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A parametric study is presented for the incompressible, zero-pressure-gradient flat-plate boundary layer perturbed by streamwise vortices. The vortices are placed near the leading edge and model the vortices induced by miniature vortex generators (MVGs), which consist in a spanwise-periodic array of small winglet pairs. The introduction of MVGs has been experimentally proved to be a successful passive flow control strategy for delaying laminar-turbulent transition caused by Tollmien–Schlichting (TS) waves. The counter-rotating vortex pairs induce non-modal, transient growth that leads to a streaky boundary layer flow. The initial intensity of the vortices and their wall-normal distances to the plate wall are varied with the aim of finding the most effective location for streak generation and the effect on the instability characteristics of the perturbed flow. The study includes the solution of the three-dimensional, stationary, streaky boundary layer flows by using the boundary region equations, and the three-dimensional instability analysis of the resulting basic flows by using the plane-marching parabolized stability equations. Depending on the initial circulation and positioning of the vortices, planar TS waves are stabilized by the presence of the streaks, resulting in a reduction in the region of instability and shrink of the neutral stability curve. For a fixed maximum streak amplitude below the threshold for secondary instability (SI), the most effective wall-normal distance for the formation of the streaks is found to also offer the most stabilization of TS waves. By setting a maximum streak amplitude above the threshold for SI, sinuous shear layer modes become unstable, as well as another instability mode that is amplified in a narrow region near the vortex inlet position.

Summary In this work, a study of the mechanism by which free-stream acoustic and vorticity disturbances interact with a boundary layer flow developing over a flat plate featuring a step excrescence located at a certain distance from a blunt leading edge is included. The numerical tool is a high-fidelity implicit numerical algorithm solving for the unsteady, compressible form of the Navier–Stokes equations in a body-fitted curvilinear coordinates and employing high-accurate compact differencing schemes with Pade-type filters. Acoustic and vorticity waves are generated using a source term in the momentum and energy equations, as opposed to using inflow boundary conditions, to avoid spurious waves that may propagate from boundaries. The results show that the receptivity to surface step excrescences is largely the result of an overall adverse pressure gradient posed by the step, and that the free-stream disturbances accelerate the generation of instabilities in the downstream. As expected, it is found that the acoustic disturbance interacting with the surface imperfection is more efficient in exciting the Tollmien–Schlichting waves than the vorticity disturbance. The latter generates Tollmien–Schlichting waves that are grouped in wave packets consistent with the wavelength of the free-stream disturbance. Copyright © 2015 John Wiley & Sons, Ltd.

The problem of freestream-turbulence-generated instability waves in the flat-plate boundary layer is solved on the basis of a nonlinear turbulence model admitting the deviation of the speed of propagation of vortex disturbances from the flow velocity. The solution obtained well describes the experimental dependence of the laminar-turbulent transition Reynolds number on the freestream turbulence degree.

An experimental investigation has been performed to study the free-stream turbulence influence on the flow over a transversely-grooved surface. Free-stream flow at 8.1 m/s with turbulence intensities of 0.5% and 4.4% have been provided over smooth and grooved surfaces. Mean velocity profiles, fluctuating velocity moments and turbulence length scales are studied using data obtained from hot-wire anemometry measurements. Introducing quasi-isotropic free-stream turbulence of 4.4% to the flow resulted in fuller velocity profiles, increased boundary layer thickness and significantly augmented streamwise turbulence intensity throughout the boundary layer in flows over both smooth and grooved surfaces. Probability density functions of the velocity fluctuation show that interaction of quasi-isotropic free-stream turbulence with wall turbulence leads to nearly-isotropic turbulence in a small portion of the boundary layer in the near-wall region; while the remaining boundary layer experiences an anisotropic turbulence. In other words, with the introduction of additional turbulence generated by the grooves, the momentum of the fluid particles is increased, resulting in better flow separation resistance. Accordingly, the flow over a grooved surface showed an increase in turbulence intensity, and hence in energy dissipation rate. This is amplified under free-stream turbulence conditions as a result of the interaction of groove-generated turbulence with free-stream turbulence in the boundary layer. The energy cascade span as indicated by the integral to Kolmogorov ratio is also widened by the grooves.

The present work predicts the formation of laminar separation bubbles at low Reynolds numbers and the related transition to turbulence by means of Implicit Large Eddy Simulations with a high-order Discontinuous Galerkin method. The flow around an SD7003 infinite wing at an angle of attack of 4° is considered at Reynolds numbers of 10 000, 22 000, and 60 000 in order to gain insight into the characteristics of the laminar and turbulent regimes. At the lowest Reynolds number studied, the flow remains laminar and two dimensional over the wing surface, with a periodic vortex shedding. For higher Reynolds numbers, the flow is unsteady over the upper wing surface and exhibits a separation bubble along which the flow transitions to turbulence. Tollmien–Schlichting (TS) waves are observed in the boundary layer, and transition is found to be caused by unstable TS modes as revealed by the growth of the stream-wise amplification factor. Copyright © 2010 John Wiley & Sons, Ltd.

