Time-resolved visualization of coherent structures during supersonic boundary layer transition

Lin HE, Xiaoge LU, Junhao HAN, Zhengbang WU, Shihe YI

College of Aerospace Science and Engineering, National University of Defense Technology, Changsha 410073, China

KEYWORDSCoherent structures;Flow visualization;Supersonic boundary layer;Time-resolved;Transition

AbstractThe coherent structures arising during flat-plate boundary layer transition at Mach number 3.4 are investigated using a custom-built hyper-rate imaging system.The evolution of transitional structures is investigated in the Eulerian and Lagrangian reference frames.The upstream evolution of transition is dominated by the generation of new hairpin structures,while the interaction among multiple structural types dominates the evolution downstream.The breakdown of the existing structure,which may be caused by interactions among multiple types of structures with similar scales, is also visualized.

1.Introduction

Laminar-turbulent transitions are important in both practical applications and fundamental research.Studying these transitions is especially vital in the development of future space vehicles operating at sustained supersonic and hypersonic speeds.The boundary layer transition can be roughly divided into two types.The first one is connected with boundary-layer instabilities and occurs when environmental disturbances are relatively small.The second one is usually called bypass transition and occurs when strong environmental perturbations are present.1In the first type, coherent structures appear during the boundary layer transition when nonlinear perturbations grow and become predominant.Then these structures break down and the laminar-turbulent transitions are completed.2.

Numerous numerical simulation studies have focused on the coherent structures and their evolution during transition.For example, the spatial evolution of the three-dimensional structures and their spatiotemporal behavior in a transitional boundary layer were investigated by Rempfer and Fasel,3using the Direct Numerical Simulation (DNS) database.The nonlinear evolution of new vortex structures during the late stage of the transition was studied by Chen et al.,4who also adopted the DNS database.Sayadi et al.5simulated the complete H-type and K-type transitions of a spatially growing boundary layer by DNS.

In experimental research,the visualization of the organized motions of coherent structures in time domain during transition, especially using time-resolved visualization methods, is a powerful approach for revealing the complicated dynamics involved in the laminar-turbulent transitions.The timeresolved methods can provide more useful information about the organized motions of coherent structures during transition,which can assist in better understanding of the transition mechanisms.For example,how the hairpin vortices evolve further downstream, what mechanisms are responsible for the flow breakdown in supersonic plate boundary layers, and how new structures are generated have not yet been sufficiently explained.Thus, visualizing the evolution of structures during transition will help to understand the generation,development and factors affecting the breakdown of the coherent structures during transition and may offer suggestions for controlling the boundary layer transition.

The majority of the existing time-resolved methods focus on the evolution of coherent structures in incompressible boundary layers, because these methods are relatively easy to adopt for low-speed flows.Lee et al.6used the hydrogen bubble method to obtain clear pictures of the formation and development of coherent structures during a low-speed boundary layer transition.Guo et al.7also employed the hydrogen bubble method to investigate the complex flow structures and their breakdown during the later stages of the boundary layer transition.Lo¨gdberg et al.8characterized the streamwise evolution of longitudinal counter-rotating vortices in a turbulent boundary layer by using both smoke visualization and three-component hot-wire measurements.Jiao et al.9investigated the evolution of wall flow structure issuing from a supersonic jet with an extended shelf using the surface oil flow infrared detection technology.In addition to the traditional flow visualization methods, a large number of investigations on the evolution of flow structures have been performed using the Particle Imaging Velocimetry(PIV) method.Schro¨der et al.10investigated the temporal evolution of coherent structures in the logarithmic region of a turbulent boundary layer using time-resolved tomographic PIV and Particle Tracking Velocimetry (PTV) at a 1 kHz sampling rate.The evolution of vortex structures during boundary layer transition induced by roughness elements was experimentally investigated using the hydrogenbubble visualization method and 2D-PIV measurements by Zhang et al.11He et al.12adopted the same methods and studied the evolution of Lagrangian coherent structures in a flat-plate boundary layer transition induced by the wake of a circular cylinder.Gao et al.13designed and implemented a moving tomographic PIV method to measure the temporal evolution of velocity fields in threedimensional volumes and to track coherent structures within a turbulent boundary layer.LeHew et al.14used timeresolved PIV to examine the swirling coherent structures and their evolution in a turbulent boundary layer along the planes parallel to the wall.Liu et al.15studied the evolution of turbulent boundary layer over a three-dimensional bump using the PIV method.The time-resolved PIV has become an important and widely used tool to investigate the evolution of flow structures.16–18.

