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Prof. Wei-Cheng Wang  ( National Tsing Hua University )

2008 - 06 - 26 (Thu.)
11:00 - 12:10
308, Mathematics Research Center Building (ori. New Math. Bldg.)

We introduce the generalized vorticity formulation for incompressible flows with a coordinate symmetry. Examples of such symmetric flows include axisymmetric flows and helical flows. In this formulation, the divergence free constraint is automatically satisfied and the equations are essentially decoupled, therefore better suited for numerical simulations. On the other hand, the axis of symmetry becomes domain boundary and is accompanied with a formal coordinate singularity. We begin with extra pole conditions to complement the regularity requirement for classical solutions. We then introduce corresponding Sobolev spaces that incorporate these pole conditions and proper weak formulation with these Sobolev spaces. These pole conditions are essential for the global regularity of axisymmetric flows. In case they are not satisfied, the singularity at the axis of symmetry will persist in time for inviscid flows. These pole conditions also play a crucial role in the design and analysis of high order numerical schemes. We will present the simulation of flow past a sphere with Reynolds number up to 100,000.