Most fluid flows are turbulent rather than laminar and the reason for this was studied. One of the earliest explanations was that laminar flow is unstable, and the linear instability theory was first developed to explore this possibility. A series of early papers by Rayleigh produced many notable results concerning the instability of inviscid flows, such as the discovery of inflectional instability. Viscosity was commonly thought to act only to stabilize the flow, and flows with convex velocity profiles appeared to be stable. The investigations that led to a viscous theory of boundary layer instability was reported. The earliest application of linear stability theory to transition prediction calculated the amplitude ratio of the most amplified frequency as a function of Reynolds number for a Blasius boundary layer, and found that this quantity had values between five and nine at the observed Ret. The experiment of Schubauer and Skramstad (1947) completely reversed the prevailing option and fully vindicated the Gottingen proponents of the theory. This experiment demonstrated the existence of instability waves in a boundary layer, their connection with transition, and the quantitative description of their behavior by the theory of Tollmien and Schlichting. It is generally accepted that flow parameters such as pressure gradient, suction and heat transfer qualitatively affect transition in the manner predicted by the linear theory, and in particular that a flow predicted to be stable by the theory should remain laminar. The linear theory, in the form of the e9, or N-factor is today in routine use in engineering studies of laminar flow. The stability theory to boundary layers with pressure gradients and suction was applied. The only large body of numerical results for exact boundary layer solutions before the advent of the computer age by calculating the stability characteristics of the Falkner-Skan family of velocity profiles are given. When the digital computer reached a stage of development which permit the direct solution of the primary differential equations, numerical results were obtained from the linear theory during the next 10 years for many different boundary layer flows: three dimensional boundary layers; free convention boundary layers; compressible boundary layers; boundary layers on compliant walls; a recomputation of Falkner-Skan flows; unsteady boundary layers; and heated wall boundary layers.

Boundary-Layer Linear Stability Theory

Parabolized stability equations (PSE) have opened new avenues to the analysis of the streamwise growth of linear and nonlinear disturbances in slowly varying shear flows such as boundary layers, jets, and far wakes. Growth mechanisms include both algebraic transient growth and exponential growth through primary and higher instabilities. In contrast to the eigensolutions of traditional linear stability equations, PSE solutions incorporate inhomogeneous initial and boundary conditions as do numerical solutions of the Navier-Stokes equations, but they can be obtained at modest computational expense. PSE codes have developed into a convenient tool to analyze basic mechanisms in boundary-layer flows. The most important area of application, however, is the use of the PSE approach for transition analysis in aerodynamic design. Together with the adjoint linear problem, PSE methods promise improved design capabilities for laminar flow control systems.

Parabolized stability equations

Based on the hypothesis that patterns of skin-friction lines and external streamlines reflect the properties of continuous vector fields, topology rules define a small number of singular points (nodes, saddle points, and foci) that characterize the patterns on the surface and on particular projections of the flow (e.g., the crossflow plane). The restricted number of singular points and the rules that they obey are considered as an organizing principle whose finite number of elements can be combined in various ways to connect together the properties common to all steady three dimensional viscous flows. Introduction of a distinction between local and global properties of the flow resolves an ambiguity in the proper definition of a three dimensional separated flow. Adoption of the notions of topological structure, structural stability, and bifurcation provides a framework to describe how three dimensional separated flows originate and succeed each other as the relevant parameters of the problem are varied.

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Topology of Three-Dimensional Separated Flows

