Breakdown in a boundary layer exposed to free-stream turbulence

Transcription

1 Experiments in Fluids (2005) 39: DOI /s RESEARCH ARTICLE J. Mans Æ E. C. Kadijk Æ H. C. de Lange A. A. van. Steenhoven Breakdown in a boundary layer exposed to free-stream turbulence Received: 23 December 2004 / Revised: 5 July 2005 / Accepted: 17 August 2005 / Published online: 11 October 2005 Ó Springer-Verlag 2005 Abstract The natural secondary breakdown modes of a flat plate boundary layer exposed to a free-stream turbulence level of 6.7% are characterised by experimental visualisations. The used experimental set-up is a water channel and no external triggering of the instabilities is used. The visualisations show the presence of two secondary instability modes, a sinuous (antisymmetric) and a varicose (symmetric) mode. The amplitudes of both instabilities increase, according to a non-linear profile, in upstream direction. The amplitudes also experience temporal growth. When a critical amplitude is reached, roll-up structures appear. These structures develop and finally merge, resulting in a turbulent spot. The development of the amplitude is similar in both modes. However, the amplitude of the varicose instability varies between 0.38 and 0.63 [2 p A/ k] and the sinuous amplitude between 0.11 and 0.23 [2 p A/ k]. The propagation velocities of the sinuous and varicose instabilities are 0.91 [u/u blas ] and 0.87 [u/u blas ] respectively. This shows that both modes are located in a low speed streak. 1 Introduction The boundary layer transition process is a fundamental problem within the fluid mechanics community, the more so because, as Morkovin (1994) showed in his route map to turbulence, the transition process can proceed through several paths. Despite the different paths, all transition processes end with breakdown into J. Mans (&) Æ E. C. Kadijk Æ H. C. de Lange A. A. van. Steenhoven Department of Mechanical Engineering, Technische Universiteit Eindhoven, P.O. Box 513, 5600 Eindhoven, The Netherlands j.mans@tue.nl patches of turbulence. These so called turbulent spots were discovered by Emmons already in Schubauer and Klebanoff (1955) performed an extensive experimental study on the turbulent spots. With the increasing knowledge on the physics behind a turbulent spot, the interest in the transition process increased. Traditionally, the focus has been on the classical transition scenario. This classical Tollmien Schlichting scenario has its origin in the linear theory. There the Orr-Sommerfeld and squire equations describe the evolution of a disturbance, linearised around a base velocity profile. The first solutions of unstable waves were obtained by Tollmien and Schlichting. They showed that above a critical Reynolds number, the unstable waves experience modal exponential growth. Eventually, a secondary instability leads to breakdown, see Herbert (1988). However, experiments revealed an unstable boundary layer under conditions that correspond to stable Tollmien Schlichting waves. A re-examination of the Orr Sommerfeld and squire equations proved that the eigenfunctions are nonnormal. This means that besides exponential growth, other disturbances may also grow in the form of the so called nonmodal growth. These disturbances undergo an algebraic growth followed by a viscous decay. This transient disturbance growth is a key phenomenon in several transition paths, see Saric et al. (2002). It is generally accepted, despite the different paths, to refer to these transition processes with one term, bypass transition. Physically the bypass mechanism is based on the presence of streamwise vorticity in the boundary layer. The vorticity will transfer low momentum fluid away from the wall and high-momentum fluid towards the wall. This results in streaks of low and high streamwise velocity. A secondary instability of these streaks will then cause the transition to turbulence, see Schmid and Henningson (2001). The actual transition scenario occurring in a boundary layer depends on the initial conditions. A common technique used to establish a bypass transition scenario, is the use of free-stream turbulence. In experimental

2 1072 studies, the bypass transition scenario has been elucidated by flow visualisations and velocity measurements. By measuring low- frequency oscillations in hot-wire signals, which were attributed to the slow spanwise motions of streaks, Klebanov (1971) was the first to identify the presence of the high and low speed streaks. Kendall (1985) studied the influence of the streaks, which he denoted as Klebanoff modes, on the mean flow profile. He showed that the maximum of the streamwise velocity fluctuations is located in the middle of the boundary layer. Matsubara and Alfredsson (2001) concluded in their review article on several years of experiments that the initial growth of the streaks can be successfully described by nonmodal theory. They also showed that close to the leading edge, the spanwise scale of the streamwise streaks is large compared to the boundary-layer thickness. However, as the streaks develop in streamwise direction the scale reaches