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.     

Swept-Wing Boundary Layer Receptivity to a Steady Free-Stream Vortex Disturbance

A single trailing vortex developed behind a micro-wing immersed in a free stream was used to study the vortex receptivity of a swept-wing boundary layer. As a result of the interaction, longitudinal-velocity disturbances develop in the boundary layer. On the swept wing, disturbance transformation occurs near the leading edge and is accompanied by the formation of a wave packet consisting of waves typical of cross-flow instability. Disturbances with other characteristics are also detected. These disturbances may be attributable to distributed boundary-layer receptivity to the free-stream vortex disturbance considered.     

等热流密度加热垂直下降液膜的研究

利用等热流密度加热条件下降膜流动的三维模型方程进行线性稳定性分析和数值模拟。线性稳定性分析表明,模型方程在小到中等Reynolds数下都适用,并且流向不稳定性增长率随着Reynolds数和Marangoni数增加而增加,展向不稳定性增长率则随着Marangoni数增加而增加,随着Reynolds数增加而减小,流向和展向对扰动波数都存在一个不稳定区间。三维数值模拟表明,在等热流密度加热条件下,液膜在随机扰动的情况下最终会形成带孤立波的三维溪流状结构,液膜与气体的换热也因溪流状结构的出现而加强;在随机扰动的基础上引入占优势地位的展向最不稳定扰动会使得换热增强,液膜会提前破裂;在随机扰动的基础上引入占优势地位的流向最不稳定扰动时,液膜的换热会增强,但不会提前破裂;在随机扰动的基础上同时引入占优势地位的流向和展向最不稳定扰动时,换热会加强且液膜会提前破裂。     

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.     

Excitation of unsteady Görtler vortices by localized surface nonuniformities

A combined theoretical and numerical analysis of an experiment devoted to the excitation of Görtler vortices by localized stationary or vibrating surface nonuniformities in a boundary layer over a concave surface is performed. A numerical model of generation of small-amplitude disturbances and their downstream propagation based on parabolic equations is developed. In the framework of this model, the optimal and the modal parts of excited disturbance are defined as solutions of initial-value problems with initial values being, respectively, the optimal disturbance and the leading local mode at the location of the source. It is shown that a representation of excited disturbance as a sum of the optimal part and a remainder makes it possible to describe its generation and downstream propagation, as well as to predict satisfactorily the corresponding receptivity coefficient. In contrast, the representation based on the modal part provides only coarse information about excitation and propagation of disturbance in the range of parameters under investigation. However, it is found that the receptivity coefficients estimated using the modal parts can be reinterpreted to preserve their practical significance. A corresponding procedure was developed. The theoretical and experimental receptivity coefficients are estimated and compared. It is found that the receptivity magnitudes grow significantly with the disturbance frequency. Variation of the span-wise scale of the nonuniformities affects weakly the receptivity characteristics at zero frequency. However, at high frequencies, the efficiency of excitation of Görtler vortices depends substantially on the span-wise scale.     

Receptivity of hypersonic boundary layer due to fast-slow acoustics interaction

The objective of receptivity is to investigate the mechanisms by which external disturbances generate unstable waves. In hypersonic boundary layers, a new receptivity process is revealed, which is that fast and slow acoustics through nonlinear interaction can excite the second mode near the lower-branch of the second mode. They can generate a sum-frequency disturbance though nonlinear interaction,which can excite the second mode. This receptivity process is generated by the nonlinear interaction and the nonparallel nature of the boundary layer. The receptivity coefficient is sensitive to the wavenumber difference between the sumfrequency disturbance and the lower-branch second mode.When the wavenumber difference is zero, the receptivity coefficient is maximum. The receptivity coefficient decreases with the increase of the wavenumber difference. It is also found that the evolution of the sum-frequency disturbance grows linearly in the beginning. It indicates that the forced term generated by the sum-frequency disturbance resonates with the second mode.     

Receptivity of the Blunt-Leading-Edge Plate Boundary Layer to Unsteady Vortex Perturbations

The response of the boundary layer on a plate with a blunt leading edge to frozen-in vortex perturbations whose vorticity is normal to the plate surface is found. It is shown that these vortices generate an inhomogeneity of the streamwise velocity component in the boundary layer. This inhomogeneity is analogous to the streaky structure developing as the degree of free-stream turbulence increases. The dependence of the amplitude and shape of the boundary layer inhomogeneity on the distance from the leading edge, the streamwise and spanwise scales, and other parameters is found for periodic and local initial perturbations. It is shown that the receptivity of the boundary layer decreases with increase in the frequency and with decrease in the streamwise perturbation scale.     

