Abstract:
In this talk I'll discuss 2-Dimensional flows for the "shallow-water" equations. The model describes the motion of a finite volume of fluid taking place in container whose bottom is described by a paraboloidal like surface of the form:
z = (αx^{2} +βy^{2})/2, α>0, β>0
or more generally
z = a(x, y)
where a tends to infinity as (x^{2} + y^{2}) tends to infinity. The model includes gravity, coriolis, and viscous forces.
I'll discuss a set of a-priori estimates and other key facts about the solution of the viscous shallow-water system. I'll also present an interesting exact Rotating-Pulsating solution to this system.
Such Rotating-Pulsating solutions are well known for inviscid shallow-water equations.
I'll also present a Lagrangian reformulation of the
shallow-water system. This formulation holds in a fixed spatial domain which corresponds to the region in space which is ?wet? at time t = 0. It represents a somewhat more involved system of PDE's than the original Eulerian formulation but with this formulation we avoid having to explicitly track the unknown free-boundary where
the height of the water column vanishes.
The Lagrangian model induces a natural computational model. I'll discuss briefly the parallel implementation of the computational model and present performance results and timing comparisons for the parallel implementation
of the computational model; our speedups are impressive.
I'll conclude with two simulations which demonstrate that Lagrangian computational model produces steady-state or long-time solutions like the exact Rotating-Pulsating solutions discussed earlier. These results validate the effectiveness of the Lagrangian reformulation of the problem. |