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A theoretical study is described of the effects of a passive porous wall on boundary-layer instability. The passive porouswall isconceived of as a thin porous sheet stretched over a plenum chamber. When disturbancesin theform of Tollmien‐Schlichting waves propagate along the boundary layer, the e uctuating pressure forces air in and out of the plenum chamber. The basic approach is based on classic linear stability theory for the e at-plate boundary layer with modie ed wall boundary conditions. The wall response is represented by a complex admittance (ratio of e uctuating e ow rate per unit area to wall pressure ) and, therefore, applies to a general class of passive porous, and other interactive, walls. The effects of a wide range of admittance values are studied. A stabilizing effect is obtained when the admittance phase is close to o/2, and an optimum value of admittance magnitude is also found. A theoretical model is introduced and used to calculate the admittance of the porous panels used in a parallel experimental study. It is shown that it may be possible to manufacture stabilizing porous panels. The stabilizing mechanism is due to the production of a near-wall region of negative Reynolds stress.

Small-amplitude wave systems interacting nonlinearly can produce 0(1) amplitude streamwise vortex structures through the vortex-wave interaction mechanism described, for example, by [1-3]. The key feature of the interaction is that the spanwise velocity component of a vortex is small as compared to the streamwise component so that a nonlinear wave system driving the spanwise velocity component through Reynolds stresses can provoke a 0(1) response of the vortex. The wave system can correspond to either a Rayleigh or Tollmien-Schlichting wave disturbance, but previous work on the initiation of the process has been confined to Rayleigh waves (see, for example, [5, 6]). Here, we address the nonlinear initial value problem for Tollmien-Schlichting wave-vortex interactions in channel flows. The evolution of the disturbances is accounted for using the phase equation approach of [7]. We determine the circumstances, if any, under which the finite amplitude vortex-wave equilibrium states of [4] are generated. Our discussion of the nonlinear evolution of a wave system points toward a possible mechanism for the experimentally observed breakup of three-dimensional instabilities into shorter streamwise scales.

We study the nonlinear development of two-dimensional Tollmien-Schlichting waves in an incompressible flat-plate boundary layer at asymptotically large values of the Reynolds number. Attention is restricted to the far-downstream lower-branch' regime where a multiple-scales analysis is possible. It is supposed that to leading-order the waves are inviscid and neutral, and governed by the [Davis-Acrivos-]Benjamin-Ono equation. This has a three-parameter family of periodic solutions, the large-amplitude (soliton) limit of which bears a qualitative resemblance to the 'spikes' observed in certain K-type'transition experiments. The variation of the parameters over slow length- and timescales is controlled by a viscous sublayer. For the case of a purely temporal evolution, it is shown that a solution for this sublayer ceases to exist when the amplitude reaches a certain finite value. For a purely spatial evolution, it appears that an initially linear disturbance does not evolve to a fully nonlinear stage of the envisaged form. The implications of these results for the soliton theory of spike formation are discussed

The paper presents a review of results of some recent (mainly experimental) studies devoted to a quantitative investigation of the problem of receptivity of the 2D and 3D boundary layers with respect to various 3D (in general) external perturbations. The paper concentrates on the mechanisms of excitation and development of stationary and travelling instability modes in a 3D boundary layer on a swept wing, as well as in 2D boundary layers including the Blasius flow and a self-similar boundary layer with an adverse pressure gradient. In particular, the following problems of the boundary-layer receptivity are discussed: (i) receptivity to localized 3D surface roughness, (ii) receptivity to localized 3D surface vibrations, (iii) acoustic receptivity in presence of 3D surface roughness, and (iv) acoustic receptivity in the presence of 3D surface vibrations. All experiments described in the paper were conducted using controlled disturbance conditions with the help of simulation of the stationary and non-stationary perturbations by means of several disturbance generators. This approach gives us the possibility to obtain quantitative results which are independent of any uncontrolled background perturbations of the flow and the experimental model. In contrast to the data obtained at “natural” environmental conditions these results can be directly compared with calculations without any significant assumptions about the physical nature of the disturbances under investigation. The complex (amplitude and phase) coefficients of the boundary-layer receptivity to external perturbations, obtained as functions of the disturbance frequency and the spanwise wavenumber (or the wave propagation angle), represent the main results of the experiments described. These results can be used for the evaluation of the initial amplitudes and phases of the instability modes generated by various external perturbations, as well as for quantitative verification of linear receptivity theories. Several examples of the comparison of experimental results with calculations are also presented in this paper. A brief analysis of the state-of-art in the field is performed and some general properties of different receptivity mechanisms are discussed.

An experimental investigation was undertaken to reveal the characteristic flow structure in the regime of nonlinear boundary layer instability and onset of turbulence. A controlled Tollmien-Schlichting wave was introduced into a two-dimensional boundary layer over a flat plate to study the growth and evolution of the controlled disturbances using both hot film measurements and an improved hydrogen bubble visualization technique. The results demonstrate that the actual breakdown of the laminar boundary layer and the breakdown of the \ensuremath{\Lambda} vortices do not imply the immediate onset of turbulence. Rather, the onset occurs later with the breakdown of the long streaks. Therefore, an alternative transitional scenario was developed.

The nonlinear interaction between Gortler vortices and two-dimensional Tollmien–Schlichting (TS) waves is studied with a spatial, nonparallel model based on the parabolized stability equations. The effect of the TS waves on the development of the vortices is accounted for, showing that TS wave amplitudes of the same order of magnitude as the vortices result in significant nonlinear interaction. The range of governing parameters that has been studied so far is extended and the main effects of Gortler number, spanwise wave number, and initial amplitudes are identified. The study shows that the relative growth rates and initial amplitudes are the two most significant parameters.