However, there have been only a few experimental studies on the evolution of structures in the compressible boundary layers, especially in supersonic or hypersonic boundary layers.This is because experimental investigations of the evolution of supersonic or hypersonic flows present significant challenges.The phenomena of interest often occur on timescales of microseconds in supersonic flows, requiring highresolution time-resolved images at very fast (kHz or even MHz) rates to track the evolution of structures in highspeed flows.In addition, capturing the instantaneous flow features requires very short exposure time.Therefore, maintaining a meaningful spatial resolution while satisfying the temporal resolution requirements of high-speed imaging is challenging.

McIntyre and Settles19first acquired four time-resolved schlieren images with different time separations ranging from 1 to 24 μs to study the evolution of a supersonic shear layer.Smith and Smits20used high-speed schlieren movies to reveal the evolution of large-scale structures in flat-plate zeropressure-gradient supersonic boundary layers.Ben-Yakar and Hanson21used schlieren imaging with a new ultra-fast camera system to study the time evolution of jets in supersonic crossflows.Laurence et al.22described the propagation of instability waves in a hypersonic boundary layer via timeresolved high-speed schlieren cinematography.They also investigated the evolution of instability waves within the boundary layer on a slender cone under high Mach number conditions using the same method.23Wang et al.24used a time-resolved schlieren system with a 384 × 128 pixel resolution and a 3.7 μs shooting interval to explore the evolution of the interaction between a transverse plasma jet and a shock wave induced by a ramp.

Because the spatial integration of the schlieren method limits its application to the investigation of the threedimensional features of coherent structures, the laser sheet technique was developed.Smith et al.20performed flow visualization with laser sheet illumination of acetone droplets to reveal the structural evolution in supersonic turbulent boundary layers.Huntley et al.25used a high-speed imaging system to visualize elliptic cone boundary layer transition at Mach number 8.Volumetric images of the centerline of the cone revealed the hairpin structures’characteristic of the early stages of subsonic turbulent spot formation.However, their hyper-rate cameras had a limited resolution of 180 × 180 pixels at a framing rate of 1 MHz with a field of view of 3 cm × 3 cm.Lu et al.26used a recently developed Nanotracer Planar Laser Scattering (NPLS) technique with a resolution of 2048 × 2048 pixels (a spatial resolution of 0.056 mm/pixel) at a time interval of 12 μs to study the evolution of unsteady structures of the interactions between a shock wave and a turbulent boundary layer.

In this study, time-resolved measurements were taken with a multiple-pulsed laser and multiple cameras during a very short time interval allowing high-resolution recording of rapid changes in the coherent structures during the transition of a supersonic boundary layer.This study focused in particular on the motion of structures in the streamwise wall-normal plane.To that end, a custom-built hyper-rate imaging system was employed to enable high-resolution time-resolved visualization of coherent structures during the transition of a supersonic boundary layer.The new information obtained from this study will assist in better understanding of the boundary layer transition and breakdown into turbulence.

The rest of this paper is organized as follows.Section 2 describes the experimental facility, test model, and visualization technique employed.The evolution of transitional structures investigated in the Eulerian and Lagrangian reference frames is presented in Section 3.In Section 4, examples of the generation of new structures and breakdown of the existing structure during transition were provided.A summary of the work and its major conclusions are presented in Section 5.

2.Experiments

2.1.Supersonic wind tunnel

The experiments were conducted in a vacuum-indraft supersonic wind tunnel located at the aerodynamics laboratory of the National University of Defense Technology.The test section is 100 mm wide, 120 mm high and 500 mm long.The two sidewalls were equipped with 250 mm × 140 mm ×25 mm (length × height × thickness) transparent glass windows (i.e., higher than the height of the test section), and the top and bottom walls with 250 mm × 140 mm × 25 mm(length × height × thickness) transparent glass windows.Thus, all four sides of the test section had transparent glass windows,which was facilitated flow visualizations from different directions.A photograph of the facility is shown in Fig.1.

In the present experiments, the Mach number Ma was 3.4(free-stream velocity U∞=650 m/s), with a free-stream unit Reynolds number of 5.8 × 106m-1.The free-stream turbulence level was measured below 0.5 % using a highfrequency pressure transducer (Kulite XCE-62).The detailed flow parameters are given in Table 1.

