Basic experiments on transition in three-dimensional boundary layers dominated by crossflow instability

Progress in Aerospace Sciences 35 (1999) 363—412

Basic experiments on transition in three-dimensional boundary layers dominated by crossflow instability H. Bippes* DLR, Institute for Fluid Mechanics, Bunsenstr. 10, D-37073 Go¨ ttingen, Germany

Abstract The objective of this review is to summarize the results of basic experiments on the transition process in threedimensional boundary layers governed by crossflow instability and to inform on the present state of knowledge. In particular, the intent is to give a detailed description of the essential physical features rather than a complete survey on all the experiments known so far. For convenience, emphasis is placed on the experiments carried out at DLR in Go¨ttingen, however, essential contributions of other experiments which complete the physical understanding of the problem or lead to different conclusions are also discussed. The DLR experiments were specifically designed to study the transition process dominated by crossflow instability and to create the database for theoretical and numerical analyses of the disturbance development up to the final breakdown of laminar flow and for transition prediction. These analyses should take into account the physics of transition in a more comprehensive manner than the e,-method allows. It will be shown why, in contrast to transition dominated by Tollmien—Schlichting instability, details of the upstream conditions and the related nonlinear development cannot be ignored. The possibilities for their inclusion are discussed. In this regard, it is of special interest to verify that modern stability analyses enable the quantitative description of the linear and nonlinear disturbance development. High emphasis is placed on comparisons between experiment and nonlinear PSE and DNS approaches.  1999. Published by Elsevier Science Ltd. All rights reserved.

Contents 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1. Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2. Earlier work and scope of the present review . . . . . . . . . . . . . . . . . . . . . 1.3. Some fundamentals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Design concepts of the main basic experiments currently performed . . . . . . . . . . . 2.1. The ASU experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2. The ITAM experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3. The NAL experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4. The DLR experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. On the physical features of the transition process governed by crossflow instability . . 3.1. Identification of the instabilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2. Disturbance growth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3. Dependence of the disturbance development upon the environmental conditions

* Tel.: 0049 5 517 092 406; fax: 0049 3 517 092 829; e-mail: [email protected]. 0376-0421/99/$ — see front matter  1999. Published by Elsevier Science Ltd. All rights reserved. PII: S 0 3 7 6 - 0 4 2 1 ( 9 9 ) 0 0 0 0 2 - 0

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364 364 364 365 367 367 368 369 370 371 372 375 379

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3.4. 3.5. 3.6.

Nonlinear mechanisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Final breakdown of laminar flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Effect of surface curvature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6.1. Convex curvature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6.2. Concave curvature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4. Experiment and theoretical prediction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1. Linear theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2. Nonlinear approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3. Secondary instability and final breakdown of laminar flow . . . . . . . . . . . . . . . . . . 4.4. Effect of surface curvature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.1. Convex curvature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.2. Concave curvature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5. Receptivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1. Receptivity to surface roughness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2. Receptivity to free stream fluctuations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6. Transition prediction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7. Possibilities of stabilisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

379 384 388 388 390 392 392 393 396 396 397 397 398 399 400 402 404 407 408 408 412

1. Introduction

1.2. Earlier work and scope of the present review

1.1. Motivation

First experiments on transition in swept wing flow were initiated at the British Royal Aircraft establishment. In flight tests, Gray [2, 3] perceived that the boundary layer of a swept wing became turbulent much closer to the leading edge than on the corresponding unswept wing. China clay patterns displayed regularly spaced striations in the streamwise direction prior to transition. They were interpreted as due to the action of stationary vortices. These observations gave rise to the classical basic research work by Gregory et al. [4]. In an experimental and theoretical study on stability and transition of rotating disk flow, they identified the underlying instability mechanism as inflectional instability due to the crossflow and revealed the characteristic features of this instability. In particular, Stuart [4] found the access to the theoretical treatment of the stability problem occurring in accelerated three-dimensional flows by considering the mean flow components in the direction of disturbance propagation. With this idea, he established the basis for the later work. Basic experimental investigations on transition in swept wing flows were performed only 30 years later by Poll [5] at the RAE on a swept cylinder, by Arnal et al. [6] and Michel et al. [7], respectively, on a swept wing, and by Saric and Yeates [8] on a swept flat plate where crossflow was generated by a bump at the tunnel wall opposite to the flat plate. These experiments confirmed the physical explanation of the instability in the accelerated region of swept wing flows as derived by Stuart in [4] for rotating disk flow. A rather complete review of the

The prediction and control of transition in shear flows is of high practical relevance. It effects not only skin friction and flow separation, i.e. the aerodynamic forces, but also heat and mass transfer. Furthermore, the reliability of Navier—Stokes computations cannot be better than the location of transition assigned. Thus, it is not a surprise that a considerable amount of research work in fluid mechanics was devoted to the transition problem. Already with the early work of Reynolds [1], it became obvious that the solution of this problem is extremely difficult. For this reason, past research in transition concentrated on two-dimensional or axisymmetric flows although in nature flows are rather three-dimensional. The present report concentrates on transition dominated by crossflow instability as appearing in the accelerated region of swept wing flows. Since crossflow instability is an inflectional instability, it is highly amplified and can occur at very low Reynolds numbers. In swept wing flows, it may lead to transition very close to the leading edge. The main impetus to tackle this problem came from the study of laminar wing design concepts for drag reduction purposes at the British Royal Aircraft Establishment around 1970. Laminar wing design requires not only the prediction of transition in dependence on the disturbance environment, but also the detailed knowledge of the disturbance development with focus on the possibilities and the efficiency of passive and active flow control.

H. Bippes / Progress in Aerospace Sciences 35 (1999) 363—412

work on the stability problem of three-dimensional accelerated boundary-layer flows performed at that time is given by Reed and Saric [9]. Currently, four main long-term experimental programs are being performed for the detailed basic study of the transition process in swept wing flows: (1) by Saric et al. at Arizona State University (ASU) in Tempe, USA, on a swept wing; (2) by Kachanov et al. at the Institute of Theoretical and Applied Mechanics (ITAM) in Novosibirsk, USSR, on a swept flat plate using the same design as Saric and Yeates [8]; (3) at the German Aerospace Center (DLR) in Go¨ttingen, Germany, on various models with flat, concave and convex surfaces; and (4) at the National Aerospace Laboratory (NAL) in Tokyo, Japan on a swept cylinder. Main results obtained up to 1993—1994 are reviewed by Saric [10]. Although in the mean time significant progress was reported, the present review is not sufficiently complete to be taken as an update of the review by Saric [10]. Instead, special emphasis is given to the description of essential physical features of the transition process not reviewed so far. For simplicity’s sake, main regard is paid to the work at DLR. However, relevant results of the other basic experiments which contradict or supplement the results obtained at DLR, or which assure the interpretations are also included. The work at DLR is part of a theoretical and experimental program aimed at the prediction of transition in swept wing flows on a comprehensive physical basis. In order to allow realistic comparison, the experiments are designed to simulate flows where theory is based upon rather than those common in practical applications. 1.3. Some fundamentals Crossflow is an inherent property of three-dimensional boundary-layer flows (see [11]). It is necessarily connected with the in-plane curvature of the streamlines. This in-plane curvature causes centrifugal forces. Outside the boundary layer, the centrifugal forces are balanced by the pressure forces. Inside the boundary layer, the centrifugal forces decrease towards the wall corresponding to the velocity whereas the pressure in first approximation remains constant. The resulting excess of the pressure forces generates crossflow and a three-dimensional velocity distribution normal to the wall (Fig. 1a). Such a flow can become unstable to oblique waves. In the limit of zero frequency, these waves have lines of constant phase roughly parallel to the streamlines of the inviscid flow and are stationary with respect to the surface (Fig. 1b). A comprehensive discussion of the related three-dimen-

 In steady flow vortex lines are parallel to the streamlines.

365

sional stability problem and its theoretical solution is given by Mack [12]. The most simple access is provided by the temporal approach and the parallel flow approximation. It allows the transformation of the three-dimensional stability equations to two-dimensional stability equations for two-dimensional waves. Thus, in a threedimensional boundary layer, the eigenvalues of an oblique wave can be obtained from the two-dimensional stability analysis for the profile of the velocity component in the direction of the wave number vector (Fig. 1c). It is the key result of the classic work by Stuart (in [4]) that it is this profile, usually called directional profile, that governs the instability. Mack [12] stated that even in the spatial approach without introducing any transformation, the governing real velocity profile is the directional profile. Fig. 1c reveals that among the variety of directional profiles those close to the crossflow direction (t"90°) become inflectional. Since according to Rayleigh [13] inflectional velocity profiles are unstable at infinite Reynolds number and moreover, according to Tollmien [14], inflection points in the profiles of channel and boundary layer flows are a necessary and sufficient condition for instability, the crossflow in three-dimensional boundary layers leads to an inflectional instability. Hence, in contrast to two-dimensional boundary-layer flows, in three-dimensional boundary-layer flows a negative pressure gradient has a destabilizing effect. This fact elucidates the severe problem of laminar wing design for large transport aircraft. Practical interest is centered on flows along swept wings and turbine blades. The treatment of the latter is further complicated due to Coriolis and centrifugal forces caused by system rotation and concave surface curvature, respectively. Furthermore, the high disturbance level in the oncoming flow may lead to a bypass of the amplification process as predicted by linear theory. In order to limit the scope of the present review, those aspects are not taken into account. Neither, the particular case of the three-dimensional flow on rotating discs is considered, even though the early systematic work favoured this particular case of three-dimensional flow rather than swept wing flow. Certainly, rotating disc flow grants some simplifications: It does not have a pressure gradient parameter or a variable sweep angle, the boundary-layer profiles and the boundary-layer thickness are independent of the radius, but it has diverging streamlines and is subject to Coriolis forces which also influence boundary-layer stability. It still seems to be unclear how these specific properties affect the stability problem. Thus, up to date, no clear evidence has been found for the possible appearance of an absolute instability in swept wing flows, as derived theoretically and verified experimentally by Lingwood [15, 16] for rotating disk flow. For a more detailed discussion, it is referred to Taylor and Peake [17].

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Fig. 1. Characteristics of swept wing flows. (a) The boundary layer profile Q (ºM ,»M are the streamwise and cross components of the    mean flow); (b) expected types of instabilities; (c) the directional profiles QW (W is the angle between the direction of the outer flow Q and  the plane where the three-dimensional profile is projected on).

H. Bippes / Progress in Aerospace Sciences 35 (1999) 363—412

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Fig. 2. Instability mechanisms possibly acting on swept wing configurations.

Our interest will focus on crossflow instability as appearing on swept wing flows. It must be noted, however, that under conditions of practical relevance on swept wings, various instability mechanisms will be acting as illustrated in Fig. 2. These mechanisms may interact and thus obscure the link of the observed phenomena to the instability mechanisms involved. In the present review we will restrict ourselves to boundary layers unstable solely to crossflow instability in the case of small disturbance excitation. Only for the discussion of the effect of surface curvature, some insight is provided into flows where centrifugal instability is also present. In order to further limit the volume of the paper, all the numerous experiments aimed at establishing empirical or semi-empirical transition criteria are not reported. The focus is set on the experimental work devoted to the improvement of the basic physical understanding of the transition process governed by crossflow instability. The characteristics of the most unstable disturbances and the dominating features of the transition process are identified in experiments performed under natural conditions of transition in a variety of disturbance environments. For detailed quantitative studies of the downstream disturbance evolution and of nonlinear processes, experiments under controlled disturbance excitation are conducted. The results presented stem from unclassified publications. The actual references are assigned. No reference is given for own unpublished results.

2. Design concepts of the basic experiments currently performed Parallel to the work at DLR, main basic experimental work on crossflow instability is being carried out at the Arizona State University (ASU) in Tempe by W. S. Saric and co-workers, at the Institute of Theoretical and Applied Mechanics (ITAM) in Novosibirsk by Y. S. Kachanov and co-workers, and at the National Aero-

space Laboratory (NAL) in Tokyo by Takagi and coworkers. In order to enable comparison of the results and for the correct interpretation of differences, relevant and published details of the design concepts and the environmental conditions are summarized in the following. 2.1. The ASU experiment W. S. Saric and his co-workers performed their experiments on an airfoil with a NLF(2)-0415 profile that was designed to favour natural laminar flow. For the generation of crossflow, it was set at the sweep angle U "45°.  The small leading edge radius makes the flow stable to attachment-line instability. Negative angles of incidence up to a"!4° provide an extended region of accelerated flow only unstable to crossflow instability, i.e. stable to TS instability. Wall liners contoured like stream surfaces approximate infinite sweep conditions even though the aspect ratio of the airfoil is relatively small. The model and pressure distribution for a"!4° are shown in Fig. 3. The tests are being performed in the low turbulence ASU wind tunnel. In the velocity range of the tests Q "15—30 m/s the turbulence level Tu "  SY 0.038%—0.067% and Tu "Tu "0.032%—0.015%, reTY UY spectively (2—1000 Hz band-pass filtered). According to the details given in Saric et al. [19] and the very small values Tu "Tu , the largest part of Tu must be due to TY UY SY low-frequency pressure fluctuations. Since travelling crossflow modes are excited mainly by the vortical content of the free stream turbulence (see Section 5.2) in the ASU wind tunnel, the effective free stream turbulence is smaller than in the T-324 ITAM tunnel, the NAL and the DLR facilities. The experiments are carried out under natural conditions and with controlled stationary disturbance stimulation. As disturbance generator, a spanwise array of rub-down dots is used [20]. The height of the dots and their spanwise spacing determine the initial amplitudes and wavelengths of the stationary instability modes. The

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H. Bippes / Progress in Aerospace Sciences 35 (1999) 363—412

Fig. 3. The model of the ASU experiment by Saric and co-workers. (a) Profile and chordwise pressure distribution; (b) schematic of the model with the wall liners. Dagenhart et al. [18], Figs. 1 and 2.

efficiency of the disturbance control is favoured by an extremely smooth surface (down to 0.12 lm rms finish). For the visualization of the skin-friction lines and for tracing transition, the naphthalene technique is used. The flow field is scanned with the aid of hot-wire and hot-film anemometry. The use of single and crosswire probes, as well as micro-thin multi-hot-film elements allows the detailed resolution of the flow field. For the closer description of the measuring and data processing techniques, it is referred to Dagenhart et al. [18] and Radeztsky [20]. The low turbulence environment favours the growth of the stationary vortices as it is expected from free flight conditions. Hence, the investigations focus on this instability mode and its effect on transition. A further goal is the receptivity to surface roughness. 2.2. The ITAM experiment Kachanov and Tararykin [21] have chosen the same experimental design as Saric and Yeates [8]. The model is a swept flat plate where crossflow is provided by the sweep angle U "25° and by a pressure gradient im posed from outside by means of a bump in the tunnel wall opposite to the flat plate. In contrast to the other basic experiments on crossflow instability, a moderate wing sweep has been chosen. Moreover, the pressure gradient, i.e. three-dimensional flow, starts only 0.3—0.4 m downstream of the leading edge (estimate from Fig. 1 in [22]). Thus, the streamwise profiles should be closer to Tollmien—Schlichting instability than in the other basic experiments. The experimental design is reproduced in Fig. 4. According to Saric and Yeates [8], there are some shortcomings in this arrangement. The streamwise pressure decrease well beyond the x -position  where the wall bump is of constant thickness indicates

that the flow separates at the upstream corner of the wall bump (cf. Figs. 1 and 3 in [21]). Since separation on such a configuration is unsteady, the induced pressure gradient should have some unsteady content. Furthermore, as discussed in detail by Saric and Yeates [8], the small aspect ratio of the plate needs contoured side walls to eliminate spanwise pressure gradients. Unfortunately, the spanwise mean flow distributions given by Kachanov and Tararykin [21] are limited to small distances. The tests are performed in the low turbulence wind tunnel T-324 with closed test section (1 m;1 m). Kachanov and Tararykin [21] quote the turbulence level Tu(0.02% without further specification. In a later paper, Gaponenko et al. [23] note Tu"0.065% in the frequency range f'1 Hz without further specification. In order to create controlled conditions, stationary and nonstationary disturbances are excited using various types of disturbance generators as described by Kachanov [22]. Their position with respect to the leading edge and to the position of neutral instability is not given so that comparisons with other experiments are unsure. For the velocity measurements, the hot-wire anemometry is employed. Two- and three-wire probes in »- and X-arrangement allow the correct determination of the velocities in the three-dimensional boundary layer [21]. The flow parameters chosen favour the travelling disturbances. In contrast to the ASU and DLR experiments, stationary crossflow vortices do not develop on the smooth surface (the surface finish of the model is not given). Even under controlled conditions, only very small amplitudes are reported. The work concentrates on the determination of the complete characteristics of travelling instability modes under controlled conditions in the linear range of amplification and the comparison with related stability theory.

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369

Fig. 4. Design of the experiment by Kachanov and co-workers, Kachanov [22], Fig. 1. (1) flat plate, (2) traversing mechanism, (3) wall bump, (4) wall of the test section, (5) flap, (6) disturbance source.

Fig. 5. The model used by Takagi and co-workers, Takagi and Itoh [25], Fig. 2. The models span the tunnel walls. The diameters D range from 138 to 500 mm and the aspect ratios S/D from 4.96 to 6.01.

