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

SIMULTANEOUS VELOCITY AND TEMPERATURE MEASUREMENTS IN
THE LAMINAR BOUNDARY LAYER OF A MELTING VERTICAL ICE FACE

Pamoda Herath
Saurabh Pathak
Department of Mechanical Engineering Department of Mechanical Engineering
The University of Melbourne
Victoria 3010 Australia
jherath@student.unimelb.edu.au

The University of Melbourne
Victoria 3010 Australia
pathak@unimelb.edu.au

Bishakhdatta Gayen
Joseph Klewicki
Department of Mechanical Engineering Department of Mechanical Engineering
The University of Melbourne
Victoria 3010 Australia
CAOS
Indian Institute of Sciencebishakhdatta.gayen@unimelb.edu.au

The University of Melbourne
Victoria 3010 Australia
klewicki@unimelb.edu.au

Jimmy Philip
Department of Mechanical Engineering
The University of Melbourne
Victoria 3010 Australia
jimmyp@unimelb.edu.au

ABSTRACTinterface and ambient saline water, a buoyancy driven convective boundary layer flow forms next to the ice water interface.
The nature of this boundary layer flow is determined by thetemperature and salinity of the ambient water. Usually boundary layer flow starts as a laminar flow and becomes turbulent as
it flows along the interface. The ice melting process is mainly
driven by the heat and salt transport to the ice interface (Josberger & Martin, 1981; Kerr & McConnochie, 2015; Gayen
et al., 2016) across the boundary layer. Therefore, heat and salt
conservation at the interface determine ice ablation rate. The
heat flux balance at the interface is given by

The present experimental investigation is motivated byan aim to understand the processes by which the Antarctic
ice shelves melt, under convective flow conditions. The density driven boundary layer flow that forms next to a melting
ice-water interface has a direct influence on the heat and salt
transfer to the ice that further determines the ice ablation rate.
High resolution velocity and temperature measurements in thisboundary layer are thus useful to understand and estimate themelting process of a vertical ice face. As a starting point, in thepresent paper we consider a relatively small vertical ice block
resulting in a laminar melt boundary layer, unlike the field situation where the boundary layer is turbulent. Single component
molecular tagging velocimetry (1c-MTV) and Molecular tagging thermometry (MTT) are employed to simultaneously measure the boundary layer vertical flow velocity and temperature
close to the phase-change ice-water interface in a laboratory
setup for different ambient water temperatures and salinities.
Also, the ice face ablation rate is estimated by determining theposition of the ice water interface as a function of time. The
experimental findings are compared with idealised direct numerical simulations (DNS) that solve the three-dimensional andnon-hydrostatic Navier-Stokes equations under similar conditions as well as with the similarity solution for an isothermal
wall.

𝐻
𝐻
𝑄 𝑖𝑐𝑒
− 𝑄𝑤
= 𝑄𝑚,

(1)

𝐻 ) and in the ice
relating the heat flux in to the water (𝑄 𝑤
𝐻𝐻=
(𝑄 𝑖𝑐𝑒 ) to the latent heat of ice melting (𝑄 𝑚 ), where 𝑄 𝑤
𝜌 𝑤 𝑐 𝑤 𝜅𝑇 𝜕𝑇/𝜕𝑥 and 𝑄 𝑚 = 𝜌𝑖 𝐿𝑉. Here 𝜌 𝑤 , 𝑐 𝑤 and 𝜅𝑇 are
the density, specific heat and thermal diffusivity of the water,whereas 𝜌𝑖𝑐𝑒 , 𝐿 and 𝑉 are the density, specific latent heat andablation velocity of the ice, respectively. The direction perpendicular to the ice-face and into water is 𝑥 (c.f., figure 1) and
𝑇 is the temperature. The heat flux into the ice is negligible,𝐻 ≈ 0 (Holland & Jenkins, 1999), and therefore the heat
𝑄 𝑖𝑐𝑒balance at the interface is given by

Introduction
When an ice face is in contact with saline water, due to thedensity difference between cold fresh water released from the𝜌𝑖𝑐𝑒 𝐿𝑉 = −𝜌 𝑤 𝑐 𝑤 𝜅𝑇

1

𝜕𝑇
.
𝜕𝑥

(2)