The stability of pressure-driven flow in a rectangular channel with deformable neo-Hookean viscoelastic solid walls is analysed for a wide range of Reynolds numbers (from Re ≪ 1 to Re ≫ 1) by considering both sinuous and varicose modes for the perturbations. Pseudospectral numerical and asymptotic methods are employed to uncover the various unstable modes, and their stability boundaries are determined in terms of the solid elasticity parameter Γ = Vη/(ER) and the Reynolds number Re = RV ρ/η; here V is the maximum velocity of the laminar flow, R is the channel half-width, η and ρ are respectively the viscosity and density of the fluid and E is the shear modulus of the solid layer. We show that for small departures from a rigid solid, wall deformability could have a destabilizing or stabilizing effect on the Tollmien–Schlichting (TS) instability (a sinuous mode) depending on the solid-layer thickness. Upon further increase in solid deformability, the TS mode coalesces with another unstable mode (absent in rigid channels) giving rise to a single unstable mode which extends to very low Reynolds number (Re = 0), there is a short-wave instability of both sinuous and varicose modes arising due to the discontinuity of the first normal stress difference across the fluid–solid interface. For both sinuous and varicose modes, it is shown that inclusion of inertia is important even for Re ≪ 1, wherein a new class of long-wavelength unstable modes are predicted which are absent at Re = 0. These unstable modes are a type of shear waves in an elastic solid which are destabilized by the flow. These long- and short-wave instabilities are absent if a simple linear elastic model is used for the solid. At intermediate and high Re, upstream and downstream travelling waves of both sinuous and varicose modes become unstable. We show that sinuous and varicose modes become critical in different parameter regimes, thereby demonstrating the importance of capturing all the unstable modes. Inclusion of dissipative effects in the neo-Hookean model is generally shown to play a stabilizing role on the instabilities due to both sinuous and varicose modes. The predicted instabilities will be important for the flow of liquids (with viscosity ≥ 10−3 Pa s) in deformable channels of width ≤1 mm, and with shear modulus ≤ 105 Pa.

This is a review of the literature on turbulence transition in free stream boundary layer flows. The transition to turbulence is induced by the classical method of introducing an external perturbation into the flow in the form of a vibrating ribbon. Three defining aspects of the subsequent transition phenomenon are considered: its physics, mathematics, and numerical simulation. The physics describes the physical nature of turbulence with depiction of the generation and interaction of disturbance waves and the formation of turbulent flow structures. The mathematics converts the physical problem into its governing equations and boundary conditions. The numerical simulation then solves the governing equations to realize the flow behavior. An example simulation of two dimensional boundary layer flow in the linear instability stage demonstrates wave amplification and damping dynamics.

The nonlinear stability of a laminar boundary layer that flows at high Reynolds number (Re) above a plane surface covered by a liquid film is investigated. The basic flow is considered to be nearly parallel and the simulations are based on triple deck theory. The overall interaction problem is solved using the finite element methodology with the two-dimensional B-cubic splines as basis functions for the unknowns in the boundary layer and the film and the one-dimensional B-cubic splines as basis functions for the location of the interface. The case of flow above an oscillating solid obstacle is studied and conditions for the onset of Tollmien–Schlichting (TS) waves are recovered in agreement with the literature. The convective and absolute nature of TS and interfacial waves is captured for gas-film interaction, and the results of linear theory are recovered. The evolution of nonlinear disturbances is also examined and the appearance of solitons, spikes and eddy formation is monitored on the interface, depending on the relative magnitude of Froude and Weber numbers (Fr,We), and the gas to film density and viscosity ratios (ρ/ρw, μ/μw). For viscous films TS waves grow on a much faster time scale than interfacial waves and their effect is essentially decoupled. The influence of interfacial disturbances on short-wave growth in the bulk of the boundary layer bypassing classical TS wave development is captured. For highly viscous films for which inertia effects can be neglected, e.g. aircraft anti-icing fluids, soliton formation is obtained with their height remaining bounded below a certain height. When water films are considered interfacial waves exhibit unlimited local growth that is associated with intense eddy formation and the appearance of finite time singularities in the pressure gradient.

The interaction of discrete and continuous Orr–Sommerfeld modes in a boundary layer is studied by computer simulation. The discrete mode is an unstable Tollmien–Schlichting wave. The continuous modes generate jet-like disturbances inside the boundary layer. Either mode alone does not cause transition to turbulence; however, the interaction between them does. The continuous mode jets distort the discrete modes, producing Λ shaped vortices. Breakdown to turbulence is subsequent. The lateral spacing of the Λ ’s is sometimes the same as the wavelength of the continuous mode, sometimes it differs, depending on the ratio of wavelength to boundary layer thickness.

A series of swept laminar separation bubbles is used to investigate the principle applicability of the e N method to laminar separation bubbles in swept configurations. To this end the effect of sweep and of the propagation direction of disturbance waves on the accuracy of linear stability theory and solutions of the parabolized stability equations is studied systematically. Direct comparisons of spatial linear stability theory and linear parabolized stability equation solutions to results of direct numerical simulations allow for a qualitative and quantitative evaluation of their performance in the presence of sweep, flow separation, and local backflow. A variety of Tollmien-Schlichting waves as well as the most amplified stationary crossflow vortex is analyzed within a sweep angle range between 0 and 45 deg. It turns out that even though linear stability theory works satisfactorily, parabolized stability equations are clearly preferable in terms of accuracy, especially for very oblique modes or larger sweep angles.

