The Tip of Dendritic Crystal in an Inclined Viscous Flow

Abstract

We study the flow around the tip of a dendritic crystal by an inclined stream of viscous incompressible liquid. The tip shape is chosen accordingly to recent theory [Phil. Trans. R. Soc. A 2020, 378, 20190243] confirmed by a number of experiments and computations [Phil. Trans. R. Soc. A 2021, 379, 20200326]. Our simulations have been carried out for a 0, 30, 60, and 90-degree flow slope to the dendrite axis. We show that the stream inclination has a significant effect on the hydrodynamic flow and shear stress. In particular, a transition from laminar to turbulent currents on the upstream side of the dendritic crystal may occur in an inclined hydrodynamic flow. This leads to the fact that the heat and mass transfer mechanisms on the upstream and downstream sides of a growing dendritic crystal may be different.

1. Introduction

It is well known that the dendritic form is one of the most common forms of crystals growing in supercooled melts and supersaturated solutions. As this takes place, the dendritic growth rate and the dispersion of the material microstructure are determined by heat and mass transfer processes and melt hydrodynamic currents near the liquid–solid interface. Recently a number of new theoretical results extending the understanding of dendritic crystal growth dynamics have been obtained [1,2,3,4,5,6,7,8]. Thus, the problem of stable growth of the free-growing dendrite tip as well as the problem of the influence of convective melt flow on dendrite tip growth and lateral branch formation are relevant and under active investigation [9,10,11]. For example, a plane-parallel flow of supercooled melt flowing around the dendrite essentially influences the characteristics of the stable mode of dendritic growth. Namely, the tip velocity of the dendritic crystal and selection parameter strongly increase with increasing hydrodynamic flow rate, while the tip radius of the crystal decreases [12,13].

Natural convection (e.g., thermogravitational in the Earth’s gravity field or thermocapillary in microgravity) as well as forced convective flows of the melt (e.g., in electromagnetic levitation facilities) drastically change the mechanisms of heat and mass transfer near the interfaces of dendritic growth. Namely, convection substantially changes local temperature and impurity concentration gradients and thus changes the dynamics of interfacial crystal growth (e.g., the development of morphological instability and formation of secondary dendrite branches) [14,15,16,17]. Experimental data and theoretical estimates [18,19] as well as phase-field simulations [20,21,22] indicate a significant influence of fluid convection on the formation of dendritic material microstructure. So, for example, secondary dendritic branches predominantly grow on the side of the main dendrite flow, while on the opposite side they grow considerably less [22,23]. As this takes place, melt convection gives rise to the amplitude and frequency of dendritic side branches [21]. In addition, experimental data on the cast structure of an alloy Al 99.5 solidified in the mechanical stirrer demonstrate that the grains are inclined into the upstream direction [24]. Moreover, experiments on dendritic structure of AI-4% Si alloy showed that the fluid flow modified the dendrite spacing from large passing small to large [25]. Thus, convective flow affects phase selection [26,27], grain structure refinement [28,29], and generally the formation of solid structure of crystallizing materials [30]. In addition, intense convective melt flows near interfaces may change the mechanism of heat and mass transfer, i.e., instead of conductive boundary conditions of heat and mass transfer, the corresponding convective boundary conditions take place [31,32,33,34,35].

Given the importance of melt flow in dendritic growth theory, in this paper we study the influence of the tilt angle on the hydrodynamic flow around a dendritic crystal. In doing so, an important feature of the study is the consideration of the real shape of the dendritic tip, recently derived analytically by Alexandrov and Galenko in Ref. [36] and then confirmed by comparing the theory with experiments and numerical simulations (using the phase-field and enthalpy methods) in Refs. [37,38].

