12 - New Emerging Methods in Numerical Analysis: Applications to Fluid Mechanics

Summary

Summary

Our object in this article is to present some aspects of new emerging methods in numerical analysis and their application to Computational Fluid Dynamics. These methods stem from Dynamical Systems Theory.

The attractor describing a turbulent flow is approximated by smooth manifolds and by projecting the Navier-Stokes equations onto these manifolds we obtain new algorithms, the Inertial Projections. These algorithms have proven to be stable, efficient and well suited for long time integration of the equations. They can be implemented with all forms of spatial discretizations, spectral methods, finite elements and finite differences (possibly also wavelets).

Introduction

Algorithms that have been introduced at a time of scarce computing resources may not be well adapted to supercomputing and to the more difficult problems that are tackled at the present time or that we foresee for the near future.

For incompressible fluid mechanics (or thermal convection), by using the full Navier-Stokes equations (NSE) we can, at present time, compute flows at the onset of turbulence: in particular the permanent regime is not anymore a stationary flow. The flow can be time periodic if a Hopf bifurcation has occurred or the permanent regime can be an even more complicated flow. Examples of time dependent flows have been numerically computed in the case of the driven cavity (see e.g. [15], [16], [20], [4] and [34]) and for other types of flows (see e.g. [14], [33] and [30]). Section 12.2 of this article is a brief survey, for the CFD practitioner, of some basic and relevant concepts in Dynamical Systems Theory (behavior for large time of the solutions of the NSE).