper reports the first local quantitative measurements in compressible
turbulence in the Variable Density and Speed of Sound Vessel (VDSSV), along
with details of its design principles and capabilities. We generate
compressible flow states from which we extract turbulence quantities of
interest such as correlations on large scales, turbulent Mach number profiles,
Kolmogorov constants, spectra, and their evolution with Mach number. We then
compare our results with previous work.

##  2 Apparatus and Methods

To raise the Mach number at fixed Reynolds number and for fixed boundary
conditions, we compared experiments performed in air with ones performed in
SF6. The chief advantage of SF6 is that its speed of sound is approximately
2.5 times lower than the one for air. In contrast to air-breathing facilities
running at similar Reynolds and Mach numbers, SF6 experiments operate at lower
speeds that are easier to resolve temporally, produce larger scales that are
easier to resolve spatially, generate lower forces that ease mechanical
design, consume less power and so facilitate electrical design, and can be run
continuously in contrast to blow-down or burst-disk experiments so that
statistics better converge. Finally, measurements from high (SF6) and low
(air) Mach number experiments can be made with the same probes operating at
the same frequencies.

An alternate strategy of raising the Mach number of a gas flow at fixed
Reynolds number and geometry is to increase the flow speed while
simultaneously decreasing the pressure at constant temperature. This is so
since the Reynolds number is proportional to changes in the gas density, while
the speed of sound and dynamic viscosity are insensitive to it. This strategy
was employed in the 1950’s in the Variable Density High Speed Cascade Wind
Tunnel at the Deutsche Forschungsansfalt, in which the gas pressure was varied
between 0.1 and 1 bar and the mean flow Mach number between 0.2 and 1.1 in
order to characterize the performance of compressor blade cascades
(Schlichting, 1956). This strategy, however, comes at the cost of increasing
the frequency of the fluctuations that need to be detected (in proportion to
the Mach number), so that faster instruments are needed at higher Mach
numbers. To compare high and low Mach number measurements therefore, the
frequency response of the instruments needs to be understood (Hutchins et al.,
2015). Even with the improvements in measurement technology since Schlichting
(1956), we need to bring down the speed of the flow while holding constant its
Mach number.

###  2.1 Pressure vessel

We conducted experiments in a 16 bar pressure vessel (Fig. 1 and Fig. 2)
consisting of two 0.6 m long cylindrical sections with 1 m diameter and two
rounded end caps made of hot-rolled steel. Each half of the vessel is mounted
onto a cart that rolls on rails in order to pull the halves apart and to open
the full cross section of the vessel to the experimenter. The length of the
pressure vessel, now 1.7 m, can be extended to 3 m with the addition of
annular sections between the two halves. Acrylic windows bolted to 30 cm
diameter portholes welded onto the shell provide optical access up to gas
pressures of about 3 bar (Gloutak, 2018). These acrylic windows can be
exchanged with smaller sapphire windows (not shown) to extend this access up
to 16 bar.

To run experiments with SF6, the VDSSV was first evacuated with a vacuum pump
down to an absolute pressure of 0.01 bar after ensuring that no leaks were
present per Rottländer et al. (2016), keeping the leak rate below
1.6$\times$10-1 mbar L/s. The facility was then charged to a set pressure from
an external tank using a system of pumps and compressors (Enervac’s GRU-4 SF6
Recovery Unit). After each experiment, the gas was stored and recycled. At a
pressure of 0.165 bar, the density of SF6 is such that the kinematic
viscosity, $\nu=\mu/\rho$, is the same as the one for air, and consequently
the Reynolds numbers are matched.

