Algebraic growth in boundary layers: optimal control by blowing and suction at the wall

The upstream perturbations that maximise the spatial energy growth in a boundary layer are called optimal perturbations. The optimal perturbations correspond to streamwise vortices and the downstream response corresponds to streamwise streaks.The aim of the present paper is to find a control by blowing and suction at the wall that zeros the energy of perturbation, when the initial disturbance is itself optimal. We shall also address the question: which kind of blowing and suction at the wall is most effective in controlling optimal disturbances?The problem is examined by a method of receptivity analysis based on a numerical solution of a system of equations adjoint to the linearised boundary layer equations. We shall investigate both cases of a flat and a concave wall.     

Experiments on localized disturbances in a flat plate boundary layer. Part 2. Interaction between localized disturbances and TS-waves

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

Near-field measurements and development of a new boundary layer over a flat plate with localized suction

Suction on a turbulent boundary layer is applied through a narrow strip in order to understand the effects suction can have on the boundary layer development and turbulent structures in the flow. Detailed two-component laser Doppler velocimetry (LDV) and laser-induced fluorescence (LIF) based measurements have been undertaken in regions close to the suction strip and further downstream. The region close to the strip involves a flow reversal accompanied by a change in sign for the Reynolds shear stress and strong gradients in the flow variables. The mean streamwise velocity after suction remains larger than its corresponding no-suction value. Relative to the no-suction case, the velocity fluctuations first decrease with suction followed by a slow recovery which may involve a slight overshoot. LIF visualizations indicate that compared to the no-suction case, the low-speeds streaks stay closer to the wall and exhibit a smaller amount of spanwise and wall-normal oscillations with suction. The visualization results are consistent with two-point velocity correlation measurements. The streamwise and spanwise correlation measurements indicate that the structures are disrupted or removed from the boundary layer due to suction suggesting that the original boundary layer has been strongly influenced by suction. The results are explained by the development of a new inner layer that forms downstream of the suction strip.     

Algebraic growth in a Blasius boundary layer: Nonlinear optimal disturbances

The three-dimensional, algebraically growing instability of a Blasius boundary layer is studied in the nonlinear regime, employing a nonparallel model based on boundary layer scalings. Adjoint-based optimization is used to determine the “optimal” steady leading-edge excitation that provides the maximum energy growth for a given initial energy. Like in the linear case, the largest transient growth is found for inlet streamwise vortices, that yield streamwise streaks downstream. Two different definitions of growth are employed, providing qualitatively similar results, although the spanwise wavenumbers of optimal growth differ by up to 20% in the two cases. The wavelength of the most amplified optimal disturbance increases with the initial amplitude. For large input amplitudes, significant deformations of the mean velocity field are found; in such cases it is reasonable to expect that nonlinear streaks may break down through a secondary instability.     

Velocity statistics and flow structures observed in bypass transition using stereo PTV

It is known from smoke visualizations that in a transitional boundary layer subjected to free-stream turbulence, streaks appear and eventually break down to turbulence after wavy motions. In order to observe the streaky structures directly, a stereo particle-tracking velocimetry system using hydrogen bubbles in a water channel has been developed and validated against laser Doppler velocimetry. Mean flow statistics show good agreement with previous results. With the developed measurement system, the instantaneous spanwise distribution of the streamwise and wall-normal velocities can be measured fast enough to resolve the time development of the streaky structures. Measurements of instantaneous spanwise distributions of the streamwise and wall-normal velocity disturbances show strong negative correlation between the wall-normal and streamwise velocities in the streaks. Published online: 19 November 2002     

Experiments on localized disturbances in a flat plate boundary layer. Part 1. The receptivity and evolution of a localized free stream disturbance

The receptivity of a laminar boundary layer to free stream disturbances has been experimentally investigated through the introduction of deterministic localized disturbances upstream of a flat plate mounted in a wind tunnel. Hot-wire measurements indicate that the spanwise gradient of the normal velocity component (and hence the streamwise vorticity) plays an essential role in the transfer of disturbance energy into the boundary layer. Inside the laminar boundary layer the disturbances were found to give rise to the formation of longitudinal structures of alternating high and low streamwise velocity. Similar streaky structures exist in laminar boundary layers exposed to free stream turbulence, in which the disturbance amplitude increases in linear proportion to the displacement thickness. In the present study the perturbation amplitude of the streaks was always decaying for the initial amplitudes used, in contrast to the growing fluctuations that are observed in the presence of free stream turbulence. This points out the importance of the continuous influence from the free stream turbulence along the boundary layer edge.     

