多孔纺织材料中多相多组分流模型的数值计算与分析

Multiphase and multi-component flow model in porous textile materials attracts considerable attention in recent years. This model arises from the design of functional clothing in textile industries. Since the different physical performance of air and vapor in the model, the movement of air and vapor can be seen as a multi-component flow of binary gas mixutre. The different distribution of temperature, gas mixture concentration and pressure gradient result in heat conduction/convection and gas mixture diffusion/convection coupled with complex phase changes. The phase changes, condensation/evaporation,absorpsion,freezing and heat radiation influence the performance of clothing dierctly. The model can be described by a system of nonlinear, coupled parabolic partial difference equations. Since the strong nonlinearity and coupling of this system, complex geometries, discontinuous coefficients and long time simulation of practicle interests, an efficient numerical method is needed to handle these complicated situations. In this project, the main research goal is: propose an efficient finite difference method for the three-dimensional multiphase and multi-component flow model and show the optimal error estimate for proposed method. This is a cross subject of engineering application and numerical methods for PDEs, its achievements will not only enrich the theoretical knowledge of related disciplines but also have broad application prospects.

多孔纺织材料中多相多组分流模型的研究是近年来的热点问题，该模型来自于纺织品工业中功能型服装的设计。由于模型中空气和水蒸气的不同物理表现，空气、水蒸气的运动被看做二元混合气体的多组分流运动。温度、混合气体浓度和压力梯度的不同分布导致了热的传导、对流以及混合气体的扩散、对流并伴随有复杂的相位变化。冷凝/蒸发、吸水、结冰和热辐射等相位变化会直接影响服装的性能。模型可由一组非线性、耦合的偏微分方程组来描述。由于非线性、强耦合性、复杂几何、间断系数和长时间模拟的实际需求，我们需要设计有效的数值方法处理这些复杂的情况。本项目的主要研究目标是：提出有效的有限差分算法求解多相多组分流的三维模型并对算法进行分析以得到算法的最优误差估计。本项目属于工程应用与偏微分方程数值方法的交叉研究课题，所得的成果不仅能丰富相关学科的理论知识，而且有广阔的应用前景。

本项目研究多孔纺织材料中的多相多组分流模型，该模型来自于纺织品工业中功能型服装的设计。本项目着重考查水-水蒸气混合气体的多组分流运动以及伴随存在的复杂的相位变化，如冷凝/蒸发、纤维的吸水、纤维层的结冰等。本项目建立了一个非线性耦合的数学模型使模型能够科学的模拟复杂的物理过程和各种相位变化。其次，为了处理模型本身具有的较强的耦合性、非线性以及复杂几何、间断系数和长时间模拟的实际需求，我们用一组非线性、耦合的偏微分方程组来模拟模型并运用分离的Crank-Nicolson格式的有限差分算法求解模型。Crank-Nicolson格式的有限差分算法在时间方向上具有较高的精度，也能够处理模型较强的非线性和耦合性。本项目对上述分离的有限差分算法进行分析，并得到算法没有时间步长限制的最优误差估计，理论结果也得到了较大的加强。