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.     

Turbulent boundary layer flow with a step change from smooth to rough surface

A direct numerical simulation (DNS) dataset of a turbulent boundary layer (TBL) with a step change from a smooth to a rough surface is analyzed to examine the characteristics of a spatially developing flow. The roughness elements are periodically arranged two-dimensional (2-D) spanwise rods, with the first rod placed 80θ_in downstream from the inlet, where θ_in denotes the inlet momentum thickness. Based on an accurate estimation of relevant parameters, clear evidence for mean flow universality is provided when scaled properly, even for the present roughness configuration, which is believed to have one of the strongest impacts on the flow. Compared to previous studies, it is shown that overshooting behavior is present in the first- and second-order statistics and is locally created either within the cavity or at the leading edge of the roughness depending on the type of statistics and the wall-normal measurement location. Inspection of spatial two-point correlations of the streamwise velocity fluctuations shows a continuous increase of spanwise length scales of structures over the rough wall after the step change at a greater growth rate than that over smooth wall TBL flow. This is expected because spanwise energy spectrum shows presence of much energetic wider structures over the rough wall. Full images of the DNS data are presented to describe not only predominance of hairpin vortices but also a possible spanwise scale growth mechanism via merging over the rough wall.     

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.     

Görtler Vortices with System Rotation

The steady primary instability of Görtler vortices developing along a curved Blasius boundary layer subject to spanwise system rotation is analysed through linear and nonlinear approaches, to clarify issues of vortex growth and wavelength selection, and to pave the way to further secondary instability studies.A linear marching stability analysis is carried out for a range of rotation numbers, to yield the (predictable) result that positive rotation, that is rotation in the sense of the basic flow, enhances the vortex development, while negative rotation dampens the vortices. Comparisons are also made with local, nonparallel linear stability results (Zebib and Bottaro, 1993) to demonstrate how the local theory overestimates vortex growth. The linear marching code is then used as a tool to predict wavelength selection of vortices, based on a criterion of maximum linear amplification.Nonlinear finite volume numerical simulations are performed for a series of spanwise wave numbers and rotation numbers. It is shown that energy growths of linear marching solutions coincide with those of nonlinear spatially developing flows up to fairly large disturbance amplitudes. The perturbation energy saturates at some downstream position at a level which seems to be independent of rotation, but that increases with the spanwise wavelength. Nonlinear simulations performed in a long (along the span) cross section, under conditions of random inflow disturbances, demonstrate that: (i) vortices are randomly spaced and at different stages of growth in each cross section; (ii) upright vortices are the exception in a universe of irregular structures; (iii) the average nonlinear wavelengths for different inlet random noises are close to those of maximum growth from the linear theory; (iv) perturbation energies decrease initially in a linear filtering phase (which does not depend on rotation, but is a function of the inlet noise distribution) until coherent patches of vorticity near the wall emerge and can be amplified by the instability mechanism; (v) the linear filter represents the receptivity of the flow: any random noise, no matter how strong, organizes itself linearly before subsequent growth can take place; (vi) the Görtler number, by itself, is not sufficient to define the state of development of a vortical flow, but should be coupled to a receptivity parameter; (vii) randomly excited Görtler vortices resemble and scale like coherent structures of turbulent boundary layers.A.Z. has been supported, during his stay at EPFL, by an ERCOFTAC Visitor Grant. A.B. acknowledges the Swiss National Fund, Grant No. 21-36035.92, for travel support associated with this research. This work was also supported by the Swedish Board of Technical Development (NUTEK), the Swedish Technical-Scientific Council (TFR), and an ERCOFTAC Visitor Grant, through which the stay of B.G.B.K. at the EPFL was made possible. Cray-2 computing time for this research was generously provided by the Service Informatique Centrale of EPFL.     

Distributed Roughness Effects on Transitional and Turbulent Boundary Layers

A numerical investigation is carried out to study the transition of a subsonic boundary layer on a flat plate with roughness elements distributed over the entire surface. Post-transition, the effect of surface roughness on a spatially developing turbulent boundary layer (TBL) is explored. In the transitional regime, the onset of flow transition predicted by the current simulations is in agreement with the experimentally based correlations proposed in the literature. Transition mechanisms are shown to change significantly with the increasing roughness height. Roughness elements that are inside the boundary layer create an elevated shear layer and alternating high and low speed streaks near the wall. Secondary sinuous instabilities on the streaks destabilize the shear layer promoting transition to turbulence. For the roughness topology considered, it is observed that the instability wavelengths are governed by the streamwise and spanwise spacing between the roughness elements. In contrast, the roughness elements that are higher than the boundary layer create turbulent wakes in their lee. The scale of instability is much shorter and transition occurs due to the shedding from the obstacles. Post-transition, in the spatially developing TBL, the velocity defect profiles for both the smooth and rough walls collapsed when non dimensionalized in the outer units. However, when compared to the smooth wall, deviation in the Reynolds stresses are observable in the outer layer; the deviation being higher for the larger roughness elements.     

