(303a) Data-Driven Development of Approximate Inertial Forms and Closures for Coarse-Scale Modeling of Multiphase Flows

Learning closures for coarse-graining fine scale models in multiscale systems is relevant when the macroscopic physics must be accurately described without having to explicitly resolve the fine scales [1]. Traditional constitutive modeling to close the equations and solve for the variables that describe the relevant fields is now evolving due to machine learning techniques for data-driven modeling [2][3]. This modeling approach can also take advantage of further reduced latent spaces, based on the idea of approximate inertial manifolds [4], learning reduced order nonlinear representations that are not easily physically interpretable. The aim of this work is to learn coarse grained PDEs as well as reduced order models of coarse-scale PDEs for multiphase flows using a data-driven approach. In the first portion of this work, we train a neural network to learn an approximate inertial form: a few ODEs for the system behavior projected onto a few proper orthogonal decomposition (POD) modes obtained from the fine scale simulations [5]: From 2D Direct Numerical Simulations of a multiphase bubbly flow in a vertical channel, we average in the direction parallel to the overall flow to create a dataset of one-spatial-dimension, time-dependent profiles, corresponding to the vertical velocity of the liquid and the void fraction fields [6]. We perform POD to reduce the high-dimensional averaged snapshot data to a truncated set of 10 leading-mode amplitude coefficients, and further reduce these to two latent coordinates (through an autoencoder) which are one to one with the first two POD modes, as illustrated in Fig.1(a).

We train a neural network to approximate the continuous-time dynamics of the system in terms of the amplitudes of the first two, "determining" POD modes. Since the higher POD modes are functions (on our data) of the first two "determining" ones, that parametrize our approximate inertial manifold, we reconstruct the full solution from these two time series through a network that has learned the manifold, i.e. the higher 8 POD coefficients as a function of the 2 "determining" ones, as depicted in Fig.1(b).

In the second portion of this work, we attempt to learn the right-hand-side operator of the averaged PDE through (a) a black box model and (b) a learned "grey-box" model that exploits the known parts/structure of the operator: everything else but the closure. To evolve the relevant fields, a pair of unknown closure terms must be modelled, F_l (x, t), G_l (x, t) = C_l (system state (x, t)) for the wall-normal liquid flux and summed dissipative terms, respectively. These terms are learned from coarse evolution data [7]: we train a network to approximate these closure terms C_l away from the wall, using only x-local information. Close to the wall, we learn a smooth non-decreasing function that corrects the effect of the drag. The wall function is zero in a region close to the wall, and asymptotes to one far from it. Before applying this closure-learning method to the data derived from the full 2D multiphase flow, we validate the approach on a simplified version of the problem with an explicitly known closure function (which we show we can accurately recover).

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