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Experiments in Fluids (2021) 62:2

https://doi.org/10.1007/s00348-020-03088-1

RESEARCH ARTICLE

On thepotential andchallenges oflaser‑induced thermal acoustics

forexperimental investigation ofmacroscopic ﬂuid phenomena

ChristophSteinhausen1 · ValerieGerber1· AndreasPreusche2· BernhardWeigand1· AndreasDreizler2·

GraziaLamanna1

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

1 3

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

(−)

P1,2

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 etal. (2002)

(

m2s−1

)

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 etal. (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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Page 3 of 16 2

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

𝜆∕2

-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 etal. 2016; Müller etal.

2016; Baab etal. 2016, 2018; Crua etal. 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

etal. 2017; Lamanna etal. 2018; Steinhausen etal. 2019;

Stierle etal. 2020; Nomura etal. 2020; Lamanna etal.

2020; Qiao etal. 2020). Microscopic investigations by

Santoro and Gorelli (2008), Simeoni etal. (2010) as well

as Bencivenga etal. (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 etal. 2010; Bencivenga

etal. 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 etal. 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 etal.

(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 etal. (2018)

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Experiments in Fluids (2021) 62:2

1 3

2 Page 4 of 16

review on gas-phase diagnostics using laser-induced tran-

sient grating spectroscopy (LIGS) is provided by Stampa-

noni-Panariello etal. (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 etal.

(2018) has shown. Kimura etal. (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 etal. (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 etal. (2005a) and for

liquid solvents by Kimura etal. (2005b). An investigation

of acoustic damping rates in pure gases was presented by Li

etal. (2002) analysing the temporal behaviour of transient

grating spectroscopy. The investigations included diﬀerent

gases at pressures up to

25 atm

at room temperature. Li etal.

(2002) compared these ﬁndings with classical acoustic the-

ory and derived a linear pressure dependency for the meas-

ured acoustic damping rate. Li etal. (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

etal. (2016), Baab etal. (2018), and Förster etal. (2018)

already showed the capability of acquiring quantitative

speed of sound data in jet disintegration. Especially to be

emphasised are the investigations by Baab etal. (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

ontherelevance oflaser‑induced thermal

acoustic insupercritical 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 etal. (2005)

for binary mixtures of monoatomic species. Using tran-

sient grating spectroscopic Li etal. (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

etal. (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 etal. 2002). Using

the theoretical description by Hubschmid etal. (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 etal. (2002):

(1)

𝛤

=1

2𝜚

[

4

3𝜇s+𝜇v+(𝛾−1)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 etal. (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 etal. (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 etal. (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,

𝛽−1

is the characteristic

(2)

𝛤

c=1

2𝜚

[

4

3𝜇s+(𝛾−1)𝜅

c

p].

(3)

𝛤m,Ar =𝛤c

(

1∕30 patm +1

)

𝛤m,N2

=𝛤

c(

1∕6p

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 {−𝛽t−i𝜐t}+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 etal. (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 etal.

(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 etal. (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 andmeasurement

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 etal. (2016) and Förster etal. (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 250KM) 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 etal. (1995) and Stampanoni-Panariello

etal. (2005a). Note that the analytical expression presented

by Stampanoni-Panariello etal. (2005a) is only valid for

inﬁnite beam sizes, whereas Cummings etal. (1995) takes

ﬁnite beam sizes into account. In the limit of inﬁnite beam

sizes, both theories merge. Schlamp etal. (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 etal. (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

thirdinput 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 etal. (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

𝜆∕2

-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

𝜆∕2

-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

𝛩≈1◦

. 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 etal. (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

etal. (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

1 3

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 byHemmerling 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 etal.

(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 etal. (1999) and Cummings etal. (1995). The

used assumptions are categorized and listed in the appen-

dix of this work. As discussed in more detail by Stampa-

noni-Panariello etal. (2005a), the temporal shape of the

excitation laser pulse is estimated using a

𝛿

-function at

t0

.

