(751d) Applying Data-Driven Dimension Reduction Techniques to Constitutive Model Formulation for Gas-Particle Flows

Introduction: Gas-particle flows in fluidized beds and circulating fluidized beds are inherently unstable and manifest structures on a wide range of length and time scales. Researchers have sought to develop filtered models to simulate large flow structures without having to resolve fine-scale flow structures, which are computationally very expensive to resolve [1,2]. Highly resolved Euler-Euler (two-fluid model) and Euler-Lagrange simulations (such as CFD-DEM) of gas-particles flows are being used to generate the computational data to infer the level of complexity of the filtered models that one must employ and associated constitutive equations [2,3,4]. Systematic filtering of the results from highly resolved flow simulations yields a large dataset, which form the basis for the sub-filter (commonly referred to as sub-grid) constitutive models. Formulation of these constitutive models is hampered by the fact that the optimal set of filtered variables and sub-grid correlations one should employ to formulate the constitutive model is not known. In literature studies, plausible sets of variables to employ in constitutive models are identified through a trial and error procedure [2,3,4]; while such approaches have led to significant progress, their further improvements are prohibitively tedious. In the present study, we have employed machine learning techniques, in particular data-driven dimension reduction, to identify optimal set of filtered variables and sub-grid correlations.

Methodology: A nonlinear dimension reduction technique, namely diffusion map [5], was found to provide fast and accurate identification of the intrinsic dimension of datasets. The diffusion map algorithm exploits a low dimensional geometric embedding of a high dimensional dataset. Starting from a dataset organized in the form of an n_samples x n_variables matrix, the diffusion map technique reduces the dimension to n_samples x m, where m is less than or equal to n_variables. If and when m is smaller than n_variables, the diffusion map technique allows us to identify the smaller set of variables we need for the modeling process; further analysis of the low dimensional diffusion map coordinates can provide us with insight into their correlation to the original variables. This dimension reduction process can simplify constitutive model formulation process and make further model fitting methods such as neural networks computationally affordable.

Results:A dataset was first obtained by filtering the results from highly resolved Euler-Lagrange simulations; the dataset takes the form of a collection of filtered variables, sub-filter scale correlations and the drag correction for which we seek a model. Diffusion map analysis of the dataset revealed the intrinsic dimension to be two, and helped establish a new two-dimensional representation of the dataset. This finding implies that the drag correction can be parametrized by two independent (sub-filter scale) correlations, and points the way to further refinement of sub-filter scale constitutive models beyond what has been achieved in the literature.

Summary:Data-driven dimension reduction techniques have shed more light on constitutive model formulation for filtered models for gas-particle flows.

References:

[1] Agrawal, K., Loezos, P. N., Syamlal, M., & Sundaresan, S. (2001). The role of meso-scale structures in rapid gas–solid flows. Journal of Fluid Mechanics, 445, 151-185.

[2] Ozel, A., Fede, P., & Simonin, O. (2013). Development of filtered Euler–Euler two-phase model for circulating fluidised bed: high resolution simulation, formulation and a priori analyses. International Journal of Multiphase Flow, 55, 43-63.

[3] Igci, Y., Andrews, A. T., Sundaresan, S., Pannala, S., & O'Brien, T. (2008). Filtered two‐fluid models for fluidized gas‐particle suspensions. AIChE Journal, 54(6), 1431-1448.

[4] Ozel, A., Kolehmainen, J., Radl, S., & Sundaresan, S. (2016). Fluid and particle coarsening of drag force for discrete-parcel approach. Chemical Engineering Science, 155, 258-267.

[5] Lafon, S. and Lee, A.B. (2006). Diffusion maps and coarse-graining: a unified framework for dimensionality reduction, graph partitioning, and data set parameterization. IEEE Transactions on Pattern Analysis and Machine Intelligence, 28(9), 1393-1403.