Mathematical Sciences: Computational Analysis of Wave Propagation in the Presence of Multiple Scales

9304406 Hagstrom研究者对多时空尺度下非线性波传播的数值模拟进行了研究。该项目涉及新型数值方法的开发、分析和实施，以及它们在流体动力学中具有挑战性问题的应用。多尺度问题在计算上是困难的，因为最小尺度的统一分辨率将导致近似解中有大量的自由度。它们也是科学技术上重要的流体流动现象的共同特征。在数学上，它们通常与控制方程的奇异摄动有关，并且可以用渐近技术进行分析。研究者使用渐近分析在一些方法来指导和补充数值计算。需要详细研究的具体问题包括在微可压缩流动中，声波与涡量浓度的产生和相互作用，复杂反应和扩散机制对火焰动力学的影响，以及圆柱形不可压缩流体中的非线性波。微可压缩流具有两种不同的时间尺度，快速尺度与声波传播有关，缓慢尺度与不可压缩流效应有关。例如，在空气声学中，理解这些不同尺度上的动力学之间的相互作用是很重要的。所考虑的数值问题包括在长时间计算的人工边界处建立精确的辐射边界条件和开发有效的时间步进程序。针对燃烧问题，研究人员尝试了多种求解零马赫数反应流动方程的高阶差分法。利用自适应方法捕捉薄火焰前缘，他对化学动力学和输运系数的各种模型进行了详细的研究。对于圆柱流的模拟，他开发了有效的切比雪夫伪谱方法，用于在圆柱和螺旋坐标系中使用坐标映射的微分算子来求解高雷诺数的内部临界层。一个应用将是解决管道中向湍流过渡的长期问题。波传播科学有着丰富的历史，它为分析不同的物理现象提供了统一的方法。先进的数值方法与现代计算技术相结合，是全面研究非线性波动力学的有力工具。然而，当存在不同的尺度时，计算方法会遇到困难。这些可以是时间尺度，不同的波以非常不同的速度传播，或者是空间尺度，某些物理量在相对较薄的层中变化很大。即使是最强大的计算机，要在整个物理领域中完全解决最精细的尺度也太费力了。因此，必须在对解的结构进行仔细的数学分析的基础上，发展专门的方法。这里考虑的具体问题显示了多个时间尺度（声波-涡旋相互作用）和薄层（火焰中的反应区）。从物理学的角度来看，它们是基本问题，也为新的计算技术提供了一个试验台。＊＊＊

9304406 Hagstrom The investigator conducts research on the numerical simulation of nonlinear wave propagation in the presence of multiple temporal and spatial scales. The project addresses both the development, analysis, and implementation of novel numerical methodologies, and their application to challenging problems in fluid dynamics. Multiple scales problems are computationally difficult because the uniform resolution of the smallest scales present would lead to a prohibitively large number of degrees of freedom in the approximate solution. They are also a common feature of scientifically and technologically important fluid flow phenomena. Mathematically, they are generally associated with singular perturbations of the governing equations, and can be analyzed by asymptotic techniques. The investigator uses asymptotic analysis in a number of ways to guide and complement the numerical computations. Particular problems to be studied in detail include the generation and interaction of sound waves with concentrations of vorticity in slightly compressible flows, the effects of complex reaction and diffusion mechanisms on flame dynamics, and nonlinear waves in incompressible fluids in cylindrical geometries. Slightly compressible flows have two distinct time scales, the fast scale being associated with the propagation of sound waves and the slow with incompressible flow effects. Understanding the interaction between the dynamics on these disparate scales is of importance, for example, in aeroacoustics. Numerical problems considered include the construction of accurate radiation boundary conditions at artificial boundaries for long time computations and the development of efficient time-stepping procedures. For the combustion problems, the investigator experiments with various high order difference methods for solving the zero Mach number reacting flow equations. Using adaptive methods to capture the thin flame front, he undertakes a detailed study of various models for the chemical kinetics and transport coefficients. For the simulation of cylindrical flows, he has developed efficient Chebyshev pseudospectral methods for differential operators in cylindrical and helical coordinates with coordinate mappings to resolve internal critical layers for high Reynolds numbers. One application will be to the long-standing problem of transition to turbulence in pipes. The science of wave propagation has a rich history and provides a unified approach to the analysis of diverse physical phenomena. Advanced numerical methods, in concert with modern computing technology, are a powerful tool for the comprehensive study of the dynamics of nonlinear waves. The computational approach runs into difficulties, however, when disparate scales are present. These can be time scales, where different waves propagate at very different speeds, or spatial scales, where certain physical quantities vary greatly in relatively thin layers. To fully resolve the finest scales throughout the physical domain is far too taxing for even the most powerful computers. Therefore, one must develop specialized methods, based on a careful mathematical analysis of the solution structure. The specific problems considered here display both multiple time scales (sound wave - vortex interactions) and thin layers (reaction zones in flames). They are fundamental problems from the physical point of view, and also provide a testbed for new computational techniques. ***