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On the potential and challenges of laser-induced thermal acoustics for experimental investigation of macroscopic fluid phenomena

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Abstract

Mixing and evaporation processes play an important role in fluid injection and disintegration. Laser-induced thermal acoustics (LITA) also known as laser-induced grating spectroscopy (LIGS) is a promising four-wave mixing technique capable to acquire speed of sound and transport properties of fluids. Since the signal intensity scales with pressure, LITA is effective in high-pressure environments. By analysing the frequency of LITA signals using a direct Fourier analysis, speed of sound data can be directly determined using only geometrical parameters of the optical arrangement no equation of state or additional modelling is needed at this point. Furthermore, transport properties, like acoustic damping rate and thermal diffusivity, are acquired using an analytical expression for LITA signals with finite beam sizes. By combining both evaluations in one LITA signal, we can estimate mixing parameters, such as the mixture temperature and composition, using suitable models for speed of sound and the acquired transport properties. Finally, direct measurements of the acoustic damping rate can provide important insights on the physics of supercritical fluid behaviour. Graphic Abstract
Vol.:(0123456789)
1 3
Experiments in Fluids (2021) 62:2
https://doi.org/10.1007/s00348-020-03088-1
RESEARCH ARTICLE
On thepotential andchallenges oflaser‑induced thermal acoustics
forexperimental investigation ofmacroscopic fluid phenomena
ChristophSteinhausen1 · ValerieGerber1· AndreasPreusche2· BernhardWeigand1· AndreasDreizler2·
GraziaLamanna1
Received: 5 June 2020 / Revised: 19 October 2020 / Accepted: 22 October 2020
© The Author(s) 2020
Abstract
Mixing and evaporation processes play an important role in fluid injection and disintegration. Laser-induced thermal acous-
tics (LITA) also known as laser-induced grating spectroscopy (LIGS) is a promising four-wave mixing technique capable to
acquire speed of sound and transport properties of fluids. Since the signal intensity scales with pressure, LITA is effective in
high-pressure environments. By analysing the frequency of LITA signals using a direct Fourier analysis, speed of sound data
can be directly determined using only geometrical parameters of the optical arrangement no equation of state or additional
modelling is needed at this point. Furthermore, transport properties, like acoustic damping rate and thermal diffusivity, are
acquired using an analytical expression for LITA signals with finite beam sizes. By combining both evaluations in one LITA
signal, we can estimate mixing parameters, such as the mixture temperature and composition, using suitable models for
speed of sound and the acquired transport properties. Finally, direct measurements of the acoustic damping rate can provide
important insights on the physics of supercritical fluid behaviour.
Graphic Abstract
Extended author information available on the last page of the article
List of symbols
Latin characters
AP1,P2
Complex amplitudes of the acoustic waves 
()
AT
Complex amplitudes of the thermal grating 
()
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Experiments in Fluids (2021) 62:2
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2 Page 2 of 16
B Amplitude of harmonic oscillation 
()
C Amplitude of harmonic oscillation 
()
DT
Thermal diffusivity 
(
m2s
1
)
Eexc
Pulse energy of the excitation
beams 
(
kg m2s
2
)
I Signal intensity 
()
Complex parameter to compute
𝛹(t)
;
P1,2 =AP1,P2𝛴P1,P2

()
Pint
Power of the interrogation beam 
(
kg m2s
3
)
T Complex parameter to compute
𝛹(t)
;
T=AT𝛴T

()
Tch
Fluid temperature in measurement
chamber 
(K)
Tmix
Local mixing temperature 
(K)
U𝛩
Dimensionless modulation depth of thermaliza-
tion grating 
()
UeP
Dimensionless modulation depth of electrostric-
tion grating 
()
cp
Specific isobaric heat capacity 
(
m2s
2K
1)
cs
Speed of sound 
(
ms
1)
f Focal length 
(m)
j Indicator related to fluid behaviour;
j=1
: reso-
nant;
j=2
: non-resonant 
()
p Pressure 
(
kg m
1
s
2)
q Magnitude of the grating vector;
q=2
𝜋
𝛬

(
m
1)
s Specific entropy 
(
m2s
2K
1)
t Time 
(s)
t0
Time of laser pulse
(s)
v Fluid velocity component in
y-direction 
(
ms
1
)
w Fluid velocity component in
z-direction 
(
ms
1
)
xFl
Local mole fraction of a fluid in a mixture 
()
x Cartesian coordinate 
(m)
y Cartesian coordinate 
(m)
z Cartesian coordinate 
(m)
Greek characters
𝛥y
Beam distance in front of lens 
(m)
𝛥zL
Distance between the interrogation and excitation
beams in front of lens 
(m)
𝛤
Acoustic damping rate 
(
m
2
s
1)
𝛤c
Classical acoustic damping rate; bulk viscosities
are neglected 
(
m
2
s
1)
𝛤m
Pressure corrected classical acoustic damping rate
according to the theoretical and empirical consid-
erations of Li etal. (2002) 
(
m2s1
)
𝛬
Grid spacing of the optical interference
pattern 
(m)
𝛷
Crossing angle of interrogation beam 
(rad)
𝛹(t)
Time-dependent dimensionless diffraction effi-
ciency of a LITA signal 
()
𝛴P1,P2
Complex parameter related to the damping of
oscillations 
()
𝛴T
Complex parameter related to the damping of the
signal 
()
𝛩
Crossing angle of excitation beam 
(rad)
𝛼
Calibration constant 
(
s
2
kg
1
m
2)
𝛽
Decay rate 
(
s
1
)
̄𝜂
Misalignment length scales in y-direction 
(m)
𝛾
Specific heat ratio 
()
𝛾n𝛩
Rate of excited-state energy decay not caused by
thermalization 
(
s
1
)
𝛾𝛩
Rate of excited-state energy decay caused by
thermalization 
(
s
1
)
𝜅
Thermal conductivity 
(
kg m s
3
K
1)
𝜆
Wavelength 
(m)
𝜇v
Dynamic bulk viscosity 
(
kg m
1
s
1)
𝜇s
Dynamic shear viscosity 
(
kg m
1s
1
)
𝜈
Dominating frequency of the LITA
signal 
(
s
1
)
𝜔
Gaussian half-width of excitation beams 
(m)
𝜚
Mass density 
(
kg m
3
)
𝜎
Gaussian half-width of interrogation beam 
(m)
𝜏
Laser pulse length 
(s)
𝜐
Angular frequency 
(
s
1)
𝜐0
Natural angular frequency associated with speed
of sound 
(
s
1)
̄
𝜁
Misalignment length scales in z-direction 
(m)
Subscripts
Ar Related to fluid: argon
BP Related to beam profiler measurement
DFT
Related to calculations using a direct Fourier
transformation
LITA
Related to measurement using LITA
N2 Related to fluid: nitrogen
NIST
Related to theoretical calculations using NIST
database by Lemmon etal. (2018)
atm Unit of pressure used: atmospheres
c Related to properties at the critical point
cal Related to calibration
ch Related to condition in measurement chamber
exc Related to excitation beam
int Related to interrogation beam
mix Related to local condition of mixture
r Reduced properties scaled with the properties
related to the critical point
th Related to theoretical calculations using data sheet
specifications
Miscellaneous characters
O(
)
Order of magnitude
(
)
Complex conjugate
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Experiments in Fluids (2021) 62:2
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Abbreviations
DFT Direct Fourier transformation
GLP Glan–Laser polarizer
LAR Least-Absolute Residuals
LIGS Laser-induced (transient) grating spectroscopy
LITA Laser-induced thermal acoustics
MM Multi-mode fibre with diameter of
25 𝜇m
NIST National Institute of Standards and Technology
PBS Polarizing beam splitter
SBS Stimulated Brillouin scattering
SM Single-mode fibre with diameter of
4𝜇m
STBS Stimulated thermal Brillouin scattering
STRS Stimulated (thermal) Rayleigh scattering
WP
𝜆2
-wave plate
1 Introduction
Fluid injection, disintegration, and subsequent evaporation
are of high importance for a stable and efficient combus-
tion. Especially for high pressures exceeding the critical
value of the injected fluids, mixing and evaporation pro-
cesses as well as fundamental changes in fluid behaviour
are not yet fully understood. The latter have received
increased attention in the past decade, as the recently pub-
lished literature shows (Falgout etal. 2016; Müller etal.