Instantaneous fluctuation velocity and skewness distributions upstream of transition onset

The development of streamwise orientated disturbances through the boundary layer thickness prior to transition onset for zero-pressure gradient boundary layer flow under the influence %Tu = 4.2 is presented. The analysis concentrates on the development of the maximum positive and negative of the fluctuation velocity in order to gain further insight into the transition process. The average location of the peak negative fluctuation velocity over a range of Reynolds numbers was measured in the upper portion of the boundary layer at y/δ ≈ 0.6, whereas the location of the peak positive value was measured at y/δ ≈ 0.3. The disturbance magnitude of the negative fluctuation velocity increased beyond that of the positive as transition onset approached. The distribution and disturbance magnitude of the maximum positive and negative fluctuation velocities indicate that the initiation of transition may occur on the low-speed components of the flow that are lifted up to the upper region of the boundary layer. This is in qualitative agreement with recent direct numerical simulations on the breakdown of the flow on the lifted low-speed streaks near the boundary layer edge. The results presented in this investigation also demonstrate the increased physical insight gained by examining the distributions of the maximum positive and negative of the streamwise fluctuation velocity component associated with the low- and high-speed streaks, compared to time-averaged values, in determining what structures cause the breakdown to turbulence.     

A vorticity stretching diagnostic for turbulent and transitional flows

Vorticity stretching in wall-bounded turbulent and transitional flows has been investigated by means of a new diagnostic measure, denoted by Γ, designed to pick up regions with large amounts of vorticity stretching. It is based on the maximum vorticity stretching component in every spatial point, thus yielding a three-dimensional scalar field. The measure was applied in four different flows with increasing complexity: (a) the near-wall cycle in an asymptotic suction boundary layer (ASBL), (b) K-type transition in a plane channel flow, (c) fully turbulent channel flow at Re _τ = 180 and (d) a complex turbulent three-dimensional separated flow. Instantaneous data show that the coherent structures associated with intense vorticity stretching in all four cases have the shape of flat ‘pancake’ structures in the vicinity of high-speed streaks, here denoted ‘h-type’ events. The other event found is of ‘l-type’, present on top of an unstable low-speed streak. These events (l-type) are further thought to be associated with the exponential growth of streamwise vorticity in the turbulent near-wall cycle. It was found that the largest occurrence of vorticity stretching in the fully turbulent wall-bounded flows is present at a wall-normal distance of y ⁺?=?6.5, i.e. in the transition between the viscous sublayer and buffer layer. The associated structures have a streamwise length of ~200–300 wall units. In K-type transition, the Γ-measure accurately locates the regions of interest, in particular the formation of high-speed streaks near the wall (h-type) and the appearance of the hairpin vortex (l-type). In the turbulent separated flow, the structures containing large amounts of vorticity stretching increase in size and magnitude in the shear layer upstream of the separation bubble but vanish in the backflow region itself. Overall, the measure proved to be useful in showing growing instabilities before they develop into structures, highlighting the mechanisms creating high shear region on a wall and showing turbulence creation associated with instantaneous separations.     