A theory is described for propagation of vortical waves across alternate rigid and compliant panels. The structure in the fluid side at the junction of panels is a highly vortical narrow viscous structure which is idealized as a wave driver. The wave driver is modelled as a ‘half source cum half sink’. The incoming wave terminates into this structure and the outgoing wave emanates from it. The model is described by half Fourier–Laplace transforms respectively for the upstream and downstream sides of the junction. The cases below cutoff and above cutoff frequencies are studied. The theory completely reproduces the direct numerical simulation results of Davies & Carpenter (J. Fluid Mech., vol. 335, 1997, p. 361). Particularly, the jumps across the junction in the kinetic energy integral, the vorticity integral and other related quantities as obtained in the work of Davies & Carpenter are completely reproduced. Also, some important new concepts emerge, notable amongst which is the concept of the pseudo group velocity.

The interaction between free-stream disturbances and the boundary layer on a body with a rounded leading edge is considered in this paper. A method which incorporates calculations using the parabolized stability equation in the Orr–Sommerfeld region, along with an upstream boundary condition derived from asymptotic theory in the vicinity of the leading edge, is generalized to bodies with an inviscid slip velocity which tends to a constant far downstream. We present results for the position of the lower branch neutral stability point and the magnitude of the unstable Tollmien–Schlichting (T-S) mode at this point for both a parabolic body and the Rankine body. For the Rankine body, which has an adverse pressure gradient along its surface far from the nose, we find a double maximum in the T-S wave amplitude for sufficiently large Reynolds numbers.

Boundary layers exposed to an adverse pressure gradient are known to be very unstable with respect to traveling wave instability. This instability can be eliminated through the application of wall suction. It is shown that when suction distribution is not uniform, another type of instability may occur, giving rise to streamwise vortices. It is further shown that an increase of uniform suction is not an effective method for control of this instability. Finally, it is shown that nonuniformities of surface blowing can induce similar vortex-type instability.

Within the framework of the triple-deck theory, it is shown that the growth of the relative wall velocity in the Poiseuille-Couette flow leads to perturbations splitting into two wave packets, of which one grows faster and propagates at a higher velocity.

This paper is concerned with the nonlinear interaction between a planar and a pair of oblique Tollmien–Schlichting (T-S) waves which are phase-locked in that they travel with (nearly) the same phase speed. The evolution of such a disturbance is described using a high-Reynolds-number asymptotic approach in the so-called ‘upper--branch’ scaling regime. It follows that there exists a well-defined common critical layer (i.e. a thin region surrounding the level at which the basic flow velocity equals the phase speed of the waves to leading order) and the dominant interactions take place there. The disturbance is shown to evolve through several distinctive stages. In the first of these, the critical layer is in equilibrium and viscosity dominated. If a small mismatching exists in the phase speeds, the interaction between the planar and oblique waves leads directly to super-exponential growth/decay of the oblique modes. However, if the modes are perfectly phase-locked, the interaction in the first instance affects only the phase of the amplitude function of the oblique modes (so causing rapid wavelength shortening), while the modulus of the amplitude still evolves exponentially until the wavelength shortening produces a back reaction on the modulus (which then induces a super-exponential growth). Whether or not there is a small mismatch or a perfect match in the phase speeds, once the growth rate of the oblique modes becomes sufficiently large, the disturbance enters a second stage, in which the critical layer becomes both non-equilibrium and viscous in nature. The oblique modes continue to experience super-exponential growth, albeit of a different form from that in the previous stages, until the self-interaction between them, as well as their back effect on the planar mode, becomes important. At that point, the disturbance enters a third, fully interactive stage, during which the development of the disturbance is governed by the amplitude equations with the same nonlinear terms as previously derived for the phase-locked interaction of Rayleigh instability waves. The solution develops a singularity, leading to the final stage where the flow is governed by fully nonlinear three-dimensional inviscid triple-deck equations. The present work indicates that seeding a planar T-S wave can enhance the amplification of all oblique modes which share approximately its phase speed.

Perturbations in a boundary layer when there is unsteady free viscous-inviscid interaction at transonic velocities are investigated using a modified “triple-deck” model. Two problems are considered: the effect of acoustic perturbations arriving from outside and of perturbations excited from a streamline surface with a vibrator situated on it. The modification of the model consists of taking into account singular terms of the transonic expansion, which enables the Lin–Reissner–Tsien equation to be improved and enables non-stationary and non-linear phenomena in the flow to be described more correctly. It is shown that the modified model enables additional perturbations, overlooked when using the classical triple-deck model, to be taken into account.

An iterative implicit fractional step method is developed and employed for the simulation of transitional boundary layer over compliant surfaces. The three-dimensional perturbation Navier–Stokes equations are discretized by curvilinear finite volumes on a collocated grid system. A multigrid procedure is used for computations associated with the pressure-Poisson equation, while simulation is carried out by MPI-based parallel computation with domain decomposition. Results in the literature for oblique linear waves and the non-linear breakdown of a wave triad over a rigid wall are repeated to check the accuracy of the codes that had been developed. Oblique linear TS (Tollmien–Schlichting) waves over finite-length compliant membranes are generally found to coexist with CIFI (compliance induced flow instability) or FISI (flow induced surface instabilities) waves. The latter waves usually possess longer wavelengths and thus propagate at a larger oblique wave angles than the TS waves. Simulation reveals that compliant surfaces may slow down the development of secondary instabilities during the early stages of laminar–turbulent transition. However, during the later stages of fundamental wave breakdown, interactions with CIFI and edge-generated waves may increase the amplitudes of the original 2D and 3D TS waves, leading to an earlier breakdown on compliant surfaces. Linear interaction between the flow and compliant membranes has been assumed.

