The stability of a supersonic boundary layer above a flexible surface is considered in the limit of large Reynolds number and for Mach numbers O(1). Asymptotic theory of viscous-inviscid interaction has been used for this purpose. We found that for a simple elastic surface, for which deflections are proportional to local pressure differences, the boundary-layer flow remains stable as it is for a rigid wall. However, when either damping or surface inertia is included the flow becomes unstable. Moreover, in a certain range of wave numbers the boundary layer develops more then one unstable mode. It is interesting that these modes are connected to one another via saddle points in the complex-frequency plane. A more complex Kramer-type surface is also analysed and in some parameter ranges is found to permit the evolution of unstable Tollmien-Schlichting waves. The neutral curves are found for a variety of situations related to the parameters associated with the flexible surface.

Stability of the supersonic compression ramp flow is studied based on the numerical solution of viscous-inviscid interaction equations. These are valid when the Reynolds number Re is large and Mach number is an order one quantity. Assuming that the ramp angle θ = O(Re), the fluid motion may be described by the well-known triple-deck theory. For relatively small values of the scaled ramp angle α = θRe, steady stable solutions can be obtained. However, it is shown, that when a recirculation zone is present near the corner point and the ramp angle α is sufficiently large, the flow in the recirculation zone is susceptible to convective instabilities when perturbations are introduced there. At still larger values of the scaled ramp angle α, an absolute instability is shown to occur that leads to a violent local breakdown of the boundary layer.

High-speed incompressible flow past a thin airfoil in a uniform stream is considered. When the angle of attack for a solid airfoil exceeds a certain critical value, the boundary layer in the leading-edge region separates in a process known to lead to dynamic stall. Here suction near the leading edge is studied as a means of controlling separation and thereby inhibiting dynamic stall. First, steady boundary-layer solutions are obtained to determine the nature of suction distributions required to suppress separation on an airfoil at an angle of attack beyond the critical value (for a solid wall). Unsteady boundary-layer solutions are then obtained, using a combination of Eulerian and Lagrangian techniques, for an airfoil at an angle of attack exceeding the critical value; the effects of various parameters associated with the finite-length suction slot, its location and the suction strength are considered. Major modifications of the Lagrangian numerical method are required to account for suction at the wall. It is determined that substantial delays in separation can be achieved even when the suction is weak, provided that the suction is initiated at an early stage.

A three-dimensional inviscid velocity distribution is selected to mimic the flow produced at the wall by a three-dimensional vortex convected above an infinite wall. The main interest is in determining the viscous response of the boundary layer on the wall to the imposed pressure distribution. It is demonstrated that complex effects occur in the boundary layer (including the formation of zones of apparent recirculation) that eventually lead to separation. Solutions are obtained in both a conventional Eulerian frame of reference and with a three-dimensional Lagrangian method. The separation takes the form of a sharply focused eruptive tongue of fluid in a manner consistent with modern asymptotic theories of three-dimensional separation, which indicate the boundary-layer solution will develop a singularity in the absence of interaction with the external flow. The unsteady separation structure is similar to that in two dimensions when viewed in the appropriate direction. The possible relevance of the results to the dynamics of turbulent boundary layers is described.

Physical situations where a viscous boundary layer breaks down and interacts strongly with an effectively inviscid external flow are common place. For large Reynolds numbers, viscous effects are normally confined to thin boundary layers on all solid surfaces for the majority of any observation time. In most practical situations, exposure of such layers to an adverse pressure gradient is inevitable and in this circumstance, a sequence of events commences near the wall that culminates in an eruption and a strong viscous-inviscid interaction with the external flow. The events leading up to eruption are known as the Van Dommelen–Shen process and the eruption itself is referred to as boundary-layer separation; here the term ‘separation’ denotes the first process of interaction between a hitherto thin boundary layer and the external flow. The event is sufficiently complicated that extraordinary measures are needed to compute its evolution. In most situations, the onset of separation is subtle and hard to detect and thus development of rational control procedures is a challenging task. Here recent calculations of unsteady separation events are discussed for two- and three-dimensional flows. The phenomena involved are generic but leading-edge separation on airfoils and rotorcraft blades is emphasized. Recent studies on various control mechanisms are described, which are found to have the effect of slowing down and/or weakening the separation process. For some control processes, it has proved possible to eliminate separation entirely.

Nonlinear flow-induced oscillations of a thin rectangular plate are studied using a spectral method to calculate the vertical and tangential displacements. When the plate is subjected to forcing near the first linear resonance frequency, a complex motion with amplitude comparable to the plate thickness occurs in the form of a modulated high-frequency vibration. Numerical solutions for two- and three-dimensional plates are obtained to establish the nature of these oscillations. The numerical results are supported by an asymptotic analysis describing nonlinear resonance for rectangular plates.

