There is a paper titled "Lagrangian/Hamiltonian formalism for description of Navier-Stokes fluids" in PRL. After reading the paper, the question arises how far can we investigate turbulence with this approach? With all the mathematical machinery available to us in solving classical field theory and QFT, will it be of any help if we start with a Lagrangian density in taming the turbulence problem?
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Look into Smooth Particle Hydrodynamics for a lagrangian approach to fluid simulation.