Mathematical Sciences: Internal Layers in Fluid Flow and Solid Deformation

DMS 9732876 James Glimm摘要混合层、内部层和锋面是许多重要的流体流动和固体变形问题的主要特征。这些提出者在这些领域取得了惊人的进展，并在此建议为他们的研究系统地发展一种有效的理论和计算方法。研究层面和锋面的中心智力挑战涉及具有不同长度尺度的物理过程的复杂相互作用。一个层内的关键微观过程的长度尺度通常比嵌入该层的环境流体的长度尺度小几个数量级，但它们可能对流体流动的宏观结构产生深远的影响。因此，直接数值模拟需要特殊的算法，以便在合理的计算资源下，准确地求解重要的层组织。理论上的挑战是双重的，因为这个问题需要放大技术来理解宏观层面上微观过程的影响，然后需要方法来表达在可比长度尺度上运行的不同物理过程的非线性耦合。虽然对层和锋面的研究至少是一个两个尺度的问题，但这里感兴趣的是实际应用，这通常涉及一个层内的多个长度尺度。该项目将提供(A)用于模拟复杂流体流动和固体变形的新一类生产程序的理论和算法基础；(B)理解湍流多相流和流体混合的理论基础；(C)通过探索与物理实验一致的高分辨率模拟中的参数模式来验证湍流和混合模型；以及(D)关于非线性双曲波相互作用的新的数学理论，包括对在某些情况下可以防止或克服模拟程序表现不佳的现象的理解。当两种不同材料之间的接触表面不稳定时，表面上的小随机扰动将发展成相互渗透的结构，如手指、射流、液滴和气泡。这些结构复杂而混乱，它们共同构成了一个“混合层”。混合层在超新星爆炸等自然现象中以及在惯性约束聚变(ICF)等技术应用中都是至关重要的，在这些应用中，它们被认为是ICF胶囊性能的主要限制因素。通常，混合层被嵌入到大规模流体或固体流动中，在这种情况下，如果要使用计算机代码定期进行定量预测，则通过计算机模拟对混合层进行精确的分辨率是不切实际的(如果不是不可能的话)。因此，需要一种科学的理论来捕捉混合层中无法分辨的微观结构，使之符合计算机模拟的宏观框架。本项目的目标是发展这一理论，并通过与计算机模拟和物理实验的比较来证明其有效性。正如本提案所述，这项工作将需要新的想法，并将有助于建立科学的基本知识基础，对广泛的科学现象具有重要意义。

DMS 9732876 James Glimm Abstract Mixing layers, internal layers, and fronts are the primary features of many important fluid flow and solid deformation problems. The proposers have established striking progress in these areas, and propose here to develop systematically an effective theoretical and computational methodology for their study. The central intellectual challenge in the study of layers and fronts involves the complex interactions of physical processes with disparate length scales. The length scales of key microscopic processes within a layer are typically orders of magnitude smaller than those of the ambient fluid in which the layer is embedded, yet they may have a profound effect on the macroscopic structure of the fluid flow. For this reason, direct numerical simulation requires special algorithms to resolve the important layer microstructure accurately with reasonable computational resources. The theoretical challenge is two-fold, as the problem requires scale-up techniques to understand the effects of the microscopic processes at the macroscopic level, and then methods to express the nonlinear coupling of distinct physical processes operating on comparable length scales. While the study of layers and fronts is at least a two-scale problem, the interest here is in practical applications, which usually involve multiple length scales within a layer. This project will provide (a) the theoretical and algorithmic basis for a new class of production codes for the simulation of complex fluid flow and solid deformation; (b) the theoretical basis for understanding turbulent multiphase flow and fluid mixing; (c) validation of turbulence and mixing models, through exploration of parameter regimes in highly resolved simulations that are in step with physical experiment; and (d) a new mathematical theory for the interaction of nonlinear hyperbolic waves, including an understanding of phenomena which can in some cases prevent or overcome the poor perform ance of simulation codes. When the surface of contact between two distinct materials is unstable, small random disturbances on the surface will develop into interpenetrating structures, such as fingers, jets, droplets, and bubbles. These structures are complex and chaotic, and together they comprise a "mixing layer." Mixing layers are of fundamental importance in natural phenomena such as supernova explosions, and in technological applications such as inertial confinement fusion (ICF), where they are known to be a major limiting factor in the performance of ICF capsules. Typically a mixing layer is embedded in a large-scale fluid or solid flow, and in this case the precise resolution of a mixing layer by computer simulation is impractical (if not impossible) if the computer code is to be used to make quantitative predictions on a regular basis. For this reason, a scientific theory is required to capture the unresolvable microstructure of mixing layers in a macroscopic framework which is amenable to computer simulation. The goal of this project is to develop such a theory and to demonstrate its validity by comparison with computer simulation and physical experiment. This work will require new ideas, as described in this proposal, and will contribute to the fundamental knowledge base of science, of significance to a broad range of scientific phenomena.