Efficient Numerical Methods for Viscous Incompressible Flows

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The investigator proposes to continue the development of efficient and accurate numerical methods for unsteady incompressible flow, with particular emphasis on (i) finding new equivalent formulations of Navier-Stokes equations, often inspired by physical considerations, better suited to numerical computation; (ii) proper handling of boundary conditions, a key difficulty in the subject since important physical interactions frequently occur near the boundary; (iii) rapid computation of three dimensional flows and generalization of schemes for flows in complex geometries; (iv) development of efficient schemes for complex fluids such as magneto-hydrodynamics, liquid crystal polymers, geodynamo, climate modeling, and large eddy turbulence simulations; (v) design of numerical schemes that preserves numerically conserved quantities such as energy and helicity for inviscid flow, and accurately captures their dissipation rates at high Reynolds number. It is widely believed that different dissipation rates for energy and helicity is the key mechanism leading to the formation of large coherent structures of turbulent flows.The numerical simulation of incompressible flows, which plays an important role in numerous scientific and industrial applications of current interest, is a challenging task for both numerical analysts and computational fluid dynamicists. The proposed fast and accurate numerical methods for incompressible flow is expected to become an important tool for simulation and analysis of complex turbulent flow phenomena including vortex breakdown, (massive) flow separation, vortex shedding, transient jets in cross-stream, wake-body interaction, high-swirl flow, etc. It is also an essential tool for the design of advanced flow control mechanisms used, for example, to reduce flow-induced noise and vibration, and to improve lift/drag performance at reduced energy consumption rates. Examples include flow over bluff bodies such as ground or under-water vehicles; in engines; in/around rotating machinery or in data storage units with rotating and moving parts.