Direct Numerical Simulations of the Navier-Stokes Alpha Model

We explore the utility of the recently proposed alpha equations in providing a subgrid model for fluid turbulence. Our principal results are comparisons of direct numerical simulations of fluid turbulence using several values of the parameter alpha, including the limiting case where the Navier-Stokes equations are recovered. Our studies show that the large scale features, including statistics and structures, are preserved by the alpha models, even at coarser resolutions where the fine scales are not fully resolved. We also describe the differences that appear in simulations. We provide a summary of the principal features of the alpha equations, and offer some explanation of the effectiveness of these equations used as a subgrid model for three-dimensional fluid turbulence

Stochastic Fluid Dynamic Model and Dimensional Reduction

International audienceThis paper uses a new decomposition of the fluid velocity in terms of a large-scale continuous component with respect to time and a small-scale non continuous random component. Within this general framework, a stochas-tic representation of the Reynolds transport theorem and Navier-Stokes equations can be derived, based on physical conservation laws. This physically relevant stochas-tic model is applied in the context of the POD-Galerkin method. In both the stochastic Navier-Stokes equation and its reduced model, a possibly time-dependent, inhomoge-neous and anisotropic diffusive subgrid tensor appears naturally and generalizes classical subgrid models. We proposed two ways of estimating its parametrization in the context of POD-Galerkin. This method has shown to be able to successfully reconstruct energetic Chronos for a wake flow at Reynolds 3900, whereas standard POD-Galerkin diverged systematically

Consequences of Symmetries on the Analysis and Construction of Turbulence Models

Since they represent fundamental physical properties in turbulence (conservation laws, wall laws, Kolmogorov energy spectrum, ...), symmetries are used to analyse common turbulence models. A class of symmetry preserving turbulence models is proposed. This class is refined such that the models respect the second law of thermodynamics. Finally, an example of model belonging to the class is numerically tested.Comment: Published in SIGMA (Symmetry, Integrability and Geometry: Methods and Applications) at http://www.emis.de/journals/SIGMA

Fluid flow dynamics under location uncertainty

We present a derivation of a stochastic model of Navier Stokes equations that relies on a decomposition of the velocity fields into a differentiable drift component and a time uncorrelated uncertainty random term. This type of decomposition is reminiscent in spirit to the classical Reynolds decomposition. However, the random velocity fluctuations considered here are not differentiable with respect to time, and they must be handled through stochastic calculus. The dynamics associated with the differentiable drift component is derived from a stochastic version of the Reynolds transport theorem. It includes in its general form an uncertainty dependent "subgrid" bulk formula that cannot be immediately related to the usual Boussinesq eddy viscosity assumption constructed from thermal molecular agitation analogy. This formulation, emerging from uncertainties on the fluid parcels location, explains with another viewpoint some subgrid eddy diffusion models currently used in computational fluid dynamics or in geophysical sciences and paves the way for new large-scales flow modelling. We finally describe an applications of our formalism to the derivation of stochastic versions of the Shallow water equations or to the definition of reduced order dynamical systems