Linear stability of confined flow around a 180-degree sharp bend

Abstract

This study seeks to characterise the breakdown of the steady 2D solution in the flow around a 180-degree sharp bend to infinitesimal 3D disturbances using a linear stability analysis. The stability analysis predicts that 3D transition is via a synchronous instability of the steady flows. A highly accurate global linear stability analysis of the flow was conducted with Reynolds number $Re<1150$ and bend opening ratio (ratio of bend width to inlet height) $0.2\leq\beta\leq5$. This range of $Re$ and $\beta$ captures both steady-state 2D flow solutions as well as the inception of unsteady 2D flow. For $0.2\leq\beta\leq1$, the 2D base flow transitions from steady to unsteady at higher Reynolds number as $\beta$ increases. The stability analysis shows that at the onset of instability, the base flow becomes three-dimensionally unstable in two different modes, namely spanwise oscillating mode for $\beta=0.2$, and spanwise synchronous mode for $\beta \geq 0.3$. The critical Reynolds number and the spanwise wavelength of perturbations increase as $\beta$ increases. For $1<\beta\leq2$ both the critical Reynolds for onset of unsteadiness and the spanwise wavelength decrease as $\beta$ increases. Finally, for $2<\beta\leq5$, the critical Reynolds number and spanwise wavelength remain almost constant. The linear stability analysis also shows that the base flow becomes unstable to different 3D modes depending on the opening ratio. The modes are found to be localised near the reattachment point of the first recirculation bubble

Wedge shaped protrusions in duct flows for heat transfer enhancement

“…Forward and adjoint eigenvalue problems are solved via an implicitely restated Arnoldi iteration method [12]. The solver has been validated for numerous flow simulations [3,15] and stability analysis problems [14,13].…”

“…Lanzerstorfer and Kuhlmann [11] from their analysis for a FFS concluded that a lift-up mechanism along with flow decelaration was the cause of the instability. For flow around a 180 o bend [14] an elliptic instability mechanism in a re-circulation region behind the corner was responsible for the instability. The velocity components of the perturbation indicates a lift-up mechanism [9] responsible for the instability where a small transverse perturbation moves the fluid to a high velocity region leading to the formation of streaks, which decays downstream.…”

Mixed baroclinic convection in a cavity

“…where q ′ = (u ′ , T ′ , p ′ ) contains all perturbation fields and k is the wavenumber along the homogeneous direction ez. Further, the absence of the third component of the velocity field in the base flow allows a single phase of the complex Fourier mode to be considered, following [5], and others [4,44]. The two-dimensionality of the base allows us to reduce the three-dimensional perturbation field to a family of two-dimensional fields parametrised by wavenumber k, and computed on the same two-dimensional domain as the base flow.…”

“…Stuart-Landau model We now seek to characterise the bifurcation associated with the instabilities identified in the previous section by means of a truncated Stuart-Landau equation. This model has been widely applied to find the nature of bifurcations in a number of fluid flows, for example, flow past a circular cylinder (Provansal, Mathis & Boyer 1987;Dušek, Le Gal & Fraunié 1994;Schumm, Berger & Monkewitz 1994;Albarède & Provansal 1995;Thompson & Le Gal 2004), staggered cylinder (Carmo et al 2008) and rings (Sheard et al 2004), and the flow confined around a 180 • sharp bend (Sapardi et al 2017;Pothérat & Zhang 2018). The principle traces back to the equation proposed by Landau (1944) to describe the transition to turbulence, and later used by Stuart (1958Stuart ( , 1960 to understand the behaviour of the plane Poiseuille flow.…”

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

“…Behind unconfined cylinders, the periodic vortex shedding organised in a von Kàrman street appears at Re = 46, but in sharp bends and BFS, both the critical Reynolds number and the nature of the critical mode heavily depend on the opening ratio, β between the minimum and maximum channel width. At low values of β, a jet-like instability with oscillatory critical modes takes place, whereas for β near unity and beyond, the instable mode is localised within the bubble itself, with no oscillatory component ( [19,25]). Crucially, these results were obtained in configurations without side walls, for which the instability develops on a two-dimensional base state.…”