Anisotropy measurements in the boundary layer over a flat plate with suction

Laser Doppler velocity measurements are carried out in a turbulent boundary layer subjected to concentrated wall suction (through a porous strip). The measurements are taken over a longitudinal distance of 9× the incoming boundary layer thickness ahead of the suction strip. The mean and rms velocity profiles are affected substantially by suction. Two-point measurements show that the streamwise and wall-normal autocorrelations of the streamwise velocity are reduced by suction. It is found that suction alters the redistribution of the turbulent kinetic energy k between its components. Relative to the no-suction case, the longitudinal Reynolds stress contributes more to k than the other two normal Reynolds stresses; in the outer region, its contribution is reduced which suggests structural changes in the boundary layer. This is observed in the anisotropy of the Reynolds stresses, which depart from the non-disturbed boundary layer. With suction, the anisotropy level in the near-wall region appears to be stronger than that of the undisturbed layer. It is argued that the mean shear induced by suction on the flow is responsible for the alteration of the anisotropy. The variation of the anisotropy of the layer will make the development of a turbulence model quite difficult for the flow behind suction. In that respect, a turbulence model will need to reproduce well the effects of suction on the boundary layer, if the model is to capture the effect of suction on the anisotropy of the Reynolds stresses.     

Optimal disturbances above and upstream of a flat plate with an elliptic-type leading edge

Adjoint-based iterative methods are employed to compute linear optimal disturbances in a spatially growing boundary layer around an elliptic leading edge. The Lagrangian approach is used where an objective function is chosen and constraints are assigned. The optimisation problem is solved using power iterations combined with a matrix-free formulation, where the state is marched forward in time with a standard direct numerical simulation solver and backward with the adjoint solver until a chosen convergence criterion is fulfilled. We consider the global and, more relevant to receptivity studies, the upstream localised optimal initial condition leading to the largest possible energy amplification at time T. We find that the two-dimensional initial condition with the largest potential for growth is a Tollmien–Schlichting-like wave packet that includes the Orr mechanism and is located inside the boundary layer downstream of the leading edge. Three-dimensional optimal disturbances induce streaks by the lift-up mechanism. Requiring the optimal initial condition to be localised upstream of the plate enables us to better study the effects of the leading edge on the boundary layer receptivity mechanisms. Two-dimensional upstream disturbances are inefficient at triggering unstable eigenmodes, whereas three-dimensional disturbances induce streamwise streaks with significant growth.     

Subcritical spatial transition of swept Hiemenz flow

It is known from experimental investigations that the leading-edge boundary layer of a swept wing exhibits transition to turbulence at subcritical Reynolds numbers, i.e. at Reynolds numbers which lie below the critical Reynolds number predicted by linear stability theory. In the present work, we investigate this subcritical transition process by direct numerical simulations of a swept Hiemenz flow in a spatial setting. The laminar base flow is perturbed upstream by a pair of stationary counter-rotating vortex-like disturbances. This perturbation generates high- and low-speed streaks by a non-modal growth mechanism. Further downstream, these streaky structures exhibit a strong instability to secondary perturbations which leads to a breakdown to turbulence.The observed transition mechanism has strong similarities to by-pass transition mechanisms found for two-dimensional boundary layers. It can be shown that transition strongly depends on the amplitude of the primary perturbation as well as on the frequency of the secondary perturbation.     

Numerical simulation on evolution of subharmonic low-speed streaks in minimal channel turbulent flow

The evolution of low-speed streaks in the turbulent boundary layer of the minimum channel flow unit at a low Reynolds number is simulated by the direct numer- ical simulation （DNS） based on the standard Fourier-Chebyshev spectral method. The subharmonic sinuous （SS） mode for two spanwise-aligned low-speed streaks is excited by imposing the initial perturbations. The possibilities and the physical realities of the turbulent sustaining in the minimal channel unit are examined. Based on such a flow field environment, the evolution of the low-speed streaks during a cycle of turbulent sus- taining, including lift-up, oscillation, and breakdown, is investigated. The development of streamwise vortices and the dynamics of vortex structures are examined. The results show that the vortices generated from the same streak are staggered along the streamwise direction, while the vortices induced by different streaks tilt toward the normal direction due to the mutual induction effect. It is the spatial variations of the streamwise vortices that cause the lift-up of the streaks. By resolving the transport dynamics of enstrophy, the strength of the vortices is found to continuously grow in the logarithmic layer through the vortex stretching mechanism during the evolution of streaks. The enhancement of the vortices contributes to the spanwise oscillation and the following breakdown of the low-speed streaks.     

