Modeling the wall shear stress in large-eddy simulation using graph neural networks

Abstract

As the Reynolds number increases, the large-eddy simulation (LES) of complex flows becomes increasingly intractable because near-wall turbulent structures become increasingly small. Wall modeling reduces the computational requirements of LES by enabling the use of coarser cells at the walls. This paper presents a machine-learning methodology to develop data-driven wall-shear-stress models that can directly operate, a posteriori, on the unstructured grid of the simulation. The model architecture is based on graph neural networks. The model is trained on a database which includes fully developed boundary layers, adverse pressure gradients, separated boundary layers, and laminar–turbulent transition. The relevance of the trained model is verified a posteriori for the simulation of a channel flow, a backward-facing step and a linear blade cascade.

Using graph neural networks for wall modeling in compressible anisothermal flows

“…This value implies that edge and node features that are within a distance up to N = 4 neighbors away from a given target wall node will influence the prediction. This value was found to be sufficient to discriminate non-equilibrium regions in Dupuy et al (2023b). The multilayer perceptrons f V ε , f E ε , f V π , and f E π are composed of n ℓ = 2 layers while the multilayer perceptron f V δ is composed of n ℓ + 1 hidden layers.…”

“…Demonstrating an ability to learn from such an heterogeneous database is crucial for the further development of machinelearning wall models with much larger datasets. The six incompressible isothermal simulations are the same as found in Dupuy et al (2023b), as summarized in Table 1: two fully developed channel flows at friction Reynolds number Re τ = 180 (CF1, (Agostini and Vincent, 2020) and Re τ = 950 (CF2, (Del Álamo and Jiménez, 2003;Lozano-Durán and Jiménez, 2014;Lozano-Durán and Jiménez, 2015); a three-dimensional diffuser corresponding to the geometry "Diffuser 1" of Cherry et al (2008) (3DD, (Ercoftac, 2022); a backward-facing step (BFS, (Pouech et al, 2019(Pouech et al, , 2021; a curved backward-facing step (APG, (Ercoftac, 2022); and a NACA 65-009 blade cascade on a flat plate such as studied experimentally by Ma et al (2011) and Zambonini et al (2017) at an incidence angle of 4°and 7°(N65) (Dupuy et al, 2023b). The five compressible anisothermal simulations are the simulations of two fully developed asymmetrically cooled/heated channel flows at friction Reynolds number Re τ = 180 (AC1, (Dupuy et al, 2018) and Re τ = 395 (AC2, (Dupuy et al, 2019), with a temperature ratio of 2 between the two walls; two fully developed symmetrically cooled channel flows at friction Reynolds number Re τ = 320 (SC1, Appendix A) and Re τ = 1150 (SC2, Appendix A), with a temperature ratio between the bulk flow and the walls of 1.1 and 3 respectively; and a cooled high-pressure turbine blade which corresponds to the test case MUR235 of Arts et al (1990) (L89, (Dupuy et al, 2020).…”

“…The edges of the graph are given by the edges of the mesh, or in other words the mesh connectivity. The present graph neural network wall model uses the Encode-Process-Decode architecture of Battaglia et al (2018) and Dupuy et al (2023b), represented schematically in Figure 2. This architecture has been selected for its ability to directly operate on the unstructured data of the simulation, without requiring the prior interpolation of the inputs.…”

“…The decoded output of the model at node is given by . Following Dupuy et al (2023b), the size of the latent space is at both nodes and edges and the number of processing steps is in the present study. This value implies that edge and node features that are within a distance up to neighbors away from a given target wall node will influence the prediction.…”

“…Data-driven wall-stress models were developed and assessed for various incompressible flow configurations, including fully developed wall turbulence and separated turbulent flows (Huang et al, 2019;Yang et al, 2019;Bae, 2020, 2022;Bhaskaran et al, 2021;Radhakrishnan et al, 2021;Zangeneh, 2021;Zhou et al, 2021;Bae and Koumoutsakos, 2022;Dupuy et al, 2023a). For complex configurations, Dupuy et al (2023b) introduced a machine-learning wall model that can directly operate on the unstructured grid of a LES, based on a graph neural network (GNN) architecture (Battaglia et al, 2018;Pfaff et al, 2020;Zhou et al, 2020). The relevance of the approach was demonstrated a priori and a posteriori for the modeling of the wall shear stress in incompressible isothermal flows.…”