The Lagrangian Averaged Navier-Stokes Equations with Applications to Turbulence Modeling

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0105004ShkollerThis three year research effort is founded upon the author's recent development of a novel Lagrangian averaging procedure for the Euler and Navier-Stokes equations of fluid dynamics. The method is based on expressing the exact Lagrangian fluid flow as a composition of a smooth deterministic volume-preserving Lagrangian flow and a rough stochastic flow consisting of volume-preserving near-identity transformations. This decomposition is asymptotically expanded about the identity, substituted into the variational principle, and then averaged. The resulting deterministic system of equations is termed the Lagrangian averaged Navier-Stokes (LANS) equations in the presence of viscosity, and the Lagrangian averaged Euler (LAE) equations in the ideal case when viscosity is absent. Both the LAE and LANS models are parameterized by a small spatial scale alpha, and are derived in such a fashion as to accurately reproduce the dynamics of the Euler and Navier-Stokes equations at spatial scales larger than alpha, while averaging (or homogenizing) the fluid motion at scales smaller than alpha. Unlike current approaches such as Reynolds averaged Navier-Stokes (RANS) or Large Eddie Simulation (LES) models which add artificial dissipation to the system to remove subgrid scales, the LAE/LANS equations preserve the underlying structure of the inviscid dynamics, namely, energy, helicity, and circulation, by instead using a geometric, nonlinear dispersive mechanism. As a result, our LANS model, unlike RANS or LES, does not artificially suppress intermittency, a fundamental feature of fluid turbulence. The resulting system is a set of dynamically coupled partial differential equations for the mean velocity field and covariance tensor. This system will be the backbone of a massive analytic and computational assault on the modeling and understanding of fluid turbulence.Although heavily studied by numerous researchers for over a century, incompressible fluid turbulence elusively remains one of the last great challenges of modern scientific exploration. Its understanding is of paramount importance in a wide range of engineering and physical applications, ranging from the design of airplanes and automobiles, to daily weather forecasts and global climate prediction. Roughly speaking, a flow becomes turbulent when all of the spatial scales in the fluid are activated, or in other words, when the fluid is moving so chaotically, as to create smaller and ever smaller vortices. In such a flow regime, the trajectory of each fluid particle appears unpredictable, yet the challenge is to derive a mathematical set of equations which can describe this unpredictable motion. About 150 years ago, the Navier-Stokes equations were introduced for this very purpose, and although it is now generally accepted that these equations do indeed provide a remarkable physical model of reality, it remains a mathematical mystery as to whether or not unique solutions to these equations exist for all time. Moreover, even numerical approximations of these equations on the world's fastest supercomputers are incapable of modeling the small-scale structures and patterns which are formed in a turbulent regime -- the computer simply runs out of memory long before it can simulate the prohibitively small vortical motion. The LANS model, described above, is intended to alleviate these fundamental difficulties, and make the computational simulation of turbulent flows feasible.