Bifurcations in curved duct flow based on a simplified Dean model

TL;DR

This paper introduces a simplified dynamical model for flow in curved ducts, capturing various flow regimes and bifurcations, and accurately reproducing bifurcation points observed in more complex systems.

Contribution

The study presents a minimal, yet effective, model that reproduces bifurcation phenomena in curved duct flow, providing insights into flow dynamics with reduced computational complexity.

Findings

Identification of stationary, periodic, aperiodic, and chaotic flow regimes.

Reproduction of bifurcation points consistent with full Navier-Stokes solutions.

Flow oscillates between symmetric steady states via heteroclinic cycles.

Abstract

We present a minimal model of an incompressible flow in square duct subject to a slight curvature. Using a Poincar\'e-like section we identify stationary, periodic, aperiodic and chaotic regimes, depending on the unique control parameter of the problem: the Dean number (De). Aside from representing a simple, yet rich, dynamical system the present simplified model is also representative of the full problem, reproducing quite accurately the bifurcation points observed in the literature. We analyse the bifurcation diagram from De = 0 (no curvature) to De = 500, observing a periodic segment followed by two separate chaotic regions. The phase diagram of the flow in the periodic regime shows the presence of two symmetric steady states, the system oscillates around these solutions following a heteroclinic cycle. In the appendix some quantitative results are provided for validation purposes, as well as the python code used for the numerical solution of the Navier-Stokes equations.