On Tollmien–Schlichting-like waves in streaky boundary layers

The linear stability of the boundary layer developing on a flat plate in the presence of finite-amplitude, 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 role in the observed stabilisation. The possible relevance of the streak stabilising role for applications in boundary layer transition delay is discussed.     

The Effect of Pressure Gradient on Boundary Layer Receptivity

This paper presents numerical results for the receptivity of three laminar boundary layers with zero (ZPG), adverse (APG) and favourable (FPG) pressure gradients. Each boundary layer is subjected to a series of simple freestream waveforms which can be considered as constituent parts of either an isotropic or a non-isotropic turbulent freestream. Each freestream waveform has a single frequency in each spatial direction and is divided into two mutually perpendicular components. The first component has a zero spanwise velocity and hence lies in the streamwise normal plane whereas the second component lies in a plane which is perpendicular both to this plane and the spatial frequency vector. High boundary layer receptivities are only obtained for a minority of these waveforms and so only the resulting flow structures for these waveforms are considered in detail. The dominant flow structures are identified as either Tollmien Schlichting (T-S) waves or streaky structures. The streaky structures can be induced by both freestream components, but the response to the second component, which results in streamwise vortices in the freestream, is considerably stronger and occurs over a much larger streamwise frequency range. The boundary layer is only receptive to a relatively narrow band of spanwise wavelengths ranging from approximately one to four times the local boundary layer thickness. The APG leads to receptivities which are more than double those for the FPG case. The ratio of the freestream fluctuation streamwise wavelength to the distance from the plate leading edge is identified as an important influential parameter for receptivity leading to streaks. Significant T-S activity is only observed for APG, but is also detected for ZPG.     

On the inviscid acoustic-mode instability of supersonic shear flows

The same methods used previously to study acoustic-mode instability in supersonic boundary layers are applied to free shear layers, and new calculations are made for boundary layers with cooling and suction. The objective is to obtain additional information about acoustic-mode instability, and to find what features of the instability are common to boundary layers and free shear flows. Acoustic modes exist whenever there is an embedded region of locally supersonic flow relative to the phase speed of the instability wave. Consequently, they can be found in boundary layers, wakes, and jets, but not in mixing layers unless the flow is confined. In this first part of a two-part paper, attention is directed principally to two-dimensional waves. The linear, inviscid stability theory is used to calculate spatial amplification rates at Mach number 3 for the sinuous and varicose modes of a single wake flow and a single jet flow, each made up of the same mixing-layer profile plus a central region of uniform flow. Along with sequences of sinuous and varicose unstable modes clearly identifiable as acoustic modes, both of these flows, unlike the boundary layer, have a lowest sinuous mode that is the most unstable. The unstable modes include both subsonic and radiating disturbances with large amplification rates. The latter phenomenon is also found for highly cooled boundary layers with suction. In these boundary layers, suction is generally stabilizing for nonradiating acoustic disturbances, but destabilizing for radiating disturbances.The work described in this paper was carried out at the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration (NASA). Support from the Aerodynamics Division of the Office of Aeronautics and Exploration Technology is gratefully acknowledged. A preliminary version of this paper was presented at the Fourth Symposium on Numerical and Physical Aspects of Aerodynamic Flows, California State University, Long Beach, CA, 16–19 January 1989.     

Elementary solutions for streaky structures in boundary layers with and without suction

