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.     

Disturbance evolution in the leading-edge region of a blunt flat plate in supersonic flow

The spatio-temporal dynamics of small disturbances in viscous supersonic flow over a blunt flat plate at freestream Mach number M_∞=2.5 is numerically simulated using a spectral approximation to the Navier–Stokes equations. The unsteady solutions are computed by imposing weak acoustic waves onto the steady base flow. In addition, the unsteady response of the flow to velocity perturbations introduced by local suction and blowing through a slot in the body surface is investigated. The results indicate distinct disturbance/shock-wave interactions in the subsonic region around the leading edge for both types of forcing. While the disturbance amplitudes on the wall retain a constant level for the acoustic perturbation, those generated by local suction and blowing experience a strong decay downstream of the slot. Furthermore, the results prove the importance of the shock in the distribution of perturbations, which have their origin in the leading-edge region. These disturbance waves may enter the boundary layer further downstream to excite instability modes.     

Optimal Control of a Vortex Trapped by an Airfoil with a Cavity

An airfoil with a cavity traps a vortex; the lift increases but the vortex shows great receptivity to upstream disturbances. A simple potential flow model confirms that the vortex stability basin is of a reduced extent. In this paper we present a control technique stabilizing the vortex position based on a potential flow model. The actuators are sources/sinks at the wall and the suction/blowing law is obtained by the adjoint optimization method. This revised version was published online in July 2006 with corrections to the Cover Date.     

波纹壁对高超声速平板边界层稳定性的影响

高超声速边界层转捩会使飞行器表面热流和摩阻增加3～5倍,极大影响高超声速飞行器的性能.波纹壁作为一种可能的推迟边界层转捩的被动控制方法,具有较强的工程应用前景.文章研究了不同高度和安装位置的波纹壁对来流马赫数6.5的平板边界层稳定性的影响.采用直接数值模拟(DNS)得到层流场,并在上游分别引入不同频率的吹吸扰动以研究波纹壁对扰动演化的作用.对于不同位置的波纹壁,探究了其与同步点相对位置对其作用效果的影响,与相同工况下光滑平板的扰动演化结果进行了对比,发现当快慢模态同步点位于波纹壁上游时,波纹壁会对该频率的第二模态扰动起到抑制作用.当同步点位于波纹壁之中或者下游时,波纹壁对扰动的作用可能因为存在两种不同的机制而使得结果较为复杂.对于不同高度波纹壁,发现高度较低的波纹壁,其作用效果强弱与波纹壁高度成正相关,而更高的波纹壁则会减弱其作用效果.与DNS结果相比,线性稳定性理论可以定性预测波纹壁对高频吹吸扰动的作用,但在波纹壁附近的强非平行性区域误差较大.     

Non-Parallel Acoustic Receptivity of a Blasius Boundary Layer Using an Adjoint Approach

Localized and non-localized acoustic receptivity for a Blasius boundary layer is investigated using the adjoint Parabolized Stability Equations. The scattering of an acoustic wave onto a hump, a rectangular roughness or a wall steady blowing and suction is described. Comparisons with local approaches, triple deck theory, direct numerical simulations and experiments are successfully shown. Non-parallel effects are discussed. For comparable parameters, the non-localized receptivity problem produces amplitudes one order of magnitude larger than for the case of localized receptivity. This revised version was published online in July 2006 with corrections to the Cover Date.     

On the Control of the Fastest Growing Görtler Vortex

The response of the fastest growing linear G?rtler vortex to wall suction or blowing is investigated. The fastest growing mode occurs in flows of high local curvature and we show how its growth may be inhibited using very small amounts of suction or blowing. We show how the suction or blowing should be distributed in order to minimize the growth of a disturbance over a specified interval. Received 3 July 1996 and accepted 10 November 1998     

Sensitivity Analysis Using Adjoint Parabolized Stability Equations for Compressible Flows

