## Lagrangian averaging for compressible fluids

H. S. Bhat, R. C. Fetecau, J. E. Marsden, K. Mohseni, and M. West

*Multiscale Modeling and Simulation* **3**(4), 818-837, 2005.

This paper extends the derivation of the Lagrangian averaged Euler (LAE-α) equations to the case of barotropic compressible flows. The aim of Lagrangian averaging is to regularize the compressible Euler equations by adding dispersion instead of artificial viscosity. Along the way, the derivation of the isotropic and anisotropic LAE-α equations is simplified and clarified.

The derivation in this paper involves averaging over a tube of
trajectories η^{ε} centered around a given
Lagrangian flow η. With this tube framework, the Lagrangian
averaged Euler (LAE-α) equations are derived by following a
simple procedure: start with a given action, Taylor expand in terms of
small-scale fluid fluctuations ξ, truncate, average, and then model
those terms that are nonlinear functions of ξ. Closure of the
equations is provided through the use of *flow rules*, which
prescribe the evolution of the fluctuations along the mean flow.

DOI: 10.1137/030601739

Full text: BhFeMaMoWe2005.pdf