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Most of the literature is targeting a special stochastic version of the deterministic Navier-Stokes equation without giving a derivation of the considered equation.

I'm searching for such a derivation. Moreover, the derivation should be based on the assumption that the trajectories of the fluid particles are perturbed (in contrast to the assumption that the velocity fluid is perturbed - which leads to additive noise).

I've searched for a long time and I've read many papers, but the only derivation I've found is Stochastic Navier-Stokes Equations for Turbulent Flows. However, the derivation is somehow vague and they impose special assumptions on the deterministic resolved velocity.

So, I would be very happy, if someone knows a book or paper with a detailed derivation.

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