Data from a surface hot-film array on the outlet stator of a 1.5-stage axial compressor are analyzed to look for direct evidence of natural transition phenomena. An algorithm is developed to identify instability waves within the Tollmien–Schlichting (T–S) frequency range. The algorithm is combined with a turbulent intermittency detection routine to produce space-time diagrams showing the probability of instability wave occurrence prior to regions of turbulent flow. The paper compares these plots for a range of blade loading, with free-stream conditions corresponding to the maximum and minimum inflow disturbance periodicity produced by inlet guide vane clocking. Extensive regions of amplifying instability waves are identified in nearly all cases. The implications for transition prediction in decelerating flow regions on axial turbomachine blades are discussed.

A Blasius laminar boundary layer and a steady turbulent boundary layer on a flat plate in an incompressible fluid are considered. The spectral characteristics of the Tollmien—Schlichting (TS) and Squire waves are numerically determined in a wide range of Reynolds numbers. Based on the spectral characteristics, relations determining the three–wave resonance of TS waves are studied. It is shown that the three–wave resonance is responsible for the appearance of a continuous low–frequency spectrum in the laminar region of the boundary layer. The spectral characteristics allow one to obtain quantities that enter the equations of dynamics of localized perturbations. By analogy with the laminar boundary layer, the three–wave resonance of TS waves in a turbulent boundary layer is considered.

Laminar-turbulenttransitionmechanismsinducedbyaharmonicpointsourcedisturbanceinae at-plateboundary layerwithadversepressuregradientareinvestigated by fourth-orderaccuratespatialdirectnumericalsimulation based on the complete three-dimensional Navier ‐Stokes equations for incompressible e ow. The disturbance is introduced into thetwo-dimensionalbasee owbytime-periodicsimultaneousblowing andsuction within acircular spot at the wall to quietly mimic the momentum input by an active loudspeaker below a hole in the surface in respective experiments. Thus a wave train consisting of pure Tollmien ‐Schlichting waves of a singlefrequency and a large number of obliqueness angles is stimulated, and its downstream evolution in both physical and spectral space is investigated. A breakdown scenario dominated entirely by oblique modes is observed that shows a spanwise peak/valley amplitude splitting with the valley plane at the centerline of the wave train. Dominant vorticity structures develop off centerline, and a clear-cut -shaped structure is formed in e nal stages related to the wall shear being a footprint of a e rst pair of K -vortices in the e ow.

Localized and non-localized acoustic receptivity for a Blasius boundary layer is investigated using the adjoint Parabolized Stability Equations. The scattering of an acoustic wave onto a hump, a rectangular roughness or a wall steady blowing and suction is described. Comparisons with local approaches, triple deck theory, direct numerical simulations and experiments are successfully shown. Non-parallel effects are discussed. For comparable parameters, the non-localized receptivity problem produces amplitudes one order of magnitude larger than for the case of localized receptivity.

It is known that in adiabatic boundary layer flow over a curved surface the detailed structure of the spanwise periodic Gortler vortex instability varies markedly over the range of spanwise wavelength. At short wavelengths the modes tend to be concentrated in a well-defined thin zone located within the boundary layer. As the vortex wavenumber diminishes so the region of vortex activity is first driven to the bounding wall but subsequently expands to cover the entire boundary layer at which stage the modes take on a principally inviscid form. At yet longer wavelengths the vortices are given by the solution of an interactive multi-deck structure which has some similarities with that for Tollmien-Schlichting waves. In this work we investigate how the application of wall cooling affects the above scenario. It is shown how cooling both restricts the range of mode types and gives rise to two new structures. The first, for moderate cooling and which relates to longer wavelengths, is interactive in nature. Here the viscous-inviscid interaction between an essentially inviscid Gortler problem, albeit for an effective basic flow which in its general form has a non-standard near-wall structure, and a viscous sublayer is provided by novel boundary conditions. Shorter wavelength vortices are largely unaffected by wall cooling unless this is quite severe. However when this degree of cooling is applied, the vortices take on a fully viscous form and are confined to a thin region next to the bounding wall wherein the basic flow assumes an analytic form. Numerical solutions are obtained and we provide evidence as to how the two new structures are related both to each other and to the previously known uncooled results.

An experimental investigation was carried out of the incompressible boundary layer flow along a flat plate, in the presence of an adverse pressure gradient that corresponds to a linear retardation of the free stream velocity. The turbulence level in the free-stream was 0·12% and transition occurred with the laminar boundary layer being close to separation. For three values of the Reynolds number (2·56, 3·11 and 4·09 million based on the reference length that is defined as the reciprocal of the nondimensional-velocity gradient) the laminar, transitional and turbulent regions were studied by single (normal) hot-wire surveys at several streamwise positions. This allows the fluctuations of the streamwise velocity component to be followed from the amplification of laminar instability waves, through breakdown in the intermittency region, and the subsequent development towards a more or less developed turbulence structure. The study reveals that within the transition region fluctuation levels are reached throughout a large part of the boundary layer that are significantly higher than those in fully developed turbulent flow, which is partly a direct consequence of the intermittent character of the flow. For the highest Reynolds number additional cross-wire surveys were carried out in the turbulent region to observe the development of the turbulent stresses following transition. The data are interpreted in terms of structural coefficients, eddy viscosity and mixing length. Also, these results indicate that the transition process can be associated with turbulence levels well in excess of those occurring in fully developed turbulence, and reveal the relaxation of the outer region turbulence structure.