this value. Their visualisation results showed that just before breakdown the streaks undergo a streamwise waviness of relatively short wavelength. This oscillation develops further into a turbulent spot. This secondary instability of the streaks is one of the important stages in the transition process which is still not fully understood. By direct numerical simulations, Andersson et al. (2001) studied the nonlinear evolution of streaks. The study considers in detail a fundamental sinuous and varicose mode and the subharmonic equivalents of these modes. The paper also reports that the sinuous mode, either fundamental or subharmonic, is the most critical disturbance. This mode has a critical amplitude of 26% of the streamwise velocity. A value of 37% is found for the varicose mode, what shows that this mode is more stable. In a secondary instability experiment, Asai et al. (2002) visualised in air the sinuous and varicose modes. In the experiment a controlled trigger initiates both modes. Note that in this experiment the varicose mode is the most unstable one. Most research on the secondary breakdown process utilises such a forced breakdown process. A natural, unforced, secondary instability breakdown process during bypass transition was studied by Brandt (2003). DNS results published in Brandt et al. (2004) show the velocity field during a sinuous and a varicose breakdown. Matsubara and Alfredsson (2001) characterised the length scales and the growth of the streaks in the secondary instability. The motivation of the present study is to experimentally determine the natural breakdown modes during by-pass transition and to characterise this breakdown process. This is done by visualisation experiments. The experimental set-up used is a water channel, this in contradiction to most other experiments in which a wind tunnel is used. The big advantage of a water channel is that the time and length scales present in the bypass transition process are much larger. This enables a more detailed analysis. The first results show the turbulent spots in the boundary layer, their origin and development. In the next part, the focus is on the instability modes present in this natural bypass transition process. A study on the velocities of the instabilities elucidates the role of the high and low speed streaks in the process. By describing the amplitude development of the instabilities, a characterisation of the breakdown process is given. 2 Experimental description 2.1 Set-up The experiments are performed in a water channel at the Technische Universiteit Eindhoven, see Fig. 1. The Reynolds number scaled dimensions of the Ludwieg tube set-up used by Schook (2001) founded the design of this water channel. Fig. 1 The water channel without pumps and pipes network

3 1073 Fig. 2 The Blasius profile with the measured velocity at Re x = u/u Blasius solution Experimental solution η = y * (U /ν*x) 1/2 The water channel set-up consists of two cylindrical reservoirs (height= 2 m, diameter= 1.2 m, with a glass measuring section (optical accessible) placed in between. The measuring section is 2.7 m long, 0.57 m wide and 0.45 m high. A flat plate with the same width as the channel, a length of 1.8 m and a thickness of m is positioned parallel to the bottom in the channel. The distance between the bottom and the flat plate is m. The distance between the flat plate leading edge and the start of the measuring section is 0.9 m. Pumps and a network of pipes provide a channel flow. Two straightening units, consisting of a honeycomb aligner and a wire mesh, are placed between the inflow reservoir and the measuring section to ensure a laminar and uniform main flow. The honeycomb aligner has an opening width of m and an opening height of m. The mesh size of the wire mesh is m, the thickness of thread is mto avoid separation at the leading edge, a suction unit creates under-pressure below the leading edge. This ensures the development of a laminar boundary layer and the position of the stagnation line at the leading edge (Fig. 2). The free-stream turbulence is generated by means of a static grid. The grid is constructed of cylindrical rods with a diameter of m. The mesh size of the grid is m. The distance between the grid and the leading edge of the plate is 0.8 m. This distance equals 20 times the mesh size of the grid, which is sufficient to obtain a fairly homogeneous turbulence structure at the leading edge of the plate. 2.2 Measurement technique In the experiments, a dye visualisation method is used. Through a spanwise row of holes with an intermediate distance of m and hole diameter of m the fluorescent dye Rhodamine B is inserted in the boundary layer (Fig. 3). The dye inlet is positioned 0.19 m from the leading edge of the flat plate. In absence of the grid, the inserted dye forms steady streaklines close to the plate. Excited by an Nd:YAG laser sheet (wavelength, k, 532 nm) the streaklines emit fluorescent light. This is used to visualise the flow. The sheet is realised with a negative lens. The light from the emitting streaklines is recorded by a CCD camera (Kodak Megaplus pixel, 10 bits). The described measuring Fig. 3 Global view of the measuring configuration

