Machine Learning Methods for Turbulence Closure Modelling

Abstract

In the aerospace, automotive, chemical, nuclear, hydroelectric, and wind industries, numerical simulations of turbulent flows are relied upon to design safe and efficient systems. However, the nature of the equations which give rise to turbulence makes them staggeringly expensive to solve numerically. Over the past decades, various techniques and modelling approaches have been proposed for the practical simulation of turbulent flows. The core modelling difficulty is the turbulence closure problem: additional unknowns appear when averaged quantities are used in the governing equations. Reynolds-averaged Navier-Stokes (RANS) is the modern "workhorse" in many industries. RANS makes many simplifications and approximations in order to enable a computationally practical technique. In particular, a major error source in most RANS approaches is the use of an eddy viscosity approximation for the turbulence closure problem.

While turbulence closure modelling for RANS has been an ongoing research area for decades, recent advancements in the field of deep learning have renewed interest in this area. The main value proposition of deep learning in turbulence closure modelling is the additional performance unlocked via the ability to infer complex functional relationships from data. Machine learning offers a promising method to augment intuition, heuristics, and simple physical arguments that have been used to traditionally construct turbulence closure models. Though RANS is considered the most popular industrial approach for simulating turbulent flow, countless examples remain where existing turbulence models are unable to capture industrially relevant physics. While the idea sounds simple, the details of how exactly to exploit machine learning for turbulence modelling is an ongoing research area, attracting substantial attention in the field.

This thesis presents two directions for augmenting turbulence closure models via machine learning. The first direction utilizes machine learning to train a highly expressive closure mapping which corrects the Reynolds stress anisotropy tensor, a major source of error for turbulence models. The corrected anisotropy tensor is injected back into the momentum equation, thereby producing corrected mean fields. The mapping is generated by training a machine learning model to predict high-fidelity closure terms from low-fidelity input features. While formulating an appropriate training procedure and model architecture for constructing this mapping receives significant attention in this thesis, another critical issue is injecting the model predictions back into a numerical simulation. Feeding the outputs from a machine learning model into a coupled set of partial differential equations is a nontrivial process, which requires attention to the stability and conditioning of the numerical solution.

This thesis consists of several published articles that address numerous issues in the broad areas of training, model architecture, and injection of data-driven anisotropy mappings. The central novelties in this area are: the creation of the first curated dataset for the purpose of training these anisotropy mappings; the proposal of two stable and well-conditioned injection frameworks; the formulation of several anisotropy mappings; the proposal of specialized neural network architectures for anisotropy mappings; a detailed investigation into the generalizability of data-driven anisotropy mappings; and a new type of physics-informed loss function, termed "realizability-informed" learning, which embeds additional physics-based preferences into the learned anisotropy mapping.

While the majority of this thesis is focused on machine learning for generating anisotropy mappings, the second augmentation direction proposed is the calibration of turbulence model coefficients via Bayesian optimization. Traditional turbulence closure models contain several coefficients that can be used to tune their performance. Though tuning these coefficients can significantly enhance the performance of turbulence closure models, this tuning is not widely done. This thesis proposes a straightforward and automated procedure for tuning these coefficients, employing Bayesian optimization to efficiently locate their optimal values. Termed "turbo-RANS", the proposed calibration algorithm is demonstrated to efficiently tune coefficients within a standard turbulence closure model. A specialized objective function for the purpose of calibrating turbulence closure models is proposed. This objective function is data-flexible in that it can handle a mixture of dense, sparse, and integral parameter reference data from a variety of sources.

The recommended augmentation pathway depends on the type and availability of reference data. While machine learning augmented anisotropy mapping techniques are highly expressive, they require computationally expensive reference data. In the event that only sparse or integral parameter reference data is available, the proposed turbo-RANS algorithm can be used. Ultimately, the techniques proposed in this thesis provide a flexible set of options for leveraging data to enable accurate numerical simulations of industrially relevant turbulent flows.