arXiv subjects

Pratik P. Aghor

Publications and source records attributed to Pratik P. Aghor.

3 recordsLinked to original sources

arXiv2026

Algebraic models of plane Couette equilibria

Recent computations of weakly unstable equilibria, traveling waves, and periodic orbits in transitional shear flows suggest a spatiotemporal, dynamical-systems approach to low-Reynolds turbulence. Many invariant solutions have been computed precisely using high-dimensional direct numerical simulations, but little is known about how many solutions exist, how they are organized, or which sets of solutions best characterize the flow. In this paper we present a framework for addressing these questions in a low-dimensional context. Using classical approximation methods and exploiting symmetries and kinematic constraints, we derive ordinary differential equation models of plane Couette flow whose equilibria are governed by systems of quadratic algebraic equations. Solutions of these algebraic systems approximate known equilibria of plane Couette flow in as few as 17 dimensions and converge toward the known solutions as dimension increases. Searches over the systems produce sixteen distinct equilibrium solution branches in seven different symmetry groups. These results suggest that the equilibrium and traveling-wave solutions of closed shear flows are organized by the algebraic structure of systems of quadratic equations. Additionally, the differential equations and divergence-free basis provide explicit, closed-form, and convergent dynamical-systems representations of plane Couette flow.

physics.flu-dyn↗

arXiv2024

Fourier neural operators for spatiotemporal dynamics in two-dimensional turbulence

High-fidelity direct numerical simulation of turbulent flows for most real-world applications remains an outstanding computational challenge. Several machine learning approaches have recently been proposed to alleviate the computational cost even though they become unstable or unphysical for long time predictions. We identify that the Fourier neural operator (FNO) based models combined with a partial differential equation (PDE) solver can accelerate fluid dynamic simulations and thus address computational expense of large-scale turbulence simulations. We treat the FNO model on the same footing as a PDE solver and answer important questions about the volume and temporal resolution of data required to build pre-trained models for turbulence. We also discuss the pitfalls of purely data-driven approaches that need to be avoided by the machine learning models to become viable and competitive tools for long time simulations of turbulence.

physics.flu-dyn↗

arXiv2024

Symmetry groups and invariant solutions of plane Poiseuille flow

Equilibrium, traveling-wave, and periodic-orbit solutions of the Navier-Stokes equations provide a promising avenue for investigating the structure, dynamics, and statistics of transitional flows. Many such invariant solutions have been computed for wall-bounded shear flows, including plane Couette, plane Poiseuille, and pipe flow. However, the organization of invariant solutions is not well understood. In this paper we focus on the role of symmetries in the organization and computation of invariant solutions of plane Poiseuille flow. We show that enforcing symmetries while computing invariant solutions increases the efficiency of the numerical methods, and that redundancies between search spaces can be eliminated by consideration of equivalence relations between symmetry subgroups. We determine all symmetry subgroups of plane Poiseuille flow in a doubly-periodic domain up to translations by half the periodic lengths and classify the subgroups into equivalence classes, each of which represents a physically distinct set of symmetries and an associated set of physically distinct invariant solutions. We calculate fifteen new traveling waves of plane Poiseuille flow in seven distinct symmetry groups and discuss their relevance to the dynamics of transitional turbulence. We present a few examples of subgroups with fractional shifts other than half the periodic lengths and one traveling wave solution whose symmetry involves shifts by one-third of the periodic lengths. We conclude with a discussion and some open questions about the role of symmetry in the behavior of shear flows.

physics.flu-dyn↗