2.1. Governing non-dimensional numbers

For a single-component flow, say, driven only by the difference between ambient and interface temperature ( $T_\infty$ and $T_i$ , respectively), the relevant non-dimensional parameters are the Prandtl and Grashof numbers:

(2.1) \begin{equation} \textit{Pr} = \frac {\nu }{\kappa _T}\quad \textrm {and}\quad \textit{Gr}_z = \frac {z^3 g\alpha (T_\infty -T_i)}{\nu ^2}, \end{equation}

where $\nu$ is the kinematic viscosity of the fluid, $z$ is the vertical distance with origin at the bottom (see figure 1 ai and 1 aii), $g$ is the gravitational acceleration and $\alpha \equiv -(1/\rho )(\partial \rho /\partial T)$ is the thermal expansion coefficient. It is also common to define $\textit{Gr}_H$ based on the total height of the ice face $H$ . Based on $H=8.5$ cm for our set-up, we have $ 3.6\times 10^6 \lessapprox \textit{Gr}_H \lessapprox 9.7\times 10^7$ and $7 \lessapprox \textit{Pr} \lessapprox 13$ . Also, it is interesting to represent $\textit{Gr}_H$ as a ratio of viscous to buoyancy time scales: $\sqrt {\textit{Gr}_H} = (H^2/\nu )/\sqrt {H/g'}$ , where reduced gravity $g':=g\alpha (T_\infty -T_i)$ , and $\sqrt [4]{\textit{Gr}_H} = H/\sqrt [4]{\nu ^2 H/g'}$ as a ratio of length scales (where the denominator is the viscous-gravity length scale $l_T$ in (A6) without the numerical factor and $z \mapsto H$ ). For example, in our case $1900\lessapprox \sqrt {\textit{Gr}_H} \lessapprox 9850$ and $45 \lessapprox \sqrt [4]{\textit{Gr}_H} \lessapprox 100$ , which further highlights the importance of buoyancy.

Addition of a second component, say salinity, to the existing temperature adds two dimensional quatities: $S_\infty$ or the difference of far-field to interface salinity $(S_\infty -S_i)$ and $\kappa _S$ , giving rise to two more non-dimensional groups:

(2.2) \begin{equation} {\varLambda } \equiv \frac {\alpha \rho _0 (T_\infty -T_i)}{\beta \rho _0(S_\infty -S_i)}=\frac {\Delta \rho _T}{\Delta \rho _S}\quad\textrm {and}\quad \textit{Sc} = \frac {\nu }{\kappa _S}\quad\textrm {or}\quad \textit{Le} = \frac {\kappa _T}{\kappa _S}. \end{equation}

These are, respectively, the ratio of density changes owing to temperature and salinity, and Schmidt and Lewis numbers, where $\beta \equiv (1/\rho )(\partial \rho /\partial S)$ is the haline contraction coefficient and $\rho _0$ is reference density. For our cases ${\varLambda }\approx 1/40$ , $\textit{Sc}\approx 2000$ and $\textit{Le} \approx 200$ .

For nonlinear density variations, Gebhart & Mollendorf (Reference Gebhart and Mollendorf1977) and Carey & Gebhart (Reference Carey and Gebhart1982b ) introduced another non-dimensional number:

(2.3) \begin{equation} R = \frac {T_\textit{md}(S_\infty )-T_\infty }{T_i-T_\infty }, \end{equation}

where $T_\textit{md}$ is the temperature where the maximum density occurs. Parameter $R$ is a stronger function of salinity at lower $S$ (see figure 1 b and 1 c). This is because, at low $S$ , the value of $T_\textit{md}$ deviates significantly from $T_i$ as $S$ decreases and it provides an indirect measure of how significant the nonlinear effects of temperature are on the density for a given $T_i$ and $T_\infty$ . Further, $R$ is useful to predict the direction of flow for different ambient conditions when interface conditions are known (Gebhart & Jaluria Reference Gebhart and Jaluria1988), especially for low-salinity cases. These flow regimes are now discussed.

2.2. Flow regimes

When an ice face is in contact with saline water, the water adjacent to the ice–water interface is subjected to cooling and dilution. Cooling induces a thermal BL within which the temperature varies from the freezing point temperature $T_\textit{fp}(S_i)$ (shown in figure 1 b) to the ambient water temperature ( $T_\infty$ ). Owing to dilution, a salinity BL forms where water gets less saline closer to the ice due to freshwater released from the melting ice. These thermal and saline BLs are depicted in figure 1(ai) and 1(aii). Importantly, since salt has very low diffusivity compared with heat ( $\textit{Le} =\kappa _T/\kappa _S \approx 200$ ), the saline BL thickness ( $\delta _S$ ) is about 14 times smaller (assuming $\delta _S/\delta _T\approx \textit{Le}^{-1/2}$ , which is discussed in § 6) compared with the thickness of the thermal BL ( $\delta _T$ ). Although the contribution of salinity to the change in density is large compared with the temperature ( ${\varLambda }\approx 1/40$ ), owing to the low diffusivity of salt, density variations due to salt only affect a very thin layer of water closest to the interface. On the other hand, despite having a relatively small contribution to the density variation, the thermal BL affects density variations across a broader domain than the salinity BL.

Josberger & Martin (Reference Josberger and Martin1981) used experiments and existing similarity solutions to map the BL forming next to a melting vertical ice face. The mapping was carried out on a temperature–salinity ( $T_\infty ,\,S_\infty$ ) diagram like that shown in figure 1(b). Although the map was prepared mostly for turbulent flows, as discussed later, some features are relevant for laminar flows. Importantly, we draw connections between this map on the ( $T_\infty ,\,S_\infty )$ plane and the more informative non-dimensional $({\varLambda },\,\textit{Le})$ plane description of Nilson (Reference Nilson1985).

In figure 1(b), the temperature at maximum density for each salinity is the line $T_\textit{md}$ (in blue colour, where $R=0$ ; equation (2.3)) and the freezing point temperature line is $T_\textit{fp}$ (in black colour) that separates the ice and water states. Depending on where $T_\infty$ and $S_\infty$ lie on the T–S diagram, Josberger & Martin (Reference Josberger and Martin1981) report different BL regimes. When the ambient conditions are in region I, enclosed by $T_\textit{md}$ and $T_\textit{fp}$ , $\varLambda$ is negative. This is because $\Delta \rho _T := \alpha \rho _0 (T_\infty -T)$ is negative as $\alpha$ is negative in this region, and $\Delta \rho _S:= {\beta } \rho _0 (S_\infty -S)$ is always positive because freshwater is always less dense relative to saltwater at the same temperature. That is, both salt and temperature produce less dense water, favouring an upward uni-directional BL flow. Due to this uniformity of the flow, similarity solutions exist for the laminar BL equations in this region (see (A1a ) and (A1b )).