Unsteady separation processes at large finite, Reynolds number, Re, are considered, as well as the possible relation to existing descriptions of boundary-layer separation in the limit Re → ∞. The model problem is a fundamental vortex-driven three-dimensional flow, believed to be relevant to bursting near the wall in a turbulent boundary layer. Bursting is known to be associated with streamwise vortex motion, but the vortex/wall interactions that drive the near-wall flow toward breakdown have not yet been fully identified. Here, a simulation of symmetric counter-rotating vortices is used to assess the influence of sustained pumping action on the development of a viscous wall layer. The calculated solutions describe a three-dimensional flow at finite Re that is independent of the streamwise coordinate and consists of a crossflow plane motion, with a developing streamwise flow. The unsteady problem is constructed to mimic a typical cycle in turbulent wall layers and numerical solutions are obtained over a range of Re. Recirculating eddies develop rapidly in the near-wall flow, but these eddies are eventually bisected by alleyways which open up from the external flow region to the wall. At sufficiently high Re, an oscillation was found to develop in the streamwise vorticity field near the alleyways with a concurrent evolution of a local spiky behaviour in the wall shear. Above a critical value of Re, the oscillation grows rapidly in amplitude and eventually penetrates the external flow field, suggesting the onset of an unstable wall-layer breakdown. Local zones of severely retarded streamwise velocity are computed which are reminiscent of the low-speed streaks commonly observed in turbulent boundary layers. A number of other features also bear a resemblance to observed coherent structure in the turbulent wall layer.

A fundamental flow problem of unsteady wind-up of a spanwise vortex is studied in this theoretical work on deepening dynamic stall and transition in a boundary layer, internal layer or related unsteady motion. It examines the nonlinear evolution of the spanwise vortex produced when the local wall pressure develops a maximum or minimum, subsequent to the finite-time break-up of an interacting layer and the impact of normal pressure gradients. The evolution is controlled by an inner–outer interaction between the effects of the normal pressure gradient and the momentum jumps across and outside the vortex, which is situated near the strong inflexion point induced in the mean flow. Although the work concentrates on a particular internal-flow context, many of the flow properties found are generic and in particular apply for a more general case including external flows. Analysis and associated computations point to two main distinct trends in the vortex response, depending to a large extent on a parameter gauging the relative strengths of the above effects. The response is either an explosive one, provoking enhanced wind-up, growth and pressure in the vortex, or it is implosive, causing the vortex to shrink and virtually empty itself through unwinding, leaving little local pressure variation. A further discussion includes the after-effects of this vortex response and some of the connections with experiments and direct computations on deepening stall and transition.

In this chapter, asymptotic analysis is used to consider various aspects of turbulent boundary layers in the limit of large Reynolds number. Turbulent wall-bounded shear flows are common in engineering practice and, although such motions can be very complex, generic trends are exhibited over a wide range of Reynolds numbers. In such circumstances, asymptotic theory is an essential tool for revealing the critical aspects of boundary-layer structure, as well as the dominant physical processes in the turbulence. In the following six sections, specific issues related to both the prediction and physics of turbulent shear flows near walls will be addressed. In §2, some of the classical results for two-dimensional incompressible flows will be reviewed and extended; special emphasis is placed on the minimum information and the numerical algorithms that are required to structure a prediction scheme for such flows; this chapter forms a basis for the more complicated types of boundary layers considered in subsequent sections. In §3, a model for the mean flow profile in the near-wall region of the boundary layer is described; this model is based on the observed coherent structure of the wall-layer flow and provides a simple alternative to conventional mixing-length formulations. In §4, the case of incompressible two-dimensional flow with heat transfer at the wall is addressed; the asymptotic theory constrains the types of models that can be used in the energy equation and provides an effective way to determine the heat transfer at the surface in a prediction method.

Following the finite-time collapse of an unsteady interacting boundary layer (step 1), shortened length and time scales are examined here in the near-wall dynamics of transitional-turbulent boundary layers or during dynamic stall. The next two steps are described, in which (step 2) normal pressure gradients come into operation along with a continuing nonlinear critical-layer jump and then (step 3) vortex formation is induced typically. Normal pressure gradients enter in at least two ways, depending on the internal or external flow configuration. This yields for certain internal flows an extended KdV equation with an extra nonlinear integral contribution multiplied by a coefficient which is proportional to the normal rate of change of curvature of the velocity profile locally and whose sign turns out to be crucial. Positive values of the coefficient lead to a further finite-time singularity, while negative values produce a rapid secondary instability phenomenon. Zero values in contrast allow an interplay between solitary waves and wave packets to emerge at large scaled times, this interplay eventually returning the flow to its original, longer, interactive, boundary-layer scales but now coupled with multiple shorter-scale Euler regions. In external or quasi-external flows more generally an extended Benjamin–Ono equation holds instead, leading to a reversal in the roles of positive and negative values of the coefficient. The next step, 3, typically involves the strong wind-up of a local vortex, leading on to explosion or implosion of the vortex. Further discussion is also presented, including the three-dimensional setting, the computational implications, and experimental links.