2.3. The NAL experiments In recent years, Takagi and co-workers [24—26] started an experiment on a swept circular cylinder. The design is equal to the one chosen by Poll [5] (Fig. 5). The models span the closed test sections. Infinite sweep conditions are approximated due to large aspect ratios of the models which range from 4.96 to 6.01. Main emphasis is on the basic theoretical and experimental studies of the effect of surface and streamline curvature in the presence of crossflow instability. Due to its considerable convex

curvature, the model used at NAL is excellently suited to contribute to the solution of this restive problem. Hence, the work at NAL provides an important supplement to the other basic work described in the present paper. The work further benefits from the disposability of three wind tunnels with turbulence levels of Tu"0.18, 0.05, and 0.046% (without further specification). This enables to study the influence of the environmental disturbances on the same model and allows to acquire further data for modelling of receptivity. Velocities and disturbance characteristics are measured by means of

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hot-wire anemometry using one-wire probes. In order to identify the existence of streamline curvature instabilities, a disturbances generator is used [26]. 2.4. The DLR experiments The DLR experiments are part of a research program for the study of transition in three-dimensional boundary layers. Therefore, they mainly aim at supporting theoretical work and at providing the basis for realistic comparison between experiment and theory. Correspondingly, the models were designed to simulate most closely the flow conditions chosen for the theoretical approaches. In this regard, a swept wing could not be the first choice. The various instability mechanisms possibly acting on such geometries (Fig. 2) may obscure the link of the observed phenomena to the instability mechanisms involved. Therefore, models were designed which enable the isolated study of all these instability mechanisms. The mechanisms leading to the disturbance excitation or amplification in the leading edge region appearing under the actual environmental conditions are studied on a wing shaped model with large nose radius. It is of the same cross section as the model used by Poll [5] (Fig. 6a). The effect of oscillations of the stagnation line is studied on an oscillating cylinder (Fig. 6b). For the isolated investigation of crossflow instability, the swept flat plate arrangement shown in Fig 2.4c has been chosen. This model favours the study of crossflow instability in absence of surface curvature and, correspondingly, comparison with instability codes neglecting surface curvature. In order to get some idea on the effect of convex and concave surface curvature on crossflow instability, comparative experi-

ments are performed on the models shown in Fig. 6d and e. In particular, the former allows the study of threedimensional centrifugal instability. The models chosen for investigating the instabilities of swept wing boundary-layer flows are furnished with small leading edge radius in order to avoid attachmentline instabilities. Small negative angles of attack shift the stagnation line further towards the upper side of the plate, to eliminate suction peaks with adjacent positive pressure gradient possibly causing Tollmien—Schlichting instability (e.g. see the pressure distribution along the swept flat plate). The sweep angles are chosen such that the transition process is governed by crossflow instability. Infinite sweep conditions are approximated with the use of contoured end plates, aligned with the streamlines at the edge of the boundary layer. The pressure gradient normal to the model surface is neglected. In particular in the leading edge region, this needs some adjustments in order to avoid significant deviations. Most of the measurements have been performed on the swept flat plate. This set-up proved to be most adequate for the principle study of crossflow instability isolated from other instabilities and it offers most realistic comparison with theory. In particular, the infinite swept wing conditions assumed in stability theory are approximated satisfactorily [27]. In addition, the pressure distribution (Fig. 7) leads to a base flow that can be approximated by a Falkner—Scan—Cooke solution favouring parametric studies [29]. The experiments are performed under both natural and controlled conditions. Stationary vortices are excited by a spanwise array of heating or roughness elements [30], while travelling waves are excited by a spanwise array of oscillators [31]. For the stabilisation

Fig. 6. DLR models for basic investigations of instability mechanisms possibly appearing on swept wing configurations. Model for the investigation of (a) leading edge phenomena; (b) the stability of oscillating stagnation flow; (c) crossflow instability; (d) the effects of concave surface curvature; and (e) the effects of convex surface curvature.

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371

Fig. 7. Pressure distribution along the swept flat plate (Fig. 6c) for various free stream velocities and chordwise variation of the local sweep angle U , Lerche [28]. 

Table 1 Facilities used for the DLR experiments and some free stream characteristics. Tu"100/Q (1/3(u#v#wN ) is measured in the  frequency range 2 Hz(f(2 kHz at wind tunnel speeds of Q "17—25 m/s. Tu "100 u/Q , Tu +Tu , Tu results from long SY  UY TY distance correlation Facility

Test section

NWG 1 MK 1 MK with additional screen NWB

3m;3m open 1m;0.7m open 1m;0.7 closed 5.6;2.8 m open

Water towing tank

1.1 m;1.1 m

Free stream fluctuations Vortical content Tu Tu Tu Tu Tu Tu SY TY TY TSY TTY 0.70 0.15 0.24 0.08

of the boundary layer by means of surface suction, the swept flat plate was equipped with a suction device. In order to study the influence of the environmental conditions, the experiments on the swept flat plate were performed in different wind tunnels and in a water towing tank. In the latter, the model is moved through the water at rest. This closely simulates free flight conditions. The experimental facilities used and their main specifications are listed in Table 1. In addition, the surface finish of the swept flat plates was varied to have peak to peak values ranging from 1.8 to 50 lm. Like in the ASU, the ITAM and the NAL experiments, hot-wire anemometry was employed for the determination of the downstream development of the unsteady velocity field. The flow velocities were measured using two-wire probes in V-arrangement to account for the varying flow direction in the three-dimensional boundary layer. Such probes neglect the normal to wall components but allow measurements closest to the surface. Under natural conditions, wave number vectors were determined with the aid of a rotatable array of surface

0.44 0.12 0.20 0.06

0.80 0.16 0.25 0.10

0.58 0.13 0.17 0.07

0.36 0.09 0.24 0.02

0.67 0.15 0.22 0.09

Tu

Acoustic content Tu Tu Y Y

0.38 0.06 0.09 0.04

0.46 0.08 0.11 0.04

0.25 0.05 0.09 0.03

approximates free flight conditions

hot-films. For flow visualization, oil-flow and sublimation techniques were used in the wind tunnel tests and the hydrogen bubble technique was used in the water towing tank tests. Details on the data acquisition and processing were described by Deyhle et al. [32] and Deyhle and Bippes [33]. The definition of the flow quantities chosen for the presentation of the results are given in the appendix.

3. On the physical features of the transition process governed by crossflow instability The first step for the basic study of an instability mechanism is the identification of the most amplified instabilities and the inherent amplification processes leading to transition. The experiments performed under natural conditions of transition in varied disturbance environments are the appropriate means to explore all relevant instability features possibly acting in practical situations. In the following, the results of those experiments are discussed.

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3.1. Identification of the instabilities In order to provide a first insight into the nature of the disturbance motion developing due to crossflow instability, a series of velocity profiles was measured at various downstream locations x /c"const. The single profiles A were closely spaced in the spanwise direction to resolve the disturbance motion. In favour of a more illustrative presentation, the data measured in a non-equidistant 19;23 measurement grid were spline interpolated first in the z- then in the y -direction to form a 40;40 grid. The A left-hand side of Fig. 8 exhibits the cross component of the mean flow profiles »M (for definition see Eq. (A.2) in  the appendix) measured at four downstream locations x /c. Already in the most upstream position, a stationary A spanwise variation superimposed upon the laminar flow is observed. In the downstream direction (Fig. 8b—d), this spanwise variation increases rapidly, whereby the periodic character becomes more and more evident. This suggests that it is due to a stationary instability. On the right-hand side of Fig. 8, the crossflow component of the r.m.s. fluctuations is plotted (for definition see Eq. (A.7) in the appendix). At the most upstream position, the fluctuations inside the boundary layer have a maximum at the wall distance z+d/4 and are still smaller than the fluctuations outside the boundary layer. Further downstream, both the fluctuations and stationary perturbations increase rapidly in amplitude. This indicates that primary unsteady disturbances are amplified as well. Additionally, a spanwise periodic variation is observed starting at x /c"0.6. It has the same wavelength as the spanwise A mean flow variation and can therefore be seen as first indication of an interaction between stationary and nonstationary disturbances. For a closer insight into the structure of the perturbations, the isotachs of the wall parallel components of the velocity, measured in the cross section at x /c"0.7, are A displayed in Fig. 9. They represent the situation in an advanced state of the instability process. The upper figures show the isotachs of the mean flow Q s"[ºM , »M ]. In   the undisturbed laminar flow, both components would yield horizontal lines. Their deformation is due to the stationary instabilities. The isotachs of the crossflow component have the structure of corotating vortices. In sublimation and oil-flow visualizations, these vortices cause the regular streamwise streaks closely aligned with the inviscid streamlines as first shown by Gray [2, 3] (both in free flight and wind tunnel tests) and later more clearly by Saric and Yeates [8] and others [5, 6, 34]. The figures in the middle exhibit the spanwise periodic content of the mean flow q s ( y , z) " x "const. (for definition see A A Eq. (A.3) in the appendix). Within the limits of experimental accuracy and of the linear range of amplification,

 Lines of constant velocity.

[uN , vN ] can be seen as the wall parallel components of the   stationary vortices. They have the structure of counterrotating vortices with axes apparently in the streamwise direction corresponding to the normal mode solutions of stability theory. In the lower figures, the r.m.s. content of the unsteady disturbance motion is shown. Although it gives no further information on the nature of unsteady instabilities, the periodic spanwise variation having the wavelength of the stationary modes should be pointed out again. It is always observed as soon as the stationary vortices become of clearly measurable size. For the upstream conditions of the appertaining test, this variation starts at x /c"0.6 (cf. Fig. 8). As it will be shown below, A the disturbance interaction is a dominating phenomenon in the transition process governed by crossflow instability. In contrast to transition due to Tollmien—Schlichting instability, no efficient transition criterion is conceivable in the crossflow case which neglects the nonlinear development. In this report, main emphasis will be therefore placed on the nonlinear regime. Under natural conditions of transition, irregularly spaced stationary vortices are usually observed. This may be illustrated by a typical spanwise mean velocity variation in the boundary layer which is due to the action of stationary disturbances, see Figs. 10a and b. The corresponding wave number spectrum in Fig. 10c reveals that the most amplified modes appear in a relatively narrow band. Average wavelengths j (aligned with y ) A A measured by Arnal and Juillen [35] and Dagenhart et al. [18] on a swept wing and by Mu¨ller [27] (see also [36]) and Kachanov and Tararykin [21] on a swept flat plate are compared in Fig. 11. In all the experiments, the most amplified modes are in the range predicted by linear theory. In the ONERA/CERT [6], DLR and ITAM (for a vortex generated by an isolated roughness element, Figs. 9 and 10 in [21]) experiments j /d turned out to A scale with the local boundary layer thickness by a factor of approximately 4. The nature of the unsteady disturbances is derived from the analysis of the velocity fluctuations. The instantaneous hot-wire signal in Fig. 12a is a representative example. The related power spectrum reveals a relatively broad band of almost equally amplified frequencies typically starting at very low values (Fig. 12b). Wavelengths, phase velocity, and propagation direction of the naturally developing disturbances were determined by Deyhle et al. [32]. Analysing the correlated signals of a rotatable hot-film array (Fig. 13), they found that the naturally developing unsteady disturbances are oblique travelling waves in a band of frequencies with propagation angles depending on the frequency. It may be worthwhile to note that even under natural conditions their correlation length was found to be larger than 23 mm, the length of the hot-film array. The result of these identification tests can be summarized as follows: In the presence of crossflow instability,

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Fig. 8. Spanwise variation of the cross components of the velocity profiles measured at various downstream locations x /c. »M and v are A   the cross components of the mean velocity and the r.m.s. velocity fluctuations, respectively, Deyhle and Bippes [33]. Q "20.5 m/s,  U "43.5°, Tu"0.24%, RM "5 lm. (a) x /c"0.4; (b) x /c"0.5; (c) x /c"0.6; (d) x /c"0.9.  X A A A A

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Fig. 9. Isotachs of the wall parallel components of the mean velocity ºM , »M , the stationary disturbances uN , vN , and the r.m.s. fluctua    tions u , v in a cross section (y , z-plane) at x /c"0.7. Q "20.5 m/s, U "43.5°, Tu"0.24%, RM "5 lm, Deyhle and      A   X Bippes [33].

the naturally developing disturbances are oblique travelling waves. The closer to the crossflow direction they propagate, the smaller are their frequencies. In the limiting case, their frequency become zero and we call them stationary vortices. Then, their lines of constant phase or their axes become aligned with the local flow direction which is close to the direction of the local inviscid streamlines. The spanwise modulation of the r.m.s. amplitudes of the unsteady disturbances observed in Figs. 8 and 9 suggests an interaction between stationary and non-

stationary instability modes. It begins far upstream of transition. Since Lingwood has demonstrated theoretically [15] and experimentally [16] that rotating disc flow can become unstable to an absolute instability, the question

 The author has some doubts that Lingwood’s experiments really provide the unequivocal evidence for this statement.

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375

Fig. 10. Spanwise variation of the mean velocity component ºM due to the presence of stationary vortices measured in the boundary  layer at z/d"0.25 and x /c"0.9 in the 1MK, (a) without (Tu"0.15%, RM "5 lm) and with additional screen upstream of the tunnel A X exit (Tu"0.24%, RM "5 lm); (b) related wave number spectra. Q "20.5 m/s, U "43.5°. X  

arose whether swept wing flow having a similar type of base flow profiles could also be subject to an absolute instability. This question was studied experimentally on the swept flat plate in a water towing tank. Streamwise vortices were stimulated artificially using a spanwise array of localized heating elements spaced at the wavelength of the most amplified stationary vortices. After the vortices arrived at a steady state, the disturbance generating heating was switched off. At lower velocities, when the vortices did not reach the nonlinear range of amplification, they were convected downstream. Thereafter, the flow state existing before the vortex stimulating heating was switched on became restored, as expected from a convective instability. Only when the velocity was increased until the amplitude of the stationary vortices approached saturation (with amplitudes exceeding 15—20% of the local external flow) and naturally excited vortices came also into play, results became difficult to interpret and less unequivocal. Hence, it cannot be excluded that somewhere in the nonlinear regime profiles develop which may be susceptible to an absolute instability. However, up to this state in the transition process, the

instability occurring in the swept flat plate flow proved to be convective. Unfortunately, no other experiments to identify an absolute instability in swept wing flows are known. 3.2. Disturbance growth Since crossflow instability appearing on the swept flat plate turned out to be convective, only the spatial growth was considered. Precise measurements require the downstream tracing of individual disturbances. Under natural conditions, where the instabilities are initiated irregularly in space and time, this is extremely difficult. Hence, most of the tests were conducted under controlled conditions. However, in order to assure that the transition process studied includes all the essential features occurring in practical situations, measurements performed under natural conditions of transition are of specific interest. The procedure adopted at the DLR experiments to determine the downstream development of the disturbance motion was to measure series of velocity profiles

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Fig. 11. Wavelengths of stationary vortices measured, (a) at ONERA/CERT on a swept wing (Arnal et al. [35], Fig. 5); (b) at ASU on a swept wing (Dagenhart et al. [18], Fig. 11); and (c) at the DLR on a swept flat plate (Bippes et al. [36]).

Fig. 12. The unsteady disturbance motion developing under natural conditions of transition on the swept flat plate. Q "20.5 m/s,  U "43.5°, x /c"0.9, z/d"0.25, Tu"0.15%, RM "6 lm. (a) Instantaneous hot-wire signal; (b) related power spectrum; (c)  A X amplification rates u for various wave propagation angles W resulting from local linear theory by Dallmann and Bieler [29]. G

(cf. Fig. 8) spaced in span less than 1/6 of the disturbance wavelength apart at a number of successive chord positions. A typical result is shown in Fig. 14a. Both steady and unsteady disturbances, uNK and u , respectively, are  

amplified approximately to the same extent. Comparison with local linear theory shows that in contrast to the wave number vectors and frequencies of the most amplified modes, their growth is not correctly predicted. On

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Fig. 13. Characteristics of the f"60, 100, and 140 Hz waves obtained by means of cross-spectral analysis of the instantaneous signals recorded on a 10-element sensor array under natural conditions of transition. Q "20.5 m/s, U "43.5°, x /c"0.7, Tu"0.15%,   A RM "5 lm. (a) Rotatable hot-film array; (b) direction of wave propagation; (c) magnitude of the phase velocity; (d) wavelength. Deyhle X et al. [32].

Fig. 14. Disturbance growth measured on the swept flat plate, (a) under natural conditions, Q "20.5 m/s, U "43.5° and (b) with the   stationary modes artificially excited by spanwise periodic surface heating. Q "15 m/s, U "43.5°. Here u and u are the r.m.s.       amplitudes measured at spanwise locations of ºM and ºM , respectively, s"1/l  »M dz is the crossflow Reynolds number, the      numbers at the curves give the disturbance amplitudes at the beginning and at the end of the measuring domain, Tu"0.15%, RM "6 X lm, Bippes [37].

the way downstream, the initial growth attenuates more and more until it ends up at saturation prior to transition. Both in the ASU and in the DLR experiments, this proved to be a main feature on the path to transition.