A linear and weakly nonlinear theory of stability of a laminar viscous fluid wall jet is considered. At large Reynolds numbers (calculated from the characteristic jet length) the undisturbed stationary flow is concentrated in a narrow region adjacent to the rigid surface. Nonstationary perturbations of this flow can be described by equations similar to the equations used in the theory of free interaction of a boundary layer. Such a description is made possible by introducing asymptotic order relations between the Reynolds number and the perturbation amplitudes and wavelengths. The set of solutions of the dispersion relation for the linear stability problem includes the eigenvalues of the phase velocities and wave numbers corresponding to the neutral and unstable perturbation modes. For finite fluctuation amplitudes, the evolution of the wave fields obeys the Korteweg-de Vries equations.

We study the linearized Navier–Stokes (LNS) equations in channel flows from an input–output point of view by analysing their spatio-temporal frequency responses. Spatially distributed and temporally varying body force fields are considered as inputs, and components of the resulting velocity fields are considered as outputs into these equations. We show how the roles of Tollmien–Schlichting (TS) waves, oblique waves, and streamwise vortices and streaks in subcritical transition can be explained as input–output resonances of the spatio-temporal frequency responses. On the one hand, we demonstrate the effectiveness of input field components, and on the other, the energy content of velocity perturbation components. We establish that wall-normal and spanwise forces have much stronger influence on the velocity field than streamwise force, and that the impact of these forces is most powerful on the streamwise velocity component. We show this using the relative scaling of the different input–output system components with the Reynolds number. We further demonstrate that for the streamwise constant perturbations, the spanwise force localized near the lower wall has, by far, the strongest effect on the evolution of the velocity field. In this paper, we analyse the dynamical properties of the Navier–Stokes (NS) equations with spatially distributed and temporally varying body force fields. These fields are considered as inputs, and different combinations of the resulting velocity fields are considered as outputs. This input–output analysis can in principle be done in any geometry and for the full nonlinear NS equations. In such generality, however, it is difficult to obtain useful results. We therefore concentrate on the geometry of channel flows, and the input–output dynamics of the linearized Navier–Stokes (LNS)

An experimental study was performed to analyze the stability of localized streaky structure in a Blasius boundary layer. An artificial streaky structure was created by using suction or blowing through a thin spanwise slot at the wall. The velocity gradient generated by the suction or blowing was controlled by a damper. The Reynolds number based on the displacement thickness δ1 was % MathType!Translator!2!1!AMS LaTeX.tdl!TeX -- AMS-LaTeX! % MathType!MTEF!2!1!+- % feaafeart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaciOuaiaacw % gadaWgaaWcbaGaeqiTdq2aaSbaaWqaaiaaigdaaeqaaaWcbeaaaaa!3A72! $$\operatorname{Re} _{\delta _1 }$$ =280 at the slot. The behavior of the artificial streaky structure was scrutinized by damping the velocity gradient. It was found that the local streamwise and spanwise velocity gradients play a significant role in the formation of different types of instability. Artificial Tollmien–Schlichting (T–S) wave packets were created by a loudspeaker to elucidate the interaction of the streaky structure with the T–S wave packets. The T–S wave packets imposed on the streaky structure become unstable when the frequency of the T–S wave packets exceeds a certain critical frequency. The development of the T–S wave packets was investigated on the basis of the neutral stability curve.

The evolution of small two-dimensional perturbations of boundary flow in the region preceding the laminar-turbulent transition zone is investigated using a four-deck asymptotic model. It is assumed that the main flow is a two-dimensional boundary layer flow distorted by a wave of finite amplitude that is periodic in time and with a longitudinal coordinate. The problem is reduced to investigating the solutions of a linearized Benjamin-Ono equation, which describes secondary perturbations of a periodic Benjamin wave. The explicit expressions obtained for the structure of the perturbations confirm the neutral stability of the main flow with respect to two-dimensional perturbations. An exact solution of the problem of the passage of a Tollmien-Schlichting wave through a region perturbed by a solitary Benjamin wave is constructed by analysing the stability of the periodic.

Hot-wire measurements in the boundary layer developing on a concave surface of 2.0 m radius of curvature in the presence of forced wavelength Gortler vortices have been conducted for a free-stream velocity of 3.0 m/s. The wavelengths of vortices were preset by vertical perturbation wires of 0.2 mm diameter located 10 mm upstream of the concave surface leading edge. The velocity contours in the cross-sectional planes at several streamwise locations show the growth and breakdown of the vortices that are similar to those found in the transitional flow field. It shows the occurrence of the second instability mode that is indicated by the formation of small horseshoe eddies generated between the two neighboring vortices traveling in the streamwise direction to form mushroom-like structures as a consequence of the nonlinear growth of the Gortler vortices. The breakdown of these structures before the boundary-layer flow becomes turbulent is also shown to qualitatively predict the start of the transition in the f...

