﻿ 基于润滑理论的二维积冰数值模拟
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Numerical simulation of two dimensional ice accretion based on lubrication theory
Hou Shuo , Cao Yihua
School of Aeronautic Science and Engineering, Beijing University of Aeronautics and Astronautics, Beijing 100191, China
Abstract:Water film-ice layer model based on lubrication theory was extended to two-dimensional orthogonal curvilinear coordinate system on arbitrary sections. Two governing partial differential equations of water film flow and ice accretion on curved sections were obtained. The numerical methods for solving these unsteady governing equations were presented. An implicit-explicit discrete scheme was applied to obtain the algebraic form of governing equations. Examples of numerical icing calculation were verified on NACA0012 airfoil and circle cylinder section in typical aero icing and structural icing environment. The results of calculation using current model were compared with ice shapes of simulation using traditional Messinger model and results from ice wind tunnel tests. Ice shapes on airfoil obtained using this numerical method were close to icing curves calculated by Messinger model in low temperature glaze icing conditions. In relatively high temperature icing conditions, more accurate results were obtained than Messinger model compared with results of ice tunnel experiments. Ice accretion on transmission line wires which cannot be predicted properly by traditional Messinger model was also effectively predicted by current model in structural icing conditions.
Key words: lubrication theory     film flow     ice accretion     airfoil     cable     numerical simulation

1 数学模型 1.1 润滑理论对表面水膜流动的近似处理

1) 水膜连续分布,满足连续介质的条件.

2) 水膜的厚度H相对于横向尺度L的比率(长宽比)ε(H/L)以及消减雷诺数ε2Re都是可以略去的充分小量(Re为水膜流动的雷诺数).

 图 1 二维截面的正交曲线坐标系图示Fig. 1 Diagram of orthogonal curvilinear coordinate system on two-dimensional section

1.2 积冰方程

1) 水膜的长宽比ε(H/L)是一个充分小量.

2) 水膜的Peclet数也是一个充分小量,Peclet数代表对流项与热传导项的量纲之比,表明水膜能量方程中起支配作用的是热传导.

1) 冰层的厚度与冰层的横向尺度之比ε(B/L)是一个充分小量；

2) 忽略初始时刻冰晶和水膜的形成过程中复杂的非稳态传热和瞬时成核现象.

1.3 水膜-空气界面的能量传递

2 数值求解方法 2.1 水膜流动方程的数值求解方法

2.2 积冰方程的数值求解方法

1) 首先假设型线表面全部为明冰,利用式(25)计算出新时刻的水膜厚度.

2) 当h>hp时表明该单元的明冰假设合理,用式(28)计算新时刻的冰层厚度;当h<hp时表明该单元属于霜冰,重新令h=hp并采用式(29)计算霜冰厚度.

3 模型输入参数的计算 3.1 空气压强、壁面切应力和表面传热系数的计算

3.2 水滴壁面收集系数和撞击速度的计算

 编号 弦长/m 自由流速率/(m/s) 静温/K 攻角/(°) Qlwc/(g/m3) MVD/μm 时间/min r401 0.533 4 102.8 265.37 4 0.55 20 7 062791.002 0.533 4 58.1 269.19 4 1.3 20 8

 图 2 算例r401的7×1 min多步推进过程Fig. 2 Example r401,multi-stepping advanced process of 7×1 min

 图 3 翼型积冰算例最终冰形的对比图Fig. 3 Comparison of final ice shapes on airfoil

4.2 传输线积冰

 编号 直径/m 自由流速率/(m/s) 静温/K LWC/(g/m3) MVD/μm 时间/min case4 0.034 9 10 268.15 1.8 26 30 case5 0.019 05 5 268.15 1.8 33 30

 图 4 算例case5的3×10 min多步推进过程Fig. 4 Example of case5,multi-stepping advanced process of 3×10 min

 图 5 传输线积冰算例最终冰形的对比图Fig. 5 Comparison of final ice shapes on transmission line

 图 6 CFD计算的圆柱上表面流动分离图像Fig. 6 Vectors image of separated flow on upper part of circle cylinder calculated by CFD
5 结 论

1) 水膜-冰层模型具有明显的物理意义和严格的数学形式,在二维积冰问题上的应用是可行的.

2) 采用水膜-冰层模型计算的冰形能够较准确地预测出冰层在翼型上下表面延伸的极限位置以及冰角的位置和形状.

3) 以冰风洞实验冰形为参照,低温环境下基于水膜-冰层模型的翼型冰形曲线是能够与传统Messinger模型比拟的,相对温暖环境下水膜-冰层模型的计算结果要优于Messinger模型的模拟冰形,完全可以满足工程计算的需要.

4) 通过与实验和文献公布的数据比较,水膜-冰层模型同样可以有效预测传输线积冰,计算结果与圆柱体积冰的实验结果相当吻合;这种类型的积冰采用Messinger模型计算十分困难.水膜-冰层模型在空气切应力不占支配作用的低速积冰条件下同样有效,体现了模型的广泛适用性.

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#### 文章信息

Hou Shuo, Cao Yihua

Numerical simulation of two dimensional ice accretion based on lubrication theory

Journal of Beijing University of Aeronautics and Astronsutics, 2014, 40(10): 1442-1450.
http://dx.doi.org/10.13700/j.bh.1001-5965.2013.0625