Stochastic Parametric Forcing in Hydrodynamics

Arizona State University, Scottsdale AZ

Investigators

Abstract

To study noise in the Navier-Stokes equations directly is a formidable task, and so lower-dimensional reduced models are often sought. The investigators introduce a new computational framework for the study of high-dimensional nonlinear stochastic partial differential equations, and apply this to hydrodynamic problems for which there are well-controlled precision experiments with which to compare. The main characteristics of the approach are: (1) the noise is introduced in a physically motivated manner; (2) the stochastic problem is reduced to a problem for the mean flow and another for its variance; and (3) an efficient implementation is developed involving state-of-the-art spectral discretization and stochastic integration numerical techniques. The equations governing hydrodynamics, the Navier-Stokes equations, have been known for well over a century and have been successful in describing many observed physical flows. Nevertheless, the transition to complex flow and turbulence remains an outstanding challenge; it is not clear what role noise plays here. The investigators develop a novel framework for studying the effect of noise on the properties, particularly the stability and transition to turbulence, of dynamical systems modeling fluid flows. The combined use of mathematical modeling of stochastic processes, design of numerical algorithms, computer simulations of physical problems and direct comparison with experimental observations provides an enriching experience for the students directly involved in the project. The ideas are also incorporated into existing and new courses.