Concerning marginal singularities in the boundary-layer flow on a downstream-moving surface

The formation of separation singularities in solutions of the classical boundary-layer equations is studied numerically and analytically for the case of a two-dimensional incompressible steady flow near a solid surface moving in the direction of the main stream. Unlike the previously studied regime of the incipient separation located at the maximum point in the external pressure distribution, the breakdown in this work occurs under an adverse pressure forcing and involves a regular flow field upstream of the Moore-Rolt-Sears point with an algebraic non-analyticity downstream. Small deviations from the precisely regular approach to the singular point are shown to result in an exponential amplification of linear disturbances; in the subsequent nonlinear stage the solution terminates in a finite-distance blow-up singularity or, alternatively, continues in a regular fashion across the singular station. The case of asymptotically small slip velocities is considered and a connection with marginal separation on a fixed wall is discussed

Instabilities in a high-Reynolds-number boundary layer on a film-coated surface

A high-Reynolds-number asymptotic theory is developed for linear instability waves in a two-dimensional incompressible boundary layer on a flat surface coated with a thin film of a different fluid. The focus in this study is on the influence of the film flow on the lower-branch Tollmien-Schlighting waves, and also on the effect of boundary-layer/potential flow interaction on interfacial instabilities. Accordingly, the film thickness is assumed to be comparable to the thickness of a viscous sublayer in a three-tier asymptotic structure of lower-branch Tollmien-Schlichting disturbances. A fully nonlinear viscous/inviscid interaction formulation is derived, and computational and analytical solutions for small disturbances are obtained for both Tollmien-Schlichting and interfacial instabilities for a range of density and viscosity ratios of the fluids, and for various values of the surface tension coefficient and the Froude number. It is shown that the interfacial instability contains the fastest growing modes and an upper-branch neutral point within the chosen flow regime if the film viscosity is greater than the viscosity of the ambient fluid. For a less viscous film the theory predicts a lower neutral branch of shorter-scale interfacial waves. The film flow is found to have a strong effect on the Tollmien-Schlichting instability, the most dramatic outcome being a powerful destabilization of the flow due to a linear resonance between growing Tollmien-Schlichting and decaying capillary modes. Increased film viscosity also destabilizes Tollmien-Schlichting disturbances, with the maximum growth rate shifted towards shorter waves. Qualitative and quantitative comparisons are made with experimental observations by Ludwieg & Hornung (1989)

Planar flows past thin multi-blade configurations

Two-dimensional steady laminar flows past multiple thin blades positioned in near or exact sequence are examined for large Reynolds numbers. Symmetric configurations require solution of the boundary-layer equations alone, in parabolic fashion, over the successive blades. Non-symmetric configurations in contrast yield a new global inner-outer interaction in which the boundary layers, the wakes and the potential flow outside have to be determined together, to satisfy pressure-continuity conditions along each successive gap or wake. A robust computational scheme is used to obtain numerical solutions in direct or design mode, followed by analysis. Among other extremes, many-blade analysis shows a double viscous structure downstream with two streamwise length scales operating there. Lift and drag are also considered. Another new global interaction is found further downstream. All the interactions involved seem peculiar to multi-blade flows

On 'spot' evolution under an adverse pressure gradient

The unsteady travelling 'spots' or spot-like disturbances are produced, in an otherwise planar boundary layer, by an initial impulse/blip, from wall forcing or from nearby external forcing. Theory and computations are described for the evolving spot-like structure, yielding initial-value problems for inviscid spot-like disturbances, commencing near the onset of an adverse pressure gradient. A transient stage incorporates the initial conditions, following which adverse pressure gradient effects become significant. Leading and trailing critical layers then form, which confine and define the spot-like disturbance, and these depart from the wall downstream accompanied by disturbance amplification and mean flow distortion. The interplay of adverse pressure gradient effects with three-dimensionality, nonlinearity and non-parallelism is considered in turn.Three-dimensional effects provoke a universal closed planform of spot-like disturbance, which has a different side behaviour from the zero-gradient case. Nonlinear interactions eventually change the internal structure, particularly at the spot-like disturbance leading edge, while pointing to the mean-flow alteration underhanging the spot-like disturbance and to a pressure-feedback alteration for the region behind the spot-like disturbance. These two alterations offer complementary mechanisms for describing the calmed region trailing a spot-like disturbance, in which an attached thinned wall layer is identified. Non-parallel effects lead to enhanced spot-like disturbance growth and larger-scale/shorter-scale interactive behaviour downstream. The approach to separation is also considered, yielding maximal growth for small spot-like disturbances at 5/6 of the way from the minimum pressure position to the separation position. Links with recent experiments on adverse-gradient spot-like disturbances and with findings on calmed region properties are investigated, as well as the unsteady forcing effects from an incident relatively thick vortical wake outside the boundary layer