Figure 2.

(a) Relative buoyancy–diffusivity diagram ( $\varLambda {-}\textit{Le}$ ) showing flow regimes. Here () $\varLambda \textit{Le}= 1$ , () $\varLambda \textit{Le}^{1/2} = 1$ and () $\varLambda \textit{Le}^{1/3} = 1$ . Contours represent $\varLambda \textit{Le}^{1/2}$ . Symbols: filled $\diamond$ and filled $\vartriangleleft$ show experimental data for cases 5 and 6, respectively. Also shown are laminar bi-directional experimental results of Josberger & Martin (Reference Josberger and Martin1981) in filled yellow $\triangledown$ and of Carey & Gebhart (Reference Carey and Gebhart1982a ) in filled green $\triangle$ for comparison. Analytical values of $\varLambda$ obtained using (2.6) are shown in the colour bar for ambient temperatures $T_\infty =-2^{\,\circ} \text{C}$ to $4^{\,\circ} \text{C}$ (blue to red colour) expected in the context of Antarctic ice melting with higher (lower) temperatures in red (blue). (b) Zoomed-in view of (a). (c) Temperature $T_i$ obtained from solving (2.6) for different ambient conditions (labels on the curves denote salinity). Also shown are the experimental results of Carey & Gebhart (Reference Carey and Gebhart1982a ) (filled green $\triangle$ ) and numerical results of Carey & Gebhart (Reference Carey and Gebhart1982b ) (open green $\triangle$ ) for $S_\infty = 10$ g kg⁻¹. The experimental data of Josberger & Martin (Reference Josberger and Martin1981) are indicated by filled yellow $\triangledown$ (for $S_\infty = 34.4$ g kg⁻¹). The DNS results of this study for case 5 and case 6 are shown as open $\diamond$ and $\vartriangleleft$ , respectively (both for $S_\infty = 34$ g kg⁻¹).

When the ambient condition shifts into region II, having a high salinity and relatively low temperature, $\Delta \rho _T$ gradually becomes negative, resulting in ${\varLambda } \gt 0$ . In this region, buoyancy forces from salinity (closer to the wall) and temperature (farther from the wall) oppose each other. This can result in bi-directional or counter-current flows. However, Carey & Gebhart (Reference Carey and Gebhart1982b ) showed through numerical calculations that even with opposing buoyancy forces, uni-directional upward flow can exist for some cases within the region where the salinity buoyancy force dominates over the thermal buoyancy force. This shows that the limits of counter-current flow are less apparent on a $T_\infty {-} S_\infty$ diagram. Nilson (Reference Nilson1985) obtained these limits on the $({\varLambda },\,\textit{Le})$ plane as shown in figure 2(a). Starting with a dominant upwards salinity-driven flow, an incipient temperature-driven downward flow (i.e. an overall bi-directional flow) appears when ${\varLambda } \textit{Le}\approx 1$ . If, however, there is a dominant temperature-driven downward flow an incipient salinity-driven upward flow adjacent to the wall will result when ${\varLambda } \textit{Le}^{1/3} \approx 1$ . Appendix A provides a derivation of these limits starting from the BL equations. These limits of upward and downward flows are presented in figure 2(a). They divide the $({\varLambda },\,\textit{Le})$ plane into three regions: up-flow; counter-flow or bi-directional flow; and down-flow.

An estimate of the demarcating line between the upward saline-buoyancy-dominated and the downward thermal-buoyancy-dominated regimes can be obtained quite straightforwardly by evaluating the relative buoyancy strengths. Net buoyancy is not governed only by $\Delta \rho _T$ and $\Delta \rho _S$ , i.e. $\varLambda$ , rather by the density difference acting over the respective BL thickness $\delta _T$ and $\delta _S$ . Therefore, it is the ratio

(2.4) \begin{equation} \frac {\Delta \rho _T \,\delta _T}{\Delta \rho _S\,\delta _S}\sim {\varLambda } \textit{Le}^{1/2} \end{equation}

that determines the relative strength of temperature and salinity within the BL. Not surprisingly, the line ${\varLambda } \textit{Le}^{1/2}$ bisects the bi-directional flow region in figure 2(a) as shown in the blue dashed line. In figure 2(a), we compare the experimental results of Josberger & Martin (Reference Josberger and Martin1981) and Carey & Gebhart (Reference Carey and Gebhart1982a ), where laminar bi-directional or marginally bi-directional flows were observed, in the $\varLambda$ – $\textit{Le}$ plane. As shown in the figure, these data fall within the region corresponding to counter-flow, providing confidence in the validity of the formulation.

Since natural convection adjacent to Antarctic ice shelves is one of the motivations of the present work, in the following we estimate the locations of Antarctic data within the $({\varLambda },\,\textit{Le})$ plane in figure 2(a).

2.3. Laminar flow at Antarctic conditions

Only $T_\infty$ and $S_\infty$ conditions are known for Antarctic conditions. If, however, one can estimate $T_i$ and $S_i$ , $\varLambda$ and $\textit{Le}$ can be estimated. Towards this end, we assume that within the laminar BL both temperature and salinity vary linearly across the respective BLs, and, therefore, (1.1) and (1.2) can be written in terms of interface and far-field conditions:

(2.5) \begin{equation} \rho _{\textit{ice}} L V \approx \rho _w c_w \kappa _T \frac {(T_\infty -T_i)}{\delta _T}\quad \textrm {and}\quad \rho _{\textit{ice}} S_i V \approx \rho _w \kappa _S \frac {(S_\infty -S_i)}{\delta _S}. \end{equation}

Dividing the above two equations, substituting $({\delta _T}/{\delta _S})^2 = {\kappa _T}/{\kappa _S}$ and replacing $S_i$ with freezing point depression $T_i/a_s$ results in

