Title: Mathematical Architecture for Models of Fluid Flow Phenomena

Speaker: Alexander Labovsky, Florida State University

Date/Time: Wednesday, February 9, 2011, 10:00A.M. to 11:00A.M.        

Location: CSRI Building/Room 90 (Sandia NM)

Brief Abstract: Turbulence modeling for NSE and coupled NSE systems (namely, magneto hydrodynamics) will be discussed in the first part of the talk. Magnetically conducting fluids arise in important applications including climate change forecasting, plasma confinement, controlled thermonuclear fusion, liquid-metal cooling of nuclear reactors, electromagnetic casting of metals, MHD sea water propulsion. In many of these, turbulent MHD (Magneto Hydrodynamic) flows are typical. The difficulties of accurately modeling and simulating turbulent flows are magnified many times over in the MHD case. They are evinced by the more complex dynamics of the flow due to the coupling of Navier-Stokes and Maxwell equations via the Lorentz force and Ohm’s law.

We consider the family of approximate deconvolution models (ADM) for the simulation of the large eddies in turbulent viscous, incompressible, electrically conducting flows. We prove existence and uniqueness of solutions, we prove that the solutions to the ADM-MHD equations converge to the solution of the MHD equations in a weak sense as the averaging radii converge to zero, and we derive a bound on the modeling error. We prove that the energy and helicity of the models are conserved, and the models preserve the Alfv´en waves. We provide the results of the computational tests, that verify the accuracy and physical fidelity of the models.

In the second half of the talk I will discuss an efficient numerical method for high dimensional Stochastic PDEs. Realistic simulations of complex systems governed by nonlinear partial differential equations must account for “noisy” features of modeled phenomena, such as material properties, coefficients, domain geometry, excitations and boundary data. “Noise” can be understood as uncertainties in the specification of the physical model. In real life applications one often knows only the statistical properties of the problem’s parameters, which results in the usage of stochastic partial differential equations. Also, in many of the applications the number of random variables is high; therefore one has to represent and integrate functions on a high dimensional set of parameters. An elegant way of representing such functions is the Analysis of Variance expansion (ANOVA), also used in different reports under the name HDMR (high-dimensional model representation).

CSRI POC: Jim Stewart, 505-844-8630