2.2.Flat plate

In the present experiments, a flat plate was positioned 30 mm above the bottom floor of the test section and parallel to the oncoming uniform flow (see Fig.2).This flat plate had been optimally designed and experimental results27had proven that it did not affect the flow-field quality of the supersonic wind tunnel or the boundary layer transition on the flat plate.The leading edge of the plate was located inside the diamondshaped uniform flow region of the nozzle section.The tested boundary layer was developed along this flat plate with a nominally zero-pressure gradient condition.

In order to minimize the influence of scattered light on the visualization of near-wall boundary layer,a glass window was embedded in the middle of the metal flat plate.Care was taken to make the plate and glass surfaces flush with each other to avoid any potential disturbance from surface discontinuity on the boundary layer transition.The flat-plate surface was considered to be a smooth surface, as the roughness of glass is in the submicron region.The plate was 5 mm thick and 330 mm long, with a sharp nose.The width of the flat plate was 99 mm, while the width of the test section was 100 mm to avoid the flow under the plate to interfere with the boundary layer transition above the plate as much as possible.A coordinate system was adopted, which had the origin at the center of the leading edge of the plate, and the coordinates x,y, and z corresponded to the streamwise, wall-normal, and spanwise directions,respectively.In this paper,the streamwise wall-normal plane is referred to as the xy-plane for short.Images of the tested boundary layer were obtained in the xyplane, with the laser sheet perpendicular to the wall along the centerline of the plate, and the Charge Couple Device(CCD) cameras set perpendicular to the laser sheet.A schematic of the overall experimental arrangement is shown in Fig.3.

Table 1 Flow conditions.

Fig.2 Arrangement of flat plate inside wind tunnel.

2.3.NPLS technique

To visualize the evolution of coherent structures during the boundary layer transition, the NPLS technique was used.The NPLS technique is based on the traditional planar laser scattering technique,but uses nano-sized rather than the usual micron-sized particles as the tracer, which provides enhanced ability to follow supersonic or even hypersonic flows.Furthermore, nanoparticles with cross-sections larger than molecule can be easily visualized using an un-intensified CCD camera.This technique has been successfully applied in the visualizations of different flows, such as a boundary layer,27compression ramp28and shock wave/boundary layer interaction29.The details of this technique are available in Ref.30.

In the current experiments,a custom-built eight-pulsed Nd:YAG laser was used instead of a traditional double-cavity Nd:YAG for illumination, and eight CCD cameras instead of a single camera placed side by side were used as the imaging system, which were able to capture eight sequential images in a short time interval.Each camera was equipped with a Nikon f = 105 mm F/2.8 lens and a Tilt-Shift (TS) mechanism to make sure that the perspectives of the eight cameras almost overlapped and the entire field of view could be clearly in focus.For details of the multi-camera system with the TS mechanism,one can refer to Ref.31.The laser and imaging system were controlled by a synchronizer with an accuracy of 250 ps.Fig.4 shows a schematic of the NPLS system.

Fig.1 Photograph of supersonic wind tunnel.

Fig.3 Schematic of experimental arrangement.

2.4.Eight-pulsed laser technique

A new custom-built eight-pulsed laser system was developed,which utilized the traditional pulsed Nd:YAG laser.A photo of the eight-pulsed laser system is shown in Fig.5.Unlike the traditional PIV laser system with a dual-cavity laser, here,eight cavities of individual Nd:YAG lasers at 532 nm wavelength were combined.Each pulse width was 8 ns with a nominal beam diameter of 10 mm.The maximum laser energy was 300 mJ per pulse,with the rms of the energy variation less than 3 %.Eight laser beams were combined co-linearly through a beam-combining optical system,which used ?-waveplates,Second Harmonic Generation (SHG) KD*P, dichroic mirrors,and linear polarizers.The co-linear beams were delivered by an articulated arm to the test section, and subsequently transformed into laser sheets using optical lens,resulting in at least 95%overlap among the eight laser sheets at the field of view.The laser system can produce eight independent pulses with an arbitrary time interval between lasers.However,the minimum time interval between lasers is limited to 100 ns to avoid the heating up of the optical system by the laser beam during a very short time interval.

Fig.4 Schematic of NPLS system.

Fig.5 Photo of eight-pulsed laser system.