(2.6) \begin{equation} T_i^2\left (\frac {\textit{Le}^{1/2} c_w}{a_s L}\right ) -T_i\! \left (\frac {1}{a_s} +\frac {\textit{Le}^{1/2} c_w T_\infty }{a_s L}\right ) + S_\infty =0. \end{equation}

The above quadratic equation can be solved for $T_i$ (and hence $S_i$ ) by knowing the other quantities. In passing we note that a quadratic equation for $S_i$ for turbulent flow (using different assumptions) is obtained by McPhee, Morison & Nilsen (Reference McPhee, Morison and Nilsen2008). The approximate values of $\kappa _T=1.4\times 10^{-7}$ (Nayar et al. Reference Nayar, Panchanathan, McKinley and Lienhard2014) and $\kappa _S=7\times 10^{-10}$ (Caldwell Reference Caldwell1974) give $\textit{Le}=200$ . The roots of (2.6) between $0$ and $a_s S_\infty$ are plotted in figure 2(c) along with some existing data for different salinities (Josberger & Martin Reference Josberger and Martin1981; Carey & Gebhart Reference Carey and Gebhart1982a , Reference Carey and Gebhartb ). These show a reasonable agreement between the solution of (2.6) and the data. Note that this root is always zero for the freshwater case ( $S_\infty =0$ ). For saltwater, $T_i$ increases with increasing $T_\infty$ . Further, for a given $T_\infty$ , $T_i$ decreases with increasing $S_\infty$ . We should mention that the validity of the assumption $\delta _T / \delta _S = \textit{Le}^{1/2}$ leading to (2.6) may not be theoretically accurate under certain ambient conditions where the flow becomes bi-directional resulting in two distinct velocity scales within the BL, as discussed further in § 6. However, as shown in figure 2(c), the formulation in (2.6) is able to reproduce both numerical and experimental observations, even when the flow becomes bi-directional (because the two aforementioned velocity scales are of similar magnitude; see § 6).

At Antarctic ice shelves, the water temperature can range from approximately $2 ^{\,\circ} \text{C}$ to $-2^{\,\circ} \text{C}$ (Jenkins et al. Reference Jenkins, Dutrieux, Jacobs, McPhail, Perrett, Webb and White2010; Boehme & Rosso Reference Boehme and Rosso2021). The values of $\varLambda$ estimated for this condition are plotted in figure 2(a) using colour ranging from blue to red corresponding to $T_\infty =-2^{\,\circ} \text{C}$ to $4 ^{\,\circ} \text{C}$ . A zoomed-in view is given in figure 2(b). The values lie in the inner-dominated bi-directional or counter-flow region for this entire temperature range. Also, as $T_\infty$ decreases, the data deviate from the ${\varLambda } \textit{Le}^{1/2}$ line and become more inner-dominated. Note that figure 2(a) also shows two of our laboratory experimental data in symbols (see table 1, cases 5 and 6).

We now proceed to present these laboratory experiments, and especially the modifications made to the MTV/MTT technique allowing us to measure temperature and velocity in the ice-melt BL.

3.1. Experimental set-up

The experiments are performed in a small water tank with $9$ cm depth and a square base of $18$ cm in length and width (figure 3 a). One sidewall of the tank consists of a fused silica window to allow the ultraviolet (UV) laser beam needed for MTV to enter the tank. At the beginning of the experiments, an 85 mm high and 35 mm thick ice block spanning the width of the tank is placed at one end of the tank (figure 3 b), and magnets are used to secure the ice block to the endwall.

Table 1.

Experimental cases. The far-field conditions ( $T_\infty$ and $S_\infty$ ) and environments for which these experiments are performed, as well as symbols used for each case.

Figure 3.

(a) Schematic diagram of the experimental set-up and the arrangement of its components. (b) Schematic of the water tank and ice block.

As schematically shown in figure 3(a), the full experimental set-up mainly consists of the water tank, a pulsed UV laser and an intensified gated camera. A digital delay generator is used to synchronise the camera and the laser. The laser beam is guided to the fused silica window using an optics arrangement consisting of UV mirrors, and a pinhole arrangement is used to obtain a thin laser line. Images captured by the camera are logged using a host computer. Experiments with low-temperature conditions (below $4^\circ\, \textrm{C}$ ) are performed in a temperature-controlled room (not shown here) where temperature can be maintained at a constant value ranging from $-15^{\,\circ} \text{C}$ to $25^{\,\circ} \text{C}$ .

3.2. Flow measurement techniques

Measuring the velocity and temperature in the BL flow that forms next to the phase-change ice–water interface with high resolution is challenging for several reasons. Firstly, the thickness of this BL flow is small (of the order of a millimetre), and measurements have to be within (and adjacent to) this BL. Secondly, this BL flow is formed as a result of a free convection process, and, consequently, the velocities are small, about a few mm s⁻¹. This enforces stringent requirements on the measurement system if we acknowledge that simultaneous velocity and temperature measurements are required.

Figure 4.

(ai) An image pair captured by the camera with $11$ ms interframe delay. First image with 1 ms and second image with 2 ms camera exposure. (aii) Timing chart for phosphorescence intensity ( $I$ ) decay and image pair acquisition. Here, $t_0$ is the time between the laser pulse and first image acquisition, $\Delta t$ is interframe delay, $\delta t_1$ is first image exposure and $\delta t_2$ is second image exposure with $t_0 \ll \Delta t$ . (b) Molecular tagging melt-rate measurements – tagged line intensities.

Considering the above requirements in this study, MTV and MTT are selected. The main advantage of these molecular tagging techniques is that they use molecules as tracers (e.g. Hu & Koochesfahani Reference Hu and Koochesfahani2006), enabling them to be used to obtain measurements close to the interface with high resolution. Furthermore, in this study, the standard MTV/MTT technique is modified to obtain temperature profiles in the BL flow for simultaneously measuring small velocities. Modifications to the standard MTV/MTT implementation adopted in this study are presented later in this section.

