  Long-term predictions of nonlinear dynamics of three-dimensional (3D)
turbulence are very challenging for machine learning approaches. In this paper,
we propose an implicit U-Net enhanced Fourier neural operator (IU-FNO) for
stable and efficient predictions on the long-term large-scale dynamics of
turbulence. The IU-FNO model employs implicit recurrent Fourier layers for
deeper network extension and incorporates the U-net network for the accurate
prediction on small-scale flow structures. The model is systematically tested
in large-eddy simulations of three types of 3D turbulence, including forced
homogeneous isotropic turbulence (HIT), temporally evolving turbulent mixing
layer, and decaying homogeneous isotropic turbulence. The numerical simulations
demonstrate that the IU-FNO model is more accurate than other FNO-based models
including vanilla FNO, implicit FNO (IFNO) and U-Net enhanced FNO (U-FNO), and
dynamic Smagorinsky model (DSM) in predicting a variety of statistics including
the velocity spectrum, probability density functions (PDFs) of vorticity and
velocity increments, and instantaneous spatial structures of flow field.
Moreover, IU-FNO improves long-term stable predictions, which has not been
achieved by the previous versions of FNO. Besides, the proposed model is much
faster than traditional LES with DSM model, and can be well generalized to the
situations of higher Taylor-Reynolds numbers and unseen flow regime of decaying
turbulence.
