CAREER: Optimized Computational Fluid Dynamics -- Towards Exact Numerical Methods for Conservation Equations

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This proposal outlines a research and education plan focused on computational fluid dynamics (CFD). The research component addresses one of the fundamental weaknesses of current numerical methods - dissipation and/or dispersion errors due to the spatial discretization of the first-order terms in the governing conservation equations. In marked contrast to the current state of the art, we propose a methodology for obtaining optimized numerical formulations, based on minimization of an objective function that provides an estimate of the local discretization error. The proposed optimization strategy will for the first time allow a truly adaptive numerical methodology that yields a "best case" solution for a particular problem using a particular grid mesh. The end result of this effort will be a complete, fully documented, and fully validated numerical framework for application to the governing equations of incompressible fluid flow, both steady and unsteady. Once developed, future extensions of the methodology to compressible flows and to other conservation equations will be straightforward. The educational component addresses a fundamental need for the nation and for the state of Mississippi - the attraction of high-school students to science and engineering careers. Using a coordinated team comprised of university researchers, industrial experts, outreach administrators and high-school teachers, we will implement a CFD project module into the Physics curriculum of four to six high schools in Mississippi. It is expected that the visual, interactive nature of computational simulation will have a positive impact on the students' learning, and more importantly on the student's attitude toward science and engineering. The five-year program will be assessed to determine its effectiveness in improving conceptual understanding of mechanics among high-school seniors, and in encouraging students to pursue science and/or engineering careers after high school. Intellectual Merit. The primary research focus in numerical methods for (general) computational fluid dynamics over the past four decades has been mitigation of discretization errors arising from the approximation of the first-order (convective) terms. While progress has been substantial, it is telling that it remains the primary focus to this day. A truly non-incremental, step-change advancement over the current state of the art requires an entirely new framework, and forms the motivation for this proposal. To date, all numerical formulations have been based on explicit prescriptions of the numerical derivatives as functions of the dependent variable field. These are often complex, involving higher-order reconstructions, limiters, etc., but they do not allow the numerical approximations to be adapted in response to estimates of the local or global numerical error. The proposed strategy employs general, adaptive forms of the numerical approximations, which are iteratively optimized concurrent with the numerical solution itself. In effect, the numerical discretization is prescribed using feedback control to provide an optimized solution that minimizes the numerical error. Preliminary results indicate that the proposed methodology has the potential to reduce numerical error by several orders of magnitude versus current approaches, and in some cases to yield solutions with essentially zero numerical error. Broader Impacts. The impact of the research component will be substantial, potentially influencing every scientific and engineering discipline that currently makes use of computational fluid dynamics. It is also believed that development of the new framework will spawn future research efforts into optimized numerical methods that will impact computational techniques in fields beyond CFD. The educational impact will also be substantial. The stated goal is the increased participation of graduating high-school students in science and engineering careers. Participants will be selected from high schools in rural Mississippi school districts, which educate disproportionately high percentages of disadvantaged and under-represented groups. This program will allow these students the opportunity to interact with university researchers and to utilize exciting scientific tools in ways that they currently cannot. The program will also foster an awareness of and an appreciation for science and technology that will have a positive, long-term impact regardless of their career choices. Additional impacts will arise from the participation of graduate and undergraduate students, including at least one female Ph.D. student who is already working with the PI as an undergraduate researcher and has committed to pursue graduate study in his research group.