The immersed interface method (IIM) for interfaces immersed in fluids

该奖项是根据2009年美国复苏和再投资法案（公法111-5）资助的。本研究的主要目标是将浸入式界面法（IIM）的强度、分层网格和水平集方法相结合，以准确、高效地模拟界面问题。特别是，研究者和他的学生开发了流-固界面的层次网格IIM和流-流体界面的水平集IIM。分层网格为在固体中解析流动和几何图形提供了良好的分辨率，并允许处理开放远场边界条件的大域。分层网格IIM特别适合在移动帧中模拟自然界的飞行者或游泳者。在水平集方法中，流体-流体界面被捕获为由偏微分方程（PDE）演化的标量函数的水平集。水平集方法对于捕获受拓扑变化影响的接口具有很强的鲁棒性。跳跃条件发生在流体-流体界面上。当固体通过δ函数表示为集中在界面上的（奇异）力时，它们也出现在流固界面上。IIM的主要思想是直接将必要的跳跃条件合并到数值格式中。如果可用的跳跃条件有限，如何实现高精度？如何结合跳跃条件，如果他们是耦合的？如何确定强制移动的刚性实体运动的力？在这项研究中，研究者和他的学生回答了这些问题。特别是，他们应用广义泰勒展开式来构造具有有限跳跃条件的高阶有限差分，使用增广变量方法来实现耦合跳跃条件，并开发了一种边界条件捕获方法来模拟自由移动的刚性物体。本研究的智力价值体现在三个方面：各种数值方法的全面发展和实施，流体中界面计算建模的新方法的产生，以及本研究中产生的思想的可扩展性，以广泛的其他界面问题。流固和流固界面问题具有非常丰富的性质和技术应用。昆虫飞行是流固界面问题的一个很好的例子。昆虫飞行之所以吸引人，不仅因为它的美丽，而且因为它的非常规空气动力学具有重要的技术意义，特别是在帮助设计扑翼微型飞行器（MAVs）方面。一个众所周知的流体-流体界面问题的例子是在采油过程中。这个想法是通过注水将困在地下的石油挤出来。寻找抑制油水界面指状失稳的方法是采油的技术难题。为了研究这类界面问题，计算流体力学（CFD）已经发挥了非常重要的作用。CFD可以提供非常详细的流动数据，这些数据很难或不可能通过实验获得。CFD数据有助于理解流动物理，并有助于构建和验证简化的分析模型。由于流动的多样性和相关的复杂性，CFD技术的发展和改进仍然有很高的需求。在这项研究中，研究者和他的学生开发了CFD技术来实现各种界面问题的高保真模拟。他们改进了一些现有的CFD方法，结合了每种方法的优点，并开发了新的CFD方法。这项研究对揭示自然界飞行者和游泳者的非常规空气动力学和流体动力学具有重要影响。对润滑运输、气动液冷、激光焊接也有影响。它的教育影响包括：(1)将与研究相关的主题融入现有课程和新课程；(2)培养研究生进行基于仿真的研究；(3)吸引本科生参与研究。特别是，构建一个用户输入模块来模拟自然界的飞行者和游泳者，对于本科生来说是一个很好的夏季研究机会。

This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5).The key objective of this research is to combine the strength of the immersed interface method (IIM), hierarchical grids, and the level set method for accurate and and efficient simulation of interface problems.In particular, the investigator and his students develop the hierarchical-grid IIM for fluid-solid interfaces and the level set IIM for fluid-fluid interfaces. Hierarchical grids provide fine resolution for resolving flow and geometry at a solid and allow a large domain for treating open far-field boundary conditions. The hierarchical-grid IIM is particularly suitable for simulating nature's flyers or swimmers in moving frames. In the level set method, a fluid-fluid interface is captured as a level set of a scalar function which is evolved by a partial differential equation (PDE).The level set method is very robust for capturing interfaces subject to topological changes. Jump conditions occur at a fluid-fluid interface. They also appear at a fluid-solid interface when the solid is represented as a (singular) force concentrating at the interface through the delta function. The main idea of the IIM is to directly incorporate necessary jump conditions into a numerical scheme. How to achieve high accuracy if available jump conditions are limited?How to incorporate jump conditions if they are coupled? How to determine the force to enforce the motion of a moving rigid solid?In this research, the investigator and his students answer these questions. In particular, they apply generalized Taylor expansions to construct high-order finite differences with limited jump conditions, use augmented-variable approaches to implement coupled jump conditions, and develop a boundary condition capturing approach to model free-moving rigid objects. The intellectual merit of this research is reflected in three aspects: the comprehensive development and implementation of various numerical methods, the generation of new approaches for computational modeling of interfaces in fluids, and the extensibility of the ideas produced in this research to a broad range of other interface problems.Fluid-solid and fluid-fluid interface problems are very rich in nature and technological applications. Insect flight is a beautiful example of fluid-solid interface problems. Insect flight is fascinating not only because it is beautiful to human eyes but also because its unconventional aerodynamics has great technical importance, especially in helping design flapping-wing micro air vehicles (MAVs). One well-known example of fluid-fluid interface problems is in oil recovery. The idea is to push out the oil trapped in the ground by flooding with water. A technical problem for oil recovery is to find means to suppress the fingering instability at water-oil interfaces. To study such interface problems, computational fluid dynamics(CFD) has become a very important role. CFD can provide very detailed flow data, which are difficult or impossible to obtain experimentally. The CFD data help understand flow physics and help construct and validate reduced analytical models.Because of the variety of flows and associated complexity, development and improvement of CFD techniques is still in high demand. In this research, the investigator and his students develop CFD techniques to achieve high-fidelity simulation of various interface problems. They improve some existing CFD methods, combine the strength of each, and develop new ones.This research has impact on unveiling unconventional aerodynamics and hydrodynamics of nature's flyers and swimmers. It also has impact on lubricated transport, air-driven liquid cooling, and laser welding. Its educational impact includes: (1) blending research-related topics into existing and new curricula; (2) training graduate students for simulation-based research; and (3) engaging undergraduate students in the research. In particular, building a user input module for simulating nature's flyers and swimmers is a nice summer research opportunity for undergraduate students.