A new mechanism is proposed for the generation of Tollmien–Schlichting (T–S) waves by free-stream turbulence. For definiteness and self-consistency, the mechanism is described mathematically by using a triple-deck formalism. The free-stream turbulence is represented by convecting gusts consisting of the so-called vortical and entropy waves of small amplitude. We show that suitable convecting gusts can interact with sound waves in the free stream to produce a forcing that has the same time and length scales as those of the T–S waves, thereby exciting such waves in the vicinity of the lower branch of the neutral stability curve. The T–S waves so produced have the order of magnitude of ε2R5/16, where ε is the amplitude of the free-stream disturbance and R the global Reynolds number. The scale conversion is achieved without resorting to any non-homogeneity on the wall, and hence the mechanism operates in a flat boundary layer. Furthermore, the T–S waves so generated do not undergo any immediate decay, as they may do in some other receptivity processes. For homogeneous isotropic free-stream turbulence, the spectrum of the T–S waves is obtained. The efficiency of the receptivity mechanism is assessed by parametric studies.

Nonlinear evolution of high-amplitude periodic disturbances in a boundary layer on a flat plate for Mach numberM=2 is studied. An anomalous downstream evolution of the disturbances is found, quasi-two-dimensional disturbances being most unstable. The obtained phase velocities of the waves are 30–40% greater than the phase velocities of the Tollmien-Schlichting waves. The nonlinear evolution of vortex waves is accompanied by an increase in steady disturbances from the source of controlled vibrations. High-frequency disturbances decay, and a periodic wave train degenerates downstream into a quasiharmonic wave train.

Asymptotic and scaling analyses at high Reynolds numbers are used to investigate channel flow over a wavy wall. Flow is divided into a core away from the boundaries and two viscous wall layers adjacent to the boundaries. The spanwise dependent amplitudes of the Tollmien–Schlichting (TS) and boundary waves contribute to a source term in the system for the longitudinal vortices. The presence of this source term requires streamwise variation of either the TS wave or the boundary wave. For the case where the initial TS wave does not exist, the boundary wave, due to the wavy wall, controls the existence, scales, variations and shape of the longitudinal vortices. For the case where the initial TS wave is present, the wavy wall affects the TS wave and the vortex motion through the boundary wave. The TS wave is subjected to secondary three-dimensional instability for favorable boundary waves due to particular forms of the wavy wall, while the secondary instability can be suppressed by special boundary waves due to some other particular wavy walls.

An experimental and theoretical investigation has been carried out to understand the tonal noise generation mechanism on aerofoils at moderate Reynolds number. Experiments were conducted on a NACA0012 aerofoil section in a low-turbulence closed working section wind tunnel. Narrow band acoustic tones were observed up to 40 dB above background noise. The ladder structure of these tones was eliminated by modifying the tunnel to approximate to anechoic conditions. High-resolution flow velocity measurements have been made with a three-component laser-Doppler anemometer (LDA) which have revealed the presence of strongly amplified boundary-layer instabilities in a region of separated shear flow just upstream of the pressure surface trailing edge, which match the frequency of the acoustic tones. Flow visualization experiments have shown these instabilities to roll up to form a regular Kármán-type vortex street.A new mechanism for tonal noise generation has been proposed, based on the growth of Tollmien–Schlichting (T–S) instability waves strongly amplified by inflectional profiles in the separating laminar shear layer on the pressure surface of the aerofoil. The growth of fixed frequency, spatially growing boundary-layer instability waves propagating over the aerofoil pressure surface has been calculated using experimentally obtained boundary-layer characteristics. The effect of boundary-layer separation has been incorporated into the model. Frequency selection and prediction of T–S waves are in remarkably good agreement with experimental data.

This paper is concerned with the effect of free-stream turbulence on the pretransitional flat-plate boundary layer. It is assumed that either the turbulent Reynolds number or the downstream distance (or both) is small enough that the flow can be linearized. The dominant disturbances in the boundary layer, which are of the Klebanoff type, are governed by the linearized unsteady boundary-region equations, i.e. the linearized Navier–Stokes equations with the streamwise derivatives neglected in the viscous and pressure-gradient terms. The turbulence is represented as a superposition of vortical free-stream Fourier modes and the corresponding Fourier component solutions to the boundary-region equations are obtained numerically. The results are then superposed to compute the root mean square of the fluctuating streamwise velocity in the boundary layer produced by the actual free-stream turbulence. It is found that the disturbances computed with isotropic free-stream turbulence do not reach the levels measured in experiments. However, good quantitative agreement is obtained with the relatively low turbulent Reynolds number data of Kendall when the measured strong anisotropy of the low-frequency portion of his spectrum is accounted for. Data at higher turbulent Reynolds numbers are affected by nonlinearity, which manifests itself through the generation of small spanwise length scales. We attempt to model this within the context of the linear theory by choosing a free-stream spectrum whose energy is concentrated at larger transverse wavenumbers and achieve very good agreement with the data. The results suggest that even small deviations from pure isotropy can be an important factor in explaining the large amplitudes of the Klebanoff modes in the pre-transitional boundary layer, and also point to the importance of nonlinear effects. We discuss some additional effects that may need to be accounted for in order to obtain a complete description of the Klebanoff modes.

We consider nonlinear wave motions in thermally stratified Poiseuille flow. Attention is focused on short wavelength wave modes for which the neutral Reynolds number scales as the square of the wave number. The nonlinear evolution of a single monochromatic wave is governed by a first harmonic/mean-flow interaction theory in which the wave-induced mean flow is comparable in size to the wave component of the flow. An integrodifferential equation is derived which governs the normal variation of the wave amplitude. This equation admits finite-amplitude solutions which bifurcate supercritically from the linear neutral point(s).