2.5.Imaging system and image calibration

In order to capture multiple images during a very short time interval, eight cameras placed side by side were adopted as the imaging system, with a minimum exposure time of 2 μs.After the eight cameras were placed side by side, the field of view and angle of each camera were slightly different and deformed.In order to ensure the consistency of the imaging area of each camera,an image calibration was conducted using checkerboard images as calibration targets.Fig.6 shows the original calibration target images taken by different cameras,and Fig.7 shows the images after calibration and clipping.

Fig.8 shows the errors of each checkerboard calibration point of the corrected images from seven cameras relative to Camera 5(i.e., one of the cameras in the middle of the camera row).The maximum error of each point on the checkerboard after calibration relative to Camera 5 was less than 1.2 pixels,and the mean error of all the points relative to Camera 5 was about 0.68 pixels, proving the accuracy of image calibration procedure adopted here.

3.Temporal evolution of structures during boundary layer transition

Under the current experimental conditions,due to the low turbulence level of the incoming flow in the wind tunnel, the boundary layer developed over the flat plate in the experimental section (x = 80–330 mm) was essentially in a laminar or early unstable state, and no clear boundary layer transition structures could be observed.To remedy this, a 10 mm wide rough belt was placed at x=50 mm to promote the boundary layer transition.

Fig.6 Original checkerboard images from different cameras.

Fig.7 Checkerboard images of different cameras after calibration and clipping.

Fig.8 Scatter plot of re-projection errors of eight cameras.

3.1.Structures of boundary layer transition in xy-plane

The instantaneous structures of a Ma = 3.4 flat-plate boundary layer in the xy-plane are shown in Fig.9,where the field of view extends from x=160 mm to x=280 mm with a spatial resolution of 0.082 mm/pixel, and the flow was from left to right.It should be noted that the actual units instead of the non-dimensional units with respect to the thickness of boundary layer were adopted here,because the local thickness of the boundary layer varied along the flow direction.

The spatial development of the boundary layer from laminar to turbulent flow can be identified clearly.In the upstream laminar region,the linear growth of the boundary layer can be observed and no coherent structures can be identified.Then,large-scale wave-like structures arise and soon evolve into hairpin-like structures downstream in the transitional region.Later, these large-scale structures breakdown into smallerscale structures and turbulent flow develops.As the upstream turbulence and the disturbances that come from the rough belt, as well as the turbulent boundary layers on the two sidewalls of the test section, affect the boundary layer transition,the transition location and sequence will change differently with time.

3.2.Evolution of coherent structures in Eulerian reference frame

Fig.10 shows eight consecutive images of the coherent structures in a Eulerian reference frame, with 5 μs between everytwo adjacent images.The field of view extends from x = 200 mm to x = 290 mm, and the flow was from left to right, here t0represents the moment of the first image.It can be observed that during the 35 μs documented in Fig.10, the flow structures mostly moved quickly downstream, while changing their shape and size.However, the rate of structural changes was much less than the velocity of downstream travel.

Fig.9 Instantaneous structures of Ma = 3.4 flat-plate boundary layer in xy-plane.

Fig.10 Evolution of coherent structures in Eulerian reference frame.

The most striking dynamic features depicted in these eight images are the generation of new coherent structures and the evolution of the existing coherent structures.Here two typical structures were selected as examples for detailed analysis.The first is an individual vortex structure demarcated with the blue circle and referred to as Vortex A, which illustrates the generation of a new hairpin vortex.The second is an existing largescale vortex structure shown with the red circle and referred to as Vortex B,which is an example of the existing structure evolution.Vortex A was generated from the upstream wave-like structures, then it rolled up with the increase of its height,length and inclination angle, and finally a typical hairpin vortex was formed.Moreover, an oblique, upward stretching structural deformation away from the wall along the flow direction can be identified from the consecutive images, such as those of the structures just downstream from Vortex A.In Fig.10(a), Vortex B, which can also be treated as a hairpin vortex, already existed, and then it grew in size with time.New smaller structures were generated on the edge of the inclined shear layer of Vortex B,and moved along the inclined shear layer, instead of stretching like the structures observed upstream.More complicated structures were visualized inside Vortex B, which in fact may be treated not as a single vortex but rather a packet of multiple small vortices.The detailed analysis of the evolution of Vortex B will be presented in the next section.