Table 1: Gas properties. $P$ is the static pressure, $\mu$ is the dynamic viscosity, $\gamma$ is the specific heat ratio, $c$ is the speed of sound, $T$ is the mean gas temperature, $\rho$ is the mean gas density.  Gas |  $P$ [bar] |  $\mu|_{1bar}$ [Pa $\cdot$ s] |  $c$ [m/s] |  $\rho$ [kg/m3] |  $T$ [K]
---|---|---|---|---|---
Air | 1.0 | 1.845 $\times 10^{-5}$ | 345 | 1.17 | 300
SF6 | 0.165 | 1.549 $\times 10^{-5}$ | 136 | 0.97 | 300

###  2.2 Turbulence generation

A 12-blade 90 mm diameter ducted fan produced a turbulent jet which expanded
down the centerline of the pressure vessel (see Fig. 2). The fan drew up to
2.4 kW of electrical power, part of which was converted into mechanical energy
in the flow, and all of which was ultimately converted into heat. Fan exit
flow velocities, measured by a Pitot tube, ranged from 10 to 90 m/s,
corresponding to jet Mach numbers up to approximately 0.7. This is well into
the compressible regime, which is conventionally taken to prevail when
$M_{j}>0.3$. From the mean rotation rate of a streamer in the jet, we
estimated that the swirl number, $S=U_{\theta}/U_{j}$, which compares the the
azimuthal to axial components of momentum, was smaller than 0.08 over the full
range of fan speeds. Swirl tends to increase jet spreading rates when it is
larger than 0.6 (Farokhi et al., 1989; Gilchrist and Naughton, 2005;
Vanierschot and Van den Bulck, 2008), which is the opposite of the expected
effect of compressibility. When the turbulent jet impinges on the pressure
vessel wall opposite to that of the fan, it generates a return flow near the
outer walls of the pressure vessel (Fig. 2) that is incompressible at every
jet Mach number, since the cross section available to the return flow is on
the order of 100 times larger than the cross section of the jet.

We classified our observations in terms of the Taylor microscale Reynolds
number, $R_{\lambda}={u^{\prime}\lambda}/{\nu}$ and the turbulent Mach number,
$M_{t}=\sqrt{3}\,u^{\prime}/c$ (Samtaney et al., 2001). The factor of
$3^{1/2}$ has its origin in the isotropic assumption that $u^{\prime}\approx
v^{\prime}\approx w^{\prime}$, which we retain though jet turbulence is
anisotropic, particularly at its margins. As shown below, the jet comprises
velocity fluctuations up to approximately $u^{\prime}=15\,m/s$, corresponding
to $R_{\lambda}$ up to 2000 and $M_{t}$ up to 0.2.

###  2.3 Diagnostic apparatus

We used constant temperature anemometry (CTA) with hot-wire probes that
resolved inertial-range statistics. The VDSSV produces a subsonic,
compressible jet whose Kolmogorov scales can be as small as 10 microns. The
probes were made of Wollaston wire, and measured 0.6 microns in diameter and
about 120 microns in length, which corresponds to a spatial resolution of
between 1 and 12 Kolmogorov scales. Hot wires measure a cooling rate due to
heat transfer rate from the wire to the flow, which in this paper we interpret
as a cooling velocity, $u$. In the compressible flow regime, hot wires respond
to both mass flux and temperature fluctuations. The balance between these two
types of fluctuations depends on the Mach number (Quadros et al., 2016a) and
on the wire temperature. In the present experiment, the probes were operated
at an overheat ratio of 1.4, so that the probes responded predominantly to
mass flux fluctuations. Assuming that the Strong Reynolds Analogy holds
(Gatski and Bonnet, 2013), density fluctuations were approximately 0.04 $\%$
of the mean at an axial position of 9 fan diameters, where turbulence
measurements are reported. Hence, density fluctuations were considered small
enough compared to the velocity fluctuations, which at the jet center achieved
values of 25 $\%$ of the mean.

Resolving diminutive scales is challenging not only due to the limited spatial
resolution of modern probes, but also due to the proximity to the limit
between the continuum regime and the slip-flow regime. Measurements in this
regime were performed, for instance, by Kokmanian et al. (2019), where a
modified version of the state-of-the-art nanoscale thermal anemometry probes
(NSTAPs) were used to measure turbulence at supersonic conditions, and found a
linear relationship between Reynolds number and Nusselt number due to the
closeness of the probe diameter to the mean free path of molecules (Knudsen
number, $Kn~{}O(1)$). In our experiment, the mean free path in SF6 is
approximately 200 nm, corresponding to a wire-diameter based Knudsen number of
0.33, and a micro-structure Knudsen number, based on the Kolmogorov scale, of
0.017. The former Knudsen indicates that our probes are not strictly operating
on a continuum flow regime. The latter Knudsen number suggests that, at low
speeds of sound, the smallest flow scales produced in the experiment are
marginally within the continuum limit. In fact, the micro-structure Knudsen
number increases linearly with $M_{t}$ and with the inverse square root of
$R_{\lambda}$ (Gatski and Bonnet, 2013). Hence, in the limit of very low
Reynolds numbers and very high Mach numbers, turbulent eddies may interfere
with molecular motion (Tennekes and Lumley, 2018). Nevertheless, we discard
this possibility in our experiment, as it produces Reynolds numbers high
enough to enable experimentation for a broad range of Mach numbers where the
motion of gas molecules is in statistical equilibrium and molecular transport
effects can be represented by transport coefficients such as the viscosity.