2016; Baab etal. 2016, 2018; Crua etal. 2017). Since the
main objectives are evaporation and disintegration pro-
cesses of liquid fluids at pressures and temperatures either
close to or exceeding their critical points, quantitative
data for validation of numerical simulations have recently
become a research concern with increasing interest (Bork
etal. 2017; Lamanna etal. 2018; Steinhausen etal. 2019;
Stierle etal. 2020; Nomura etal. 2020; Lamanna etal.
2020; Qiao etal. 2020). Microscopic investigations by
Santoro and Gorelli (2008), Simeoni etal. (2010) as well
as Bencivenga etal. (2009) made it possible to distinguish
various regions above the critical pressure, as is depicted
in Fig.1. At supercritical pressures, the region between the
critical isotherm and the Widom line, which is character-
ized by the maximum in specific isobaric heat capacity, is
identified as liquid-like. Indeed, it preserves large densities
and sound dispersion (Simeoni etal. 2010; Bencivenga
etal. 2009), while exhibiting the molecular structure of a
gas (Santoro and Gorelli 2008). In contrast, regions with
supercritical temperatures right of the Widom line are
gas–like, as propagation of sound waves at the adiabatic
speed of sound is recovered (Simeoni etal. 2010). In this
context, the area between the critical isotherm and the
Widom line can be denoted as the supercritical region,
because it exhibits dynamical and physical properties
intermediate between gas and liquid states. The relevance
of these microscopic findings on the dynamic behaviour of
supercritical fluids at macroscopic scale remains till today
poorly understood. In Sect.2, it is shown how the acoustic
damping rate enables to disclose the interrelated nature of
sound dispersion at microscopic and macroscopic scales.
At this stage, it is important to point out that the current
macroscopic description of supercritical states is mainly
focused on the selection of accurate equation of states. The
latter are capable to describe the continuous fluid trans-
formation in terms of density changes and the singulari-
ties in terms of some physical properties (heat capacity,
isothermal compressibility, etcetera) across the Widom
line. This approach, however, may not be sufficient for a
correct description of the dynamical behaviour of super-
critical fluids, as currently suggested by the microscopic
investigations.
The previous consideration provides the motivation for
the present work, where emphasis is placed on the measure-
ment of speed of sound (
cs
), thermal (
DT
), and viscous (
𝛤
)
relaxation constants, identified as key parameters to enable
macroscopic investigations of these different fluid regions.
In addition, these macroscopic fluid properties, such as
speed of sound, acoustic damping, and thermal diffusivity,
enable a quantitative comparison of injection studies with
analytical and numerical data, as has been shown for speed
of sound measurements in high-pressure jets by Baab etal.
(2018). The speed of sound data of the mixture was acquired
using homodyne laser-induced thermal acoustics (LITA).
A detailed description of the used experimental setup can
be found in the work of Förster (2016). A comprehensive
Fig. 1 Thermodynamic fluid states based on microscopic investi-
gations; p
r=
p
p
c
: reduced pressure;
sr=
s
s
c
: reduced entropy;
reduced properties are scaled with the properties related to the critical
point; Widom line: line of maxima in specific isobaric heat capacity;
thermodynamic data are taken from Lemmon etal. (2018)
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Experiments in Fluids (2021) 62:2
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2 Page 4 of 16
review on gas-phase diagnostics using laser-induced tran-
sient grating spectroscopy (LIGS) is provided by Stampa-
noni-Panariello etal. (2005b).
LIGS, LITA or similar techniques are mostly used to
determine transport properties in quiescent environments,
where high spatial resolution is not the focus in the inves-
tigation. However, LITA is indeed sensitive to small-scale
processes, as investigation of the speed of sound data in
multi-component jet mixing at high pressures by Baab etal.
(2018) has shown. Kimura etal. (1995) measured transport
properties of high-pressure fluids, namely carbon dioxide
and trifluoromethane, using LIGS. The study focused mainly
on determination of thermal diffusion, mass diffusion, and
sound propagation in the vicinity of the critical point. In the
same research group, thermal diffusion and sound propaga-
tion of binary mixtures of carbon dioxide and a hexafluoro-
phosphate were investigated by Demizu etal. (2008). Latzel
and Dreier (2000) investigated heat conduction, speed of
sound data, as well as viral coefficients of gaseous mixtures
at pressures up to
50 MPa
by analysing the acoustic oscil-
lations and the long-term decay of a near-infrared LIGS
signal. Vibrational energy relaxation of azulene was stud-
ied in super-critical states by Kimura etal. (2005a) and for
liquid solvents by Kimura etal. (2005b). An investigation
of acoustic damping rates in pure gases was presented by Li
etal. (2002) analysing the temporal behaviour of transient
grating spectroscopy. The investigations included different
gases at pressures up to
25 atm
at room temperature. Li etal.
(2002) compared these findings with classical acoustic the-
ory and derived a linear pressure dependency for the meas-
ured acoustic damping rate. Li etal. (2005) later proposed
a binary mixture model to determine the acoustic damping
rate for binary atomic species. Note that all the previously
reviewed studies measure transport properties using an opti-
cal arrangement with unfocused beams for grid excitation.
This leads to a measurement volume with an order of mag-
nitude
O(
101mm
)
in diameter and
O(
102mm
)
in length and
hence a poor spatial resolution.
To utilize LITA as a reliable tool for experimental inves-
tigation in jet disintegration or droplet evaporation studies,
a high spatial resolution is imperative. Studies by Baab
etal. (2016), Baab etal. (2018), and Förster etal. (2018)
already showed the capability of acquiring quantitative
speed of sound data in jet disintegration. Especially to be
emphasised are the investigations by Baab etal. (2018),
which demonstrated the potential of acquiring speed of
sound data for multi-component jet mixing at high pres-
sures in the near nozzle region. The purpose of this study is
to present the calibration and validation processes needed
for the extraction of speed of sound data, acoustic damp-
ing rates, as well as thermal diffusivities using LITA with a
spatial resolution with an order of magnitude
O(
10
1
mm
)
in diameter and
O(
100mm
)
in length in a high-pressure and
high-temperature environment for resonant and non-resonant
fluids.
2 Theoretical consideration
ontherelevance oflaser‑induced thermal
acoustic insupercritical mixture studies
The LITA (or LIGS) technique provides an excellent oppor-
tunity to measure independently and simultaneously speed
of sound data and acoustic damping rates. The implications
of these measurements are twofold. First, measuring acoustic
damping rates allows to assess whether sound dispersion in
supercritical fluids is significant and provides the possibil-
ity to indirectly measure bulk viscosities, which are mainly
responsible for sound dispersion. Additionally, bulk viscosi-
ties will enable the improvement of models for the stress
tensor and the kinetic energy dissipation in supercritical
fluid flow simulations. Second, if both speed of sound and
acoustic damping rate are measured, a set of independent
equations can be derived to extract local mixing parameters,
like temperature and composition of a binary mixture.
The possibility to estimate mixing parameters using laser
induced thermal acoustics was shown by Li etal. (2005)
for binary mixtures of monoatomic species. Using tran-
sient grating spectroscopic Li etal. (2005) were able to
derive the mole fraction of a Helium–Argon mixture. As
we will later show in the post-processing section of this
work (Sect.3.2.2), analysing the temporal evolution of a
detected LITA signal enables the determination of three
transport properties, namely the speed of sound
cs
, the
acoustic damping rate
𝛤
, and, in case of a resonant fluid
behaviour, the thermal diffusivity
DT
. Whereas thermal dif-
fusivity and speed of sound are well-known transport prop-
erties and accessible using the NIST database by Lemmon
etal. (2018), acoustic damping rates have to be modelled in
more detail. Dissipation of a sound waves energy is mainly
caused by internal friction and heat conduction. The damp-
ing rate of an acoustic wave is, therefore, dependent on the
viscosity and thermal conductivity (Li etal. 2002). Using
the theoretical description by Hubschmid etal. (1995), the
acoustic damping rate
𝛤
depends on both shear viscosity
𝜇s
and bulk viscosity
𝜇v
, and can be modelled as:
where
𝜚
is the fluid density,
𝛾
the specific heat ratio,
𝜅
the
thermal conductivity, and
cp
the specific isobaric heat capac-
ity. At atmospheric conditions, bulk viscosities are neglecta-
ble compared to shear viscosities; using this assumptions,
we can calculate the classical acoustic damping rate
𝛤c
as
used by Li etal. (2002):
(1)
𝛤
=1
2𝜚
[
4
3𝜇s+𝜇v+(𝛾1)𝜅
c
p],
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Experiments in Fluids (2021) 62:2
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Page 5 of 16 2
𝛤c
predicts accurate damping rates for monoatomic sub-
stances at low pressures. For pressures up to
2.5 MPa
at room temperature, Li etal. (2002) estimated the pres-
sure dependence of measured acoustic damping rate with
respect to the classical solution. A linear dependence with
a fluid–dependent slope was found. For nitrogen and argon,
the measured acoustic damping rate is expressed in (3).
