On bifurcations and traction forces on an obstacle in incompressible flow

TL;DR

This paper investigates bifurcations and traction forces in 2D incompressible flow past a circular cylinder, revealing qualitative links between steady and unsteady flow behaviors and proposing efficient methods for detecting critical Reynolds numbers.

Contribution

It introduces a systematic numerical approach combining duality, deflation, and stability analysis to study bifurcations and traction profiles in flow past a cylinder, offering a computationally cheaper detection method.

Findings

Bifurcations include symmetry breaking, oscillations, multiple solutions.

Qualitative correspondence between steady traction profiles and unsteady flow bifurcations.

Proposed strategies enable efficient detection of critical Reynolds numbers.

Abstract

We present a systematic numerical investigation of bifurcations in the two-dimensional incompressible Navier-Stokes flow past a confined circular cylinder. The results indicate that there is a qualitative correspondence between changes in the traction profiles of the steady Navier-Stokes equations and bifurcations of the long-time behavior of the unsteady Navier-Stokes equations. The bifurcations include the appearance of symmetry breaking, oscillations, and multiple steady solutions. The well-known planar Sch\"afer-Turek benchmark is considered with Reynolds numbers up to 1000. For the analysis of bifurcations and traction profiles, several numerical strategies are applied, including a duality method for computing traction profiles, deflation methods, and linear stability analysis. Long-time flow behavior is often explored through direct numerical simulation of the unsteady equations; an approach that is computationally demanding. The relations presented here indicate the possibility of a computationally inexpensive strategy to detect critical Reynolds numbers.