4 1074 technique is utilised in a configuration where the streaklines are excited by a horizontal laser sheet as shown in Fig. 3. The CCD camera is mounted above the measuring section. streaklines. Also, the passing turbulent spots with the swirled streaklines are clearly visible. When compared with the result from Matsubara and Alfredsson (2001), a good resemblance is observed. 3 Turbulent spots results 3.1 The transitional boundary layer A typical visualisation result is shown in Fig. 4. The figure shows an area of m 2 in the centre of the channel and a streamwise position for which 1.13<x<1.69 m. The visualisation height is slightly lower than d *, the boundary layer displacement thickness for a laminar boundary layer based on the length from the leading edge. The flow is from left to right. The freestream velocity is set to 0.11 m/s, resulting in a Reynolds number range of <Re x < Notice that the transitional Reynolds number based on the streamwise coordinate is of the order 10 5, which is almost a factor of 10 lower than the value valid for classical transition. The Reynolds numbers based on the mesh size and rod diameter of the grid are very similar to the Reynolds numbers of grid E in Matsubara and Alfredsson (2001). As a result, the turbulence level of 6.7% at the leading edge, determined by a PIV measurement, is very close to the value reported by Matsubara and Alfredsson (2001). The Taylor length scale for the streamwise velocity component of the free-stream turbulence was 6 mm as determined from PIV measurements. Figure 4 clearly shows the emitted fluorescence light by the streaklines. The laminar parts in the boundary layer are clearly distinguishable by the straight aligned 3.2 Turbulent spot features Two image sequences with developing turbulent spots are shown in Figs. 5 and 6. The experimental conditions in both sequences are the same as the conditions described at Fig. 4. Only the streamwise position differs. Figure 5 shows the breakdown of a spanwise oscillation and its development into a triangular turbulent spot. Figure 6 clearly shows the streamwise and spanwise growth of a spot as it convects downstream. The calmed region behind a turbulent spot is also visible in this sequence. The growth in streamwise direction is analysed by determining the leading and trailing edge velocities of a spot. The development of the location of the spanwise edge provides information about the spanwise growth. Of course the boundary of a turbulent spot is an arbitrary definition. Here obviously the interior of a spot is defined by the swirl extent of the streaklines. The interior of a spot is determined manually, so no mathematical criterion is used. There is an estimated error of ±3 [x/d * 0 ] in the location of the edges of a spot, where d * 0 is a scaling parameter which is defined by Brandt (2003) as Re d ¼ 300: 0 The result of the analysis is shown in Figs. 7 and 8. They show the development of the leading and trailing edge of a spot as function of time and the development of the spanwise edge as function of the streamwise position, respectively. From this, both the trailing and leading edge velocity of a turbulent spot and its spreading angle are determined. The breakdown of the experimental boundary layer exposed to free-stream turbulence leads to turbulent spots which have a leading edge velocities of 0.82 U, a trailing edge velocity of 0.5 U and a spreading angle of 11. Singer (1996) and Schubauer and Klebanoff (1955) report a value of 0.94 and 0.88 U respectively for the leading edge velocity. For the trailing edge velocity, they repor.63 and 0.5 U, respectively. Singer (1996) found a spreading angle of 6.4 while Schubauer and Klebanoff (1955) report a value of 11. Comparing these values with the presented experimental results shows that there is a good agreement. These results confirm that the transition process established in the water channel is a by-pass transition process in which the boundary layer flow breaks down into turbulent spots which behave as described in literature. Fig. 4 Flow visualisation of instabilities in a flat boundary layer subjected to a free-stream turbulence level of 6.7%. The flow is from left to right. U =0.11 m/s and <Re x < Secondary instabilities Figure 5 shows that a turbulent spot is initiated by a spanwise oscillation. This oscillation is studied in the

5 1075 Fig. 5 Sequence of streak breakdown, sequence rate 3.75 Hz and U =0.11 m/s <Re x < Fig. 6 Sequence of streak breakdown, sequence rate 3.75 Hz and U =0.11 m/s <Re x <