Briefly, MTV uses phosphorescent molecules excited by a UV laser as a Lagrangian tracer, and velocities are determined by two images separated by a specified delay time as acquired by a gated intensified camera. On the other hand, MTT leverages the fact that the decay rate of the phosphorescence is uniquely temperature-dependent. In this study for both MTV and MTT, a phosphorescent supramolecular complex is used as a molecular tracer. This complex consists of 1-bromonaphthalene (1-BrNp), a certain alcohol (cyclohexanol) and an aqueous solution of maltosyl-β-cyclodextrin (Mβ-CD). In this triplex, 1-BrNp acts as the luminophore while cyclohexanol and cyclodextrin insulate the luminophore from oxygen quenching and thus generating long-lived phosphorescence (Gendrich et al. Reference Gendrich, Koochesfahani and Nocera1997). For the present study, $1 \times 10^{-5}\,\rm M$ molar concentration of 1-BrNp is used while the molar concentrations of cyclohexanol and cyclodextrin are $0.05 \,\rm M$ and $3 \times 10^{-4}\,\rm M$ , respectively. A 308 nm excimer laser is used for excitation for the higher-temperature cases, and a 266 nm Nd:YAG laser for temperatures of $4^\circ$ C or below for the cold-room experiments. The choice is merely due to convenience; and although a 308 nm excimer laser was preferred owing to higher power, it is not portable and could not be located closer to the cold room. A Princeton Instruments PI-max4 1024i intensified gated camera that captures five frames per second (2.5 image pairs per second) with an image resolution of $1024\times 1024$ pixels is used for image acquisition.

Figure 4(ai) shows a sample phosphorescent image pair used to acquire velocity and temperature. The laser enters from the right of the image and stops after impinging on the ice surface close to the left edge. The vertical pixel displacement of the maximum intensity pixels along the line is used to calculate the Lagrangian displacement of the tracer molecules. From the knowledge of displacement and time delay, the vertical velocity profile is obtained.

Once phosphorescent molecules are excited by the UV laser beam, phosphorescent intensity decays exponentially with time, as shown in figure 4(aii). This decay can be characterised by the time constant of the decay, i.e. phosphorescent lifetime $(\tau )$ , which depends on the temperature. In the standard MTT technique $\tau$ is calculated from phosphorescence signals collected from the two images taken a time $\Delta t$ apart: $\tau = \Delta t/(\ln (S_1/S_2))$ , where $S_1$ and $S_2$ are image 1 and 2 intensities taken with the same exposure time $\delta t$ . In our case, where velocities are small ( $\lessapprox 8$ mm s⁻¹), we require larger $\Delta t$ for sufficient pixel displacement (at least 5 pixels) to obtain velocity in MTV. This implies that the second image intensity $S_2$ will be too small (within the noise) to be useful as the phosphorescence intensity decays exponentially. To overcome this, the $S_2$ camera exposure time ( $\delta t_2$ ) is increased compared with the $S_1$ exposure time ( $\delta t_1$ ) needed to obtain velocities. This, however, implies that we cannot use the standard MTT methodology, because $\delta t_1 \neq \delta t_2$ . Thus, the MTT equation is modified as follows. A phosphorescence signal $S$ , collected by a camera for a shutter opening at time $t_0$ (after the laser is fired), is exposed for a time $\delta t$ given by $S=\int _{t_0}^{t_0+\delta t} I_0 {\rm e}^{-t/\tau } \,$ d $t$ , where $I_0$ is the initial light intensity at time $t_0$ . This yields $S= I_0 \tau (1-{\rm e}^{-\delta t/\tau }){\rm e}^{-t_0/\tau }$ . Using the two signals ( $S_1$ and $S_2$ ) from the two images separated by the interframe delay, $\Delta t$ , the phosphorescent lifetime ( $\tau$ ) is then obtained using

(3.1) \begin{equation} \frac {S_2}{S_1}= \left (\frac {1-{{\rm e}}^{{-\delta t_2}/{\tau } }}{1-\textrm {e}^{{-\delta t_1}/{\tau } }}\right ) {{\rm e}}^{{-\Delta t}/{\tau } }, \quad \textrm {where} \quad \delta t_2 \gt \delta t_1. \end{equation}

See also figure 4(aii), where $t_0$ is the time between the laser pulse and first image acquisition, $\Delta t$ is interframe delay, $\delta t_1$ is first image exposure and $\delta t_2$ is second image exposure with $t_0 \ll \Delta t$ . Note that when $\delta t_2 = \delta t_1$ , (3.1) reverts to the standard MTT equation discussed above. Equation (3.1) is numerically solved to determine $\tau$ . These $\tau$ values are then converted to the temperature profile using a calibration curve, where MTT images are obtained across a range of known temperatures. This calibration procedure is described in Appendix C.

3.3. Making bubble-free ice

The ice is created in an acrylic mould using slightly heated water (of approximately $70^{\,\circ} \text{C}$ to reduce bubble formation) and keeping it in a freezer typically overnight. Once the ice is frozen, small layers of sufficiently diluted phosphorescent chemical solution are poured on the ice face (where melting happens) and allowed to freeze again. This ensures that when the ice melts, the melt plume solution emits sufficiently strong MTV and MTT signals during the experiments. Since the MTV chemical molecules can lower the freezing point of ice, a freezing point depression calculation is carried out as detailed in Appendix B. This shows that the MTV chemicals do not cause freezing point depression of more than $0.1^{\,\circ} \text{C}$ . Furthermore, freezing thin layers of water reduces the chances of entrapped air bubbles as the freezing predominantly takes place from the bottom and the thickness of the layers cannot accommodate large bubbles. However, introducing these layers with MTV chemicals makes the ice block slightly opaque as can be seen in figure 12(ci) and 12(cii). After removing the ice block from the mould, a slightly warm (approximately $50^{\,\circ} \text{C}$ ) metal plate is used as a planer to obtain a smooth ice face by melting off irregularities.

3.4. Flow measurement procedure

Experiments are conducted under six different ambient water temperatures $T_\infty$ and salinity $S_\infty$ conditions as listed in table 1. This table also contains information about the lasers used and the environments in which experiments were performed, as well as the symbols for the plots in later sections. The temperature and salinity of the water everywhere in the tank are maintained at these specific values before placing the ice block inside the water. A pinhole of diameter $0.3$ mm is used to obtain a thin laser line (see figure 3 a). The vertical location of the tagged line from the bottom of the tank is also listed in table 1 for each case (see figure 4 ai). These locations are chosen such that the measurement location is nominally two-thirds of the total ice–water interface height (85 mm) in the flow direction. Note that for $S_\infty =0$ and $T_\infty \gt 4^{\,\circ} \text{C}$ , the flow is downwards. For all other cases the flow is more complicated but there is an upward flow close to the melting ice surface.