Hot-wire voltages were acquired at streamwise positions between ${x}/{D_{f}}=$
1.9 and 9.1, which we call the near- and far-fields, and sampled with a
digital acquisition card (16-bit NI-USB 6221) at a frequency, $f_{s}$, of 200
kHz for approximately 10 s (corresponding to about $1.7\times 10^{4}$ to
$1.7\times 10^{5}$ integral time scales), and low-pass filtered at the Nyquist
frequency, $f_{Nyq}$, of 100 kHz to prevent aliasing.

The probes were calibrated in situ against a Pitot tube while varying the fan
speed. We employed King’s Law to relate mean voltages and speeds (King, 1914),
along with a temperature correction (Bruun, 1996). Along the axis of a jet,
the velocity distribution is approximately Gaussian (Anselmet et al., 1984).
Chanal et al. (2000) exploit this fact to determine a calibration from the
differences between measured hot-wire voltage distribution and a Gaussian
distribution modified to account for probe behavior at near-zero speeds, such
that

$P(u)\propto u^{2}\exp\left(\frac{\left(u-U\right)^{2}}{2\sigma^{2}}\right)$
(1)

where $u(x,t)$ is the flow speed, $U$ = $\langle u\rangle$ is its mean, and
$\sigma^{2}$ = $\langle u^{2}\rangle$ is its variance. We used this method to
extend the calibration given by King’s law to velocities beyond those for
which we made Pitot tube measurements. The spatial structure of turbulence was
investigated by invoking Taylor’s Hypothesis of frozen turbulence (Taylor,
1937; Tong and Goldburg, 1988; Pinton and Labbé, 1994; Tennekes and Lumley,
2018). We estimated the turbulent kinetic energy dissipation rate, $\epsilon$,
from the inertial-range scaling of second-order velocity structure functions
(Pearson et al., 2002). Large scale turbulence statistics were computed as
follows. The amplitude of velocity fluctuations was $u^{\prime}=\langle
u^{2}\rangle^{1/2}$. The integral scale was taken as the area under the
velocity autocorrelation function, $L=\int_{0}^{\infty}\\!f(r)\,dr$ with
$f(r)=\langle u(x+r)u(x)\rangle/\langle u^{2}\rangle$, where the tails for
correlations below 0.1 were calculated from integrals of exponential fits to
the autocorrelation functions between values of $1/e$ and 0.1 (Bewley et al.,
2012). The Taylor microscale was computed as
$\lambda=u^{\prime}\sqrt{{15\nu}/{\epsilon}}$, and the Kolmogorov length scale
as $\eta=\left({\nu^{3}}/{\epsilon}\right)^{1/4}$.

Figure 1:  Top: The facility includes the VDSSV, a compressed SF6 storage
tank, and the SF6 gas-handling system. Windows provide optical access through
five 30 cm portholes. The VDSSV is mounted on carts and rails, so that its two
halves can be separated to access its interior. Bottom: A view upstream along
the axis of the VDSSV, once it has been separated along its midline, shows the
fan that produces the turbulence characterized in this paper. The fan diameter
is $D_{f}$ = 90 mm. The red cages house loudspeakers for acoustic experiments
not discussed here.  Figure 2:  A sketch (not to scale) of the interior of the
VDSSV shows the configuration of the fan and probes. The distance between the
fan and the wall upstream of it is 25 cm. The distance along the axis of the
fan from its outlet to the probe ($x$), can be varied between 0.1 and 1.3 m (1
and 15 fan diameters). Orthogonal to the $x$-direction, $r$ is the radial
distance outward from the centerline.

