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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)

The flow around infinite swept wings is computed using a Reynolds-averaged Navier-Stokes method coupled to a boundary layer and a transition prediction method based on the e N approach. This approach applies the two N factor e N strategy, which allows the differentiation between excited Tollmien-Schlichting waves of constant frequency and stationary crossflow waves of constant spanwise wave number. The applicability is documented by comparing computations with infinite swept-wing experiments on four configurations, where one configuration was tested in two different wind tunnels. The limiting N factors for both types of waves are a priori unknown. Hence, firstly the limiting N factors, which represent the stability limit, are deduced from the measured transition locations. Subsequently, iterative computations are executed to determine the transition location using the coupled system, Navier-Stokes, boundary-layer, and two N factor e N method in combination with the stability limit

The laminar breakdown of a three-dimensional flat-plate boundary-layer flow with favourable and ensuing adverse pressure gradient (APG), generic for an infinite swept wing, is investigated in detail by means of spatial direct numerical simulations. Emphasis is on transition mechanisms in the region of adverse pressure gradient where the undistorted laminar base flow also becomes unstable with respect to Tollmien– Schlichting (TS) waves that are most amplified if two-dimensional with respect to the local streamwise direction. The influence of finite-amplitude crossflow vortices, coming from the region of favourable pressure gradient, on the TS instability properties is investigated. It turns out that crossflow-vortex-induced secondary instabilities are the most amplified disturbances even for low-amplitude vortex modes. The TS waves act as generators of fundamental low-frequency secondary modes, but are neither important for their growth nor for breakdown. As for active disturbance control, any method aiming at attenuating two-dimensional TS waves must fail. On the other hand, the passive upstream-flow-deformation technique also delays transition in the APG region.

Experiments are described in which well-defined weak Free Stream Nonuniformity (FSN) is introduced by placing fine wires upstream of the leading edge of a flat plate. Large amplitude spanwise thickness variations form in the boundary layer as a result of the interaction between the steady laminar wakes from the wires and the leading edge. The centerline of a region of elevated layer thickness is aligned with the centerline of the wake in the freestream and the response is shown to be remarkably sensitive to the spanwise length-scale of the wakes. The region of elevated thickness is equivalent to a long narrow low speed streak in the layer. Elevated Free Stream Turbulence (FST) levels are known to produce randomly forming arrays of long narrow low speed streaks in laminar boundary layers. Therefore the characteristics of the streaks resulting from the FSN are studied in detail in an effort to gain some insight into bypass transition that occurs at elevated FST levels. The shape factors of the profiles in the vicinity of the streak appear to be unaltered from the Blasius value, even though the magnitude of the local thickness variations are as large as 60% of that of the undisturbed layer. Regions of elevated background unsteadiness appear on either side of the streak and it is shown that they are most likely the result of small amplitude spanwise modulation of the layer thickness. The background unsteadiness shares many of the characteristics of Klebanoff modes observed at elevated FST levels. However, the layer remains laminar to the end of the test section (Rx≈1.4×106) and there is no evidence of bursting or other phenomena associated with breakdown to turbulence. A vibrating ribbon apparatus is used to examine interactions between the streak and Tollmien-Schlichting (TS) waves. The deformation of the mean flow introduced by the streak is responsible for substantial phase and amplitude distortion of the waves and the breakdown of the distorted waves is more complex and it occurs at a lower Reynolds number than the breakdown of the K-type secondary instability that is observed when the FSN is not present.

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.

The linear stability of the boundary layer developing on a flat pl ate in the presence of finite-a mplitude, steady and spanwise periodic streamwise streaks is investigated. The streak amplitudes considered here are below the threshold for onset of the inviscid inflectional instability of sinuous perturbations. It is found that , as the amplitude of the streaks is increased, the most unstable viscous waves evolve from two-dimensional Tollmien–Schlichting waves into three-dimensional varicose fundamental modes which compare well with early experimental findings. The analysis of the growth rates of these modes confirms the stabilising effect of the streaks on the viscous instability and that this stabilising effect increases with the streak amplitude. Varicose subharmonic modes are also found to be unstable but they have growth rates which typically are an order of magnitude lower than those of fundamental modes. The perturbation kinetic energy production associated with the spanwise shear of the streaky flow is found to play an essential rol e in the observed stabilisation. The possibl e relevance of the streak stabilising role for applications in boundary layer transition delay is discussed.  2004 Elsevier SAS. All rights reserved.

