Session ZC30: Low-Order Modeling and Machine Learning for Turbulence II

Applying and optimizing Gene Expression Programming (GEP) applied to URANS modelling of cloud cavitating flows

Cavitation is characterized by formation of vapor bubbles when the liquid pressure falls below the vapor pressure. These bubbles travel with high velocities and burst upon exiting the low-pressure region, generating performance-degrading effects like shock, noise and vibration. Alternatively, cavitation has also been utilized for non-invasive surgical procedures and jet-drilling for the hydrocarbon industry. Thus, cavitation needs to be studied more using both experiments and numerical methods. Modelling an unsteady flow like cloud cavitating flows (where clouds of cavitating bubbles form and detach from the wall periodically) requires coupling of a cavitation and turbulence model, generally a URANS model due to its lower computational costs. However, recent studies show URANS models having considerable discrepancies when the turbulence properties are compared with experiments on a local scale. To overcome these drawbacks, Gene Expression Programming (GEP), a branch of machine learning based on an iterative survival-of-the-fittest concept is applied here. GEP is utilized specifically to correct the Boussinesq approximation, a standard assumption to compute Reynolds stress tensors in a URANS model. Here the Reynolds stress tensors are computed as a function of time-averaged velocities in X and Y directions, the void fraction and the Reynolds stress tensor values provided by experiments and traditional URANS models. However, GEP itself has several underlying factors like the population size, the number of generations, mutation index etc. that increase the variability of the "ideal" solution and its uncertainty. A dataset of solutions provided by GEP, separated by coefficients is created and a regularized linear regression technique is applied to it. This optimizes the solution coefficients to reduce the dependence on the inherent GEP factors and thus provide a stable, general relation to ameliorate the approximation. Employing this approach substantially improves the Reynolds stress modeling as compared to the URANS cases.   

Synthetic Lagrangian Turbulence by Generative Diffusion Models

Lagrangian turbulence is central to numerous applied and fundamental problems concerning the physics of dispersion and mixing across engineering, bio-fluids, atmosphere, oceans, and astrophysics. Despite exceptional theoretical, numerical, and experimental efforts over decades, no current models are capable of accurately reproducing statistical and topological properties of particle trajectories in turbulence. We propose a machine learning approach [1], based on a state-of-the-art Diffusion Model, to generate single-particle trajectories in three-dimensional turbulence at high Reynolds numbers, thereby bypassing the need for direct numerical simulations or experiments to obtain reliable Lagrangian data. Our model demonstrates the ability to quantitatively reproduce all relevant statistical benchmarks over the entire range of time scales, including the presence of fat tails distribution for the velocity increments, anomalous power law, and enhancement of intermittency around the dissipative scale. Surprisingly, the model exhibits good generalizability for extreme events, achieving unprecedented intensity and rarity. This paves the way for producing synthetic high-quality datasets for pre-training various downstream applications of Lagrangian turbulence.   

A dynamic recursive neural-network-based subgrid-scale model for large eddy simulation

One approach to developing a subgrid-scale (SGS) model for large eddy simulation involves obtaining the SGS stresses and resolved flow variables from filtered direct numerical simulation (fDNS) data and inserting them into a neural network (NN). However, due to the limitation of neural networks in extrapolation, previous NN-based SGS models were unable to be applied to untrained flows. To overcome this drawback, we have devised a recursive NN-based SGS model. This model is trained with forced homogenous isotropic turbulent (HIT) flows only, yet it demonstrates satisfactory results in turbulence statistics when applied to decaying HIT flows. Additionally, a dynamic approach akin to that of Germano et al. (1991) is implemented in this recursive NN-based SGS model to ensure that the SGS stresses are adequately reduced in laminar and near-wall turbulent flows. To assess the performance of the present SGS model, LESs of 3D Taylor-Green vortex flow, turbulent channel flow at Re τ =178 and turbulent boundary layer flow at Re θ =1410 are conducted. The results show that our dynamic recursive NN-based SGS model can accurately predict the turbulence statistics of these flows, despite being trained solely with forced HIT flows.   

Physics-guided deep learning for reconstructing small-scale structures in turbulent flows

Deep learning has garnered significant attention in fluid turbulence modeling and analysis. One such application involves using super-resolution (SR) algorithms to reconstruct small-scale structures from their larger counterparts in turbulent flows. Current SR algorithms are limited by the requirement of supervised training or unpaired high-resolution reference data, making them difficult to implement for practical fluid flow scenarios. Consequently, the development of physics-guided models that can take advantage of the multi-scale nature of turbulence becomes crucial. To address these challenges, we present a physics-guided self-supervised workflow based on deep neural networks for reconstructing small-scale structures in homogeneous isotropic turbulence. Through evaluation using various statistical metrics like spectra, structure functions, and probability density functions, we demonstrate the quality of the reconstruction, showing promising agreement with the ground truth data, even though the latter was not included during training. Our work opens up possibilities for reconstructing small-scale structures from large-eddy simulation data, providing prospects for further advances in this field.   

Large eddy simulation of flow over a circular cylinder using a neural-network-based subgrid-scale model and its application to complex turbulent flows.