##  3 Results

We survey mean flow characteristics as well as turbulence statistics in the
turbulent jet and compare the data with those available in the literature. A
description of the parameter space covered by the present experiment is
followed by a characterization of the near ($\approx 2D$) and far ($\approx
9D$) fields of the turbulent jet, and their evolution with increasing Mach
number, all of which anchor our analysis of second-order turbulence statistics
such as autocorrelation functions and spectra.

###  3.1 Parameter Space

Figure 3:  The $R_{\lambda}-M_{t}$ parameter space achieved in various
experiments (closed symbols) and simulations (open symbols). In the VDSSV
$M_{t}$ and $R_{\lambda}$ can be adjusted independently by changing between
working gases that have different sound speeds (closed blue and red circles
for air and SF6, respectively), and by adjusting mean jet velocities (moving
up and down along the blue and red curves). Different shapes correspond to
different flow type as follows: jets (circles), boundary layers (hexagrams),
mixing layers (triangles), homogeneous shear (diamonds), and homogeneous
isotropic flows (squares). The sources of the experimental data are as
follows. Circles: light blue (Wygnanski and Fiedler, 1969), orange (Hussein et
al., 1994), magenta (Biagioni and D’Agostino, 1999), yellow (Narayanan et al.,
2002). Hexagrams: green (Spina and Smits, 1987). Triangles: magenta (Barre et
al., 1994), dark gray (Bowersox and Schetz, 1994). Squares: black (Mydlarski
and Warhaft, 1996), light blue (Zwart et al., 1997), magenta (Honkan and
Andreopoulos, 1992), light green (Briassulis et al., 2001), gray (Bodenschatz
et al., 2014). An abundance of approximately incompressible turbulence
experiments not shown lie below $M_{t}$ = 0.01 (e.g. Sinhuber et al., 2015).
The DNS are as follows. Squares (isotropic): green (Samtaney et al., 2001),
red (Wang et al., 2011), gray (Wang et al., 2012), dark blue (Donzis and
Jagannathan, 2013), orange (Donzis and John, 2019). Diamonds (homogenous
shear): gray (Blaisdell et al., 1993), light blue (Chen et al., 2018), black
(Wang et al., 2018).

The parameter space of the Reynolds number ($R_{\lambda}$) and turbulent Mach
number ($M_{t}$) accessed in a representative collection of experiments and
numerical simulations drawn from the literature is shown in Fig. 3. Data from
simulations, most of which are DNS, span a range of turbulent Mach numbers up
to approximately 1.0 and Taylor Reynolds numbers rarely exceeding 100. At
higher Reynolds numbers, the simulation of compressible turbulence becomes
prohibitive in terms of computational cost and the small-scale flow physics
need to be modeled. Additionally, low Mach number simulations are difficult
due to the separation of the acoustic and turbulence time scales.

Previous experiments, in contrast to simulations, easily reach Reynolds
numbers $R_{\lambda}>100$, a regime in which turbulence is well developed in
the sense that an inertial subrange emerges where turbulence is only weakly
affected by boundaries and friction (Pope, 2001). Attaining high Mach numbers
in experiments poses challenges both from the power requirement and
measurement resolution perspectives. Turbulent Mach numbers in excess of 0.2
are reached in some boundary layer experiments (e.g. Spina and Smits, 1987),
where the mean flows are typically too fast to enable high resolution
measurements.

Our experiments in the VDSSV (gray trapezoid) bridge a sparsely populated
space between previous experiments that tend to occupy the low Mach number,
high Reynolds number region, whereas numerical simulations reside in the high
$M_{t}$, low $R_{\lambda}$ region. To the left and right, the space is bounded
by the lowest (0.05 bar) and highest (15 bar) gas pressures in the VDSSV. To
the top right, the space is bounded by the power of the fan, where $M_{t}\sim
R_{\lambda}^{-1}$ since the maximum power of the fan is $P_{max}\sim
R_{\lambda}^{2}M_{t}^{2}$, which assumes $P_{max}\sim\rho U_{j}^{3}$ and
$u\sim U_{j}$. Two representative trajectories followed by our experiments for
1 bar air and 0.165 bar SF6 (blue and red filled circles, respectively, in
Fig. 3), show that the Mach number grows approximately as the square of the
Reynolds number (black line: $M_{t}\sim R_{\lambda}^{2}$) while increasing the
fan speed and holding all else constant. This quadratic scaling holds if the
flow velocity changes while its length scales do not, since $M_{t}\sim
u^{\prime}$ and $R_{\lambda}\sim u^{\prime 1/2}$.