Note that the unit of pressure used by Li etal. (2002) is
atmospheres:
As mentioned in the Introduction, for mixing processes,
e.g., high-pressure turbulent jets, transport properties can
be used to derive the local mixing state of macroscopic fluid
phenomena. These mixing states are defined by the local
temperature of the mixture
Tmix
, the local mole fraction of
the fluid
xFl,mix
, and the overall pressure in the measurement
chamber
pch
. The local transport properties speed of sound
cs,mix
, acoustic damping rate
𝛤mix
, and thermal diffusivity
DT,mix
of the mixture can be expressed in the following way:
Using a controlled environment where the pressure in the
measurement chamber
pch
is known, the local mixing tem-
peratures
Tmix
and local mole fraction of the fluid
xFl,mix
are
the only unknown fluid properties of the investigated mix-
ture. After careful validation in well-known binary gas–fluid
mixtures, this should enable us to determine the desired
mixing parameters when coupling them to non-ideal mix-
ture data, e.g., Lemmon etal. (2018). By measuring LITA
signals using an optical arrangement with focused beams, it
would, therefore, be possible to determine the local mixing
temperature and mole fraction simultaneously. This would
enable us to study mixing and evaporation processes in high-
pressure jet mixing or droplet evaporation in the vicinity of
the critical point.
To evaluate whether the phenomenon of sound dispersion
in supercritical fluids is significant, we first have to understand
the behaviour of the relaxation of an electrostriction grating.
The latter can be assimilated to a damped harmonic oscillator,
which admits a general solution of the following type:
where I denotes the signal intensity, t denotes the time, B
and C are dimensionless amplitudes,
𝛽1
is the characteristic
(2)
𝛤
c=1
2𝜚
[
4
3𝜇s+(𝛾1)𝜅
c
p].
(3)
𝛤m,Ar =𝛤c
(
130 patm +1
)
𝛤m,N2
=𝛤
c(
16p
atm
+1
)
.
(4)
cs,mix
(
pch,Tmix ,xFl,mix
)
=cs,LITA
𝛤mix
(
pch,Tmix ,xFl,mix
)
=𝛤LITA
DT,mix(
p
ch
,T
mix
,x
Fl,mix)
=D
T,LITA.
(5)
I(t)=Bexp {𝛽ti𝜐t}+Cexp {𝛽t+i𝜐t},
decay time of the oscillation’s amplitude, and
𝜐
is the angular
frequency. Note that the decay rate
𝛽
is directly proportional
to the damping constant, like the acoustic damping rate for
acoustic waves. With reference to laser-induced gratings,
it was found by Stampanoni-Panariello etal. (2005a) that
𝛽=q2𝛤
. Here, q denotes the magnitude of the grating
vector. The frequency of the counter-propagating acoustic
waves can be, therefore, expressed as (see Hubschmid etal.
(1996)):
where
𝜐0
is the natural frequency, which is associated with
the adiabatic speed of sound
cs
in the following way:
For a system with a small damping constant (
𝜐0≫𝛽
),
it follows that the frequency of oscillation is close to the
undamped natural frequency
𝜐=𝜐0=const.
. The implica-
tions of Eq.6 are twofold. First, it shows that the local sound
speed depends indeed upon the acoustic damping rate. Only
if the latter is negligible, we recover the well-known condi-
tion that sound waves propagate at the adiabatic speed of
sound for low-pressure gases. Second, it follows that the
local speed of sound is a function of the excitation grat-
ing vector. This effect is known as sound dispersion and
is commonly observed in liquids, as has been shown by
Mysik (2015). For supercritical fluids, Simeoni etal. (2010)
demonstrated that a significant sound dispersion could be
observed in the region comprised between the critical iso-
therm and the Widom line. However, the probing length
scale was much smaller (X-ray scattering), thus resulting in
larger q values and, therefore, larger frequency dispersions.
Following the procedure adopted by Mysik (2015) for
liquids, the bulk viscosity can be measured as deviation
between the measured acoustic damping
𝛤
in Eq.1 and the
classical model
𝛤c
in Eq.2. In liquids, high values of the
bulk viscosity are mainly responsible for the observed sound
dispersion. LITA measurements, therefore, will enable to
verify whether this behaviour is valid also for supercritical
fluids.
3 Experimental facility andmeasurement
technique
Investigations for this study are performed in well-controlled
quiescent conditions. Three different atmospheres, namely
nitrogen with a purity of
99.999 %
, argon with a purity of
99.998 %
, and carbon dioxide with a purity of
99.995 %
, are
studied. The optical setup is adapted from the one described
in Baab etal. (2016) and Förster etal. (2015).
(6)
𝜐
=
𝜐2
0
q4𝛤2
,
(7)
𝜐0=qcs.
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Experiments in Fluids (2021) 62:2
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2 Page 6 of 16
3.1 Pressure chamber
Experimental investigations are performed using a heat-
able high-pressure, high-temperature chamber. The latter is
designed for phenomenological as well as statistical inves-
tigations of free-falling droplets in a near-critical environ-
ment. For the presented study, the droplet generator on top
of the chamber is replaced with a closed lid. The operating
condition for nitrogen range between
pch =2
and
8 MPa
for
temperatures up to
Tch =700 K
. For argon and carbon diox-
ide, the operating pressures vary between 0.5 and
8 MPa
for
temperatures up to
600 K
. Before each set of experiments,
the chamber is carefully evacuated to ensure no contamina-
tion from previous investigations. The experimental setup
is operated as a continuous-flow reactor. The mass flow into
the chamber is hereby controlled using a heat-capacity-based
mass flow controller (Bronkhorst) for nitrogen and argon
as well as a Coriolis-based mass flow controller for carbon
dioxide. Note that carbon dioxide is pressurized beforehand
using a pneumatic-driven piston compressor. Pressurized
fluids are supplied from two sides on top of the chamber
through an annular orifice. The pressure inside the chamber
is controlled using a pneumatic valve at the system exhaust
(Badger Meter). Since the derivation of the present equa-
tions relies on the assumption of negligible flow velocities,
it is important to emphasise that the used mass flow does not
exceed
2.5 kg/h
for carbon dioxide and
1.25 kg/h
for argon
and nitrogen, which leads to flow velocities below
0.04 m/s
.
The chamber is constructed of heat-resistant stainless
steel (EN-1.4913). Eight UV-transparent quartz windows at
two different heights are placed at an angle of
90
to each
other ensuring optical accessibility. Eight heating cartridges
are vertically inserted in the chamber body. Additionally, a
heating plate with four cartridges is placed below the cham-
ber. All heaters are controlled using type-K thermocouples
in the chamber body as well as the heater cartridges. The
chamber encloses a cylindrical core with a diameter of
40 mm
and a height of
240 mm
. For thermal insulation, a
mineral-based silicate (SILCA 250KM) is used. The bottom
of the heating plate is insulated using a vermiculite plate.
Vertical and horizontal sectional drawings of the chamber
are depicted in Fig.2. For pressure measurement inside the
chamber, a temperature–compensated pressure transducer
(Keller 35 X HTC) with an uncertainty rated at
±0.1 MPa
is
chosen. The pressure transducer is located at the chamber
exhaust. Temperature measurements inside the chamber take
place at three different heights with miniaturized resistance
thermometers penetrating the metal core. Since the uncer-
tainty of these resistance thermometers is temperature-
dependent, the measurement uncertainties are calculated for
each condition separately. Both temperature and pressure are
logged continuously.
