Session A29: Data-Driven Approaches to Fluid Dynamics

Unraveling Turbulent Pipe Flow Dynamics: Data-Driven Models at Re = 2500

Fluid flows can be computationally demanding due to the need for a large number of degrees of freedom (a high state-space dimension) for accurate simulation. However, despite this requirement, certain flows exhibit a phenomenon where their long-term behaviour can be effectively represented using a much smaller number of dimensions. This behaviour is due to the rapid dissipation of small-scale features through viscous diffusion, leading to the emergence of a finite-dimensional invariant manifold on which the long-time dynamics lie. In this work, we explore the minimal flow unit turbulent pipe flow at Re = 2500, where a fully resolved solutions requires O(10⁵) degrees of freedom. Using only data from this simulation we build models with fewer than 30 degrees of freedom that quantitatively capture key characteristics of the flow. Advanced autoencoders are used for dimension reduction and we apply both a function space approach (Koopman) and state space approach (neural ODE) to build these data-driven models.  Our results demonstrate that both frameworks can be used to generate data-driven dynamic models of spatiotemporally chaotic solutions to PDEs in manifold coordinates.   

Data-driven resolvent analysis of turbulent channel flow

Resolvent analysis of a turbulent flow (McKeon & Sharma, JFM, 2010) identifies the most responsive inputs, their gains, and the most receptive outputs for the system according to the linear part of its dynamics, which is forced by nonlinearity. Data-driven resolvent analysis (Herrmann et al., JFM, 2021) performs this task on linear flows relying only on time-resolved snapshot data and Dynamic Mode Decomposition (DMD). Application of the data-driven technique to turbulent flows requires specific treatment of the nonlinearity, which may otherwise contaminate the linear operator approximated by DMD. In this work, we explore the use of recently developed data-driven methods to address this challenge in the context of high-dimensional and nonnormal nonlinear dynamical systems, such as wall-bounded turbulent flows. Specifically, we investigate the use of ResDMD (Residual DMD, Colbrook et al., JFM, 2023) to quantify the error made by DMD when producing a linear model from recordings of dynamics that are inherently nonlinear. Furthermore, we discuss the application of LANDO (Linear and Nonlinear Disambiguation Optimization, Baddoo et al., PRSA, 2022) that leverages kernel regression to simultaneously fit models to the linear and nonlinear contributions to the dynamics in a dataset. Both approaches are tested on numerical simulation data of a turbulent channel flow in a minimal box unit.   

Low-dimensional data-driven modeling through temporally parameterized neural networks for geophysical forecasting

The dynamics of high-dimensional systems can often be described by low-dimensional models, as the presence of dissipation induces long-time dynamics to collapse onto a finite-dimensional manifold. Data-driven modeling though neural networks is a powerful tool for constructing such models, creating a nonlinear mapping between the high-dimensional state space and a low-dimensional latent space and forecasting the dynamics in the reduced dimensional state. For systems with time-dependent external forcing, there exist inherent difficulties in generating minimal dimensional models, as the data-driven models must learn the dynamics of both the system and the forcing. We overcome the need to learn the dynamics of the external forcing through the use of temporally parameterized neural networks, data-driven models that learn to supplement a predefined time-dependent parametrization of the dynamical system. Here, we apply our temporally parametrized neural networks the problem of forecasting sea surface temperatures, a system with inherent time dependent forcing due to seasonal variations from the orbit of the Earth around the Sun. We develop data-driven low-dimensional models using temporally parameterized neural networks and compare the dimension reduction and forecasting capabilities to those modeled with standard neural networks. We show that the temporally parameterized neural networks can reproduce the sea surface temperatures fields with a much smaller latent dimension than standard neural networks, while capturing the dynamics of the sea surface temperatures with comparable accuracy.   

Assessment of Sonic Boom Loudness of Supersonic Aircraft using Reduced Order Modeling

The development of novel supersonic passenger aircraft is dependent on the mitigation of sonic boom loudness to acceptable levels. However, the propagation of the pressure signature through the atmosphere requires access to the precise aerodynamic field around the aircraft, which can be computationally expense to acquire. This challenge is further exacerbated in multi-query problems, such as design space exploration and uncertainty quantification, as the simulation cost becomes prohibitive. This study introduces a reduced order modeling strategy to predict the near-field pressure signature of the aircraft and rapidly evaluate the sonic boom loudness. The performance of this approach is assessed for inviscid supersonic flow over a low-boom aircraft configuration at varying Mach numbers and atmospheric conditions using several integrated and field-level error metrics. The results of the study illustrate the robustness and efficiency of the methodology in sparse data sets and predicting high-dimensional fields generated during supersonic aircraft design.   

