  In this paper, we propose normalizing flows (NF) as a novel probability
density function (PDF) turbulence model (NF-PDF model) for the
Reynolds-averaged Navier-Stokes (RANS) equations. We propose to use normalizing
flows in two different ways: firstly, as a direct model for the Reynolds stress
tensor, and secondly as a second-moment closure model, i.e. for modeling the
pressure-strain and dissipation tensor in the Reynolds stress transport
equation. In classical PDF closure models, a stochastic differential equation
has to be modeled and solved to obtain samples of turbulent quantities. The
NF-PDF closure model allows for direct sampling from the underlying probability
density functions of fluctuating turbulent quantities, such that
ensemble-averaged quantities can then be computed. To illustrate this approach
we demonstrate an application for the canonical case of homogeneous shear
turbulence. Training data are extracted from high-fidelity direct numerical
simulations (DNS).