3.2 Laser‑induced thermal acoustics
LITA, also more generally referred to as LIGS, is discussed
in detail in literature. A theoretical approach describing the
generation of the laser-induced grating as well as the inher-
ent phonon–photon and thermon–photon interaction can be
found in Cummings etal. (1995) and Stampanoni-Panariello
etal. (2005a). Note that the analytical expression presented
by Stampanoni-Panariello etal. (2005a) is only valid for
infinite beam sizes, whereas Cummings etal. (1995) takes
finite beam sizes into account. In the limit of infinite beam
sizes, both theories merge. Schlamp etal. (1999) extended
Fig. 2 Horizontal and vertical sections of the high-pressure cham-
ber. Left: vertical section through pressure chamber. Centre (section
B—B): horizontal section through pressure chamber at centre of first
window. Right (section C—C): horizontal section through pressure
chamber at fluid inlet and annular orifice. A: annular orifice; F: fluid
inlet; G: graphite gaskets; H: heating cartridges; I: thermal insulation;
O: Willis O-rings; T: resistance thermometer; V: vermiculite plate;
W: quartz windows
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Experiments in Fluids (2021) 62:2
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Page 7 of 16 2
the theory presented by Cummings etal. (1995) to account
for beam misalignment and flow velocities.
LITA occurs due to the non-linear interaction of mat-
ter with an optical interference pattern. The latter is intro-
duced by two short-pulsed excitation laser beams, which
are crossed using the same direction of linear polarization
to produce a spatially periodic modulated polarization/light
intensity distribution. The resulting changes in the optical
properties of the investigated fluids are interrogated using a
thirdinput wave. The third wave originates from a second
laser source and is scattered by the spatially periodic pertur-
bations within the measurement volume. Depending on the
absorption cross-section of the investigated fluid, changes
in optical properties result from different processes. For
non-resonant substances, pure electrostriction is observed,
whereas in resonant substances, simultaneously, an addi-
tional thermal grating is produced. Eichler etal. (1986) dis-
tinguishes three dominant forms of light scattering impor-
tant for LITA. Light scattering from a non-resonant grating
can be referred to as stimulated Brillouin scattering (SBS),
whereas scattering from a resonant thermal grating depends
on the thermalization time. In case of fast energy exchange,
stimulated thermal Brillouin scattering is observed (STBS).
For slow energy exchange, stationary density modulations
emerge, which are referred to as stimulated (thermal) Ray-
leigh scattering (STRS).
3.2.1 Optical setup
The optical arrangement used for the presented investiga-
tions is depicted in Fig.3. For excitation, a pulsed Nd:YAG
laser (Spectra Physics QuantaRay:
𝜆exc
=1064 nm
,
𝜏pulse =10 ns
,
30 GHz
line width) is used. To ensure stable
and reproducible conditions, the excitation laser is set to
a pulse energy of
150 mJ
, which is continuously measured
by a pyroelectric sensor (D3). The energy of the excitation
pulse is subsequently controlled using a
𝜆2
-wave plate
(WP) together with a Glan–Laser polarizer (GLP) and con-
tinuously observed by a pyroelectric sensor (D4; Thorlabs).
The pulse energy used for investigation is adjusted to val-
ues between 18 and
50 mJ
. The GLP additionally ensures
polarization of the excitation beam, which is split by a beam
splitter (T1) into two excitation beams.
The interrogation laser source is provided using a contin-
uous-wave DPSS laser (Coherent Verdi V8,
𝜆int =532 nm
,
5 MHz
line width). The power of the interrogation laser is
adjusted to ensure a good signal-to-noise ratio and varies
from 0.1 to
8.5 W
. Note that, to ensure stable power output
at low power settings, the beam power is reduced using a
polarizing beam splitter (PBS) together with a
𝜆2
-W P.
A forward folded BOXCARS configuration is used to
arrange all beams and achieve phase matching. An AR-
coated lens (
f=1000 mm
at
532 nm
) is utilized to focus
all beams into the measurement volume. With an excitation
beam distance of
𝛥yexc 36 mm
, the crossing angle yields
𝛩1
. Based on the laser specifications, the Gaussian half-
width of the excitation beams in the focal point is estimated
to be
𝜔th =312 𝜇m
. Due to the Gaussian beam profile and
the beam arrangement the optical measurement volume is
an ellipsoid elongated in x-direction. Using the modelling
proposed by Schlamp etal. (1999), the size of the inference
pattern is estimated to be approximately
8.6 mm
in length
and
312 𝜇m
in diameter. This optical interference pattern
has a Gaussian intensity profile with a grid spacing
𝛬
modu-
lated in y-direction, see Siegman (1977). In this context, it
is crucial to mention that the direction of propagation of the
acoustic waves is normal to the beam direction. Hence, the
extension of the effective measurement volume in x-direc-
tion is smaller than the length of the elliptical interference
pattern. The spatial resolution in x-direction is, therefore,
higher than the optical interference pattern suggests. Evalu-
ating the speed of sound radial profile data provided by Baab
etal. (2018) together with the provided shadowgram, we
Fig. 3 Optical setup of the LITA system. BE: beam expander; BS:
beam sampler; BT: beam trap; C: coupler; D: detector (D1: Ava-
lanche detector; D2: photo diode; D3, D4: pyroelectric sensor; D5:
thermal sensor); F: fibre; F1: neutral density filter wheel with orifice;
GLP: Glan–Laser polarizer; L: lens; M: mirror; PBS: polarized beam
splitter; T: beam splitter; WP: wave plate
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Experiments in Fluids (2021) 62:2
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2 Page 8 of 16
estimate the spatial resolution in beam direction to be less
than the jet diameter at the measurement location. This leads
to a spatial resolution in the present report to of approxi-
mately
312 𝜇m
in diameter and less than
2 mm
in length
in the x-direction. An avalanche detector (D1; Thorlabs
APD110) serves for detection of the scattered signal beam.
The latter is previous spatially and spectrally filtered using
a coupler and single-mode/multi-mode fibres. The detectors
voltage signal is logged with
20 GS/s
by a
1 GHz
bandwidth
digital oscilloscope (LeCroy, Waverunner 610Zi).
3.2.2 Post‑processing
The simplest and most common approach to extract speed
of sound from a LITA signal is a direct Fourier transforma-
tion (DFT). It is imperative to mention that the speed of
sound data is directly obtained from the frequency domain
of the temporal LITA signal, involving only the geometrical
parameters of the optical arrangement. No equation of state
or modelling assumptions are necessary at this point. Using
the theoretical considerations byHemmerling and Kozlov
(1999), the speed of sound
cs
of the probed fluid can be
estimated as follows:
The dominating frequency of the LITA signal is hereby
denoted by
𝜈
. The constant j indicates if the fluid shows reso-
nant behaviour at the wavelength of the excitation beam. In
case of non-resonant fluid behaviour
j=2
, whereas in case
of resonant fluid behaviour,
j=1
. The grid spacing
𝛬
of the
optical interference pattern is a calibration parameter for the
optical setup. Without a mixing model, thermometry can
only be performed at known gas composition and pressure,
using a suitable model for the speed of sound. However, the
temperature is then only indirectly determined by applying a
suitable model for the fluid under consideration. This can be
challenging in supercritical or high-pressure states.
Using an analytical approach for finite beam sizes for
the evaluation of LITA signals proposed by Schlamp etal.
(1999), it should be possible to extract the speed of sound,
the acoustic damping rate, as well as the thermal diffusiv-
ity from the shape of the LITA signal. In the following,
we will summarize the essential parts of the mathemati-
cal derivation necessary for this study, as proposed by
Schlamp etal. (1999) and Cummings etal. (1995). The
used assumptions are categorized and listed in the appen-
dix of this work. As discussed in more detail by Stampa-
noni-Panariello etal. (2005a), the temporal shape of the
excitation laser pulse is estimated using a
𝛿
-function at
t0
.