The effect of high levels of free-stream turbulence on the transition in a Blasius boundary layer is studied by means of direct numerical simulations, where a synthetic turbulent inflow is obtained as superposition of modes of the continuous spectrum of the Orr–Sommerfeld and Squire operators. In the present bypass scenario the flow in the boundary layer develops streamwise elongated regions of high and low streamwise velocity and it is suggested that the breakdown into turbulent spots is related to local instabilities of the strong shear layers associated with these streaks. Flow structures typical of the spot precursors are presented and these show important similarities with the flow structures observed in previous studies on the secondary instability and breakdown of steady symmetric streaks.Numerical experiments are performed by varying the energy spectrum of the incoming perturbation. It is shown that the transition location moves to lower Reynolds numbers by increasing the integral length scale of the free-stream turbulence. The receptivity to free-stream turbulence is also analysed and it is found that two distinct physical mechanisms are active depending on the energy content of the external disturbance. If low-frequency modes diffuse into the boundary layer, presumably at the leading edge, the streaks are induced by streamwise vorticity through the linear lift-up effect. If, conversely, the free-stream perturbations are mainly located above the boundary layer a nonlinear process is needed to create streamwise vortices inside the shear layer. The relevance of the two mechanisms is discussed.

The propagation of an instability wave over the junction region between rigid and compliant panels is studied theoretically. The problem is investigated using three different methods with reference to flow in a plane channel containing sections with elastic walls. Within the framework of the first approach, using the solution of the problem of flow receptivity to local wall vibration, the problem considered is reduced to the solution of an integro-differential equation for the complex wall oscillation amplitude. It is shown that at the junction of rigid and elastic channel walls the instability-wave amplitude changes stepwise. For calculating the step value, another, analytical, method of investigating the perturbation propagation process, based on representing the solution as a superposition of modes of the locally homogeneous problem, is proposed. This method is also applied to calculating the flow stability characteristics in channels containing one or more elastic sections or consisting of periodically alternating rigid and compliant sections. The third method represents the unknown solution as the sum of a local forced solution and a superposition of orthogonal modes of flow in a channel with rigid walls. The latter method can be used for calculating the eigenvalues and eigenfunctions of the stability problem for flow in a channel with uniformly compliant walls.

The self-sustained oscillations arising in subsonic transitional flow over a flexible finite panel are investigated numerically. The flowfields are obtained by solving the full compressible Navier–Stokes equations employing a recently developed sixth-order implicit code which is capable of retaining its superior accuracy on deforming meshes. The panel mechanics is described by means of the nonlinear von Karman plate equations which are solved with a previously validated structural module. An iterative implicit coupling of the fluid and structural solvers is provided in order to eliminate lagging errors at the fluid/solid interface. Computations are performed for two freestream Mach numbers (M∞=0.2,0.8), and for a range of Reynolds number and dynamic pressure. At both flow speeds, self-excited oscillations develop in the absence of an applied external forcing. At M∞=0.2, the origin of the instability appears to be of an aeroelastic nature (i.e., flutter) and is found to persist even at very low Reynolds numbers. The fluctuating surface also generates boundary-layer instability modes downstream of the panel trailing edge. At the higher Mach number, static divergence occurs followed by the onset of travelling wave flutter (TWF), with frequency and wavelength compatible with those of Tollmien–Schlichting (T–S) waves. The observed limit-cycle oscillations appear to originate from the coupling of T–S waves with the higher-mode flexural waves on the elastic panel, and therefore subside below a critical value of Reynolds number. The TWF also generates significant acoustic radiation above the fluttering panel. The previous instabilities are observed for panels with either pinned or fully clamped edge conditions. Sensitivity of the solutions to the prescribed cavity pressure underneath the panel is noted. This dependence arises due to the different response of the boundary layer to the favorable or adverse pressure gradient induced by the upward or downward panel deflections, respectively.

For spiral Poiseuille flow with radius ratios η ≡ R i /R o = 0.77 and 0.95, we have computed complete linear stability boundaries, where R i and R o are the inner and outer cylinder radii, respectively. The analysis accounts for arbitrary disturbances of infinitesimal amplitude over the entire range of Reynolds numbers Re for which the flow is stable for some range of Taylor number Ta, and extends previous work to several non-zero rotation rate ratios μ ≡ Ω o /Ω i , where Ω i and Ω o are the (signed) angular speeds. For each combination of μ and η, there is a wide range of Re for which the critical Ta is nearly independent of Re, followed by a precipitous drop to Ta = 0 at the Re at which non-rotating annular Poiseuille flow becomes unstable with respect to a Tollmien-Schlichting-like disturbance. Comparison is also made to a wealth of experimental data for the onset of instability

This paper investigates the stability properties of a Blasius boundary layer perturbed by a small-amplitude steady three-dimensional distortion, which may be isolated or periodic along the spanwise direction. It is shown that once the strength of the distortion exceeds a threshold, the perturbed flow becomes inviscidly unstable. An isolated distortion which features a dominant low-speed streak induces a localized mode, while a periodic distortion supports spatially quasi-periodicmodes through a parametric resonance. The frequencies and growth rates of both types of mode are much higher than those of the viscous Tollmien–Schlichting (T–S) waves. For moderate distortions, the instability modes can be viewed as kinds of modified T–S waves, which amplify at rates in excess of the viscous instability. For a localized distortion, these modes do not reduce to the usual T–S wavesin the zero-distortion limit. The nonlinear development of the inviscid modes is also studied, and is foundto be governed by a slightly modified version of the evolution equation derived by Wu (1993). A numerical study suggests that nonlinearity has a strong destabilizing effect, and ultimately leads to an explosive growth in the form of a finite-distance singularity. The present theoretical model is found to capture qualitatively some key experimental observations.