An input/output framework is used to analyze the sensitivity of two- and three-dimensional disturbances in a compressible boundary layer for changes in wall and momentum forcing. The sensitivity is defined as the gradient of the kinetic disturbance energy at a given downstream position with respect to the forcing. The gradients are derived using the parabolized stability equations (PSE) and their adjoint (APSE). The adjoint equations are derived in a consistent way for a quasi-two-dimensional compressible flow in an orthogonal curvilinear coordinate system. The input/output framework provides a basis for optimal control studies. Analysis of two-dimensional boundary layers for Mach numbers between 0 and 1.2 show that wall and momentum forcing close to branch I of the neutral stability curve give the maximum magnitude of the gradient. Forcing at the wall gives the largest magnitude using the wall normal velocity component. In case of incompressible flow, the two-dimensional disturbances are the most sensitive ones to wall inhomogeneity. For compressible flow, the three-dimensional disturbances are the most sensitive ones. Further, it is shown that momentum forcing is most effectively done in the vicinity of the critical layer. This revised version was published online in July 2006 with corrections to the Cover Date.     

Viscoelastic fluid flow impinging on a wall with suction or blowing

The oblique flow of a viscoelastic fluid impinging on a porous wall with suction or blowing is studied. It is found that when suction is applied the fluid penetrates the wall while blowing causes the shifting of the stagnation point. It is also found that this shifting depends upon the magnitude of the blowing and upon the Weissenberg number.     

Numerical modeling of the disturbances of the separated flow in a rounded compression corner

Numerical modeling of the time-dependent supersonic flow over a compression corner with different roundness radii is performed on the basis of the solution of the two-dimensional Navier-Stokes equations in the regimes corresponding to local boundary layer separation. The development of unstable disturbances generated by local periodic injection/suction in the preseparated boundary layer is calculated. The results are compared with those of similar calculations for a flat plate. It is shown that the natural oscillations of the boundary-layer second mode stabilize in the separation zone and grow intensely downstream of the reattachment point. The acoustic modes excited within a separation bubble are studied using numerical calculations and an asymptotic analysis.     

Receptivity mechanisms for Görtler vortex modes

The receptivity problem for Görtler vortices induced by wall roughness or freestream disturbances is reviewed. To date, receptivity studies for this problem have been exclusively linear in character and here we show how the roughness and freestream disturbance mechanisms can each play dominant or inconsequential roles as possible routes to transition. The importance of each process is dependent on the exact situation at hand. For example, distributed wall-roughness elements tend to be more important in the generation of O(1) wave-number vortices than are isolated roughness patches whilst variations in the freestream velocity can easily provoke high wave-number disturbances on which roughness distributions typically have negligible effect.It has only been in recent times that the influence a spanwise component of the underlying basic boundary-layer flow may have on the Görtler mechanism has come to be appreciated. In some new computations we show that the imposition of such a spanwise component can lead to an increase in the coupling coefficient (a measure of the efficiency of a generating process) for modes provoked by wall roughness. However, such crossflow tends to reduce the global amplification rate of the most unstable mode so has the overall effect of restricting vortex growth downstream of any roughness element. We also demonstrate how the nonparallelism of Görtler vortices implies that conclusions concerning vortex receptivity properties can only be drawn after taking full account of upstream conditions and the precise form of the generating mechanism but it appears that for a large class of flows distributed wall forcing is more important in the provocation of modes than are either isolated roughness or freestream disturbances.This work was completed whilst APB was on study leave at the School of Mathematics, University of New South Wales, Sydney. He is indebted to the Royal Society of London and the Australian Research Council without whose grants (the latter to Dr. Peter Blennerhassett) his visit would not have been possible. In addition he is grateful to the staff and students of New College, UNSW for their provision of a Visiting Fellowship and to the School of Mathematics for their hospitality.     

Three-to-one internal resonance in multiple stepped beam systems

In this study, the vibrations of multiple stepped beams with cubic nonlinearities are considered. A three-to-one internal resonance case is investigated for the system. A general approximate solution to the problem is found using the method of multiple scales （a perturbation technique）. The modulation equations of the amplitudes and the phases are derived for two modes. These equations are utilized to determine steady state solutions and their stabilities. It is assumed that the external forcing frequency is close to the lower frequency. For the numeric part of the study, the three-to-one ratio in natural frequencies is investigated. These values are observed to be between the first and second natural frequencies in the cases of the clamped-clamped and clamped-pinned supports, and between the second and third natural frequencies in the case of the pinned-pinned support. Finally, a numeric algorithm is used to solve the three-to-one internal resonance. The first mode is externally excited for the clamped-clamped and clamped-pinned supports, and the second mode is externally excited for the pinned-pinned support. Then, the amplitudes of the first and second modes are investigated when the first mode is externally excited. The amplitudes of the second and third modes are investigated when the second mode is externally excited. The force-response, damping-response, and .frequency- response curves are plotted for the internal resonance modes of vibrations. The stability analysis is carried out for these plots.     

