Wavy Ice Patterns as a Result of Morphological Instability of an Ice–Water Interface with Allowance for the Convective–Conductive Heat Transfer Mechanism

Abstract

In this research, the wavy ice patterns that form due to the evolution of morphological perturbations on the water–ice phase transition interface in the presence of a fluid flow are studied. The mathematical model of heat transport from a relatively warm fluid to a cold wall includes the mechanism of convective–conductive heat transfer in liquid and small sinusoidal perturbations of the water–ice interface. The analytical solutions describing the main state with a flat phase interface as well as its small morphological perturbations are derived. Namely, the migration velocity of perturbations and the dispersion relation are found. We show that the amplification rate of morphological perturbations changes its sign with variation of the wavenumber. This confirms the existence of two different crystallization regimes with (i) a stable (flat) interfacial boundary and (ii) a wavy interfacial boundary. The maximum of the amplification rate representing the most dangerous (quickly growing) perturbations is found. The theory is in agreement with experimental data.

1. Introduction

A mathematical description of phase transformations in the presence of convection in liquid is a complex task with a moving boundary, the position of which is determined by solving the problem. In addition, such problems are strongly complicated by various peculiarities of the hydrodynamic flow near a phase transition boundary, leading to a mixed convective–conductive mechanism of heat and mass transfer. Examples of such events are the processes of water freezing in rivers, lakes, and seas, the solidification of molten metal in a mold and magma in a magmatic chamber, and the growth of stenoses in blood vessels leading to limited or complete cessation of blood flow (see, among others, [1,2,3,4,5,6,7]).

The nonlinearity of such a phase transformation problem arises for several reasons. Firstly, convective terms appear in the heat and mass transfer equations in the moving coordinate system connected with the growing/melting interfacial boundary [8,9,10,11]. This is due to the fact that the ordinary derivative with respect to time can be expanded into local and convective components. Secondly, nonlinearity may be initially involved in the boundary conditions at the moving crystallization front when the dependence of phase transformation temperature on impurity concentration, front curvature, and atomic kinetics is considered [12,13]. Thirdly, in the case of turbulent fluid flow, additional diffusion terms of turbulent heat and mass fluxes appear in the equations of heat and mass transfer [14,15,16]. The presence of such terms makes nonlinear the heat and mass transport in liquid due to the strong mixing of liquid and the occurrence of complex circulation currents near the interphase boundary [17]. Fourth, a nonlinearity of the problem arises due to the morphological instability of the phase transformation boundary and the formation of new periodic and/or irregular crystallization scenarios [18,19,20,21,22,23,24].

The presence of convection in liquid strongly affects the heat and mass transfer near a phase transformation boundary. For example, convection mixes the liquid phase and partially levels the temperature and concentration distributions. This, in particular, leads to the appearance of convective components in heat and mass fluxes at the liquid–solid interface. Thus, the heat and mass fluxes at this interface become of a mixed type (conductive–convective), i.e., they have both conductive and convective contributions [25,26,27]. Taking this fact into account, a complete mathematical model of the directional crystallization process of a binary liquid has recently been formulated with allowance for the conductive–convective heat and mass transfer mechanism [28]. In this paper, a linear analysis of the morphological instability of a flat crystallization front in the presence of convection was also carried out and a new criterion for constitutional supercooling origination was derived, which demonstrated the existence of four crystallization scenarios. The theory developed in [28] also showed a significant influence of convection on the crystallization process as a whole and on the stability criterion, which was thoroughly studied by various scientific groups after the classical theory by Mullins and Sekerka [18] was published. Below, we investigate the influence of convective heat transfer on the morphological stability of the liquid–solid interface, which leads to the appearance of so-called wavy ice layers (Figure 1) [29,30,31,32,33,34].

