Numerical study of transitions in lid-driven flows in semicircular cavities

TL;DR

This paper numerically investigates three-dimensional lid-driven flows in semicircular cavities, identifying critical Reynolds numbers for flow oscillations and analyzing the oscillation modes, revealing similarities and differences with shallow cavity flows.

Contribution

It introduces a combined numerical methodology to study flow bifurcations in semicircular cavities and details the flow oscillation modes near critical Reynolds numbers.

Findings

Critical Reynolds numbers for oscillatory flow are determined.

Flow oscillation modes are characterized and compared to shallow cavities.

Flow patterns near bifurcation points show consistent behavior across different cavity geometries.

Abstract

In this article, three-dimensional (3D) lid-driven flows in semicircular cavities are studied. The numerical solution of the Navier-Stokes equations modeling incompressible viscous fluid flow in cavities is obtained via a methodology combining a first-order accurate operator-splitting scheme, a fictitious domain formulation, and finite element space approximations. The critical Reynolds numbers (Re_{cr}) for having oscillatory flow (a Hopf bifurcation) are obtained. The associated oscillating motion in a semicircular cavity with length equal to width has been studied in detail. Based on the averaged velocity field in one period of oscillating motion, the flow difference (called oscillation mode) between the velocity field and averaged one at several time instances in such period shows almost the same flow pattern for the Reynolds numbers close to Re_{cr}. This oscillation mode in a semicircular cavity shows a close similarity to the one obtained in a shallow cavity, but with some difference in a shallow cavity which is triggered by the presence of two vertical side walls and downstream wall.