LEAPS-MPS: Exploring various subgrid scale turbulence models via convergence analysis, data assimilation and deep learning

Towson University, Towson MD

Investigators

Abstract

Turbulence is a set of complex nonlinear phenomena showing unsteady, irregular, seemingly random and chaotic characteristic motions of fluids. It happens in various systems involving the atmosphere, ocean, aerodynamics, and technology. The study of turbulent fluid flow is well known to be highly important and challenging. Since some issues of existence and uniqueness of solutions of the three-dimensional Navier-Stokes equations are still not yet fully established despite century-long efforts, turbulence modeling currently provides the best qualitative and, in many cases, even quantitative measures for many problems in applications. Recently, generating through an averaging process, various subgrid scale turbulence models were developed with good success. These models not only capture the large-scale dynamics of the flow, but also provide “unresolved” small-scale representations of the physics of fluids as well as reliable closure models to the averaged equations. Moreover, they have nice analytical, empirical and computational properties, such as global regularity and good matching with empirical data collected from examples such as turbulent channels and pipes. Therefore, studying these models will be beneficial and effective as far as both mathematical rigors and real-world applications are concerned. This project aims to study these subgrid turbulence models from the point of view of both basic mathematical research and applications. The project will have significant impacts on both undergraduate and graduate students, particularly those from underrepresented groups, through their participation in accessible research projects. This will also establish a strong research agenda for the PI via working with different career-stage researchers, building the research capability and curricular offerings of the Department of Mathematics to fulfill regional needs for data science expertise and offering educational experiences in the local community. In this project, a synthesizing effort of convergence analysis, data assimilation algorithm and deep learning computation will be made to study various subgrid scale turbulence models. The PI will first explore the relationship between, and the emergence of, these models in association with the Navier-Stokes equations. This will help us explore their intriguing connections to the global regularity problem. Next, the PI will apply the data assimilation algorithm to these subgrid scale models. The PI plans to build a data assimilation system for these models, prove the existence of solutions, and show convergence of the data-assimilated solutions of these models to the weak solutions of the Navier-Stokes equations on a three-dimensional domain. As a target, the PI will conduct parameter estimation via the determining map and its computation using deep learning. The PI aims to develop a rigorous framework via the determining map and computing it using neural networks. Exploiting the computational advantages of recent deep learning techniques, the PI and her team hope to be able to treat some cases where the traditional numerical methods face hurdles such as the curse of dimensionality and complex geometries. This project will help provide an illuminating link between various subgrid scale turbulence models and the Navier-Stokes equations and improve our understanding of turbulence. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.