The study of flow evolution in complex geometries and challenging flow regimes (e.g., high speed, turbulent, and/or featuring chemical reactions) rely on experiments and, moreover, on detailed computational simulations.

As experiments and computations become more advanced, they generate ever-increasing amounts of data. The availability of these huge datasets creates new challenges: how to manipulate and analyze such massive data to find meaningful results and physical insights. This is leading to data-driven methods based on mathematical techniques and/or on artificial intelligence that can take a dataset and characterize it in meaningful ways with minimal guidance.

Dynamic Mode Decomposition

Dynamic Mode Decomposition (DMD) is a mathematical method able to decompose flow field data into coherent modes and determine their oscillatory frequencies and growth/ decay rates, allowing for the investigation of unsteady and dynamic phenomena unlike conventional statistical analyses. The decomposition can be applied for the analysis of data having a broad range of temporal and spatial scales since it identifies structures that characterize the physical phenomena independently from their energy content.

A DMD algorithm has been specifically created [1] for the analysis of massive databases generated by three-dimensional Direct Numerical Simulations.

The focus of this investigation is the identification of the most important modes and frequencies for the physical phenomena, specifically heat release and turbulence, governing the flow field evolution.

Genetic Expression Programming

Gene Expression Programming (GEP), a class of evolutionary algorithms. GEP uses a complex tree structure that learns and adapts by changing its size, shape, and composition, much like a living organism, to create and evolve a population of candidate models until a satisfactory model emerges. Mimicking survival of the fittest in biological systems, the fitness of a model is measured by how well it approximates the training data, with fitness increasing its chance of survival.

GEP implements survival of the fittest to arrive at some lowest-error/best fit solution to a problem. A population of candidate solutions, denoted as individuals, compete against one another in tournaments to eliminate the weakest solutions from the candidate pool. The remaining individuals then survive to the next generation, where the population is repopulated through randomly genetically altering the individuals. This process then repeats until a solution converges or an allocated number of iterations has been completed.

An advantage of utilising GEP algorithms to meet these modelling needs is that GEP provides some syntactically correct functional form as output. This eliminates the opacity inherent with artificial neural network approaches, which can act as a black box form of modelling. This should allow for a more clearly interpretable relation between inputs and outputs in modelling. The functional form output enables easier implementation of models across various simulations, which should also provide a method for stress testing the validity of models in an a posteriori setting. The functional form of the GEP models may help to provide some insight into the physics of the flame.

Generative adversarial networks

Generative adversarial networks (GANs) are deep-learning frameworks composed of a generator and a discriminator competing against another. The generator creates an output that is judged by the discriminator to be real (genuine) or fake (created by the generator). Super-resolution GANs (SRGANs) have typically been applied to image enhancement, where the input to the generator is a low-resolution image the model learns to enhance and resolve with more detail.

Similarly, in the context of turbulence, deep convolutional neural networks and GANs can be used to generate high-wavenumber detail of lower resolution simulations [5]. To accurately resolve the multi-scale behavior of turbulence in computational fluid dynamics directly, highly-resolved grids must be used with short time-steps to capture the smallest length scale of turbulence – the Kolmogorov scale – as well as the shortest timescales of turbulence.

The ESRGAN has been extended to a 3-D Turbulence Super-Resolution GAN (TSRGAN) [6, 7],

as shown in Figure 1, will be used for reconstruction in the work proposed here. The TSRGAN is able to deal with 3-D sub-boxes of the filtered DNS data (scalar and vector fields) as input and employs physics-based loss functions for training of the network.

References

[1] T. Grenga and M. E. Mueller. Dynamic mode decomposition: A tool to extract structures hidden in massive datasets. In Data Analysis for Direct Numerical Simulations of Turbulent Combustion, pages 157–176. Springer, 2020.

[2] T. Grenga, J. F. MacArt, and M. E. Mueller. Dynamic mode decomposition of a direct numerical simulation of a turbulent premixed planar jet flame: convergence of the modes. Combustion Theory and Modelling, 22(4):795–811, 2018.

[3] Ma, M. C. (2020). High-fidelity simulations and data-driven modelling of turbulent premixed jet flames (Doctoral dissertation). University of Melbourne.

[4] Schoepplein, M., Weatheritt, J., Sandberg, R., Talei, M., & Klein, M. (2018). Application of an evolutionary algorithm to LES modelling of turbulent transport in premixed flames. Journal of Computational Physics, 374, 1166-1179

[5] M. Bode, M. Gauding, J. H. Göbbert, B. Liao, J. Jitsev, and H. Pitsch. Towards prediction of turbulent flows at high Reynolds numbers using high performance computing data and deep learning. In International Conference on High Performance Computing, pages 614–623. Springer, 2018.

[6] M. Bode, M. Gauding, K. Kleinheinz, and H. Pitsch. Deep learning at scale for subgrid modeling in turbulent flows: Regression and reconstruction. In International Conference on High Performance Computing, pages 541–560. Springer, 2019.

[7] M. Bode, M. Gauding, Z. Lian, D. Denker, M. Davidovic, K. Kleinheinz, J. Jitsev, and H. Pitsch. Using physics-informed super-resolution generative adversarial networks for subgrid modeling in turbulent reactive flows. arXiv preprint arXiv:1911.11380, 2019.