Introduction and problem formulation History, background, and rationale Initial-value concepts and stability bases Classical treatment: modal expansions Transient dynamics Asymptotic behavior Role of viscosity Geometries of relevance Spatial stability bases Temporal stability of inviscid incompressible flows General equations Nondimensionalization Mean plus fluctuating components Linearized disturbance equations Recourse to complex functions Three-dimensionality Squire transformation Kelvin-Helmholtz theory Interface conditions Piecewise linear profile Unconfined shear layer Confined shear layer Inviscid temporal theory Critical layer concept pagex xix xxi

New classes of exact solutions of the incompressible Navier-Stokes equations are presented. The method of solution has its origins in that first used by Kelvin ( Phil. Mag . 24 (5), 188-196 (1887)) to solve the linearized equations governing small disturbances in unbounded plane Couette flow. The new solutions found describe arbitrarily large, spatially periodic disturbances within certain two- and three-dimensional ‘ basic ’ shear flows of unbounded extent. The admissible classes of basic flow possess spatially uniform strain rates; they include two- and three- dimensional stagnation point flows and two-dimensional flows with uniform vorticity. The disturbances, though spatially periodic, have time-dependent wavenumber and velocity components. It is found that solutions for the disturbance do not always decay to zero ; but in some instances grow continuously in spite of viscous dissipation. This behaviour is explained in terms of vorticity dynamics.

The equations of an incompressible fluid are linearized for small perturbations of a basic parallel flow. The initial‐value problem is then posed by use of Fourier transforms in space. Previous results are systematized in a general framework and used to solve a series of problems for prototypical examples of basic shear flow and of initial disturbance. The prototypes of shear flow are (a) plane Couette flow bounded by rigid parallel walls, (b) plane Couette flow bounded by rigid walls at constant pressure, (c) unbounded two‐layer flow with linear velocity profile in each layer, (d) a piecewise linear profile of a boundary layer on a rigid wall. The prototypes of initial perturbation are the fundamental ones: (i) a point source of the field of the transverse velocity (represented by delta functions), (ii) an unbounded sinusoidal field of the transverse velocity, (iii) a point source of the lateral component of vorticity, (iv) a sinusoidal field of the lateral vorticity. Detailed solutions for an inviscid fluid are presented, but the problem for a viscous fluid is only broached.

The growth of small disturbances in a two-dimensional incompressible wake has been investigated theoretically and experimentally. The theoretical analysis is based upon inviscid stability theory wherein small disturbances are considered from both temporal and spatial reference frames. Through a combined stability analysis, in which small disturbances are permitted to amplify in both time and space, the relationship between the disturbance characteristics for the temporal and spatial reference frames is shown. In these analyses a quasi-uniform assumption is adopted to account for the continuously varying mean-velocity profiles that occur behind flat plates and thin airfoils. It is found that the most unstable disturbances in the wake produce transverse oscillations in the mean-velocity profile and correspond to growing waves that have a minimum group velocity.Experimentally, the downstream development of the wake of a thin airfoil and the wave characteristics of naturally amplifying small disturbances are investigated in a water tank. The disturbances that develop are found to produce transverse oscillations of the mean-velocity profile in agreement with the theoretical prediction. From the comparison of the experimental results with the predictions for the characteristics of the most unstable waves via the temporal and spatial analyses, it is concluded that the stability analysis for the wake is to be considered solely from the more realistic spatial viewpoint. Undoubtedly, this conclusion is also applicable to other highly unstable flows such as jets and free shear layers.In accordance with the disturbance vorticity distribution as determined from the spatial model, a description of the initial development of a vortex street is put forth that contrasts with the description given by Sato & Kuriki (1961).