Iterative control of Görtler vortices via local wall deformations

Görtler vortices develop along concave walls as a result of the imbalance between the centrifugal force and radial pressure gradient. In this study, we introduce a simple control strategy aimed at reducing the growth rate of Görtler vortices by locally modifying the surface geometry in spanwise and streamwise directions. Such wall deformations are accounted in the boundary region equations by using a Prandtl transform of dependent and independent variables. The vortex energy is then controlled via a classical proportional control algorithm for which either the wall-normal velocity or the wall shear stress serves as the control variable. Our numerical results indicate that the control algorithm is quite effective in minimizing the wall shear stress.     

Optimal disturbances in suction boundary layers

A well-known optimization procedure is used to find the optimal disturbances in two different suction boundary layers within the spatial framework. The maximum algebraic growth in the asymptotic suction boundary layer is presented and compared to previous temporal results. Furthermore, the spatial approach allows a study of a developing boundary layer in which a region at the leading edge is left free from suction. This new flow, which emulates the base flow of a recent wind-tunnel experiment, is herein denoted a semi-suction boundary layer. It is found that the optimal disturbances for these two suction boundary layers consist of streamwise vortices that develop into streamwise streaks, as previously found for a number of shear flows. It is shown that the maximum energy growth in the semi-suction boundary layer is obtained over the upstream region where no suction is applied. The result indicates that the spanwise scale of the streaks is set in this region, which is in agreement with previous experimental findings.     

Separation control by vortex generator devices in a transonic channel flow

An experimental study was conducted in a transonic channel to control by mechanical vortex generator devices the strong interaction between a shock wave and a separated turbulent boundary layer. Control devices—co-rotating and counter-rotating vane-type vortex generators—were implemented upstream of the shock foot region and tested both on a steady shock wave and on a forced shock oscillation configurations. The spanwise spacing of vortex generator devices along the channel appeared to be an important parameter to control the flow separation region. When the distance between each device is decreased, the vortices merging is more efficient to reduce the separation. Their placement upstream of the shock wave is determinant to ensure that vortices have mixed momentum all spanwise long before they reach the separation line, so as to avoid separation cells. Then, vortex generators slightly reduced the amplitude of the forced shock wave oscillation by delaying the upstream displacement of the leading shock.     

Turbulent time-events in channel with rough walls

This brief communication quantifies the time-events that contribute to the dynamics of wall-bounded flows with rough walls. Lumley’s Proper Orthogonal Decomposition (POD) methodology has been used to extract the energetic modes of the flow. We have used the concept of entropy, a representation of lack of organization in the flow, to represent the extent of spread of turbulent kinetic energy to higher modes. The rough-wall dynamics is dominated by fast activity (short time period) propagating modes and slow activity (long time period) roll modes. A single dominant timescale has been captured for all the propagating modes in flows over smooth walls; multiple dominant timescales representing various vortex shedding events are captured for rough walls. Variable-interval time averaging technique has been used to obtain the bursting frequency. The bursting frequency of rough-wall turbulence is higher compared to smooth-wall turbulence, suggesting that roughness enhances turbulence production activity. Another insightful observation for rough walls revealed by our study is that the vortex shedding frequency of roughness elements is much higher compared to the bursting frequency of rough-wall turbulence. POD provides a straightforward method to extract the natural frequency of shed vortices due to roughness, an important dynamical activity in rough-wall turbulent boundary layers.     

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.     

虚拟边界法研究正交双圆柱及串列双圆球绕流

把Goldstein等人提出的虚拟边界法推广到三维情况,研究了 Re=150时不同间距下正交双圆柱绕流,和Re=250时不同间距下串列双 圆球绕流流场. 对于正交双圆柱绕流,当间距比大于3,下游圆柱对上游圆柱尾流的影响只 限定在下游圆柱的尾流所扫过的范围之内;当间距比小于等于3,下游圆柱对上游圆柱尾流 的影响扩大,下游圆柱尾流扫过区上下出现两排三维流向二次涡结构. 对于串列圆球绕流, 研究发现,在小间距比(L/D≈ 1.5)的情况下,由于上下游圆球尾流区的相互抑 制消除了压力不稳定性,整个流场呈现稳 态轴对称特征;间距比为2.0时,周向压力梯度诱发出流体的周向输运,流场呈现稳态非对 称性,但流场中存在特定的对称面;间距比增大到2.5后,绕流场开始周期振荡,原有的对 称面依旧存在;在间距比3.5时下游圆球下表面的涡结构强度有所减弱,导致占优频率发生 交替;间距比增至7.0时,整个流场恢复稳态特征,两圆球尾部同时出现双线涡,这时流场 对称面的位置发生了变动.     

