A transformer-based synthetic-inflow generator for spatially developing turbulent boundary layers

Abstract

This study proposes a newly developed deep-learning-based method to generate turbulent inflow conditions for spatially developing turbulent boundary layer (TBL) simulations. A combination of a transformer and a multiscale-enhanced super-resolution generative adversarial network is utilised to predict velocity fields of a spatially developing TBL at various planes normal to the streamwise direction. Datasets of direct numerical simulation (DNS) of flat plate flow spanning a momentum thickness-based Reynolds number, $Re_\theta = 661.5\unicode{x2013}1502.0$ , are used to train and test the model. The model shows a remarkable ability to predict the instantaneous velocity fields with detailed fluctuations and reproduce the turbulence statistics as well as spatial and temporal spectra with commendable accuracy as compared with the DNS results. The proposed model also exhibits a reasonable accuracy for predicting velocity fields at Reynolds numbers that are not used in the training process. With the aid of transfer learning, the computational cost of the proposed model is considered to be effectively low. Furthermore, applying the generated turbulent inflow conditions to an inflow–outflow simulation reveals a negligible development distance for the TBL to reach the target statistics. The results demonstrate for the first time that transformer-based models can be efficient in predicting the dynamics of turbulent flows. They also show that combining these models with generative adversarial networks-based models can be useful in tackling various turbulence-related problems, including the development of efficient synthetic-turbulent inflow generators.

β-Variational autoencoders and transformers for reduced-order modelling of fluid flows

“…Furthermore, the transformer has been used in the context of temporal predictions of turbulent flows in Ref. [31]. The three model architectures have been tuned to obtain the lowest mean-squared error over the validation data.…”

“…In particular, due to their ability to capture long-range dependencies, transformers are particularly well suited to model dynamic systems [30]. In fact, transformers are able to represent the multi-scale character of turbulence in long temporal sequences [31]; this can only be captured by LSTMs when separately predicting modes of different ranges of frequencies [32]. In these applications, the goal is to learn a lowdimensional representation of the system that captures the underlying dynamics, which can then be used to make predictions or generate new trajectories.…”

Modelling and controlling turbulent flows through deep learning

“…In these studies the focus is on predicting the temporal dynamics of the flow, a task that can be achieved with quite some success with data-driven methods involving long-short-term-memory (LSTM) networks [27] (which are deep-learning architectures capable of exploiting the temporal patterns in the data to perform time-series predictions) and also Koopman-based frameworks with nonlinear forcing [15]. Other promising approaches for such temporal predictions include reservoir computing [11] (which has been shown to effectively capture extreme events in the time series) and transformers [75] (which have the potential to perform accurate instantaneous predictions over longer time horizons than other data-driven methods). Here it is important to note that, although after a certain time horizon the various data-driven approaches start to deviate with respect to the original time series, the temporal dynamics of the system is well represented as illustrated via e.g.…”

“…This datadriven approach has also been able to predict the flow from sparse measurements, as described by Sitte & Doan [57]. An alternative way to perform fluid-flow simulations at a reduced computational cost is to reduce the size of the computational domain through data-driven inflow conditions [21,47,75] and far-field distributions, which can be obtained e.g. via Gaussian-process regression [43].…”

Predicting the wall-shear stress and wall pressure through convolutional neural networks

“…Recently, machine-learning-based approaches have been increasingly used in the study of turbulent flows for a variety of tasks related to flow prediction (Duraisamy et al, 2019;Brunton et al, 2020;Guastoni et al, 2021Guastoni et al, , 2022, prediction of temporal dynamics (Srinivasan et al, 2019;Eivazi et al, 2021;Borrelli et al, 2022), extraction of flow patterns (Jiménez, 2018;Eivazi et al, 2022;Martínez-Sánchez et al, 2023), generation of inflow conditions (Fukami et al, 2019;Yousif et al, 2023) or flow control (Rabault et al, 2019;Guastoni et al, 2023) to name a few. In addition, neural network models have started offering new interesting opportunities to formulate efficient data-driven wall models, as highlighted in Vinuesa and Brunton (2022).…”