The model suggested by Schlamp etal. (1999) can be sim-
plified using two key assumptions proposed by Cummings
(8)
c
s=
𝜈𝛬
j
.
etal. (1995), namely the limit of fast thermalization and
negligible damping over a wave period. Correspondingly,
the amplitudes of the acoustic waves
AP1,P2
and the ampli-
tudes of the thermal grating
AT
in the upcoming modelling
equation (13) of the LITA signal simplify to the expres-
sions in equation (9). The real part of
AP1,P2
indicates
the influence of thermalization or STBS on the damping
oscillation of the LITA signal, while the imaginary part
expresses the electrostrictive contribution or SBS. Con-
sequentially,
AT
represent the weight of thermalization on
the signal damping:
The quantities
U𝛩
and
UeP
denote the approximate modula-
tion depth of thermalization and electrostriction gratings,
respectively (Cummings etal. 1995), which are used as fit-
ting constants. Note that we further assume instantaneous
release of absorbed laser radiation into heat, as proposed
by Stampanoni-Panariello etal. (2005a). In case of reso-
nant fluid behaviour, both thermalization and electrostric-
tion gratings must be considered. On the other hand, when
non-resonant fluid behaviour is expected, the grid genera-
tion process is purely electrostrictive. Therefore, the ther-
mal modulation depth
U𝛩
is negligible leading to a further
simplified model. Considering small beam crossing angles
and negligible bulk flow velocities, parameters related to
the damping of oscillations
𝛴P1,P2
and the damping param-
eter
𝛴T
can be expressed as given in Eq. (10) referred in
Schlamp etal. (1999). Note that, due to negligible bulk
flow velocities in the chamber only, beam misalignment in
horizontal y-direction
̄𝜂
has an effect on the time history of
the LITA signal (Schlamp etal. 1999). Hence, after careful
beam alignment through the quartz windows before each
measurement resulting in a maximized signal, all other pos-
sible misalignments are neglected:
DT
denotes the thermal diffusivity,
𝛤
the acoustic damping
rate, t the time,
t0
the time of the laser pulse,
𝜔
, and
𝜎
is
the Gaussian half–width of the excitation and interrogation
beam in the focal point, respectively. The magnitude of the
grating wave vector q depends on the grid spacing
𝛬
, which
is a function of the crossing angle
𝛩
of the excitation beams
and the wavelength of the excitation pulse
𝜆exc
, see Eq.(11)
and(12) referred in Stampanoni-Panariello etal. (2005a):
(9)
A
P1,P2
=12U
𝛩
±i2UeP
A
T
=−1U
𝛩
.
(10)
𝛴
P1,P2=exp
𝛤q2tt0
2
̄𝜂 ±cs
tt0
2
𝜔2+2𝜎2
exp ±iqcstt0
𝛴T=exp
DTq2
tt0
2̄𝜂 2
𝜔2+2𝜎2
.
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Experiments in Fluids (2021) 62:2
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Using the simplifications explained above and summarized
in the appendix, the time-dependent diffraction efficiency
𝛹(t)
of a detected LITA signal can be expressed as:
where the parameters
P1
, T, etc. are calculated using
P1=AP1𝛴P1
,
T=AT𝛴T
, etc., and
denotes the complex
conjugate. Based on a curve fitting of a LITA signal using
modelling Eq.(13), an estimation of the thermodynamic
variables
cs
,
DT
and
𝛤
is possible. Based on these trans-
port properties, experimental investigations using LITA in
the vicinity of the critical point and in the vicinity of the
Widom line of a pure fluid disclose the possibility to study
transitions between the supercritical fluid states depicted in
Fig.1 on a macroscopic level. Additionally, given a suitable
thermodynamic model for these parameters, the knowledge
of local transport properties enables us, as shown in Eq.(4),
to extract local mixture quantities with a spatial resolution
O(
10
1
mm
)
in diameter and
O(
10
0
mm
)
in length.
4 Results
Extraction of transport properties from LITA signals, using
the approach explained above, requires a thorough cali-
bration of the optical setup as well as a validation of the
acquired transport properties. Both calibration and valida-
tion are presented in the following section. The uncertainty
analysis of the operating conditions as well as the Fourier
analysis of the LITA signal, the calibration, and validation
of the grid spacing
𝛬
is performed according to the Guide
to the expression of uncertainty in measurement by the Joint
Committee for Guides in Metrology (2008). For values taken
from a database, uncertainties are acquired using sequential
perturbation as presented by Moffat (1988).
Since the purpose of this study is the feasibility to extract
transport properties from LITA signals, the presented
uncertainties of the acoustic damping rates gained by the
(11)
q
=
2𝜋
𝛬
(12)
𝛬
=
𝜆
exc
2 sin (𝛩2).
(13)
𝛹
(t)∝exp
8𝜎2
𝜔2𝜔2+2𝜎2
cs
tt0
2
2
P1+P2T+P
1+P
2T
+exp 8𝜎2
𝜔2
𝜔2+2𝜎2
cstt02
P
1
P
2
+P
1
P
2
+
P
1
P
1
+P
2
P
2
+TT
.
curve fitting algorithm are based on the confidence interval
computed by the algorithm. These confidence intervals are
estimated using the inverse R factor from the QR decom-
position of the Jacobian, the degrees of freedom, as well
as the root-mean-squared error. Hence, the uncertainties of
the acoustic damping rates are only a representation of the
statistical error margin of the curve fitting at this point and
do not take the uncertainties of the fitted data and the input
parameters into account. The uncertainties of speed of sound
data extracted using curve fitting are estimated based on the
results for the DFT analysis. All uncertainties are presented
within a confidence interval of
95 %
.
4.1 Calibration ofoptical setup
Modelling of LITA signals requires a deep understanding
of non-linear optical processes, and phonon–photon as well
as thermon–photon interaction inherent to the LITA meas-
urement technique (Cummings etal. 1995). Additionally,
Eq.(13) is highly dependent on the beam waist of the excita-
tion beam
𝜔
as well as the magnitude of the grating vector
q, which depends on the spacing of the optical grid
𝛬
. Both
values are highly vulnerable to distortions due to turbulence
and convective transport processes if they occur in a similar
time scale. However, averaging the signal over a high num-
ber of laser pulses smears the signal and minimizes the effect
of shot to shot variations in turbulence and convective pro-
cesses as well as laser noise, jitter, and drift. An independent
study to quantify these effects, however, is highly complex.
Using Eq.(8) and/or(13) to measure speed of sound, acous-
tic damping rate or thermal diffusivity requires, therefore, a
careful and thorough calibration of the optical measurement
volume, specifically the spacing of the optical grid
𝛬
and the
Gaussian half-width
𝜔
of the excitation beam.
The calibration is done in well-known quiescent con-
ditions. When assuming non-resonant behaviour, the grid
spacing of the optical measurement volume can be charac-
terized by the excitation wavelength
𝜆exc
:
A beam profiling camera (DataRay) is used for beam align-
ment and to measure the beam arrangement in the foci of the
excitation and the interrogation beams at atmospheric condi-
tions. Using the collected geometrical data, the grid spacing
is estimated to be
𝛬BP =29.6 ±7𝜇m
with a beam distance
of
𝛥yexc,BP =36.9 ±3.8 mm
. Despite the agreement of the
geometrical calibration using the beam profiling camera and
the known parameters of the used optical setup, the meas-
urement uncertainties of the calibration are unacceptable.
Therefore, a new calibration procedure has been developed
to calibrate the optical setup. The grid spacing is estimated
for conditions up to
700 K
and
8 MPa
in non-resonant fluids,
(14)
𝛬
(𝛥yexc,fexc ,𝜆exc )=2
c
s
𝜈.
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Experiments in Fluids (2021) 62:2
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2 Page 10 of 16
namely nitrogen and argon. Using the dependencies of the
grid spacing
𝛬
in Eq.(14) together with the speed of sound
cs
extracted from Lemmon etal. (2018), we are able to cal-
culate a mean grid spacing
𝛬
. It is important to emphasise
that the grid spacing is, as shown by Li etal. (2002), solely
dependent on the geometrical and optical parameters of the
optical setup (f,
𝛥yexc
,
𝜆exc
). Hence, it is independent of the
refractive index of the probed environment. The dominating
frequency of the LITA signal
𝜈
is estimated using a DFT
together with a von Hann window and a band-pass filter.
This ensures the correct extraction of the frequencies even
for noisy signals in gas, gas–like, and compressed liquid
states. Note that, for each condition, the acquired frequen-
cies are averaged over at least 5000 samples. The calibration
procedure yields a grid spacing of the measurement vol-
ume of
𝛬cal =29.33 ±0.14 𝜇m
. Note that the measurement
uncertainties using the new calibration procedure are more
than one order of magnitude below the calibration using the
beam profiling camera.