Constrained large-eddy simulation of turbulent flow over inhomogeneous rough surfaces

In this work we extend the method of the constrained large-eddy simulation(CLES) to simulate the turbulent flow over inhomogeneous rough walls. In the original concept of CLES, the subgrid-scale(SGS) stress is constrained so that the mean part and the fluctuation part of the SGS stress can be modelled separately to improve the accuracy of the simulation result. Here in the simulation of the rough-wall flows, we propose to interpret the extra stress terms in the CLES formulation as the roughness-induced stress so that the roughness inhomogeneity can be incorporated by modifying the formulation of the constrained SGS stress. This is examined with the simulations of the channel flow with the spanwise alternating high/low roughness strips. Then the CLES method is employed to investigate the temporal response of the turbulence to the change of the wall condition from rough to smooth. We demonstrate that the temporal development of the internal boundary layer is just similar to that in a spatial rough-tosmooth transition process, and the spanwise roughness inhomogeneity has little impact on the transition process.     

有限长正方形棱柱绕流的双稳态现象

运用大涡模拟(large eddy simulation, LES) 方法对有限长正方形棱柱绕流进行了研究. 棱柱高宽比为5,一端固定于平板上,另一端为自由端. 平板表面边界层厚度可忽略不计. 基于自由流速度和棱柱宽度的雷诺数为3 900. 由计算结果发现,有限长棱柱的尾流在自由端后下扫流的作用下具有显著的三维性. 尾流中会交替出现两种典型的流动状态:一种是柱体两侧所脱落的涡在尾流中形成类似卡门涡街的交错排列状态;另一种是柱体尾流中涡团对称分布且无明显的周期性. 两种典型流动状态的交替出现,对有限长棱柱气动力特性有直接的影响.     

Wind-Up of a Spanwise Vortex in Deepening Transition and Stall

A fundamental flow problem of unsteady wind-up of a spanwise vortex is studied in this theoretical work on deepening dynamic stall and transition in a boundary layer, internal layer or related unsteady motion. It examines the nonlinear evolution of the spanwise vortex produced when the local wall pressure develops a maximum or minimum, subsequent to the finite-time break-up of an interacting layer and the impact of normal pressure gradients. The evolution is controlled by an inner–outer interaction between the effects of the normal pressure gradient and the momentum jumps across and outside the vortex, which is situated near the strong inflexion point induced in the mean flow. Although the work concentrates on a particular internal-flow context, many of the flow properties found are generic and in particular apply for a more general case including external flows. Analysis and associated computations point to two main distinct trends in the vortex response, depending to a large extent on a parameter gauging the relative strengths of the above effects. The response is either an explosive one, provoking enhanced wind-up, growth and pressure in the vortex, or it is implosive, causing the vortex to shrink and virtually empty itself through unwinding, leaving little local pressure variation. A further discussion includes the after-effects of this vortex response and some of the connections with experiments and direct computations on deepening stall and transition. Received 22 February 1999 and accepted 28 March 2000     

Mixing Intensification on the Edge of a Jet by Means of Longitudinal Vortices

The action of an artificially generated spanwise flow in the form of periodical longitudinal vortices on a plane turbulent mixing layer is investigated. It is shown that the disturbances result in a significant increase in the thickness of the mixing region. For two kinds of spanwise flow, namely, vortices whose centers lie in the plane separating the streams and vortices located above this plane, the dependence of the mixing layer thickness on the vortex amplitude and vertical dimension and on the longitudinal coordinate is found.     