6 1076 x/δ * 0 [ ] /15 +3/15 +5/15 +7/15 +9/15 Fig. 7 The streamwise position of the leading and trailing edge of a turbulent spot as function of time z/δ * 0 [ ] visualisation results presented previously; and in results from new experiments, performed under similar conditions as the previous ones but using a smaller recording area. The instabilities in the analysed results fulfil the requirement that they are surrounded by a laminar boundary layer, they are not initiated by a nearby turbulent spot. The study reveals the presence of two types of instability modes, a sinuous and a varicose mode. The two modes will be discussed by means of two image sequences. These sequences were chosen from the larger set of measured instabilities. The behaviour of the presented instabilities is representative for all the observed instabilities. Furthermore, it is believed that as time goes on, the most unstable wavelengths eventually emerge. The characteristic behaviour of these emerging unstable t [s] x/δ * 0 [ ] Fig. 8 The spanwise position of the spanwise edge of a turbulent spot as function of the streamwise position wave-packets will be discussed. Note that an integrating visualisation method is used. This means that small perturbations are not detected. This confirms that the presented instabilities consist of the final unstable wavelengths. In Fig. 9, an image sequence shows the sinuous instability mode, while Fig. 10 shows the varicose mode. Figure 9 clearly shows that at each streamwise position the sinuous mode has an uni-directional spanwise velocity. spatially, the velocity varies between a positive and a negative spanwise velocity. The varicose mode has no uni-directional spanwise velocity at each streamwise position. In fact, the spanwise velocity field is symmetric around the centreline of the instability mode. At each side of the centreline, the spanwise velocity varies spatially again between a positive and a negative velocity. Nevertheless, the two sequences show that both instability modes have some resemblances. The figures clearly show that the amplitude of the instability spatially increases in upstream direction. However, the last period deviates from this trend. The instability utilises this period to adapt to the laminar surrounding. Note that the beginning of the instability is defined by the most downstream location where the spanwise motion is visible in the dye streak. This means that both the sinuous and varicose mode are localised instabilities which travel with a certain streamwise velocity. Notice that in this natural breakdown process, the amplitude of the instability spatially increases in upstream direction. Furthermore, the last period is used to adapt to the laminar surrounding, while in the results of Brandt et al. (2004) and Asai et al. (2002) the opposite seems to take place. The amplitude increases spatially in downstream direction and by means of the first period, the instability adapts to the laminar surrounding. The reason for this may be related to the fact that Asai et al. (2002) used a controlled trigger to initiate an instability. Another reason could be that Asai et al. (2002) utilises a steady fence generated low speed streak. This streak differs from the free-stream turbulence induced streaks and could therefore behave differently. The two sequences also reveal that when the amplitude passes a critical value, roll-up structures appear. These structures continue to develop and finally they interact and merge resulting in a triangular shaped turbulent spot. In the sinuous mode, these roll-up structures appear in a staggered pattern on the flanks of the low speed streaks as observed in the DNS by Brandt and Henningson (2002). In the varicose mode, the roll-up structures seem to arise at one side of the instability. With the development in streamwise direction, the structures finally affect the whole instability. Another remarkable resemblance between the two modes is that both consist of 4 periods. However, we should remark that some other instabilities consist of a slightly different number of periods. Besides the resemblances some differences exist between the modes. A difference lies in the length scales of

7 1077 Fig. 9 Sequence of streak breakdown, sequence rate 7.5 Hz and U =0.13 m/s <Re x < Fig. 10 Sequence of streak breakdown, sequence rate 5 Hz and U =0.11 m/s <Re x <

8 1078 the instabilities. Figure 11 shows a zoomed view of one period and its spanwise and streamwise length scale from the sinuous and varicose mode. The periods shown are those where the first roll-up structures appear. The sinuous instability has a characteristic streamwise length scale of 28d 0 * and a spanwise scale of 2.5d 0 *. The scales of the varicose instability are 19d 0 * and 5.5d 0 *, respectively. These length scales are very similar to the scales from DNS as shown in Brandt (2003). The streamwise and spanwise scales for the sinuous mode resulting from DNS calculations are respectively 15 d 0 * and 3 d 0 * and for the varicose mode 12 d 0 * and 5 d 0 *. However, it should be noted that the length scales reported here are deduced from an integrating visualisation method. 5 Detailed analysis As pointed out, the sinuous and varicose instabilities travel with a certain streamwise velocity. In relation to the appearance of low and high speed streaks during bypass transition a study on the group and phase velocities of the instabilities is interesting. It is observed that the roll-up structures arise as the amplitude of the instability modes reaches a certain critical value. These roll-up structures finally result in a triangular shaped turbulent spot. From this it can be concluded that the roll-up structures are the first sign of a turbulent spot. The point where the amplitude reaches the critical value represents therefore, the required conditions for turbulent breakdown. By studying the development of the amplitude and wavelength of the instability and relating this to the appearance of the roll-up structures, the required conditions can be clarified. 