Experimental cases 1, 2 and 3 are performed under relatively higher temperature conditions (see table 1), and the data are acquired after 60 s from the time the ice block was placed in the water tank. This initial time allows for any disturbances caused during the placement of the ice block in the water to settle, and ensures that the BL velocity and ice ablation velocity reach a steady state. The DNS findings of Gayen et al. (Reference Gayen, Griffiths and Kerr2016) indicate that the ice ablation velocity achieves a statistically steady state after ${\approx} 20$ s. Furthermore, even at the lowest BL velocity (of $\approx$ 0.2 cm s⁻¹), the BL flow can traverse more than the entire height of the ice block within the initial 60 s period, demonstrating that this time is sufficient for the flow to stabilise. It was observed that the ice block temperature was approximately $-8^{\,\circ} \text{C}$ before being placed in water, and it dropped further to $-6^\circ \, \text{C}$ before data acquisition began. The sampling frequency of data acquisition is 2.5 image pairs per second (5 f.p.s.). This means that after the initial period, velocity and temperature profiles are acquired every 0.4 s for about 3 minutes. During the experiment, ambient water temperatures are recorded using two thermistors placed 15 and 100 mm from the interface to verify that the ambient water temperature was approximately constant at the specified value everywhere in the tank. In this study, the reported velocity and temperature profiles were obtained between 60 and 120 s after placing the ice block in water.

The procedure just described is followed for the low-temperature cases (cases 4, 5 and 6) but the data acquisition interval is 120–240 s. Note that in the low-temperature cases the establishment of a steady BL flow takes longer, and the temperature in the tank remains approximately constant at specified temperatures for a longer period. We note that, since most of these experiments are conducted below the dew point of air, condensation occurs whenever the water tank and optics inside the cold room are exposed to the outside air. This significantly disrupts the experiments, as the presence of condensation on the cold-room window, water tank or any optics leads to an unacceptable degradation in the quality of measurements. This presented a major challenge that is mitigated by scheduling the experiments on days with very low dew points and minimising the number of times the cold-room door is opened.

Before and after conducting the ice block experiments, in situ pre-calibration and post-calibration image pairs are obtained for the specified temperature values. This is done because the calibration curve may have shifted slightly from the original calibration due to the ageing of the solution. These post- and pre-calibration points enable us to obtain temperature measurements from a normalised calibration curve that is immune to ageing effects as described in Hu & Koochesfahani (Reference Hu and Koochesfahani2006). See Appendix C for further details of the MTT calibration procedure.

Once obtained, the phosphorescent images are processed using Matlab. The image pre-processing procedure is outlined in Appendix D. In the phosphorescent images (as shown in figure 4 ai), the MTV/MTT tagged line is parallel to horizontal lines of image pixels. The pixel with maximum intensity in each vertical pixel line is considered for obtaining the vertical location of undeformed and deformed tagged lines. In order to obtain the location of the line with subpixel accuracy, a second-order polynomial is fitted in the neighbourhood of the maximum intensity pixel, and the location of the peak of that polynomial is taken as the vertical location of the tagged line in the image. Once the location of the tagged line is found in both images of the image pair, the pixel displacement of the tagged line is obtained. This is then converted to actual displacement using the pixel-to-length ratio obtained during the experiments using a ruler placed in the water tank. Since the timing parameters of imaging are known, flow velocity is calculated from displacement.

In MTT, the pixel with maximum intensity in each vertical pixel line is considered in both images of the image pair and the intensity ratio is obtained along the tagged line. Once the intensity ratio is obtained, (3.1) is used to obtain phosphorescent lifetime ( $\tau$ ), and then a calibration curve (as explained in Appendix C) is used to obtain temperature profiles.

3.5. Melt-rate measurements

Separate experiments are carried out to measure the ice melting rate for high-ambient-water-temperature cases (cases 1, 2 and 3) that have significantly higher melt rates compared with low-temperature cases. This involved the same set-up used for the velocity and temperature measurements, but no laser is used. Rather, an intense white light source is used to illuminate the melting ice face. A grid paper is placed on the back wall, and the camera image depth of field is increased to obtain images where the ice face and the grid paper are visible. Images of melting ice are obtained at a frame rate of 5 Hz for about 5 min. The receding position of the ice–water interface is therefore obtained as a function of time (using image analysis), and the ice melt rate (reduction of the ice-face thickness per unit time) is estimated. In accordance with the MTV/MTT experiments, the melt-rate data are also obtained within the $60{-}120$ s window to ensure steady-state melting. The melt-rate measurements for the low-temperature cases could not, however, be obtained with sufficient accuracy using this method because the melt rates are very small, and interface movement was difficult to detect.

To address this challenge, a different method that enables melt-rate measurements simultaneously with velocity and temperature measurements is developed. Here, the first image of each phosphorescent image pair is used, and the location of the tagged line is obtained. The ice–water interface is detected from the large gradient of phosphorescent intensity observed when the tagged line passes from water to ice, as shown in figure 4(ai). Once the interface location is known in each image, the ice melt rate is calculated with the timing information, i.e. slope of the black line in figure 4(b). Further details of melt-rate measurements using both methods are presented in Appendix E.

5.1. Freshwater ( $S_\infty =0$ ) cases: effect of $T_\infty$ and nonlinear density

Cases 1–4 are for freshwater. Figure 1(c) shows where these four freshwater cases lie on a T–S diagram. Cases 1, 2 and 3 ( $T_\infty =20$ , $15$ and $10^{\,\circ} \text{C}$ ) lie at the edge of region II (see figure 1 b) where bi-directional flows could be observed, but which is less likely for zero ambient salinity and sufficiently away from $4^{\,\circ} \text{C}$ . In figure 5(a) and 5(b), respectively, velocity and temperature profiles from experiments are presented along with nonlinear EoS DNS profiles. Cases 1–3 in figure 5(a) show a uni-directional downward flow with velocity magnitude increasing with increasing far-field temperature. This increase is due to the increase in density difference between far field and interface. Case 4 shows a uni-directional upward flow. The experimental velocity and temperature data show a reasonable agreement with those of the DNS, with the differences primarily attributable to the recirculation present in the experiments and to the estimation of the virtual origin of the BL. The effects of this recirculation are most significant in case 1, where the flow velocity is highest.