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022
Similarly, for salt the flux balance,𝑆
𝑄 𝑖𝑐𝑒− 𝑄 𝑆𝑤 = 𝑄 𝑆 ,A direct numerical simulation (DNS) model (Gayen et al.,2014) that solves the three-dimensional and non-hydrostaticNavier-Stokes equations, along with the incompressiblity condition and heat and salinity conservation equations is used tosimulate the ice melting and boundary layer formation process
in the water tank. The DNS model used either linear or non
linear equations of state (EoS). The DNS results are hereincompared to the experimentally measured data.
The Navier–Stokes, mass, heat and salt conservation equations that are used in the DNS model are(3)

𝑆 is the salt flux
where 𝑄 𝑆𝑤 is the salt flux in to the water, 𝑄 𝑖𝑐𝑒
𝑆 isin the ice and 𝑄 𝑆 is the brine rejection flux. Here 𝑄 𝑖𝑐𝑒
negligible and the salt balance is then given by𝜌𝑖𝑐𝑒 𝑆𝑖 𝑉 = −𝜌 𝑤 𝜅 𝑆

𝜕𝑆
,
𝜕𝑥

(4)

where 𝜅 𝑆 is the salt diffusivity in water, and this determines thesalinity at the interface (𝑆𝑖 ). 𝑆𝑖 determines the interface temperature (𝑇𝑖 ) according to freezing point depression equationof salt water,

𝑇𝑖 = 𝑚𝑆𝑖 ,∇ · 𝑢˜ = 0,

(6)

𝜕𝒖
−∇𝑝 ∗
𝜌∗
+ (𝒖 · ∇)𝒖 =+ 𝜈∇2 𝒖 − 𝑔,
˜
𝜕𝑡
𝜌0
𝜌0

(7)

𝜕𝑇 ∗
+ (𝒖 · ∇)𝑇 ∗ = 𝜅𝑇 ∇2𝑇 ∗ ,
𝜕𝑡

(8)

𝜕𝑆 ∗
+ (𝒖 · ∇)𝑆 ∗ = 𝜅 𝑆 ∇2 𝑆 ∗ .
𝜕𝑡

(9)

(5)

where 𝑚 = −6 × 10−2◦ 𝐶/‰(Josberger & Martin, 1981).
Through various experimental and numerical studies (Josberger & Martin, 1981; Kerr & McConnochie, 2015; McConnochie & Kerr, 2016; Gayen et al., 2016) it has been foundthat the nature of the boundary layer flow that forms next tophase changing interface directly affects the melt rate of theice-face. In order to study the efficiency of the heat transport
across the boundary layer, which is ultimately important toquantify the melt-rate of ice, studying the convective boundary
layer flow is important. Full temperature and velocity profiles
in the melt plume are only available through Direct Numerical
Simulations (Gayen et al., 2016; Mondal et al., 2019), and todate no experimental methodology has been developed to obtain high resolution velocity and temperature measurements inthis phase change boundary layer. The present study focuses ondeveloping an experimental procedure to obtain simultaneous
velocity and temperature measurements in a laminar boundary
layer flow close to a vertically melting ice face. The longer
term aim is to understand the Antarctic ice-shelf melting process and boundary layer dynamics that are associated with aturbulent boundary layer flow.
Single component molecular tagging velocimetry (1cMTV) is employed to estimate the boundary layer vertical flow
velocity close to the phase-change ice-water interface in a laboratory setup for different ambient water temperatures in freshand saline environments. In the MTV technique, phosphorescent molecules excited by an ultra-violet laser are used as a
Lagrangian tracer, and velocities are determined by two images separated by a specified delay time as acquired by a gatedintensified camera (Hu & Koochesfahani, 2006; Agrawal et al.,2021). Here we acquire velocity data within and outside of
the melt plume. Simultaneous temperature measurements inthe boundary layer are also carried out by molecular tagging
thermometry (MTT). MTT leverages the fact that the decayrate of the phosphorescence is uniquely temperature dependent (Gendrich et al., 1997). The standard MTT methodology
is modified in the present case by employing different camera exposure times for the two required images. This enables
larger inter-frame delays, which makes it possible to measuretemperature and relatively small magnitudes of velocity simultaneously. In addition to flow measurements, the ice ablation
velocity is also measured separately.

Here 𝜌 ∗ , 𝑇 ∗ and 𝑆 ∗ are changes in density, temperature andsalinity. 𝜌0 is reference density, 𝜈 is kinematic viscosity, 𝒖 =(𝑢, 𝑣, 𝑤), 𝒈 = (0, 0, 𝑔) and 𝑝 ∗ is the pressure deviation from its
reference value.
The change in density ( 𝜌 ∗ ) for linear EoS DNS is given by,𝜌 ∗ = 𝜌0 (𝛽𝑆 ∗ − 𝛼𝑇 ∗ ).