哈特曼边界层的初级稳定性分析

导电流体在法向外置磁场的作用下,在贴近壁面处会形成哈特曼边界层.哈特曼边界层的稳定性研究对电磁冶金过程和热核聚变反应冷却系统等相关设备的设计和运行都有着十分重要的意义.本文采用非正则模态稳定性分析方法,对两无限大绝缘平行平板内导电流体流动的稳定性进行了研究.通过在时间上迭代求解扰动变量的控制方程组和伴随控制方程组,获得了在磁场作用下初级扰动的增长情况及其空间分布形式,分析了磁场强度对最优扰动增长倍数G_max、最优展向波数β_opt和最优时刻t_opt的影响,并考察了上下两个哈特曼边界层之间的相互作用.结果表明,最优初始扰动的空间分布形式为沿着流场方向的漩涡,关于法向方向对称或者反对称.当哈特曼数Ha较大时(Ha>10),对称漩涡和反对称漩涡形式的初始扰动增长倍数基本相等;上下两个哈特曼边界层可以认为是彼此独立的,不会相互影响,此时最优扰动增长倍数G_max与局部雷诺数R的平方成正比,相应的最优展向波数β_opt和最优时刻t_opt均正比于哈特曼数Ha.当哈特曼数Ha较小时(Ha<10),反对称漩涡形式的初始扰动更为不稳定,其增长倍数大于对称漩涡的增长倍数,且上下两个边界层之间存在着一定的相互作用,并对整个流场的稳定性产生一定的影响.      

Effects of two-dimensional V-shaped grooves on turbulent channel flow

Detailed Laser Doppler velocimeter (LDV) measurements have been carried out in a turbulent rectangular channel flow with one rough wall. The roughness elements of two-dimensional spanwise 120° V-shaped grooves are periodically arranged with different depths and pitches. The Reynolds number based on the centerline velocity, and the channel half height ranges from 2,740 to 20,000. The comparisons of turbulence statistics over smooth and rough walls indicate that the present roughness leads to a significant change in the turbulence both in the inner and in the outer flow. Particularly, the distribution density of the grooves is a key parameter to evaluate the effect of roughness. The low-Reynolds-number dependence of turbulence statistics is also observed. The rough walls with the same pitch-to-depth ratio exhibit the equivalent roughness function under the corresponding Reynolds numbers. The disagreement of velocity defect profiles between smooth and rough walls challenges the defect universal law. The variations of the turbulence stresses and Reynolds shear stress decomposition in the outer layer suggest that the turbulent motions may be modified by the present grooves. The importance of sweep events for the present groove-roughened walls is reflected by the differences in relative contribution to Reynolds shear stress from each quadrant and the higher-order moments over smooth and rough walls.     

Development of a turbulent boundary layer after a step from smooth to rough surface

The flow developing downstream of a step change from smooth to rough surface condition is studied in the light of Townsend’s wall similarity hypothesis. Previous studies seem to support the hypothesis for channel and pipe flows, but there are considerable controversies about its application to boundary layers and in particular to surface roughness formed by spanwise bars. It has been suggested that this controversy arises from insufficient separation of scales between the boundary layer thickness and the roughness length scale. An experimental investigation has therefore been undertaken where the flow evolves from a fully developed smooth wall boundary layer at high Reynolds numbers over a step in surface roughness (Re _θ = 13,400 at the step). The flow is mapped through the development of the internal layer until the flow is fully developed over the rough wall. The internal layer is found to grow as δ ∼ X ^0.73, and after about 15 boundary layer thicknesses at the step, the internal layer has reached the outer edge of the incoming layer. At the last rough wall measurement station, the Reynolds number has grown to Re _θ ≈ 32,600 and the ratio of boundary layer to roughness length scales is δ/k ≈ 140. The outer layer differences between the smooth and the rough wall data were found to be sufficiently small to conclude that for this setup the Townsend’s wall similarity hypothesis appears to hold.     

Structure of the turbulent boundary layer over a rod-roughened wall

Turbulent coherent structures near a rod-roughened wall are scrutinized by analyzing instantaneous flow fields obtained from direct numerical simulations (DNSs) of a turbulent boundary layer (TBL). The roughness elements used are periodically arranged two-dimensional spanwise rods, and the roughness height is k/δ = 0.05 where δ is the boundary layer thickness. The Reynolds number based on the momentum thickness is varied in the range Re_θ = 300–1400. The effect of surface roughness is examined by comparing the characteristics of the TBLs over smooth and rough walls. Although introduction of roughness elements onto the smooth wall affects the Reynolds stresses throughout the entire boundary layer when scaled by the friction velocity, the roughness has little effect on the vorticity fluctuations in the outer layer. Pressure-strain tensors of the transport equation for the Reynolds stresses and quadrant analysis disclose that the redistribution of turbulent kinetic energy of the rough wall is similar to that of the smooth wall, and that the roughness has little effect on the relative contributions of ejection and sweep motions in the outer layer. To elucidate the modifications of the near-wall vortical structure induced by surface roughness, we used two-point correlations, joint weighted probability density function, and linear stochastic estimation. Finally, we demonstrate the existence of coherent structures in the instantaneous flow field over the rod-roughened surface.