A further characteristic feature is the splitting of the growth of the unsteady modes in u and u if     measured along streamwise zones of ºM and   ºM (for definition see Eq. (A.8) in the appendix) in the  

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Fig. 15. Spanwise ºM and u variation due to stationary vortices artificially stimulated at the most amplified wavelength j "  

  13.3 mm. s " 1/l  »M dz"115 (crossflow Reynolds number), Q "10 m/s, U "43.5°, x /c"0.9 and z/d"0.25, Tu"0.15%,   A   RM "6 lm, Bippes [37]. X

stationary vortex field. Both the saturation and the splitting of the growth of the unsteady modes indicate the nonlinear development. It should be stressed that in all our experiments, this nonlinear development starts well upstream of transition as soon as the naturally stimulated disturbances become of measurable size [36]. The measurements at ASU also conducted under natural conditions of transition show the same tendency [38], especially the early nonlinear development and the overestimation of the disturbance growth by various linear approaches. In order to provide data for the linear growth, additional measurements were performed in a flow state closer to neutral stability, i.e. at reduced free stream velocity. Under these conditions, the stationary disturbances became measurable only if they were artificially excited by means of local heating [37]. Now, the travelling modes are more amplified than the stationary modes corresponding to local linear theory, but still the growth is overpredicted (Fig. 14b). The latter may be attributed to the irregularity of the flow field which is due to the arbitrary excitation of the instabilities and to disturbance interaction to be observed as soon as the stationary vortices are of measurable size. In Fig. 15, this becomes evident by the significant spanwise variation of the r.m.s. fluctuations having the wavelength of the stationary modes although the amplitudes of those do not exceed 2.5% of Q . A similar result is reported by Radeztsky  et al. [30]. He measured the growth of stationary vortices stimulated by a spanwise array of single roughness elements with the swept wing set at zero angle of incidence (a"0°). Comparison with various theoretical approaches shows that even at amplitudes of 1—3% the growth is not correctly predicted (Fig. 16). Reibert [39]

Fig. 16. Growth of stationary disturbances, stimulated by a spanwise array of single 0.112 lm height roughness elements. Swept wing experiment at ASU (Radeztsky et al. [30], Fig. 19). U "45°, a"0°, Re "3.0;10, Tu "0.038%.  A SY

argued that the large roughness used may have caused the linear receptivity regime to be bypassed, and indeed, in later experiments at ASU, with the wing set at a"!4° to increase crossflow instability and at DLR, also using the stationary disturbance stimulation technique of Radeztsky [20], linear conditions became accessible. The results will be presented in Section 4 where the applicability of theoretical approaches is discussed in more detail. Here, the large streamwise extent of the nonlinear regime should be pointed out again. It is in contrast to the observations in transition processes governed by Tollmien—Schlichting instability and may be a main reason for the failure of the e,-method based on

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local linear theory to predict transition in the case of crossflow instability. For a more detailed discussion the reader is referred to Section 6. 3.3. Dependence of the disturbance development upon the environmental conditions The results discussed so far make it obvious that the boundary-layer flow unstable to crossflow instability exhibits the features of nonlinear dynamic systems. These are known to strongly depend on the initial conditions. Hence, in order to provide a more complete view on the transition process dominated by crossflow instability, the initial conditions were varied. They are given by the environmental disturbances and the boundary-layer receptivity. Therefore, in order to change them, Mu¨ller [27] (see also [36]) and Deyhle [40] (see also [33]) performed additional tests on the same model in different wind tunnels and in a water towing tank (Table 1). They observed dramatic changes in the disturbance development. In a low turbulence environment (Fig. 17a), the stationary modes dominated. They underwent a larger growth and arrived at saturation amplitudes twice as large as in the tests with increased free stream turbulence (Fig. 17b). The splitting of the growth of the travelling modes u (u in the figure) occurred further upstream 

  and became more distinct. In addition, the frequency analysis performed by Mu¨ller [41] revealed that the frequency band of the most amplified frequencies shifted from around 70 Hz to around 135 Hz. In further tests by Deyhle and Bippes [33] in a water towing tank which closely simulates free flight conditions, the maximum amplitudes for the travelling modes were also found to be in the 135 Hz frequency range. In a very large turbulence environment, the nonstationary disturbances dominated (Fig. 17c) and the stationary vortices were even damped in the final stage of the transition process. Further experi-

379

ments, at the constant free stream turbulence Tu" 0.15% and with a varied surface roughness RM revealed X that at small RM the travelling modes clearly dominated. X However, with increasing RM the stationary modes X ended up at larger saturation amplitudes and finally dominated as in low turbulence environment (Fig. 18). The saturation levels of the unsteady modes were not affected significantly. Only the splitting in the u growth 

 occurred earlier and was more pronounced. It was accompanied by a stronger spanwise variation (Fig. 15) and by a shift of the most amplified frequencies from around 70 Hz to around 135 Hz (Fig. 19) as at low free stream turbulence. All these observations point to a stronger nonlinear development. For a more complete description of the dependence of the disturbance growth upon the environmental conditions the reader is referred to Deyhle and Bippes [33]. The results can be summarized as follows: An increase in free stream turbulence decreases the saturation amplitudes of the stationary modes and finally leads to the domination of the travelling modes. On the other hand, an increase in surfaces roughness increases the saturation amplitudes of the stationary modes and finally leads to their domination whereby the saturation amplitudes of the unsteady modes are only weakly affected. The domination of stationary modes favours the nonlinear development. For practical purposes, it is important to note that the different disturbance developments linked to the environmental conditions considerably affect transition. Next, the underlying nonlinear mechanisms will be elucidated to deepen the basic physical understanding and to give hints for the proper modelling of nonlinear theoretical approaches. 3.4. Nonlinear mechanisms The deformation of the mean flow under the action of the stationary vortices and the spanwise modulation

Fig. 17. Disturbance growth measured on the swept flat plate with RM "6 lm surface roughness in three different wind tunnels. (a) 1 MK X (open test section), Tu"0.15%, Tu "0.13%; (b) NWB (closed test section), Tu"0.08%, Tu "0.07%; (c) NWG (open test section), T T Tu"0.7%, Tu "0.58. Q "20.5 m/s, U "43.5°, Deyhle and Bippes [33]. T  

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Fig. 18. Disturbance growth measured in the 1 MK at Tu"0.15%, Tu "0.13% and RM varied. (a) RM "1.8 lm; (b) RM "6 lm; (c) T X X X RM "10 lm (single roughness elements (SR), equivalent to 50 lm distributed roughness). Q "20.5 m/s, U "43.5°, Deyhle and Bippes X   [33].

Fig. 19. Frequencies of the most amplified travelling waves measured in different disturbance environments existing in the DLR facilities (Table 1). Re "Q x /l is varied with Q , x /c"0.9 and z/d"0.3, Deyhle and Bippes [33]. V  A  A

of the travelling modes exhibited in Figs. 8 and 9 suggest that the stability characteristics change significantly and that secondary developments are present simultaneously with disturbance interactions. Experiments for the detailed study of nonlinear mechanisms are being conducted at ASU, ITAM, and DLR. Deterministic single-mode excitation at given initial amplitudes enables the study of the complex nonlinear phenomena under well-defined conditions. Whereas at DLR both stationary and travelling disturbance modes were stimulated to allow for the study of their mutual interaction [28, 31], the ASU tests concentrated on the investigation of forced stationary vortices of different wavelengths and initial amplitudes [30, 42]. The low turbulence environment of the ASU wind tunnel enabled the study of the transition process only marginally effected by primary unsteady modes, as it is expected from transition under free flight conditions. Hence, the ASU and DLR experiments supplement one

another. The ITAM experiments were performed in a flow which was obviously only very weakly susceptible to stationary crossflow vortices (see also Section 2.2), because these did not appear on the smooth surface [22] in contrast to the experiments by Saric and Yeates [8] (on an equal experimental arrangement), the flight tests by Gray [2, 3], and the early experiments by Poll [5] and Arnal et al. [6]. In all these experiments, the model was set at a larger sweep angle. Therefore, and because in addition the turbulence level of the ITAM tunnel is relatively low, it is suggested that in the ITAM experiment crossflow instability did not dominate. This could explain why the results do not compare with those obtained at ASU and DLR. Since the DLR experiments were specifically aimed at explaining the dependence of the nonlinear disturbance developments on the initial conditions as reported by Deyhle and Bippes [33], the experiment of Lerche [28]

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Fig. 20. Disturbance generator used for the controlled initiation of instabilities, Lerche and Bippes [31].

(see also [31, 43]) considered three different cases: (1) Case (0,1) where only a stationary mode was initiated; (2) case (1,1) where only a travelling mode was initiated; and (3) the mixed case (0,1)#(1,1). In this notation, the first number in the brackets denotes multiples of the frequency of the artificially excited primary instability mode, and the second stands for multiples of its spanwise wave number. As depicted in Fig. 20, a spanwise array of rub-down dots [30] was used for the stimulation of the steady modes, and a spanwise array of oscillating membranes which allows to deliberately choose amplitude, frequency, and propagation angle [31] for the stimulation of the unsteady modes. In the limiting case (0,1) where only stationary vortices were initiated, the disturbance interaction mainly lead to the generation of higher harmonics (0,2), (0,3), etc., as it can be seen in the wave number spectrum of Fig. 21. According to this figure, their amplitudes did not exceed 25% of those of the primary modes. However, it must be noted that especially the modes of order higher than (0,2) were weakly resolved due to the finite width of the hotwire probe. A remarkable result is reported by Reibert [39] studying the same case in lower turbulence environment. He measured much larger amplitudes for the higher-order modes when he set the distance of the disturbance generating roughness elements to be three times the wavelength of the most amplified primary mode. In the other limiting case where only an oblique travelling wave was excited, case (1,1), again mainly higher harmonics (2,2), (3,3), etc., were generated. The corresponding increase of their growth rates further confirms that they are due to the interaction of the excited primary wave (Fig. 22). Some further characteristic features become obvious comparing the growth of the primary travelling mode forced at oscillation amplitudes of the membranes varying between RK  "10, 15, and 20 lm. For X small amplitudes the amplification rate is independent from the initial amplitudes as it is expected in the linear

Fig. 21. Wave number spectra for the stationary modes measured on the swept flat plate in case (0,1) at various chord locations x /c. Q "20.5 m/s, U "43.5°. A  

Fig. 22. Disturbance growth measured in case (1,1) where the primary travelling mode was excited by the membranes oscillating with amplitudes of RK  "10, 15, and 20 lm. 䉭 excited X primary mode (F ); 䉫 first harmonic (F ); second harmonic (F );    * naturally developing stationary mode. Q "12.5 m/s,  U "43.5°, Lerche [28]. 

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Fig. 23. Mean velocity profiles ºM measured in the cases (0,1), upper figures, and (0,1)#(1,1), lower figures, at three downstream locations  x /c ranging from the beginning of the nonlinear regime up to the saturation state. Q "16 m/s, U "43.5°, Lerche [28]. A  

range of amplification. The main difference is that for larger excitation rates the saturation state is reached earlier and the higher harmonics appear further upstream. Moreover, it is interesting to note that the primary disturbances arrive roughly at the same saturation level as it has also been shown for stationary modes by Reibert [39]. An instructive insight into the nonlinear features and their dependence on the initial conditions is provided by the velocity profiles. As an example Fig. 23 compare spanwise series of mean flow profiles for three subsequent chord positions ranging from the beginning of the nonlinear development up to the saturation state in the cases (0,1) and (0,1)#(1,1). The profiles of each series, plotted one over the other, were measured at small distances apart in span, so that the spreading indicates the presence of stationary vortices and the spreading width their amplitudes. At the most upstream position, which is located at the beginning of the nonlinear regime, the width of the spreading, i.e. the amplitude of the stationary vortices is almost the same in both cases. However, at the two downstream positions a considerable difference is observed. The travelling mode additionally excited in case (0,1)#(1,1) nonlinearly attenuates the growth of the stationary mode so that it ends up at a considerably lower saturation level. This feature will prove to be crucial for the onset of the final breakdown of laminar flow (see Section 6). It explains the difference in the nonlinear

developments and the saturation amplitudes as measured on the same model in different disturbance environments (Figs. 17 and 18). The larger the initial amplitudes of the unsteady modes compared with those of the travelling modes, the smaller are the saturation amplitudes of the stationary modes. This dependence of the disturbance development on the initial conditions may also explain why Poll [5] established the formation of stationary vortices on his swept cylinder whereas on an equal model, Takagi and Itoh [25] observed only travelling waves. Finally, it is interesting to note that this nonlinear feature qualitatively confirms the results of the nonlinear parametric study of swept Hiemenz flow by Malik et al. [44] as well as the nonlinear PSE and DNS calculations by Bertolotti [45] and Mu¨ller et al. [46], respectively, but it is in contrast to the observations in the ITAM experiment by Kachanov [22] where travelling waves, excited by a vibrating ribbon, increase the growth of stationary vortices by an order of magnitude. Another aspect of the difference in the disturbance evolution becomes most obvious considering the unsteady disturbance profiles in cases (1,1) and (0,1)#(1,1). In Fig. 24 the spanwise variation of the unsteady disturbances is compared both for the modes having the frequency of the excited primary travelling wave, and for the nonlinearly generated modes having the frequency of the first harmonic. In contrast to case (1,1), in the mixed case (0,1)#(1,1) a strong spanwise variation occurs

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Fig. 24. Spanwise variation of the disturbances having the frequency of the excited primary travelling mode ( f "82 Hz) and the first  harmonic ( f "164 Hz). x /c"0.93 (saturated state) in cases (1,1) and (0,1)#(1,1), Lerche [28].  A

Fig. 25. Mean velocity profiles measured in a downstream location of saturation for the cases (1,1), (0,1)#(1,1), and (0,1). Shadowing indicates the width of the spanwise spreading of the mean flow profiles. (- - - ) base flow profile, ( ) spanwise averaged mean flow profile, (——) selected profile. Q "16 m/s, U "43.5°, Lerche [28].  

which was observed to be associated with the appearance of the stationary vortices, as pointed out in Section 3.1 (Figs. 8 and 9). It suggests the existence of various modes having the same frequency. Additionally, in a more advanced stage of the nonlinear development, an interaction between the disturbances and the base flow was observed. It manifested itself by the deformation of the spanwise averaged mean velocity profiles. This becomes obvious in Fig. 25 which compares the mean flow profiles for the three cases measured at a chord position where the primary instabilities

have arrived at saturation. The shaded area displays the width of the spanwise spreading which is a measure of the intensity of the stationary vortices. The most deformed profiles are specifically selected for comparison with the undisturbed laminar boundary layer and with the spanwise averaged mean flow. We note that the undisturbed laminar boundary layer profile, which is used in linear theory, is clearly different from the spanwise averaged mean flow. In case (0,1), where only a steady mode is stimulated, the spanwise spreading, and correspondingly the deformation of the spanwise averaged mean flow

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profile at saturation, is by far largest. The additional excitation of a travelling mode (1,1) reduces the width of the spanwise spreading considerably. Finally, in case (1,1) the spanwise spreading is only very weak. However, a mean flow distortion is observed in all three cases. The resulting difference between the spanwise averaged mean and base flow profiles is described in nonlinear theories as due to the interaction between the base flow and the disturbances and is called (0,0) mode. The largest (0,0) mode is found in case (0,1). The fact that the phase relations of the unsteady motion are known under controlled conditions allowed Lerche [28] to apply the spatial and temporal Fourier decomposition to determine the amplitudes, frequencies, and wave number vectors. It turned out that, especially in the mixed case (0,1)#(1,1), the disturbances of each of the two frequencies (shown in Fig. 24) comprised Fourier modes of different wave number vectors. An overview on the Fourier modes identified by Lerche [43] for the three cases studied are depicted in the scheme of Fig. 26 which was originally used by Mu¨ller et al. [46] to present the results of their DNS computations on the transition process of the DLR swept flat plate experiment. In each case higher harmonics of the stimulated primary instability had developed, but in the mixed case (0,1)#(1,1) additional modes of equal frequencies, but different spanwise wave numbers were generated so that the number of Fourier components having measurable size was much larger. The spanwise modulation of the travelling modes and the deformation of the mean flow under the action of the stationary vortices exhibited in Figs. 8, 23 and 25 suggest that the stability characteristics had changed significantly and it is this phenomenon which inspired Reed [47], Fischer and Dallmann [48], and Malik et al. [44] to perform secondary stability analyses (Section 4.2).

A most striking feature of the mean flow profiles was the formation of inflection points first appearing at some spanwise location in the stationary vortex field, both for the streamwise and crosswise components. This suggests that they became secondarily unstable and indeed, connected with the formation of these inflectional profiles, a high-frequency secondary instability appeared which lead to the onset of the final breakdown of the laminar flow as discussed in the next section. The results of these controlled experiments enables us now to interpret the difference in the disturbance development measured in wind tunnels with different free stream fluctuation content and on models with different surface finish as described in Section 3.3. Different disturbance environments generated different initial conditions corresponding to the cases (0,1), (0,1)#(1,1) and (1,1). How the disturbances enter the boundary layer and fix the initial conditions is a question of receptivity. This will be discussed in Section 5. The crucial question to be tackled next is the mechanism initiating the final breakdown of laminar flow and the influence of the initial conditions. 3.5. Final breakdown of laminar flow In Section 3.2 it has been shown that the disturbances grow until they become saturated (Fig. 14). This state can extend over a considerable downstream distance before the laminar flow finally breaks down. Thus, in order to enable the prediction of transition, we need to find the mechanism that initiates the final breakdown after the disturbances have arrived at saturation. Arnal et al. [6] and Poll [5] gave the first experimental hint. In the instantaneous hot-wire signal traced on the swept wing and on the swept cylinder, respectively, they observed the occasional appearance of higher-frequency disturbances

Fig. 26. Schematic of the nonlinearly generated higher order modes obtained in the DNS computations for the DLR swept flat plate experiment by Mu¨ller et al. [46] and the higher order modes experimentally identified by Lerche [28]. The excited primary instabilities are shaded black and the experimentally identified higher modes are shaded grey.