Both experimental and theoretical studies have beenconsidered on flat plate boundary layers as well as on wakesbehind porous cylinders. The main thread in this work iscontrol, which is applied passively and actively on boundarylayers in order to inhibit or postpone transition toturbulence; and actively through the cylinder surface in orderto effect the wakecharacteristics. An experimental set-up for the generation of the asymptoticsuction boundary layer (ASBL) has been constructed. This studyis the first, ever, that report a boundary layer flow ofconstant boundary layer thickness over a distance of 2 metres.Experimental measurements in the evolution region, from theBlasius boundary layer (BBL) to the ASBL, as well as in theASBL are in excellent agreement with boundary layer analysis.The stability of the ASBL has experimentally been tested, bothto Tollmien-Schlichting waves as well as to free streamturbulence (FST), for relatively low Reynolds numbers (Re). For the former disturbances good agreement is foundfor the streamwise amplitude profiles and the phase velocitywhen compared with linear spatial stability theory. However,the energy decay factor predicted by theory is slightlyoverestimated compared to the experimental findings. The latterdisturbances are known to engender streamwise elongated regionsof high and low speeds of fluid, denoted streaks, in a BBL.This type of spanwise structures have been shown to appear inthe ASBL as well, with the same spanwise wavelength as in theBBL, despite the fact that the boundary layer thickness issubstantially reduced in the ASBL case. The spanwise wavenumberof the optimal perturbation in the ASBL has been calculated andis β = 0.53, when normalized with the displacementthickness. The spanwise scale of the streaks decreases withincreasing turbulence intensity (Tu) and approaches the scale given by optimalperturbation theory. This has been shown for the BBL case aswell. The initial energy growth of FST induced disturbances hasexperimentally been found to grow linearly as Tu2Rexin the BBL, the transitional Reynolds numberto vary as Tu-2, and the intermittency function to have a relativelywell-defined distribution, valid for all Tu. The wake behind a porous cylinder subject to continuoussuction or blowing has been studied, where amongst other thingsthe Strouhal number (St) has been shown to increase strongly with suction,namely, up to 50% for a suction rate of 2.5% of the free streamvelocity. In contrast, blowing shows a decrease ofStof around 25% for a blowing rate of 5% of the freestream velocity in the considered Reynolds number range. Keywords:Laminar-turbulent transition, asymptoticsuction boundary layer, free stream turbulence,Tollmien-Schlichting wave, stability, flow control, cylinderwake.

Many aquatic species such as dolphins and whales have fins, which can be modeled as swept wings. Some of these fins, such as the dorsal fin of a dolphin, are semi-rigid and therefore can be modeled as a rigid swept wing with a compliant surface. An understanding of the hydrodynamics of the flow past swept compliant surfaces is of great interest for understanding potential drag reduction mechanisms, especially since swept wings are widely used in hydrodynamic and aerodynamic design. In this paper, the flow past a swept wing with a compliant surface is modeled by an attachment-line boundary layer flow, which is an exact similarity solution of the Navier-Stokes equations, flowing past a compliant surface modeled as an elastic plate. The hydrodynamic stability of the coupled problem is studied using a new numerical framework based on exterior algebra. The basic instability of the attachment line boundary layer on a rigid surface is a traveling wave instability that propagates along the attachment line, and numerical results show that the compliance results in a substantial reduction in the instability region. Moreover, the results show that, although the flow-field is three-dimensional, the qualitative nature of the instability suppression is very similar to the qualitative reduction of instability of the two-dimensional Tollmien-Schlichting modes in the classical boundary-layer flow past a compliant surface.

The ine uence of two-dimensional humps and steps on the stability characteristics of a two-dimensional laminar boundary layer is investigated by means of direct numerical simulations (DNS). The localized surface irregularity ishereby modeled within theCartesian grid by assigning body forces oversurfacesthat need notcoincidewith grid lines.Comparedtotheuseofabody-e ttedcoordinatesystem,thismethodsavesmemoryandcomputationtime.The method is validated by grid ree nement tests as well as by a comparison with water channel experiments. The DNS results for the steady e ow over a rectangular hump, as well as for an instability wave traveling over a hump, show good agreementwith theexperimental ones. Simulation resultsshow thata localized hump destabilizesthelaminar boundary layer, whereas a forward facing step stabilizes it. The destabilization is stronger when the height or the width of the localized hump are increased. A rounded shape of the hump is less destabilizing than a rectangular shape with sharp corners. The parameter that shows the strongest ine uence on the stability characteristics of the boundary layer is clearly the height of the localized hump.

A laminar boundary layer separates in a region of adverse pressure gradient on a flat plate and undergoes transition. Finally the turbulent boundary layer reattaches, forming a laminar separation bubble (LSB). Laminar-turbulent transition within such a LSB is investigated by means of Laser-Doppler-Anemometry (LDA), Particle Image Velocimetry (PIV), and direct numerical simulation (DNS). The transition mechanism occurring in the flow-field under consideration is discussed in detail. Observations for the development of small disturbances are compared to predictions from viscous linear instability theory (Tollmien-Schlichting instability). Non-linear development of these disturbances and their role in final breakdown to turbulence is analyzed.