5.1 The data sets In the study on the velocity, development of wavelength and development of amplitude of an instability use is made of a set of images taken from the sequences in Figs. 9 and 10. Each set consists of the first image of the sequence till the image where the first roll-up structures appear. For the sinuous mode, this results in a set of four images. The time between two successive images is 2/15 s, the time lapse is therefore 6/15 s. A set of 5 images with a time step of 1/5 s and time lapse of 4/5 s is used in the analysis of the varicose mode. For practical reasons, the recording area also differs between the two sets. The recording area in the varicose set is much bigger than the area of the sinuous set. This means that the sinuous set contains the first two periods of the total instability, while the varicose set contains the whole instability. Again due to practical reasons, there is also a difference in the visualisation height. In the sinuous set, a visualisation height of y= m is used. A height of y= m is used in the varicose set. These heights are related to the boundary layer displacement thickness, d *, in the centre of an image. This gives that the sinuous height corresponds to 1.6d * and the varicose height to 0.97d *. The wavelengths and amplitudes of the oscillations of the instability modes are manually determined. The scale of the amplitude of the modes is one order lower than the scale of the wavelength, as given by Fig. 11. To simplify the manual determination of the amplitude, the figures used in the analysis are stretched in the spanwise direction. Figure 12 shows a selection from the stretched sinuous and varicose modes together with the manually determined valleys, peaks and nodes (stars) of the oscillation. The distance used for a wavelength, k, is also denoted in this figure. In the figure, the distance used for the Fig. 11 A period and its characteristic length scales from the sinuous (left) and varicose mode (right) Fig. 12 A selection form the sinuous (left side) and varicose (right side) instability stretched in the spanwise direction, together with the manually determined valleys peaks and nodes. Also the standards used for an amplitude and a wavelength are stated

9 1079 wavelength in the sinuous mode is given between two valleys. However, the distance between two peaks is also used. Therefore, the wavelength is determined for each half period for the sinuous mode. In the varicose mode a wavelength is determined only for each full period. In the figure, the distance used for an amplitude, A is also stated. The distance used for the amplitude in the sinuous mode is represented between a peak and a valley. Therefore, the amplitude is similar to the wavelength determined for each half period for the sinuous mode. In the varicose mode an amplitude is determined for each full period only. Wavelength [ δ 0 * ] /5 [s] +2/5 [s] +3/5 [s] +4/5 [s] 5.2 The wavelength of the instabilities In each image from the data sets, the wavelength of each period is determined. The development of the wavelength is analysed. The result of the analysis is shown in Figs. 13 and 14. For the sinuous instability, Fig. 13 shows the spatial development of the wavelength in each image from the set of 4 images. Each symbol represents the wavelength of a period at a specific point in time. The solid lines represent the fitted profiles for the spatial development of the wavelength. The spatial development of the wavelength in each image from the varicose set together with the fitted profiles are shown in Fig. 14. As stated already at Fig. 11, both figures again clearly show that the streamwise length scale of the sinuous instability mode is larger than the length scale of the varicose mode. The wavelength of the sinuous mode varies from around 18d 0 * to 32d 0 *, while the varicose wavelength varies between 13d 0 * and 20d 0 *. The estimated error in the wavelength of the sinuous mode is ±2%. For the varicose mode the error is ±4%. In Fig. 13, the wavelengths determined for period 1.5 clearly deviate from the trend present in the figure. In each image, the wavelength seems to be underestimated. Wavelength [ δ 0 * ] /15 [s] +4/15 [s] +6/15 [s] period 0.5 period 1 period 1.5 period 2 Fig. 13 The spatial development of the wavelength of the sinuous instability at four points in time period 1 period 2 period 3 period 4 Fig. 14 The spatial development of the wavelength of the varicose instability at five points in time The explanation for this phenomena is that period 1.5 is already influenced by the roll-up process. This means that the peak is shifted in streamwise direction which results in an