A method for solving equations that describe the dynamics of wave packets of the Tollmien–Schlichting waves in the boundary layer is proposed. The method of splitting the initial problem into the linear and nonlinear parts at each time step is used. The linear part is resolved by using an equation for spectral components of the wave packet with a subsequent Fourier transform from the space of wavenumbers to the physical space. A system of ordinary differential equations is solved in the physical space. The Fourier transform is performed by means of the library procedure of the fast Fourier transform. As examples, the problems solved were the linear dynamics of the wave packet concentrated in the vicinity of the instability region (i.e., a set of wave vectors in the space of wavenumbers for which the imaginary part of the eigenfrequency of the Tollmien–Schlichting waves is positive) and the nonlinear dynamics of the wave packet overlapping the instability region.

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

For spiral Poiseuille flow with radius ratio η ≡ R i /R o = 0.5, we have computed complete linear stability boundaries for several values of the rotation rate ratio μ ≡ Ω o /Ω i , where R i and R o are the inner and outer cylinder radii, respectively, and Ω i and Ω o are the corresponding (signed) angular speeds. The analysis extends the previous range of Reynolds number Re studied computationally by more than eightyfold, and accounts for arbitrary disturbances of infinitesimal amplitude over the entire range of Re for which spiral Poiseuille flow is stable for some range of the Taylor number Ta

A flat plate laminar boundary layer subjected to free-stream turbulence, generated by grids upstream of the leading edge, has been studied. Correlations between a wall wire and a hot wire in the boundary layer have been obtained. The hot wire has been traversed to 328 positions for each of the five wall-wire positions studied. The length, height, and width of the correlation distributions is seen to increase in the downstream direction being self-similar in coordinates scaled with the boundary-layer length scale. The propagation speeds of the structures are obtained from the data and its variation in the streamwise and wall-normal directions is found to agree with what is obtained from a simple kinematic model.

We explore the possibility of radically decreasing the hydrodynamic drag by changing the external flow so that the boundary layer is replaced by some flow with a small velocity gradient.

The temporal stability of swept attachment-line boundary layer flow based on a swept Hiemenz flow model is studied. Starting from the global stability problem and motivated by analytical free-stream solutions, a Hermite expansion is employed in the chordwise coordinate direction which results in coupled local stability problems. A complete study of the temporal spectrum is presented and the discrete and continuous modes are classified according to their symmetry, chordwise polynomial order and asymptotic decay. Uniform, Gortler-Hammerlin and higher-order modes are described in detail. Estimates are given for the location of the continuous spectrum, and bounds are derived for the validity of the linear approximation

This paper describes investigations on active attenuation of naturally occurring Tollmien–Schlichting (TS) instabilities on an unswept wing. TS disturbances are canceled in their linear stage by superimposition with artificially generated counter waves in order to shift the laminar–turbulent transition downstream. In this method, the need for energy input is considerably lower than the stabilization by manipulation of the local mean velocity profile (e.g. boundary layer suction). An optimum TS wave cancellation is achieved by a fast digital signal processor performing an adaptive control algorithm. The oncoming traveling TS disturbances are detected by a reference sensor upstream of an actuator. The reference signal is directly calculated into an actuator signal by means of a linear transfer function. The transfer function is continuously adapted to minimize the error signal measured downstream of the actuator. Previous investigations with a single control system have shown the successful TS wave damping by an adaptive control algorithm. These investigations led to an improvement in the set up of sensors and actuators used in the field of TS wave control. As a follow up of these experiments, recent investigations focus on the application of multi-channel control systems. This approach is essential to attenuate three-dimensionally dominated boundary layer instabilities by spanwise arranged sensor–actuator systems. Furthermore, the employment of multiple sensor–actuator systems allows a streamwise repeated damping to extend the delay of laminar–turbulent transition considerably.

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 work presented extends previous research on linear controllers in temporal channel flow to spatially evolving boundary layer flow. The flows studied are those on an infinite swept wedge described by the Falkner–Skan–Cooke (FSC) velocity profiles, including the special case of the flow over a flat plate. These velocity profiles are used as the base flow in the Orr–Sommerfeld–Squire equations to compute the optimal feedback control through blowing and suction at the wall utilizing linear optimal control theory. The control is applied to a parallel FSC flow with unstable perturbations. Through an eigenvalue analysis and direct numerical simulations (DNS), it is shown that instabilities are stabilized by the controller in the parallel case. The localization of the convolution kernels for control is also shown for the FSC profiles.Assuming that non-parallel effects are small a technique is developed to apply the same controllers to a DNS of a spatially evolving flow. The performance of these controllers is tested in a Blasius flow with both a Tollmien–Schlichting (TS) wave and an optimal spatial transiently growing perturbation. It is demonstrated that TS waves are stabilized and that transient growth is lowered by the controller. Then the control is also applied to a spatial FSC flow with unstable perturbations leading to saturated cross-flow vortices in the uncontrolled case. It is demonstrated that the linear controller successfully inhibits the growth of the cross-flow vortices to a saturated level and thereby delays the possibility of transition through secondary instabilities. It is also demonstrated that the controller works for relatively high levels of nonlinearity, and for stationary as well as time-varying perturbations.