Due to the high spatial resolution required for investiga-
tions in the wake of free-falling evaporation droplets and jet
disintegration, the optical arrangement uses focused beams
with a Gaussian beam profile. Hence, calibration of the beam
waist is crucial for the correct modelling of the LITA signal
and robust extraction of the acoustic damping rate as well as
the thermal diffusivity. Calibration is performed using LITA
signals with different operating conditions, as listed in Table1.
All signals are averaged over
nLITA
laser shots. To acquire the
Gaussian half-width
𝜔
, the experimentally detected and aver-
aged LITA signals presented in Table1 are curve fitted using
the simplified model by Schlamp etal. (1999) expressed by
Eq.(13), which is presented in Sect. 3.2.2. A robust non-linear
least-absolute fit using the Levenberg–Marquardt algorithm
is utilized by using the non-linear fit in Matlab (MathWorks)
with the robust option Least-Absolute Residuals (LAR). This
method optimizes the fit by minimizing the absolute differ-
ences of the residual rather than the squared differences. We
have chosen this option instead of an approach using bisquare
weights, since signals averaged over more than 5000 laser
shots are fitted, which leads to few outliers. The Gaussian
beam width
𝜔
, the modulation depths
UeP
,
U𝛩
, the beam mis-
alignment
̄𝜂
, the speed of sound
cs
, and the temporal offset
t0
are hereby output parameters, whereas the remaining param-
eters are input parameters, which were kept constant during the
curve fitting. The grid spacing is set to the calibrated value and
the Gaussian half-width of the interrogation beam to the value
measured by beam profile camera
𝜎BP
. Transport properties are
estimated using Lemmon etal. (2018) for thermal diffusivity
and shear viscosity. Acoustic damping rates are assessed by
the model proposed by Li etal. (2002), see Eq.(3). The curve
fitting of the last case listed in Table1 is exemplarily depicted
in Fig.4.
Assuming an exponential dependence with negative expo-
nent of pulse energy and Gaussian half beam width
𝜔
of the
excitation beam,
𝜔
can be expressed as:
where
𝛼exc
is the calibration constant,
𝜔BP =190 𝜇m
is the
Gaussian half-width measured with the beam profiling cam-
era, and
𝜔th =312 𝜇m
is the Gaussian half-width estimated
using the laser specifications. Note that for high pulse ener-
gies
𝜔
approaches
𝜔th
, whereas for low pulse energies
𝜔
can be approximated by the investigations using the beam
profiling camera. A robust non-linear least square fit using
the Levenberg–Marquardt algorithm with bisquare weights
is used to acquire
𝛼exc
.
(15)
𝜔cal
=𝜔
th
(
𝜔
th
𝜔
BP)
exp
{
𝛼
exc
E
exc},
Table 1 Overview of the operating conditions, input parameters, and
results of the calibration of the Gaussian half-width of the excita-
tion beam
𝜔
; operating conditions:
Tch
: fluid temperature;
pch
: fluid
pressure;
nLITA
: number of laser shots used for averaging;
Eexc
: pulse
energy of excitation beams;
Pint
: power of interrogation beam; SM:
single-mode fibre with diameter of
4𝜇m
; MM: multi-mode fibre with
diameter of
25 𝜇m
; curve fitting input parameters: calibrated grid
spacing
𝛬cal
=
29.33 𝜇m
; measured Gaussian half-width of the inter-
rogation beam
𝜎BP
=
192 𝜇m
;
DT,NIST
: thermal diffusivity;
𝛤m,NIST
:
pressure corrected acoustic damping rates; curve fitting results:
cs,LITA
: speed of sound;
𝜔LITA
: Gaussian half-width of the excitation
beam
Gas
pch
in
MPa
Tch
in
K
Eexc
in
mJ
Pint
in
W
Fibre
nLITA
cs,LITA
in
ms
1
𝛤m,NIST
in
mm2s
1
DT,NIST
in
mm2s
1
𝜔LITA
in
𝜇m
N2
2.0 ±0.1
295.6 ±1.8
40.0 8.0 SM 6057
356 ±4
3.25 1.08
277 ±2
Ar
2.0 ±0.1
294.7 ±0.6
22.5 8.0 SM 9901
313 ±2
1.37 1.01
259 ±2
Ar
2.0 ±0.1
295.3 ±0.6
22.5 6.0 SM 9334
317 ±2
1.38 1.02
251 ±2
Ar
2.0 ±0.1
295.2 ±0.6
22.5 4.0 MM 10713
320 ±2
1.38 1.02
241 ±2
Ar
2.0 ±0.1
293.9 ±0.6
27.0 4.0 MM 5283
324 ±2
1.37 1.01
301 ±1
Ar
0.5 ±0.1
295.4 ±0.6
32.4 5.0 MM 8650
316 ±2
3.78 4.13
209 ±2
Ar
1.0 ±0.1
295.2 ±0.6
32.4 4.0 MM 10100
315 ±2
2.17 2.05
222 ±2
Ar
2.0 ±0.1
295.2 ±0.6
32.4 1.5 MM 10602
316 ±2
1.38 1.02
216 ±1
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4.2 Validation andanalysis ofmeasurement
uncertainties
Due to the availability of speed of sound data at a wide range
of pressure and temperature, the validation of the optical
grid spacing
𝛬
using Eq.(8) is possible in the whole operat-
ing range. However, since acoustic damping rates depend
on both bulk and shear viscosities, available data for high-
temperature and high-pressure environments are rare. Hence,
validation is performed in the following section using only
acoustic damping rates of argon at room temperature.
4.2.1 Experimental speed ofsound data
Figure5 depicts the validation of the calibration process
using the grid spacing to characterize the measurement
volume. The speed of sound was calculated by Eq.(8),
where the non-resonant frequency was estimated using a
DFT together with a von Hann window and a band-pass
filter. A relative distribution is shown in Fig.6. For clarity,
we omit the distinction between the different non-resonant
fluids argon and nitrogen in the distribution. Both distribu-
tions show a Gaussian shaped distribution. For non-resonant
cases, the skewness of the distribution is
0.3
with a kurtosis
of 3.3. In case of resonant fluid behaviour, skewness is
1
with a kurtosis of 3.7.
Validation of the calibration process shows good agree-
ment between the non-resonant measurements of nitrogen
and argon, and theoretical values extracted from Lemmon
etal. (2018) (NIST database). The relative measurement
uncertainty of the acquired speed of sound for all investi-
gated fluids is below
2%
. For argon and carbon dioxide,
the uncertainties of measurement and NIST database are
in the same order of magnitude, which indicates the good
Fig. 4 Measured LITA signal with non-resonant fluid behav-
iour in pure argon with curve fitting result used for calibration.
The signal is averaged over 10602 laser shots. Operating condi-
tions:
pch =2±0.1 MPa
;
Tch =295.2 ±0.6 K
;
Eexc =32.4 mJ
;
Pint =1.5 W
; Curve fitting input parameters:
𝛬=𝛬cal =29.33 𝜇m
;
𝜎=𝜎BP =192 𝜇m
;
𝛤
=𝛤
m,NIST
=1.38 mm
2
s
1
;
DT
=D
T,NIST
=1.02 mm
2
s
1
; Curve fitting results:
𝜔LITA =216 ±1𝜇m
;
cs,LITA
=316 ±2ms
1
Fig. 5 Absolute comparison of speed of sound data for
pch =2
to
8 MPa
and temperatures up to
Tch =700 K
(nitrogen) and
pch =0.5
to
8 MPa
and temperatures up to
Tch =600 K
(argon and carbon dioxide)
using a grid spacing of
𝛬=𝛬cal =29.33 ±0.14 𝜇m
. The speed of
sound is calculated using a DFT with Eq.(8). Thermodynamic data
for validation are taken from Lemmon etal. (2018)
Content courtesy of Springer Nature, terms of use apply. Rights reserved.
Experiments in Fluids (2021) 62:2
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2 Page 12 of 16
precision of the LITA setup. However, the distribution in
Fig.6 shows a width of approximately
3%
for non-resonant
fluids argon and nitrogen. These differences result from una-
voidable misalignments of the excitation beams as well as
beam steering effects due to the high-temperature and high-
pressure environment.