Receptivity of boundary layers to three-dimensional disturbances

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 β_H=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.     

Effect of streamwise and spanwise electric fields on transient growth in a two-fluid channel flow

A linear model of a two-fluid channel flow under streamwise/spanwise electric field is built. Both the fluids are assumed to be incompressible, viscous and perfectly dielectric. The effect of the streamwise and spanwise electric fields on transient behavior of small three-dimensional disturbances is studied. The numerical result shows that the streamwise electric field suppresses transient growth of the disturbance with spanwise uniform wave number. The spanwise electric field diminishes transient growth of the disturbance with streamwise uniform wave number. Two peaks of optimal growth are detected in the wave number plane. The peak at relatively large spanwise wave number is dominated by the lift-up mechanism and little influenced by electric field. Differently, the peak at relatively small wave number is associated with the characteristic of the interface and possibly influenced by electric field. The effect of the Weber number, the Reynolds number and the relative electrical permittivity on optimal growth is studied as well. A scaling law is obtained for relatively small Weber numbers and relatively large Reynolds numbers.     

Boundary-layer receptivity due to distributed surface imperfections of a deterministic or random nature

Acoustic receptivity of a Blasius boundary layer in the presence of distributed, two-dimensional surface irregularities is investigated analytically. It is shown that, out of the entire spatial spectrum of the surface irregularities, only a narrow band of Fourier components can lead to an efficient conversion of the acoustic input at any given frequency to an unstable eigenmode of the boundary-layer flow. The location and the width of this most receptive band of wave numbers is fixed by the requirement of a relative detuning of O(R _inf1.b. ^sup–3/8 ) or less with respect to the instability wave number at the lower-branch station for the frequency under consideration. Surface imperfections in the form of discrete-mode waviness in this range of wave numbers then lead to initial instability amplitudes which are larger by a factor of O(R _inf1.b. ^sup3/8 ) than the amplitudes resulting from a single, isolated roughness element of streamwise extent comparable with the instability wavelength at the lower-branch location. In contrast, random irregularities which are spatially homogeneous in nature, and also possess a continuous spectrum in the streamwise direction, lead to instability amplitudes that are intermediate to those caused by the periodic and isolated irregularities, respectively, being, in fact, of the same order as the geometrical mean of the amplitudes in the latter two cases. A physical explanation for these asymptotic scalings is given, in addition to providing an analytical expression for the expected value of the instability amplitude for an ensemble of statistically irregular surfaces with random phase distributions. The duality between the localized and distributed receptivity analyses is also discussed.Financial support for this work was provided by the Theoretical Flow Physics Branch, Fluid Mechanics Division, NASA Langley Research Center, Hampton, VA, under contract NAS1-19299.     

Flat-Plate Boundary Layer Receptivity to a Steady Free-Stream Vortex Disturbance

A single trailing vortex developed behind a micro-wing immersed in a free stream was used to study the vortex receptivity of the boundary layer on a flat plate. As a result of the interaction, in the boundary layer there develop longitudinal-velocity disturbances which grow almost linearly in the longitudinal coordinate. The parameters of the excited steady disturbances agree with the data of previous experiments performed under natural conditions and dealing with an indirect scenario of laminar-turbulent transition at high free-stream turbulence. It is shown that the leading edge of the plate does not play a decisive role in the mechanism of growth of disturbances of this kind and the receptivity is non-local in nature.     

Unsteady compressible boundary layer flow over a circular cone near a plane of symmetry

An analysis has been performed to study the unsteady laminar compressible boundary layer governing the hypersonic flow over a circular cone at an angle of attack near a plane of symmetry with either inflow or outflow in the presence of suction. The flow is assumed to be steady at time t=0 and at t>0 it becomes unsteady due to the time-dependent free stream velocity which varies arbitrarily with time. The nonlinear coupled parabolic partial differential equations under boundary layer approximations have been solved by using an implicit finite-difference method. It is found that suction plays an important role in stabilising the fluid motion and in obtaining unique solution of the problem. The effect of the cross flow parameter is found to be more pronounced on the cross flow surface shear stress than on the streamwise surface shear stress and surface heat transfer. Beyond a certain value of the cross flow parameter overshoot in the cross flow velocity occurs and the magnitude of this overshoot increases with the cross flow parameter. The time variation of the streamwise surface shear stress is more significant than that of the cross flow surface shear stress and surface heat transfer. The suction and the total enthalpy at the wall exert strong influence on the streamwise and cross flow surface shear stresses and the surface heat transfer except that the effect of suction on the cross flow surface shear stress is small.     

