磁场作用下导电流体流动与传热问题的高精度数值方法与数值模拟

The flow and heat transfer of the electrically conductive fluids in the presence of magnetic field can be widely found in new energy, new materials and other emerging industries. The studies for the flow and heat transfer under the magnetic fields, especially the research on its numerical methods, are the advanced topics for magnetohydrodynamics (MHD) and its application fields, which have attracted the interest of an increasing number of researchers in view of its wide applications in many emerging industries. Analysis of this class of flows involves solving Maxwell equations of electrodynamics and incompressible Navier-Stokes equations of flow fluid. Incompressible MHD equations are computationally challenging and are technologically important in areas of metal casting, nuclear cooling, and deposition processes. This project will first establish and develop an accurate, efficient and robust finite difference method for the solution of the whole MHD system of equations based on the high-order compact finite difference schemes proposed successfully in the project for incompressible streamfunction-velocity formulation of Navier-Stokes equations. Then, accurate numerical investigations for the electrically conductive fluids under the influence of an external magnetic field will be performed under different nondimensional parameters, boundary conditions and geometrical figures, and mechanism analysis will also be given by the qualitative and quantitative comparison of flow structures and heat transfer characteristics. This study will contribute to extend the research idea of high-order compact method, and enrich research methods in Computational Fluid Dynamics, Numerical Heat Transfer and Computational Magnetohydrodynamics. Moreover, the project will help to exhibit deeply the flow and heat and mass transfer characteristics and their mechanism of the electrically conducting metal fluid.

磁场作用下导电流体流动与传热问题广泛存在于新能源、新材料等新兴工业产业中，对该问题的深入研究，是磁流体动力学（MHD）及其应用领域在当今世界非常重要的前沿性课题。本项目拟以不可压流动流函数-速度形式Navier-Stokes方程的高阶紧致差分方法研究为基础，逐步推广建立精确、高效、稳健的求解MHD控制方程和能量方程的高精度数值方法。在此基础上，对几个典型磁场作用下导电流体流动与传热问题进行精细数值模拟研究，定性与定量地模拟研究不同物性参数、边界条件以及几何形状对流场结构及传热特性影响的机理。本项目一方面可扩展高精度紧致差分方法的研究思路，丰富计算流体力学、数值传热学和计算磁流体力学的研究手段和内容；另一方面也有利于深刻揭示磁场对导电液态金属双扩散对流行为的影响，从而促进相关技术在工程中的应用。

利用数值方法深入研究磁场作用下导电流体流动与传热问题是磁流体动力学（MHD）在当今世界非常重要的前沿性课题，具有工程意义的MHD问题往往具有强磁场、多组分、大温差、非线性和多物理场耦合的诸多特点，由此给数值算法带来了巨大的挑战。 为实现不可压MHD问题的准确数值模拟，尤其是为保证完整模型下MHD问题的磁场散度为零的自然约束条件，本项目组开展了基于不可压流函数-速度形式Navier-Stokes(NS)方程的紧致差分格式研究，并将其推广到不可压MHD控制方程中。通过本项目支持，课题组完成与项目研究内容相关的学术论文17篇（其中已发表10篇），完成研究生学位论文5篇。本项目取得的主要研究成果如下：建立了针对直角坐标系和任意曲线坐标系的不可压流函数-速度形式NS方程2阶迎风型紧致格式，并在此基础上构造了更高阶的紧致差分格式；利用上述研究过程中的新思路，构造了一种可求解三维不可压纯流函数形式NS的高阶紧致差分格式；对于不可压MHD完整模型，本项目组也分别建立了基于直角坐标系和极坐标系的紧致差分格式，并对比研究了完整模型下与低Rem数简化模型的圆柱绕流问题，发现两者仅在特定条件下才能获得比较一致的结果；利用构造的算法，进一步开展了MHD自然对流和双扩散对流等问题的数值模拟，解释了其中的一些流动与传热机理。. 本项目建立和发展的非定常不可压MHD问题的基于流函数-速度形式控制方程的差分算法，可使非定常不可压MHD流动与传热传质问题的数值模拟工作在计算准确性、稳健性和高效性方面显著提高；通过探讨磁场作用下导电流体流动与传热传质系统中几何外形效应、流动稳定性以及传热传质特性与无量纲参数或物性参数的依赖关系，能促进计算流体力学、数值传热学和磁流体力学等相关学科的发展。