Measurements with resonant fluid behaviour observed for
carbon dioxide, however, show a consistent deviation, result-
ing in an offset between the measurements and theoretical
values taken from Lemmon etal. (2018). In addition to the
observed offset in validation for carbon dioxide at tempera-
tures up to
600 K
and pressures up to
8 MPa
, a resonant fluid
behaviour is observed. However, based on the absorption
cross-section of carbon dioxide, resonant fluid behaviour can
be a priori excluded. One and the most probable explana-
tion for these resonant behaviours is residual moisture in
the experimental setup. With the absorption cross-section of
water at the excitation wavelength, residual moisture would
cause resonant fluid behaviour as has already been observed
by Cummings (1995). The constant offset even for low mole
fractions of water also demonstrates quite nicely the sensi-
tivity of the LITA signal on the concentration of a mixture.
This sensitivity is essential to extract the mixing temperature
as well as the mole fraction using Eq.(13) together with the
relations shown in(4).
To ensure the robustness of the optical setup, addi-
tional investigations in carbon dioxide are conducted. The
objective of these experiments is twofold. First, experimen-
tal investigation in an open loop setup with carbon dioxide
after drying the experimental setup using elevated tempera-
ture and vacuum shows no influence of residual moisture
on the LITA signal and the calibration procedure. Hence,
the origin of residual moisture is most likely due to purity
of the used carbon dioxide. Second, the authors propose an
intensity study, in which the energy of the excitation laser
is systematically varied. This could shed some light on two
other possible explanations for the observed resonant fluid
behaviour, namely spontaneous Raman scattering or an opti-
cal breakdown of carbon dioxide. The authors hypothesize
that due to the high pulse energy of up to
32 mJ
used for
LITA measurements in carbon dioxide an optical breakdown
of the fluid might occur, which would cause changes in the
fluid properties. This would affect the formation of the den-
sity grating and, therefore, the measured LITA signal. A
detailed description can be found in the work of Stampanoni-
Panariello etal. (2005a). Additionally, the dependencies of
the intensity of the detectable LITA signal on the excita-
tion beam pulse energy are different for non-resonant LITA
and spontaneous Raman scattering. Note that non-resonant,
purely electrostrictive LITA signals are caused by stimulated
Brillouin scattering (SBS), whereas resonant, LITA signals
are caused by the combination of SBS, stimulated thermal
Brillouin scattering (STBS), and/or stimulated thermal Ray-
leigh scattering (STRS). With reference to the theoretical
work by Stampanoni-Panariello etal. (2005a), the signal
intensity of the detectable LITA signal for both resonant and
non-resonant fluid behaviour shows a quadratic dependence
on the pulse energies of the excitation beams. Spontaneous
Raman scattering experiences, however, a linear dependence
on the incident intensity, see Powers (2013). Starting from
low-energy pulses, a systematic increase of the pulse energy
should, therefore, indicate when an optical breakdown of
carbon dioxide occurs and if spontaneous Raman scattering
is the cause of the resonant fluid behaviour, by favouring one
of the mentioned effects over the other.
4.2.2 Comparison ofexperimental andtheoretical LITA
signals
Acoustic damping rate, speed of sound, and thermal dif-
fusivity are gained using curve fitting of experimentally
detected, averaged LITA signals with the simplified model
by Schlamp etal. (1999) expressed by Eq.(13) presented
in Sect.3.2.2.
Similar to the curve fit in the calibration procedure, a
robust non-linear least-absolute curve fit is used. The lat-
ter utilizes a Levenberg–Marquardt algorithm. Besides the
desired speed of sound
cs
, acoustic damping rate
𝛤
, thermal
diffusivity
DT
, the modulation depths
UeP
,
U𝛩
, the beam
misalignment
̄𝜂
, and the temporal offset
t0
, all parameters
Fig. 6 Relative distribution of the comparison of speed of sound
data for
pch =2
to
8 MPa
and temperatures up to
Tch =700 K
(nitrogen) and
pch =0.5
to
8 MPa
and temperatures up to
Tch =600 K
(argon and carbon dioxide) using a grid spacing of
𝛬=𝛬cal =29.33 ±0.14 𝜇m
. The speed of sound is calculated using
a DFT with Eq. (8). Thermodynamic data for validation are taken
from Lemmon etal. (2018). For carbon dioxide, the skewness of the
distributions is
1
with a kurtosis of 3.7; for nitrogen, argon skewness
is acquired to be
0.3
with a kurtosis of 3.3
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Experiments in Fluids (2021) 62:2
1 3
Page 13 of 16 2
are input parameters, which were held constant during the
curve fitting. The grid spacing
𝛬
and Gaussian half-width
of the excitation beam
𝜔
are set to the calibrated value. The
Gaussian half-width of the interrogation beam
𝜎BP
is held
constant at the value measured by the beam profile camera.
A curve fit for a LITA signal in pure argon at a pressure of
4 MPa
and a temperature of
295 K
is shown in Fig.7. The
signal is averaged over 10991 shots. Curve fitting yields
cs,LITA
=319 ±2m s
1
and
𝛤LITA
=1.2 ±0.1 mm
2
s
1
,
which compared to theoretical estimations using Eq.(3)
together with the NIST database by Lemmon etal. (2018)
(
cs,NIST
=324 m s
1
and
𝛤m,NIST
=1.0 mm
2
s
1
) show good
agreement. The feasibility to extract acoustic damping
rates in a high-pressure, high-temperature environment for
a resonant fluid is presented in Fig.8. A carbon dioxide
atmosphere with residual moisture at pressure of
8 MPa
and
a temperature of
502.5 K
is shown. Note that, due to the
low modulation depth of the thermal grating compared to
the electrostrictive grating
U𝛩,LITAUeP,LITA <0.08
, the fit-
ted thermal diffusivity does not reveal physically realistic
results.
Figure 9 depicts acoustic damping rate ratios
𝛤LITA𝛤c,NIST
for pure argon at room temperature for various
pressures up to
8 MPa
. Values are compared to the experi-
mental and theoretical investigations by Li etal. (2002). For
pressures up to
4 MPa
, our experimental investigation shows
good consensus with data by Li etal. (2002). The points
at same temperatures and pressures indicate experiments
with similar operating conditions, which are investigated
on different days. The deviation between those points is
most likely caused by beam steering effects of the excitation
beams due to staining of the quartz windows resulting from
Fig. 7 Measured LITA signal with non-resonant fluid behaviour in
pure argon with curve fitting result used to acquire transport proper-
ties. The signal is averaged over 10991 laser shots. Operating con-
ditions:
pch =4±0.1 MPa
;
Tch =295.0 ±0.5 K
;
Eexc =22.5 mJ
;
Pint =2.5 W
; curve fitting input parameters:
𝛬=𝛬cal =29.33 𝜇m
;
𝜎=𝜎BP =192 𝜇m
;
𝜔=𝜔cal =238 𝜇m
; curve fitting results:
cs,LITA
=319 ±2ms
1
;
𝛤LITA =
1.2
±
0.1 mm
2
s
1
; theoretical esti-
mations using Eq. (3) together with the NIST database by Lemmon
etal. (2018) give:
cs,NIST
=324 m s
1
and
𝛤m,NIST
=1.0 mm
2
s
1
Fig. 8 Measured LITA signal with resonant fluid behaviour in car-
bon dioxide with curve fitting result used to acquire transport prop-
erties. The signal is averaged over 11030 laser shots. Operating
conditions:
pch =8±0.1 MPa
;
Tch =502.5 ±4.9 K
;
Eexc =18 mJ
;
Pint =1W
; curve fitting input parameters:
𝛬=𝛬cal =29.33 𝜇m
;
𝜎=𝜎BP =192 𝜇m
;
𝜔=𝜔cal =230 𝜇m
; curve fitting results:
cs,LITA
=356 ±2ms
1
;
𝛤LITA =
41
±
1 mm
2
s
1
Fig. 9 Acoustic damping rate ratio
𝛤LITA𝛤c,NIST
over chamber pres-
sure
pch
for pure argon at room temperature. Classical acoustic damp-
ing rates are estimated using NIST database by Lemmon etal. (2018).
Experimental and theoretical data are taken from Li etal. (2002)
Content courtesy of Springer Nature, terms of use apply. Rights reserved.