The evolution of Vortex B is different from that of Vortex A, because Vortex A can be treated as an individual vortex,while there are multiple vortices inside Vortex B.Moreover,Vortex A is in the early stage of boundary layer transition,while Vortex B is in the later stage of the transition.Although it is difficult to accurately quantify the velocity and rate of change of coherent structures only from images in the xyplane because of the inherent three-dimensional characteristics of turbulence, the evolution of flow structures in different regions during boundary layer transition can still be qualitatively analyzed to understand their differences.In the most upstream region of boundary layer transition, hairpin vortices were the most common, accompanied by a clear oblique upward stretching trend.In the downstream of boundary layer transition, the interactions between multiple structures dominated the evolution process instead of individual hairpin vortices.

3.3.Evolution of coherent structures in Lagrangian reference frame

In order to study the evolution of structures without the need to consider convection, a view moving with the flow, i.e., a Lagrangian reference frame was adopted.Vortex B in Fig.10 was selected as an example, and it is shown in Fig.11.The fields of view (Δx = 20 mm and y = 0–10 mm)were selected so that Vortex B was nearly still in each image,allowing the evolution of Vortex B to be observed more clearly without the effect of convection.

In addition to the evolution of Vortex B itself, the generation of a new small-scale structure (referred to as Vortex C)and its movement along the edge of the inclined shear layer of Vortex B can be observed more clearly in the Lagrangian reference frame, as shown in Fig.11(b)-11(h).The generation and movement of Vortex C can be explained by the ideal hairpin vortex model by the induced motion of the inclined shear layer toward the surrounding flow.As a result, another new small-scale structure (referred to as Vortex D) was generated again soon after,and also moved along the inclined shear layer of Vortex B,as shown in Fig.11(g)and(h).Although it cannot be fully confirmed from the eight images in Fig.11 that a third similar structure seems to have emerged, as can be seen in Fig.11(h).

Another interesting finding is that a small-scale vortex under the head of Vortex B can be seen in Fig.11(e).This vortex later became fully integrated into the head of Vortex B, as shown in Fig.11(f), making the head of Vortex B larger and more complicated.Although the presented eight consecutive images cannot provide more information about the next steps in the evolution of Vortex C, it can be reasonably assumed to continue moving along the edge of the head of Vortex B, due to the attraction of the head of Vortex B.When it moved under the head of Vortex B, it was swallowed by Vortex B,similar to Fig.11(g) and 11(h).However, the premise of this scenario is that Vortex B still maintained a complete hairpin vortex structure while swallowing Vortex C.In an alternative evolution scenario, it can be assumed that vortex C moves to the head of vortex B and changes its structure.When the cumulative change caused by multiple small vortices similar to Vortex C is large enough,the organized motion of the hairpin vortex cannot be maintained, and Vortex B will break down.This scenario will be further discussed in Section 4.2.It should be noted that the evolution of the small vortex just upstream of Vortex B was different from that of Vortex B during the same time interval,because this vortex was smaller and closer to the wall than Vortex B.Moreover, the evolution of Vortex B may be treated as the interaction of multiple vortices rather than that of an individual vortex itself.Compared to the evolution of individual structures, the interaction among multiple vortices would make the evolution more complicated and faster.The random distribution of multiple vortices and their interactions would lead to a variety of instantaneous changes in flow structures during boundary layer transition, of which Fig.11 is just one example.

4.Generation and breakdown of coherent structures

Two questions about the boundary layer transition are very important and need to be answered.The first one is how the coherent structures are generated, and the second one is how large-scale structures break down.As the present imaging system can obtain eight images within a very short time, this will provide useful information to address these two questions.

4.1.Generation of new structures

Although the instantaneous evolution of different flow structures varies,similar dynamics can also be discerned in the visualizations.Here, a typical example is selected to describe the process of new structure generation in the flat-plate boundary layer transition when environmental disturbances are relatively small.