Distributed forcing of the flow past a blunt-based axisymmetric bluff body

In this paper, we address the influence of a blowing-/suction-type distributed forcing on the flow past a blunt-based axisymmetric bluff body by means of direct numerical simulations. The forcing is applied via consecutive blowing and suction slots azimuthally distributed along the trailing edge of the bluff body. We examine the impact of the forcing wavelength, amplitude and waveform on the drag experienced by the bluff body and on the occurrence of the reflectional symmetry preserving and reflectional symmetry breaking wake modes, for Reynolds numbers 800 and 1,000. We show that forcing the flow at wavelengths inherent to the unforced flow drastically damps drag oscillations associated with the vortex shedding and vorticity bursts, up to their complete suppression. The overall parameter analysis suggests that this damping results from the surplus of streamwise vorticity provided by the forcing that tends to stabilize the ternary vorticity lobes observed at the aft part of the bluff body. In addition, conversely to a blowing-type or suction-type forcing, the blowing-/suction-type forcing involves strong nonlinear interactions between locally decelerated and accelerated regions, severely affecting both the mean drag and the frequencies representative of the vortex shedding and vorticity bursts.     

Effect of local forcing on a turbulent boundary layer

An experimental study is performed to analyze flow structures behind local suction and blowing in a flat-plate turbulent boundary layer. The local forcing is given to the boundary layer flow by means of a sinusoidally oscillating jet issuing from a thin spanwise slot at the wall. The Reynolds number based on the momentum thickness is about Re _θ =1700. The effects of local forcing are scrutinized by altering the forcing frequency (0.011 ≤ f ⁺≤ 0.044). The forcing amplitude is fixed at A ₀=0.4. It is found that a small local forcing reduces the skin friction and the skin friction reduction increases with the forcing frequency. A phase-averaging technique is employed to capture the large-scale vortex evolution. An organized spanwise vortical structure is generated by the local forcing. The cross-sectional area of vortex and the time fraction of vortex are examined by changing the forcing frequency. An investigation of the random fluctuation components reveals that turbulent energy is concentrated near the center of vortical structures. Received: 17 March 2000/Accepted: 3 April 2001     

Heat and mass transfer in a visco–elastic fluid flow over an accelerating surface with heat source/sink and viscous dissipation

 In this paper we present a mathematical analysis of heat and mass transfer phenomena in a visco–elastic fluid flow over an accelerating stretching sheet in the presence of heat source/sink, viscous dissipation and suction/blowing. Similarity transformations are used to convert highly non-linear partial differential equations into ordinary differential equations. Several closed form analytical solutions for non-dimensional temperature, concentration, heat flux, mass flux profiles are obtained in the form of confluent hypergeometric (Kummer's) functions for two different cases of the boundary conditions, namely, (i) wall with prescribed second order power law temperature and second order power law concentration (PST), and (ii) wall with prescribed second order power law heat flux and second order power law mass flux (PHF). The effect of various physical parameters like visco–elasticity, Eckert number, Prandtl number, heat source/sink, Schmidt number and suction/blowing parameter on temperature and concentration profiles are analysed. The effects of all these parameters on wall temperature gradient and wall concentration gradient are also discussed. Received on 23 March 2000 / Published online: 29 November 2001     