The temporal evolutions of small, streamwise elongated disturbances in the asymptotic suction boundary layer (ASBL) and the Blasius boundary layer (BBL) are compared. In particular, initial perturbations localized (δ-functions) in the wall-normal direction are studied, corresponding to an axi-symmetric jet coming out of a plane parallel to the flat plate. Analytical solutions are presented for the wall-normal and streamwise velocities in the ASBL case whereas both analytical and numerical methods are used for the BBL case. The initial position of the perturbation and its spanwise wave number are varied in a parameter study. We present results of maximum amplitudes obtained, the time to reach them, their position and optimal spanwise scales. Free-stream disturbances are shown to migrate towards the wall and reach their (negative) optimum inside the boundary layer. The migration is faster for the ASBL case and a larger amplitude is reached than for the BBL. For perturbations originating inside the boundary layer the amplitudes are overall larger and show the phenomenon of overshoot, i.e. positive amplitudes moving out of the boundary layer. The overall largest amplitudes are obtained for the BBL case, as in other studies, but it is shown that for free-stream disturbances initiated somewhere downstream the leading edge streak growth may be amplified due to suction since in the BBL the disturbance mainly advects above the boundary layer.     

Response of the boundary layer developing over a blunt-nosed flat plate to free-stream non-uniformities

The response of the boundary layer on a flat plate with blunt nose to infinitesimally small non-uniformity in the freestream velocity along the span has been studied. The non-uniformity was shown to excite boundary-layer disturbances similar to streaks or Klebanoff modes generally observed in experiments conducted with a high level of free-stream turbulence. The boundary layer disturbances have a predominantly streamwise velocity component and exhibit transient growth. In contrast to streaks generated by streamwise vortices impinging on the sharp nose of a plate, the disturbances produced by free-stream non-uniformity interaction with a blunt nose have a different level of growth. Their maximal amplification scales with the Reynolds number, based on the size of nose bluntness and is almost independent of the spanwise period of disturbances. This difference was shown to be caused by additional amplification of disturbances via vortex lines stretching around the leading edge.     

Streamwise streaks and secondary instability in a two-dimensional bent channel

Streamwise streaks generated from a pair of oblique waves and secondary instability of the streaks are studied in a two-dimensional bent channel. Nonlinear parabolized stability equations (NPSE) are employed to investigate streamwise streaks and vortices. A pair of oblique waves from linear stability analysis is imposed as initial disturbances. Generation of streamwise streaks and vortices and subsequent development are described in detail. The case of plane channel is also studied to provide comparable data. Through comparison, the effect of bent region is clearly highlighted. Results of parametric studies to examine the effect of Reynolds number, radius of curvature, and bent angle are also given and discussed in detail. Secondary instability analysis for the modified mean flow due to the streamwise streaks is carried out by solving a two-dimensional eigenvalue problem. Several unstable modes which can be classified into fundamental and subharmonic mode of secondary instability are identified. Among several unstable modes, two modes are turned out to be dominant modes. Details on these two modes including generation mechanism, typical pattern, and dependency on wave number and streak amplitude are discussed. It is found that the presence of bent channel can lead to early oblique-mode breakdown via strong growth of the streamwise streaks due to the curved section. Such large amplitude of streaks and its secondary instability eventually could trigger transition even for small amplitude oblique waves at subcritical channel Reynolds numbers.     

The Onset and Nonlinear Development of Vortex Instabilities in a Horizontal Forced Convection Boundary Layer With Uniform Surface Suction

We consider the flow and heat transfer caused by a strong external flow passing over a hot surface with uniform surface suction. When the Péclet number based on the external velocity is sufficiently large, the resulting thermal boundary layer develops in a nonsimilar manner until it attains an asymptotic state which is independent of the streamwise coordinate, x, when it is dominated by the surface suction. For sufficiently large, but moderate, values of the Darcy–Rayleigh number this boundary layer becomes unstable to streamwise vortex disturbances. We employ a parabolic solver to determine how such disturbances, when placed very close to the leading edge, evolve with distance downstream. Neutral stability is then defined to be when a suitable energy functional ceases to decay/grow as x increases. Thus a neutral curve may be mapped out based upon the behaviour of this functional. Given that the uniform asymptotic state is well known to admit subcritical instabilities, our linearised analysis is extended into the nonlinear domain and the effect of different magnitudes of disturbance is ascertained. It is found that a surprisingly rich variety of vortex pattern emerges which is sometimes sensitively dependent on the values of the governing parameters. These patterns include wavy vortices and abrupt changes in perceived wavelength.     

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.