﻿ 六角冰晶生长过程的相场模拟
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 哈尔滨工程大学学报  2019, Vol. 40 Issue (9): 1537-1542  DOI: 10.11990/jheu.201806067 0

### 引用本文

HAN Duanfeng, WANG Yongkui, JU Lei, et al. Phase field simulation of the hexagonal ice crystal growth process[J]. Journal of Harbin Engineering University, 2019, 40(9): 1537-1542. DOI: 10.11990/jheu.201806067.

### 文章历史

Phase field simulation of the hexagonal ice crystal growth process
HAN Duanfeng , WANG Yongkui , JU Lei , WANG Qing , WANG Chunyang
College of Shipbuilding Engineering, Harbin Engineering University, Harbin 150001, China
Abstract: In order to study the microscopic mechanism of ice formation after airborne supercooled water droplets adhere to ship superstructure, numerical simulation of ice crystal growth was carried out in this paper. Based on the classical Kobayashi model in the phase domain method, this paper used the nine-point difference scheme in the finite difference method to discretize the partial differential equation, and realized the simulation and visualization of ice crystal growth with the aid of the Python programming tool. Furthermore, through sensitivity analysis of the main parameters, it was found that some parameters, such as the mean of the interface width and melting temperature, has different influences on ice crystal growth and final morphology. The simulation process of dendrite growth in this paper was consistent with the results of a microscopic experiment, and was similar to the true snowflake profile in nature. This verified the reliability of the numerical method used. Numerical simulation of the ice crystal growth process used in this study will further reveal the micro-mechanism of the ice deposition process by supercooled water droplets.
Keywords: superstructure    icing    droplets    ice crystals    phase field method    finite difference method    numerical simulation    parameter influence

1 Kobayashi相场模型

 $\begin{array}{l} \tau \frac{{\partial P}}{{\partial t}} = - \nabla \left( {{{\left| {\nabla P} \right|}^2}\varepsilon \frac{{\partial \varepsilon }}{{\partial {\mathit{\pmb{n}}}}}} \right) + \nabla ({\varepsilon ^2})\nabla P + \\ \;\;\;\;\;\;\;\;\;\;{\varepsilon ^2}{\nabla ^2}P + {\rm{ }}P\left( {1 - P} \right)\left( {P - 0.5 + m\left( T \right)} \right) \end{array}$ (1)

 $\frac{{\partial T}}{{\partial t}} = {\nabla ^2}T + {\rm{ }}K\frac{{\partial P}}{{\partial t}}$ (2)

2 数值求解过程 2.1 离散形式

 $\frac{{\partial P}}{{\partial t}} = \frac{{{P_{i, j, n + 1}} - {P_{i, j, n}}}}{{\Delta t}} + o(\Delta t)$ (3)

 $\begin{array}{l} {\nabla ^2}P = (2({P_{i + 1, j}} + {P_{i - 1, j}} + {P_{i, j + 1}} + {P_{i, j - 1}}) + \\ \;\;\;\;\;\;\;\;\;\;{P_{i + 1, j + 1}} + {P_{i + 1, j - 1}} + {P_{i - 1, j + 1}} + {P_{i - 1, j - 1}} - \\ \;\;\;\;\;\;\;\;\;\;12{P_{i, j}}){(^2}\Delta {x^2}) - 1 \end{array}$ (4)

2.2 初始条件及边界条件

 ${x^2} + {y^2} \le {r^2}:P = 1, T = 1$ (5)
 ${x^2} + {y^2} > {r^2}:P = 0, T = 0$ (6)
 Download: 图 1 晶核设置示意 Fig. 1 A schematic diagram of nucleation

 Download: 图 2 冰晶计算程序基本流程 Fig. 2 Basic process of ice crystal calculation program
2.3 数值模拟结果及对比 2.3.1 冰晶生长过程

 Download: 图 3 冰晶生长过程 Fig. 3 The growth process of ice crystals

 Download: 图 4 模拟冰晶生长过程 Fig. 4 A schematic diagram for simulating the growth process of ice crystals
2.3.2 模拟冰晶与自然界雪花对比

Libbrecht[18]展现了其团队拍摄到的形式多样的雪花照片，真实呈现了大自然界中广泛存在的雪花(见图 5(b)), 本文保持上述其他参数不变，划分网格区域为1 000×1 000，迭代1.0 s后获得了如图 5(a)所示的模拟结果，由图 5可见，本文模拟的六角冰晶形貌与大自然中真实存在的雪花是相似的，两图中各主支均生长出或长或短的侧分支，且生长较长的侧分支又生长出新的分支，有效说明了本文相场模型的真实性和准确性。

 Download: 图 5 数值模拟结果与真实雪花对比 Fig. 5 Comparison between numerical simulation results and real snowflakes
3 参数敏感性分析

Kobayashi相场模型中的弛豫时间、各向异性强度、潜热常值等物性参数会影响冰晶生长过程(包括尖端速度、尖端半径等)和冰晶形貌(侧枝大小、取向等)[19]，本文将对模型中各主要参数进行敏感性分析，与真实冰晶形貌进行对比分析，进一步提高冰晶生长模拟的准确性。

3.1 弛豫时间τ

 Download: 图 6 不同弛豫时间τ模拟结果 Fig. 6 Simulation results of different relaxation time
3.2 界面宽度均值ε

 Download: 图 7 不同界面宽度均值ε模拟结果 Fig. 7 Simulation results of different mean interface width ε
3.3 融化温度TM

 Download: 图 8 不同融化温度TM模拟结果 Fig. 8 Simulation results of different melting temperature TM
3.4 各向异性模数j

Kobayashi模型表述界面宽度关于角度的函数时引入了各项异性模数j，其能够影响晶体分支数目及最终形态。本文保持其他参数不变，依次设定j为4、6、8，迭代时间为0.30 s，由图 9可见，模数增加，侧枝的尖端速度及半径也会增加。

 Download: 图 9 不同各向异性模数j模拟结果 Fig. 9 Simulation results of different anisotropic modulus j
3.5 各向异性强度δ

 Download: 图 10 不同各向异性强度δ模拟结果 Fig. 10 Simulation results of different anisotropic strength δ
3.6 潜热常值K

 Download: 图 11 不同潜热常值K模拟结果 Fig. 11 Simulation results of different latent heat constants
4 结论

1) 潜热常值作为影响冰晶生长的最主要参数，其象征着界面扩散层的热量释放，值越大，则能量释放越多，六角冰晶生长越明显，弛豫时间越小则冰晶生长越快，各向异性模数决定了冰晶生长的分支数目，广义上印证了凝固材料杂质的存在，各向异性强度影响冰晶侧分支生长，界面宽度均值则影响着冰晶生长尖端速度及半径。

2) 本文采用的相场模型不仅可以通过耦合相场与外场方程将微观与宏观结合起来，还能模拟分析晶体生长过程中的物性参数(如界面宽度、各向异性强度等)对冰晶生长的影响。

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