Incremental unknowns, multilevel methods and the numerical simulation of turbulence

The purpose of this monograph is to describe the dynamic multilevel (DML) methodology applied to the numerical simulation of turbulence. The general setting of the Navier-Stokes equations is recalled, and also a number of basic notions on the statistical theory of turbulence. The practical limitations of direct numerical simulation (DNS) and the needs for modeling are emphasized. The objective of this article is to discuss the modeling and the numerical simulation of turbulent flows by multilevel methods related to the concept of approximate inertial manifolds (AIM). This mathematical concept stemming from the dynamical systems theory is briefly presented; AIM are based on a decomposition of the velocity field into small and large scale components; they give a slaving law of the small scales as a function of the large scales. The novel aspect of the work presented here is the time adaptative dynamical implementation of these multilevel methods. Indeed the DML methodology is based on the properties of this decomposition of the velocity field and on numerical arguments. Numerical analysis of multilevel methods for a simple system is proposed. DML methods, with some numerical and physical justifications, are described for homogeneous turbulence. The numerical results obtained are discussed. (C) Elsevier Science S.A.