underestimation of the wavelength. This process is also present in period 2. However, the valley from this period is closer to a dye streakline. Due to the higher dye concentration, the base oscillation at this period can be followed. The expression base wave refers to the spanwise oscillations without roll up structures. From both figures, it is evident that in both the sinuous and the varicose instability, the wavelength spatially increases. The sinuous instability clearly shows a linear increase. Initially, the varicose instability also shows a linear spatial development. However, with time, the linear profile changes into a non-linear profile. This development is described by an increase with time of the wavelengths of the first two periods. The wavelength of the third period initially increases but finally decreases with time and the wavelength of the last period clearly decreases with time. This change in the spatial profile is not present in the sinuous case. The wavelengths in the sinuous instability do not vary significantly with time. Therefore, the spatial development of the sinuous wavelength is represented with one profile in Fig The velocity of the instabilities The phase velocities of the different periods in an instability are estimated by tracking the positions of the different peaks and valleys in different images. The peaks and valleys from the wavelength analysis is utilised. The result of this analysis is shown in Figs. 15 and 16, in which the relative position of peaks and valleys as function of time are shown for the sinuous and varicose instability, respectively. The solid line in the figures represents a fit through the data points. The relative

10 1080 ε [δ * 0 ] valley 3 valley 2 valley 1 peak 3 peak 2 peak 1 +2/15 [s] +4/15 [s] +6/15 [s] Fig. 15 The relative position,, of peaks and valleys in the sinuous instability as function of time ε [δ * 0 ] node 1 node 2 node 3 node 4 node 5 +1/5 [s] +2/5 [s] +3/5 [s] +4/5 [s] Fig. 16 The relative position, e, of peaks and valleys in the varicose instability as function of time position,, of a valley or peak at a certain point in time is determined as: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ ðx x t0 Þ 2 þðz z t0 Þ 2 ð1þ where x is the streamwise position, x t_0 the streamwise position at t=, z is the spanwise position, z t0 the spanwise position at t=. The estimated error in the relative position,, is ±3% in both modes. Figure 15 shows that in the sinuous instability, almost all nodes travel with approximately the same velocity. This velocity, represented by the fitted solid line, can be interpreted as an estimate for the group velocity. The influence of 3D effects and the wavelength variations are then neglected. Furthermore, it is assumed that this velocity represents the velocity of the envelop of the instability. The analysis gives that the nodes in the instability travel with an average velocity of 0.75 [u/u ]. Note that this propagation velocity of the sinuous instability is very close to the value of 0.8 reported by Brandt et al. (2003). Furthermore, the study of Brandt et al. (2003) revealed that there is a strong correlation between the basic velocities and the characteristics velocities of an instability. The Blasius velocity at the visualisation height of 1.6d * is 0.82 [u/u ]. This value is higher than the propagation velocity of the sinuous instability. This difference indicates that, due to the strong correlation between the basic velocity and this propagation velocity, the instability is located in a low speed streak. This is also expressed by the ratio of the sinuous propagation velocity and the Blasius velocity at the visualisation height, of 0.91 [u/u blas ]. The figure also shows that the node located in the third peak of the sinuous instability represented by the diamonds deviates from the trend. The velocity of this node is 0.82[u/U ]. Probably, the reason for this deviation is that due to the increasing amplitude of the instability the node of the third peak moves out of the low speed streak and enters the high shear region between a low and high speed streak. Of course the velocity in this region is of the same order as the Blasius velocity. Recall that the roll-up process influences the third peak, section 5.2. The consequence of the information on the location of this peak is that the roll-up structures also must be generated in this high shear region. Another remarkable phenomenon is that the nodes of the valleys tend to have a lower velocity than the nodes of the peaks. This indicates that the initial part of the sinuous instability is located at one side of a low speed streak. In Fig. 16, it is shown that the difference between the phase velocities of the nodes is larger in the varicose instability than in the sinuous instability. Still a mean velocity is determined, represented by the solid line. This mean velocity can again be interpreted as an estimate for the group velocity of the varicose instability. The mean velocity of the nodes is 0.47 [u/u ]. At the visualisation height of 0.97d *, the Blasius velocity is 0.54 [u/u ]. Therefore the