An investigation of the local receptivity of a Blasius boundary layer to a harmonic vortical disturbance is presented as a step towards understanding boundary-layer receptivity to free-stream turbulence. Although there has been solid experimental verification of the linear theory describing acoustic receptivity of boundary layers, this was the first experimental verification of the mechanism behind local receptivity to a convected disturbance. The harmonic wake from a vibrating ribbon positioned upstream of a flat plate provided the free-stream disturbance. Two-dimensional roughness elements on the surface of the plate acted as a local receptivity site. Hot-wire measurements in the boundary layer downstream of the roughness confirmed the generation of Tollmien–Schlichting (TS) instability waves by an outer-layer interaction between the long-wavelength convected disturbance and the short-scale mean-flow distortion due to the roughness. The characteristics of the instability waves were carefully measured to ensure that their behaviour was correctly modelled by linear stability theory. This theory was then used to determine the immeasurably small initial wave amplitudes resulting from the receptivity process, from wave amplitudes measured downstream. Tests were performed to determine the range of validity of the linear assumptions made in current receptivity theories. Experimental data obtained in the linear regime were then compared to theoretical results of other authors by expressing the experimental data in the form of an efficiency function which is independent of the free-stream amplitude, roughness height and roughness geometry. Reasonable agreement between the experimental and theoretical efficiency functions was obtained over a range of frequencies and Reynolds numbers.

Weakly nonlinear modulation of short Tollmien–Schlichting waves in a boundary–layer flow on a flat surface covered with a liquid film of different fluid is studied theoretically in the limit of large Reynolds numbers. The analysis is based on a triple–deck flow scheme with the interface positioned in the lower viscous layer. Two main flow regimes considered are, firstly, for a general choice of the flow parameters, with the wave modulation governed by a passive, phase–affecting type of nonlinearity, and, secondly, for the case of a linear Tollmien–Schlichting/capillary–wave resonance, the effect of the nonlinear modulation then being a strong stabilisation of disturbances. Secondary instability of inviscid nature is found to arise for a sufficiently strong primary wave, and much sooner than in the case of a homogeneous flow, when the stabilising influence of surface tension (or gravity) is weak. Intermediate parametric stages between the two key modulation regimes arealso studied.

In many fluid flows of practical importance transition is caused by the linear growth of wave instabilities, such as Tollmien–Schlichting waves, which eventually grow to a finite size at which stage secondary instabilities come into play. If transition is to be delayed or even avoided in such flows, then the linear growth of the disturbances must be prevented since control in the nonlinear regime would be a considerably more difficult task. Here a strategy for active control of two-dimensional incompressible and compressible Tollmien–Schlichting waves and its use in controlling the more practically relevant problem of crossflow instability which arises in swept-wing flows is discussed. The control is through an active suction/blowing distribution at the wall though the same result could be achieved by variable wall heating. In order to control the instability it is assumed that the wall shear stress and pressure are known from measurements. It is shown that, certainly at finite Reynolds numbers, it is sufficient to know the flow properties at a finite number of points along the wall. The cases of high and finite Reynolds numbers are discussed using asymptotic and numerical methods respectively. It is shown that a control strategy can be developed to stop the growth of all two-dimensional Tollmien–Schlichting waves at finite and large Reynolds numbers. Some discussion of nonlinear effects in the presence of active control is given and the possible control of other instability mechanisms investigated.

The linear stability of a boundary layer flow with a spanwise-periodic nonuniformity in the velocity profile is investigated. This flow can be considered as a model of a streaky structure occurring in the boundary layer at a high freestream turbulence level. It is shown that for a small nonuniformity amplitude symmetric modes similar to Tollmien-Schlichting waves are the most unstable. At higher nonuniformity amplitudes, antisymmetric modes, qualitatively different from Tollmien-Schlichting waves and having a larger phase velocity, are the most amplified.

Previous studies on boundary layer transition at moderate levels of free stream turbulence (FST) have shown that the transition process can be promoted by the introduction of Tollmien-Schlichting (TS) waves. In the present work the interaction between localized boundary layer disturbances and controlled TS-waves is studied experimentally. The localized disturbances are generated either from a controlled free stream perturbation, or by means of suction or injection through a slot in the flat plate surface. Both methods result in boundary layer disturbances dominated by elongated streamwise streaks of high and low velocity in the streamwise component. A strong interaction is observed preferably for high frequency TS-waves, which are damped when generated separately, and the interaction starts as a local amplification of a wide band of low-frequency oblique waves. The later stages of the transition process can be identified as a non-linear interaction between the oblique structures, leading to regeneration of new and stronger streamwise streaks.

Turbulence measurements were obtained in a three-dimensional supersonic turbulent boundary layer. A 20° curved fin was used to generate a three-dimensional compression of a boundary layer at Mach 3 in the absence of shock waves. Data include hot-wire measurements of five components of the Reynolds stress tensor. The results are interpreted in terms of the mean flow field history of the turbulence. It is demonstrated that in-plane curvature can have a strong stabilizing effect on the turbulence.

Experiments are conducted to study the longitudinal vortices that develop in the boundary layer on the upper surface of an inclined, heated plate. An isothermal plate in water is inclined at angles ranging from 20 to 60 degrees (from the vertical) while the temperature difference is varied from 2 to 23°C. A double-pass Schlieren system is used to visualize the vortices and particle image velocimetry (PIV) is used to measure velocities. In addition, a unique method is developed such that both the Schlieren visualization and PIV can be performed simultaneously. The wavelengths of the vortices and the critical modified Reynolds numbers (R˜) for the onset, merging, and breakup of the vortices are determined from Schlieren images for Pr=5.8. The critical values for R˜ and the critical wavelengths are compared to results of previous experiments and stability analyses. The spatial growth rates of vortices are determined by using the PIV measurements to determine how the circulation in the vortices grows with distance from the leading edge. This is the first time that the growth rate of the vortices have been found using velocity measurements. These spatial growth rates are compared to the results of Iyer & Kelly (1974) and found to be in general agreement. By defining a suitable circulation threshold, the critical R˜ for the onset of the vortices can be found from the growth curves.