The model suggested by Schlamp etal. (1999) can be sim-

pliﬁed using two key assumptions proposed by Cummings

(8)

c

s=

𝜈𝛬

j

.

etal. (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 etal. 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 etal. (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 etal. (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 etal. 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 etal. (2005a):

(9)

A

P1,P2

=1∕2U

𝛩

±i∕2UeP

A

T

=−1U

𝛩

.

(10)

𝛴

P1,P2=exp

−𝛤q2t−t0−

2

̄𝜂 ±cs

t−t0

2

𝜔2+2𝜎2

exp ±iqcst−t0

𝛴T=exp

−DTq2

t−t0

−2̄𝜂 2

𝜔2+2𝜎2

.

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Experiments in Fluids (2021) 62:2

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Page 9 of 16 2

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 (𝛩∕2).

(13)

𝛹

(t)∝exp

−8𝜎2

𝜔2𝜔2+2𝜎2

cs

t−t0

2

2

P1+P2T∗+P∗

1+P∗

2T

+exp −8𝜎2

𝜔2

𝜔2+2𝜎2

cst−t02

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 ofoptical 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 etal. 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 etal. (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 etal. (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 Table1.

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 Table1 are curve ﬁtted using

the simpliﬁed model by Schlamp etal. (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 etal. (2018) for thermal diﬀusivity

and shear viscosity. Acoustic damping rates are assessed by

the model proposed by Li etal. (2002), see Eq.(3). The curve

ﬁtting of the last case listed in Table1 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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Page 11 of 16 2

4.2 Validation andanalysis ofmeasurement

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 ofsound data

Figure5 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

etal. (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 etal. (2018)

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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 etal. (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 etal. (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 etal. (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 ofexperimental andtheoretical 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 etal. (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 etal. (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 etal. (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,LITA∕UeP,LITA <0.08

, the ﬁt-

ted thermal diﬀusivity does not reveal physically realistic

results.

Figure 9 depicts acoustic damping rate ratios

𝛤LITA∕𝛤c,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 etal. (2002). For

pressures up to

4 MPa

, our experimental investigation shows

good consensus with data by Li etal. (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

etal. (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

𝛤LITA∕𝛤c,NIST

over chamber pres-

sure

pch

for pure argon at room temperature. Classical acoustic damp-

ing rates are estimated using NIST database by Lemmon etal. (2018).

Experimental and theoretical data are taken from Li etal. (2002)

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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 etal. (2004) and Meier

etal. (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 etal. (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

etal. (2005). We further emphasise that bulk viscosities

are neglected in the theoretical ﬁt by Li etal. (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 etal. (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 etal. (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 etal. (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 etal. (1999). The assumptions are based on the

theoretical considerations by Cummings etal. (1995) and

Stampanoni-Panariello etal. (2005a):

– Cummings etal. (1995) proposed that in the limit of

fast thermalization

DT≪1

, where

DT

denotes the ther-

mal diﬀusivity.

– Cummings etal. (1995) proposed that, due to negligible

damping over a wave period

𝛤≪1

, where

𝛤

denotes the

acoustic damping rate.

– Stampanoni-Panariello etal. (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 etal. (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 etal. (2005a), this parameter is denoted as

𝜓

).

– Note that, using the above assumptions, Eq.(15b) in the

work of Schlamp etal. (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 etal. (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

etal. 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 etal. (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 etal. (1999)

can be further simpliﬁed to Eq. (13) proposed in this work.

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Transf 151:119450. https ://doi.org/10.1016/j.ijhea tmass trans

fer.2020.11945 0

Publisher’s Note Springer Nature remains neutral with regard to

jurisdictional claims in published maps and institutional aﬃliations.

Aliations

ChristophSteinhausen1 · ValerieGerber1· AndreasPreusche2· BernhardWeigand1· AndreasDreizler2·

GraziaLamanna1

* Christoph Steinhausen

christoph.steinhausen@itlr.uni-stuttgart.de

1 Institute ofAerospace Thermodynamics (ITLR), University

ofStuttgart, Pfaﬀenwaldring 31, 70569Stuttgart, Germany

2 Institute Reactive Flows andDiagnostics (RSM),

Technical University ofDarmstadt, Otto-Berndt-Str. 3,

64287Darmstadt, Germany

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