Fig.12 shows the entire process of the generation of a new hairpin vortex referred to as Vortex F.The field of view extends from x =230 mm to x =290 mm,individual images were taken every 5 μs, and the flow is from left to right.In Fig.12(a)–12(d), only wave-like structures can be observed.In addition, a small convex structure can be observed in the upper part of the existing downstream wave-like structure in Fig.12(d), which contributed to the generation of Vortex F.However, in Fig.12 (e), this wave-like structure has rapidly evolved into two parts: the upstream part corresponding to Vortex F and the downstream part which was a new vortex structure.Subsequently, Vortex F is rolled up continuously,with its length and height simultaneously increasing.The head of Vortex F has an obvious oblique upward stretching trend along the downstream direction, which is consistent with the evolution of Vortex A shown in Fig.10.Although the locations of Vortex E in Fig.12 are nearly the same as those of Vortex B in Fig.10(corresponding to the same local Reynolds number), no evolution process similar to that observed for Vortex B, where small scale structures were generated and interacted with the existing large-scale structures, can now be seen.In Fig.12(h), three structures are generated side by side along the flow direction and maintain clear ideal hairpin characteristics, without obvious interactions between one another.

Fig.11 Evolution of coherent structures in Lagrangian reference frame.

There are two points worth noting here.The first point is the evolution and development rate of vortex structures.The time between every-two adjacent images is 5 μs, but it can be concluded from Fig.12(a)–12(c) that the change in the shape of the existing wavy structures is not obvious within a time interval of 10 μs.However, in Fig.12(c)–12(e), two hairpin vortex structures were formed during the same time interval of 10 μs, especially in the 5 μs between the instances when Fig.12(d) and 12(e) were taken.Unfortunately, the evolution of the structure within 5 μs cannot be analyzed here.In Fig.12(f)-12(h),the evolution of the same hairpin vortices during the same 10 μs interval can be clearly seen.

Fig.12 Generation of new structure during early stage of transition.

Using the eight consecutive images shown in Fig.12, the generation of new structures can be divided into three stages.In the first stage (Fig.12(a)–12(d)), the structures evolved at a very slow rate.On the other hand, in the second stage(Fig.12(d) and 12(e)), the structures evolved rapidly and formed hairpin vortices.In the third stage (Fig.12(e)–12(h)),the evolution slowed down, but was still faster than that in the first stage.Similar, but not equally clear, observations can also be made in Fig.10(d)and 10(e).The reason for these behaviors is not clear, but the wave-like structures in the first stage can be regarded as quasi-stable with a slow evolution rate.However, due to unknown reasons, the original balance was disturbed, the hairpin structures formed rapidly, and the flow entered a new quasi-stable state.The second stage can be regarded as a rapid transition from the wave-like structures to the hairpin structures, and the strong shear stress along the direction normal to the wall may be one of the reasons for this to happen, which needs to be verified by additional velocity measurements.The evolution rate in the third stage was faster than that in the first stage, indicating that the evolution of hairpin structures was faster than that of the wave-like structures.

The second point is that there are differences in the generation process of different vortex structures at the same time.As shown from Fig.12(d)–12(h),a new hairpin vortex(Vortex F)was formed within 20 μs.However,Vortex E just upstream of Vortex F was formed earlier, as shown in Fig.12(a).The environmental conditions around Vortices E and F seem to be essentially the same, but Vortex E formed earlier and further upstream than Vortex F.Besides, the shape change of Vortex E within 35 μs (Fig.12(a)–12(h)) is not as obvious as that of Vortex F during 20 μs(Fig.12(d)–12(h)).This indicates that the evolution rate of upstream Vortex E was significantly slower than that of downstream Vortex F.Furthermore, the closer to the downstream region, the more obvious the rotation characteristics of the hairpin head structure.Conversely,the closer to the upstream direction, the more obvious the effect of the hairpin head being stretched along the flow direction.One possible explanation is that in the upstream region the boundary layer is thinner and the velocity gradient along the direction normal to the wall is larger, resulting in a stronger shear effect along the flow direction.Therefore, in the upstream region, the hairpin head is more easily stretched along the flow direction, while in the downstream region, the velocity shear effect decreases with the increase in boundary layer thickness.As a result, the effect of stretching the hairpin head along the flow direction is weaker, and the induction effect of the hairpin head itself is enhanced.The rotation characteristics of hairpin heads and their oblique upward stretching away from the wall are more obvious further downstream.

Although the evolution processes of Vortices E and F are different, they are still very consistent with the hairpin vortex model.Vortices E and F, and another hairpin vortex downstream of Vortex F were positioned side by side along the flow direction, and the spacing between one another was markedly smaller than that between the upstream wavy-like structures.There was thus essentially no interaction between them, i.e.,each vortex can be regarded as an individual hairpin vortex rather than a hairpin packet discussed by Adrian et al.32.