Vortex/inflectional-wave interactions with weakly three-dimensional input

The subtle impact of the spanwise scaling in nonlinear interactions between oblique instability waves and the induced longitudinal vortex field is considered theoretically for the case of a Rayleigh-unstable boundary-layer flow, at large Reynolds numbers. A classification is given of various flow regimes on the basis of Reynolds-stress mechanisms of mean vorticity generation, and a connection between low-amplitude non-parallel vortex/wave interactions and less-low-amplitude non-equilibrium critical-layer flows is discussed in more detail than in previous studies. Two new regimes of vortex/wave interaction for increased spanwise lengthscales are identified and studied. In the first, with the cross-scale just slightly larger than the boundary-layer thickness, the wave modulation is governed by an amplitude equation with a convolution and an ordinary integral term present due to nonlinear contributions from all three Reynolds-stress components in the cross-momentum balance. In the second regime the cross-scale is larger; and the wave modulation is found to be governed by an integral/partial differential equation. In both cases the main-flow non-parallelism contributes significantly to the coupled wave/vortex development

Singular modes in Rayleigh instability of three-dimensional streamwise-vortex flows

The upper-branch neutral modes of inviscid instability in a boundary-layer flow with significant longitudinal vortices present are shown to possess typically a logarithmically singular, non-inflectional, critical layer. This contrasts with previous linear and nonlinear suggestions implemented in vortex-wave interaction and secondary instability theories, which are re-examined. The analysis here is based first on perturbation techniques applied to a Rayleigh unstable planar motion supplemented by a vortex centred around the inflection level, followed by the extension to more general cases. Flows with order one and larger spanwise scales are considered. Multiple solutions, their limit properties and parametric continuations are illustrated with concrete examples

On the starting process of strongly nonlinear vortex/Rayleigh-wave interactions

An oncoming two-dimensional laminar boundary layer that develops an unstable inflection point and becomes three-dimensional is described by the Hall-Smith (1991) vortex/wave interaction equations. These equations are now examined in the neighbourhood of the position where the critical surface starts to form. A consistent structure is established in which an inviscid core flow is matched to a viscous buffer-layer solution where the appropriate jump condition on the transverse shear stress is satisfied. The final result is a bifurcation equation for the (constant) amplitude of the wave pressure. A representative classical velocity profile is considered to illustrate solutions of this equation for a range of values of the wave-numbers

Multiple Deck Theory

This chapter is an introduction to the asymptotic theory of laminar viscous flows at large Reynolds numbers. Starting with a derivation of the governing Navier–Stokes equations, the theory is presented as a sequence of worked examples of increasing complexity including the classical boundary layer, the triple-deck flow with viscous-inviscid interaction and the double-deck interactive flow in a wall jet, briefly touching upon supersonic and condensed flow regimes. The theory is used to study the phenomena of upstream influence, flow induced by a local roughness, short-scale boundary layer separation near an irregularity in the wall geometry, self-induced separation as part of a global picture, and instability in the flow

On-wall and interior separation in a two-fluid boundary layer

A two-fluid boundary layer is considered in the context of a high Reynolds number Poiseuille–Couette channel flow encountering an elongated shallow obstacle. The flow is laminar, steady and two-dimensional, with the boundary layer shown to have the pressure unknown in advance and a specified displacement (a condensed boundary layer). The focus is on the detail of the flow reversal triggered by the obstacle. The interface between the two fluids passes through the boundary layer which, in conjunction with the effects of gravity and distinct densities in the two fluids, leads to several possible topologies of the reversed flow, including a conventional on-wall separation, interior flow reversal above the interface, and several combinations of the two. The effect of upstream influence due to a transverse pressure variation under gravity is mentioned briefly