Investigation and inference of soft material deformation mechanisms 

Energy transfer across scales is fundamental in fluid dynamics, linking large-scale flow motions to small-scale turbulent structures in engineering and natural environments. Triadic interactions among three wave components form complex networks across scales, challenging understanding and model reduction. We introduce Triadic Orthogonal Decomposition (TOD), a method that identifies coherent flow structures optimally capturing spectral momentum transfer, quantifies their coupling and energy exchange in an energy budget bispectrum, and reveals the regions where they interact. TOD distinguishes three components—a momentum recipient, donor, and catalyst—and recovers laws governing pairwise, six-triad, and global triad conservation. TOD reveals networks of triadic interactions with forward and backward energy transfer across frequencies and scales within fluid flows.

Investigation and inference of soft material deformation mechanisms 

We Leverage advances in modeling and data assimilation to identify material parameters and the physical models and theory that underpin them. We hypothesize that data assimilation, integrated into forward multiphase flow models, Multi-component Flow Code (MFC) and Inertial microcavitation rheometry (IMR), will describe the fast mechanical behavior of materials with quantifiable uncertainty and inform us when the models need augmentation.  Our goal is to integrate experimental and computational methods to identify soft material behavior with optimal performance. 

Mesh-free methods for flow simulations and hydrodynamic stability

In the past two decades, radial basis functions (RBF)-based discretizations have emerged as a viable alternative to established approaches for computational fluid dynamics (CFD). RBF-based methods are often referred to as mesh-free as they facilitate the discretization of partial differential operators directly on a set of scattered nodes, i.e., without the need of local elements. The main promise of RBF-based methods is that they can combine the ease of implementation and high order of accuracy of finite differences (FD) with the geometrical flexibility of finite volume (FV), element, and discontinuous Galerkin (DG) methods. In this project, we propose the use of RBF-based finite difference (RBF-FD) discretization to 

1. develop a semi-implicit fractional-step scattered-but-staggered incompressible Navier-Stokes solver (Chu & Schmidt 2023, JCP)

2. solve the large eigenvalue problems arising in hydrodynamic stability analyses of flows (Chu & Schmidt 2024, JCP)

in complex domains. Polyharmonic splines (PHS) with polynomial augmentation (PHS+poly) are used to construct the global differentiation matrices. A systematic parameter study identifies combinations of stencil size, PHS exponent, and polynomial degree that minimize the truncation error for a wave-like test function on scattered nodes. These sets of parameters guarantee stability without the need for hyperviscosity or other ad hoc regularizations, while balancing accuracy and computational efficiency.

Stochastic SPOD-based reduced-order models

The use of spectral proper orthogonal decomposition (SPOD) to construct reduced-order models (ROMs) for broadband turbulent flows is explored. Leveraging the optimality, convergence, and space-time coherence properties of SPOD modes for statistically stationary flows, we establish two model order-reduction techniques, namely the operator-based Galerkin projection and the data-driven time-delay Koopman approach. For broadband turbulent flows, these two approaches yield the following stochastic ROMs: 

• Stochastic two-level SPOD-Galerkin model (Chu & Schmidt 2021, TCFD)

• Stochastic Low-dimensional Inflated Convolutional Koopman (SLICK) model 

Following the core concept of Koopman theory that an infinite-dimensional linear operator can describe the nonlinear dynamics, we inflate the linear state with an exogenous forcing to account for the nonlinear interactions and background turbulence. Closure is achieved by modeling the remaining residue as stochastic noise. The result models accurately predict the initial transient dynamics and reproduce the second-order statistics of broadband turbulent flows.

The dynamics of Buoyant vortices

A comprehensive investigation has been conducted for vortical flows, taking into account the complex interplay of buoyancy effects, including gravity, density variations, and surface tension. In particular, we consider  the dynamics of:

• Incompressible vortex filaments with helical symmetry (Chu & Llewellyn Smith 2021, RCD)

• Point vortices within the weakly compressible regime (Llewellyn Smith, Chu & Hu 2022, PTRSA)

Equations for contour dynamics for helical vortices and equations of motion for weakly compressible point vortices are derived. Although distinct in their focus, these two sub-projects are intricately intertwined, offering synergistic insights into the broader spectrum of vortical fluid flow phenomena.