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

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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. ***