  We study the Lagrangian flow associated to velocity fields arising from
various models of fluid mechanics subject to white-in-time, $H^s$-in-space
stochastic forcing in a periodic box. We prove that in many circumstances,
these flows are chaotic, that is, the top Lyapunov exponent is strictly
positive. Our main results are for the Navier-Stokes equations on $\mathbb T^2$
and the hyper-viscous regularized Navier-Stokes equations on $\mathbb T^3$ (at
arbitrary Reynolds number and hyper-viscosity parameters), subject to forcing
which is non-degenerate at high frequencies. As an application, we study
statistically stationary solutions to the passive scalar advection-diffusion
equation driven by these velocities and subjected to random sources. The
chaotic Lagrangian dynamics are used to prove a version of anomalous
dissipation in the limit of vanishing diffusivity, which in turn, implies that
the scalar satisfies Yaglom's law of scalar turbulence -- the analogue of the
Kolmogorov 4/5 law. Key features of our study are the use of tools from ergodic
theory and random dynamical systems, namely the Multiplicative Ergodic Theorem
and a version of Furstenberg's Criterion, combined with hypoellipticity via
Malliavin calculus and approximate control arguments.