The linear stability of incompressible boundary-layer flow of dusty gas on a semiinnite flat plate is considered. The particles are assumed to be under the action of the Stokes drag only. The problem is reduced to the solution of the modied Orr{Sommerfeld equation (Saman 1962). This is solved numerically using two approaches: directly by orthonormalization method, and by perturbation method at small particle mass content. The stability characteristics are calculated for both monoand polydisperse particles. The dust suppresses the instability waves for a wide range of the particle size. The most ecient suppression takes place when the relaxation length of the particle velocity is close to the wavelength of the Tollmien{Schlicting (TS) wave. The reduction in growth rate per unit dust content is approximately ten times greater than the characteristic value of the growth rate for a clean gas. For monosized dust the complex frequency of the TS wave changes in a discontinuous way. As a result a domain in the space of independent parameters arises where two discrete TS modes exist and a domain where no TS mode may exist. For polydisperse dust with a discrete distribution in particle size the number of breaks in the dependence equals the number of particle sizes. For the continuous distribution in particle size the dependence of the complex-frequency on Reynolds number and wavenumber is continuous. The eigenfunction becomes a non-smooth function of the normal coordinate in this case. Some comments are made about the role of the lift force acting on the particles for the problem in question.

Results from computational fluid dynamics experiments of internal wave reflection from sloping boundaries are presented. In these experiments the incident wave lies in a plane normal to the slope. When the angle of wave energy propagation is close to the bottom slope the reflection causes wave breakdown into a quasi-periodic, turbulent boundary layer. Boundary layer energetics and vorticity dynamics are examined and indicate the importance of the three-dimensional turbulence. The boundary layer exhibits intermittent turbulence: approximately every 1.2 wave periods the boundary layer mixes energetically for a duration of about one-third of a wave period, and then it restratifies until the next mixing event. Throughout the wave cycle a strong thermal front is observed to move upslope at the phase speed of the incident waves. Simulations demonstrate that the net effects of turbulent mixing are not confined to the boundary layer, but are communicated to the interior stratified fluid by motions induced by buoyancy effects and by the wave field, resulting in progressive weakening of the background density gradient. Transition to turbulence is determined to occur at Reynolds numbers of approximately 1500, based upon the wavelength and maximum current velocity of the oncoming wave train. The boundary layer thickness depends on the Reynolds number for low Richardson numbers, with a characteristic depth of approximately one-half of the vertical wavelength of the oncoming wave.

Instability modes that satisfy the triad resonance conditions in three-dimensional compressible (subsonic) boundary layer flow over a swept wing are investigated using theoretical and computational methods. Detuning parameters are used for the wave numbers and frequencies for the triad resonance conditions in space and time. Different resonant triads comprised of crossflow and Tollmien–Schlichting modes are investigated for both the incompressible and compressible cases. The effects of compressibility on various flow features, instability modes, and wave interactions are determined.

SUMMARY In this paper we apply a finite volume method, together with a cost-effective segregated solution algorithm, to solve for the primitive velocities and pressure in a set of incompressible Navier‐Stokes equations. The well-categorized workshop problem of lid-driven cavity flow is chosen for this exercise, and results focus on the Reynolds number. Solutions are given for a depth-to-width aspect ratio of 1:1 and a span-to width aspect ratio of 3:1. Upon increasing the Reynolds number, the flows in the cavity of interest were found to comprise a transition from a strongly two-dimensional character to a truly three-dimensional flow and, subsequently, a bifurcation from a stationary flow pattern to a periodically oscillatory state. Finally, viscous (Tollmien‐Schlichting) travelling wave instability further induced

Interactions of oblique waves have recently been investigated theoretically and numerically and found to give rise to rapid transition in flows subcritical to linear wave disturbances. The transition scenario consists of the formation and transient growth of streamwise streaks of high and low velocity and later a rapid growth of high-frequency disturbances leading to breakdown. The present study is the first extensive experimental investigation of oblique transition. The experiments were carried out in a plane Poiseuille flow air channel in which the oblique waves were generated, one at each wall, by vibrating ribbons and the development of the flow was mapped with hot-wire anemometry. The experiments consist both of low- and high-amplitude wave disturbances; in both cases streaky structures are created. For the low-amplitude case these structures decay, whereas for the high amplitude the flow goes towards breakdown. This study has confirmed and extended previous theoretical and numerical results showing that oblique transition may be an important transition scenario.

When the Reynolds number is treated as an asymptotically large number in a boundary-layer stability analysis, it is possible to identify Reynolds number scalings at which different terms in the governing equations balance. These balances determine which physical mechanisms are operating under which circumstances and enable a systematic treatment of the various parameter regimes to be carried out. Linear waves can grow by viscous mechanisms if the velocity profile is noninflexional and both viscous and inviscid instabilities are present for inflexional profiles. When disturbances become nonlinear the resulting dynamics depend upon the type of instability and, in particular, on whether the critical layer lies within the viscous wall layer or is separate from it. Therefore, the classification of a wave as viscous or inviscid is important to the theory of transition. However, at finite Reynolds numbers the boundaries separating different types of instability become blurred. Moreover, certain asymptotic theories are known to give poor quantitative agreement with experiment while others remain untested by detailed experimental comparison. This paper is concerned with identifying the domains where the different asymptotic theories are most relevant so as to facilitate their comparison with experiment. Numerical solutions of the Orr-Sommerfeld equation are presented that indicate that in wind-tunnel experiments the instability driving transition is esssentially viscous even for adverse pressure gradient boundary layers, and that inviscid instability waves would be difficult to observe. For an inviscid wave near the upper branch there could be a lower frequency viscous wave at the same point in the flow with amplitude 50 times larger in a typical practical situation.