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Fig. 27. Hydrogen bubble pattern of the transitional flow on the swept flat plate (top view on the surface). The hydrogen bubble generating wire (¼) is positioned parallel to the leading edge at x /c"0.4, z/d"0.25, B"turbulent burst. U "45° (geometric sweep A  angle), Re "0.75;10. A

superimposed on the fundamental wave prior to transition. These higher-frequency disturbances had frequencies one order of magnitude larger than those of the fundamental waves. Further evidence was established in the ASU and DLR experiments. Like Arnal et al. [6], Kohama et al. [49] as well as Bippes [37] and Deyhle and Bippes [33] identified a high-frequency band in the frequency spectrum prior to transition. As in the experiments by Arnal et al. [6] and Poll [5], the higher frequencies were one order of magnitude larger than those of the pre-existing primary band. Kohama et al. [49] found the appearance of the high-frequency disturbances to be confined to certain spanwise locations where the spanwise mean flow profiles become inflectional under the action of the stationary crossflow vortices and interpreted them as secondary instability. They described them as corotating vortices with axes in the crossflow direction and moving in the chord direction. Moreover, they stressed that it is neither a ¹ollmien— Schlichting wave nor any sort of travelling wave. In flow visualizations performed in the DLR water towing tank by Bippes [37], i.e. in a disturbance environment similar to free flight conditions, stationary vortices dominate up to the very end of the transition process, but then, unsteady modes undergo an almost explosive growth that

 Probably to be read as streamwise direction.

immediately leads to transition. In the towing tank tests, this looks like a bursting of individual stationary vortices (Fig. 27). In order to establish further details on the characteristics of this secondary instability, Lerche [43], Bippes and Lerche [50] (for details see [28]) conducted controlled experiments. They studied two different cases. In the first case, (1,1), only a travelling wave was initiated and in the second, (1,1)#(0,1), both a travelling wave and a stationary vortex, each in the range of the most amplified primary modes. These experiments allowed the phase locked tracing of the instantaneous velocity field and therewith the spatial and temporal analysis of the disturbance motion. Like Arnal et al. [6] (see their Fig. 21) and Poll [5] (see his Fig. 14), Lerche identified the high frequency secondary instability in the hot-wire records. A typical example is shown in Fig. 28a. There, the lowfrequency wave represents the excited primary travelling wave and the superimposed high-frequency oscillations indicate the occurrence of the secondary instability. Some inherent features become evident. First of all, the high-frequency oscillations appear in each cycle of the excited primary travelling wave. This implies that the secondary instability is correlated to the primary travelling wave. In the wall distance where the secondary instability first appears, we find it in the phase range of decreasing velocity. The instantaneous profiles forming at each cycle of the excited primary travelling wave (Fig. 28b) and the phase interval in each cycle where the

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Fig. 28. Flow characteristics at the onset of the high frequency disturbances in case (1,1) at a spanwise position of their first occurrence. Q "16 m/s; U "43.5°, z/d"0.7, x /c"0.9. (a) Hot-wire record (over 4 cycles ¹ of the excited primary wave); (b) instantaneous   A profiles; (c) related power spectrum. The shaded area labels the high frequency disturbances.

high-frequency oscillations occur (shaded areas in Fig. 28a,b) reveal the conditions which lead to this instability. It originates during the phase interval where the instantaneous profiles become inflectional and is centred around the inflection point with larger jº(z, t)/jz which is located closer to the edge of the boundary layer. This infers that the secondary high-frequency instability is an inflectional instability of the nonlinearly disturbed flow as suggested by Kohama et al. [49]. However, it is not necessarily due to the inflection point in the normal to wall jº (z)/jz (Fig. 25) or spanwise jº (y)/jy (see Fig. 15) mean flow profiles but rather to that one of the instantaneous normal to wall profiles (Fig. 28b). Moreover, it is neither riding over the primary stationary vortices nor has it the structure of corotating vortices as described by Kohama et al. [49] but it is moving with the primary travelling wave confined to the interval with instantaneous inflectional profiles. In order to localize the origin of the secondary instability in the stationary vortex field, we consider it in the mixed case (1,1)#(0,1) with saturation levels of the stationary modes '10% but (20—22%. This origin can be recognized most clearly in the isotach patterns of the stationary vortices in a plane normal to the plate

chord. A typical result is depicted in Fig. 29. In this figure the area where the secondary instability is present during one cycle of the primary travelling wave is bounded by solid lines. It turns out that it is not centred in the region of high spanwise shear as found by Kohama et al. [49] but somewhere in the outer portion of the stationary vortices where the streamwise component of the stationary disturbance uN is negative or more precisely near the  spanwise positions of º (y , z)"min º (y , z).   A W  A Hence, in the controlled experiments at DLR the secondary instability originates at the inflection points of the instantaneous velocity profiles which first appear along streamwise zones in the stationary vortex field near spanwise locations of ºM (y , z).   A The experimental evidence established in the controlled experiment at DLR can be summarized as follows: The onset of the secondary high-frequency instability which initiates the final breakdown of laminar flow is associated with the formation of inflectional profiles of

 See the appendix for definition of the velocities and Fig. 1a for the orientation of the coordinate systems.

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Fig. 29. Isotachs of the stationary vortices in a cross section of the mean flow and locations of the spanwise periodic appearance of secondary high frequency instabilities in case (0,1)#(1,1). These secondary instabilities are centred in the areas bounded by the solid lines. Q "16 m/s; "43.5°; z/d"0.7; x /c"0.9, Lerche [28].  A

Fig. 30. Streamwise and cross component of mean velocity profiles ºM (z), » (z). º , » are profiles at spanwise positions of   QG

I QG

I maximum deformation in the streamwise or cross components, ºM , »M spanwise averaged mean flow profiles, º , » base flow profiles,     j boundary layer thickness of the streamwise flow. Q "20.5 m/s, U "43.5°.   

the velocity distribution normal to wall. Obviously, the velocity gradient j/jz at the inflection point has to exceed some critical value. This critical value is first achieved only instantaneously with the periodicity of the primary travelling wave near spanwise locations of ºM (y , z). Only in the limiting case where the flow is   A primarily disturbed exclusively by stationary vortices, the secondary high-frequency instability is expected to appear permanently. High saturation amplitudes of the stationary vortices advance the formation of the critical value and thus the breakdown of laminar flow. However, it should be borne in mind that inflectional velocity profiles, due to mean flow distortion, were observed for all initial conditions between the limiting cases (1,1) and (0,1), as shown in Fig. 25. This may explain why the high-frequency instability appeared independently of the strength of the stationary crossflow vortices.

Up to date, no attempt has been made to establish the existence of a critical gradient jº (z, t)/jz at the inflection point. In searching for such value, it has to be borne in mind that we considered only the streamwise component of the three-dimensional velocity profiles, but other directional profiles QW (Fig. 1) between º and $»   (Fig. 1c) may have larger gradients at the inflection point. This may become evident in Fig. 30 where the mean flow profiles of the streamwise and cross components are compared for various spanwise positions. According to the early work by Stuart in Gregory et al. [4], in the linear range of amplification within the limits of a localized consideration, the instability waves are governed by the velocity components in their propagation direction. If such an argument also applies to localized regions in the nonlinear range, the propagation direction of the secondary high-frequency waves would give some hint whether

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it is, in fact, the three-dimensional profile that has to be considered. Unfortunately, the propagation direction of the high-frequency waves has not yet been unequivocally established. Some evidence was found that it points between º and !» .   In the above discussions of the experimental results on the high-frequency secondary instability reported from the ASU and DLR experiments, considerable discrepancies become obvious. Thus, the question arises whether in fact two different breakdown scenarios were observed, or whether the results were simply not well understood. Unfortunately, the conclusions by Kohama et al. [49] are partly deduced from sophisticated interpretations of flow visualisations, that are streak patterns of the unsteady disturbance motion. Measurements are scarce, so that direct comparisons are very difficult. Besides that, the saturation rates of the stationary vortices in the ASU and DLR experiments were different. In the ASU tests they amounted to 28% of the local inviscid flow (20% r.m.s.) whereas in the DLR tests they were ranging from 20% down to 2% and in the NAL tests they were not even clearly present. The large amplitudes observed in the ASU experiment may also explain why Kohama et al. [49] traced the high-frequency disturbances at the spanwise high shear layer and not at the inflection point of the normal to wall profile at the upper third of the boundary layer. Further experiments are needed to overcome the discrepancies in the interpretations. Concluding this section, it should be noted that the high-frequency secondary instability does not have to be the only mechanism initiating the final breakdown of laminar flow. At the DLR swept wing experiment (Fig. 6e), Po¨thke [51] established evidence for at least one other breakdown scenario. It appeared somewhere in

the saturation state of the primary disturbances and manifested itself as an accelerating augmentation of unsteady modes. This became most obvious in the gradual downstream widening of the frequency spectra (Fig. 31) and did not allow to localize transition. The conditions for its appearance are still rather obscure. It was observed beyond the pressure minimum where the crossflow changes direction. The related profiles were only weakly or not inflectional, i.e., (0,0) modes were small even though streamwise vortices of 12% amplitude were measured. Further experiments are needed to establish the conditions for the appearance of this breakdown mechanism.

3.6. Effect of surface curvature 3.6.1. Convex curvature Stability analyses predict a damping effect of convex curvature. However, up to date no experiments are known specifically aimed at demonstrating the effect of convex curvature in the presence of dominating crossflow instability. Indirect information may be obtained comparing the experiments on weakly curved or flat surfaces at ONERA/CERT [6], ASU, ITAM, and DLR with these on the swept cylinders by Poll [5] and at NAL. Unfortunately, the different environmental conditions under which the experiments were performed do not allow precise statements (see Section 3.3). Thus, even experiments on equal models as conducted by Poll [5] and Takagi and Itoh [25] yield different results. Whereas in the former, the formation of stationary vortices was visualized, no such disturbances were observed in the latter, although the flow was highly unstable to crossflow instability and stable to Tollmien—Schlichting instability. This

Fig. 31. Frequency spectra measured by Po¨thke, [51] in the breakdown stage of the flow on the DLR swept wing for increasing values Q . U "30°.  

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may mainly be due to the exceptional surface finish of the models of Takagi and Itoh [25] with roughness levels between 1.3 and 0.07 lm r.m.s. Unfortunately, the surface finish of Poll’s model is not known. However, since the free stream turbulence in the wind tunnel used by Poll is rather larger than in the wind tunnels used by Takagi and Itoh [25] and thus favouring unsteady modes, it might be conjectured that on significantly curved convex surfaces, also in the presence of high crossflow instability, stationary vortices are less amplified than on flat or weakly curved surfaces. Some further evidence is supplied by the experiment of Bippes [52] on a wing shaped model (Fig. 6d) where a profile segment of concave curvature is followed by

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a segment of convex curvature (Fig. 32). On the model in swept position, stationary vortices were strongly amplified along the concave segment of the model whereas they were damped on the convex segment in spite of large negative pressure gradient, i.e. large crossflow. In contrast to that, the unsteady modes keep growing (Fig. 33a). In a current experiment at DLR on an airfoil with a profile of 16% thickness ratio set at a sweep angle of 45° (Fig. 6e), the saturation amplitudes of the stationary vortices differed considerably from those observed on the swept flat plate in the same wind tunnel. On the considerably curved upper side of the airfoil, they only amounted to 12% instead to 20% on the swept flat plate, although the amplification rates of the stationary

Fig. 32. Profile of the model, chordwise pressure distribution c ((䉭) upper side, ()) lower side), velocity of the inviscid flow N Q (*) measurement, (- - - ) boundary layer calculation), and local Go¨rtler parameter G "2 (Q d /l )d /R along the chord x /c.  V C V  V A d "x l /Q and R"650 mm (radius of curvature of the concave segment). V A  C

Fig. 33. (a) Chordwise growth of stationary vortices uª and travelling modes u (u ) measured under natural conditions of transition on     the model in Fig. 32 and comparison with local linear stability theory with and without surface curvature. Q "15 m/s, U "30°.   (b) Free stream velocity leading to transition at fixed chord positions measured on the model in swept (U "30°) and unswept (U "0°)   positions, a"angle of incidence.

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vortices theoretically predicted (without taking surface curvature into account) are rather larger than those predicted for the DLR swept flat plate and although both models had the same surface finish. For practical purposes, the main question is aimed at the influence of surface curvature on transition. Again, the answer cannot be given directly. However, according to Deyhle and Bippes [33] larger saturation amplitudes advance transition (see Section 6). Thus, the smaller saturation amplitudes measured on the swept wing compared with those traced on the swept flat plate under equal environmental conditions suggest that convex curvature rather delays transition. This statement may further be substantiated by the tests on the swept model with significant concave and convex curvature where the free stream velocity for transition in dependence upon the chord position was determined (Fig. 33b). Comparison of the results obtained under swept and unswept conditions reveals that in contrast to the concave model segment, wing sweep advances transition only moderately on the convex segment, even with a much larger negative pressure gradient (see Fig. 32) and much larger crossflow than on the concave segment. If it is taken into account that under swept conditions the instabilities were much more amplified than in the unswept case when arriving at the segment with convex curvature, this becomes even more obvious. The experimental observations may be summarized as follows: No hint was found that convex curvature changes the physics of the transition process up to the final breakdown of laminar flow, but it influences the disturbance amplification. Hence, in transition criteria for swept airfoil flow under flight conditions which becomes unstable close to the highly curved nose region, surface curvature should be taken into account. It is worth to

note that up to date the e,-methods used in practical applications still neglect surface curvature. 3.6.2. Concave curvature In the past, numerous theoretical and experimental work on centrifugal instability in two-dimensional boundary layers was carried out. Much less has been done in three-dimensional boundary layers. The main difference in the stability conditions is that in the threedimensional case the flow along concave walls is unstable not only to centrifugal instability, but also to crossflow instability. To date, only some theoretical analyses are known (Section 4.3). The first basic experiments have been started at DLR by Bippes [52, 53]. The profile of the model and the pressure distribution has already been shown in Fig. 32. The sweep angle was set at 30° and the angle of incidence a"!4°. Infinite sweep conditions were approximated by contoured end plates. At the free stream velocity chosen for the tests, the flow became unstable at x /c+0.05. At this chord position, the pressure gradient A was almost zero (Fig. 32) so that centrifugal instability should have dominated. Further downstream the pressure gradient increased and arrived at rather large values towards the end of the concave segment so that crossflow instability became more and more effective. Under natural conditions of transition, the main feature observed was that all along the concave segment, the flow was most susceptible to stationary vortices. The wave number spectrum in Fig. 34 reveals that at x /c"0.34, i.e. toA wards the end of the concave segment, a certain wavelength had carried through. Controlled experiments, where the stationary vortices were excited artificially in the region of neutral instability with the wave number of the most amplified mode observed under

Fig. 34. Wave number spectrum of the stationary instabilities uª measured under natural conditions of transition at various chord  positions and comparison with the amplification rates a at x /c"0.34 of local linear theory (- - - -) by Hein [54]. A

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natural conditions of transition, provide a better insight (Fig. 35). In this case, only the forced mode peaks in the wave number spectrum up to x /c"0.24 (Fig. 35a). A However, at the subsequent location, x /c"0.31, besides A of the wave numbers of the harmonics additional peaks at smaller and larger wave numbers are found. There, the amplitude of the mode with smaller wave number is larger. At the subsequent measuring position, the pronounced peak at smaller wave number has completely disappeared and the peak at the larger wave number has considerably been reduced, only the first harmonic has further increased. For comparison with theory, it is worth to note that the changes in the wave number spectrum look different if the nondimensional quantities are plotted (Fig. 35b), as it is usually practised in theoretical work. Now, the wave number first dominating disappears and a smaller wave number originating further downstream carries through. In order to detect structural changes which may identify the eventual simultaneous appearance of Go¨rtler and crossflow vortices, changes in the mode shapes may be of special interest (Fig. 36). Surprisingly, the smallest change is found between x /c"0.24 and 0.31 where the addiA

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tional peaks in the wave number spectrum (Fig. 35b) have become most distinct while between x /c"0.31 and A 0.36, where the forced mode has carried through, the difference is most pronounced. Thus, in the range of Go¨rtler parameters appearing in the experiment (see Fig. 32 for the local Go¨rtler parameters) and at the sweep angle U"30°, no clear evidence could be established that both Go¨rtler and crossflow modes can be amplified as it is theoretically predicted by Bassom and Hall [55] and Zurigat and Malik [56]. For a more detailed analysis of the disturbance motion, additional measurements with much higher spatial resolution are needed. However, since on one hand, on the model set at U"0°, Go¨rtler vortices developed only very weakly, and on the other hand on the model set at U"30°, stationary vortices exhibited larger amplitudes than in the same wind tunnel on the swept flat plate having the same surface finish than the curved model, the stationary vortices traced may be seen as crossflow instabilities where the growth is favoured by concave curvature. A further striking difference between two- and threedimensional centrifugal instability was observed in the primary instability modes. Whereas in the two-dimensional

Fig. 35. Wave number spectrum of the stationary instabilities uª measured under controlled conditions at various chord positions. (a) in  dimensional and (b) in non-dimensional quantities. Q "15 m/s; U "30°.  

Fig. 36. Mode shapes of the stationary vortices measured under controlled conditions at 3 successive chord positions for the streamwise and spanwise components. Q "15 m/s; U "30°.  

