Vassilios Theofilis

Advances in global linear instability analysis of nonparallel and three-dimensional flows

Abstract A summary is presented of physical insights gained into three-dimensional linear instability through solution of the two-dimensional partial-differential-equation-based nonsymmetric real or complex generalised eigenvalue problem. The latter governs linear development of small-amplitude disturbances upon two-dimensional steady or time-periodic essentially nonparallel basic states; on account of this property the term BiGlobal instability analysis has been introduced to discern the present from earlier global instability methodologies which are concerned with the analysis of mildly inhomogeneous two-dimensional basic flows. Alternative forms of the two-dimensional eigenvalue problem are reviewed, alongside a discussion of appropriate boundary conditions and numerical methods for the accurate and efficient recovery of the most interesting window of the global eigenspectrum. A number of paradigms of open and closed flow systems of relevance to aeronautics are then discussed in some detail. Besides demonstrating the strengths and limitations of the theory, these examples serve to demarcate the current state-of-the-art in applications of the theory to aeronautics and thus underline the steps necessary to be taken for further progress to be achieved.

On the origins of unsteadiness and three-dimensionality in a laminar separation bubble

We analyse the three–dimensional non–parallel instability mechanisms responsible for transition to turbulence in regions of recirculating steady laminar two–dimensional incompressible separation bubble flow in a twofold manner. First, we revisit the problem of Tollmien–Schlichting (TS)–like disturbances and we demonstrate, for the first time for this type of flow, excellent agreement between the parabolized stability equation results and those of independently performed direct numerical simulations. Second, we perform a partial–derivative eigenvalue problem stability analysis by discretizing the two spatial directions on which the basic flow depends, precluding TS–like waves from entering the calculation domain. A new two–dimensional set of global amplified instability modes is thus discovered. In order to prove earlier topological conjectures about the flow structural changes occurring prior to the onset of bubble unsteadiness, we reconstruct the total flowfield by linear superposition of the steady two–dimensional basic flow and the new most–amplified global eigenmodes. In the parameter range investigated, the result is a bifurcation into a three–dimensional flowfield in which the separation line remains unaffected while the primary reattachment line becomes three dimensional, in line with the analogous result of a multitude of experimental observations.

Viscous linear stability analysis of rectangular duct and cavity flows

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

Structural changes of laminar separation bubbles induced by global linear instability

A\in[0.5,4]

The Extended Görtler-Hämmerlin Model For Linear Instability of Three-Dimensional Incompressible Swept Attachment-Line Boundary Layer Flow

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

Modal Analysis of Fluid Flows: An Overview

\phi\in(0,{\pi}/{2})

Spatial stability of incompressible attachment-line flow

with the homogeneous duct direction, is shown to be linearly unstable above a Reynolds number

Linear instability analysis of low-pressure turbine flows

\Rey\,{=}\,800

On Linear and Nonlinear Instability of the Incompressible Swept Attachment-Line Boundary Layer

(based on the lid velocity and the duct length/height) at all

Transient growth analysis of the flow past a circular cylinder

\phi