On the wave-cancelling nature of boundary layer flow control

This work deals with the feedforward active control of Tollmien–Schlichting instability waves over incompressible 2D and 3D boundary layers. Through an extensive numerical study, two strategies are evaluated; the optimal linear–quadratic–Gaussian (LQG) controller, designed using the Eigensystem realization algorithm, is compared to a wave-cancellation scheme, which is obtained using the direct inversion of frequency-domain transfer functions of the system. For the evaluated cases, it is shown that LQG leads to a similar control law and presents a comparable performance to the simpler, wave-cancellation scheme, indicating that the former acts via a destructive interference of the incoming wavepacket downstream of actuation. The results allow further insight into the physics behind flow control of convectively unstable flows permitting, for instance, the optimization of the transverse position for actuation. Using concepts of linear stability theory and the derived transfer function, a more efficient actuation for flow control is chosen, leading to similar attenuation of Tollmien–Schlichting waves with only about 10% of the actuation power in the baseline case.     

Global two-dimensional stability measures of the flat plate boundary-layer flow

The stability of the two-dimensional flat plate boundary-layer is studied by means of global eigenmodes. These eigenmodes depend both on the streamwise and wall-normal coordinate, hence there are no assumptions on the streamwise length scales of the disturbances. Expanding the perturbation velocity field in the basis of eigenmodes yields a reduced order model from which the stability characteristics of the flow, i.e. the initial condition and forcing function leading to the largest energy growth, are extracted by means of non-modal analysis. In this paper we show that, even when performing stability analysis using global eigenmodes, it is not sufficient to consider only a few of the least damped seemingly relevant eigenmodes. Instead it is the task of the optimization procedure, inherent in the non-modal analysis, to decide which eigenmodes are relevant. We show that both the optimal initial condition and the optimal forcing structure have the form of upstream tilted structures. Time integration reveals that these structures gain energy through the so called Orr mechanism, where the instabilities extract energy from the mean shear. This provides the optimal way of initiating Tollmien–Schlichting waves in the boundary layer. The optimal initial condition results in a localized Tollmien–Schlichting wavepacket that propagates downstream, whereas the optimal forcing results in a persistent Tollmien–Schlichting wave train.     

Nonlinear Instability in the Region of Laminar-Turbulent Transition in Supersonic Three-Dimensional Flow over a Flat Plate

Direct numerical simulation is applied to obtain laminar-turbulent transition in supersonic flow over a flat plate. It is shown that, due to the nonlinear instability, Tollmien–Schlichting waves generated in the boundary layer lead to the formation of oblique disturbances in the flow. These represent a combination of compression and expansion waves, whose intensities can be two orders higher than that of external harmonic disturbances. The patterns of the three-dimensional flow over the plate are presented and the structures of the turbulent flat-plate boundary layers are described for the freestream Mach numbers M = 2 and 4.     

The effect of wall cooling on compressible Görtler vortices

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

Three-dimensional instability analysis of boundary layers perturbed by streamwise vortices

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.     

Characterization of noise amplifiers with global singular modes: the case of the leading-edge flat-plate boundary layer

This article deals with the linear dynamics of a transitional boundary layer subject to two-dimensional Tollmien–Schlichting instabilities. We consider a flat plate including the leading edge, characterized by a Reynolds number based on the length of the plate equal to Re = 6 × 10⁵, inducing a displacement thickness-based Reynolds number of 1,332 at the end of the plate. The global linearized Navier–Stokes equations only display stable eigenvalues, and the associated eigen-vectors are known to poorly represent the dynamics of such open flows. Therefore, we resort to an input–output approach by considering the singular value decomposition of the global resolvent. We then obtain a series of singular values, an associated orthonormal basis representing the forcing (the so-called optimal forcings) as well as an orthonormal basis representing the response (the so-called optimal responses). The objective of this paper is to analyze these spatial structures and to closely relate their spatial downstream evolution to the Orr and Tollmien–Schlichting mechanisms. Analysis of the spatio-frequential diagrams shows that the optimal forcings and responses are respectively localized, for all frequencies, near the upstream neutral point (branch I) and the downstream neutral point (branch II). It is also shown that the spatial growth of the dominant optimal response favorably compares with the e ^N method in regions where the dominant optimal forcing is small. Moreover, thanks to an energetic input–output approach, it is shown that this spatial growth is solely due to intrinsic amplifying mechanisms related to the Orr and Tollmien–Schlichting mechanisms, while the spatial growth due to the externally supplied power by the dominant optimal forcing is negligible even in regions where the dominant optimal forcing is strong. The energetic input–output approach also yields a general method to assess the strength of the instability in amplifier flows. It is based on a ratio comparing two quantities of same physical dimension, the mean-fluctuating kinetic energy flux of the dominant optimal response across some boundary and the supplied mean external power by the dominant optimal forcing. For the present boundary-layer flow, we have computed this amplification parameter for each frequency and discussed the results with respect to the Orr and Tollmien–Schlichting mechanisms.     

BY-PASS MECHANISM OF TRANSITION TO TURBULENCE

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.     