Experiments in Fluids (2021) 62:2
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2 Page 14 of 16
a slightly excessive pulse energy. These stains could change
the Gaussian beam width of the excitation beams without
changing the beam crossing angle. Acoustic damping rate
ratios at higher pressure do not show a linear dependence
on pressure. Considering the critical pressure of argon
pc
=4.9 MPa
, deviations are expected, since shear and bulk
viscosity show non-linear behaviour in the vicinity of the
critical point as indicated by Meier etal. (2004) and Meier
etal. (2005).
The authors hypothesize that the high values of acoustic
damping rates in Fig.9 result from thermodynamic anoma-
lies in the vicinity of the critical point. These anomalies are
an exponential increase in shear viscosity above the critical
pressure as shown by Meier etal. (2004) and peak values
in bulk flow velocities in the vicinity of the critical pressure
as has been simulated for Lennard–Jones fluids by Meier
etal. (2005). We further emphasise that bulk viscosities
are neglected in the theoretical fit by Li etal. (2002), and
that the fit is only validated for pressures up to
2.53 MPa
.
The comparison in Fig.9 indicates the capability of spa-
tially high-resolved LITA measurements to extract speed of
sound and acoustic damping rates in fluids. However, for a
more concise calibration, theoretical or experimental data
of acoustic damping rate at high pressure and temperatures
are necessary. Hence, further investigations of theoretical
approximations and experimental data of bulk viscosities
are essential to validate the presented post processing curve
fitting algorithm for high pressure.
5 Conclusion
In this study, the challenges as well as the potential of
laser-induced thermal acoustics for small-scale macro-
scopic fluid phenomena occurring in jet disintegration or
droplet evaporation are presented. By applying LITA with
an optical arrangement using focused beams, we can suc-
cessfully acquire transport properties using an elliptical
measurement volume with a spatial resolution of
312 𝜇m
in diameter and less than
2 mm
in length. The speed of
sound is measured in a high-pressure and high-temperature
environment for various fluids using LITA together with
a direct Fourier analysis. To validate these measurements
and access their measurement uncertainties, a comparison
with the NIST database by Lemmon etal. (2018) is imple-
mented. Using a confidence interval of
95 %
, the relative
uncertainties for the speed of sound data are within
3%
of
the acquired values. Furthermore, acoustic damping rates
are acquired by curve fitting experimental LITA signals
to a simplified analytical expression based on the model
by Schlamp etal. (1999). Validation using pure argon at
elevated pressures shows promising results and, hence,
confirm the capability of LITA to simultaneously measure
transport properties in small-scale fluid phenomena. The
importance of these transport properties measured using
LITA is twofold. Investigations in pure fluids in the vicin-
ity of their critical point and across their Widom line
enable us to study transitions between supercritical fluid
states on a macroscopic level. Moreover, by applying suit-
able models for speed of sound, acoustic damping rates,
as well as thermal diffusivities in fluid mixtures, it is pos-
sible to determine mixing parameters in macroscopic fluid
phenomena on a small scale.
Acknowledgements The authors gratefully acknowledge the finan-
cial support by the Deutsche Forschungsgemeinschaft (DFG, Ger-
man Research Foundation)—Project SFB–TRR 75, Project number
84292822.
Funding Open Access funding enabled and organized by Projekt
DEAL. This study was funded by the Deutsche Forschungsgemein-
schaft (DFG, German Research Foundation)—Project SFB–TRR 75,
Project number 84292822.
Data availability statement The experimental data are not yet publicly
available.
Compliance with ethical standards
Conflict of interest The authors declare that they have no conflict of
interest.
Code availability Calibration, validation, and curve fitting codes are
implemented using MATLAB (MathWorks). The code is not publicly
available.
Open Access This article is licensed under a Creative Commons Attri-
bution 4.0 International License, which permits use, sharing, adapta-
tion, distribution and reproduction in any medium or format, as long
as you give appropriate credit to the original author(s) and the source,
provide a link to the Creative Commons licence, and indicate if changes
were made. The images or other third party material in this article are
included in the article’s Creative Commons licence, unless indicated
otherwise in a credit line to the material. If material is not included in
the article’s Creative Commons licence and your intended use is not
permitted by statutory regulation or exceeds the permitted use, you will
need to obtain permission directly from the copyright holder. To view a
copy of this licence, visit http://creat iveco mmons .org/licen ses/by/4.0/.
Appendix
The temporal shape of the excitation laser pulse can be
estimated using a
𝛿
-function at
t0
. As discussed by Stam-
panoni-Panariello etal. (2005a), this assumption is valid
since the laser pulse length
𝜏
:
is small compared to the reciprocal of the acoustic fre-
quency
(
𝜏≪1
(
c
s
q
))
;
Content courtesy of Springer Nature, terms of use apply. Rights reserved.
Experiments in Fluids (2021) 62:2
1 3
Page 15 of 16 2
– is small compared to the reciprocal of the acoustic
decay rates
(
𝜏≪1
(
q
2
𝛤
))
and
(
𝜏≪1
(
q
2
D
T))
The following assumptions are used to derive Eq.(9) in
this work from Eqs.(6d), (6e) and (15b) in the work of
Schlamp etal. (1999). The assumptions are based on the
theoretical considerations by Cummings etal. (1995) and
Stampanoni-Panariello etal. (2005a):
Cummings etal. (1995) proposed that in the limit of
fast thermalization
DT1
, where
DT
denotes the ther-
mal diffusivity.
Cummings etal. (1995) proposed that, due to negligible
damping over a wave period
𝛤≪1
, where
𝛤
denotes the
acoustic damping rate.
Stampanoni-Panariello etal. (2005a) proposed that, in
case of instantaneous release of absorbed laser radiation
into heat
csq≪𝛾
𝛩+𝛾n𝛩
, where
cs
denotes the speed of
sound, q denotes the magnitude of the grating wave vec-
tor, and
𝛾𝛩
denotes the rate of excited-state energy decay
caused by thermalization (in the work of Stampanoni-
Panariello etal. (2005a), this parameter is denoted as
𝜆
)
and
𝛾n𝛩
denotes the rate of excited-state energy decay not
caused by thermalization (in the work of Stampanoni-
Panariello etal. (2005a), this parameter is denoted as
𝜓
).
Note that, using the above assumptions, Eq.(15b) in the
work of Schlamp etal. (1999) yields
AD=0
, which is
not shown in Eq.(9) in this work.
Equation(10) in this work is derived from Eqs. (3b), (13b),
(14d) in the work of Schlamp etal. (1999) using the follow-
ing assumptions and simplifications based on the specific
conditions of presented experimental investigations:
Considering the small beam crossing angles in the optical
setup, we assume
cos (𝛩)1
and
cos (𝛷)1
.
Due to the low mass flow rates during the investigations
and the vanishing flow velocities, bulk flow velocities can
be neglected:
v=0
and
w=0
.
Due to negligible bulk flow velocities in the chamber
only beam misalignment in horizontal y-direction,
̄𝜂
has
an effect on the time history of the LITA signal (Schlamp
etal. 1999). Hence, after careful beam alignment through
the quartz windows before each measurement resulting in
a maximized signal, the beam misalignment in z-direc-
tion is neglected
̄
𝜁=0
.
Note that, due to the simplified version of Eq. (9) in this
work,
AD=0
; therefore, the value
D=AD𝛴D=0
and
it is not necessary to calculate
𝛴D
in Eq.(14d) from the
work of Schlamp etal. (1999).
Following the simplifications and assumptions summarized
above and considering only the temporal evolution of the
LITA signal Eq. (17) in the work of Schlamp etal. (1999)
can be further simplified to Eq. (13) proposed in this work.
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... A less complex nonlinear technique that does not require fitting to a spectrally resolved signal is laser-induced grating scattering (LIGS), also known as laser-induced thermal acoustics (LITA) [16][17][18]. LIGS/LITA experiments measure the speed of sound in a gas directly, and models of the LIGS signal evolution can be used to determine transport properties such as the acoustic damping rate and thermal diffusivity, of key interest for studies of fundamental fluid physics [19]. Previous studies have demonstrated the capabilities of the LIGS technique for temperature measurements, which can readily achieve sub-1% precision and accuracies of order of 1% [20][21][22][23][24]. ...
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