Unsteady boundary-layer development over moving walls in the limit of infinite Reynolds number is investigated using both the Eulerian and Lagrangian formulations. To illustrate general trends, two model problems are considered, namely the translating and rotating circular cylinder and a vortex convected in a uniform flow above an infinite flat plate. To enhance computational speed and accuracy for the Lagrangian formulation, a remeshing algorithm is developed. The calculated results show that unsteady separation is delayed with increasing wall speed and is eventually suppressed when the speed of the separation singularity approaches that of the local mainstream velocity. This suppression is also described analytically. Only ‘upstream-slipping’ separation is found to occur in the model problems. The changes in the topological features of the flow just prior to the separation that occur with increasing wall speed are discussed.

Limit cycle oscillations are complex vibrations of airframe surfaces for which the cause-and-effect relationships are poorly understood. This fundamental investigation of oscillations of a Ž nite two-dimensional  exible plate considers unforced vibrations, as well as situations where motion is induced by a convected vortex, a pressure wave, or an unsteady boundary-layer separation on the moving plate. It is found that very complex vibration patterns can occur and, with increasing nonlinearity, essentially chaotic motion can develop. When the plate is forced at frequencies close to the natural frequencies of the linear problem, a nonlinear resonance phenomenon occurs leading to a complex limit cycle behavior.

At high Reynolds numbers, the process leading to dynamic stall on airfoils initiates in the leading-edge region. For thin airfoils, the local motion near rounded leading edges can be represented as flow past a parabola and when the mainstream flow is at an angle of attack to the airfoil, a portion of the boundary layer will be exposed to an adverse pressure gradient. Once the angle of attack exceeds a certain critical value, it is demonstrated that unsteady boundary-layer separation will occur in the leading-edge region in the form of an abrupt focused boundary-layer eruption. This process is believed to initiate the formation of the dynamic stall vortex. For impulsively-started incompressible flow past a parabola, a generic behavior is found to occur over a range of angles of attack, and a limit solution corresponding to relatively large angles is found. The separation in the leading-edge region develops in a zone of relatively limited streamwise extent over a wide range of angles of attack. This suggests that localized control measures (such as suction) may possibly be effective at inhibiting separation.

A mathematical model and numerical algorithm are developed to predict electrode-position rates from a 2-D jet of electrolyte that impinges on a flat surface The principal situations of interest are for applied voltages that produce current densities below the limiting current. The motion is assumed to be at high speed with the jet inducing a thin laminar boundary layer on the surface; a progressively thinner concentration layer and an electrochemical double layer near the surface ave accounted for. Two cases, corresponding to a submerged and unsubmerged jet, are considered. A boundary integral method is used to compute the current density along the plate in a general iterative numerical procedure coupled to the solution of the hydrodynamic, concentration and electrochemical boundary layers. The results show that relatively high deposition rates occur near the point of impingement and that altering the jet angle relative to the surface influences local electrodeposition rates significantly.

The process of unsteady two-dimensional boundary-layer separation at high Reynolds number is considered. Solutions of the unsteady non-interactive boundary-layer equations are known to develop a generic separation singularity in regions where the pressure gradient is prescribed and adverse. As the boundary layer starts to separate from the surface, however, the external pressure distribution is altered through viscous-inviscid interaction just prior to the formation of the separation singularity; hitherto this has been referred to as the first interactive stage. A numerical solution of this stage is obtained here in Lagrangian coordinates. The solution is shown to exhibit a high-frequency inviscid instability resulting in an immediate finite-time breakdown of this stage. The presence of the instability is confirmed through a linear stability analysis. The implications for the theoretical description of unsteady boundary-layer separation are discussed, and it is suggested that the onset of interaction may occur much sooner than previously thought.

An asymptotic analysis of the equations describing supersonic turbulent flow over an adiabatic wall is carried out for high Reynolds numbers, Re, and mainstream Mach numbers, Me=O(1). A general expression for the adiabatic-wall temperature is derived. The asymptotic theory constrains the types of turbulence models that are suitable to represent the effects of viscous dissipation. A simple algebraic turbulence model is proposed and comparisons with measured total enthalpy profile data show good agreement, capturing the overshoot observed in total enthalpy near the boundarylayer edge.

The effect of wall cooling on hypersonic boundary-layer separation near a compression ramp is considered. Two cases are identified corresponding to the value of the average Mach number $\overline{M}$ across the upstream boundary layer approaching the compression ramp. The flow is referred to as supercritical for $\overline > 1$ and subcritical for $\overline{M} < 1$. The interaction is described by triple-deck theory, and numerical results are given for both cases for various ramp angles and levels of wall cooling. The effect of wall cooling on the absolute instability described recently by Cassel, Ruban & Walker (1995) for an uncooled wall is of particular interest; a stabilizing effect is observed for supercritical boundary layers, but a strong destabilizing influence occurs in the subcritical case. Wall cooling also influences the location and size of the separated region. For supercritical flow, progressive wall cooling reduces the size of the recirculating-flow region, the separation point moves downstream, and upstream influence is diminished. In contrast for the subcritical case downstream influence is reduced with increased cooling. In either situation, a sufficient level of wall cooling eliminates separation altogether for the ramp angles considered. The present numerical results closely confirm the strong wall cooling theory of Kerimbekov, Ruban & Walker (1994).