Inserting machine-learned virtual wall velocity for large-eddy simulation of turbulent channel flows

We propose a supervised-machine-learning-based wall model for coarse-grid wall-resolved large-eddy simulation (LES). Our consideration is made on LES of turbulent channel flows with a first grid point set relatively far from the wall ($\sim$ 10 wall units), while still resolving the near-wall region, to present a new path to save the computational cost. Convolutional neural network (CNN) is utilized to estimate a virtual wall-surface velocity from $x-z$ sectional fields near the wall, whose training data are generated by a direct numerical simulation (DNS) at ${\rm Re}_{\tau}=180$. The virtual wall-surface velocity is prepared with the extrapolation of the DNS data near the wall. This idea enables us to give a proper wall condition to correct a velocity gradient near the wall. The estimation ability of the model from near wall information is first investigated as a priori test. The estimated velocity fields by the present CNN model are in statistical agreement with the reference DNS data. The model trained in a priori test is then combined with the LES as a posteriori test. We find that the LES can successfully be augmented using the present model at both the friction Reynolds number ${\rm Re}_{\tau}=180$ used for training and the unseen Reynolds number ${\rm Re}_{\tau}=360$ even when the first grid point is located at 5 wall units off the wall. We also investigate the robustness of the present model for the choice of sub-grid scale model and the possibility of transfer learning in a local domain. The observations through the paper suggest that the present model is a promising tool for recovering the accuracy of LES with a coarse grid near the wall.