Biglobal linear stability analysis for the flow in eccentric annular channels and a related geometry

Abstract

Recently, it has been observed that simple geometry characterized by a low level of symmetry present interesting peculiarities in the process of transition from laminar Poiseuille flow to turbulent flow. Examples of this type of geometry are eccentric channels and, more generally, parallel channels containing a narrow gap. In the present work, a global linear stability analysis for the flow in this class of geometry has been performed. The problem is discretized through spectral collocation and the eigenvalue problem has been solved with the Arnoldi-method based algorithms and the QZ algorithm. Since no numerical studies of this type have yet been performed to address the issue of transition in this geometry, the codes have been validated toward results obtained in simplified geometries ͑e.g., concentric annular channel and square channel͒. The eigenvalue spectra of the Poiseuille flow in eccentric channels and a U-shaped channel have then been computed and analyzed for a wide range of geometric parameters. After comparison with spectra typical of channel flow and pipe flow it is shown that an additional linear mechanism of instability is present, related to the spanwise variation of the laminar velocity profile.

BiGlobal linear stability analysis on low-Re flow past an airfoil at high angle of attack

“…We perform the BiGlobal stability analysis on this duct Poiseuille flow with parameters Re = 100, β = 1, and A = 1 and 2 to be consistent with the works by Theofilis et al 40 and Merzari et al 21 A uniform mesh is used for this verification, with the spatial resolution ranging from 32 2 to 256 2 for the square duct (A = 1), and from 64 × 32 to 512 × 256 for the rectangular one (A = 2). The eigenvalue spectra for the plane Poiseuille flow (PPF) at Re = 100 is computed by solving the Orr-Sommerfeld equation corresponding to the extreme case A → ∞.…”

“…The BiGlobal stability analysis method is proposed to take the advantage of this feature by assuming the perturbation in the form of a normal mode in the spanwise direction, thus the primary perturbation characteristics in the cross-sectional (x-y) plane are preserved and the computational cost is significantly reduced. The BiGlobal linear stability analysis has been employed for a number of physical problems, such as the flow in a channel, [19][20][21][22] over a backward-facing step [23][24][25] or a bluff body. 26 It has also been employed for the problem of flow past an isolated airfoil.…”

Proper orthogonal decomposition of the flow in geometries containing a narrow gap

“…Additional results concerning the POD of computational data for this geometry are available in Merzari et al [3]. In particular, it is remarkable that the most energetic mode of turbulence (k = 2, m = 1) is very similar in shape to the most unstable eigenmode found by the linear stability analysis of Merzari et al [4].…”

“…At higher Reynolds numbers coherent structures are observed on the border of the gap. This behaviour might be related to a bifurcation of the laminar steady state that has been observed both computationally and experimentally [4,5]. In fact, experimental results [5] have shown that the onset of the transition to turbulence in geometries containing a narrow gap is characterized by a strong oscillatory behavior of the cross velocity in the gap region [6].…”

“…In fact, experimental results [5] have shown that the onset of the transition to turbulence in geometries containing a narrow gap is characterized by a strong oscillatory behavior of the cross velocity in the gap region [6]. Moreover, the bifurcation can be predicted by linear stability analysis as shown by Merzari et al [4]. At higher Reynolds numbers, the oscillations begin to be associated to counter-rotating eddies located near the gap-edge [7].…”