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Fig. 37. Frequency spectrum of the unsteady instabilities u measured under natural conditions of transition and comparison with the   amplification rates a of local linear theory by Hein [54]. Q "15 m/s; U "30°, x /c"0.31.   A

case, i.e. in the Go¨rtler case, the flow was primarily first unstable to stationary vortices, in the three-dimensional case both stationary and travelling modes were amplified as shown in Fig. 33a. The broad band of amplified frequencies that includes very small frequencies (Fig. 37) is typical for crossflow instability. The crucial question in which way concave curvature affects transition could be answered only indirectly by comparing the location of transition appearing on the model in swept and unswept positions. Most obvious was the increase in the amplification of both stationary vortices and travelling waves. Thus the result depicted in Fig. 33 reveals that along the concave segment, the sweep angle promoted transition considerably. No observation was made that clearly indicated the presence of a Go¨rtler mechanism competing with the crossflow instability mechanism.

4. Experiment and theoretical prediction In the previous sections, the experimental observations on the physical features of the transition process governed by crossflow instability were described. In the following, the ability of stability analyses to predict the observed developments will be addressed by comparison with experiment. For an overview on the available approaches, on their effectiveness and limitations, we refer to Haynes et al. [57]. Some qualitative comparison with results from linear and nonlinear analyses have already been made in Section 3. Thus, in the following most emphasis will be placed on quantitative comparison. Although it is evident that for these purposes only controlled experiments yield reliable data, experiments performed under natural conditions are of specific interest since in practice these are the conditions under which transition occurs. Hence, wherever it is possible, besides

experimental results obtained under controlled conditions, those obtained under natural conditions are also used for quantitative comparison. 4.1. Linear theory As it has been pointed out in Section 3, linear local and nonlocal approaches correctly predict the wavelengths of the most amplified stationary modes [6, 18, 29] as well as the frequency and wave number vectors of the most amplified nonstationary modes [29, 32] experimentally traced under natural conditions (Figs. 11—13). Much less agreement is found comparing the disturbance growth ([30, 41, 58], see Figs. 14 and 16). In [37], experimental evidence was established which implies that besides the inherent problem to measure the growth of naturally developing disturbances, this may be due to the early nonlinear development which starts as soon as the stationary disturbances become measurable (Fig. 15). Controlled experiments yielding precise quantitative comparison with theory are only known from ITAM, ASU, and DLR. (The NAL work was focused on streamline curvature instability and will be discussed below.) At ITAM, Kachanov [22, 59] compared the amplification rates with the results of local linear theory. He found excellent agreement. However, following the introduction of the parabolized stability equations (PSE) by Herbert and Bertolotti [60] nonparallelism and streamline curvature can be easily accounted for (for details see [45, 61, 62]). This approach has been matured in numerous applications and has proved to be superior to local analysis in many regards (for a comprehensive discussion on its benefits and the experiences in recent applications, we refer to the review by Herbert [62]). In particular, it is generally agreed that in flows dominated by crossflow instability, PSE and local approaches yield significantly different amplification rates (see also [63]). In

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comparative computations on the ASU swept wing, Haynes [64] found the large differences between local and nonlocal linear approaches precisely in the range of neutral stability where Kachanov [59] performed his measurements. Moreover, in this range, accurate measurements require the forcing of large amplitude disturbances which may surpass the limits of linear behaviour. Hence, the agreement between experiment and local linear theory stated by Kachanov [59] must be seen as an exceptional result. At ASU and DLR, the results of controlled experiments are compared with PSE computations which take nonparallelism into account. A further crucial advantage of PSE approaches is that they allow for the direct comparison of the disturbance amplitudes if the initial conditions are known. At DLR, the growth of amplified travelling and stationary crossflow modes was measured in a controlled experiment on the swept flat plate [31]. In order to get a crude idea on the extension of the linear regime in the presence of small stationary vortices, an amplified travelling crossflow mode was excited at various small amplitudes. In the most upstream measuring domain, this led simply to a parallel shift of the growth as expected from linear behaviour (Fig. 22). The appearance of the higher harmonics indicates that the nonlinear regime started at r.m.s.-amplitudes less than 1% of Q . Besides the stimu lated travelling modes, stationary vortices were also present. In the linear regime, they range from 0.2 to (0.5% of Q . Since hot wire-measurements of such small mean  velocities may not be precise enough, the comparison between experiment and the PSE computations by Bertolotti (in [65]) in Fig. 38 is restricted to a travelling crossflow mode. It should be pointed out that for the determination of the initial amplitudes, Bertolotti modelled the generation of the instability by the spanwise array of oscillators in order to make use of the full benefit of the PSE approach. Haynes [64] and Reibert and Saric [66] compared the growth of stationary vortices and found very good agreement with PSE computations that take surface curvature into account, whereas parallel theory was found to underestimate the growth (Fig. 39). Thus, evidence established so far indicates that only stability approaches that account for nonparallelism and streamline curvature are able to correctly predict the linear growth of crossflow instabilities. Moreover, the work by Haynes [64] and Reibert and Saric [66] implies that curvature has to be taken into account even if it is very weak. An interesting linear stability analysis was recently performed by Itoh [67, 68] with specific regard to the effect of streamline curvature. In addition to crossflow instability he found an instability caused by streamline curvature. This streamline curvature instability yields instability modes that are characteristically different from crossflow modes. Stability calculations for the flow on the NAL swept cylinder predicted this instability to

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Fig. 38. Spatial growth of a travelling instability mode artificially excited using a spanwise array of membranes (O) in the presence of a weak stationary mode and comparison with PSE computations (receptivity model included). Q "12.5 m/s;  U "43.5°, Wu et al. [65]. 

occur at very small distances from the stagnation region and to give the lowest critical Reynolds numbers. Further downstream crossflow instability becomes more amplified. This may be the reason why instability modes resulting from Itoh’s analysis are not reported in PSE computations. However, Takagi et al. [26] established some evidence for the existence of such an instability with the aid of controlled experiments. In other experiments, streamline curvature instabilities were not identified. The reason may be that they resemble the crossflow modes and propagate roughly in the same direction [68]. 4.2. Nonlinear approaches Inspired by the experiment of Saric and Yeates [8], Reed [47] introduced a secondary theory that predicts the appearance of superharmonic stationary vortices having three times the amplitude of the primary mode as a result of disturbance-wave interactions. Fischer and Dallmann [48] considered the stability of a flow containing stationary vortices. They demonstrated that the flow becomes susceptible to secondary instability waves even at very small amplitudes of the primary stationary vortices, as observed in the experiment (Figs. 8 and 15). Unfortunately, the frequencies of these secondary instabilities are only slightly larger than those of the primary modes so that the experiment gives no clear answer whether the predicted secondary mechanism is present and in which way it contributes to the disturbance growth. However, the predicted amplification of secondary wave combinations qualitatively explains the spanwise amplitude modulation of the unsteady disturbances and the splitting in their growth observed in the experiments (Figs. 8, 9, 15, 17 and 18).

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Fig. 39. Spatial growth of stationary crossflow vortices on a swept wing artificially excited using a spanwise array of roughness elements [39] and comparison with PSE computations by Haynes [64] (Fig. 5.111) that take surface curvature into account. Re "2.4;10,  U "45°. 

Fig. 40. Qualitative comparison of the spanwise variation of unsteady disturbance profiles between theory and experiment. (a) experiment by Deyhle and Bippes [33]; (b) DNS computation by Meyer and Kleiser [69] (subscript v denotes the coordinate system aligned with the axes of the stationary vortices). Q "20.5 m/s; U "43.5°.  

DNS computations by Meier and Kleiser [69], Wagner [70], Wintergerste and Kleiser [71] and Mu¨ller et al. [46] and nonlinear PSE computations by Malik et al. [44], Bertolotti [45] and Haynes [64] have already demonstrated that the nonlinear processes and their dependence on the initial conditions as observed in the experiments are qualitatively correctly described (see Section 3.4 and Figs. 26 and 40 for comparison between DNS and experiment). The crucial question is now whether the disturbance development in the nonlinear

range of amplification can also be predicted quantitatively. The answer to this question requires the comprehensive quantitative comparison between theory and experiment. For this purpose experiments have to be carried out which allow to introduce well-defined individual disturbances of small initial amplitudes and to trace their downstream growth together with the connected nonlinear developments. To date, only two attempts to validate PSE codes are known. At DLR, Bertolotti [45] performed nonlinear

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Fig. 41. Spatial growth of stationary crossflow vortices artificially excited on the swept flat plate using a spanwise array of roughness elements and comparison with PSE computations (receptivity model included) by Bertolotti [45] for various excitation intensities t of the primary travelling modes. Q "20.5 m/s; U "43.5°.  

PSE computations to predict the growth of stationary vortices and their saturation amplitudes for the swept flat plate flow. In order to determine the initial amplitudes, he modelled the generation of the stationary vortices by the spanwise array of roughness elements used in the experiment for the disturbance excitation. Neglecting unsteady modes the saturation amplitudes of the stationary vortices became overpredicted (Fig. 41). However, since in the experiment, that was performed in 0.15% turbulence environment, unsteady modes were also excited, Bertolotti repeated his calculations with the initial amplitudes of the unsteady modes set nonzero. With this

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procedure, he could not only match the saturation amplitudes with those of the experiment but also explain the disturbance interactions described in Section 3 (see also Figs. 23 and 25). In the other attempt to validate a nonlinear PSE code, Haynes [64] calculated the growth and saturation of the stationary vortices for the flow on the ASU swept wing. In order to fit his computations to the experiment by Reibert [39], Haynes chose the initial conditions to match computational and experimental amplitudes at some chord position. If he took surface curvature into account, he found an excellent agreement (Fig. 42). This is a surprising result since unsteady modes were neglected in the computations. It suggests that for turbulence levels not larger than those of the ASU wind tunnel, there is no significant interaction between stationary vortices and travelling modes as in the DLR swept flat plate experiment so that the determination of the saturation amplitudes of the stationary vortices becomes reduced to the steady case in Fig. 41. Since the free stream turbulence under free flight conditions is expected to be even smaller than in the ASU wind tunnel, and because of the correlation between the saturation amplitudes of the stationary vortices and transition (see Section 6), it would be of high practical interest to further substantiate this suggestion. Certainly, a final statement on the efficiency of the PSE codes to quantitatively predict the disturbance growth and saturation needs more comprehensive comparison

Fig. 42. Spatial growth of stationary crossflow vortices artificially excited on a swept wing using a spanwise array of roughness elements [39] and comparison with nonlinear PSE computations by Haynes [64] (Fig. 6.24). Re "2.4 ) 10, U "45°. A 

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with controlled experiments. However, the results in [45, 64] suggest that the saturation amplitudes of the stationary vortices can be correctly predicted provided that the initial conditions are known. In this regard the introduction of PSE approaches is an essential step forward in transition prediction. Attempts to validate DNS codes are not known so far. 4.3. Secondary instability and final breakdown of laminar flow The deformation of the mean flow under the action of stationary vortices and the appearance of high-frequency waves prior to breakdown were the motivator for the secondary instability analyses by Fischer et al. [72] and Malik et al. [44]. They studied the instability of the flow primarily disturbed by stationary vortices using a temporal local approach. Both approaches lead to similar results. They yield a secondary instability to travelling waves with frequencies one order of magnitude larger than those of the primary modes. These secondary high-frequency disturbances originate in the upper third of the boundary layer where the streamwise mean velocity profiles are secondarily inflectional due to the interaction between the primary disturbances and the base flow, as exhibited in Fig. 25 (see also Section 3.4). These features agree with the experimental observations by Lerche [43] (for details see [28]). In their later work Malik et al. [73] found three different secondary instability modes, two high-frequency modes with growth rates roughly of the same order and a lowfrequency mode. A similar result was obtained by Koch et al. [74] studying the linear, temporal secondary instability of the stationary equilibrium solutions for the DLR swept flat plate flow. The amplitude variation of the equilibrated crossflow vortices allowed Koch et al. [74] to determine possible secondary instability modes and the location in the stationary vortex field where they originate. The most amplified mode was found to appear at the inflection points of the spanwise mean velocity profile. However, high threshold amplitudes for the stationary vortices are necessary. In the ASU experiment with the model set at the angle of incidence a"!4°, those amplitudes are present. This may explain why Kohama et al. [49] traced the high-frequency secondary instability indeed at the inflection points of the spanwise mean velocity profile although the frequencies agree with those of mode II in the computations by Malik et al. [73] for the ASU experiment which originates at the inflection points of the normal to wall profile. In the DLR experiments, a breakdown initiated by the secondary high frequency instability is also observed at much smaller saturation amplitudes. They indeed originate at the inflection points of the normal to wall profile corresponding to the computations by Koch et al. [74].

In his controlled experiment, Lerche [28] traced inflectional profiles and high-frequency waves even in case (1,1) where the primary travelling mode dominated and the stationary vortices arrived at saturation amplitudes less than 2%. However, he clearly exhibited that it is not the inflectional mean flow profile but the inflectional instantaneous profile which first becomes unstable to the secondary high-frequency instability (Section 3.5). This may confirm another result of Malik et al. [73] for the other limiting case where the flow is primarily disturbed only by travelling crossflow modes. It demonstrates that this flow also becomes secondarily unstable to a high-frequency disturbance. Thus, the secondary instability approaches by Fischer et al. [72], Malik et al. [44] and by Koch et al. [74] cover only the limiting cases that the flow is primarily disturbed either by stationary vortices or by travelling waves. In practical cases, however, both stationary and travelling modes will be present. This suggests that secondary theories have to be developed for flows primarily disturbed both by stationary and travelling modes. Furthermore, it has to be confirmed that it is in fact the streamwise component and not a component in any other direction which governs the secondary high-frequency instability. According to the DLR experiments, the secondary high-frequency waves propagate in a direction that is oriented between º and !» .   It should be noted that to date nonlinear PSE or DNS approaches do not offer the resolution to capture this secondary high-frequency instability. On the other hand, in the DNS calculations, Mu¨ller [75] and Wintergerste and Kleiser [71] found the onset of the final breakdown process initiated by small secondary streamwise vortices originating close to the wall. This in turn is not observed in the experiments. On the other hand, at the DLR swept wing experiment (Fig. 6e), Po¨thke [51] established some evidence of another breakdown process. It originated somewhere in the saturation state of the primary disturbances and manifested itself as an accelerating augmentation of unsteady modes. This became most obvious in the gradual widening of the frequency spectra with increasing Q (Section 3.5 and Fig. 31). Such kind of  development is obtained both in DNS and nonlinear PSE computations. As an example, Fig. 43 shows the downstream birth and growth of nonlinearly generated higher modes due to the interaction between a stationary and travelling primary mode. It is a result of a parametric study of swept wing flow by Stolte [76] using a nonlinear PSE code. 4.4. The effect of surface curvature As described in Section 3.6, the experimental observations gave no hint indicating the appearance of any additional physical features due to surface curvature. The most obvious effect was the influence on the disturbance

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amplification. Hence, the following discussions concentrate on that.

lower side where the flow was investigated was very weak, and correspondingly, the metric coefficients in the stability equations were only slightly affected. This suggests that on modern transport aircraft where the flow becomes unstable already in the strongly curved nose region, the damping effect on the vortex growth should not be neglected. Unfortunately, it is exactly at this region where d/R is expected to be larger than O(1/Re) so that the results of the PSE computations are unsure. Thus, Hein [54] found in his comparative stability computations for the swept cylinder of Poll [5] (nose radius R"0.15 m), that the difference in the disturbance growth resulting from analyses with metric terms included or neglected (Fig. 44) is not as large as expected from the result for the extremely weakly curved airfoil portion considered by Haynes [64]. To date no quantitative comparison between experiment and PSE computations on the disturbance growth on such highly curved surfaces is known. Measurements in the nose region of airfoils are extremely difficult because of the high velocity needed and the extremely thin boundary layers associated. The DLR experiment on the considerably curved upper side of the swept wing is aimed to provide further evidence and data for code validation.

4.4.1. Convex curvature Unlike the local approaches [77], the PSE analyses do not have any specific difficulty with incorporating curvature effects, provided that d/R is O(1/Re). On most airfoils this condition will be satisfied except possibly for the nose region. The excellent agreement between the experimental N-factors measured by Reibert [39] and the PSE computations by Haynes [64] (Fig. 42) may be seen as valuable confirmation. It is important to note that the difference to the results with neglected surface curvature was significant, although the profile curvature on the

4.4.2. Concave curvature In the case of concave curvature, nonparallel approaches based on asymptotic analyses by Hall [78], Bassom and Hall [55], Otto and Denier [79], as well as the parallel approach restricted to the exclusive consideration of stationary modes by Zurigat and Malik [56] demonstrate that concave curvature has a destabilising effect on crossflow instability (also shown in the parallel approach by Collier and Malik [80] and that crossflow stabilises Go¨rtler instability. Moreover, Bassom and Hall [55] and Zurigat and Malik [56] determined limiting

Fig. 43. Spatial growth of a primary stationary vortex (0, !1, 1) and a primary travelling wave (1,1,0) and nonlinearly generated higher-order modes as a result of the parametric study on swept wing flow by Stolte [76] using a nonlinear PSE code.

Fig. 44. Spatial growth of a most unstable (a) travelling and (b) stationary crossflow mode resulting from stability computations by Hein [54], using different approaches, for the flow on the swept cylinder of Poll [5]. N"xV a dx (a is the chordwise amplification rate), G  G Q "28.9 m/s; U "63°.  