The ‘flow reversal’ in case 4, where the density variation is purely from temperature, is a symptom of the nonlinear variation of water density. Here, the flow is upward because water has a density maximum at $4^{\,\circ} \text{C}$ and the melt-water released from the ice interface is less dense than ambient water. These flow behaviours can be understood by referencing the density variation of freshwater with temperature (see figure 1 c) and the average thermal expansion coefficient $\alpha _a \equiv -(\rho _\infty -\rho _i)/(\rho _0(T_\infty -T_i))$ values in table 2, where $\alpha _a$ becomes negative in case 4, indicating that the average density within the BL is lower than that of the ambient water at this ambient temperature. Since all these cases are for freshwater, one does not observe any upward inner layer, and the flow is uni-directional downward. Case 4 lies on the boundary between regions I and II. This suggests that freshwater flow reversal happens at an ambient temperature between $4$ and $10^{\,\circ} \text{C}$ , and hence line $T_\textit{md}$ (in figure 1 b) is not the exact boundary between upward and downward flows. From figure 5(a) it is also apparent that the peak location of the velocity profiles moves slightly further from the interface and the peak velocity magnitude decreases with decreasing $T_\infty$ . Consistently, in figure 5(b) the thermal BL thickness (say, $\delta _T$ ) increases with decreasing $T_\infty$ . This is a result of the decreasing heat transfer rate that reduces the temperature gradient ( $\partial T/\partial x$ ) as $T_\infty$ decreases as expected for a laminar BL. In fact, the peak vertical velocity $w_{\textit{max}} \sim [g\alpha (T_\infty - T_i)z]^{1/2}$ and the BL thickness $\delta \sim \delta _T \sim [\nu \kappa _T z /(g \alpha (T_\infty - T_i))]^{1/4}$ , which come from the order-of-magnitude matching of viscous diffusion and buoyancy force in the vertical momentum equation, $\nu w_{\textit{max}}/\delta ^2 \sim g \alpha (T_\infty - T_i)$ , and the vertical advection and thermal diffusion in the temperature equation, $w_{\textit{max}}/z \sim \kappa _T/\delta ^2$ (see also Appendix A and Ostrach Reference Ostrach1952; Kuiken Reference Kuiken1968; Josberger & Martin Reference Josberger and Martin1981; Carey & Gebhart Reference Carey and Gebhart1982b ). The data in figure 5(a) show expected trends of $w_{\textit{max}}$ and $\delta$ with $(T_\infty - T_i)$ . The variation with $z$ is discussed in the context of melt-rate scaling in § 6.

Figure 6.

Comparison of experimental (symbols), DNS with linear EoS (dashed lines) and DNS with nonlinear EoS (solid lines). (a,d,g) Velocity ( $w$ ) profiles for cases 3 , 2 and 1 , respectively. (b,e,h) Comparison of non-dimensional experimental and DNS (linear and nonlinear EoS) velocity ( $\tilde {w}^*\equiv (w z_1/2\nu )\textit{Gr}_{z_1}^{-1/2}$ ) profiles. (c,f,i) Non-dimensional temperature ( $\tilde {T}\equiv {(T_\infty -T)}/{( T_\infty -T_i)}$ ; see Appendix A, (A2)) profiles along with the analytical solution for $\textit{Pr}=10$ (black dashed lines) for cases 1, 2 and 3. The DNS results are presented for $z$ values of $35$ , $55$ and $75$ mm, and the colour gets darker as $z$ increases.

From figure 5, it is evident that both the velocity and temperature measurements are obtained very close to the ice–water interface (within 1 mm), and ambient velocity and temperature conditions are captured and show a reasonable agreement with DNS. Because of the higher temperature difference and limited size of the tank, in case 1, the temperature drops slightly from the specified ambient temperature over the duration of the experiments. This effect is apparent in figure 5(b).

Averaged velocity and temperature profiles obtained experimentally for cases 1, 2 and 3 are compared separately with the DNS in figure 6 to investigate the nonlinear effects further. Here, the DNS is run separately, first with a linear and then a nonlinear EoS relating temperature and salinity to density. The left-hand column of figure 6 shows dimensional vertical velocities whereas the middle and right-hand columns present non-dimensional velocity and temperature; rows top to bottom represent, in order, cases 1–3. Figure 6(a,d,g) shows a comparison between the experimental and DNS vertical velocity profiles. Note that nonlinear EoS DNS results are not shown for case 1. Here, the dashed (grey) lines are for the DNS using a linear EoS, while the nonlinear EoS results are in solid (blue) lines. The figure shows that the measured data are well within the range of velocities obtained from the DNS. We note that the precise velocity magnitudes are sensitive to the accurate choice of the $z$ location (virtual origin effect). As such, in figure 6 we show DNS velocity and temperature profiles at three different $z$ locations. It is well known that the BL does not start precisely at the ‘leading edge’ or the free surface, and a correction in terms of virtual origin is usually required.

Table 3.

Far-field conditions and corresponding non-dimensional parameters (experimental) $R$ and $\varLambda$ that characterise resulting BL flow.

Both the experimental and the DNS profiles are non-dimensionalised, and compared with the analytical similarity solution of Ostrach (Reference Ostrach1952) of the BL equations (see (A1) with $S=0$ ) for natural convection along an isothermally cooled vertical flat plate at a fixed Prandtl number of 10 and a fixed thermal expansion coefficient ( $\beta$ ) (i.e. linear EoS) in figure 6(b), 6(e) and 6(h) for cases 1, 2 and 3, respectively. Note that Ostrach (Reference Ostrach1952) does not account for the melting on the boundary conditions that can influence the near-wall behaviour (Carey & Gebhart Reference Carey and Gebhart1982b ) and also neglects the nonlinearity of the EoS, which can affect the convergence of similarity solutions if BL flow becomes bi-directional (Gebhart & Mollendorf Reference Gebhart and Mollendorf1977; Carey & Gebhart Reference Carey and Gebhart1982a , Reference Carey and Gebhartb ). The present DNS, however, accounts for both the melting BL as well as the nonlinearity.