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.

A method is presented for estimating the laminar/turbulent transition location in three-dimensional boundary layers for computational fluid dynamics (CFD) applications requiring numerous transition estimates with no user intervention. Given the Reynolds number and the Cp distribution, the location of transition is estimated based on the attachment-line state, the potential for relaminarization, the occurrence of laminar separation, and the growth of instabilities. Transition caused by instability is estimated based on N factors calculated for Tollmien-Schlichting waves and for stationary crossflow instabilities. A neural network is used (in place of solving the Orr-Sommerfeld stability equation) for determining the instability growth rates. The current version of the method assumes incompressible flow. The boundary-layer flow and instabilities are calculated based on an infinite-sweep (strip boundary layer) approximation; the instability calculations also employ the parallel-flow approximation

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 by-pass mechanism of transition for a wall-bounded shear layer is explained for the case when an infinite row of convecting vortices migrate over a boundary layer at a specific speed range. Such a mechanism is important for noisy flows over bluff bodies, flows inside turbomachinery and flows over helicopter rotor blades. By solving the Navier–Stokes equation, it is shown that this by-pass transition is a consequence of vortices migrating at convection speeds that are significantly lower than the free-stream speed. This situation is commonly found in flows that are affected by the presence of periodic wakes. Whenever the speed of migrating vortices is in a certain critical range, there is a local instability of the underlying shear layer with a very high-growth rate as compared to the growth of pure Tollmien–Schlichting waves created by wall excitation. The above interpretation is supported by solving the linearized and full Navier–Stokes equation for disturbance quantities under the parallel flow approximation in two dimensions. Various ramifications of such a by-pass route of transition are discussed in this paper.

The flow stability in a boundary layer with an inhomogeneous spanwise-periodic velocity profile modeling the streaky structure that develops at a high level of turbulence of the incident flow is analyzed in the three-dimensional formulation for perturbations with an arbitrary transverse period. It is shown that in the presence of inhomogeneity the dispersion relation for the Tollmien-Schlichting waves is split into two branches periodic in the transverse wave number, which correspond to symmetric and antisymmetric modes. The solution for the packet of inhomogeneous-flow modes generated by localized time-periodic fluid injection/ejection is found. The shape of this packet corresponds qualitatively to the shape of the Tollmien-Schlichting wave packet, but the fine perturbation structure inside it is sharply different.

In this study, numerical simulation was performed with the SIMPLER method on the amplification of Tollmien-Schlichting (T-S) wave in the boundary layer over a flat plate and its suppression with a wall motion actuator. The T-S wave was induced by the interaction between sound wave introduced by a speaker and a protuberance on the surface of the plate, and amplified in the unstable region of the boundary layer. The vertical vibration of a wall motion actuator was used to control the T-S wave. The influence of the amplitude and phase of actuator vibration on the control effectiveness was investigated in the simulation. The results show that the T-S wave can be effectively suppressed by the motion of the actuator surface if the phase and amplitude of the actuator are appropriately adjusted.

The 3D receptivity of 2D laminar boundary layers to localized surface vibrations has been investigated both experimentally and theoretically for two types of basic flow: (i) the Blasius boundary layer and (ii) a boundary layer with a negative streamwise pressure gradient (Hartree parameter βH=0.10). For the boundary-layer excitation, a specially designed surface vibrator was used. The development of the excited wave-trains was measured by means of hot-wire anemometry and decomposed into oblique normal Tollmien–Schlichting-modes. The initial spectra of the excited perturbations at the position of the vibrator was obtained by two different techniques. The first used an additional source which was mounted upstream and provided the amplification curves for the instability modes in the vicinity of the vibrator, the second was based on linear stability calculations. The receptivity coefficients were defined as the ratio of the initial wavenumber spectrum of the excited TS-waves and the corresponding resonant spectrum of the surface vibrations. They were determined for each fixed frequency as a function of the spanwise wavenumber. The boundary value problem for the disturbance produced by the vibrating membrane was solved theoretically for the same conditions as in the experiments in the framework of the classical hydrodynamic stability theory. The Navier–Stokes equations were linearized around a incompressible basic flow described by a solution of the Falkner–Skan equation. Comparisons of the theoretical and experimental results on the 3D receptivity show good quantitative agreement. It is concluded that the favorable pressure gradient increases the boundary-layer receptivity to surface vibrations.

