In this study, the freezing of water inside the channel with constant wall temperature was studied numerically. At the entrance of the channel, the velocity profile is assumed. The thermophysical properties of the fluid, including density, thermal capacity, thermal expansion coefficient and viscosity, are function of the temperature and the volume fraction of ice. Nonlinear equations of continuity, momentum and energy are solved in a general form and solved with finite volume method. The SIMPLEC algorithm is employed to handle the pressure-velocity coupling for the continuity and momentum equations. The enthalpy-porosity method has been used to model the phase-change problem. In this technique, the interface between solid and liquid is called the mushy zone. In the mushy zone, due to the high viscosity of this zone, a Laplace equation for velocity is used to solve the equation of momentum. It was observed that with the advance along the channel, the continuous effect of the cooling of the walls created and increased the thickness of the ice layer. Creating the ice layer causes fluid flow obstruction and pressure drop, with the highest pressure drop in Fourier number 12.0. Investigating the thickness of the ice layer, it was shown that in the Re = 1800, when the wall temperature difference and the input fluid temperature is about 7 degrees, ice is not formed. However, when this difference reaches 12 degrees, the ice layer thickness in the steady state covers 20% of the channel diameter. By increasing the wall temperature difference and the input fluid to 17 degrees, this number is increased to about 35%. Also, the effect of gravity on the freezing of water inside the channel was investigated and it was observed that gravity has negligible affect on the freezing of water and can be discarded.

中文翻译：

---

重力作用下壁温恒定的层流水流瞬态冻结方程数值解

在这项研究中，数值研究了恒定壁温下通道内部水的冻结。在通道的入口处，假定速度分布。流体的热物理性质，包括密度，热容量，热膨胀系数和粘度，是温度和冰体积分数的函数。连续性，动量和能量的非线性方程以一般形式求解，并通过有限体积法求解。SIMPLEC算法用于处理连续性和动量方程的压力-速度耦合。焓-孔隙率方法已被用来模拟相变问题。在这种技术中，固体和液体之间的界面称为糊状区域。在糊状区域，由于该区域的高粘度，拉普拉斯速度方程用于求解动量方程。观察到，随着沿通道的前进，壁冷却的连续作用产生并增加了冰层的厚度。形成冰层会导致流体流动受阻和压力下降，其中最高的压力下降为傅立叶数12.0。研究冰层的厚度，发现在Re = 1800时，当壁温差和输入流体温度约为7度时，不会形成冰。但是，当该差达到12度时，稳态下的冰层厚度将覆盖通道直径的20％。通过将壁温差和输入流体增加到17度，该数字将增加到大约35％。也，