In these figures, the abscissa ( $x$ ) is non-dimensionalised as $x/l_T = (x/z_1)(\textit{Gr}_{z_1}/4)^{1/4}$ , where $l_T := [4 \nu ^2 z_1/(g \alpha (T_\infty -T_i))]^{1/4}$ is here defined with $z_1=H-z$ rather than $z$ (as in (A6)) since the flow is downwards starting from the top of the ice face, and the vertical velocity $w$ on the ordinate is normalised as $\tilde {w}^*\equiv w/(4z_1 \nu /l_T^2) = (w z_1/2\nu )\textit{Gr}_{z_1}^{-1/2}$ . This analytical solution is used as a reference profile to observe the changes in velocity profiles due to changing $T_\infty$ and the nonlinear effect of temperature on density. It can be observed that with a virtual origin shift of 28 mm, non-dimensionalised experimental velocity profiles nearly coincide with the nonlinear EoS DNS profiles.

When non-dimensional peak velocities are compared across the three cases in figures 6(b), 6(e) and 6(h), the difference between peak velocities from measurements and linear analytical solution (as well as linear EoS DNS) increases with decreasing $T_\infty$ . As shown in table 3, the non-dimensional parameter $R$ (see (2.3)) also decreases with decreasing $T_\infty$ for these cases, indicating increasing nonlinear effects. This reduction of peak velocity demonstrates how the effects of nonlinearity of density variation become more significant as ambient temperature approaches the temperature corresponding to the density maximum ( $4^{\,\circ} \text{C}$ for freshwater). This is also apparent in the ( $\rho , T$ ) diagram of figure 1(c). For the low-temperature cases, the density difference between ambient water and water released from ice melting is smaller than what the linear EoS assumes. Hence, the linear EoS overestimates the velocity magnitude. Further, it is also noticeable that the peak velocity location of the nonlinear EoS DNS velocity profiles is in better agreement with the measured profile than the linear EoS DNS results or the similarity solution. This highlights the role of the nonlinear EoS in momentum transport. These differences are, however, less apparent in the temperature BL profiles shown in figures 6(c), 6( f) and 6(i) where the experimental and DNS profiles show a closer agreement. Within the thermal BL (say at $x/l_T \approx 1$ ) in figures 6(c) and 6( f), however, careful inspection reveals that the nonlinear data (solid lines) compare more favourably with the experimental data. This further highlights the apparent importance of the nonlinear EoS in numerical simulations. Nonetheless, as mentioned before, even for case 1, the differences in $\alpha$ and $\alpha _a$ suggest that one could improve the linear approximations with a different definition of $\alpha$ .

5.2. Effects of salinity: $T_\infty =4$ ( $S_\infty = 0$ and $34$ ‰)

Figure 7.

Experimental (a) velocity and (b) temperature BL profiles of case 4 : $S_\infty =0$ g kg⁻¹, $T_\infty =4^{\,\circ} \text{C}$ ; and case 5 : $S_\infty =34$ g kg⁻¹, $T_\infty =4^{\,\circ} \text{C}$ . Here solid lines represent nonlinear EoS DNS results.

In this section, velocity and temperature profiles obtained for cases 4 and 5 are compared to study the effects of $S_\infty$ . Here, $T_\infty$ is held constant at $4^{\,\circ} \text{C}$ . As depicted in figure 7, after introducing a salinity of 34 ‰ to freshwater, the velocity profile and melt BL dynamics are completely altered. The combined effect of both salinity and temperature governs the density in the far field as well as within the BL, causing the apparent complex behaviour of the BL flow. Instead of the fully upward flow observed for case 4, case 5 shows a bi-directional flow where a thin inner upward flow is accompanied by a broader outer downward flow.

To understand the combined effect of temperature and salinity that drives this BL, the DNS velocity profile for case 5 of figure 7(a) is re-plotted in figure 8(a); the corresponding density difference ( $\Delta \rho = \rho _\infty - \rho$ ) is presented in figure 8(b); and the salinity–temperature profile is plotted in figure 8(c). In all these profiles, points are marked at different distances from the interface and increasing marker size represents the increasing distance from the ice face. When the segment between the first and third markers from the interface in the velocity profile is considered, flow is predominantly upward close to the interface. This is apparent from the density profile in figure 8(b), where $\Delta \rho$ increases markedly as the wall is approached. When the same region is considered in the T–S diagram (figure 8 c), it is clear that this density change is principally due to the variation of salinity within that region. Although temperature changes slightly within that segment, its contribution to density change is negligible compared with salinity variation, as the density contours are mostly vertical in this region of the T–S diagram. Since the density close to the interface is smaller than that in the far field, an inner upward flow is observed within this segment. The flow is, however, mainly downward between the third and last markers. The reason for this is not obvious in the $\Delta \rho$ profile in figure 8(b) as the density variation is quite small owing to negligible salinity change in this region (outside the salt BL). If, however, this segment of the profile is enlarged, the T–S diagram (in figure 8 c) shows that the slightly curved density contours within this region induce a density increase due to decreasing temperature. Note that at $S_\infty =34$ ‰, the density monotonically decreases with increasing temperature. This is unlike the freshwater case where $\rho$ peaks at ${\approx} 4^{\,\circ} \text{C}$ . Although this density variation is quite small compared with the density variation due to salinity, it is present over a broader region and, as described in § 2.2, this leads to a downward outer flow.

Figure 8.

(a) Velocity ( $w$ ) and (b) $\Delta \rho = \rho _\infty -\rho$ from nonlinear DNS for case 5. Blue and red dashed lines represent, respectively, $\beta \rho _0(S_\infty -S)$ and $10\,\alpha \rho _0(T_\infty -T)$ . Here, the value of $\alpha \rho _0(T_\infty -T)$ is multiplied by a factor 10 for visualisation only. (c) Temperature–salinity curve of case 5 plotted on a T–S diagram with contours denoting the water density. In all these profiles, points are marked at different distances from the interface and increasing marker size represents increasing distance from the ice face.