2. Morphological Stability

A wavy ice layer, as shown in Figure 1, results from the development of liquid–solid interface instability in the hydrodynamic flow [35]. At the initial moment of time $t=0$, a groove was made on the ice surface (see Figure 8 in [35]), which transformed over time and led to the formation of the wavy ice surface. In the course of time, the downstream side of the groove melts to form a longer groove and the ice downstream of the groove becomes slightly wider. Approximately 2–3 h later, a second groove begins to form downstream about 15 cm from the initial groove. Over time, this sinusoidal-like wave propagates downstream, leading to the formation of several waves. As this takes place, the amplitudes of these waves also increase until the first wave downstream of the initial groove develops a sharp crest after about $t=3$ h. Subsequent waves also develop sharp crests, after which further changes in wave amplitudes cease and a so-called wavy or ripple ice surface is formed (wave crests are placed in a perpendicular to the flow). The waves continue to migrate slowly downstream. In the fluid flow downstream of the crest of each wave, there is a region of separated flow. An important point is that the ice surface is stable if it is reverted to a planar liquid–solid surface some time after the groove has been made. On the other hand, in the case of instability, the initial groove leads to the formation of a wavy surface, as described above.

Let us formulate below a mathematical model of heat transfer describing the crystallization process with a wavy water–ice interface shown in Figure 2. The phase interface position $s_i(x,t)=s_0\left(t\right)+\eta (x,t)$ is considered to be a sum of two contributions: the growing ice layer $s_0$ and its perturbation $\eta (x,t)$ of the form

$$\begin{array}{c}\hfill \begin{array}{c}\hfill \eta (x,t)=Asin\left[k(x-ct)\right]exp\left(\alpha t\right),\end{array}\end{array}$$

(1)

where x and t are the spatial and time variables, respectively; A is the amplitude of perturbations; $\alpha$ and k are their amplification rate and wavenumber, respectively; and c is the migration velocity of the surface wave. Here, the water–ice interface perturbation is assumed to be small, i.e., $\left|\eta (x,t)\right|\ll s_0\left(t\right)$.

At the interfacial boundary, fluid currents result in a conductive–convective heat flux at the fluid side [25,26,27,28]. In addition, the temperature continuity condition is fulfilled at this boundary. Taking this into account, we have the following heat balance and temperature continuity boundary conditions

$$\begin{array}{c}\hfill \begin{array}{c}\hfill {\rho }_{i}L\frac{\partial s_i}{\partial t}=k_i\frac{\partial T_i}{\partial y}-k_w\frac{\partial T_w}{\partial y}-h(T_{\infty }-T_w),y=s_i,\\ \hfill T_w=T_i=T_f,y=s_i.\end{array}\end{array}$$

(2)

Here, $T_i$ and $T_w$ are the temperatures of ice and water, respectively, $k_i$ and $k_w$ are their thermal conductivities, respectively, ${\rho }_i$ is the density of ice, L is the latent heat parameter, h is the heat-transfer coefficient, $T_{\infty }$ is the temperature in water far from the water–ice phase interface, and $T_f$ is the temperature at the phase interface. Note that the simultaneous convective–conductive heat transfer flux at the boundary $y=s_i$ represents the main novelty of the model under consideration.

In the case of slow motion of the water–ice interface, the temperature profile of the ice is described by the stationary heat conduction equation [36,37]

$$\begin{array}{c}\hfill \begin{array}{c}\hfill {\nabla }^2{T}_i=0,0\le y\le s_i(x,t).\end{array}\end{array}$$

(3)

The temperature is fixed at the cold wall (bottom) $y=0$, and we have the boundary condition

$$\begin{array}{c}\hfill \begin{array}{c}\hfill T_i=T_{wall},y=0.\end{array}\end{array}$$

(4)

The temperature distribution in ice with a perturbed interfacial boundary $y=s_i(x,t)$ can be found using the perturbation method [38]. To do this, we represent the temperature in ice $T_i$ as the sum of mean temperature $T_{i0}$ and its perturbation $T_{i1}$, i.e.,

$$\begin{array}{c}\hfill \begin{array}{c}\hfill T_i(x,y,t)=T_{i0}\left(y\right)+T_{i1}(x,y,t),0\le y\le s_i(x,t).\end{array}\end{array}$$

(5)

In addition, we assume that [38] $|T_{i1}|\ll |T_0|$.