###  3.2 Jet profile characteristics

The jet at its center is shear free and exhibits velocity fluctuations up to
15 m/s. Velocity measurements near the fan were performed to assess the flow
structure of the jet in its initial development. Turbulence measurements were
performed in the developed region of the jet and past a wake region that
extends to approximately $x/D_{f}$ = 5, where shear layers emanating from the
fan become well-mixed with the wake and form a jet with an approximately self-
similar profile.

####  3.2.1 Near-field similarity

Figure 4:  Radial mean velocity profiles close to the outlet of the fan
($x/D_{f}=1.9$) normalized by the centerline mean velocity ($U_{c}$) in air
(blue) and SF6 (red) are approximately invariant with respect to changes in
Mach number. The diameter of the fan is $2R_{f}=D_{f}$ = 90 mm. At large $r$
the differences between the measured profiles and zero are not significant
since the uncertainty is large at small velocities. Data from Narayanan et al.
(2002) at $M_{j}=0.6$ (triangles, open and closed symbols at $x/D_{f}$ = 1 and
4, respectively) are shown for reference, for which $R_{f}$ is the half-width
of the jet ($R_{1/2}$) and not the radius of the jet orifice. Inset:
Centerline mean velocity, $U_{c}/U_{j}$, as a function of jet Reynolds number,
$Re_{j}$ = $U_{j}D_{f}/\nu$.

At $x/D_{f}$ = 1.9, near the fan outlet, the mean velocity profiles were
approximately invariant with respect to changes in the Mach number (Fig. 4).
The profiles were measured at six representative pairs of jet Mach numbers
ranging from $M_{j}$ = 0.04 to 0.72, with each pair at a fixed Reynolds
number. There is a velocity deficit in the central part of the jet out to
0.5$R_{f}$ from the centerline, which corresponds to a wake whose width is
approximately the same as the fan motor diameter. Normalization of radial
positions by the fan radius collapses all of the data out to at least
$r/R_{f}\approx$ 1, consistent with the width being determined by the size of
the orifice in the near field. The inset shows the jet centerline velocity,
normalized by the jet exit velocity, as a function of jet Reynolds number,
which approaches a constant value of approximately 0.8, which is lower
compared to an compressible jet in Narayanan et al. (2002) due to the
influence of the wake.

Data from Narayanan et al. (2002) at $M_{j}$ = 0.6 and $x/D$ = 1 and 4 are
shown for comparison with our data, which displays a top-hat profile
characteristic of near-orifice jet profiles at one diameter from the orifice.
These near-field profiles compare favorably with our experiment in the sense
that the top-hat profile quickly decays toward a rounded shape, and at four
diameters from the orifice the profile from Narayanan et al. (2002) is not
unlike the one we observe at two diameters.

####  3.2.2 Far-field similarity

Figure 5:  The profiles collapse when scaled by their half-widths, $R_{1/2}$.
Data from Narayanan et al. (2002) at $M_{j}$ = 0.6 (black triangles) are shown
for reference, with open and closed symbols corresponding to streamwise
positions $x/D_{f}$ = 6 and 10 respectively. The error bars represent
uncertainty in the calibration of the hot wire, which is largest at low speeds
and far from the jet axis. Different markers correspond to Mach numbers as in
the legend for Fig. 4. Inset: Jet centerline speed, $U_{c}/U_{j}$, variation
with jet Mach number with error bars representing uncertainty in calibration
(many of them smaller than the symbols).

