Mathematical Sciences: Well-Posed Numerical Calculations of Free-Surface Flows

数学科学：自由表面流的适定数值计算

• 批准号: 9308075
• 金额: $ 3.25万
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1993
• 资助国家: 美国
• 起止时间: 1993-07-15 至 1994-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9308075&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Well; Posed; Numerical

9308075 Baker The investigator studies the motion of free surfaces in fluid flows by investigating the development of singularities in a related analytic problem. In many free-surface flows, the effects of viscosity and surface tension appear to be so small that they are neglected. However, attempts to solve the motion of the free surface by numerical techniques, boundary integral techniques for example, soon run into difficulties due to the formation of curvature singularities. Only when one of the fluids is effectively a vacuum do numerical calculations indicate that the motion of the free-surface is well-posed. Obviously, the effects of surface tension and viscosity, no matter how small the coefficients are, become very important close to the time of singularity development. Precisely how these regularizing agents prevent singularity formation, and what the nature of the subsequent motion is, are the main thrusts of this proposal. To achieve this objective, the investigator uses a simple mathematical view that describes singularity formation through the analytic extension of the boundary integrals into the complex arclength variable (or other suitable parametrization variable). In this complex plane, singularities are born, depending on the details of the initial conditions. These singularities then track through the complex plane, and may reach the real axis in finite time. At that point they become physically relevant. Preliminary studies of the influence of surface tension show that the singularities slow down and fail to reach the real axis in finite time, but they get extremely close. The close proximity of these singularities to the real axis plagues standard numerical methods, because of the high resolution needed to capture the extremely distorted free-surface. Instead, the investigator uses a method that represents any singularities in the complex plane explicitly, so that the rest of the behavior of the free-surface is analytic. In particular, this analytic part can be represented by Fourier series very accurately. The first task is to use this approach on a simpler classical free-surface flow problem, the fingering instability in a Hele-Shaw cell. The second is to explore the method on the Kelvin-Helmholtz instability of an interface between two incompressible fluids of equal density and in the presence of surface tension. Surfaces between immiscible liquids and gases occur abundantly in nature and technology. Some examples include the rise of bubbles in chemically reacting units, falling rain-drops, spray jets, and oil and water flowing in the ground. Mathematical models that attempt to describe the motion of surfaces typically exhibit great sensitivity to the details of the motion. Consequently, computer simulations can suffer from inaccuracies unless extraordinary care is taken. The investigator develops a conceptually new mathematical approach that will allow severely deformed geometries to be tracked reliably. In particular, the approach will be able to track the long term behavior of water penetrating oil in a Hele-Shaw Cell, a mathematically similar model for the motion of oil and water during secondary stage recovery from an oil-field. Once completely understood on this important problem, the approach will be adapted to the study of rising bubbles and dendritic formation during crystal growth. ***

小行星9308075 研究人员通过研究相关分析问题中奇点的发展来研究流体流动中自由表面的运动。 在许多自由表面流动中，粘性和表面张力的影响似乎很小，以至于可以忽略不计。 然而，试图解决的运动的自由表面的数值技术，边界积分技术，例如，很快就遇到困难，由于形成的曲率奇点。 只有当其中一种流体实际上是真空时，数值计算才表明自由表面的运动是适定的。 显然，表面张力和粘度的影响，无论多么小的系数，变得非常重要的奇异性发展的时间接近。 这些正则化因子究竟如何阻止奇点的形成，以及随后运动的性质是什么，是这个提议的主要推动力。 为了实现这一目标，研究人员使用一个简单的数学观点，通过分析扩展到复杂的弧长变量（或其他合适的参数化变量）的边界积分描述奇点的形成。 在这个复平面中，奇点诞生了，这取决于初始条件的细节。 然后这些奇点穿过复平面，并可能在有限时间内到达真实的轴。 在这一点上，他们成为物理相关。 对表面张力影响的初步研究表明，奇异点会减慢速度，在有限时间内无法到达真实的轴，但它们会变得非常接近。 这些奇点与真实的轴的紧密接近困扰着标准的数值方法，因为需要高分辨率来捕捉极度扭曲的自由表面。 相反，研究人员使用一种方法，明确表示任何奇点在复杂的平面，使其余的行为的自由表面是解析的。 特别地，该解析部分可以非常精确地由傅立叶级数表示。 第一个任务是使用这种方法对一个简单的经典自由表面流动问题，指状不稳定性的Hele-Shaw细胞。 第二部分是探讨等密度不可压缩流体界面在表面张力作用下的Kelvin-Helmholtz不稳定性。 不混溶的液体和气体之间的表面在自然界和技术中大量存在。 一些例子包括在化学反应单元中气泡的上升、下落的雨滴、喷射的射流以及在地下流动的油和水。 试图描述表面运动的数学模型通常对运动细节表现出极大的敏感性。 因此，除非特别小心，否则计算机模拟可能会有不准确之处。 研究人员开发了一种概念上新的数学方法，可以可靠地跟踪严重变形的几何形状。 特别是，该方法将能够跟踪水渗透油在Hele-Shaw单元中的长期行为，Hele-Shaw单元是一种数学上类似的模型，用于从油田二次开采期间的油和水的运动。 一旦完全理解这个重要的问题，该方法将适用于研究晶体生长过程中上升的气泡和树枝状晶的形成。 ***