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conditions under which the Go¨rtler mechanism is completely suppressed. Itoh [81] studied the influence of centrifugal instability on the critical conditions and on the most unstable instability modes for Falkner— Scan—Cooke flow in dependence on the pressure gradient. He specified the parameter ranges where the flow first becomes unstable to Tollmien—Schlichting, Go¨rtler or crossflow instability. The experimental evidence is rather scarce. On the one hand, only our experiments [37, 52, 53] are known to the author, and on the other hand, except of the paper by Collier and Malik [80], the above references are parametric studies partly in the large Reynolds number limit [55, 78, 79]. They do not allow direct comparison. Moreover, they concentrate on the conditions where both the Go¨rtler and the crossflow instability mechanisms are operable and competing. In the available experiment no such processes could be identified. However, the evidence is not conclusive because the conditions for the suppression of Go¨rtler instability theoretically predicted [55, 56] were already achieved at x /c+0.2—0.3 (Fig. 32). At A this chord position the stationary disturbances were still relatively small (Fig. 35) so that depending on the initial conditions either the Go¨rtler mode or the crossflow mode could had grown up to measurable size. These initial conditions resulted from the environmental disturbances and the receptivity. According to experimental observations it is suggested that Go¨rtler instabilities are most sensitive to free stream turbulence [82, 83], whereas crossflow instabilities are essentially excited by surface roughness (Section 3.3). However, the present knowledge of the relevant receptivity mechanisms do not allow any quantitative estimate. In our experiment we observed different peaks in the wave number spectrum, as shown in Fig. 35. According to theory [55, 56], the critical conditions for the suppression of Go¨rtler instability became satisfied at x /c A +0.2—0.3. Since the negative pressure gradient upstream of the measuring domain (x /c+0.2— 0.3) was very small A (Fig. 32), theory would expect Go¨rtler vortices to appear first. Further downstream crossflow vortices with significantly smaller wave numbers should become more amplified, and indeed, at x /c"0.31 we found a second peak A at smaller a wave number in the wave number spectrum of Fig. 35b. Further downstream this wave number remained present while the wave number dominating upstream disappeared. However, if this would really have been due to the increase in crossflow instability, the difference in the wave numbers experimentally observed would have been much smaller than predicted. Furthermore, no theoretical hint is given for the rise and decrease of the larger wave number disturbance that is not a harmonic. It is suggested that this is due to nonlinear effects. A better physical insight into the stability features observed is provided by the spectrum with the dimensional wave numbers (Fig. 35a). It reveals that the dominating

dimensional wavelength was sustained all along the measuring domain on the concave segment and that the additional wave numbers which are not harmonics are intermediate manifestations. Hence, our experiment could not identify the existence of two competitive instability mechanisms as discussed in the parametric studies referred to above, and little evidence was provided to estimate their effectiveness. The preliminary stability calculations by Hein [54] for our experimental conditions [52, 53] allow some more direct comparison between theory and experiment. They predict only one band of wave numbers, and the wave numbers observed in the experiment that was performed under natural conditions of transition are indeed centred in this band (Fig. 34). The formation of the discrete peaks observed in the controlled experiment cannot be explained. It is suggested that this is due to nonlinear effects. As to the travelling modes, the same stability calculations correctly predict the frequencies of the most amplified modes (Fig. 37). Additionally, Fig. 33a shows that the travelling modes were more amplified than the stationary modes, as suggested by Bassom and Hall [55]. Less agreement was found for the chordwise growth. It is overpredicted both for the stationary and the travelling modes (Fig. 33a). The reason for this discrepancy may be that the experiment did not really capture the linear range of amplification as it was the case for the swept flat plate experiments performed under natural conditions of transition (Section 3.2). Hence, it can be stated that linear PSE approaches which do not neglect metric terms can predict essential stability features appearing on concave surfaces of swept wing configurations, however, detailed quantitative comparison on the disturbance growth in the linear and nonlinear regimes to estimate the effectiveness of theoretical approaches are not yet made.

5. Receptivity The discussions in Sections 3.3 and 3.4 have pointed out the importance of the upstream conditions for the disturbance development and transition. Additionally, it has been demonstrated in Section 4.2 that PSE and DNS approaches are able to calculate the disturbance development up to saturation, provided that the initial conditions are known. Hence, the major remaining problem is to determine the initial conditions resulting from the environmental disturbances. This requires the knowledge of the mechanisms leading to the entrainment of external disturbances into the boundary layer and to the generation of instabilities with initial amplitudes dependent on the external disturbances. Unfortunately, in three-dimensional boundary layers even less is known on this problem than in two-dimensional boundary layers. Some evidence has been established on the effect of surface roughness and free stream fluctuations. For a synoptical

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introduction into boundary-layer receptivity and its manifestation in transition experiments, we refer to the review by Saric [10] that also includes an almost complete list of the most relevant literature. 5.1. Receptivity to surface roughness The observations by Arnal et al. [6] (supplementary private information), Nitschke-Kowsky and Bippes [34] (supplementary private information) and Mu¨ller and Bippes [84], that the vortex pattern on the model did not change from test run to test run indicated two possible sources for the generation of stationary vortices: firstly surface roughness and secondly mean flow variations superimposed on the wind tunnel flows by the screens in the settling chamber (see [85]). In order to provide evidence for the former, Mu¨ller and Bippes [84] displaced their swept flat plate in the spanwise direction between successive test runs. It turned out that the vortex pattern was also displaced, fixed to the model (Fig. 45). This observation infers the forcing of stationary crossflow vortices due to surface roughness. The effect of the mean flow variations was examined by Deyhle and Bippes [33]. They determined their structure in the wind tunnels used for the experiments (Fig. 46) and compared it with the structure of the vortex pattern in the unstable boundary layer. In contrast to two-dimensional flows along concave walls (see [82, 86]), no correlation between the spanwise mean flow variations and the stationary crossflow vortices could be detected. This result implies that surface roughness, rather than free stream vorticity, is the main initiator of the stationary vortices. For further experimental studies on the effect of the shape, size,

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location, and distribution of roughness on vortex excitation and transition, we refer to Mu¨ller and Bippes [84], Radeztsky et al. [87], Deyhle and Bippes [33], Reibert [39], and Reibert and Saric [66]. The quantitative determination of the initial amplitudes of the stationary crossflow vortices due to surface roughness can only be achieved computationally. Based on the parallel flow analyses by Fedorov [88] and Manuilovich [89], Crouch [90] and Choudhari [91] studied the receptivity to stationary crossflow vortices in three-dimensional boundary layers as caused by roughness of different geometry. Their most important result is that forcing of stationary vortices by roughness is most efficient because it does not need any rescaling. However, quantitative comparisons with experiments showing that receptivity models as described by Crouch [90] and Choudhari [91] and implemented in PSE approaches would allow to calculate the growth of stationary vortices excited by surface roughness are still missing, and for the same reason the efficiency of transition criteria as proposed by Crouch [90] cannot be estimated. To date only the attempt by Bertolotti [45] is known that deals with the specific case of a DLR experiment where the stationary vortices are excited by a spanwise array of roughness elements. He describes a receptivity model for nonparallel flow that uses the exact geometry of the roughness elements and couples it with his nonlinear PSE code to calculate the growth of the stationary vortices up to saturation. The most important result is that the initial amplitudes are correctly predicted. For the precise determination of the saturation amplitudes the excitation of the travelling modes has also to be taken into account. This is a remaining problem. Bertolotti’s

Fig. 45. Spanwise variation of the mean velocity in the three-dimensional boundary layer before and after 5 mm and 9 mm spanwise in-plane shifts of the swept flat plate. Q "20.4 m/s, U "43.5°, z/d"0.25, x /c"0.9, Mu¨ller and Bippes [84].   A

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Fig. 46. 5 successive measurements of the crosswise mean flow variation of the free stream in the 1 MK. ºM and ºM are the streamwise components of mean and the spatial average of the mean velocity, respectively, measured at traversed probe and ºM and ºM the     corresponding values measured at fixed probe, Deyhle and Bippes [33].

results as well as the studies of Bottaro and Zebib [92] and Luchini and Bottaro [93] on the receptivity of Go¨rtler flow to surface roughness suggest that the receptivity of three-dimensional boundary layers to random surface roughness is a solvable problem. It is important to note that appropriate validation measurements have to be performed in an extremely quiet wind tunnel where the excitation of travelling modes can be neglected unless the forcing of those is also modelled. 5.2. Receptivity to free stream fluctuations As discussed in Section 3.3, the increase in free stream turbulence increases the amplitudes of the primary travelling modes. This implies that they are excited by the velocity fluctuations in the free stream. From numerous studies on the initiation of Tollmien—Schlichting waves by velocity fluctuations in the free stream, synoptically reviewed by Saric [10], we know that we have to distinguish between long and short wavelength fluctuations which are also called acoustic and vortical modes, respectively. This suggests the separate study of the effect of these vortical and acoustic modes on the initiation of travelling modes in the presence of crossflow instability. The first experimental evidence on the effect of sound was established by King [94] on a circular cone in supersonic flow. He investigated the receptivity to sound waves with the model in axisymmetric position, and in addition with the model set at an angle of incidence. This allowed to study the initiation of travelling modes both in the presence of Tollmien-Schlichting and crossflow

instability on the same model and under the same environmental conditions. It turned out that sound could stimulate travelling waves on the model only in the axisymmetric position, i.e. in the presence of Tollmien— Schlichting instability. In a further experiment on a swept wing in subsonic flow, Radetzsky et al. [87] found that sound, along with two-dimensional and three-dimensional roughness, has no observable effect on transition, although sound pressure levels of up to 97 dB were introduced at all possible unstable primary and secondary frequencies. They interpreted this result as due to the dominating role of the stationary crossflow vortices in causing transition. It is interesting to note that this result would agree with that one of the numerical approach by Crouch [95] who studied the receptivity to stationary and travelling crossflow modes due to localized and nonlocalized surface perturbations. He found that under the conditions considered, stationary crossflow vortices have amplitudes approximately 100 times larger than travelling crossflow modes generated by an acoustic wave with a nondimensionalized amplitude of 0.01%. A similar result was obtained by Choudhari [91]. Within a narrow frequency band, acoustic disturbances of that size may approximately correspond to those introduced at sound pressure levels up to 97 dB as done by Radetzsky et al. [87]. Hence, according to Choudhari [91] and Crouch [95], the stationary crossflow modes should indeed dominate the transition process as observed in the experiment by Radetzsky et al. [87], although they are less amplified than the travelling crossflow modes. Radeztsky et al. [87] only report the effect of sound excitation on transition without showing or analysing the unsteady content of the

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hot-wire signal. Hence, a direct statement on the receptivity to acoustic type of free stream disturbances cannot be derived. Further evidence was obtained in the experiment by Deyhle and Bippes [33] that was performed in a wind tunnel with 0.15% free stream turbulence. In this case travelling crossflow modes had the same amplitudes as the stationary vortices so that the identification of their generation was facilitated. In order to get an idea on the effect of acoustic forcing, Deyhle and Bippes [33] introduced plane sound waves at sound pressure levels of up to 111 dB in the frequency band of the most amplified crossflow modes (55(f(200 Hz). As shown in Fig. 47, a response was found only outside the boundary layer (curves 1, 2). Inside the boundary layer (curves 3, 4), where the amplitudes of the travelling crossflow waves were much larger than the amplitudes of the acoustic waves, no difference was observed in the frequency spectra of the unsteady disturbances, neither in the amplitude level nor in the frequency range. Further tests with additional roughness elements on the model surface close to the leading edge and close to the neutral position in order to increase receptivity as well as pure sinusoidal excitation showed the same result. This suggests that sound is only a weak agent for the initiation of travelling crossflow waves. Since on the other hand, in the experiment by Deyhle and Bippes [33] the travelling crossflow modes are initiated at approximately the same amplitudes as the stationary modes, it appears that they are mainly generated by the vortical content of the free stream turbulence. In a recent numerical study Crouch [95] demonstrated that in the presence of Tollmien—Schlichting instability,

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the receptivity to compact vortical disturbances is 30—100 times smaller than to acoustic disturbances, and in addition Choudhari [96] found that the receptivity to oblique gusts is still 2—4 times smaller. If this would also apply in three-dimensional boundary layers, the acoustic content in the free stream fluctuations of the wind tunnel used by Deyhle and Bippes [33] (1 MK) should be considerably smaller than the vortical content. However, as it can be seen in Table 1, both portions are of the same order of magnitude even if it is taken into account that the unsteady crossflow instabilities propagate almost perpendicular to the outer flow so that the cross component of the vortical free stream turbulence may be the more relevant portion (due to the contraction at the wind tunnel exit the cross components of the vortical free stream disturbances are larger than the streamwise component). Hence, the observations by Deyhle and Bippes [33] indicate that in flows unstable to crossflow instability, the receptivity to vortical disturbances is much larger than to acoustic disturbances. Considerable effort spend by Kendall (see review by Saric [10]) and others [97—99] over the last two decades demonstrates that both from the theoretical and experimental point of view receptivity is a very complicated problem. However, the recent and successful comparison between theory and measurements of the response of the Blasius boundary layer to a single steady free-stream vortex by Bertolotti and Kendall [100] indicates that part (i.e. low frequencies) of the receptivity process can be modelled. Other studies based on the transient growth mechanism may also be of relevance [101, 102]. These advances in two-dimensional flows may form a stepping stone for our understanding of this receptivity

Fig. 47. Spectra of the hot-wire signals with and without artificially introduced sound (random noise at 103 dB pressure level in the frequency range 55(f(200). Curves 1 and 2 outside the boundary layer (z/d"2.5) without and with sound excitation, curves 3 and 4 inside the boundary layer (z/d"0.2) without and with sound excitation, Deyhle and Bippes [33].

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phenomena in more complicated flows. A comprehensive modelling requires details of the turbulence structure which are very difficult to measure.

6. Transition prediction Currently, empirical or semi-empirical criteria are used for the prediction of transition in industrial flows of interest. Among those the e,-method is the most frequently applied. In transition dominated by Tollmien— Schlichting instability, this method yields reliable results. However, in flows dominated by crossflow instability, it seems not to be possible to take account of the receptivity to roughness and free stream turbulence as well as of nonlinear development and breakdown processes by just one empirical constant N. For a comprehensive discussion on the applicability of this method we refer to Arnal [103] and Arnal et al. [104, 105]. According to the intent of the present review to report basic work on transition dominated by crossflow instability, we do not intend to provide further data for the application of the e,-method, instead we want to point out which portions of the transition process can be predicted and where we still have to make use of empirical input. We only note that according to Section 6 for the comparison of wind tunnel tests, the use of the vortical content Tu or Tu T TTY (see Table 1 for the notation) of the free stream turbulence should give a better correlation than the overall value Tu or simply its streamwise component Tu . SY In the following discussions it is assumed that in swept wing flows dominated by crossflow instability, the primary disturbances saturate prior to transition. Although no other experimental observations are known to us, it cannot be excluded that flow conditions may exist where saturation does not occur. In order to elucidate our way to proceed, it may be referred to the experimental observations by Deyhle and Bippes [33]. They are summarized in Fig. 48a. There, the N-factors and Reynolds numbers for transition (N and Re , respectively)  V2    are plotted versus the saturation amplitudes of the stationary vortices measured at various environmental conditions. Thereby, it becomes obvious that transition location correlates with the saturation amplitudes of the stationary vortices. The next question is, what determines the saturation amplitudes of the stationary vortices. In Sections 3.3 and 3.4 it was shown that these are the initial conditions given by the disturbance environment and the related nonlinear developments. The relation between the environmental disturbances and the saturation amplitudes may also be seen in Fig. 48a. An increase of surface roughness increases the saturation amplitudes and an increase in free stream turbulence up to a value Tu+0.2% decreases them. This means that up to moderate values Tu, an increase of free stream turbulence delays transition, in contrast to our knowledge on

transition dominated by Tollmien—Schlichting instability where an increase in Tu always favours transition. Only values of Tu'0.2% advance transition. The dependence on the free stream turbulence can be seen more directly in Fig. 48b. Although the evidence of Fig. 48 is limited because it comprises only results of the DLR measurements and does not include results of other experiments, it suggests that in crossflow dominated flows, the prediction of the saturation amplitudes of the stationary vortices would be a considerable step forward towards the prediction of transition. Thus the question arises whether this is possible. As shown in Section 4.2 using PSE codes, excluding their inherent limitations in the stagnation area, the saturation amplitudes can be determined provided that the initial conditions are known. The major problem is therefore to determine the initial disturbance amplitudes. In the case of crossflow instability, they result from the roughness of the model surface, the vortical content of the free stream turbulence, and the boundary layer receptivity. A satisfactory estimate of the forcing of the stationary vortices is believed to be possible (see also Section 5.1). The modelling of the generation of the travelling crossflow waves is much less advanced than that of the stationary vortices. As discussed in Section 5.2, unsteady crossflow modes are mainly initiated by the vortical content of the free stream turbulence. Since related receptivity models as well as sufficient experimental data for an empirical input are not available, the application of a transition criterion based on the saturation amplitudes of the stationary vortices for general cases is not yet possible. However, the agreement between the PSE computations by Haynes [64] and the measurements by Reibert [39] suggest that in very low turbulence environment, neglecting unsteady modes the saturation rates of the stationary vortices can be determined. If this proves to be true, flight conditions where the free stream turbulence is also very low, would allow such an approach. Consequently, for swept wing flow under flight conditions most likely a transition criterion based on the saturation amplitudes of the stationary modes can be applied. Since transition occurs somewhere in the range of saturated primary instabilities, such a criterion still requires an empirical correlation between saturation amplitudes of the stationary vortices and transition. However, in contrast to the e,-method, the empirical input taking account of the disturbance forcing due to roughness and free stream turbulence, of nonlinearity and of the onset of final breakdown must no longer be subsumed under one single value N, but can be introduced separately and therewith more specifically. In addition, the nonlinear growth which as shown in Section 3.2 extends over the largest part of the transition process would be covered by theory. Certainly, from the physical point of view, such a criterion can also not be satisfactory. Experiments [5, 6, 33,

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Fig. 48. Reynolds numbers and the N-factors for transition measured on the DLR swept flat plate at different disturbance levels and surface conditions, (a) in dependence on the saturation amplitudes of the stationary vortices and (b) in dependence on Tu. *Tu"0.08% (NWB, closed test section, Tu "0.07%), (*) Tu"0.15% (1 MK, open test section, Tu "0.13%), (£) Tu"0.24% (1 MK with T T additional screen, open test section, Tu "0.17%), (䉭) Tu"0.70% (NWG, open test section, Tu "0.58%). U "43.5°, Deyhle T T  and Bippes [33].