At 9.1 diameters from the fan (Fig. 5) the data collapse with each other when
radial positions are normalized by the half-width radii of the jets
($R_{1/2}$). The profiles align with those from Narayanan et al. (2002) at
similar Mach numbers and distances from the orifice ($M_{j}=$ 0.6 and $x/D=$ 6
and 10), and are indistinguishable from the normalized profiles of an
incompressible jet at 98 diameters from the orifice (not shown) in Wygnanski
and Fiedler (1969). These results suggest an approximate self-similarity of
the jet.

####  3.2.3 Jet width development

Figure 6:  Top: Jet widths at $x/D_{f}$ = 9.1 (circles) decrease with
increasing Mach number, while near the orifice (triangles) the width does not
change. The error bars are the differences between two experiments performed
under the same conditions. Bottom: We interpret the difference between down-
and up-stream jet widths as a spreading rate, $\alpha\equiv$ $\Delta R/\Delta
x$, where $\Delta R$ = $(R_{1/2}(x/D_{f}=9.1)-R_{1/2}(x/D_{f}=1.9))$, and
$\Delta x$ = (9.1-1.9)$D_{f}$. This spreading rate decreases with increasing
jet Mach number, $M_{j}$. The data are qualitatively consistent with the
Langley curve (Slessor et al., 2000), embodied in a fit (dashed black line) of
the form $a+(1-a)\exp{(-bM^{2})}$ to the spreading rate of mixing layers
(Dimotakis, 1991), where $M$ is the convective Mach number ($M_{c}$). Here,
$M$ is the jet Mach number ($M_{j}$) and $a$ and $b$ are equal to 1/5 and 3/2,
respectively.

At low Mach numbers, the jet spreads such that its diameter is at least one-
third larger at 9 diameters downstream than near the orifice (Fig. 6, top). As
the Mach number increases, the downstream width of the jet (circles) narrows
and approaches the jet width near the orifice (triangles), and so the width of
the orifice itself. In other words, the jet approaches a “turbulent laser”
that does not diverge with increasing distance from its source at high Mach
numbers. Axial momentum transport may be inhibited in the radial direction at
higher Mach numbers by mechanisms including smaller transverse velocity
fluctuations, which are not measured here (Sarkar et al., 1989; Arun et al.,
2019; Matsuno and Lele, 2021).

We characterize the jet spreading rate by $\alpha\equiv$ $\Delta R/\Delta x$,
which is the difference between up- and down-stream diameters, $\Delta R$ =
$(R_{1/2}(x/D_{f}=9.1)-R_{1/2}(x/D_{f}=1.9))$, divided by the distance between
the up- and down-stream positions, $\Delta x$. We make the spreading rate
relative to the median of the ones for which $M_{j}$ ¡ 0.2, which is
$\alpha_{0}$, and we assumed a constant virtual origin with increasing
Reynolds number. The virtual origin is insensitive to the Reynolds number, but
depends strongly on inflow conditions and strength of disturbances (Pitts,
1991; Boersma et al., 1998; Sinhuber et al., 2015). Normalized jet growth
rates decrease with jet Mach number (Fig. 6, bottom). This trend is
qualitatively consistent with and analogous to the Langley curve (black dashed
line), where a $40\%$ decrease in normalized compressible shear layer growth
rate can be observed at a jet Mach number as low as 0.5. Our data is compared
to previous jet studies (Schadow et al., 1990; Zaman et al., 1997; Zaman,
1999; Gutmark et al., 1991; Murakami and Papamoschou, 2002). The data from
previous jet experiments measured spreading rates in the range of axial
positions of $2<x/D<30$ (Schadow et al. (1990); Gutmark et al. (1991)) or
$2<x/D<20$ (Zaman et al. (1997); Zaman (1999); Murakami and Papamoschou
(2002)), which overlaps with the distances over which spreading rates were
computed in the present experiment. The variability in the data for any given
Mach number is an indication of sensitivity to initial and boundary
conditions, and therefore of non-universality in terms of the Mach number, in
addition to measurement uncertainties and differences in practical definitions
of the spreading rate between experiments.

###  3.3 Turbulence in the far-field

Here we examine the outer scales of the turbulence in the far-field, which we
take to mean the furthest distance from the orifice at which we measured in
the present experiments ($x/D_{f}=9$).