Parametric study of pulsed thermal bumps in supersonic boundary layer

A three-dimensional numerical study is performed to explore the effect of pulsed spanwise-periodic surface thermal perturbation (also denoted as thermal bump) in a Mach 1.5 flat plate laminar boundary layer. A high-resolution upwind-biased Roe method is used with the compressive Van Leer harmonic limiter on a suitably refined mesh. The dependence of flow stability characteristics on the variation of thermal bump geometry (shape and dimension) and pulsing properties (disturbance amplitude and frequency) is assessed. It is shown that the finite-span thermal bumps generate streamwise vortices. When the thermal bump is pulsed, vortex shedding is observed, and the streamwise vorticity grows with the downstream distance. Analysis of the integrated disturbance energy indicates that the streamwise kinetic disturbance energy dominates over those associated with other two velocity and thermodynamic components. Immediately downstream of the bump, the dominant frequency corresponds to that of the imposed excitation while higher harmonic components are observed farther downstream. An analysis of parametric variation of bump shape and dimension indicates that finite bump span is important in injecting three dimensionality and that the rectangular shape results in faster disturbance growth than the circular one. The study also concludes that disturbance growth is non-linear with bump temperature and has a strong connection with pulsing frequency.     

Higher order stability effects in a natural convection boundary layer over a vertical heated wall

 The stability of a laminar boundary layer flow under natural convection on a vertical isothermally heated wall is studied analytically. The analysis is performed by using two different two-dimensional linear models: (1) The non-parallel flow model in which the steady mean flow as well as the disturbance amplitude functions can change in the streamwise direction; (2) The parallel flow model in which the effects of the mean flow and disturbance changes in the streamwise direction are neglected. The linear non-parallel stability analysis is based on the so-called parabolised stability equations (PSEs) which have been successfully applied to the stability analysis of forced convection boundary layers. In this study the PSE equations are applied to natural convection boundary layers in order to show the difference between parallel and non-parallel stability analysis. A second part of this study deals with the effects of variable properties, which are always present in natural convection flows. They are analysed by an extended version of the Orr–Sommerfeld equation (EOSE). Received on 31 May 2000     

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.     

Three-dimensional receptivity of boundary layers

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

Bypass transition receptivity modes

The prediction of bypass transition remains an important problem in many engineering applications. This is largely because there is no suitable theoretical model for bypass transition and predictions are made using empirical models. This paper presents numerical results for the receptivity of a zero pressure gradient boundary layer subjected to simple freestream waveforms which are the constituent parts of a turbulent flow field. Significant receptivities are only obtained for a minority of freestream waveforms and these lead to two types of flow structure in the boundary layer. The first type of flow structure is essentially two dimensional in nature and consists of two rows of counter-rotating spanwise vortices and is induced by freestream waves of large normal and spanwise wavelength and streamwise wavelengths approximately equal to the boundary layer thickness. The second type of flow structure are the streamwise streaks frequently observed in flow visualisation experiments. These streaks are induced by freestream waves of long streamwise and normal wavelength and spanwise wavelengths in the range of 14.5-46 θ (1.7-5.4δ). The freestream waves can be formed of velocity components in any direction, however the boundary layer is most receptive to fluctuations that lie in a plane perpendicular to the streamwise direction. The overall receptivity to a full spectrum of waves typical of freestream turbulence is considered and is shown to have similar characteristics to those from experiments.     

Pulse-width modulated blowing/suction as a flow control actuator

The effect of pulse-width modulated (PWM) blowing/suction through a hole in a flat plate, above which a Blasius boundary layer develops, is studied. A hot wire is used to measure the streamwise velocity downstream of the hole. The PWM blowing/suction is found to create amplitude-modulated variations in the streamwise velocity, which are well correlated with the signal generating the disturbance.     

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.     

Interaction between an unsteady pressure disturbance and a hypersonic boundary layer

The development of disturbances in a hypersonic boundary layer on a cooled surface is investigated in the case in which the characteristic velocity of disturbance propagation is small but greater than the flow velocity in the wall region of the three-layer disturbed zone with interaction. The nonlinear boundary value problem formulated involves a single similarity parameter that characterizes the contribution made by the main, on average either subsonic or supersonic, region of the boundary layer to the generation of the pressure disturbance. In the linear approximation, an analytical solution and an algebraic dispersion equation are derived. It is shown that only waves exponential in time and in the streamwise coordinate can propagate downstream when themain region of the undisturbed boundary layer is subsonic on average.