In this paper, the effect of surface waviness on boundary-layer transition in two-dimensional subsonic flow is investigated. The mean flow is calculated using an interacting boundary-layer procedure, thereby accounting for viscous–inviscid interaction. Then, the linear parabolized stability equations are used to compute Tollmien–Schlichting (TS) wave amplification. As expected, wall waviness is found to destabilize TS waves. This effect is quantified by examining parameters such as wave height, wave length, wave number, wave location, unit Reynolds number, compressibility and pressure gradient and then comparing the results against those obtained in the absence of waviness. From these results, an empirical equation is extracted which correlate the waviness size with the increment in the N -factor due to waviness. The present correlation is compared with the empirical criteria derived from experiments by Fage (Aeronautical Research Council, R & M 2120, 1943) and Carmichael [Northrop Corp., Report No. NOR-59-438 (BLC-123), 1959]. In agreement with the experiments, the present results show that the effect of waviness scales as h 2 / λ where h is the wave height and λ is the wavelength. Computational results indicated that the critical size of waviness below which waviness has no influence does not exist provided the waviness is located in the unstable TS region. It is also shown that the centrifugal instability introduced by a wavy wall is relatively less significant and the dominant effect of waviness on boundary-layer transition is through its effect on TS wave amplification. It is likely, however, that streamwise vortices generated due to waviness will play a role in the nonlinear breakdown process.

Hydrodynamic instabilities in wavy channels and their effect on the convective heat transfer are investigated using both linear stability analysis and integration of the time dependent Navier–Stokes and energy equations. The linear stability of the fully developed flow is determined from a generalized eigenvalue problem resulting from a Galerkin approach using divergence free Chebychev basis functions and trigonometric polynomials. Several axial periodicity lengths to geometry length ratios have been considered. For our geometry, the instability is found to set in as a Tollmien Schlichting wave, at a Reynolds number approximately equal to 90. The dynamics of the detachment and reattachment points and of temperature field for constant wall temperature are examined under the assumption of small amplitude fluctuations. It is shown that, although the average heat transfer remains almost constant, large amplitude variations of the local heat transfer coefficient can be observed, this effect is increasing with increasing Prandtl number.

This paper presents a brief review of some recent work on the the use of wall compliance for laminar-flow control. Four main topics are covered. Firstly comparisons between theory and experiment for instability and transition in flat-plate boundary layers over compliant walls are considered. Good agreement is found for the Gaster experiments in water. But the theory does not corroborate the more recent experimental study of Lee et al. (T. Lee, M. Fisher, W.H. Schwarz, Journal of Fluid Mechanics 288 (1995) 37) in air. Secondly, the results of recent numerical simulations of the divergence instability are discussed. In agreement with experimental evidence it is found that the divergence onset speed for laminar flow is much higher than for turbulent flow. This implies that previous estimates of the onset speed based on potential flow are very conservative. Thirdly, the use of multiple-panel compliant walls for laminar-flow control is reviewed. Theory and numerical simulation strongly suggest that transition can be postponed to indefinitely high Reynolds numbers by the use of such walls. Lastly, recent work is reviewed on the effects of wall compliance on other instability mechanisms, such as inflexion-point instabilities and cross-flow vortices and absolute instabilities for the three-dimensional, rotating-disc boundary layer.

The method of parabolic stability equations is used to study the laminar-turbulent transition in a boundary layer with a stationary velocity inhomogeneity concentrated in a narrow stream. The location of the transition is found as a function of the magnitude and sign of the velocity defect. It is shown that if the inhomogeneity amplitude is small, it affects only the final nonlinear stage of disturbance development. In this case, the location of the transition is independent of the sign of the velocity defect. Having a moderate amplitude, a lower-velocity inhomogeneity shifts the transition location significantly more strongly than a higher-velocity inhomogeneity of similar shape and amplitude. This is caused by amplification of unstable disturbances in the low-velocity region and, conversely, their attenuation in the high-velocity stream. The effect of disturbance amplification in the low-velocity region is shown not to be connected with inflection-type instability. Another explanation of this phenomenon is offered.

In 2001, flight tests were performed in order to learn more about boundary layer transition induced by free-stream excited Tollmien-Schlichting waves. A sixty four element hot-film array, placed on the right hand wing of DLR’s flying test-bed LFU-205, was used to measure the wall shear stress fluctuations provoked by these instabilities while propagating in the laminar boundary layer. In the present work, a procedure was developed that allows the reconstruction of the disturbed flow field above the hot-film sensors from the wall measurements by employing linear stability analysis. The reconstruction delivers the fluctuations of all flow quantities, i.e. disturbance velocity components, pressure, temperature and density. A comparison of the reconstructed longitudinal disturbance velocity component \(\mathrm{{u^\prime }}\) with hot-wire measurements shows good qualitative agreement.

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–Schlichting 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).