By expanding the second boundary condition (2) into a Taylor series on the perturbed boundary $y=s_i(x,t)$, restricting ourselves to the linear terms, and moving this boundary to the point $y=s_0$ assuming small perturbations $\left|\eta (x,t)\right|\ll s_0$, we obtain

$$\begin{array}{c}\hfill \begin{array}{c}\hfill T_{i0}+\frac{d{T}_{i0}}{dy}\eta +T_{i1}=T_f,y=s_0.\end{array}\end{array}$$

(6)

Now, equating the terms of the same order of magnitude in perturbations in (6) and considering expressions (3) and (4), we arrive at the following problems for the determination of temperature contributions $T_{i0}$ and $T_{i1}$

$$\begin{array}{c}\hfill \begin{array}{c}\hfill \frac{d^2{T}_{i0}}{d{y}^2}=0,0\le y\le s_0,\\ \hfill T_{i0}=T_f,y=s_0,\\ \hfill T_{i0}=T_{wall},y=0\end{array}\end{array}$$

(7)

and

$$\begin{array}{c}\hfill \begin{array}{c}\hfill {\nabla }^2{T}_{i1}=0,0\le y\le s_0,\\ \hfill T_{i1}=-\frac{d{T}_{i0}}{dy}\eta ,y=s_0,\\ \hfill T_{i1}=0,y=0.\end{array}\end{array}$$

(8)

Their solutions read as

$$\begin{array}{c}\hfill \begin{array}{c}\hfill T_{i0}\left(y\right)=T_{wall}+\frac{T_f-T_{wall}}{s_0}y\end{array}\end{array}$$

(9)

and

$$\begin{array}{c}\hfill \begin{array}{c}\hfill T_{i1}(x,y,t)=-(T_f-T_{wall})\frac{\eta (x,t)}{s_0}\frac{\sinh \left(ky\right)}{\sinh \left(k{s}_0\right)},\end{array}\end{array}$$

(10)

where $\eta (x,t)$ is given by expression (1). Distributions (9) and (10) determine the temperature field in ice accordingly to expression (5).

Note that the heat-transfer coefficient $h=h_0+h_1$ should be treated as the sum of the constant main value $h_0={\alpha }_h{\rho }_w{c}_w{u}_{*}$ [39,40] and its perturbation $h_1$ of the form [35]

$$\begin{array}{c}\hfill \begin{array}{c}\hfill h_1(x,t)=f{h}_0{A}^{+}sin\left[k(x-ct)+\varphi \right]exp\left(\alpha t\right),\end{array}\end{array}$$

(11)

where ${\alpha }_h$ is the convective heat transfer coefficient, ${\rho }_w$ is the density of water, $u_{*}$ is the friction velocity, f is the perturbation amplitude, $A^{+}=A{u}_{*}/{\nu }_w$, ${\nu }_w$ is the kinematic viscosity of water, and $\varphi$ is the phase shift between the heat transfer and water–ice interface perturbations.

Let us now perturb the boundary conditions (2) at the water–ice interface in accordance with the linear stability theory for small morphological perturbations [18,22,23]. Hence, assuming that the temperature in water represents the sum of a mean quantity $T_{w0}$ and its perturbation $T_{w1}$ ($|T_{w1}|\ll |T_{w0}|$)

$$\begin{array}{c}\hfill \begin{array}{c}\hfill T_w(x,y,t)=T_{w0}\left(y\right)+T_{w1}(x,y,t),y\ge s_i(x,t),\end{array}\end{array}$$

(12)

expanding the boundary conditions (2) into a Taylor series on the perturbed water–ice boundary $y=s_i(x,t)$ and moving this boundary to the point $y=s_0$, we obtain

$$\begin{array}{c}\hfill \begin{array}{c}\hfill {\rho }_{i}L\left(\frac{d{s}_0}{dt}+\frac{\partial \eta }{\partial t}\right)=k_i\frac{d{T}_{i0}}{dy}+k_i\frac{d^2{T}_{i0}}{d{y}^2}\eta +k_i\frac{\partial T_{i1}}{\partial y}-k_w\frac{d{T}_{w0}}{dy}\\ \hfill -k_w\frac{d^2{T}_{w0}}{d{y}^2}\eta -k_w\frac{\partial T_{w1}}{\partial y}-(h_0+h_1)\left(T_{\infty }-T_{w0}-\frac{d{T}_{w0}}{dy}\eta -T_{w1}\right),y=s_0,\end{array}\end{array}$$