Study of tonal noise behavior of an airfoil by using parabolized stability equations

Tonal noise or whistle noise is an aerodynamic noise known to be generated due to boundary layer instability. The relation between the instability of Tollmien–Schlichting wave and the tonal noise was dealt with, in previous studies, for rather limited cases that employed linear stability analysis or results for idealized flow configuration. To investigate the relation between the instability wave and tonal noise in a more thorough and systematic way, we employ the parabolized stability equation approach to compute details of the stability characteristics of boundary layer developed over pressure side surface of an airfoil at various angles of attack and various free-stream velocities. Discussions on the relation between the instability and the tonal noise have been given based on the comparison of the present computational results with the experimental data. We confirm that the overall U ^1.5 dependency of the noise frequency with velocity is caused by the most amplified Tollmien–Schlichting wave. Application of a simple feedback model to the stability data of the present work provides us with the results that explain well the ladder-like structure and local U ^0.8 dependency of the tonal noise. Effects of angle of attack and chord length on the tonal noise including the frequency, velocity range, and frequency difference between peaks of the noise are also examined.     

Numerical modeling of the laminar-turbulent transition control using a dielectric barrier discharge

The control of laminar-turbulent transition driven by Tollmien–Schlichting waves is studied. The control is realized by means of accelerating the boundary layer flow using a dielectric barrier discharge. As distinct from the previous studies based on the solution of the boundary layer equations, the discharge effect on the main flow and unstable disturbances are described by the Navier–Stokes equations.     

Mach Wave Effect on Laminar-Turbulent Transition in Supersonic Flow over a Flat Plate

The effect of a Mach wave (N wave) on laminar-turbulent transition induced by the first instability mode (Tollmien–Schlichting wave) in the flat-plate boundary layer is investigated on the basis of the numerical solution of Navier–Stokes equations at the freestream Mach number of 2.5. In accordance with the experiment, the N wave is generated by a two-dimensional roughness at the computation domain boundary corresponding to the side wall of the test section of a wind tunnel. It is shown that the disturbance induced by the backward front of the N wave in the boundary layer has no effect on the beginning of transition but displaces downstream the nonlinear stage of the first mode development. The disturbance induced by the forward front of the N wave displaces the beginning of transition upstream.     

Active compliant wall for skin friction reduction

In order to reduce skin friction drag, an active laminarisation method is developed. Laminar-turbulent boundary layer transition caused by Tollmien–Schlichting (TS) waves is delayed by attenuation of these convective instabilities. An actively driven compliant wall is integrated as part of a wing’s surface. Different configurations of piezo-based actuators are combined with an array of sensitive surface flow sensors. Wall-normal actuation as well as inclined wall displacement are investigated. Together with a realtime-control strategy, transition onset is shifted downstream by six average TS-wave lengths. Using the example of flow velocity, the influence of variable flow conditions on TS-damping rates was investigated. Besides, the boundary layer flow downstream of the active wall area as well as required wall deflections and the global damping effect on skin friction are presented in this paper.     

A study of the effect of step excrescences and free‐stream disturbances on boundary layer stability

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.     

Excitation by sound of Tollmien-Schlichting waves in a supersonic boundary layer

The interaction of sound with a supersonic boundary layer is considered. Because of the dependence of the main flow on the longitudinal coordinate, a sound wave generates unstable oscillations within the boundary layer. Calculations made for Mach number M = 2.0 and dimensionless frequency 2πfv_e/U_e ² = 0.91·10^?4 showed that near the lower branch of the curve of neutral stability a Tollmien—Schlichting wave can be excited with an intensity 2–3 times greater than that of the external acoustic wave.     

Large Eddy Simulation of boundary layer transition over an isolated ramp-type micro roughness element

Boundary layer transition over an isolated surface roughness element is investigated by means of numerical simulation. Large Eddy Simulation (LES) flow-modeling approach is employed to study flow characteristics and transition phenomenon past a roughness element immersed within an incoming developing boundary layer, at a height-based Reynolds number of 1170. LES numerical results are compared to experimental data from literature showing the time-averaged velocity distribution, the velocity fluctuation statistics and the instantaneous flow topology.Despite slight difference in the intensity of streamwise velocity fluctuations, the present LES results and experimental data show very good agreement. The mean flow visualization shows streamwise counter-rotating vortices pairs formation downstream of the obstacle. The primary pair induces an upwash motion and a momentum deficit that creates a Kelvin-Helmholtz type flow instability. The instantaneous flow topology reveals the formation of coherent K-H vortices downstream that produce turbulent fluctuations in the wake of the roughness element. These vortices are streched and lifted up when moving downstream. The velocity fluctuations results show that the onset of the turbulence is dominated by the energy transfer of large-scale vortices.     