Data-driven compression of plunging airfoil wakes

Plunging airfoil generates diverse wake patterns depending on its plunging frequency and plunging amplitude. Developing a reduced-order model for such high-dimensional wakes is challenging due to their strong nonlinearity but crucial for understanding the complex airfoil wakes and enabling real-time flow analysis.  In this study, we leverage an observable-augmented autoencoder to obtain a reduced-order model that captures the underlying physics of plunging airfoil wakes. Our autoencoder, which incorporates drag coefficients into latent manifold identification, compresses the plunging airfoil wake data into only three variables. While these three variables may appear minimal, they hold sufficient information to accurately reconstruct the full flow fields for a variety of plunging cases. The three-dimensional representation by the autoencoder provides physically-tractable coordinates that express characteristic modes of the wakes, distinguishing different wake types and angles of attack. These observations suggest the present approach can promote the understanding of the complex airfoil wakes in a simple and trackable manner.  Furthermore, we will show in the talk that the wake data can be reconstructed from sparse sensors with the discovered latent coordinates.   

Station-data-driven temperature forecasting in South Korea using convolutional neural networks

Numerical weather prediction (NWP) has the drawback of high computational costs, prompting researchers to explore deep learning for weather forecasting. One specific area under investigation is temperature prediction, for which diverse outcomes were produced depending on region, grid resolution, and deep learning algorithms. In this study, we developed station-observed temperature predictions using convolutional neural networks. Our approach involves two main steps: transforming irregularly distributed station temperatures to data on the regular meshes and processing mesh data using convolutional neural networks. The first step corrects observed temperatures through "mean decomposition" taking into account the yearly and daily variations and "height adjustment" through potential temperature. Next, training is performed using mesh data, with a network designed to predict temperature at time t+h based on data at time t. Our model was benchmarked against climatology, persistence models, and traditional NWP. The results demonstrate the superiority of our proposed approach, outperforming NWP up to 12 hours of lead time.   

A POD-driven Machine Learning Algorithm to predict 3D Patient-Specific Aortic flows

Data-driven techniques are emerging as a credible alternative to traditional approaches like Computational Fluid Dynamics (CFD) to support personalised clinical decision-making processes to treat cardiovascular diseases. Here, we present a novel machine learning based approach to predict spatio-temporal 3D flow fields from clinical measurements such as 4DMR images in a patient-specific aorta.

The algorithm is driven by a Reduced Order Model (ROM) based on Proper Orthogonal Decomposition (POD). It consists of two neural networks, each trained separately. The first network is trained to predict POD coefficients directly from the flow rate at the aortic inlet, while the second network uses POD coefficient sequences from the past to predict future POD coefficients. Working synergistically, these networks can accurately predict POD coefficients throughout the entire cardiac cycle, even during diastole where traditional single network approaches fail to make reliable predictions because the blood flow rate is typically lower, providing less information. The predicted coefficients can then be used to compute velocity and pressure fields inside the 3D flow domain, allowing for more detailed medical analysis. This approach also has the potential for a wide range of applications beyond medical flow fields, extending to various domains of fluid dynamics.   

Data-driven constitutive relation reveals scaling law for hydrodynamic transport coefficients

Finding extended hydrodynamics equations valid from the dense gas region to the rarefied gas region remains a great challenge. The key to success is to obtain accurate constitutive relations for stress and heat flux. Data-driven models offer a new phenomenological approach to learning constitutive relations from data. Such models enable complex constitutive relations that extend Newton's law of viscosity and Fourier's law of heat conduction by regression on higher derivatives. However, the choices of derivatives in these models are ad hoc without a clear physical explanation. We investigated data-driven models theoretically on a linear system. We argue that these models are equivalent to nonlinear length scale scaling laws of transport coefficients. The equivalence to scaling laws justified the physical plausibility and revealed the limitation of data-driven models. Our argument also points out that modeling the scaling law could avoid practical difficulties in data-driven models like derivative estimation and variable selection on noisy data. We further proposed a constitutive relation model based on scaling law and tested it on the calculation of Rayleigh scattering spectra. The result shows our data-driven model has a clear advantage over the Chapman-Enskog expansion and moment methods.