基于定常吹吸气的波浪型圆柱主动控制研究

基于定常吹吸气对波浪型圆柱近尾迹流动进行控制以增强柱体振动, 采用大涡模拟研究了亚临界雷诺数(Re = 3000)下前吹后吸和前后吸气控制方式在不同吹吸气工况对波浪型圆柱升阻力特性、时均压力系数、环量、湍动能及近尾迹流动结构的影响. 研究发现: 前吹后吸和前后吸气控制下波浪型圆柱在不同吹吸气动量系数工况脉动升力系数均显著提高, 最大较未受控直圆柱和波浪型圆柱分别提升高达636%和391%, 这主要可能归因于吹吸气控制使波浪型圆柱回流区变短, 高强度涡集中向钝体后方靠拢, 旋涡形成长度缩短, 展向涡流与顺流向涡流相互作用在波浪型圆柱下游形成的“肋状涡”变大变长, 近尾迹环量显著增大, 从而导致脉动升力系数增大, 这可能将诱导柱体产生更强的振动; 同时两种控制方式均改变了波浪型圆柱表面的压力分布, 由于在波浪型圆柱前驻点吹气使前端趋于流线型, 前吹后吸在不同吹吸气动量系数下波浪型圆柱的高压区减小, 但在后驻点吸气使得低压区增大, 而前后吸气在不同吹吸气动量系数下波浪型圆柱的高压区基本不变, 低压区增大. 研究结果可为低风速地区分布式风力俘能结构俘能效率提升提供基础理论支持.      

Numerical prediction of locally forced turbulent boundary layer

An unsteady numerical simulation was performed to analyze flow structure behind a local suction/blowing in a flat-plate turbulent boundary layer. The local forcing was given to the boundary layer flow by means of a sinusoidally oscillating jet. A version of the unsteady k––f_μ model [Fluid Dyn. Res. 26 (6) (2000) 421] was employed. The Reynolds number based on the momentum thickness was about Re_θ=1700. The forcing frequency was varied in the range 0.011f⁺0.044 with a fixed forcing amplitude A_o=0.4. The predicted results were compared and validated with the experimental data. It was shown that the unsteady locally forced boundary layer flow is predicted well by the k––f_μ model. The time-dependent numerical flow visualizations were demonstrated during one period of the local forcing. The effect of the pitch angle of local forcing on the reduction of skin friction was examined.     

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.     

Instability and Receptivity of Laminar Wall Jets

Results of eigenvalue analysis based on global and local eigenvalue considerations are presented. A collocation method with the Chebyshev polynomial approximation has been used for the global eigenvalue analysis. The results explain the appearance of a second unstable mode. In the case of real frequencies with Reynolds number R < 381 there is only one unstable mode. This mode coalesces at R≈ 381 with a stable mode. At R > 381 they become separated by interchannging their branches, then the second unstable mode occurs. The receptivity problem has been considered with respect to perturbations emanating from a wall. The results illustrate that high-frequency modes have a stronger response than low-frequency modes. It is shown that the method of expansion in a biorthogonal eigenfunction system and the method used by Ashpis and Reshotko are equivalent with regard to the receptivity problem solution. Received: 29 March 1996 and accepted 26 November 1996     

激波与转捩边界层干扰非定常特性数值分析

激波与边界层干扰的非定常问题是高速飞行器气动设计中基础研究内容之一.以往研究主要针对层流和湍流干扰,在分离激波低频振荡及其内在机理方面存在着上游机制和下游机制两类截然不同的理论解释.分析激波与转捩边界层干扰下非定常运动现象有助于进一步加深理解边界层状态以及分离泡结构对低频振荡特性的影响规律,为揭示其产生机理指出新的方向.采用直接数值模拟方法对来流马赫数2.9,24?压缩拐角内激波与转捩边界层干扰下激波的非定常运动特性进行了数值分析.通过在拐角上游平板特定的流向位置添加吹吸扰动激发流动转捩,使得进入拐角的边界层处于转捩初期阶段.在验证了计算程序可靠性的基础上,详细分析了转捩干扰下激波运动的间歇性和振荡特征,着重研究了分离泡展向三维结构对激波振荡特性的影响规律,最后还初步探索了转捩干扰下激波低频振荡产生的物理机制.研究结果表明:分离激波的非定常运动仍存在强间歇性和低频振荡特征,其时间尺度约为上游无干扰区内脉动信号特征尺度的10倍量级;分离泡展向三维结构不会对分离激波的低频振荡特征产生实质影响.依据瞬态脉动流场的低通滤波结果,转捩干扰下激波低频振荡的诱因来源于拐角干扰区下游,与流场中分离泡的收缩/膨胀运动存在一定的关联.