Global stability of multiple solutions in plane sudden-expansion flow

Abstract

The two-dimensional, incompressible flow in a plane sudden expansion is investigated numerically for a systematic variation of the geometry, covering expansion ratios (steps-to-outlet heights) from 0.25 to 0.95. By means of a three-dimensional linear stability analysis global temporal modes are scrutinized. In a symmetric expansion the primary bifurcation is stationary and two-dimensional, breaking the mirror symmetry with respect to the mid-plane. The secondary asymmetric flow experiences a secondary instability to different three-dimensional modes, depending on the expansion ratio. For a moderately asymmetric expansion only one of the two secondary flows (the connected branch) is realized at low Reynolds numbers. Since the perturbed secondary flow does not deviate much from the symmetric secondary flow, both secondary stability boundaries are very close to each other. For very small and very large expansion ratios an asymptotic behaviour is found for suitably scaled critical Reynolds numbers and wavenumbers. Representative instabilities are analysed in detail using an a posteriori energy transfer analysis to reveal the physical nature of the instabilities. Depending on the geometry, pure centrifugal and elliptical amplification processes are identified. We also find that the basic flow can become unstable due to the effects of flow deceleration, streamline convergence and high shear stresses, respectively.

The planar X-junction flow: stability analysis and control

“…The flow in the X-junction is a complex, wall-bounded, separated flow. Understanding the bifurcation mechanisms of this flow and designing control strategies to influence them is interesting in its own right, and adds to the physical understanding of a more general class of separated flows, in particular channel junctions and similar configurations (for example Oliveira et al 2012;Poole et al 2007;Chiang & Sheu 2002;Lanzerstorfer & Kuhlmann 2012). In the present manuscript we show that the critical parameter for the onset of both 2D and 3D instabilities of the X-junction is the length of the recirculation bubble(s) rather than the Reynolds number.…”

Subcritical transition to turbulence in a sudden circular pipe expansion

“…At low frequency, the periodic forcing has the advantage to select the axial position where the oscillatory flow is amplified and eventually breaks into a localised turbulent patch. In the future, it is planned to study the effect of the expansion ratio, as plane expansions studies (Lanzerstorfer & Kuhlmann 2012) have shown that the ratio leads to different transition mechanisms depending on the amplification processes: pure centrifugal or elliptical.…”

Krylov methods for large-scale dynamical systems: Application in fluid dynamics

“…This is a necessary, albeit insufficient, condition to conclude that the bifurcation is a pitchfork Moreover, nonlinear analyses (or simulations) are required in order to conclude whether it is supercritical or subcritical. Examples from fluid dynamics include the flow in a sudden expansion channel [86,146], the flow past a sphere [94,188,227], the three-dimensional cavity flow [2,151,205] or the Rayleigh-Bénard convection between two infinite plates or inside an annular loop [157].…”

“…To distinguish these analyses, which solve the linearized 2D equations, from a local stability framework relying on a (weakly-) parallel flow approximation, Theofilis et al [250] has called them (bi-)global stability analyses. Since then, the linear stability of numerous two-dimensional flow configurations have been studied: the lid-driven cavity flow [2,251], the backward-facing step [146], or the two-dimensional flow past a bump [90,91], to name just a few. Because of its importance for the development of local stability theory, the two-dimensional boundary layer flow has also been the focus of many investigations [5,12].…”