F. Jauberteau

A nonlinear Galerkin method for the Navier-Stokes equations

Abstract Modern large scale computing allows the utilization of a very large number of variables/modes for spatial discretization. Therefore the computer tends to be saturated by computations on small wavelengths that carry a small percentage of the total energy. We advocate the utilization of algorithms treating differently small wavelengths and large wavelengths and we present here an algorithm of this sort, the nonlinear Galerkin method, stemming from the dynamical system theory.

Incremental unknowns, multilevel methods and the numerical simulation of turbulence

Abstract 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.

Solution of the incompressible Navier-Stokes equations by the nonlinear Galerkin method

Our aim in this article is to study a new method for the approximation of the Navier-Stokes equations, and to present and discuss numerical results supporting the method. This method, called the nonlinear Galerkin method, uses nonlinear manifolds which are close to the attractor, while in the usual Galerkin method, we look for solutions in a linear space, i.e., whose components are independent. The equation of the manifold corresponds to an interaction law between small and large eddies and it is derived by asymptotic expansion from the exact equation. We consider here the two- and three-dimensional space periodic cases in the context of a pseudo-spectral discretization of the equation. We notice however that the method applies as well to more general flows, in particular nonhomogeneous flows.

A Dynamic Multilevel Model for the Simulation of the Small Structures in Homogeneous Isotropic Turbulence

In turbulence simulations, the small scales of motion, even if they carry only a very small percentage of the whole kinetic energy, must be taken into account in order to accurately reproduce the statistical properties of the flows. This induces strong computational restrictions. In an attempt to understand and model the nonlinear interaction between the small and large scales, a dynamic multilevel procedure is proposed and applied to homogeneous turbulence. As in large eddy simulation, filtering operators are used to separate the different scales of the velocity field. In classical models (Smagorinsky), only the large scale equation is resolved. A different approach is proposed here. Indeed, by analyzing the nonlinear interaction term in the large scale equation, we show that they locally have a very small contribution to the whole dynamic of the flow. We then propose to treat them less accurately. Specific treatments for these terms are achieved by a space and time adaptative procedure; the cut-off value (filter width) which defines the scale separation varies as time evolves. Simulations at Reλ in the range of 60 to 150 have been performed until statistical steady states are reached, i.e. over long time period. Comparisons with direct simulations (DNS) show that this numerical modeling provides an efficient resolution of the nonlinear interaction term. The multilevel algorithm is shown to be stable; the corresponding simulated flows reach a statistically steady state very close to the DNS ones. The shape of the energy spectrum functions as well as the characteristic statistical properties of the velocity and its derivatives are accurately recovered.

Influences of subgrid scale dynamics on resolvable scale statistics in large-eddy simulations

Recently, the e-expansion and recursive renormalization group (RNG) theories as well as approximate inertial manifolds (AIM) have been exploited as means of systematically modeling subgrid scales in large-eddy simulations (LES). Although these theoretical approaches are rather complicated mathematically, their key approximations can be investigated using direct numerical simulations (DNS). In fact, the differences among these theories can be traced to whether they retain or neglect interactions between the subgrid-subgrid and subgrid-resolvable scales. In this paper, we focus on the influence of these two interactions on the evolution of the resolvable scales in LES: the effectA which keeps only the interactions between the small and large scales; and, the effectB which, on the other hand, keeps only the interactions among the subgrid-subgrid scales. The performance of these models is analyzed using the velocity fields of the direct numerical simulations. Specifically, our comparison is based on the analysis of the energy and enstrophy spectra, as well as higher-order statistics of the velocity and velocity derivatives. We found that the energy spectrum and higher-order statistics for the simulations with the effectA (referred to, hereafter, as modelA) are in very good agreement with the filtered DNS. The comparison between the computations with effectB (referred to, hereafter, as modelB) and the filtered DNS, however, is not satisfactory. Moreover, the decorrelation between the filtered DNS and modelA is much slower than that of the filtered DNS and modelB. Therefore, we conclude that the modelA, taking into acciunt the interactions between the subgrid and resolvable scales, is a faithful subgrid model for LES for the range of Reynolds numbers considered.

Subgrid modelling and the interaction of small and large wavelengths in turbulent flows

Abstract Modelling the interaction of small and large eddies in a turbulent flow is an important part of the understanding of turbulence and an important task in subgrid modelling and computational fluid dynamics. In this article we describe a new approach to this problem based on dynamical systems theory: the principle is that the turbulent flow is described by a compact attractor that may be a complicated set (fractal) and we approximate the attractor by smooth finite dimensional manifolds: these manifolds provide an approximate interaction law for small and large eddies. After describing the method we report on numerical computations based on this approach. They show an improvement in stability and accuracy and a significant gain in computing time.

The nonlinear galerkin method for the two and three dimensional Navier-Stokes equations

In this article we have described numerical tests for a new discretization algorithm called the nonlinear Galerkin method. The results show that the algorithm is more stable and less time consuming than the usual (pseudo-spectral) Galerkin method. The algorithm is robust and well suited for large time integration because it takes into account the interaction between the small and large eddies, by neglecting the very small quantities. Also this algorithm can be adapted to other form of discretizations (finite elements, finite differences, wavelets...) ; this will be reported elsewhere.