Note that within the salinity BL (between the first and third markers from the interface), density changes by roughly 16 $\textrm {kg}\,\textrm {m}^{- 3}$ while within the rest of the BL where the downward flow can be observed, the density change is much smaller than 1 $\textrm {kg}\,\textrm {m}^{- 3}$ . The density difference contributions of salinity and temperature, $\beta \rho _0(S_\infty -S)$ and $\alpha \rho _0(T_\infty -T)$ , are shown in figure 8(b). The value of $\alpha \rho _0(T_\infty -T)$ is multiplied by a factor 10 for visualisation, suggesting that $\alpha \rho _0(T_\infty -T)$ values are small, i.e. $\Delta \rho _S$ $ \gg \Delta \rho _T$ , resulting in ${\varLambda }\ll 1$ (see (2.2)). Interestingly, the much weaker thermal buoyancy ( $\Delta \rho _T$ ) gives rise to a downward flow with comparable (if not larger) velocity magnitude and much larger mass flow rate. Conversely, the inner upward flow is driven by a much larger salinity buoyancy $\Delta \rho _S$ as shown in figure 7(a). As discussed in § 2.2, the reason for this difference is the relative thickness of thermal and salinity BLs ( $\delta _T/\delta _S \sim \textit{Le}^{1/2}\approx 14$ ) and the magnitude of the relative total buoyancy force ( ${\varLambda } \textit{Le}^{1/2}$ ) acting on each of these BLs. Experimental $\varLambda$ and ${\varLambda } \textit{Le}^{1/2}$ values for cases 5 and 6 are presented in table 3 and their positions on the ${\varLambda }{-}\textit{Le}$ diagram are shown in figure 2 using dashed (blue and red) lines. This figure shows that, in case 5, although $\varLambda$ is small ( $0.026$ ), ${\varLambda } \textit{Le}^{1/2}$ is an order of magnitude larger ( ${\approx}0.29$ ) because of larger thermal diffusivity relative to that of salt. This indicates the significance of the relative thickness of thermal and salinity BLs ( $\textit{Le}^{1/2}$ ) on the buoyancy force. Even relative buoyancy force does not fully explain this flow behaviour as ${\varLambda } \textit{Le}^{1/2}=0.29\lt 1$ indicates the presence of an overall larger salt buoyancy force within the BL. The appropriate explanation is that even though the salinity buoyancy force – driving the upward flow – is larger in magnitude, it is confined closer to the wall. Conversely, the buoyancy force from temperature – driving the downward flow – is well outside the effects of the salinity BL and the wall shear stress. Also, the salinity BL flow, owing to its proximity to the no-slip wall, is affected more compared with the thermal BL that is farther from the wall.

Figure 9.

Experimental (a) velocity and (b) temperature BL profiles of case 5 : $S_\infty =34$ g kg⁻¹, $T_\infty =4^{\,\circ} \text{C}$ ; and case 6 : $S_\infty =34$ g kg⁻¹, $T_\infty =2^{\,\circ} \text{C}$ . Here solid lines represent nonlinear EoS DNS results.

Figure 7 shows that the experiments capture well the features of the complex bi-directional BL flow, and is in agreement with the nonlinear DNS. Another important fact that needs to be highlighted here is the freezing point depression due to the salinity in the temperature profile in figure 7(b). In case 5, the measured interface temperature is approximately 1 degree below $0^{\,\circ} \text{C}$ , matching with the interface temperature estimated in the DNS. Note that this interface temperature ( $T_i$ ) also allows one to calculate the interface salinity: $S_i=-T_i/{0.06}^{\,\circ} \text{C}/$ ‰ $ \approx 16.7$ g kg⁻¹.

5.3. Effects of temperature at fixed $S_\infty =34$ ‰ ( $T_\infty =4$ and $2^{\,\circ} \text{C}$ )

For a fixed $S_\infty =34$ ‰, case 5 ( $T_\infty =4^{\,\circ} \text{C}$ ) and case 6 ( $T_\infty =2^{\,\circ} \text{C}$ ) are compared to study the effect of temperature in saline water. For these cases, figure 9(a) displays vertical velocities and figure 9(b) temperatures. In case 5 the downward peak velocity magnitude is slightly larger than in case 6. This is owing to the larger $\Delta \rho _T$ in case 5. Furthermore, the downward velocity peak is closer to the interface in case 5 compared to case 6. This is similar to the uni-directional flow in § 5.1, where thermal BL thickness decreases with increasing $T_\infty$ , and hence the temperature effect is more pronounced closer to the interface for case 5 compared with case 6. Another interesting observation is that the interface temperature $T_i$ for case 5 is higher than that for case 6 according to figure 9(b), showing less freezing point depression for the higher-ambient-temperature case (case 5). Therefore, according to $S_i= -T_i/{0.06}^\circ C/$ ‰, the interface salinity ( $S_i$ ) in case 5 should be less than in case 6. The reduction in $T_i$ as $T_\infty$ decreases can be calculated from the quadratic equation (2.6) as shown in figure 2(c). These calculations show that for $T_\infty =4$ and $2^{\,\circ} \text{C}$ the corresponding $T_i\approx -1$ and $-1.4^{\,\circ} \text{C}$ , which are in reasonable agreement with the experimental and DNS data presented in figure 9(b). Reduction of interface salinity in case 5 increases $\Delta \rho _S$ . This increased $\Delta \rho _s$ results in a slight velocity increase in the inner upward flow as shown in figure 9(a). This feature is more clearly observed in the DNS, although it is difficult to discern in the experimental data.

The experimental cases 5 and 6 are also presented on the $({\varLambda},\,\textit{Le})$ plane in figure 2(a) and 2(b). The figure also shows analytical results using (2.6) for $T_\infty$ in the range of $4^{\,\circ} \text{C}$ to $-2^{\,\circ} \text{C}$ . The analytical results match experiments/DNS reasonably well, providing confidence that (2.6) can be used to calculate $T_i$ , and hence the location of the flow regime on the $({\varLambda },\,\textit{Le})$ plane.

Figure 10.

Ice melt rate profiles for case 1: $(T_\infty ,\,S_\infty )$ = ( $20^{\,\circ} \text{C}$ , 0 ‰) in green; case 2: ( $15^{\,\circ} \text{C}$ , 0 ‰) in yellow; case 3: ( $10^{\,\circ} \text{C}$ , 0 ‰) in red; case 4: ( $4^{\,\circ} \text{C}$ , 0 ‰) in blue; case 5: ( $4^{\,\circ} \text{C}$ , 34 ‰) in purple; and case 6: ( $2^{\,\circ} \text{C}$ , 34 ‰) in grey. Symbols: molecular tagging images $\square$ , direct imaging $\triangledown$ and using (1.1) with MTT data $\circ$ . Here, error bars represent the uncertainty of each measurement. The DNS with linear EoS in dashed lines and nonlinear EoS in solid lines. (a) Linear–linear plot. (b) Log–log plot, where for cases 1, 2 and 3: $z_1=H-z$ ; for cases 4, 5 and 6: $z_1=z$ .