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.

In this work a glow discharge based active flow control for high flow velocities and low Reynolds numbers is presented. Unlike common plasma actuators such as dielectric barrier discharge (DBD) or spark jets, this actuator uses small impulse bits at frequencies. The actuator is optimized for frequencies up to 40 kHz to counter Tollmien Schlichting wave effects and so reduce overall air foil drag. Several measurements to prove the non-eroding effect of the actuator and the electrical properties were performed. It was found that the actuator is capable of operating at high frequencies without measurable erosion.

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.

This paper describes the three-dimensional flow structure in grooved channels with different cavity lengths at intermediate Reynolds numbers. For steady flow, the three-dimensional effects are dominant near the side walls of the channel. However, after the onset of self-sustained oscillatory flow due to Tollmien–Schlichting waves as the primary instability, a secondary instability produces a three-dimensional flow with Taylor–Geortler-like vortical structure, at the bottom of the groove. This trend becomes more significant as the cavity length increases. Furthermore, the reason for three-dimensional flow is discussed using additional numerical analysis, and it is confirmed that the source of three-dimensional instability is the groove vortices due to the presence of side walls, rather than the channel traveling wave.

In this Brief Communication, the nonlinear evolution of periodical disturbances generated by an external sound field in a swept wing boundary layer is presented. All experimental results are compared with corresponding data for a straight wing configuration. The Tollmien–Schlichting instability has been studied, and it was found that the disturbance flow field remained highly deterministic and periodic in both time and space until the latest stages of the transition. A frequency-wave number Fourier analysis shows that the disturbance spectra comprise only the fundamental wave and its higher harmonics. The K-type breakdown scenario was observed, and the nonsymmetry of flow patterns in the swept wing boundary layer was found to be due to the presence of the cross flow.

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.

Thispaperreviewsand assessestherecentprogresstoward making theuseofcompliantwallsa practicalmethod of laminar-e ow control. Three main areas are covered. First, the current understanding of the vitally important e ow-induced surface instabilities is assessed. Some new results are included. Then the optimization of multiplepanel compliant walls is considered. New numerical simulation results are included showing that short compliant panels are very effective in suppressing Tollmien ‐Schlichting waves. It is found that for marine applications appropriately designed multiple-panel compliant walls are capable of suppressing Tollmien ‐Schlichting waves to indee nitely high Reynolds numbers. Finally, the feasibility of using compliant walls for laminar-e ow control in aeronautical applications is assessed. It is found that, although there is no reason in principle why compliant walls cannot be used in air, in practice exceptionally delicate walls are required to obtain the necessary match between thee uid and structural inertias. Theresulting lack ofrobustness forsuch walls isdeemed to make them completely impractical for aeronautical applications.

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.

An experimental investigation was conducted to examine acoustic receptivity and subsequent boundary-layer instability evolution for a Blasius boundary layer formed on a flat plate in the presence of two-dimensional and oblique (three-dimensional) surface waviness. The effect of the non-localized surface roughness geometry and acoustic wave amplitude on the receptivity process was explored. The surface roughness had a well-defined wavenumber spectrum with fundamental wavenumber kw. A planar downstream-travelling acoustic wave was created to temporally excite the flow near the resonance frequency of an unstable eigenmode corresponding to kts = kw. The range of acoustic forcing levels, ε, and roughness heights, Δh, examined resulted in a linear dependence of receptivity coefficients; however, the larger values of the forcing combination εΔh resulted in subsequent nonlinear development of the Tollmien–Schlichting (T–S) wave. This study provides the first experimental evidence of a marked increase in the receptivity coefficient with increasing obliqueness of the surface waviness in excellent agreement with theory. Detuning of the two-dimensional and oblique disturbances was investigated by varying the streamwise wall-roughness wavenumber αw and measuring the T–S response. For the configuration where laminar-to-turbulent breakdown occurred, the breakdown process was found to be dominated by energy at the fundamental and harmonic frequencies, indicative of K-type breakdown.

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.