Session F10: General Fluid Dynamics: Mathematical Methods

Learning and enforcing physical structure with physics-informed dynamic mode decomposition (piDMD)

Data-driven models that respect physical laws are robust to noise, require few training samples, and are highly generalisable. Although the dynamic mode decomposition (DMD) is a principal tool of data-driven fluid dynamics, it is rare for learned DMD models to obey physical laws such as symmetries, invariances, causalities, spatial locality and conservation laws. Thus, we present physics-informed dynamic mode decomposition (piDMD), a suite of tools that incorporate physical structures into linear system identification. Specifically, we develop efficient and accurate algorithms that produce DMD models that obey the matrix-analogues of user-specified physical constraints. Through a range of examples from fluid dynamics, we demonstrate the improved diagnostic, predictive and interpretative abilities of piDMD. We consider examples from stability analysis, data-driven resolvent analysis, reduced-order modelling, control, and the low-data and high-noise regimes. Conversely, if the physical structures are unknown then, through cross-validation, piDMD can be used to discover the physical structures present in the observed system. Finally, we show that the time and memory requirements of piDMD are competitive with standard DMD approaches.   

Learning fluid dynamics using sparse physics-informed discovery of empirical relations (SPIDER)

We show how a complete mathematical description of a complicated physical phenomenon can be learned from observational data via a hybrid approach combining three simple and general ingredients: physical assumptions of smoothness, locality, and symmetry, a weak formulation of differential equations, and sparse regression. To illustrate this approach, we extract a complete system of governing equations describing flows of incompressible Newtonian fluids -- the Navier-Stokes equation, the continuity equation, and the boundary conditions -- from numerical data describing a highly turbulent channel flow in three dimensions. These relations have the familiar form of partial differential equations, which are easily interpretable and readily provide information about the relative importance of different physical effects as well as insight into the quality of the data, serving as a useful diagnostic tool. The approach described here is remarkably robust, yielding accurate results for very high noise levels, and should thus be well-suited to experimental data.   

Learning frame-independent, nonlocal constitutive relations on unstructured meshes with an embedding neural network

Constitutive models are widely used for modeling complex systems in science and engineering, where first-principle-based, well-resolved simulations are often prohibitively expensive. For example, in fluid dynamics, constitutive models are required to describe nonlocal, unresolved physics such as turbulence and laminar-turbulent transition. In particular, Reynolds stress models for turbulence and intermittency transport equations for laminar-turbulent transition both utilize convection-diffusion partial differential equations (PDEs). However, traditional PDE-based constitutive models can lack robustness and are often too rigid to accommodate diverse calibration data. We propose a frame-independent, non-local constitutive model based on an embedding neural network that can be trained with data. The learned constitutive model can predicate the closure variable at a point based on the flow information in its neighborhood. It can take any number of points arbitrarily arranged, and thus it is suitable for unstructured meshes, which are typical for finite-element and finite-volume simulations. The merits of the proposed model are demonstrated on both scalar and tensor transport PDEs on a family of parameterized periodic hill geometries.   

Memory effects in fluctuating dynamic density-functional theory: theory and simulations

This work introduces a theoretical stochastic framework to describe the dynamics of reacting multi-species fluids in and out of equilibrium. We present an ab-initio derivation of a non-Markovian Navier-Stokes-like system of equations which constitutes a generalisation of the Dean-Kawasaki model. However, such equations still depend on all the microscopic degrees of freedom. To remove this dependence on the microscopic level without washing out the fluctuation effects, which are characteristic of a mesoscopic description, we ensemble-average our generalised Dean-Kawasaki equations. This results in a set of non-Markovian fluctuating hydrodynamic equations governing the time evolution of the mesoscopic density and momentum fields. By introducing an energy functional which recovers the one used in classical density-functional theory (DFT) under the local-equilibrium approximation, we obtain a non-Markovian fluctuating dynamical DFT (FDDFT) for reacting multi-species fluids. To numerically validate our theoretical framework, we carry out a finite-volume discretisation of our non-Markovian FDDFT. Finally, we illustrate the influence of non-Markovian effects on the dynamics of non-linear chemically reacting fluid systems with a detailed study of memory-driven Turing patterns.   

Analytical calculation of hydrodynamic coefficients of an oscillating horizontal circular cylinder using the bi-polar coordinate

The added mass and wave damping of an oscillating horizontal circular cylinder are analytically calculated using the bi-polar coordinate. In the bi-polar coordinate, a point is uniquely defined by two orthogonal circles. These circles can be used to represent the mean surface of a partially- and a fully submerged oscillating circular cylinder along with the mean position of the free surface. By expressing boundary conditions on the surface of the cylinder and the free surface in terms of the bi-polar coordinate, the Laplace equation in terms of the velocity potential is analytically solved, and, thus, the added mass and wave damping are analytically obtained. Both partially/fully submerged and heave/surge cases are considered. The analytical results are compared with existing numerical studies using the boundary element method. They agree with each other very well.   

Projection-based methods for spectral analysis of data without temporal information

Extracting coherent structures and physically important features from complex systems has become the subject of great interest in fluid dynamics, motivated and enabled by a variety of modal analysis techniques. The availability of spatiotemporally resolved computational and experimental data enables the use of techniques such as dynamic mode decomposition and spectral proper orthogonal decomposition to isolate and analyze structures that correspond to specific temporal frequencies. However, in many scenarios time-resolved data remains challenging to acquire and store. Here, we explore how spectral content may be recovered from spatially-resolved data in the absence of any temporal information. This is achieved by analyzing the spectral content of reduced-order models identified from projecting the governing equations onto a subspace identified from the original data via proper orthogonal decomposition, through either a pseudospectral analysis of the identified linearized system or by considering the spectral proper orthogonal decomposition of time-resolved data generated by evolving nonlinear reduced-order models. We explore the method performance for example problems with various temporal dynamics, ranging from systems with a single dominant frequency to those with broadband frequency content. We find that the method can identify spectral content even in cases where the identified reduced-order models have limited accuracy in predicting the time-evolution of the system dynamics.