2. The Model

Let us consider a tip region of dendritic crystal submerged into a forced convective flow of viscous fluid. In addition, we consider the case of steady-state dendritic growth in a hydrodynamic melt flow. It is well known that the growing dendritic crystal after a certain time reaches the steady-state growth scenario with the tip diameter and tip shape remaining unchanged [21,22,23,39,40,41]. Keeping this in mind, the dendrite tip shape will be considered accordingly to a recently developed theory by Alexandrov and Galenko [36]. For the sake of convenience, let us be revise the main points of this theory for a three-dimensional crystal. Let the origin of Cartesian coordinate system be at the dendritic tip. The main trunk of crystal is directed in the negative ($-z$) direction while the x-axis touches the origin and passes in a perpendicular direction. In this case, the 3D tip shape of a dendritic crystal is given by (x and z are scaled by the dendrite tip diameter) [36]

$$z=\alpha z_{\mathrm{AG}}\left(x\right)=-\alpha \frac{b_S\left(x\right)x^2+b_L\left(x\right){\left|x\right|}^{3/2}}{b_S{\left(x\right)\left|x\right|}^{1/2}+b_L\left(x\right){\left|x\right|}^{-1/2}},$$

(1)

where $\alpha$ is a shape constant, $b_S\left(x\right)$ and $b_L\left(x\right)$ satisfy the following asymptotic formulas

$$\begin{array}{c}{b}_S\left(x\right)\to 0,b_L\left(x\right)\to 1\mathrm{at}x\to 0,\hfill \\ b_S\left(x\right)\to 1,b_L\left(x\right)\to 0\mathrm{at}x\gg 1.\hfill \end{array}$$

(2)

Note that the 3D dendrite tip shape (1) contains limiting transitions to previous theories. So, if $x\to 0$, $z_{\mathrm{AG}}\left(x\right)\to -x^2$ (Ivantsov parabola [4]). From the other hand, if $x\gg 1$, $z_{\mathrm{AG}}\left(x\right)\to -{\left|x\right|}^{3/2}$ (Plapp and Karma asymptotic shape [42]). The stitching functions $b_S\left(x\right)$ and $b_L\left(x\right)$ satisfying the asymptotic limiting conditions (2) can be selected in various ways. Here we chose them after Alexandrov and Galenko [36] in the form of

$$b_S\left(x\right)=exp\left(-\frac{1}{x^{2k}}\right)\mathrm{and}{b}_L\left(x\right)=exp\left(-x^{2k}\right)$$

(3)

with $k=3$.

Next we submerge the dendrite tip shape (1) into a laminar flow of a viscous incompressible fluid coming towards the dendritic tip. The hydrodynamic model under consideration is given by conservation equations for mass (Equation (4)) and momentum (Equation (5))

$$\nabla ·\mathbf{V}=0,$$

(4)

$$\begin{array}{c}{\displaystyle \rho \left(\frac{\partial \mathbf{V}}{\partial t}+\left(\mathbf{V}·\nabla \right)\mathbf{V}\right)=-\nabla P+\nabla \left(2\mu S\right).}\hfill \end{array}$$

(5)

Here t is the time, $\mathbf{V}$ is the velocity vector, $\rho$ is the constant density of fluid, P is the pressure, $2\mu S$ is the tensor of viscous stresses, S is the tensor of deformation rates, and $\mu$ is the coefficient of dynamic viscosity. We consider the succinonitrile melt as a Newtonian fluid, which means the constant dynamic viscosity $\mu$.

A 3D computational domain is considered to simulate the flow around the dendrite tip using Equations (4) and (5) (see Figure 1). At the left domain boundary, a uniform fluid flow orthogonal to the boundary with constant velocity ${\mathrm{V}}_{\mathrm{in}}$ is set. At all other domain boundaries (highlighted in dark grey), the boundary condition is defined by a zero pressure gradient. The light grey boundary sets an impermeable dendrite interface and is defined by the corresponding “no-slip” condition, where the fluid velocity at the boundary is zero. The dendrite shape (light grey colour) is determined by the Equation (1) with a tip radius $\mathrm{R}$, and the angle at which the incoming flow passes around the dendrite tip is determined by the value of $\phi$. The goals of the simulations are to determine the fluid current lines around the dendrite tip and the shear stress on the dendrite surface, which defines its growth characteristics. Simulations were performed using the FlowVision computational software for a succinonitrile melt with the parameters given in

Table 1. Parameters of the model for succinonitrile [43].

