High-order energy-stable embedded boundary method for compressible viscous flows

Auburn University, Auburn AL

Investigators

Abstract

Accurate computations of compressible (high-speed) flows in practical domains are crucial in numerous scientific and engineering applications, e.g., bridge/building vibration in hurricanes, wing flutter in aircraft, human speech/phonation, etc. These computations are challenging because handling complicated boundaries often leads to numerical errors that compromise the accuracy of the predictions. Moreover, setting up these computations (which involves meshing or grid generation) for complex domains can be time-consuming. This project will develop computational approaches (and a theory to evaluate and improve the numerical properties of those approaches) to eliminate the need for complex grid generation, while ensuring high-fidelity flow predictions. It will enable accurate simulations of fluid-structure interactions in flow situations that have been intractable so far, helping uncover the physical mechanisms that drive the interactions and devise strategies to control them. The research tasks will be paired with instructional plans to train undergraduate and graduate students in leveraging computational tools for scientific investigations. Additionally, summer research opportunities focused on flow visualization tools and algorithms will be provided to encourage undergraduate research participation. Embedded boundary (EB) methods simplify the handling of practical geometries; however, they have been restricted to low (first/second) order of accuracy when non-dissipative interior schemes are used because of EB instabilities and small-cell issues. This project will address several key challenges associated with the application of EB methods to compressible Navier-Stokes equations. First, a stability theory will be developed for the construction of high (fourth and higher) order energy stable EB schemes for multi-dimensional compressible flow equations. Second, the theory will be extended to derive shock-capturing EB schemes that are provably stable. Finally, the derived schemes will be implemented to simulate the flow-induced vibrations of an airfoil/wing in supersonic flows. The theoretical (energy or time) stability will be proven by utilizing the dimensionally-split structure of the proposed schemes, which allows the multi-dimensional scheme to be written and analyzed as combinations of one-dimensional schemes in individual directions. No-slip wall boundary conditions will be enforced via penalty terms at the EB to satisfy the energy stability constraints. The small-cell issue, commonly encountered in existing EB methods, will be addressed by constructing a dual grid, containing separate solution and flux points, where the flux-point (or cell) spacings will be constrained to remain finite when the solution point spacings vanish near the EB. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.