Numerical simulations of a large-amplitude nonlinear two-dimensional train of Tollmien-Schlichting waves are performed first and show the development of short-scaled structures or spikes. A careful description of the spiking process and its subsequent development is given describing the generation of localized maxima in the streamwise pressure distribution and associated vortices, spikes in a perturbation velocity trace and the emergence of strong wall-normal pressure gradients. A high-Reynolds-number asymptotic theory has previously been developed by two of the authors which aims to describe this spiking process. The current work is the first to give a comparison between this theory and planar Navier-Stokes computations. We give a brief description of the theory showing how normal pressure gradients become active and their role in the generation of the streamwise pressure distribution and its subsequent effects such as vortex generation and wall-layer vorticity eruptions. The presentation is given with close qualitative reference to the simulations, so giving credence to the relevance of the theoretical account and an interpretation of that account in physical terms. The comparison, although primarily qualitative, is successful in that it is possible to identify the physical processes highlighted by the theory in the computations, so clarifying the complex fluid motions and suggesting directions for further research. The paper concludes with firstly a discussion of how the results of the simulations and the theory could be used to give an understanding of similar processes seen in unsteady planar separation and secondly their relevance to the strongly three-dimensional processes at work in deep transition experiments.

The linear problem of generation of perturbations of a flat-plate boundary layer by external turbulence is solved. The turbulence is represented in the form of a set of space- and time-periodic vortex modes. It is shown that the boundary layer is most receptive to low-frequency longitudinal vorticity modes. The mean-square velocity fluctuations in the boundary layer and their spectrum are found for isotropic turbulence with a spectrum satisfying the Kolmogorov-Obukhov law.

For applications regarding transition prediction, wing design and control of boundary layers, the fundamental understanding of disturbance growth in the flat-plate boundary layer is an important ...

The optimal distribution of steady suction needed to control the growth of single or multiple disturbances in quasi-three-dimensional incompressible boundary layers on a flat plate is investigated. The evolution of disturbances is analysed in the framework of the parabolized stability equations (PSE). A gradient-based optimization procedure is used and the gradients are evaluated using the adjoint of the parabolized stability equations (APSE) and the adjoint of the boundary layer equations (ABLE). The accuracy of the gradient is increased by introducing a stabilization procedure for the PSE. Results show that a suction peak appears in the upstream part of the suction region for optimal control of Tollmien–Schlichting (T–S) waves, steady streamwise streaks in a two-dimensional boundary layer and oblique waves in a quasi-three-dimensional boundary layer subject to an adverse pressure gradient. The mean flow modifications due to suction are shown to have a stabilizing effect similar to that of a favourable pressure gradient. It is also shown that the optimal suction distribution for the disturbance of interest reduces the growth rate of other perturbations. Results for control of a steady cross-flow mode in a three-dimensional boundary layer subject to a favourable pressure gradient show that not even large amounts of suction can completely stabilize the disturbance.

Accelerated boundary layers are considered to be stable for practical purposes. It is shown that small suction nonuniformitiescaninduceaninstabilitythatresultsinthegenerationofstreamwisevortices.Additionofauniform suctionisnoteffectiveincontrollingthisinstability.Itisshownthatnonuniformitiesofsurfaceblowingmightinduce a similar instability and that the characteristics of this instability are also little affected by the magnitude of the blowing. The same nonuniformities appear to have a small effect on the traveling (Tollmien‐Schlichting ) wave instability in the range of parameters studied. It is shown that oblique waves become dominant if the amplitudes of the nonuniformities become large enough. Nomenclature Ank = coefe cient matrix [Eq. (10)] cn = arbitrary constants [Eq. (7)] D Dd=dy = derivative operator F = similarity function in the Falkner‐Skan equation N Gm D[g.m/ u ;g .m/ w ;g .m/

Linear stability analysis of flow in a channel bounded by wavy walls is considered. It is shown that wall waviness gives rise to an instability that manifests itself through generation of streamwise vortices. The available results suggest that the critical stability criteria based on the Reynolds number based on the amplitude of the waviness can be formulated.

Direct numerical simulations (DNS) of the Navier–Stokes equations are used to investigate the role of the Klebanoff-mode in laminar–turbulent transition in a flatplate boundary layer. To model the effects of free-stream turbulence, volume forces are used to generate low-frequency streamwise vortices outside the boundary layer. A suction/blowing slot at the wall is used to generate a two-dimensional Tollmien–Schlichting (TS) wave inside the boundary layer. The characteristics of the fluctuations inside the boundary layer agree very well with those measured in experiments. It is shown how the interaction of the Klebanoff-mode with the two-dimensional TS-wave leads to the formation of three-dimensional TS-wavepackets. When the disturbance amplitudes reach a critical level, a fundamental resonance-type secondary instability causes the breakdown of the TS-wavepackets into turbulent spots.

The linear problem of the stability of a boundary layer on a flexible surface with respect to small two-dimensional perturbations in the form of travelling waves is considered. A boundary-layer model with self-induced pressure (a three-layer model) is used to describe the fluid flow. The flexible surface is modelled by a thin plate lying on an elastic base. It is shown that, in the system in question two forms of loss of stability are possible: Tollmien-Schlichting instability and instability caused by the action of hydrodynamic pressure on the surface around which the flow occurs. A special feature of the latter form of instability is that there is no neutral system of perturbations or a sudden onset of instability. The solutions obtained are analysed and the conditions leading to the appearance of such instability are established.