Parameter	Value	Unit
$\rho$	982	kg/m${}^3$
$\mu$	0.00264	Pa s
$\mathrm{R}$	0.00015	m
${\mathrm{V}}_{\mathrm{in}}$	1	m/s
$\phi$	0; 30; 60; 90	grad
$\alpha$	1	-

.

3. Results

The dendrite tip was placed in a viscous fluid flow tilted to 0, 30, 60, and 90 degrees to the dendrite axis. The results of our computer simulations are shown in Figure 2, Figure 3, Figure 4 and Figure 5. In these illustrations, we show current lines in the liquid part of the system (the colour scale of fluid flow rate is shown on the left), while the shear stress is shown on the dendrite (the colour scale of shear stress is shown on the right). First of all, we note that the flow pattern around the dendrite changes drastically with a change in the flow slope. Thus, in the case of zero flow slope (Figure 2), a symmetrical flow pattern is expected, with a thin low-velocity layer, which is formed due to the no-slip condition at the boundary. A flow tilt of 30 degrees produces low-velocity regions in front of and behind the dendrite (Figure 3). When the tilt angle is increased to 60 degrees, a wide stagnation zone occurs behind the dendrite (Figure 4). As this takes place, the shear stress in the front (red colour) and the back (blue colour) regions of the streamlined dendrite surface can differ by a factor of about 10. This effect is found at flow angles of 30, 60, and 90 degrees (see Figure 3, Figure 4 and Figure 5), i.e., it is quite typical for different inclined flows. Note that in Figure 5 there are fluid flows behind the dendrite that come from the right boundary. This is explained by the fact that the incoming flow passing around the dendrite draws the melt located on dendrite’s back side. Thus, a drop in pressure occurs there and the melt begins to flow into the area behind the dendrite from the right boundary. The model under consideration assumes a single dendrite with undisturbed liquid around it. In the case of many closely spaced dendrites, the influence of the boundaries can be much more complex and unpredictable.

A boundary layer around the dendrite tip is almost a blue thin (low speed) layer for a 0, 30, and 60-degree fluid slopes, as shown in Figure 2, Figure 3 and Figure 4. When paying our attention to Figure 5, we see that an intense fluid current along the dendrite surface (turquoise colour in upstream area) appears for a 90-degree flow slope. At the same time, the current in the area behind the dendrite (downstream area) is considerably weaker or may be absent altogether. The more intense flow in the anterior region (left in Figure 5) compared to the much less intense flow in the posterior region (right in Figure 5) may indicate different heat and mass transfer mechanisms in these regions. It is well known that at strong currents, the heat ($J_h$) and impurity ($J_i$) fluxes at the boundary are determined by the difference between the temperatures (impurity concentrations) at the boundary and away from it (instead of classic Fick’s laws) [31,32,33,34,35], i.e.,

$$J_h={\alpha }_h\rho c{u}_{*}(T_b-T_{\infty }),J_i={\alpha }_i{u}_{*}(C_b-C_{\infty }),$$

(6)

where ${\alpha }_h$ and ${\alpha }_i$ represent the heat and mass (impurity) transfer coefficients, c is the specific heat of water, $u_{*}$ is the friction velocity, and $T_b$ ($C_b$) and $T_{\infty }$ ($C_{\infty }$) are the temperatures (impurity concentrations) at the dendrite boundary and away from it. Note that the ratio of coefficients ${\alpha }_h$ and ${\alpha }_i$ is defined by the diffusion coefficients for heat $D_h$ and impurity $D_i$ at that ${\alpha }_h/{\alpha }_i={\left(D_h/D_i\right)}^l$ [34] with $2/3<l<4/5$ [44,45]. The friction velocity $u_{*}$, describing the flow at the solid–liquid boundary, is characterized through the shear stress $\tau$ and liquid density $\rho$ [46] as $u_{*}=\sqrt{\tau /\rho }$. In the case of a 90-degree slope (Figure 5), $\tau \approx {10}^2$ Pa at the front of the stream and $u_{*}\approx 0.3$ m s${}^{-1}$. Therefore, the Reynolds number $\Re =u_{*}\mathrm{R}\rho /\mu \approx 17$; the system parameters here are taken from Table 1. It is significant to note that “purely laminar flow only exists up to $\Re =10$” when dealing with an object in liquid stream [47]. In other words, a transition from laminar to turbulent currents on the upstream side of the dendrite may occur. If this does happen, different heat and mass balance boundary conditions on the upstream and downstream sides of a growing dendritic crystal must be used.