49, 50] have shown that the final onset of the breakdown of laminar flow will be initiated if a secondary highfrequency instability appears. It originates somewhere in the saturation range of the primary instabilities (Section 3.5) and is highly amplified so that it immediately leads to transition [33]. Hence, the prediction of the onset of this secondary instability will be almost equivalent with the prediction of transition whenever it is initiated by the high-frequency instability. Still such prediction is possible at most for the limiting cases where the flow is only primarily disturbed either by stationary or travelling crossflow instabilities (Section 4.3), but again the agreement between the nonlinear growth measured by Reibert [39] and the PSE computations by Haynes [64] reproduced in Fig. 42 suggest that under flight conditions where the stationary modes are expected to dominate,

the secondary analysis by Malik et al. [44, 73] may provide a first approximation. Two possibilities are seen for the solution of the problem: Firstly, an extension of the secondary instability theories to flows primarily disturbed both by stationary and travelling crossflow instabilities, secondly an extension of PSE analysis to allow for the consideration of a larger number of Fourier components. For the case that the theoretical prediction of this secondary instability can be solved, which seems on a good way, the empirical input will be reduced to the formulation of the initial conditions for the PSE or DNS calculations. Thereby, as explained above, the formulation of the initial amplitudes of the unsteady disturbances is the most obstinate problem. The nonempirical formulation requires the solution of the

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Fig. 49. Suction device on the DLR swept flat plate. (The displacement body used for the generation of a pressure gradient on the flat plate is not shown).

receptivity problem to vortical disturbances in the free stream.

7. Possibilities of stabilisation The most straightforward method to stabilise the boundary layer subject to crossflow instability is surface suction because it reduces crossflow and crossflow instability, respectively. In swept wing flow on large transport aircraft, the flow becomes turbulent already in the leading edge area so that suction has to be applied up from the leading edge where the flow is subject to both stagnation flow and crossflow instability. Thus, the effect of suction on crossflow instability becomes hard to discern. For this reason, we refer in the following to the basic experiment on the DLR swept flat plate by Bippes et al. [106]. On this model the flow in the leading edge area is by far subcritical. This allows to study the effect of suction on crossflow instability unaffected by other instability mechanisms. The geometry of the suction insert in the swept flat plate is displayed in Fig. 49. The effectiveness of the suction device to reduce crossflow is illustrated in Fig. 50. The related stabilising effect becomes most obvious in spanwise velocity traces recorded downstream of the suction area at the wall distance of maximum disturbance intensity as shown in Fig. 51. There, the mean flow variations (ºM ) indicate the intensity of  the stationary modes whereas the r.m.s. amplitudes of the unsteady modes are given directly. Comparing the results

obtained with and without suction, and also the result obtained on the polished model without suction device, it turns out that suction had reduced the unsteady modes considerably, but surprisingly not the stationary modes (Fig. 51). The latter are even larger than those on the model without suction device. In order to also reduce the stationary vortices, suction had to be increased to values beyond the scope of practical interest (Fig. 52). This surprising result is attributed to the nonuniformity of the distribution in the suction velocity due to manufacturing imperfections. Manufacturing imperfections cause a spanwise variation of the suction velocity (Fig. 53) that forces stationary crossflow vortices in the same way as surface roughness. Bertolotti compared in [106] the receptivity to roughness and spanwise nonuniformity in the suction velocity. For certain spanwise wave numbers, and at the front part of the swept flat plate, he found receptivity factors even one order of magnitude larger than for surface roughness (Fig. 54). This means that with suction through the device applied, which is current technical standard for LASER perforated surfaces, the theoretically predicted damping effect is not fully exploited. Comparison of the transition Reynolds numbers in dependence on the suction rate showed, however, that transition was delayed for each suction rate. At

 As suction panel the material currently common in research and industry is applied.

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Fig. 50. Spanwise averaged mean flow profiles with and without suction. Q "20.4 m/s, U "43.5°, c "w /Q "0.1% (w is the   O    spatial average value).

Fig. 51. Spanwise variation of the mean velocity (ºM /Q ) and intensity of the unsteady disturbances (u /Q ). Comparison of the   

  disturbance intensities developing without and with suction (c "w /Q "0.1%). Q "20.4 m/s, Q "43.5°, z/d "0.25, x /c"0.9. O     A

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Fig. 52. Dependence of the effect of suction on the suction rates in the four suction compartments. Q "20.4 m/s, U "43.5°,   z/d"0.25, x /c"0.9. A

c "0.1% the transition Reynolds number increased O from Re "6.5;10 on solid polished surface to V2    Re '9;10 (larger Reynolds numbers could V2    not be realized). A further increase of the suction rate further reduced the instabilities up to c "0.5%, the O largest value realized in the experiment. An over-suction could not be observed. It should be noted that the experiments performed so far do not provide the full physical understanding of the effect of suction on the stabilisation of swept wing flow. Thus, it is not clear why in our suction experiment [106]

the primary instabilities did not saturate up to transition. Moreover, the stability analysis for swept wing flow by Itoh [68] predicts streamline curvature instability to appear prior to crossflow instability except for a narrow region close to the attachment line. Since according to another analysis on rotating disc flow by Itoh [107], the stabilising effect of suction on streamline curvature instability is much smaller than on crossflow instability, this could additionally limit the benefit of suction in swept wing flow provided that the streamline curvature instability mechanism is operable. Despite of those

H. Bippes / Progress in Aerospace Sciences 35 (1999) 363—412

407

Fig. 53. LDA measurement of the spanwise variation of the suction velocity over the perforated wall at various distances from the surface Q "0. 

uncertainties, the experiments on the swept flat plate [106] suggest that an improvement of the suction technique could increase the stabilising effect considerably. A further method of high practical interest to delay transition in flows subject to crossflow instability is reported by Reibert and Saric [66]. They found that forcing of stationary vortices at wavelength smaller than the unstable ones delays transition beyond the value observed without forcing, i.e. under natural conditions on polished surface. Experiments of active control in swept wing flow are not known to the author.

8. Conclusions

Fig. 54. Comparison between receptivity coefficients due to roughness and non-uniformity in the suction velocity calculated for swept Hiemenz flow at Re"400 by Bertolotti in [106].

Basic experiments known so far elucidated the physics of the transition process dominated by crossflow instability up to, and including, saturation of the primary instability. A most important feature is that the flow becomes primarily unstable to both stationary vortices and oblique travelling waves. This leads to an extended region of nonlinear development. Disturbance interactions are observed as soon as the steady disturbances become of

408

H. Bippes / Progress in Aerospace Sciences 35 (1999) 363—412

measurable size. They depend on the environmental conditions in a complex manner. Stationary vortices are essentially excited by surface roughness, and travelling waves by the vortical content of the free stream turbulence. Since the receptivity to surface roughness is much larger than the receptivity to vortical free stream fluctuations, the stationary vortices dominate in low turbulence environment even on polished surface and even though travelling modes were found to be more amplified. Some uncertainty still exists on the final breakdown of laminar flow after the primary instabilities have saturated. Two scenarios are identified, the appearance of a secondary high-frequency instability that undergoes an almost explosive growth so that it immediately leads to transition and a gradual widening of the Fourier spectrum of the unsteady disturbance motion. The former is most frequently observed, it occurs when the instantaneous profiles become secondarily inflectional and when the gradient jºM (z)/jz at the inflection point exceeds a certain threshold value. The latter was observed on a swept wing when transition occurred at small values of the (0,0) mode. The conditions for its appearance are mainly unknown. Surface curvature was not observed to change the physics of the transition process in the presence of crossflow instability. Convex curvature reduces the amplification of stationary vortices and favours the growth of unsteady disturbances. Concave curvature in turn favours the growth of both stationary and nonstationary disturbances. Correspondingly, convex curvature rather delays transition whereas concave curvature can advance transition considerably. Comparison with stability theory has shown that local and nonlocal approaches correctly predict frequency, wave-number vector and phase velocity of the most unstable modes whereas under natural conditions of transition their growth is overpredicted. Both PSE and DNS codes qualitatively describe the nonlinear disturbance growth and its dependence on the initial conditions as established in the experiment. Two successful attempts have been made to validate PSE computations using controlled experiments. This suggests that the disturbance growth can correctly be predicted up to saturation if the initial conditions are known. Much less successful are the nonlinear analyses to predict the secondary high-frequency instability experimentally observed to initiate the final breakdown of the laminar flow. Only for the limiting cases where the flow is primarily disturbed exclusively by stationary vortices or by travelling waves, secondary theory could predict the appearance of such an instability. Current DNS and PSE approaches obviously do not yet resolve these secondary high-frequency disturbances. The efficiency of available theoretical approaches to correctly describe the effect of convex and concave sur-

face curvature is less clear. Still the experimental evidence is rather scarce despite the influence on transition indicated by the experiments. In view of the availability of the effective PSE and DNS codes, the solution of the receptivity problem to determine the initial conditions has become a most actual problem. The modelling of the generation of stationary crossflow vortices by roughness is approaching in a promising way. Less progress has been made in modelling the disturbance excitation by free stream fluctuations. According to the experiments, acoustic modes are of minor importance, and thus the practical interest focuses on the receptivity to vortical modes. In flows dominated by crossflow instability the use of the e,-method has proved to be much less effective than in flows dominated by Tollmien—Schlichting instability. The extended nonlinear disturbance evolution with the connected complex dependence on the initial conditions does not allow to take account of the receptivity to roughness and free stream turbulence as well as of nonlinear development and breakdown process by just one empirical constant N. It is proposed to use a criterion based on the theoretical determination of the saturation amplitudes of the stationary modes. A criterion for the onset of the secondary instability would be a further step forward. For the stabilization of crossflow instability, suction has proved to be an effective means. Currently, technical imperfection in the suction techniques do not allow to take profit of the full potential. However, for a reliable estimate of the possible benefit, the physical understanding of the suction effect has to be further improved. Very interesting for practical application especially in low turbulence environment is the subcritical forcing of stationary vortices, the method explored at ASU. Acknowledgements The author is indebted to his colleagues Dr. F. Bertolotti, S. Hein, Dr. W. Koch, Dr. T. Lerche, A. Po¨thke, and A. Stolte for the valuable comments and discussions. I am especially grateful to S. Hein for supplementing parametric stability calculations which facilitated the interpretations of results and to Dr. T. Lerche and S. Hein for providing and preparing figures. References [1] Reynolds O. On the experimental investigation of the circumstances which determine whether the motion of water shall be direct or sinuous, and the law of resistance in parallel channels. Philos Trans Roy Soc 1883;174:935—82. [2] Gray WE. The effect of wing sweep on laminar flow. RAE TM Aero 256, 1952.

H. Bippes / Progress in Aerospace Sciences 35 (1999) 363—412 [3] Gray WE. The nature of the boundary layer flow at the nose of a swept wing. RAETM Aero 256, 1952. [4] Gregory N, Stuart JT, Walker WS. On the stability of three-dimensional boundary layers with application to the flow due to a rotating disc. Philos Trans Roy Soc London A 1955;248:155—99. [5] Poll DIA. Some observations of the transition process on the windward face of a long yawed cylinder. J Fluid Mech 1985;150:329—56. [6] Arnal D, Coustols E, Juillen JC. Experimental and theoretical study of transition phenomena on an infinite swept wing. Rech Aerosp No 1984-4, 1984. [7] Michel R, Arnal D, Coustols E. Juillen JC. Experimental and theoretical studies of boundary layer transition on a swept infinite wing. In: Kozlov VV, editor. LaminarTurbulent Transition, IUTAM Symp, Novosibirsk, USSR, 1984. Berlin. Springer, 1985: 553—61. [8] Saric WS, Yeates LG. Experiments on the stability of cross-flow vortices in swept wing flow. AIAA Paper 860493, 1985. [9] Reed HL, Saric WS. Stability of three-dimensional boundary layers. Ann Rev Fluid Mech 1989;21:235—84. [10] Saric WS. Low speed boundary-layer transition experiments. AIAA Educational Series in Oxford Press, 1999, To appear. [11] Crabtree LF, Ku¨chemann D, Sowerby L. In: Rosenhead L, editor. Laminar boundary layers. Oxford: Oxford University Press, 1963. [12] Mack L. Special course on stability and transition of laminar flows. AGARD Report No. 709, Paper 3, 1984. (Boundary layer linear stability theory). [13] Rayleigh L. On the stability of certain fluid motions. Scientific Papers 1880;1:474;3:17;4:203;6:197. [14] Tollmien W. Ein allgemeines Kriterium der Instabilita¨t laminarer Geschwindigkeitsverteilungen. Nach Ges Wiss Go¨ttingen, Math Phys Kl, Fachgr I, 79, 1935. Engl. Translation, in NACA TM792 (1936). [15] Lingwood RJ. Absolute instability of the boundary layer on a rotating disk. J Fluid Mech 1995;299: 17—33. [16] Lingwood RJ. An experimental study of absolute instability of the rotating disk boundary-layer flow. J Fluid Mech 1996;314:373—405. [17] Taylor M, Peake M. Causal stability analysis of incompressible swept wing boundary layers. AIAA Paper 971777, 1997. [18] Dagenhart JR, Saric WS, Mousseux MC, Stack JP. Crossflow vortex instability and transition 45-degree swept wing. AIAA Paper 89-1992, 1989. [19] Saric WS, Takagi S, Mousseux M. The ASU unsteady wind tunnel and fundamental requirements for freestream turbulence measurements. AIAA Paper 88-0053, 1988. [20] Radeztsky RH Jr. Growth and development of roughness-induced stationary vortices. PhD Thesis, Arizona State University, Tempe, 1994. [21] Kachanov YS, Tararykin OI. The experimental investigation of instability and receptivity of swept wing flow. In Arnal D, Michel R, editors. Laminar-Turbulent Transition, IUTAM Symp, Toulouse, France, 1989. Berlin: Springer, 1990:499—509.

409

[22] Kachanov YS. Generation, development and interaction of instability modes in swept wing boundary layer. In: Duck PW, Hall P, editors. Proc IUTAM Symp On Nonlinear Stability and Transition in Three-Dimensional Boundary Layers, Manchester, UK, 1994. Dordrecht: Kluwer Academic Publishers, 1995:247—56. [23] Gaponenko VR, Ivanov AV, Kachanov YS. Experimental study of crossflow instability of a swept wing boundary layer with respect to travelling waves. In: Kobayashi R, editor. Laminar-Turbulent Transition, IUTAM Symp Sendai Japan, 1994. Berlin: Springer, 1995:373—8. [24] Takagi S, Itoh N. Observation of travelling waves in the three-dimensional boundary layer along a yawed cylinder. Fluid Dyn Res 1994;14:167—89. [25] Takagi S, Itoh N. On similarity law of the crossflow instability in three-dimensional boundary-layers on yawed cylinders. In: Kobayashi R, editor. Laminar-Turbulent Transition, IUTAM Symp, Sendai, Japan, 1994 Berlin: Springer, 1995:373—8. [26] Takagi S, Itoh N, Tokugawa N. Dispersive growth of unsteady disturbances due to crossflow and streamlinecurvature instabilities in a three-dimensional boundary layer. Proc 19th Int Congress of Theoretical and Applied Mechanics, Kyoto, Japan, 25—31 August, 1996. [27] Mu¨ller B. Experimentelle Untersuchung der Querstro¨mungsinstabilita¨t im linearen und nicht-linearen Breich des Transitionsgebietes. Dr.-Ing. Diss., University Karlsruhe, Germany, 1990 (also DLR-FB 90-09). [28] Lerche T. Experimentelle Untersuchung nichtlinearer Strukturbildung im Transitionsproze{ einer instabilen dreidimensionalen Grenzschicht. Dr.-Ing. Diss., TH Hanover, Germany, 1996; published in: VDI FortschrittBericht Reihe 7 No. 310. VDI-Verlag Du¨sseldorf, Germany, 1997. (Some of the results are published in [31, 44]). [29] Dallmann U, Bieler H. Analysis and simplified prediction of primary instability of three-dimensional boundarylayer flows. AIAA Paper 87-1347, 1987. [30] Radeztsky RH Jr, Reibert MS, Saric WS. Development of stationary crossflow vortices on a swept wing. AIAA Paper 94-2373, 1994. [31] Lerche T, Bippes H. Experimental investigation of crossflow instability under the influence of controlled disturbance excitation. In: Henkes RWAM, Van Ingen JL, editors. Transitional Boundary Layers in Aeronautics, Coll. Royal Netherlands Academy of Arts and Sciences Amsterdam, 1995. Amsterdam: North-Holland, 1996: 137—44. (For details see thesis by T. Lerche [28]). [32] Deyhle H, Ho¨hler G, Bippes H. Experimental investigation of wave propagation in a three-dimensional boundary-layer flow. AIAA J 1993;31:637—45. [33] Deyhle H, Bippes H. Disturbance growth in an unstable three-dimensional boundary-layer and its dependence on environmental conditions. J Fluid Mech 1996; 316:73—113. [34] Nitschke-Kowsky P, Bippes H. Instability and transition of a three-dimensional boundary layer on a swept flat plate. Phys Fluids 1988;31(4):786—95. [35] Arnal D, Juillen JC. Three-dimensional transition studies at ONERA/CERT. AIAA Paper 87-1345, 1987.