Figure 7:  From the centerline out to about one fan radius ($r/R_{f}$ = 1),
the turbulent Mach number ($M_{t}$) is approximately homogeneous in air (blue)
and SF6 (red) at $x/D_{f}=9.1$. Inset: The turbulence Mach number on the
centerline, $M_{t,c}$, increases linearly with the jet Mach number, $M_{j}$.
Markers correspond to Mach numbers as in the legend for Fig. 4.

####  3.3.1 Homogeneity

In the far-field ($x/D_{f}$ = 9.1), the turbulence in the jet is approximately
homogeneous within one jet diameter, in the sense that the turbulent Mach
number is constant to within about 5$\%$ up to one fan radius for all
conditions (Fig. 7). It is interesting that the turbulent Mach number profiles
do not narrow as quickly with Mach number as the mean profiles, so that the
turbulence intensity, $u^{\prime}/U$, grows to larger values at the fringes of
the jet at higher Mach numbers. The inset shows that the turbulent Mach number
at the center of the jet increases linearly with the jet Mach number,
attaining a maximum value of $M_{t}$ = 0.17 when $M_{j}\approx$ 0.7. On the
center-line, the far-field turbulence intensity was between 20$\%$ and 24$\%$
of the mean velocity. This compares with upstream intensities between 8$\%$
and 14$\%$ at $x/D_{f}$ = 1.9 and on the center-line.

Figure 8:  Top: The velocity correlation functions along the centerline are
approximately self-similar with respect to changes in Reynolds and Mach
numbers, with a small systematic variation shown in the inset. The data were
acquired at $x/D_{f}$ = 9.1 and plotted as a function of separation, $\Delta
x$, normalized by their integral time scales, $L$, by invoking Taylor’s
hypothesis. Inset: The differences between pairs of correlation functions
measured in SF6 and Air at approximately the same Reynolds number, where the
Reynolds number increases from blue to red curves. Bottom: The integral length
scale, $L$, in fan diameters, $D_{f}$, at a downstream position of
$x/D_{f}=9.1$, is approximately independent of the Mach number on the center-
line (closed symbols). Length scales at the margin of the jet (open symbols)
are about 20% larger than those on the center-line. The error bars represent
variation between different definitions of $L$, including $L_{e}$, which is
the distance at which the autocorrelation function is equal to $1/e$.

Along the centerline and in the far-field, the longitudinal velocity
correlation functions, $f$, collapse approximately when rescaled by the
correlation lengths, $L$, defined in Sec. 2.3 (Fig. 8, top). These correlation
lengths, $L$, are themselves approximately independent of the Mach number
(Fig. 8, bottom). Despite the strong fluctuations, we interpret the
correlations as spatial functions of $\Delta x=U\Delta t$, according to
Taylor’s Hypothesis (Lee et al., 1992), where $\Delta t$ is the time interval
between velocity measurements, though this interpretation is not necessary
here.

A small but systematic trend in the correlation functions is magnified by
making the difference between the high (SF6) and low (Air) Mach number
correlation functions at each Reynolds number (Fig. 8, inset). At higher Mach
numbers, slightly faster decorrelations at small separations are compensated
by stronger correlations at large separations. We link the former to acoustic
fluctuations, and the latter to turbulent fluctuations. In the pseudo-sound
regime (Ristorcelli, 1997), the frequencies of compressible disturbances are
the same as the frequencies of vortical disturbances, so that $c/\Lambda\sim
u^{\prime}/L$, where $\Lambda$ is an acoustic length scale. Thus, $\Lambda\sim
L/M_{t}$, which for our experiment suggests that strong compressible
disturbances could introduce decorrelations on scales at least a factor of ten
smaller than $L$ at $M_{t}>0.1$. Although at higher Mach numbers than in our
experiment, the simulations in Lee et al. (1992), which are in agreement with
linear wave propagation theory, similarly predict decorrelation scales of
acoustic fluctuations that are significantly smaller than those of turbulent
fluctuations.