The stability of plane channel flow between compliant walls is investigated for disturbances which have the same symmetry, with respect to the channel centreline, as the Tollmien–Schlichting mode of instability. The interconnected behaviour of flow-induced surface waves and Tollmien–Schlichting waves is examined both by direct numerical solution of the Orr–Sommerfeld equation and by means of an analytic shear layer theory. We show that when the compliant wall properties are selected so as to give a significant stability effect on Tollmien–Schlichting waves, the onset of divergence instability can be severely disrupted. In addition, travelling wave flutter can interact with the Tollmien–Schlichting mode to generate a powerful instability which replaces the flutter instability identified in studies based on a potential mean-flow model. The behaviour found when the mean-flow shear layer is fully accounted for may be traced to singularities in the wave dispersion relation. These singularities can be attributed to solutions which represent Tollmien–Schlichting waves in rigid-walled channels. Such singularities will also be found in the dispersion relation for the case of Blasius flow. Thus, similar behaviour can be anticipated for Blasius flow, including the disruption of the onset of divergence instability. As a consequence, it seems likely that previous investigations for Blasius flow will have yielded very conservative estimates for the optimal stabilization that can be achieved for Tollmien–Schlichting waves for the purposes of laminar-flow control.

Resume Une etude de l'apparition des ondes de Tollmien-Schlichting a ete menee a l'aide du code de simulation numerique directe Pegase, developpe a l'ONERA. Leur existence a pu etre mise en evidence dans l'ecoulement bidimensionnel incompressible se developpant le long d'une plaque plane, munie d'une microrugosite rectangulaire de dimensions h et l , en presence d'un ecoulement exterieur legerement fluctuant U ∞ = U 0 (1ɛ cos ωt) . Une etude parametrique a permis de quantifier l'influence de ɛ , h et l sur l'amplitude de ces ondes. Ces resultats sont en tres bon accord avec des etudes experimentales et theoriques anterieures.

Weak free-stream turbulence excites modulated Tollmien–Schlichting (T–S) waves in a laminar boundary layer that grow in magnitude with downstream distance and ultimately lead to the formation of turbulent spots and then fully turbulent flow. Hot-wire experiments have indicated that the development of localized large-amplitude ‘events’ in the velocity records are the essential precursor to the eventual formation of turbulent spots in the flow field. Traditional global Fourier techniques are unable to resolve the localized nature of these events and hence provide little useful information concerning the physical processes responsible for this breakdown process.This investigation used sequences of computer-generated deterministic white noise to excite a laminar boundary layer via a loudspeaker embedded in a flat-plate model. This form of excitation generated the modulated disturbance waves of interest a short distance downstream from the source in a repeatable and deterministic manner. Further downstream the pattern of flow breakdown and subsequent generation of turbulent spots was similar to that observed in naturally excited situations. By repeatedly exciting the boundary layer with a single white-noise sequence it was possible to examine the highly nonlinear stages of ‘event’ development and breakdown with a single hot-wire probe.Two local analysis techniques, the wavelet transform (WT) and singular spectrum analysis (SSA), were used in conjunction with the white-noise excitation technique to examine the highly nonlinear flow mechanisms responsible for the localized formation of events that lead to the eventual breakdown to turbulence.

The behaviour of wall-bounded shear layers that can be represented by a Blasius profile is studied with reference to very low frequency excitation. The whole shear layer executes a heaving motion, which is referred to as the Klebanoff mode. This has been observed experimentally for a long time, but no theoretical explanation was available so far. Here some theoretical results are presented which indicate that such low frequency excitation produces three-dimensional disturbance waves whose wave lengths are very large compared to the shear-layer thickness. This, and other properties of such a wave system, explains the observed Klebanoff mode.

This paper is concerned with the nonlinear interaction and development of a pair of oblique Tollmien–Schlichting waves which travel with equal but opposite angles to the free stream in a boundary layer. Our approach is based on high-Reynolds-number asymptotic methods. The so-called ‘upper-branch’ scaling is adopted so that there exists a well-defined critical layer, i.e. a thin region surrounding the level at which the basic flow velocity equals the phase velocity of the waves. We show that following the initial linear growth, the disturbance evolves through several distinct nonlinear stages. In the first of these, nonlinearity only affects the phase angle of the amplitude of the disturbance, causing rapid wavelength shortening, while the modulus of the amplitude still grows exponentially as in the linear regime. The second stage starts when the wavelength shortening produces a back reaction on the development of the modulus. The phase angle and the modulus then evolve on different spatial scales, and are governed by two coupled nonlinear equations. The solution to these equations develops a singularity at a finite distance downstream. As a result, the disturbance enters the third stage in which it evolves over a faster spatial scale, and the critical layer becomes both non-equilibrium and viscous in nature, in contrast to the two previous stages, where the critical layer is in equilibrium and purely viscosity dominated. In this stage, the development is governed by an amplitude equation with the same nonlinear term as that derived by Wu, Lee & Cowley (1993) for the interaction between a pair of Rayleigh waves. The solution develops a new singularity, leading to the fourth stage where the flow is governed by the fully nonlinear three-dimensional inviscid triple-deck equations. It is suggested that the stages of evolution revealed here may characterize the so-called ‘oblique breakdown’ in a boundary layer. A discussion of the extension of the analysis to include the resonant-triad interaction is given.

The receptivity to fundamental and subharmonic secondary instabilities is analysed for two-dimensional boundary layers. Fundamental modes are excited by the direct scattering of Tollmien–Schlichting (TS) waves over surface variations. The excitation of subharmonic modes stems from the combined scattering of acoustic free-stream disturbances and TS waves over surface variations. The surface variations are localized in their streamwise extent and are the result of roughness or suction. The velocity field is expanded in terms of small parameters characterizing the acoustic disturbance and the surface variation. The TS wave is included as part of the base flow leading to a non-homogeneous system with periodic coefficients governing the receptivity. The receptivity amplitudes show a strong dependence on the TS-wave amplitude, and for subharmonic modes a strong dependence on the TS-wave phase at the location of the surface variation. The receptivity analysis shows a significant bias toward fundamental modes of secondary instability for larger TS-wave amplitudes – except for conditions of extremely high free-stream sound level. A combination of receptivity results and stability results suggests a bias toward subharmonic modes for TS-wave amplitudes below 0.5% and toward fundamental modes for TS-wave amplitudes above 0.5% (normalized by the local edge velocity).