A neural-network(NN)-based subgrid-scale (SGS) models are constructed for turbulent flow over a circular cylinder, for the long-term purpose of applying them to turbulent flow over/inside complex geometries. The filtered DNS data at Re_d=Ud/ν=3900 are used for training the NNs, where U is the free-stream velocity, d is the cylinder diameter, and ν is the kinematic viscosity. Various input variables and NN architectures are considered while keeping their output as the SGS stresses: for example, NN with and without fusion, test-filtered variables as well as grid-filtered variables as inputs, etc. The NN architecture with fusion shows good predictions for flow over a circular cylinder even with different grid resolutions and at higher Reynolds numbers than those of the trained conditions. We also show that the normalization of inputs and output with the free-stream velocity and cylinder diameter is not applicable to untrained geometries. Therefore, various normalizations of input variables are considered to construct general NN-based SGS models for more complex flows, and their results will be discussed at the presentation.   

Multi-agent reinforcement learning for subgrid-scale modeling of environmental turbulence

The accuracy of large-eddy simulations relies on closures that model the unresolved subgrid effects. Traditionally, such closure models are based on physical models of the structure of the subgrid-scale stress or the energy/enstrophy transfer and self-similarity.

More recently, supervised learning approaches have been extensively investigated as an alternative to traditional closure models. These approaches

learn the subgrid-scale closures from high-fidelity snapshots of the flow. Therefore, they require a large amount of data, which can be prohibitive to acquire, or non-existing, e.g., the direct numerical simulation of environmental flows of the atmosphere or the oceans.

We learn closure models using multi-agent deep reinforcement learning. This approach relies on statistics that can be calculated from a few system snapshots. The local invariants of the flow at sparsely distributed agents represent the state to match the expected long-term statistics of the system.

We demonstrate that the closure model accurately predicts probability distributions of forced two-dimensional and β-plane flows. We also evaluate the generalizability of the trained model to predict extreme events and other system parameters (e.g., at higher Re). 

Additionally, we draw interpretable statistical conclusions between the state invariant and the interscale enstrophy/energy transfers of the learned closure.   

Wall Modeling in LES of Turbulent Flows Using Reinforcement Learning

This work seeks to design a reinforcement learning (RL)-based wall models (WMs) for large-eddy simulation (LES) that can first recover the law of the wall in equilibrium flows before capturing the dynamics of non-equilibrium flows.

Actual WMs indeed fail to reproduce strong non-equilibrium effects especially when they are spatially diffused.

As a result, the potential of machine learning (ML)-based WMs becomes a promising solution.

Two essential requirements for the development of ML-based WM are the generalization to larger Reynolds numbers and the validation of fundamental physical laws.

The use of RL for WM development removes the high-fidelity data cost issues that exist for other supervised learning methods.

Furthermore, it can provide a high level of interpretability of the model behavior.

A novel RL WM, utilizing agents dispersed near the flow wall, is proposed in this study.

Initially, the model will be compared with existing ML-based WMs using equilibrium half-channel flow up to large Reynolds numbers.

The agents' states-action map will provide valuable insights into the model's behavior, thereby enhancing the interpretability of the model.

Following this, the model's performance will be evaluated against non-equilibrium half-channel flows, under medium to high pressure gradients in both the spanwise and streamwise directions.

This research is supported by the Independent Research Fund Denmark (DFF) under the Grant No. 1051-00015B.   

A Two Neural Network Subgrid Stress Model for Large Eddy Simulation

A spatial 3D Unet Convolutional Neural Network based on a tensor basis expansion of the subgrid stress tensor is designed to combine multi-scale features of turbulence while enforcing the subgrid stress tensor structure. A novel two neural network variant of this design is applied to predict the structure and the magnitude of the subgrid stress tensor separately, with a loss function that enforces physical quantities related to the subgrid stress tensor. The two neural network variant is analyzed in a priori and a posteriori settings with large eddy simulations of Forced Homogeneous Isotropic Turbulence and Channel Flow conditions. In an a priori setting, it is demonstrated that the two neural network concept is able to accurately predict the subgrid stress even when over 50 percent of the total energy (as compared to Direct Numerical Simulation) is filtered out and is an improvement over a one neural network concept. By training on different filter widths with varying size inputs, the neural network is shown to generalize to an intermediate filter width. Turbulent space-time correlations in a posteriori analysis are also conducted and compared with current subgrid stress models.   

Uncertainty quantification of a refrigeration pool utilizing k-epsilon turbulence reduced order model

The uncertainty of a three-dimensional turbulent natural convection transient is quantified in a geometry representing an idealized refrigeration pool. 

The assessment is carried out through the creation of a surrogate model, built utilizing Proper Orthogonal Decomposition and Galerkin projection applied to a K-epsilon turbulent Navier-Stokes formulation.

We discuss the uncertainties in the flow created by the heat released by an element during its cooling. Concretely, we consider the hypothesis that the heat released suffers deviations that follow the normal distribution around a certain nominal value, with different standard deviations and sample size. We study the effect of this on the mean results and their accuracy.

The expected results and its uncertainty are computed by Monte-Carlo method, calculating numerous solutions of the Initial Value Problem with the surrogate model. 

The performance of the method has been also assessed, demonstrating  the possibility of performing ten thousand calculations inside the working time of a day with a common desktop computer. Considering computational power, this methodology is usable for common practitioners.