J. D. A. Walker

The impact of a vortex ring on a wall

The flow induced by a vortex ring approaching a plane wall on a trajectory normal to the wall is investigated for an incompressible fluid which is otherwise stagnant. The detailed characteristics of the interaction of the ring with the flow near the surface have been observed experimentally for a wide variety of laminar rings, using dye in water to visualize the flow in the ring as well as near the plane surface. Numerical solutions are obtained for the trajectory of the ring as well as for the unsteady boundary-layer flow that develops on the wall. The experimental and theoretical results show that an unsteady separation develops in the boundary-layer flow, in the form of a secondary ring attached to the wall. A period of explosive boundary-layer growth then ensues and a strong viscous-inviscid interaction occurs in the form of the ejection of the secondary vortex ring from the boundary layer. The primary ring then interacts with the secondary ring and in some cases was observed to induce the formation of a third, tertiary, ring near the wall. The details of this process are investigated over a wide Reynolds number range. The results clearly show how one vortex ring can produce another, through an unsteady interaction with a viscous flow near the wall.

The boundary layer induced by a convected two-dimensional vortex

The response of a wall boundary layer to the motion of a convected vortex is investigated. The principal cases considered are for a rectilinear filament of strength –κ located a distance a above a plane wall and convected to the right in a uniform flow of speed U ∞ *. The inviscid solution predicts that such a vortex will remain at constant height a above the wall and be convected with constant speed α U ∞ *. Here α is termed the fractional convection rate of the vortex, and cases in the parameter range 0 [les ] α t * = 0 and numerical calculations of the developing boundary-layer flow are carried out for α = 0, 0.2, 0.4, 0.55, 0.7, 0.75 and 0.8. For α

Vortex-induced boundary-layer separation. Part 1. The unsteady limit problem Re ⟶ ∞

The unsteady boundary-layer flow produced by a two-dimensional vortex in motion above an infinite plane wall in an otherwise stagnant fluid is considered in the limit of infinite Reynolds number. This study is part of a continuing investigation into the nature of the physical processes that occur near the surface in transitional and fully turbulent boundary layers. The adverse pressure gradient due to the vortex leads to the development of a zone of recirculation in the viscous flow near the surface, and the boundary-layer flow then focuses rapidly toward an eruption along a band which is very narrow in the streamwise direction. The evolution of the unsteady boundary layer is posed in Lagrangian coordinates and computed using an efficient, factored ADI numerical method. The boundary-layer solution is found to develop a separation singularity and to evolve toward a terminal stage which is generic in two-dimensional unsteady flows. The computed results are compared with the results of asymptotic theory of two-dimensional boundary-layer separation and the agreement is found to be excellent.

VORTEX-INDUCED BOUNDARY-LAYER SEPARATION. PART 2. UNSTEADY INTERACTING BOUNDARY-LAYER THEORY

The unsteady boundary layer induced by the motion of a rectilinear vortex above an infinite plane wall is calculated using interacting boundary-layer methods. The boundary-layer solution is computed in Lagrangian variables since it is possible to compute the flow evolution accurately in this formulation even when an eruption starts to evolve. Results are obtained over a range of Reynolds numbers, Re. For the limit problem Re - infinity (studied in Part 1), a singularity develops in the non-interacting boundary-layer solution at finite time. The present results show that the interacting boundary-layer calculations also terminate in a singularity at a time which is earlier than in the limit problem and which decreases with decreasing Reynolds number. The computed results are compared with the length-and timescales predicted by recent asymptotic theories and are found to be in excellent agreement. See also previous abstract.

The onset of instability in unsteady boundary-layer separation

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.

Short-scale break-up in unsteady interactive layers: local development of normal pressure gradients and vortex wind-up

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.

The structure of a three-dimensional turbulent boundary layer

The three-dimensional turbulent boundary layer is shown to have a self-consistent two-layer asymptotic structure in the limit of large Reynolds number. In a streamline coordinate system, the streamwise velocity distribution is similar to that in two-dimensional flows, having a defect-function form in the outer layer which is adjusted to zero at the wall through an inner wall layer. An asymptotic expansion accurate to two orders is required for the cross-stream velocity which is shown to exhibit a logarithmic form in the overlap region. The inner wall-layer flow is collateral to leading order but the influence of the pressure gradient, at large but finite Reynolds numbers, is not negligible and can cause substantial skewing of the velocity profile near the wall. Conditions under which the boundary layer achieves self-similarity and the governing set of ordinary differential equations for the outer layer are derived. The calculated solution of these equations is matched asymptotically to an inner wall-layer solution and the composite profiles so formed describe the flow throughout the entire boundary layer. The effects of Reynolds number and cross-stream pressure gradient on the cross-stream velocity profile are discussed and it is shown that the location of the maximum cross-stream velocity is within the overlap region.

An instability in supersonic boundary-layer flow over a compression ramp

Separation of a supersonic boundary layer (or equivalently a hypersonic boundary layer in a region of weak global interaction) near a compression ramp is considered for moderate wall temperatures. For small ramp angles, the flow in the vicinity of the ramp is described by the classical supersonic triple-deck structure governing a local viscous-inviscid interaction. The boundary layer is known to exhibit recirculating flow near the corner once the ramp angle exceeds a certain critical value. Here it is shown that above a second and larger critical ramp angle, the boundary-layer flow develops an instability. The instability appears to be associated with the occurrence of inflection points in the streamwise velocity profiles within the recirculation region and develops as a wave packet which remains stationary near the corner and grows in amplitude with time.

Boundary-layer separation control on a thin airfoil using local suction

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.

The three-dimensional turbulent boundary layer near a plane of symmetry

The asymptotic structure of the three-dimensional turbulent boundary layer near a plane of symmetry is considered in the limit of large Reynolds number. A selfconsistent two-layer structure is shown to exist wherein the streamwise velocity is brought to rest through an outer defect layer and an inner wall layer in a manner similar to that in two-dimensional boundary layers. The cross-stream velocity distribution is more complex and two terms in the asymptotic expansion are required to yield a complete profile which is shown to exhibit a logarithmic region. The flow in the inner wall layer is demonstrated to be collateral to leading order; pressure gradient effects are formally of higher order but can cause the velocity profile to skew substantially near the wall at the large but finite Reynolds numbers encountered in practice. The governing set of ordinary differential equations describing a self-similar flow is derived. The calculated numerical solutions of these equations are matched asymptotically to an inner wall-layer solution and the results show trends that are consistent with experimental observations.