(13)

$$\begin{array}{c}\hfill \begin{array}{c}\hfill T_{w1}+\frac{d{T}_{w0}}{dy}\eta =0,y=s_0.\end{array}\end{array}$$

(14)

Here, $T_{w0}=T_f$ at $y=s_0$, temperatures $T_{i0}$ and $T_{i1}$ are given by distributions (9) and (10), respectively, and $\eta$ and $h_1$ are defined by expressions (1) and (11), respectively.

Equating now the terms of the same order of smallness in perturbations in Equation (13), we obtain

$$\begin{array}{c}\hfill \begin{array}{c}\hfill {\rho }_{i}L\frac{d{s}_0}{dt}=k_i\frac{T_f-T_{wall}}{s_0}-k_w{\left(\frac{d{T}_{w0}}{dy}\right)}_{y=s_0}-h_0\left(T_{\infty }-T_f\right).\end{array}\end{array}$$

(15)

$$\begin{array}{c}\hfill \begin{array}{c}\hfill {\rho }_{i}L\frac{\partial \eta }{\partial t}=-\frac{k_i(T_f-T_{wall})k\eta }{s_0\tanh \left(k{s}_0\right)}-k_w\frac{d^2{T}_{w0}}{d{y}^2}\eta -k_w\frac{\partial T_{w1}}{\partial y}-h_1(T_{\infty }-T_f),y=s_0.\end{array}\end{array}$$

(16)

Note that Formulas (10) and (14) were used when deriving expression (16).

Generally speaking, the temperature derivatives $d{T}_{w0}/dy$ and $d^2{T}_{w0}/d{y}^2$ at $y=s_0$ can be the functions of $s_0$. Therefore, expression (15) supplemented with the initial condition $s_0\left(t\right)=s_0\left(0\right)$ at $t=0$ represents the standard Cauchy problem for the determination of ice layer thickness $s_0\left(t\right)$. In the case of slow ice growth, $s_0$ is practically independent of time t and the left hand-side of Equation (15) can be omitted. In this case, expression (15) represents an algebraic equation defining the ice thickness $s_0$.

By substituting $\eta$ and $h_1$ from (1) and (11) into (16) we conclude that $\partial T_{w1}/\partial y$ at $y=s_0$ should be a similar function of x and t, i.e.,

$$\begin{array}{c}\hfill \begin{array}{c}\hfill {\left(\frac{\partial T_{w1}}{\partial y}\right)}_{y=s_0}=g_w{A}^{+}sin\left[k(x-ct)+\psi \right]exp\left(\alpha t\right),\end{array}\end{array}$$

(17)

where $g_w$ and $\psi$ stand for the perturbation amplitude and its phase shift.

Now combining (1), (11), (16), and (17), we find the migration velocity c and dispersion relation

$$\begin{array}{c}\hfill \begin{array}{c}\hfill c\left(k^{+}\right)=\frac{1}{{\rho }_{i}L{k}^{+}}\left(h_0(T_{\infty }-T_f)fsin\varphi +k_w{g}_{w}sin\psi \right),\end{array}\end{array}$$

(18)

$$\begin{array}{c}\hfill \begin{array}{c}\hfill \alpha \left(k^{+}\right)=-\frac{h_0(T_{\infty }-T_f)u_{*}{k}^{+}}{{\rho }_{i}L{\nu }_w}\left(\frac{fcos\varphi }{k^{+}}+\frac{k_i(T_f-T_{wall})}{h_0{s}_0(T_{\infty }-T_f)\tanh (k^{+}{u}_{*}{s}_0/{\nu }_w)}\right)\\ \hfill -\frac{k_w}{{\rho }_{i}L}\left({\left(\frac{d^2{T}_{w0}}{d{y}^2}\right)}_{y=s_0}+\frac{g_w{u}_{*}cos\psi }{{\nu }_w}\right),\end{array}\end{array}$$

