Study of Dispersive Waves and Development of Accurate Nonreflecting Boundary Conditions

Old Dominion University Research Foundation, Norfolk VA

Investigators

Abstract

In numerical simulations that involve an open physical boundary, the infinite domain is necessarily truncated due to the limitation of finite computational resources, creating virtual boundaries that need to be reflectionless to out-going waves. It remains a significant challenge to formulate correctly the nonreflecting boundary condition with high accuracy and efficiency for many important applications in computational sciences where the governing equations are nonlinear or have non-constant coefficients. The proposed research is aimed at developing accurate nonreflecting boundary conditions based on the Perfectly Matched Layer (PML) methodology that can be used to produce potentially reflectionless absorbing boundaries. In the past few years, considerable progresses have been made in constructing PML for the linearized Euler equations with constant coefficients. The proposed work will resolve a key issue in extending PML from constant to non-constant mean flows and lead to the development of PML for nonlinear Euler and Navier-Stokes equations. An essential element in this research is the understanding of phase and group velocities of waves supported by a bounded flow. The proposed approach is genuinely multi-dimensional and extendible to nonlinear problems. Issues about mathematical formulation, stability and well-posedness of the partial differential equations, numerical accuracy and computational advantages of the proposed method, in particular the role of dispersive waves, will be studied. Theoretical as well as computational results will be compared with existing methods. This project will also provide training of graduate students in which a more traditional applied math topic, dispersive wave analysis, is closely coupled with a core scientific computing issue. Due to ubiquitousness of nonreflecting boundaries in computational sciences, the findings of this research are expected to have a broad and significant impact on a wide range of interdisciplinary scientific and engineering applications, such as those in the study of turbulence and transition, turbulent mixing, boundary layer control, aeroacoustics, underwater acoustics and atmospheric wave modeling. The use of the proposed new method, when successful, will also improve the quality of existing computational fluid dynamics codes both in the academia and industry.