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.

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91-11267 ABSTRACTThe linear stability of compressible plane Couette flow is investigated. The correct and proper basic velocity and temperature distributions are perturbed by a small amplitude normal mode disturbance. The full small amplitude disturbance equations are solved numerically at finite Reynolds numbers, and the inviscid limit of these equations is then investigated in sene detail. It is found that instability can occur, although the stability characteristics of the flow are quite different from unbounded flows. The effects of viscosity are also calculated, asymptotically, and shown to have a stabilizing role in all the cases investigated. Exceptional regimes to the problem occur when the wavespeed of the disturbances approaches the velocity of either of the walls, and these regimes are also analyzed in some detail. Finally, the effect of imposing radiation-type boundary conditions on the upper (moving) wall (in place of impermeability) is investigated, and shown to yield results common to both bounded and unbounded flows.

The phenomenon of Tollmien-Schlichting wave generation in a boundary layer by free-stream turbulence is analysed theoretically by means of asymptotic solution of the Navier-Stokes equations at large Reynolds numbers (Re → ∞). For simplicity the basic flow is taken to be the Blasius boundary layer over a flat plate. Free-stream turbulence is taken to be uniform and thus may be represented by a superposition of vorticity waves. Interaction of these waves with the flat plate is investigated first. It is shown that apart from the conventional viscous boundary layer of thickness O(Re−1/2), a ‘vorticity deformation layer’ of thickness O(Re−1/4) forms along the flat-plate surface. Equations to describe the vorticity deformation process are derived, based on multiscale asymptotic techniques, and solved numerically. As a result it is shown that a strong singularity (in the form of a shock-like distribution in the wall vorticity) forms in the flow at some distance downstream of the leading edge, on the surface of the flat plate. This is likely to provoke abrupt transition in the boundary layer. With decreasing amplitude of free-stream turbulence perturbations, the singular point moves far away from the leading edge of the flat plate, and any roughness on the surface may cause Tollmien-Schlichting wave generation in the boundary layer. The theory describing the generation process is constructed on the basis of the ‘triple-deck’ concept of the boundary-layer interaction with the external inviscid flow. As a result, an explicit formula for the amplitude of Tollmien-Schlichting waves is obtained.

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Key Words rotating flows, spin-down, spin-over, conical flows s Abstract We consider the manner in which a container filled with viscous fluid adjusts to changes in its rotation rate. We begin with homogeneous flows involving small departures in rotation rate from an initial state of solid-body rotation in an axisymmetric container. This is followed by a summary of other more recent developments, including weakly and fully nonlinear calculations and comparison with experiment and the question of spin-down. The question of "spin-over" is addressed, followed by a brief synopsis of free-surface effects, and a discussion of nonaxisymmetric spin-up. The second part of the review focuses on the effects of stratification on the spin-up process. Linearized (low Rossby number) spin-up within a cylindrical container is described. Thereafter, both experimental and nonlinear computational results are described and compared. The final section focuses on stratified spin-up and spin-down in conical geometries, and a number of comparisons between theory and experiment are given.

The incompressible boundary layer in the corner formed by two intersecting, semi-infinite planes is investigated, when the free-stream flow, aligned with the corner, is taken to be of the form U∞F(x), x representing the non-dimensional streamwise distance from the leading edge. In Dhanak & Duck (1997) similarity solutions for F(x) = xn were considered, and it was found that solutions exist for only a range of values of n, whilst for ∞ > n > −0.018, approximately, two solutions exist. In this paper, we extend the work of Dhanak & Duck to the case of non-90° corner angles and allow for streamwise development of solutions. In addition, the effect of transpiration at the walls of the corner is investigated. The governing equations are of boundary-layer type and as such are parabolic in nature. Crucially, although the leading-order pressure term is known a priori, the third-order pressure term is not, but this is nonetheless present in the leading-order governing equations, together with the transverse and crossflow viscous terms.Particular attention is paid to flows which develop spatially from similarity solutions. It turns out that two scenarios are possible. In some cases the problem may be treated in the usual parabolic sense, with standard numerical marching procedures being entirely appropriate. In other cases standard marching procedures lead to numerically inconsistent solutions. The source of this difficulty is linked to the existence of eigensolutions emanating from the leading edge (which are not present in flows appropriate to the first scenario), analogous to those found in the computation of some two-dimensional hypersonic boundary layers (Neiland 1970; Mikhailov et al. 1971; Brown & Stewartson 1975). In order to circumvent this difficulty, a different numerical solution strategy is adopted, based on a global Newton iteration procedure.A number of numerical solutions for the entire corner flow region are presented.

The incompressible boundary layer in the corner formed by two intersecting perpendicular semi-infinite planes is investigated; the freestream flow, aligned with the corner, is of the form x * n , x * representing the streamwise distance from the leading edge. Similarity-type asymptotic solutions for large Reynolds numbers are derived and it is found that solutions do not exist for all values of the parameter n, and that for values of n where solutions do exist, non-uniqueness is a common feature; previous theoretical work has focused almost exclusively on the Blasius-type far field, corresponding to one of the n = 0 solutions.One corollary of the study is that in the case of a boundary-layer flow over a flat plate with a distant bounding wall (or other crossflow irregularity) and zero streamwise pressure gradient, in addition to the distant flow taking the form of the Blasius similarity solution, a second similarity solution also exists, exhibiting a sizeable jet-like crossflow velocity component. The present work may help partly explain some of the difficulties reported in observing corner similarity-type flows experimentally. The question of asymmetrical solutions is also alluded to.A study is also made regarding the nature of the linear local stability of the corner flow, and this indicates that the flow becomes more stable with increasing positive values of n, and less stable with increasing negative values of n. Further, the critical Reynolds number for the local flow increases with increasing distance from the corner.