(10)

Here 𝛽 is the haline contraction coefficient and 𝛼 is the thermal
expansion coefficient at the ambient water temperature andsalinity. Depending on the ambient temperature and salinity
𝛽 varies in the range of 7.6 × 10−4 to 8.2 × 10−4 kg/g while 𝛼
ranged from 0 to 2.1 × 10−4◦ 𝐶 −1 .
For non linear EoS DNS, change in density ( 𝜌 ∗ ) is evaluatedby,𝜌 ∗ = 𝜌 − 𝜌0(11)where 𝜌 = 𝑓 (𝑇, 𝑆) as given in TEOS10 (McDougall & Barker,2011).
In this study the importance of using non-linear equationof state is investigated by comparing the experiments with DNS
that used both linear and non-linear models.

Experiments
As shown in figure 1, Experiments were performed in asmall water tank with 9 cm depth and a square base of 18 cm
length and width. One sidewall of the tank has a fused silica
window to allow UV laser beam access. At the beginning of2

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022
the water tank. The length measured in the 𝑥 direction is,
however, infinite in the DNS domain. This serves to preserve
mass conservation in the domain as the ice face melts, whereas
in the experiments the melt water simply (depending on theambient conditions) accumulates at the top or at the bottom of
the tank, forming a mild recirculation over an extended period.

Figure 1. Schematic of water tank and ice block showing adownward flowing boundary layer flow.

the experiments, as show in figure 1, an ice block with 35 mm
thickness was placed at one end of the tank. Once the ice face isin contact with the water, a buoyancy driven laminar boundary
layer flow forms next to ice-water interface.
In this study, the experiments were conducted under different ambient water temperatures and salinities, and dependingon the ambient conditions the flow can be upwards or downwards. Velocity and temperature profiles were obtained at𝑧˜ = 58mm (𝑧˜ origin at top, c.f., figure 1) for downward flowing
boundary layer cases whereas at 𝑧 = 58mm (from bottom) for
upward flowing boundary layer cases. The Grashof number
along the 𝑧-direction defined as 𝐺𝑟 𝑧 ≡ 𝑔 (Δ𝜌) 𝑧3/𝜈2 𝜌0 . Here Δ𝜌 isthe difference between water density at the interface and water
density in the far field. At the given 𝑧 location 𝐺𝑟 𝑧 varies in
the range of 1 × 106 - 1.2 × 107 depending on 𝑇∞ . Note that thecritical Grashof number for transition from laminar to turbulent
flow on a solid vertical wall varies from 109 - 1010 .
The full MTV/MTT experimental setup mainly consist of
the water tank, a pulsed UV laser, and an intensified gatedcamera that can capture 5 frames per second with an image
resolution of 1024 × 1024 pixels. A digital delay generator
is used to synchronize the camera and the laser. The laser
beam is guided to the fused silica window using an opticalarrangement consisting of UV mirrors. A pinhole arrangement
is then used to obtain a thin laser line to excite phosphorescentmolecules within the water. Images captured by the camera arelogged using a host computer. The arrangement of the different
components of the experimental setup is illustrated in figure 2.
In these experiments the ice block is created in a mold usingslightly heated water (to reduce bubble formation) and keeping
it in a freezer typically for overnight. Once the ice is frozen asmall layer of sufficiently diluted MTV solution is poured on
the ice-face (where melting happens), and allowed to freeze
again. This layer of ice with the phosphorescence molecules
ensures that when the ice melts, the melt plume has sufficient
MTV signal during the experiments.
In some cases, a separate set of experiments were carriedout to measure the ice melting rate for each ambient watertemperature. Here, however, the laser was not used. Rather, anintense white light source was used to illuminate the melting
ice-face. The receding position of the ice-water interface wasobtained as a function of time using image analysis, and fromthis the melt rate of ice (reduction of the ice thickness per unittime) was estimated..
The numerical setting of the DNS model is reported in
Gayen et al. (2014). The computational domain is similar to

Figure 2. Schematic diagram of the experimental setup andthe arrangement of its components.

Results

Figure 3. Comparison of experimental (symbols), DNS withlinear equation of state (dashed lines) and DNS with non-linear
equation of state (solid lines) velocity (𝑤) profiles for 𝑇∞ =10◦ 𝐶 and 𝑆∞ = 0. Experimental results are obtained at 𝑧˜ =58mm, i.e., 𝑧 = 27mm. All DNS results are obtained at 𝑧
values of 35, 55, and 75mm and the colour gets darker as 𝑧
increases.