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

The Growth of Gortler vortices in boundary layers on concave walls is investigated. It is shown that for vortices of wavelength comparable to the boundary-layer thickness the appropriate linear stability equations cannot be reduced to ordinary differential equations. The partial differential equations governing the linear stability of the flow are solved numerically, and neutral stability is defined by the condition that a dimensionless energy function associated with the flow should have a maximum or minimum when plotted as a function of the downstream variable X. The position of neutral stability is found to depend on how and where the boundary layer is perturbed, so that the concept of a unique neutral curve so familiar in hydrodynamic-stability theory is not tenable in the Gortler problem, except for asymptotically small wavelengths. The results obtained are compared with previous parallel-flow theories and the small-wavelength asymptotic results of Hall (1982a, b), which are found to be reasonably accurate even for moderate values of the wavelength. The parallel-flow theories of the growth of Gortler vortices are found to be irrelevant except for the small-wavelength limit. The main deficiency of the parallel-flow theories is shown to arise from the inability of any ordinary differential approximation to the full partial differential stability equations to describe adequately the decay of the vortex at the edge of the boundary layer. This deficiency becomes intensified as the wavelength of the vortices increases and is the cause of the wide spread of the neutral curves predicted by parallel-flow theories. It is found that for a wall of constant radius of curvature a given vortex imposed on the flow can grow for at most a finite range of values of X. This result is entirely consistent with, and is explicable by the asymptotic results of, Hall (1982a).

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S0022112083000968

The nonlinear development of Görtler vortices in growing boundary layers

A phenomenon of boundary-layer instability is discussed from the theoretical and experimental points of view. The china-clay evaporation technique shows streaks on the surface, denoting a vortex system generated in the region of flow upstream of transition. Experiments on a swept wing are described briefly, while experiments on the flow due to a rotating disk receive much greater attention. In the latter case, the axes of the disturbance vortices take the form of equi-angular spirals, bounded by radii of instability and of transition. A frequency analysis of the disturbances shows that there is a narrow band of disturbance components of high amplitude, some frequencies within this band corresponding to disturbances fixed relative to the surface and others corresponding to moving waves. Furthermore, the determination of velocity profiles for the rotating-disk flow is described, the agreement with the theoretical solution for laminar flow being quite satisfactory; for turbulent flow, however, the empirical theories are not very satisfactory. In order to explain the vortex phenomenon just discussed, the general equations of motion in orthogonal curvilinear co-ordinates are examined by superimposing an infinitesimal disturbance periodic in space and time on the main flow, and linearizing for small disturbances. An important result is that, within the range of certain approximations, the velocity component in the direction of propagation of the disturbance may be regarded as a two-dimensional flow for stability purposes; then the problem of stability formally resembles the well-known two dimensional problem. However, it is important to emphasize that this result—namely, that the flow curvature has little influence on stability—is applicable only to the possible modes of instability in a local region. The nature of three-dimensional flows is discussed, and the importance of co-ordinates along and normal to the stream-lines outside the boundary layer is examined. In accord with the formal two-dimensional nature of the instability, there is a whole class of velocity distributions, corresponding to different directions, which may exhibit instability. The question of stability at infinite Reynolds number is examined in detail for these profiles. As for ordinary two-dimensional flows, the wave velocity of the disturbance must lie somewhere between the maximum and minimum of the velocity profile considered. The points where the wave velocity equals the fluid velocity are called critical points, of which most of the profiles considered have two. Then Tollmien’s criterion that velocity profiles with a point of inflexion are unstable at infinite Reynolds number is extended to the case of profiles with two critical points. One particular profile—namely, that for which the point of inflexion lies at the point of zero velocity—may generate neutral disturbances of zero phase velocity, corresponding to the disturbances visualized by the china-clay technique. A variational method for the solution of certain of the eigenvalue problems associated with stability at infinite Reynolds number is derived, found by comparison with an exact solution to be very accurate, and applied to the rotating disk. The fixed vortices predicted by the theory have as their axes equi-angular spirals of angle 103°, in good agreement with experiment, but the agreement between theoretical and experimental wave number is not good, the discrepancy being attributed to viscosity. Finally, the correlation between the experimentally observed and theoretically possible disturbances is discussed and certain conclusions drawn therefrom. The streamlines of the disturbed boundary layer show the existence of a double row of vortices, one row of which produces the streaks in the china clay. Application of the theory to other physical phenomena is described.

W. S. Walker

Papers

On the stability of three-dimensional boundary layers with application to the flow due to a rotating disk