(19)

where $k^{+}=k{\nu }_w/u_{*}$ is the dimensionless wavenumber. Here f, $\varphi$, $g_w$, and $\psi$ can be the functions of $k^{+}$. Experiments [35] show that f and $\varphi$ can be approximated by the following dependencies

$$\begin{array}{c}\hfill \begin{array}{c}\hfill f\left(k^{+}\right)=50.44{\left(k^{+}\right)}^{1.435},\varphi \left(k^{+}\right)=13.08+4.32ln{k}^{+}+0.41{(ln{k}^{+})}^2.\end{array}\end{array}$$

(20)

Note that expressions (19) and (20) derived with allowance for a mixed type of convective–conductive heat transfer in water represent the main result of our theory.

Let us especially highlight that the temperature field $T_{w0}\left(y\right)$ in liquid can be found from various models of heat transfer in a hydrodynamic flow [25,26,27,41,42]. In the case of a laminar flow, the temperature field is governed by the convective heat transfer equation. In the case of turbulent fluid flow, the turbulent heat flux must be taken into account in the heat transfer equation [14,15,16]. Another way to find the temperature distribution in liquid is to perform an experiment that takes into account all the peculiarities of the fluid flow. If such an experiment is performed, it is possible to fit the experimental points by means of $T_{w0}\left(y\right)$ and evaluate the derivatives $d{T}_{w0}/dy$ and $d^2{T}_{w0}/d{y}^2$ at $y=s_0$ in Equation (15) and dispersion relation (19). We use below the experimental data [35] to find the temperature distribution $T_{w0}\left(y\right)$.

3. Results and Discussions

The migration velocity $c\left(k^{+}\right)$ from (18) can be rewritten in dimensionless form as

$$\begin{array}{c}\hfill \begin{array}{c}\hfill c^{+}\left(k^{+}\right)=\frac{{\rho }_{i}Lc}{h_0(T_{\infty }-T_f)}=c_1\left(k^{+}\right)+c_2\left(k^{+}\right),\\ \hfill c_1\left(k^{+}\right)=\frac{f\left(k^{+}\right)sin\left[\varphi \left(k^{+}\right)\right]}{k^{+}},c_2\left(k^{+}\right)=\frac{k_w{g}_w\left(k^{+}\right)sin\left[\psi \left(k^{+}\right)\right]}{k^{+}{h}_0(T_{\infty }-T_f)}.\end{array}\end{array}$$

(21)

To calculate the migration velocity and dispersion relation, we first determined $s_0$ from the stationary Equation (15). As a result, the thickness of ice layer $s_0=19$ mm at $T_{wall}=-14.79$ °C, which is in full agreement with the experimental data [35]. Then, using $s_0$, we illustrate the dimensionless migration velocity $c^{+}\left(k^{+}\right)$ in Figure 3 for experimentally known range of wavenumbers $0.00075\lesssim k^{+}\lesssim 0.003$ [35]. The product $g_w\left(k^{+}\right)sin\left[\psi \left(k^{+}\right)\right]$ defining the coefficient $c_2\left(k^{+}\right)$ was fitted by the function $150c_1\left(10k^{+}\right)k^{+}-0.3$. As is easily seen, the migration rate increases with increasing the wavenumber of perturbations.

The dispersion relation (19) is shown in Figure 4 for various temperatures $T_{wall}$ at the cooled wall $y=0$. Namely, the process is stable for low temperature $T_{wall}=-11.5$ °C illustrated by the solid curve (amplification rate $\alpha$ is negative at all wavenumbers). In other words, a groove initially made on the ice surface disappears with time and does not lead to the formation of a wavy water–ice surface. However, the amplification rate of perturbations grows with decreasing the wall temperature. The dashed and dash-dotted curves cross the axis $\alpha =0$ and partially lie in the instability region (Figure 4). Such a behavior is very similar to the classical Mullins–Sekerka theory of morphological instability of a planar solidification front (see their Figure 1 in [18]). Note that the instability region (when a wavy water–ice surface is formed) increases with decreasing the temperature at the cooled wall. Figure 4 also demonstrates that the maximum of amplification rate ($\partial \alpha /\partial k^{+}=0$) approximately corresponds to the wavenumber $k_m^{+}=1.5\times {10}^{-3}$. In the case of instability ($\alpha >0$), such a wavenumber (perturbation with the corresponding wavelength ${\lambda }_m=2\pi {\nu }_w/\left(u_{*}{k}_m^{+}\right)$) leads to the development of morphological perturbations. Figure 5 shows that the experimental data of instability onset confirms this conclusion (instability occurs at a wavenumber lying close to the value of $k_m^{+}$). Moreover, Figure 5 demonstrates that the Reynolds number $R{e}_{\delta }$ does not substantially influence the origination/development of morphological instability.