Averaged velocity and temperature profiles obtained experimentally for different ambient conditions are comparedwith direct numerical simulations (DNS). DNS uses either
a linear or non-linear equation of state to relate temperatureand salinity to density. Figure 3 shows a comparison between
the experimental (in symbols) and DNS vertical velocity (𝑤)312th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022

Figure 4. Comparison of non-dimensionalized experimental
and DNS (linear and non-linear equation of state) Velocity(𝑤 ∗ ) profiles with analytical solution of natural convection overisothermal flat plate for 𝑃𝑟 = 10 (red dotted lines), for 𝑇∞ =10◦ 𝐶 and 𝑆∞ = 0.

Figure 5. Comparison of non-dimensionalized experimental
and DNS (linear and non-linear equation of state). Temperature(𝑇 ∗ ) profiles with analytical solution of natural convection overisothermal flat plate for 𝑃𝑟 = 10 (red dotted lines), for 𝑇∞ =10◦ 𝐶 and 𝑆∞ = 0.

profiles obtained for ambient water conditions of 𝑇∞ = 10◦ 𝐶
and 𝑆∞ = 0 (ambient water salinity). Here the dashed lines
are for the DNS using a linear equation of state (EoS), while
non-linear EoS results are in solid lines. From the figure, it
is apparent that the measured data are well within the range
of velocities obtained from the DNS. We note that the precise
velocity magnitudes are quite sensitive to the accurate choiceof the 𝑧-location. It is well-known that the boundary layer does
not start precisely at the ‘leading-edge’ or the free-surface, anda correction in terms of virtual origin is usually required.
Both the experimental and the DNS profiles obtained for
fresh water (𝑆∞ = 0) are non-dimensionalized and comparedwith the analytical similarity solution for natural convection
over an isothermally cooled vertical flat plate (Ostrach, 1952)
at a fixed Prandtl number of 10 and a fixed thermal expansion
coefficient (𝛽) (i.e linear equation of state) as shown in figure 4.
It can be observed that after a virtual origin shift of 28mm, nondimensionalized experimental velocity profiles nearly coincidewith the nonlinear EoS DNS profiles. From figure 3 and 4 itis evident that the peak velocity location and magnitude of thenonlinear EoS DNS velocity profiles are in better agreementwith the measured profile than the linear EoS DNS results andsimilarity solution. This suggests the role of nonlinear EoS
on momentum transport. These differences are, however, notapparent in the temperature boundary layer profiles (see figure5). Here, careful inspection reveals that close to the ice surface(say at 𝜂 ≈ 1) the non-linear (solid lines), rather than linear(dashed-lines), compare more favorably with the experimental
data. This further highlights the apparent importance of nonlinear EoS in numerical simulations.
Figures 6 and 7 show velocity and temperature profiles
from three different low ambient temperature cases; 𝑇∞ = 10◦ 𝐶
and 𝑆∞ = 0 (black), 𝑇∞ = 4◦ 𝐶 and 𝑆∞ = 0 (green) and 𝑇∞ = 4◦ 𝐶
and 𝑆∞ = 35 g/kg (purple). In the first(black) case the flow is
downward because colder water near the melting ice face isdenser than the warm ambient water. But the flow is upward in
the second(green) case because fresh water density is maximum
at 4◦ 𝐶, and colder water near the melting ice face is less dense
than the ambient water at 4◦ 𝐶. The third(purple) case has acomplex boundary layer flow that consists of a inner upward
flow and outer downward flow that is driven by the combined
effect of temperature and salinity on the water density. In allthese cases, measurements have captured not only the flow
directions but also the velocity magnitudes and peak locations
with a good agreement with the DNS.

Figure 6. (a) Velocity (𝑤) profiles for 𝑇∞ = 10◦ 𝐶 and 𝑆∞ =0 (black), 𝑇∞ = 4◦ 𝐶 and 𝑆∞ = 0 (green) and 𝑇∞ = 4◦ 𝐶 and𝑆∞ = 35 g/kg (purple) obtained at 𝑧˜ = 58mm for downwardflowing boundary layers and 𝑧 = 58mm for upward flowing
boundary layer flows. (symbols- experiments, solid lines-nonlinear DNS, dashed lines-linear DNS)

Figure 8 illustrates how ice melt-rates (estimated by locating the ice-water interface as a function of time) comparewith the melt-rates obtained from DNS for 5 different am-