In incompressible turbulence, integral scales depend weakly on the Reynolds
number. The dependence on Mach number, however, is not entirely clear. At
$x/D_{f}$ = 9.1 and the jet center, integral scales normalized by the fan
diameter $D_{f}$ remain relatively constant with jet Mach number at about
$0.45D_{f}$, closely following the fan radius (Fig. 8). This aligns with the
notion that integral scales are geometrically related to the scales of
turbulence energy injection White et al. (2002). Integral scales are slightly
larger at the jet margins ($r=R_{f}$) than at the jet center, achieving values
slightly in excess of the fan radius on average.

####  3.3.2 Energy spectra

Figure 9:  Energy spectra in air (blue) and SF6 (red) collapse at small
wavenumbers, $k$, under normalization by $u^{\prime}$ and $L$. The solid black
line is a -5/3rds power law. At large wavenumbers, the cutoff in the spectra
are determined by filters and not by the dissipation for all but the lowest
Reynolds number data. The corresponding Kolmogorov scales range from 20 $\mu
m$ to 100 $\mu m$ or between 400 and 2000 integral scales ($L$). Inset: The
Kolmogorov constant, $C_{\kappa}$, for these spectra is approximately
independent of the Mach number, $M_{t}$.

We observe that all spectra display a -5/3rds scaling region consistent with
incompressible dynamics up to the highest turbulent Mach number we reached,
$M_{t}$ = 0.17 (Fig. 9). The observed inertial subrange extends over one to
two decades for all flow conditions, indicating well-developed turbulence. At
increasing flow speeds, our probes spatially truncated the smallest scales and
did not capture them at the highest $R_{\lambda}$ and $M_{t}$. The inset shows
that the Kolmogorov constant, $C_{\kappa}$, is independent of $M_{t}$.
$C_{\kappa}$ was estimated assuming isotropy, a -5/3rds scaling between $kL$ =
2 and 20, and a constant value of one for $C_{\epsilon}=\varepsilon
L/u^{\prime 3}$. Since we did not independently measure the dissipation, it is
the trend, and not the particular value of $C_{\kappa}$, to which we draw
attention.

##  4 Summary and conclusion

We detail the main design principles and capabilities of the VDSSV, a new
facility that generates high-speed, subsonic compressible turbulent flows.
Using hot-wire anemometry capable of resolving inertial subrange statistics,
we report the first quantitative measurements in locally isotropic
compressible turbulence. We highlight the VDSSV’s unique ability to raise the
Mach number at a fixed Reynolds number by switching the gas composition, while
also attaining the highest turbulent Mach number, $M_{t}=0.17$, in a free
shear flow and in a controlled laboratory setting, to the best of our
knowledge. We quantify turbulence at high Mach numbers at flow speeds low
enough to enable high resolution measurements. Compressible flows in the
pressure vessel were produced by means of a jet whose outlet is turbulent in
SF6. The jet’s spatial development showed that downstream mean profiles were
approximately self-similar, while also spreading at slower rates at high Mach
numbers. The pressure vessel can easily be extended to access larger $x/D_{f}$
with the addition of cylindrical sections to its middle. The streamwise
turbulence in the jet was homogeneous in the radial direction within 5$\%$ up
to one jet radius. Autocorrelation differences at fixed Reynolds and variable
Mach number revealed that initially faster decorrelation in the high Mach
number cases are compensated by stronger correlations at large distances,
possibly explained by compressible motions. Spectra and their Kolmogorov
constants were unchanged by increases in the Mach number. Future experiments
are projected to reach turbulent Mach numbers as high as 0.4 with the use of
active grids (Griffin et al., 2019) and will additionally use optical
instruments to track particles (Kearney and Bewley, 2020) and to measure
density fluctuations.

###### Acknowledgements.

The authors are grateful to N. Dam and W. van de Water for the pressure vessel
and for assistance from D. Donzis, L. Mydlarski and M. Ulinski. The authors
also thank J. John, E. Liu, and H. Rivera for helpful discussions, a team of
Master’s students including E. Bair, A. Berberian, D. Cohen, D. Feng, T.
Green, B. Oster, and K. Rajsky for designing, manufacturing and testing
elements of the facility, and a team of undergraduates including Y. Atiq, S.
Bell, W. Chan, M. Chen, S. DePue, C. Kartawira, A. Ramos Figueroa, K. Roberts,
and C. Vahn for their help with data collection and experimental setup.

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