varicose instability is located in a low speed streak just like the sinuous instability. In the varicose case, the ratio of the propagation and Blasius velocity is 0.87 [u/u blas ]. Taking a closer look at the individual phase velocities of the nodes shows that the first node, represented with a (*)-symbol, has the highest velocity. The second (e) and third node (s) have approximately the same velocity. This velocity is a bit lower than the first node velocity. This while the fourth (h) and the fifth ( ) node again have a lower velocity than the second and third node. A possible explanation for this interesting phenomenon is that the varicose instability is oblique positioned compared to the flat plate. So the first node is positioned higher in the boundary layer than the second, and so on. Another possible explanation for the velocity difference between the nodes in both the

11 1081 sinuous and varicose instability is that nonlinear effect plays a role. 5.4 The amplitude of the instabilities The amplitude of each period from the two data sets is again manually determined using the same peaks and valleys as in the wavelength analysis, (Fig. 12). In sect 2.2, it was already mentioned that the used dye visualisation technique is an integrating visualisation method. For a visualised oscillation, this means that its amplitude is a function of the wavelength. To be more exact, an increase in wavelength will also lead to an increase of the amplitude. To compensate the measured amplitude for this character of the visualisation method, a corrected dye amplitude is determined as: A cor ¼ 2pA dye ð2þ k The developments of the corrected amplitudes are given in Figs. 17 and 18. The estimated error in the corrected amplitude, A cor, is ±5%. Figure 17 shows the spatial development of the amplitude in each image from the set of 4 images from the sinuous instability. Again each symbol represents a specific point in time and the solid lines show the fitted profiles for the spatial development of the corrected amplitude. In the following the corrected amplitude is referred to as amplitude. Figure 18 shows the same quantities for the varicose mode. From Fig. 17 it is clear that the amplitude of the sinuous instability shows a spatial increase. The development shows in each image a non-linear profile. Furthermore the amplitude grows from a value of 0.11 [ ] in period 0.5 in the first image to a maximum value of 0.23 [ ] in period 2 in the last image. Figure 17 also gives an indication of the amplitude growth in time of each period of the sinuous instability. An analysis on the growth showed that for each period the amplitude increases linear in time. It is remarkable that all periods approximately have the same relative growth factor, c, of 3.4 1/s. The relative growth factor is defined by: c ¼ 1 da ð3þ A 0 dt In each image, the varicose amplitude also develops according to a non-linear shaped profile (Fig. 18). Also the varicose amplitudes experience temporal growth. The amplitude grows from a value of 0.38 [ ] in period 1 in the first image to a maximum value of 0.63 [ ] in period 2 of the last image. The amplitude growth in time of the varicose instability, indicated by Fig. 18, is also analysed. The amplitude increase in time of the varicose instability is linear, just as in the sinuous mode. It is remarkable that the growth factor is not the same for each period in the varicose instability. However the second and third period have approximately the same growth factor of 1.6 1/s. The growth factors of the first Amplitude [ ] /15 [s] +4/15 [s] +6/15 [s] period 0.5 period 1 period 1.5 period 2 Fig. 17 The spatial development of the amplitude of the sinuous instability at four points in time Amplitude [ ] /5 [s] +2/5 [s] +3/5 [s] +4/5 [s] period 1 period 2 period 3 period 4 Fig. 18 The spatial development of the amplitude of the varicose instability at five points in time and last periods deviate from this value. These growth factors are 1.9 and 1.3 1/s, respectively. This shows that the growth factor of the sinuous instability is approximately two times larger than the varicose growth factor. Notice that the maximum amplitude is present in period 2 for both instability modes. However, the amplitude in the varicose mode is around a factor 3 larger than the sinuous amplitude. This remarkable observation is in accordance with the numerical results by Andersson (2001). His results demonstrate that the velocity difference between the high and low speed streaks in the varicose mode is larger than the velocity difference between the streaks in the sinuous mode. Comparing the sinuous amplitude development with the development of the first two periods from the varicose mode in Fig. 18 shows that the behaviour is similar. However, in the sinuous instability period 2 and period

