Global stability of flow past a cylinder with centreline symmetry

Abstract

Global absolute and convective stability analysis of flow past a circular cylinder with symmetry conditions imposed along the centreline of the flow field is carried out. A stabilized finite element formulation is used to solve the eigenvalue problem resulting from the linearized perturbation equation. All the computations carried out are in two dimensions. It is found that, compared to the unrestricted flow, the symmetry conditions lead to a significant delay in the onset of absolute as well as convective instability. In addition, the onset of absolute instability is greatly affected by the location of the lateral boundaries and shows a non-monotonic variation. Unlike the unrestricted flow, which is associated with von Kármán vortex shedding, the flow with centreline symmetry becomes unstable via modes that are associated with low-frequency large-scale structures. These lead to expansion and contraction of the wake bubble and are similar in characteristics to the low-frequency oscillations reported earlier in the literature. A global linear convective stability analysis is utilized to find the most unstable modes for different speeds of the disturbance. Three kinds of convectively unstable modes are identified. The ones travelling at very low streamwise speed are associated with large-scale structures and relatively low frequency. Shear layer instability, with relatively smaller scale flow structures and higher frequency, is encountered for disturbances travelling at relatively larger speed. For low blockage a new type of instability is found. It travels at relatively high speed and resembles a swirling flow structure. As opposed to the absolute instability, the convective instability appears at much lower Re and its onset is affected very little by the location of the lateral boundaries. Analysis is also carried out for determining the convective stability of disturbances that travel in directions other than along the free stream. It is found that the most unstable disturbances are not necessarily the purely streamwise travelling ones. Disturbances that move purely in the cross-stream direction can also be convectively unstable. The results from the linear stability analysis are confirmed by carrying out direct time integration of the linearized disturbance equations. The disturbance field shows transient growth by several orders of magnitude confirming that such flows act as amplifiers. Direct time integration of the Navier–Stokes equation is carried out to track the time evolution of both the large-scale low-frequency oscillations and small-scale shear layer instabilities. The critical Re for the onset of convective instability is compared with earlier results from local analysis. Good agreement is found.

Onset of global instability in the flow past a circular cylinder cascade

“…In addition, centreline perturbation velocities for the OE modes are significantly increased with increasing blockage. Kumar et al (2009) have suggested that this non-monotonic variation is because of the effect of the lateral boundaries in suppressing disturbance growth in the cross-stream direction, together with the effect of local flow acceleration in the streamwise direction.…”

“…In Jackson (1987), the finite element method was used to solve an extended set of equations for detecting Hopf bifurcation for a class of bluff bodies. The global stability approach has also been used by Morzyński & Thiele (1991), Morzyński, Afanasiev & Thiele (1999), Chen, Pritchard & Tavener (1995), Kumar & Mittal (2006), Mittal, Kottaram & Kumar (2008), Kumar et al (2009) amongst others, for discussing the instability of the flow past cylinders and related problems. Morzyński et al (1999) was one of the first papers to suggest the use of subspace iteration (see Saad 1989) for solving the eigenvalue problem of the type arising in hydrodynamic stability computations using the global stability approach.…”

“…They find that the separated shear layer loses stability at low Reynolds numbers with very small growth rates. In Kumar et al (2009), the centreline symmetry condition has also been used to investigate instabilities in geometries identical to the present study.…”

Two-dimensional compressible viscous flow around a circular cylinder

“…In recent times, variations of the Stuart-Landau equation have been used by Noack et al (2003), Sipp & Lebedev (2007), and Sengupta et al (2010) to model the spatial and temporal evolution of flow instabilities. Other modern advancements in stability analysis of two-dimensional incompressible flow around a circular cylinder include the experimental determination of σ r (Strykowski & Sreenivasan 1990), the location of both the instability core and regions of high sensitivity to perturbations (Giannetti & Luchini 2007), and the isolation of the antisymmetric modes that give rise to further instabilities at higher Re (Kumar et al 2009). As can be seen from the above discussion, the wake geometry at stable Re, the transient behaviour of the flow as it transitions from a steady closed wake to a two-dimensional vortex street, and the fully-developed laminar vortex street have been investigated quite extensively for incompressible flow.…”

Some observations regarding steady laminar flows past bluff bodies

“…Because the end of a steady RR is a stagnation point, a small vertical plate at this location will have very little effect on the flow, apart from being highly effective in suppressing this particular mode [43]. However, such 'passive' flow control approaches will be unable to control subsequent instability modes (with the initial laminar ones analysed in [44]). Flow past a two-dimensional cylinder and over backwards steps have served as test problems for several active flow control strategies, both for the finite-Re case [45,46] and for Euler (point vortex) flows [47].…”