New Methods for Reducing the Dimensions of Multiscale Fluid Dynamic Problems

 

KUMAR BOBBA
Aeronautics 205-45
California Institute of Technology
Pasadena, CA 91125-4500, USA
bobba@galcit.caltech.edu

[ poster ]

Abstract:

Central to any numerical simulation is the problem of representing a given partial differential equation by finite set of ordinary differential equations. This process is achieved through some projection technique. However these finite number of retained modes is very large and it is of considerable interest to project the dynamics of these large number of ordinary differential equations onto a proper low dimensional subspace on which most of the important dynamics evolve.

In this poster, we will introduce new techniques based on control theory for getting simple fluid models. These methods have considerable advantages like rigorous error bounds, transparent physics, etc. The main idea behind this method is deleting the weakly controllable and weakly observable states of the system after the controllability and the observability gramians of the system are aligned through a similarity transformation. The relative importance of a state in the input-output behavior of the system is given by the corresponding Hankel singular value. We will show that for Streamwise constant Navier-Stokes equations linearized about Couette flow the Hankel singular values drop very steeply. Computations done on Couette flow using spectral method, Fourier in span wise direction and Chebechev collocation in wall normal direction will be presented.

[ abstract ]