Development of slow disturbances in a hypersonic boundary layer

The mechanisms of development of slow time-dependent disturbances in the wall region of a hypersonic boundary layer are established and a diagram of the disturbed flow patterns is plotted; the corresponding nonlinear boundary value problem is formulated for each of these regimes. It is shown that the main factors that form the disturbed flow are the gas enthalpy near the body surface, the local viscous-inviscid interaction level, and the type, either subsonic or supersonic, of the boundary layer as a whole. Numerical and analytical solutions are obtained in the linear approximation. It is established that enhancement of the local viscous-inviscid interaction or an increased role for the main supersonic region of the boundary layer makes the disturbed flow by and large “supersonic”: the upstream propagation of the disturbances becomes weaker, while their downstream growth is amplified. Contrariwise, local viscous-inviscid interaction attenuation or an increased role for the main subsonic region of the boundary layer has the opposite effect. Surface cooling favors an increased effect of the main region of the boundary layer while heating favors an increased wall region effect. It is also found that in the regimes considered disturbances travel from the turbulent flow region downstream of the disturbed region under consideration counter to the oncoming flow, which may be of considerable significance in constructing the nonlinear stability theory.     

Is Helmholtz Resonance a Problem for Micro-jet Actuators?

A theoretical analysis is described that determines the conditions for Helmholtz resonance for a popular class of self-contained microjet actuator used in both synthetic- and pressure-jump (pulse-jet) mode. It was previously shown that the conditions for Helmholtz resonance are identical to those for optimizing actuator performance for maximum mass flux. The methodology is described for numerical-simulation studies on how Helmholtz resonance affects the interaction of active and nominally inactive micro-jet actuators with a laminar boundary layer. Two sets of numerical simulations were carried out. The first set models the interaction of an active actuator with the boundary layer. These simulations confirm that our criterion for Helmholtz resonance is broadly correct. When it is satisfied we find that the actuator cannot be treated as a predetermined wall boundary condition because the interaction with the boundary layer changes the pressure difference across the exit orifice thereby affecting the outflow from the actuator. We further show that strong inflow cannot be avoided even when the actuator is used in pressure-jump mode. In the second set of simulations two-dimensional Tollmien–Schlichting waves, with frequency comparable with, but not particularly close to, the Helmholtz resonant frequency, are incident on a nominally inactive micro-jet actuator. The simulations show that under these circumstances the actuators act as strong sources of 3D Tollmien–Schlichting waves. It is surmised that in the real-life aeronautical applications with turbulent boundary layers broadband disturbances of the pressure field, including acoustic waves, would cause nominally inactive actuators, possibly including pulsed jets, to act as strong disturbance sources. Should this be true it would probably be disastrous for engineering applications of such massless microjet actuators for flow control.     

A Combined Experimental/Numerical Study of Unsteady Phenomena in a Laminar Separation Bubble

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.     

On two-dimensional linear waves in Blasius boundary layer over viscoelastic layers

This work concerns the direct numerical simulation of small-amplitude two-dimensional ribbon-excited waves in Blasius boundary layer over viscoelastic compliant layers of finite length. A vorticity-streamfunction formulation is used, which assures divergence-free solutions for the evolving flow fields. Waves in the compliant panels are governed by the viscoelastic Navier's equations. The study shows that Tollmien–Schlichting (TS) waves and compliance-induced flow instability (CIFI) waves that are predicted by linear stability theory frequently coexist on viscoelastic layers of finite length. In general, the behaviour of the waves is consistent with the predictions of linear stability theory. The edges of the compliant panels, where abrupt changes in wall property occur, are an important source of waves when they are subjected to periodic excitation by the flow. The numerical results indicate that the non-parallel effect of boundary-layer growth is destabilizing on the TS instability. It is further demonstrated that viscoelastic layers with suitable properties are able to reduce the amplification of the TS waves, and that high levels of material damping are effective in controlling the propagating CIFI.     

Boundary layer sensitivity and receptivitySensibilité et réceptivité d'une couche limite

The relation between the receptivity and the sensitivity of the incompressible flow in the boundary layer over a flat plate to harmonic perturbations is determined. Receptivity describes the birth of a disturbance, whereas sensitivity is a concept of larger breath, describing the modification incurred by the state of a system as a response to parametric variations. The governing equations ruling the system's state are the non-local stability equations. Receptivity and sensitivity functions can be obtained from the solution of the adjoint system of equations. An application to the case of Tollmien–Schlichting waves spatially developing in a flat plate boundary layer is studied. To cite this article: C. Airiau et al., C. R. Mecanique 330 (2002) 259–265.     

On the absolute instability in a boundary-layer flow with compliant coatings

The question of absolute instabilities occuring in a boundary-layer flow with compliant coatings is reassessed. Compliant coatings of the Kramer's type are considered. Performing a local, linear absolute/convective stability analysis, a family of spring-backed elastic plates with damping is shown to be absolutely unstable for sufficiently thin plates. The absolute instability arises from the coalescence between an upstream propagating evanescent mode and the Tollmien–Schlichting wave. To reinforce the local, linear stability results the global stability behaviour of the system is investigated, integrating numerically the full nonparallel and nonlinear two-dimensional Navier–Stokes system coupled to the dynamical model. Injecting Gaussian-type, spatially localized flow disturbances as initial conditions, the spatio-temporal evolution of wave packets is computed. The absolute stability behaviour is retrieved in the global system, for a compliant panel of finite length. It is demonstrated numerically that the global stability behaviour of the wall, triggered by finite-end-effects, may be independent of the disturbance propagation in the flow.