It should be noted that the present analytical model and perturbation theory do not work for a larger scale and turbulent flow. For example, taking into account the turbulent flow, when fluid particles can undergo large pulsations, the smallness of temperature perturbations near the interfacial boundary is not fulfilled. In addition, the ice surface must be smooth enough to avoid creating complex circulation currents that result in fluid mixing. Another important requirement is that the deviations of the phase interface position $\eta (x,t)$ from the flat boundary $s_0\left(t\right)$ are small enough, i.e., $\left|\eta (x,t)\right|\ll s_0$. This condition can be violated for large spatial scales as well as for small crystallization times. On the other hand, the macroscopic scale of the problem in the direction of fluid flow (direction x) must be much larger than the wavelength $\lambda$ in order to disregard solid boundaries at fixed x. In general, with these limitations taken into account, the theory under consideration describes the morphological stability of wavy ice patterns with allowance for the convective–conductive heat transfer mechanism at the ice–water boundary.

An important practical application of the morphological stability theory of the water–ice interfacial boundary is the question of heat transfer at the ice surface. In the case of a wavy ice surface, the heat transfer through it is considerably higher than in the case of a flat surface [44,45,46,47]. In this case, fluid flows enhance heat transfer across the interfacial boundary, contributing to an increase in the resulting heat flux by increasing its convective component. Since the salinity of water lowers its freezing temperature and highly influences the resulting heat budget [48,49,50], an important direction for future investigations is the morphological stability analysis of a wavy ice surface in salty water taking convective–conductive heat and mass transfer into account. Another important task extending the present theory consists in accounting for the two-phase region on the boundary of pure ice [51,52,53,54,55,56]. Such a region essentially changes heat and mass transfer in the crystallizing system and influences the process of formation and evolution of morphological perturbations due to the presence of permeability and the possibility of liquid penetration through the water–ice interface.

4. Conclusions

In summary, we study the problem of morphological instability of the liquid–solid interface leading to the formation of a wavy ice surface. Since the fluid flow near a rough surface of solid phase (ice) is not laminar, convective–conductive heat transfer is taken into account. Bearing this in mind, we formulate the mathematical model for the growth of ice layer inside the liquid from a cooled wall. Our model accounts for small deviations of the water–ice interfacial boundary from a flat surface, which may be able to grow or decay with time. To determine the conditions of stable/unstable ice growth, we construct a morphological stability theory of the water–ice surface in the case of small perturbations. Within the framework of this theory, we have determined the stationary solution corresponding to the unperturbed equations with a flat interfacial boundary, as well as the solutions describing its perturbations. These solutions allowed us to find the migration velocity of perturbations and a dispersion relation determining the amplification rate as a function of perturbation wavenumber. Numerical analysis of the dispersion relation showed that there are stability and instability regions depending on the wavenumber of morphological perturbations and the temperature of a cold wall. As this takes place, the maximum value of the amplification rate is not essentially dependent of the wavenumber and is highly dependent of the temperature of a cold wall. This agrees with experimental data for ice-surface waves corresponding to the instability onset. Let us especially emphasize in conclusion that a nonlinear morphological stability analysis of the problem under consideration should be carried out to define the perturbation amplitudes of temperature and ice thickness. Such a theory representing a challenging task for future studies requires expanding the sought functions in Taylor series up to the cubic terms in the supercriticality parameter.