4.2.Breakdown of coherent structures

The time-resolved images in Fig.13 provide an illustration of the large-scale structure breakdown into multiple small-scale structures, which may help us better understand the mechanism of breakdown.The field of view in Fig.13 is the same as that in Fig.12, the time interval between images is 5 μs,and the flow is from left to right.

In Fig.13(a) and 13(b), an individual hairpin vortex (Vortex G) already existed after having developed similar to the vortices in Fig.12 during a 5 μs interval.However, two heads instead of a single one were generated in the region of the hairpin head in Fig.13(c) during the 5 μs interval, and even more small heads occurred in Fig.13(d) during the same time interval.Vortex G changed slowly between Fig.13(a) and 13(b),but a faster change can be seen in Fig.13(c) and 13(d).Then,a dramatic structural change can be identified in Fig.13(d)and 13(e)during 5 μs.Subsequently,as can be seen in Fig.13(-e)-13(h), the evolutionary rate slowed down again.It is still impossible to explain the reason for the dramatic structural changes between Fig.13(d) and 13(e), but from the other groups of the time-resolved images, the dramatic structural change caused by the increase in the actual time interval due to control errors in the time sequence can be excluded.The entire evolution of coherent structures can also be divided into three stages, similar to those found in Fig.12.

Fig.13 Example of structural breakdown during boundary layer transition.

Unlike an individual hairpin structure that developed in time, the initial individual hairpin structure finally broke into multiple small-scale structures, which is consistent with the breakdown of coherent structures.Although the exact reason why original Vortex G developed multiple heads in Fig.13(c) is unknown, it can be hypothesized that the emergence of these multiple head structures led to the breakdown of Vortex G.When multiple structures emerged, the structural destruction was accelerated due to their interactions.Unlike the interactions among multiple vortices shown in Fig.11, where a large-scale vortex played a dominant role while the other small-scale vortices had a relatively small impact on this large-scale structure, a relatively regular evolution process can be observed in Fig.13.However, in Fig.13(c), the newly generated multiple structures had similar scales, with no dominant structure.As a result, the original structures were destroyed by random influences, but a new equilibrium structure formed again quickly.This new equilibrium structure was similar to a large-scale hairpin packet containing multiple small-scale structures in the turbulent boundary layer, where individual hairpin-like structures could not survive.Although there may be many reasons behind the breakdown,it is reasonable to assume that the interaction among multiple vortices with similar scale was one of them.Considering the threedimensional characteristics of coherent structures, their interactions will be more complicated,which also increases the difficulty of identifying the process of breakdown only from the coherent structure images in the xy-plane.

5.Conclusions

In this study, using experimental flow visualizations, a flatplate boundary layer transition and the evolution of coherent structures from the laminar to turbulent states in the Eulerian and Lagrangian reference frames at Mach number 3.4 have been demonstrated.

(1) The evolutions of coherent structures in different regions vary during boundary layer transition.In the upstream region, most individual hairpin vortices are generated and developed, and there are no obvious interactions between them.In the downstream region, instead of individual hairpin vortices dominating the transition process, new small-scale structures are generated by the induction of the existing large-scale vortices and then interact with one other, forming more complex large-scale structures.Compared to the evolution of a single structure,the interactions among multiple vortices will accelerate and complicate the evolution process.

(2) The formation of new vortices can be explained by the hairpin vortex model when environmental disturbances are relatively small.However, the evolution of instantaneous structures is also related to the local flow conditions, such as the boundary layer thickness and velocity gradient.The formation of new structures can also be roughly divided into three stages.In the first stage, the structures evolve very slowly, but in the second stage very fast and rapidly form hairpin vortices.In the third stage,the evolution rate decreases,but it is still higher than that in the first stage.The first and third stages can be regarded as quasi-steady states, while the second stage can be regarded as a sudden transition from the first to the third stage.However,the reasons for this sudden change are unclear from the present results.

(3) The interactions among multiple vortices with similar scales may be one such reason for the structural breakdown,because when no structure can become the dominant one,the original structures cannot survive and break into complex multi-scale structures to achieve a new equilibrium state.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgements

This work was funded by the National Key Research and Development Program of China (No.2019YFA0405300), the Excellent Innovation Young Project of Changsha, China(No.KQ2009026) and National Natural Science Foundation of China (No.91752102).The authors would like to express their gratitude to EditSprings (https://www.editsprings.com/)for the expert linguistic services provided.