Development of explicit algebraic LES wall models using consistent CFD-driven Machine Learning

A new consistent Machine Learning methodology to develop explicit algebraic wall models for Large Eddy Simulation (LES) is presented. Crucially, the wall models are trained using quantities from wall-modeled LES as inputs, and not filtered variables from a database, thus ensuring that resulting models are consistent with their final application environment. This is achieved by implementing the candidate wall models, created by the symbolic regression technique Gene-Expression Programming (GEP), into an LES solver and using the wall-modeled CFD outputs to construct the training cost function. The resulting models are explicit algebraic expressions and human-interpretable. Another benefit of this procedure is that it does not require dense high-fidelity data sets, although training is more computationally expensive. The framework was tested and validated using the well-established periodic hills case. Two cost functions were used separately: the mean skin friction coefficient distribution and a limited number of mean velocity profiles distributed over the domain. The wall models trained by the GEP algorithm were able to reconstruct the flow even with very under-resolved grids close to the wall for both scenarios. Despite the focus of the work being the Machine Learning method, we also tested the GEP model for other cases with similar physical features to analyze the flexibility of the model. The results are encouraging and show the potential of the developed training framework. An additional direct numerical simulation (DNS) is conducted of the periodic hills case to provide filtered ‘truth’ data. Comparing the inputs the wall model is provided from wall-modeled LES or from the filtered DNS data reveals substantial differences, both in terms of absolute values and range. This further justifies the present consistent approach of in situ wall-model training. • A new Machine Learning methodology to develop wall models for Large Eddy Simulation (LES) is presented. • The wall models developed are explicit algebraic functions and therefore human-interpretable. • The method is numerically consistent and driven by in situ CFD outputs. • The proposed framework does not require dense high-fidelity data sets. • The framework can be applied to flows with separation and large pressure gradients.