An important point is that the theory developed by Ivantsov [48,49], Brener [50,51], Almgren, Dai and Hakim [52], Plapp and Karma [42], and recently generalized by Alexandrov and Galenko [36] predicts one universal dendritic tip shape no matter what undercooling is imposed. Namely, the dendrite shape is represented by parabola at the tip region (Formula (1), which transforms to parabola at $x\to 0$). When moving away from the vertex of the parabola at a distance of the order of a few tip diameters, Formula (1) works. Physically, this tip shape is controlled by the anisotropy of the surface energy. Other surface shapes that are possible in nature (flat, spherical, ellipsoidal, faceted, etc.) arise from variations in the initial conditions of pattern formation. However, if a dendritic pattern is formed, after a certain time, it reaches the steady-state growth mode, and its tip shape is described by the universal law (1). This means that the numerical simulations carried out in this paper for an inclined hydrodynamic stream flowing onto a dendritic tip with the shape (1) describe a wide class of dendritic crystals.

4. Conclusions

In summary, we have presented the results of numerical simulations concerning the influence of inclined fluid stream on the hydrodynamic flow around a dendritic crystal. As this takes place, the tip shape of dendrite is chosen according to the recently developed theory [36], which is confirmed by computer simulations and a large number of experimental data [37,38]. Our computer calculations have shown that inclined flow can have a significant effect on dendritic growth. First of all, on the downstream side of the dendrite low-velocity (stagnation) regions appear, where the shear stress is much smaller than on the upstream side. As this takes place, the greater the flow angle to the dendrite growth axis, the stronger the role of this effect is. At high attack angles there is an intense fluid flow in a narrow boundary layer on the upstream side of the dendrite. Our estimates show that the Reynolds number in this boundary layer for succinonitrile crystals is approximately 17 while “purely laminar flow only exists up to $\Re =10$” when dealing with an object in liquid stream [47]. This means that the laminar-turbulent transition may occur on the upstream side of the dendrite. This in turn means that different heat and mass transfer mechanisms take place on the upstream and downstream sides of a dendritic crystal.

Thus, our simulations indicate the need for a more detailed study of inclined flow on crystal growth in the following directions.

(i)

A numerical simulation of dendritic growth in an inclined liquid flow should be carried out, taking into account the conductive and convective heat and mass transfer on various (with respect to the flow) sides of the dendrite. Such simulations can be done using a direct solution of the heat and mass transfer problem with flow (e.g., the phase-field or enthalpy methods) [53,54,55,56], as well as using the convective boundary integral equation [57,58,59];

(ii)

Intense convective currents on the upstream side of the dendrite are a potential source of morphological instability on its surface. It is therefore necessary to study the surface stability of the dendrite crystal to small morphological perturbations on its upstream side. Such a study should be done by following Refs. [60,61,62,63], who developed methods to study the stability induced by fluid currents;

(iii)

In future computer simulations in the presence of inclined fluid flow, the real dendritic crystal morphology (e.g., taken from experiments) should be taken into account with allowance for secondary and tertiary branches. Here it is also important to perform simulations of the growth of several dendrites with overlapping branches and to consider the possibility of crystal nucleation ahead of the dendrites, i.e., to study crystallization with a nonequilibrium two-phase region in the presence of convective flow. Such computer simulations should be performed using directional and bulk solidification theories (see, among others, Refs. [64,65,66,67,68,69,70]).