410

H. Bippes / Progress in Aerospace Sciences 35 (1999) 363—412

[36] Bippes H, Mu¨ller B, Wagner M. Measurements and stability calculations of the disturbance growth in an unstable three-dimensional boundary layer. Phys Fluids A 1991;3:2371—7. [37] Bippes H. Experiments on transition in three-dimensional accelerated boundary-layer flows. Proc. R.A.S. Boundary Layer Transition and Control, Cambridge, UK, ISBN 0 903409 86 0, 1991. [38] Carillo RB Jr. Distributed roughness effects on stability and transition in swept wing boundary layers. Master’s Thesis, Arizona State University, 1996. [39] Reibert MS. Nonlinear stability, saturation, and transition in crossflow-dominated boundary layers. PhD Thesis, Arizona State University Tempe, 1996. [40] Deyhle H. Einflu{ der a¨u{eren Stro¨mungsbedingungen auf den Transitionsproze{ einer dreidimensionalen Grenzschicht. Dr.-Ing. Diss., TH Hanover 1993, published in: VDI Fortschritt-Bericht Reihe 7 No. 226, VDIVerlag Du¨sseldorf, Germany. [41] Mu¨ller B. Experimental study of the travelling waves in a three-dimensional boundary layer. In: Arnal D, Michel R, editors. Laminar Turbulent Transition, IUTAM Symp, Toulouse, France, 1989 Berlin: Springer, 1990:489—99. [42] Reibert MS, Saric WS, Carillo RB Jr, Chapman KL. Experiments in nonlinear saturation of stationary crossflow vortices in a swept wing boundary layer. AIAA Paper 96-0184, 1996. [43] Lerche T. Experimental investigation of nonlinear wave interactions and secondary instability in three-dimensional boundary-layer flow. In: Gavrilakis S, Machiels L, Monkewitz PA, editors. Advances in Turbulence IV, 6th European Turbulence Conf, Lausanne, Switzerland, 1996. Dordrecht: Kluwer Acad. Publishers, 1996:357—60. (For details see thesis by T. Lerche [28]). [44] Malik MR, Li F, Chang CL. Cross-flow disturbances in three-dimensional boundary layers: non-linear development, wave interaction and secondary instability. J Fluid Mech 1994;268:1—36. [45] Bertolotti FP. On the birth and evolution of disturbances in three-dimensional boundary layers. In: Duck PW, Hall P, editors. Proc IUTAM Symp On Nonlinear Stability and Transition in Three-Dimensional Boundary Layers, Manchester, UK, 1994. Dordrecht: Kluwer Academic Publishers, 1995:247—56. [46] Mu¨ller W, Bestek H, Fasel H. Spatial direct numerical simulation of transition in a three-dimensional boundary layer. In: Kobayashi R, editor. Laminar-Turbulent Transition, IUTAM Symp, Sendai, Japan, 1994. Berlin: Springer, 1995:397—404. [47] Reed HL. Wave interactions in swept wing flows. Phys Fluids 1987;30:3419—26. [48] Fischer TM, Dallmann U. Primary and secondary stability analysis of a three-dimensional boundary-layer flow. Phys Fluids A 1991;3:2378—91. [49] Kohama Y, Saric WS, Hoos JA. A high frequency, secondary instability of that leds to transition. Proc R A S Conf On Boundary-Layer Transition and Control, Cambridge, UK, 1991;4.1—13. [50] Bippes H, Lerche, T. Transition prediction in three-dimensional boundary layer flows unstable to crossflow instability. AIAA Paper 97-1906, 1997.

[51] Po¨thke A. Unpublished result of the DLR swept wing experiment. 1997. [52] Bippes H. Instabilities developing in the three-dimensional boundary layer on concave and convex curved surface. In: Kobayashi R, editor. Laminar-Turbulent Transition, IUTAM Symp Sendai, Japan, 1994. Berlin: Springer, 1995:315—22. [53] Bippes, H. Experiments to the influence of surface curvature on the development of streamwise vortices in threedimensional boundary-layer flows. Z Flugwiss Weltraumforsch 1995;19:S.129—38. [54] Hein S. Local stability analysis for the flow along a concave and convex curved model. Private communication, 1998. [55] Bassom AB, Hall P. Vortex instabilities in three-dimensional boundary layers:the relationship between Go¨rtler and crossflow vortices. J Fluid Mech 1991;232:647—80. [56] Zurigat YH, Malik MR, Effect of crossflow on Go¨rtler instability in incompressible boundary layers. Phys. Fluids 7, 1995, 1616—25. [57] Haynes TS, Reed HL, Saric WS. CFD Validation issues in transition modelling. AIAA Paper 96-2551, 1996. [58] Collier FS, Mu¨ller B, Bippes H. The stability of a threedimensional laminar boundary layer on a swept flat plate. AIAA Paper No. 90-1447, 1990. [59] Kachanov YS. Experimental studies of three-dimensional boundary layers. AIAA Paper 96-1978, 1996. [60] Herbert T, Bertolotti FP. Stability analyses of nonparallel boundary layers. Bull Am Phys Soc 1987; 32:2079. [61] Bertolotti FP, Herbert T, Spalart PR. Linear and nonlinear stability of the Blasius boundary layer. J Fluid Mech 1992;242:441—7. [62] Herbert T. Parabolized stability equations. Ann Rev Fluid Mech 1997;29:245—83. [63] Malik MR, Balakumar P. Linear stability of three-dimensional boundary layers: effect of curvature and non-parallelism. AIAA-Paper 93-0079, 1993. [64] Haynes TS, Reed HL. Numerical simulation of swept wing vortices using nonlinear parabolized stability equations. SAE Paper No. 971479. Fluid Mech, 1997, Submitted. (For details see Haynes, T. S. (1996) Nonlinear stability and saturation of crossflow vortices in swept-wing boundary layers. PhD Thesis, Arizona State University, Tempe.) [65] Wu Y, Lerche T, Bertolotti F. Private communication, 1998. [66] Reibert MS, Saric WS. 1997. Review of wept wing transition. AIAA Paper 97-1816, 1997. [67] Itoh N. Effects of wall and streamline curvatures on instability of 3-D boundary layers. In: Kobayashi R, editor. Laminar-Turbulent Transition, IUTAM Symp, Sendai Japan, 1994. Berlin: Springer, 1995:323—30. [68] Itoh N. Simple cases of streamline-curvature instability in three-dimensional boundary layers. J Fluid Mech 1996;317:129—54. [69] Meyer F, Kleiser L. Numerical simulation of transition due to cross-flow instability. In: Arnal D, Michel R, editors. Laminar-Turbulent Transition, IUTAM Symp, Toulouse, France, 1989. Berlin: Springer, 1990: 609—19.

H. Bippes / Progress in Aerospace Sciences 35 (1999) 363—412 [70] Wagner M. Numerische Untersuchungen zum laminarturbulenten U®bergang in zwei- und dreidimensionalen Grenzschichten. Dissertation zum Dr.-Ing. An der Universita¨t Fridericiana zu Karlsruhe, 1992 (also DLRFB 92-36, 1992). [71] Wintergerste, T, Kleiser, L. Direct numerical simulation of transition in a three-dimensional boundary layer. In: Henkes RWAM, Van Ingen JL, editors. Transitional boundary layers in aeronautics, Coll. Royal Netherlands Academy of Arts and Sciences Amsterdam, 1995. Amsterdam: North-Holland, 1996:145—53. [72] Fischer, TM, Hein S, Dallmann U. A theoretical approach for describing secondary stability features in three-dimensional boundary-layer flows. AIAA Paper 93-0080, 1993. [73] Malik MR, Li F, Chang CL. Nonlinear crossflow disturbances and secondary instabilities i swept-wing boundary layers. In: Duck PW, Hall P, editors. IUTAM Symp On Nonlinear Stability and Transition in Three-Dimensional Boundary Layers, Manchester, UK, 1994. Dordrecht: Kluwer Academic Publishers, 1996:266—77. [74] Koch W, Bertolotti F, Stolte A, Hein S. Nonlinear equilibrium solutions in a three-dimensional boundary layer and their secondary instability. J Fluid Mech, 1999, submitted. [75] Mu¨ller, W. Numerische Untersuchung ra¨umlicher Umschlagsvorga¨nge in dreidimensionalen Grenzschichtstro¨mungen. Dissertation zum Dr.-Ing. An der Universita¨t Stuttgart, 1995. [76] Stolte, A. Private communication, 1995. [77] Schrauf G. Curvature effects for three-dimensional boundary-layer stability. Z Flugwiss Weltraumforsch 1992;16:119—27. [78] Hall P. The Go¨rtler vortex instability mechanism in three-dimensional boundary layers. Proc Roy Soc London 1985;A 399:135. [79] Otto S, Denier JP. The effect of Go¨rtler vortices on crossflow. ICASE Report No. 94-19, 1994. [80] Collier FS, Malik MR. Stationary disturbances in threedimensional boundary layers. AIAA Paper No. 87-1412, 1987. [81] Itoh N. Centrifugal instability of three-dimensional boundary layers along concave walls. Trans Japan Soc Aero Space Streamline Curvature Instability 1994;37:125—38. [82] Swearingen JD, Blackwelder RD. The growth and breakdown of streamwise vortices in the presence of a wall. J Fluid Mech 1987;182:255—90. [83] Bippes H, Deyhle H. Das receptivity-problem in Grenzschichten mit la¨ngswirbelartigen Sto¨rungen. Z Flugwiss Weltraumforsch 1992;16:S.34—41. [84] Mu¨ller B, Bippes H. Experimental study of instability modes in a three-dimensional boundary layer. Proc. AGARD Symp On Fluid Dynamics of Three-Dimensional Turbulent Sheer Flows and Transition, Cesme, Turkey, AGARD C-P 438, Paper No. 13, 1988. [85] Bo¨ttcher J, Wedemeyer E. The flow downstream of screens and ist influence on the flow in the stagnation region of cylindrical bodies. J Fluid Mech 1989;204:501—22.

411

[86] Bippes H. Experimentelle Untersuchung des laminar-turbulenten Umschlags in der instabilen Grenzschicht an einer parallel angestro¨mten konkaven Wand, Sitzungsbericht Heidelberger Akad. Wiss., 3. Abhandlung, 1972. (Translated in NASA TM-75243, 1978). [87] Radeztsky RH Jr, Reibert MS, Saric WS, Tagaki S. Effect of micron-sized roughness on transition in swept-wing flows. AIAA Paper 93-0076, 1993. [88] Fedorov AV. Excitation of waves of instability of the secondary flow in the boundary layer on a swept wing. J Appl Mech Tech Phys 1989;29:643—8. [89] Manuilovich SV. Disturbances of a three-dimensional boundary layer generated by surface roughness. Fluid Dyn 1990;24:764—9. [90] Crouch JD. Receptivity of three-dimensional boundary layers. AIAA Paper No. 93-0074, 1993. [91] Choudhari M. Roughness induced generation of crossflow vortices in three-dimensional boundary layers. Theoret Comput Fluid Dyn 1994;6:1—30. [92] Bottaro A, Zebib A. Go¨rtler vortices promoted by wall roughness. Fluid Dyn Res 1997;18:343—62. [93] Luchini P, Bottaro A. Go¨rtler vortices: a backward-intime approach to the receptivity problem. J Fluid Mech 1998;363:1—23. [94] King RA. Three-dimensional boundary layer transition on a cone at Mach 3.5. Exp Fluids 1992;13: 305—14. [95] Crouch JD. Theoretical study on the receptivity of boundary layers. AIAA Paper 94-2224, 1994. [96] Choudhari M. Boundary layer receptivity to three-dimensional unsteady vortical disturbances in free stream. AIAA Paper No. 96-0181, 1996. [97] Kachanov YS, Kozlov VV, Levchenko VY. Origin of Tollmien—Schlichting waves in boundary layer under the influence of external disturbances. Izv Akad Nauk SSSR, Mekh Zhidk Gaza 1979;5:85 (Translated in Fluid Dyn 1979;13:704). [98] Kachanov YS, Kozlov VV, Levchenko VY. Beginning of turbulence in boundary layers. Technical Report, Nauka, Siberian Div, 1982. [99] Bertolotti FP. Response of the Blasius boundary layer to free stream vorticity. Phys Fluids 1997;9:2286—99. [100] Bertolotti FP, Kendal JM. Response of the Blasius boundary layer to controlled free stream vortices of axial form. AIAA Paper 97-2018, 1997. [101] Luchini P. Reynolds-number-independent instability of the boundary layer over a flat surface. JFM 1996;327:101—15. [102] Andersson P, Berggren M, Henningson D. Optimal disturbances in boundary layers. Progress in system control. Basel: Birkha¨user, 1988. [103] Arnal, D. Predictions based on linear theory. AGARD Report 793, Special Course on Progress in transition Modelling, 1993: 2-1—2-63. [104] Arnal D, Casalis G, Juillen JC. A survey of the transition prediction methods: from analytical criteria to PSE and DNS. In: Kobayashi R, editor. Laminar-Turbulent Transition, IUTAM Symp Sendai, Japan, 1994. Berlin: Springer, 1994:315—22. [105] Arnal D, Casalis G, Reneaux J, Cousteix J. Laminarturbulent transition in subsonic boundary layers: research and application in France. AIAA Paper 97-1905, 1997.

412

H. Bippes / Progress in Aerospace Sciences 35 (1999) 363—412

[106] Bippes H, Wiegel M, Bertolotti F. Experiments on the control of crossflow instability with the aid of suction through perforated walls. Proc OfFLOWCON, IUTAM Symp on Mechanics of Passive and Active Flow Control, Go¨ttingen, Germany, 7—11 September, 1998. [107] Itoh N. Streamline-curvature instability of the flow on a rotating disc with uniform suction. J Phys Soc Japan 1995;65:2485—93. [108] Crouch JD. Transition prediction and control for airplane applications. AIAA Paper 97-1907, 1997.

Similarly, the unsteady disturbances are approximated by the harmonic unsteady content of the total flow Qs (x , y , z, t): A A qs (x , y , z, t)"Qs (x , y , z, t)!Q s (x , y , z) A A A A A A

(A.6)

in many cases simply given as r.m.s. content of the velocity fluctuations q (x , y , z, t)  A A ,v ,w ] qs,r.m.s. (x , y )"[u A A 

 

 





1 "max t !t X  

Appendix. Definition of the flow quantities used for the presentation of the results


 R q (xA, yA, z) dt . R



s

(A.7)



In the linear range of amplification In theory, the disturbance motion is well defined as part of the total flow being superimposed upon the undisturbed laminar flow, usually called base flow, where both the base flow and the disturbance motion are assumed to satisfy the Navier—Stokes equations. In the experiment, both the instabilities and the base flow cannot be measured directly. Therefore, approximate quantities are determined to enable comparison between experiment and theory. In the limits of linear theory, the base flow is approximated by the spanwise averaged temporal mean of the total three-dimensional flow: 1 Q s (x , z)"[ºMI , »MI , ¼ MI ]" A   y !y A A



y

A

y

A

Q s (x , y , z) dy , A A A (A.1)

where



t  1  ]" " Q s(x , y , z, t) " dt (A.2) Q s"[º , » , ¼    A A t !t t    is the mean flow and Q s (x , y , z, t)"[º , » , ¼ ] is the A A    total flow quantity directly measured. Correspondingly, the stationary disturbances superimposed on the base flow are approximated by the subtraction of the spanwise averaged mean flow from the mean flow

q s (x , y , z)"[u  , v  , w ]"Q s(x , y , z)!Q s (x , z). (A.3) A A   A A A The amplitudes and the maximum amplitudes over the wall distance, respectively, are then 1 qª s (x , z)"[uª  , vª  , w  ]" [max Q s (x , y , z) A    A A 2 y Q s (x , y , z)] !min y   

(A.4)

and qª s (x )"[uª , vª , wª ]"max qª  s (x , z). A    A X

(A.5)

qs,r.m.s. (x , y )"qs,r.m.s. (x ) A A A in the nonlinear range of amplification qs,r.m.s. (x , y ) is A A measured at spanwise positions y where ºM (x , y , z) is A  A A ºM (x )"max [min " º (x , y , z) "] y   A A A  X and º (x )"max [max " º (x , y , z) "], y   A A A  X

(A.8)

respectively, and denoted q (x , y ) and q (x , y ),   A A   A A respectively. For the determination of the disturbance growth usually qˆ s,max"f (x ) is chosen. Such a procedure is justified A only as long as the mode shapes do not change significantly as in the linear regime. However, in the later stages drastic deformations are observed (Fig. 24, e. g.). Therefore, also integral quantities such as 1   B I" qˆ dz (A.9)  d  are also used. It should be noted that in the figures reproduced from published work different symbols may have been chosen for the same physical quantities. In these cases the reader is referred to the notation in the captions and in the text and the definitions in the appendix. Moreover, in some experiments the coordinate normal to wall is designed y in others z. In the results of the DLR experiments the velocities are given in the coordinate system aligned with the inviscid streamlines (x ,y ,z) but plotted in the model fixed coordi  nate system (x ,y ,z) so that the spanwise variation of the A A instability motion at constant Reynolds numbers is displayed. (For the coordinate systems see also Fig. 6.)