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

Figure 7. Temperature (𝑇) profiles for 𝑇∞ = 10◦ 𝐶 and 𝑆∞ =0 (black), 𝑇∞ = 4◦ 𝐶 and 𝑆∞ = 0 (green) and 𝑇∞ = 4◦ 𝐶 and𝑆∞ = 35 g/kg (purple) obtained at 𝑧˜ = 58mm for downwardflowing boundary layers and 𝑧 = 58mm for upward flowing
boundary layer flows. (symbols- experiments, solid lines-nonlinear DNS, dashed lines-linear DNS)

Figure 8. (c) Ice melt rate profiles for 𝑇∞ = 20◦ 𝐶 and 𝑆∞ = 0(Blue), 𝑇∞ = 15◦ 𝐶 and 𝑆∞ = 0 (Red), 𝑇∞ = 10◦ 𝐶 and 𝑆∞ = 0(Black), 𝑇∞ = 4◦ 𝐶 and 𝑆∞ = 0 (Green) and 𝑇∞ = 4◦ 𝐶 and 𝑆∞ =35 g/kg (Purple). Here experimental results are in symbols,
DNS with linear equation of state in dashed lines and DNS
with non-linear equation of state in solid lines.

bient conditions. It is observed that the melt-rates increase
with increasing ambient water temperature and salinity. For
downward boundary layer flows the melt-rate increases withincreasing 𝑧, whereas it decreases with increasing 𝑧 for upward
boundary layer flows. These trends can be explained using
equation 8, where ablation velocity 𝑉 is directly proportional
to 𝜕𝑇/𝜕𝑥 at the interface. Figure 7 reveals that the temperature
gradient at the interface is much higher for the 𝑇∞ = 10◦ 𝐶 casecompared to the 𝑇∞ = 4◦ 𝐶 cases due to a higher temperature
difference between the interface and ambient water. Therefore, ice melting is higher in the 𝑇∞ = 10◦ 𝐶 case compared to
𝑇∞ = 4◦ 𝐶 cases. Furthermore, due to freezing point depression(described by the equation 5), as the water salinity increases,
ice melts at temperatures below 0◦ 𝐶. As a result, this increases
the temperature difference between the interface and ambient
water and consequently increases the temperature gradient at
the interface. This is apparent when the 𝑇∞ = 4◦ 𝐶 and 𝑆∞ = 0
case is compared to the 𝑇∞ = 4◦ 𝐶 and 𝑆∞ = 35 g/kg case infigure 7. Therefore, for the 𝑇∞ = 4◦ 𝐶 and 𝑆∞ = 35 g/kg casethe melt rate is higher compared to the 𝑇∞ = 4◦ 𝐶 and 𝑆∞ = 0
case in figure 8. Further, the thermal boundary layer thickness
grows in the flow direction, decreasing 𝜕𝑇/𝜕𝑥 in that direction.
That is, although the thickness of the BL increases, the temperature difference remains constant, and thus the ice ablation
velocity decreases in the flow direction.
From figure 8 it can be observed that both experiments and
DNS results capture these trends while also showing reasonableagreement with each other.

equation of state, as compared to a linear EoS or the solid wall
similarity solution. The reason for this is ice melting process
is driven by the nonlinear variation of water density within
the boundary layer. Melt rates of the ice were quantified bytracking the interface, and these measurements also match reasonably well with the DNS meltrates. The effect of salinityof ambient water on boundary layer dynamics and ice melting
process was also presented.

REFERENCES
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Conclusion
A molecular tagging based simultaneous velocity and temperature measurement methodology was developed to experimentally measure the phase change boundary layer flow along a
melting vertical ice face. Boundary layer velocity and temperature measurements show good agreement with DNS. It can beobserved that experimental velocity and temperature measurements match better with the DNS that employed a nonlinear

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12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022
Mondal, Mainak, Gayen, Bishakhdatta, Griffiths, Ross, Kerr,
Ross et al. 2019 Ablation of sloping ice faces into polar
seawater .
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flow and heat transfer about a flat plate parallel to the direction of the generating body force. Tech. Rep.. National
Aeronautics and Space Administration Cleveland Oh Lewis
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Kerr, Ross C & McConnochie, Craig D 2015 Dissolution of avertical solid surface by turbulent compositional convection.
J. Fluid Mech. 765, 211.
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wall plume from a vertically distributed source of buoyancy.
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McDougall, Trevor J & Barker, Paul M 2011 Getting started
with teos-10 and the gibbs seawater (gsw) oceanographic
toolbox. Scor/Iapso WG 127, 1–28.

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