Dean flow and vortex shedding in a three-dimensional 180$^\circ$ sharp bend

TL;DR

This paper analyzes the complex flow dynamics in a 180° sharp bend with square cross-section, revealing how Dean vortices and recirculation bubbles interact to produce unsteady vortex shedding at moderate Reynolds numbers.

Contribution

It provides a detailed topological and dynamical analysis of flow in a 180° bend, highlighting the interaction between Dean vortices and recirculation bubbles as a novel mechanism of vortex shedding.

Findings

Unsteady vortex shedding occurs at Re ≈ 800.

Flow topology reduces to three pairs of critical points.

Interaction between Dean vortices and recirculation bubbles drives oscillations.

Abstract

We present a detailed analysis of the flow in a 180 sharp bend of square cross-section. Besides numerous applications where this generic configuration is found, its main fundamental interest resides in the co-existence of a recirculation bubble in the outlet and a pair of Dean vortices driven from within the turning part of the bend, and how their interaction may drive the flow dynamics. A critical point analysis first shows that the complex flow topology that results from this particular configuration can be reduced to three pairs of critical points in the symmetry plane of the bend (with a focus and a half-saddle each). These pairs respectively generate the first recirculation bubble, the pair of Dean vortex tubes and a third pair of vortex tubes located in the upper corner of the bend, akin to the Dean vortices but of much lower intensity and impact on the rest of the flow. The Dean flow by contrast drives a strong vertical jet that splits the recirculation bubble into two symmetric lobes. Unsteadiness sets in at $Re\lesssim800$ through a supercritical bifurcation, as these lobes start oscillating antisymmetrically. These initially periodic oscillations grow in amplitude until the lobes break away from the main recirculation. The system then settles into a second periodic state where streamwise vortices driven by the Dean flow are alternatively formed and shed on the left and right part of the outlet. This novel mechanism of periodic vortex shedding results from the subtle interaction between the recirculation bubble in the outlet and the pair of Dean vortices generated in the turning part, and in this sense, they are expected to be a unique feature of the 180$^o$ bend with sufficiently close side walls.