Fast, High Order Vortex Methods Based on Deforming Basis Functions

这个项目扩展和增强了现有的基于变形基函数的涡旋法的现有知识。涡旋方法将流动的涡量场近似为局部基函数的线性组合。这些基函数随基函数质心处的流速而移动，速度场由涡量场通过Biot-Savart积分得到。对于以孤立涡度区域为主的流动，涡旋方法提供了相当大的优势，因为它们是自然自适应的，因为计算单元专门用于具有涡度的区域。大多数涡旋法使用刚性的轴对称基函数，但也有一些例外。本项目研究涡旋方法，它使用随局部流动偏差而变形的基函数，并建立在最近发展的基于变形椭圆高斯的高空间阶涡方法的基础上。椭圆高斯分布的特殊之处在于，它们代表了相关对流扩散方程的自相似精确解。要实现这样的方案，必须开发计算椭圆高斯的Biot-Savart积分的方法，并确定整个方法的相关收敛参数。虽然已经开发了一个二维方案，但原理调查员将通过开发快速求和算法来增强这种方法。原理调查员将把这种方法扩展到三维，在涡流拉伸和涡流细丝的几何形状之间提供关键的联系。这种新方法将被用来仔细计算各种问题，包括涡偶极子碰撞和喷流瞬变。这项工作提供了一种快速而准确地计算各种问题的流体流动特性的方法，包括但不限于航空航天应用、工业过程、洋流和各种大气流动。首席调查员还将沿着新的路线扩展这些概念，以开发能够准确计算水分和污染物在非饱和土壤中的运动和扩散的方法。这种被称为“涡旋法”的方法是不寻常的，因为这些方法自然是自适应的。自然自适应方法将计算资源专门用于流的主要区域。这些方法通过计算流体中局部角动量的演化来模拟流动。角动量通常被限制在流体总体积的一小部分内。例如，天气系统是由代表角动量集中区域的一组风暴系统驱动的。使用这种方法，人们可以仅根据角动量的演变来重建整个流场，这意味着计算机只需要在很小的区域内进行计算就可以捕捉整个流场。涡旋方法还有一个额外的优势，那就是可以很容易地在多台处理器计算机上并行化，这样就可以充分利用超级计算设施和连接的计算机网络。此外，首席调查员将把这些技术应用于一种全新的问题，涉及通过干燥土壤、粘土或混凝土等非饱和多孔介质的流动。与涡旋方法类似，这种新技术利用了这样一个事实，即水分的运动是由占据整个感兴趣区域的相对较小部分的狭窄的“首选路径”主导的。这些路径是由水分-介质相互作用创建的，计算机只需将其资源专门用于那些含有水分的区域。最后，这些活动的许多方面都是优秀的本科生研究项目。这些研究活动将提高学生对数学、环境科学和高性能计算的兴趣和知识。因此，除了它的科学价值，这个项目还将给马萨诸塞大学洛厄尔大学的学生带来有意义的研究经验，并加强几个学科的合作。

9971800This project expands and enhances existing knowledge of a new category of vortex method based on deforming basis functions. Vortex methods approximate the vorticity field of a flow as a linear combination of localized basis functions. These basis functions move with the flow velocity at the basis function centroid, and the velocity field is calculated from the vorticity field through a Biot-Savart integral. For flows dominated by isolated regions of vorticity, vortex methods offer considerable advantages because they are naturally adaptive in the sense that computational elements are dedicated exclusively to regions that have vorticity. Most vortex methods use rigid, axisymmetric basis functions though there are some exceptions. This project studies vortex methods that use basis functions which deform with local flow deviations, and builds upon the recent development of a high spatial order vortex method based on deforming elliptical Gaussians. Elliptical Gaussians are special in the sense that they represent self-similar, exact solutions to the relevant advection-diffusion equation. To implement such a scheme, one must develop methods for evaluating the Biot-Savart integral for an elliptical Gaussian and identify the relevant convergence parameters for the method as a whole. Though a two-dimensional scheme has already been developed, the Principle Investigator will enhance this method by developing a fast summation algorithm. The Principle Investigator will extend this method to three dimensions, providing a crucial link between vortex stretching and the geometry of vortex filaments. The new method will be used to perform careful calculations of a variety of problems including vortex dipole collisions and jet transients. Finally, the Principle Investigator will extend this approach to moisture transport through unsaturated porous media.This work provides a means of quickly and accurately calculating fluid flow properties on a wide range of problems including but not limited to aerospace applications, industrial processes, ocean currents and atmospheric flows of all sorts. The Principal Investigator will also extend these concepts along new lines to develop methods that can accurately calculate the motion and diffusion of moisture and contaminants in unsaturated soils. This type of method, called a "vortex method", is unusual in that these schemes are naturally adaptive. Naturally adaptive methods dedicate computational resources exclusively to the dominant regions of the flow. These methods simulate flows by calculating the evolution of the local angular momentum in the fluid. Often, the angular momentum is restricted to a small fraction of the total volume of the fluid. For instance, weather systems are driven by a collection of storm systems that represent concentrated regions of angular momentum. Using such a method, one can reconstruct the entire flow field based solely on the evolution of the angular momentum which means that the computer only need perform calculations over a small area to capture the entire flow field. Vortex methods have the added advantage of being easily parallelized on multiple processor computers so that one can take full advantage of supercomputing facilities and networks of connected computers. Also, the Principal Investigator will apply these techniques to an entirely new type of problem involving flow through unsaturated porous media such as dry soils, clays or concrete. Similar to vortex methods, this new technique takes advantage of the fact that the movement of moisture is dominated by narrow "preferred paths" occupying a relatively small fraction of the total domain of interest. These paths are created by moisture-media interactions, and the computer need only dedicate its resources to those regions containing moisture. Finally, many aspects of these activities make excellent undergraduate research projects. These research activities will enhance students' interests and knowledge in mathematics, environmental science and high performance computing. Thus, in addition to its scientific merits, this project will give students at the University of Massachusetts Lowell meaningful research experiences and enhance collaboration across several disciplines.