Turbulence measurements of the stable atmospheric boundary layer made at Halley station in Antarctica are presented. the interaction between small-scale turbulence and larger solitary wave disturbances is investigated. Spectral methods are used to separate the wave and turbulence parts of the flow, and the appropriateness of this approach is discussed. Both individual wave events and an average of 18 wave events are studied. the transport of wave-induced velocity by wave-induced turbulence is found to be mostly upwards, whilst a strong downward turbulent heat flux accompanies each wave. Waves are often found to be associated with a burst of low-level turbulence which occurs later than the perturbation in the mean shear. the effect of the turbulence on the wave is shown to be small, whilst the instantaneous local effect of the wave on the turbulence appears to be approximately linear in the wave amplitude. Comparisons are made between the observations and numerical predictions. These tend to confirm the need for at least second-order closure schemes for the turbulence modelling.

The interaction between a large-scale isolated wave and turbulence is considered in the context of the stable atmospheric boundary layer. A two-dimensional model with a second-moment turbulence closure is used to simulate the interaction numerically. Results obtained using other closures are discussed. the initial form of the wave-like disturbance is suggested by observations, which are described in a companion paper. the turbulence is shown to exert only a small influence on the wave, while the turbulence itself remains close to production-dissipation equilibrium. Wave-like components of the turbulence fluxes vary roughly linearly with wave amplitude, but overall the wave is strongly nonlinear and depends heavily on the background flow. Only one term in the equations for the quadratic turbulence quantities varies in a simple way with wave amplitude; this term represents a net transfer of energy from the wave to the turbulence.

The evolution of two-dimensional Tollmien–Schlichting waves propagating along a wall shear layer as it passes over a compliant panel of finite length is investigated by means of numerical simulation. It is shown that the interaction of such waves with the edges of the panel can lead to complex patterns of behaviour. The behaviour of the Tollmien–Schlichting waves in this situation, particularly the effect on their growth rate, is pertinent to the practical application of compliant walls for the delay of laminar–turbulent transition. If compliant panels could be made sufficiently short whilst retaining the capability to stabilize Tollmien–Schlichting waves, there is a good prospect that multiple-panel compliant walls could be used to maintain laminar flow at indefinitely high Reynolds numbers.We consider a model problem whereby a section of a plane channel is replaced with a compliant panel. A growing Tollmien–Schlichting wave is then introduced into the plane, rigid-walled, channel flow upstream of the compliant panel. The results obtained are very encouraging from the viewpoint of laminar-flow control. They indicate that compliant panels as short as a single Tollmien–Schlichting wavelength can have a strong stabilizing effect. In some cases the passage of the Tollmien–Schlichting wave over the panel edges leads to the excitation of stable flow-induced surface waves. The presence of these additional waves does not appear to be associated with any adverse effect on the stability of the Tollmien–Schlichting waves. Except very near the panel edges the panel response and flow perturbation can be represented by a superposition of the Tollmien–Schlichting wave and two other eigenmodes of the coupled Orr–Sommerfeld/compliant-wall eigensystem.The numerical scheme employed for the simulations is derived from a novel vorticity–velocity formulation of the linearized Navier–Stokes equations and uses a mixed finite-difference/spectral spatial discretization. This approach facilitated the development of a highly efficient solution procedure. Problems with numerical stability were overcome by combining the inertias of the compliant wall and fluid when imposing the boundary conditions. This allowed the interactively coupled fluid and wall motions to be computed without any prior restriction on the form taken by the disturbances.

The concentration field of mixing layers subject to stabilizing and destabilizing streamwise curvature was investigated at post-mixing-transition conditions. A set of operating conditions was implemented, identical to those at which straight layers were previously investigated in the same facility, in order to compare the effects of hydrodynamic instabilities upon scalar mixing. Quantitative imaging of planar laser-induced fluorescence was used for (i) passive scalar measurements, and (ii) chemical product measurements. Similar to the straight mixing layer, the results for the curved layers show that beyond the mixing transition the layer continues to evolve, and undergoes a small change in its scalar structure. At conditions just past the mixing transition both stable and unstable layers have average mixed-fluid compositions which are uniform across the layer, and average chemical product concentration profiles which are symmetric. At more fully developed conditions, the scalar field evolved: the average mixed-fluid concentration developed a small lateral variation, while the chemical product concentration profiles became asymmetric. Similar to the straight layer, the mixture-fraction PDF is believed to be of the tilted type for the most fully developed layer examined, with the marching PDF being a poor representation. Consistent with previous investigations, the growth rate of the unstable layer was found to be higher than that of straight or stable layers. The most important result is that the measured mixing efficiency of all the layers (curved and straight) was found to be the same: both the total mixed-fluid composition, and the volume fraction of mixed fluid were the same for all unstable, stable, and straight layers. The amount of mixed fluid (and of chemical product formed) was larger for the unstable layer, but always in a fixed proportion to the layer's thickness. The lack of increase in the mixing efficiency for the unstable layer is surprising, given that previous hydrodynamic measurements had shown enhanced turbulent transport for the unstable case. Thus, for all liquid shear layers studied, the rate of scalar mixing appears to be directly proportional to the entrainment rate (which essentially determines the layer's growth rate), and not to any hydrodynamic measures.