A nonlinear model of interaction of disturbances in the regime of coupled combinatorial relations is used to explain the dynamics of unstable waves arising due to introduction of controlled high‐intensity disturbances into a supersonic boundary layer. The model includes effects of self‐action and combinatorial interaction of the waves. The second‐order model considered offers a rather accurate description of the streamwise dynamics of plane waves.

Using the method of moments in the space of wavenumbers, a class of models of a developed turbulent flow of an incompressible fluid in a flat‐plate boundary layer is proposed. The models are based on an analysis of the Navier–Stokes equations that describe the behavior of dynamic coherent structures associated with vorticity generation and also the behavior of the stochastic component. A continuum analog of dynamic equations for a coherent structure is given in an explicit form. In the general case, the stochastic component should satisfy a system of equations of the kinetic type, which reduces to one equation under certain assumptions. It is also shown that the presence of coherent structures leads to generalization of the notion of statistical homogeneity.

Prompted by the recent experiments of Dietz (1999) on boundary-layer receptivity due to a local roughness interacting with a vortical disturbance in the free stream, this paper undertakes to present a second-order asymptotic theory based on the tripledeck formulation. The asymptotic approach allows us to treat vortical perturbations with a fairly general vertical distribution, and confirms Dietz's conclusion that for the convecting periodic wake in his experiments, the receptivity is independent of its vertical structure and can be fully characterized by its slip velocity at the edge of the boundary layer. As in the case of distributed vortical receptivity, dominant interactions that generate Tollmien–Schlichting waves take place in the upper deck as well as in the so-called edge layer centred at the outer reach of the boundary layer. The initial amplitude of the excited Tollmien–Schlichting wave is determined to O(R−1/8) accuracy, where R is the global Reynolds number. An appropriate superposition formula is derived for the case of multiple roughness elements. A comprehensive comparison is made with Dietz's experimental data, and an excellent quantitative agreement has been found for the first time, thereby resolving some uncertainties about this receptivity mechanism.

Spatial development of small amplitude disturbances in plane Poiseuille flow is investigated numerically using a commercial CFD code (CFX 4.2). Small amplitude Tollmien–Schlichting (TS) waves known from an Orr–Sommerfeld analysis are superimposed as initial conditions on a fully developed laminar channel flow. The downstream spatial disturbances are calculated on the basis of the full Navier–Stokes equations (NSE). Results are compared to calculations based on the Orr–Sommerfeld equations (OSE). In a second step the thermal energy equation is solved in addition to the NSE to investigate the temperature effects and especially the development of temperature disturbances. A certain delay in the development of these disturbances is found which is due to a thermal receptivity mechanism. These results again are compared to OSE calculations.

It is shown that suction nonuniformities may induce a new instability of the Blasius boundary layer that results in the generation of streamwise vortices. Addition of a uniform suction is not effective in controlling this instability. It is shown that nonuniformities of surface blowing may induce a similar instability and that the characteristics of this instability are also little affected by the magnitude of the blowing. Finally, it is shown that suction nonuniformities appear to have a minimal effect on the traveling (Tollmien–Schlichting) wave instability.

A spatial theory is proposed for the linear transient growth of disturbances in a parallel boundary layer. Following from the consideration of a signaling problem, the spatial development of disturbances downstream of a source may be presented as a sum of decaying eigenmodes and Tollmien–Schlichting (TS) like instability modes. Therefore, the problem of optimal disturbances may be considered as an initial value problem on the subset of the decaying eigenmodes and a TS wave, and a standard optimization procedure may be applied for evaluation of the optimal transient growth. The results indicate that the most significant transient growth is associated with stationary streamwise vortices. Numerical examples illustrate that favorable pressure gradient decreases the overall amplification. Effects of compressibility and the wall cooling are investigated as well.

A highly accurate finite-difference PSE code has been developed to investigate the stability analysis of incompressible boundary layers over a flat plate. The PSE equations are derived in terms of primitive variables and are solved numerically by using compact method. In these formulations, both nonparallel as well as nonlinear effects are accounted for. The validity of present numerical scheme is demonstrated using spatial simulations of two cases; two-dimensional (linear and nonlinear) Tollmien-Schlichting wave propagation and three-dimensional subharmonic instability breakdown. The PSE solutions have been compared with previous numerical investigations and experimental results and show good agreement.

This paper investigates the receptivity of boundary layers due to distributed roughness interacting with free-stream disturbances. Both acoustic and vortical perturbations are considered. An asymptotic approach based on the triple-deck formulation has been developed to determine the initial amplitude of the Tollmien–Schlichting wave to the O(R−1/8) accuracy, where R is the global Reynolds number. In the case of vortical disturbances, we show that the dominant contribution to the receptivity comes from the upper deck as well as from the so-called edge layer centred at the outer reach of the boundary layer. It is found that for certain forms of disturbances, the receptivity is independent of their vertical structure and can be fully characterized by their slip velocity at the edge of the boundary layer. A typical case is the vortical disturbance in the form of a convecting wake, for which the same conclusion as above has been reached by Dietz (1999) on the basis of measurements. Our theoretical predictions are compared with the experimental data of Dietz (1999), and a good quantitative agreement has been found. Such a comparison is the first to be made for distributed vortical receptivity. Further calculations indicate that the vortical receptivity in general is much stronger than was suggested previously. In the case of acoustic disturbances, it is found that our first-order theory is in good agreement with experiments as well as with previous theoretical results. But the second-order theory over-predicts, and the possible reasons for this are discussed.

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

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.