12 experience a stronger temporal growth than the first two periods, whereas in the varicose mode, the first period experiences the strongest growth. Furthermore, the temporal growth (defined as the ratio of change in amplitude and change in time), in the first period of the varicose mode is of the same order as the growth in the last two periods of the sinuous mode. Recall that the varicose time lapse is 4/5 s while the sinuous time lapse is 6/15 s. The non-linear development in both the sinuous and varicose instability is an indication for the localised behaviour of the instabilities. Finally, it is to be noted that the visualisation heights in the sinuous and varicose case differ. As stated, the visualisation heights in the sinuous and varicose mode are respectively, 1.6 d * and 0.97 d *. This could have an influence on the comparison studies between the sinuous and varicose amplitude. However, in the disturbance profile resulting from optimal growth theory, Luchini (2000), has its maximum at the position 1.3d *. The used visualisation heights are positioned almost symmetric around this maximum. Due to this fact, it is assumed that the characteristics from both modes are comparable. 5.5 Breakdown As pointed out, the presence of roll-up structures is the first sign of breakdown. These roll-up structures appear as the amplitude of the instabilities reaches a certain value, the critical amplitude. By relating the profiles shown in Figs. 17 and 18 to the appearance of the rollup structures in the sequences shown in Figs. 9 and 10, the critical amplitudes in both instability modes are determined. For the sinuous and varicose instability, the critical amplitude is are 0.22[ ] and 0.59[ ], respectively. 6 Conclusion and discussion In a water channel, the natural, unforced, breakdown process during bypass transition is experimentally studied using a dye-visualisation technique. The freestream turbulence level of 6.7%, established using a static turbulence grid, results in a bypass transition mechanism in the flat plate boundary layer. The transitional regime of the boundary layer is visualised. The results clearly show the presence of turbulent spots. The spots grow in streamwise as well as spanwise direction. The results also show that the natural breakdown in a turbulent spot is initiated by a spanwise oscillation. A detailed analysis on the oscillation revealed the presence of two secondary instability modes, a sinuous and varicose mode. The sinuous mode has at each streamwise position, a uni-directional spanwise velocity while the varicose mode has a velocity field which is symmetric around the centreline of the instability. It is clear that, just as the turbulent spot, the instabilities are a local phenomenon. Besides, it is striking that both modes consist of 4 periods. Note however that other experimental results under the same conditions show instabilities with a different number of periods. Still, the number of periods remains in the range of 2 6 periods. Both modes also develop according to a similar scenario. The amplitude of the instability increases in upstream direction according to a non-linear profile and with time. When a critical amplitude is reached, the instability breaks down into a turbulent spot. The first sign of breakdown is the presence of roll-up structures in an instability. It was found that the amplitude of the varicose mode is a factor 3 larger than the amplitude of the sinuous mode. This is in accordance with the numerical results by Andersson (2001), which showed that the velocity difference between a low and high speed streak is larger in the varicose than in the sinuous case. The propagation velocity of the sinuous and the varicose instability is also examined. The sinuous propagation velocity is 0.91 [u/u blas ] and the varicose velocity is 0.87 [u/u blas ]. This shows that both modes are located in a low speed streak. References Andersson P, Brandt L, Bottaro A, Henningson DS (2001) On the breakdown of boundary layers streaks. J Fluid Mech 428:29 60 Asai M, Minagawa M, Nishiola (2002) The instability and breakdown of a near-wall low-speed streak. J Fluid Mech 455: Brandt L (2003) Numerical studies of bypass transition in the Blasius boundary layer. Doctoral thesis, Royal Institute of technology Brandt L, Henningson DS (2002) Transition of streamwise streaks in zero-pressure-gradient boundary layers. J Fluid Mech 472: Brandt L, Cossu C, Chomaz JM, Heurre P, Henningson DS (2003) On the convectively unstable natur of optimal streaks in boundary layers. J Fluid Mech 485: Brandt L, Schlatter P, Henningson DS (2004) Transition in boundary layers subject to free-stream turbulence J Fluid Mech 517: Emmons HW (1951) The laminar-turbulent transition in a boundary layer; Part I. J Aeronaut Sci 18: Herbert T (1988) Secondary instability of boundary layers. Annu Rev Fluid Mech 20: Kendall JM (1985) Experimental study of disturbances produced in a pre-transitional laminar boundary layer by weak free-stream turbulence. AIAA paper Klebanov PS (1971) Effect of free-stream turbulence on the laminar boundary layer. Bull Am Phys Soc Luchini P (2000) Reynolds number independent instability of the boundary layer over a flat surface: optimal perturbations. J Fluid Mech 404: Matsubara M, Alfredsson PH (2001) Disturbance growth in boundary layers subjected to free-stream turbulence. 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