Physics-Informed Machine Learning Approach in Augmenting RANS Models Using DNS Data and DeepInsight Method on FDA Nozzle

TL;DR

The paper tackles inaccurate Reynolds-stress and wall-shear predictions in FDA nozzle flows by augmenting RANS with DNS-derived corrections through an implicit–explicit OpenFOAM solver enhanced by machine learning. It leverages Galilean-invariant 47 Hilbert-basis invariants of $\{\mathbf{S}, \boldsymbol{\Omega}, \nabla p, \nabla k\}$ and outputs Reynolds-stress corrections via eigenvalue–eigenvector decomposition $\mathbf{\tau} = 2k(\frac{1}{3}\mathbf{I} + \mathbf{V}\Lambda\mathbf{V}^{T})$, with rotation alignment to DNS performed through quaternion-based axis–angle transformations. The approach combines an optimal turbulence viscosity $\nu_t^{L}$ derived from minimizing $\|R_{dev} + 2\nu_t S\|$ and ML mappings (MLP, Random Forests, and CNNs via DeepInsight) to predict $\Delta\tau$, $\Delta\tau^L$, and $\nu_t^L$, feeding back into the solver for improved velocity and Reynolds-stress fields that closely match DNS. Results on square-channel DNS and FDA nozzle-related flows show reduced discrepancies in secondary flows and Reynolds stresses, suggesting potential reductions in DNS cost and improved hemolysis-relevant predictions in medical devices.

Abstract

We present a data-driven framework for turbulence modeling, applied to flow prediction in the FDA nozzle. In this study, the standard RANS equations have been modified using an implicit-explicit hybrid approach. New variables were introduced, and a solver was developed within the OpenFOAM framework, integrating a machine learning module to estimate these variables. The invariant input features were derived based on Hilbert's basis theorem, and the outputs of the machine learning model were obtained through eigenvalue-vector decomposition of the Reynolds stress tensor. Validation was performed using DNS data for turbulent flow in a square channel at various Reynolds numbers. A baseline MLP was first trained at $Re=2900$ and tested at $Re=3500$ to assess its ability to reproduce turbulence anisotropy and secondary flows. To further enhance generalization, three benchmark DNS datasets were transformed into images via the Deep-Insight method, enabling the use of convolutional neural networks. The trained Deep-Insight network demonstrated improved prediction of turbulence structures in the FDA blood nozzle, highlighting the promise of data-driven augmentation in turbulence modeling.

Figures (46)

• Figure 1: Parallel architecture of convolutional neural network used for applying regression on output images from DeepInsight algorithm 10.1038/s41598-019-47765-6
• Figure 2: Secondary flow in a channel with a square section (numbers in colored columns indicate the size of the secondary velocity vector)
• Figure 3: The first 6 inputs of the neural network obtained from the converged RANS solution
• Figure 4: Difference of eigenvalues obtained from the deviatoric part of Reynolds stress of DNS and RANS data
• Figure 5: Unit and orthogonal eigenvectors of deviatoric Reynolds stress tensor