Viscous linear stability analysis of rectangular duct and cavity flows

Abstract

The viscous linear stability of four classes of incompressible flows inside rectangular containers is studied numerically. In the first class the instability of flow through a rectangular duct, driven by a constant pressure gradient along the axis of the duct (essentially a two-dimensional counterpart to plane Poiseuille flow – PPF), is addressed. The other classes of flow examined are generated by tangential motion of one wall, in one case in the axial direction of the duct, in another perpendicular to this direction, corresponding respectively to the two-dimensional counterpart to plane Couette flow (PCF) and the classic lid-driven cavity (LDC) flow, and in the fourth case a combination of both the previous tangential wall motions. The partial-derivative eigenvalue problem which in each case governs the temporal development of global three-dimensional small-amplitude disturbances is solved numerically. The results of Tatsumi & Yoshimura (1990) for pressure-gradient-driven flow in a rectangular duct have been confirmed; the relationship between the eigenvalue spectrum of PPF and that of the rectangular duct has been investigated. Despite extensive numerical experimentation no unstable modes have been found in the wall-bounded Couette flow, this configuration found here to be more stable than its one-dimensional limit. In the square LDC flow results obtained are in line with the predictions of Ding & Kawahara (1998b), Theofilis (2000) and Albensoeder et al. (2001b) as far as one travelling unstable mode is concerned. However, in line with the predictions of the latter two works and contrary to all previously published results it is found that this mode is the third in significance from an instability analysis point of view. In a parameter range unexplored by Ding & Kawahara (1998b) and all prior investigations two additional eigenmodes exist, which are both more unstable than the mode that these authors discovered. The first of the new modes is stationary (and would consequently be impossible to detect using power-series analysis of experimental data), whilst the second is travelling, and has a critical Reynolds number and frequency well inside the experimentally observed bracket. The effect of variable aspect ratio $A\in[0.5,4]$ of the cavity on the most unstable eigenmodes is also considered, and it is found that an increase in aspect ratio results in general destabilization of the flow. Finally, a combination of wall-bounded Couette and LDC flow, generated in a square duct by lid motion at an angle $\phi\in(0,{\pi}/{2})$ with the homogeneous duct direction, is shown to be linearly unstable above a Reynolds number $\Rey\,{=}\,800$ (based on the lid velocity and the duct length/height) at all $\phi$ parameter values examined. The excellent agreement with experiment in LDC flow and the alleviation of the erroneous prediction of stability of wall-bounded Couette flow is thus attributed to the presence of in-plane basic flow velocity components.

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

“…At the present Reynolds number and spanwise wavenumber, the flow is stable to three-dimensional perturbations and all eigenvalues lie to the left of the imaginary axis. The distributions of the eigenvalues are in agreement with those obtained by Theofilis et al 40 who utilized a Legendre collocation method on a 64 × 32 grid. For both ducts of different aspect ratios, the low resolution mesh is not capable to reproduce the grid-independent result towards the most-stable regime (Re(λ) → −∞) in that the eigenvalues are generally shifting along the real axis.…”

“…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 → ∞.…”

“…Consequently, the expression of the perturbation in Eq. (2) can be simplified without loss of generality by considering only the symmetric Fourier modes as 23,24,40,41 …”

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

“…The grid used is similar in structure to the one used for the eigenvalue problem. However, as shown by Theofilis et al, 12 the solution of the Poisson equation should take place on a finer grid than the eigenvalue problem mesh. Therefore, Eq.…”

An Analytical Investigation for Analysis of Fluid Flow within a Microchannel of Rectangular Cross Section

“…The 2-D temperature field and Nusselt numbers as function of aspect ratio are also predicted by Morini [9] in case of fully developed thermal region of rectangular ducts at constant wall temperature considering a laminar fully developed velocity profile. Theofilis et al [10] determined velocity distribution of a fluid flow through a rectangular channel solving the Navier-Stoke equation. They assumed a constant pressure gradient along length of the channel.…”