The laminar-to-turbulence transition of a streamwise corner flow is recognized to occur first at the corner based on the stability analysis, but there is no persuasive experimental evidence to back it up, especially for supersonic flow. In this work, natural transition in a supersonic corner boundary layer has been experimentally studied using a nanoparticle-based planar laser scattering technique. It is inspiring to observe that the natural transition position of the corner boundary layer shows a random behavior among the corner side, flat-plate side, and their combination. Based on an intermittent factor analysis, these stochastic transitions show a dominant preference for transitions occurring near the corner region.

A streamwise corner, formed by the intersection of two adjacent walls, is commonly encountered in subsonic and supersonic applications, such as wing-body junctions, rectangular air intakes, combustion chambers, and nozzles. Near the corner region, the boundary layer is referred to as a fully three-dimensional flow that attracts numerous investigations from theoretical,1,21. S. Balachandar and M. R. Malik, “ Inviscid instability of streamwise corner flow,” J. Fluid Mech. 282, 187–201 (1995). https://doi.org/10.1017/S00221120950000972. O. T. Schmidt and U. Rist, “ Linear stability of compressible flow in a streamwise corner,” J. Fluid Mech. 688, 569–590 (2011). https://doi.org/10.1017/jfm.2011.405 experimental,33. X. Xiang and H. Babinsky, “ Corner effects for oblique shock wave/turbulent boundary layer interactions in rectangular channels,” J. Fluid Mech. 862, 1060–1083 (2019). https://doi.org/10.1017/jfm.2018.983 and numerical4,54. N. Nikitin and B. Krasnopolsky, “ Turbulent flows along a streamwise external corner,” J. Fluid Mech. 940, A16 (2022). https://doi.org/10.1017/jfm.2022.2465. R. Yang, D. Modesti, Y. X. Zhao, Q. C. Wang, Z. G. Wang, and S. Pirozzoli, “ Influence of corner angle in streamwise supersonic corner flow,” Phys. Fluids 33, 056108 (2021). https://doi.org/10.1063/5.0046716 perspectives. The evolution of this corner boundary layer is sensitive to its inherent nonlinearity, wall temperature, incoming Mach number, and disturbance type,66. S. Fu and L. Wang, “ RANS modeling of high-speed aerodynamic flow transition with consideration of stability theory,” Prog. Aerosp. Sci. 58, 36–59 (2013). https://doi.org/10.1016/j.paerosci.2012.08.004 which will undergo drastic changes in flow properties, such as surface heat flux, frictional resistance, and flow quality, if the boundary layer transitions to turbulence. In its early years, the laminar-to-turbulence transition on a streamwise corner has long been recognized to be similar to that on a flat plate, which originated from the flat-plate side instead of the corner side.77. H. J. Perkins, “ The formation of streamwise vorticity in turbulent flow,” J. Fluid Mech. 44, 721–740 (1970). https://doi.org/10.1017/S0022112070002112 An opposite opinion was held by Zamir and Young88. M. Zamir and A. D. Young, “ Experimental investigation of the boundary layer in a streamwise corner,” Aeronaut. Q. 21, 313–339 (1970). https://doi.org/10.1017/S0001925900005497 and Zamir99. M. Zamir, “ Similarity and stability of the laminar boundary layer in a streamwise corner,” Proc. R. Soc. London A 377, 269–288 (1981). https://doi.org/10.1098/rspa.1981.0124 who experimentally found that a zero-pressure-gradient corner boundary layer no longer maintains a laminar state at a local Reynolds number 104, one order of magnitude smaller than that of the flat-plate boundary layer under the same conditions. Alizard et al.10,1110. F. Alizard, J.-C. Robinet, and U. Rist, “ Sensitivity analysis of a streamwise corner flow,” Phys. Fluids 22, 014103 (2010). https://doi.org/10.1063/1.329200911. F. Alizard, J.-C. Robinet, and F. Guiho, “ Transient growth in a right-angled streamwise corner,” Eur. J. Mech. B 37, 99–111 (2013). https://doi.org/10.1016/j.euromechflu.2012.07.006 gave a possible transition route by nonmodal/transient stability analysis to understand the experimental results given by Zamir and Young.8,98. M. Zamir and A. D. Young, “ Experimental investigation of the boundary layer in a streamwise corner,” Aeronaut. Q. 21, 313–339 (1970). https://doi.org/10.1017/S00019259000054979. M. Zamir, “ Similarity and stability of the laminar boundary layer in a streamwise corner,” Proc. R. Soc. London A 377, 269–288 (1981). https://doi.org/10.1098/rspa.1981.0124 Recent direct numerical simulation results further suggested that the corner boundary layer was much more sensitive to the type of inflow disturbances.1212. O. T. Schmidt and U. Rist, “ Numerical investigation of classical and bypass transition in streamwise corner-flow,” Proc. IUTAM 14, 218–226 (2015). https://doi.org/10.1016/j.piutam.2015.03.043 The presence of Tollmien–Schlichting wave perturbations rather than free-stream disturbances will cause the onset of turbulence to occur in the corner region. To some extent, this result is consistent with the previous experimental findings99. M. Zamir, “ Similarity and stability of the laminar boundary layer in a streamwise corner,” Proc. R. Soc. London A 377, 269–288 (1981). https://doi.org/10.1098/rspa.1981.0124 but conflicts with the conclusions of the linear and nonlinear stability analyses, where the preference of transition near the corner region is believed.1,2,13–181. S. Balachandar and M. R. Malik, “ Inviscid instability of streamwise corner flow,” J. Fluid Mech. 282, 187–201 (1995). https://doi.org/10.1017/S00221120950000972. O. T. Schmidt and U. Rist, “ Linear stability of compressible flow in a streamwise corner,” J. Fluid Mech. 688, 569–590 (2011). https://doi.org/10.1017/jfm.2011.40513. W. D. Lakin and M. Y. Hussaini, “ Stability of the laminar boundary layer in a streamwise corner,” Proc. R. Soc. A 393, 101–116 (1984). https://doi.org/10.1098/rspa.1984.004814. S. J. Parker and S. Balachandar, “ Viscous and inviscid instabilities of flow along a streamwise corner,” Theor. Comput. Fluid Dyn. 13, 231–270 (1999). https://doi.org/10.1007/s00162005011715. M. R. Dhanak and J. E. Enderby, “ On the instability of flow in a streamwise corner,” Proc. R. Soc. London Ser. A 441, 201–210 (1993). https://doi.org/10.1098/rspa.1993.005716. I. Galionis and P. Hall, “ Spatial stability of the incompressible corner flow,” Theor. Comput. Fluid Dyn. 19, 77–113 (2005). https://doi.org/10.1007/s00162-004-0153-117. O. T. Schmidt and U. Rist, “ Viscid-inviscid pseudo-resonance in streamwise corner flow,” J. Fluid Mech. 743, 327–357 (2014). https://doi.org/10.1017/jfm.2014.3118. O. T. Schmidt, S. M. Hosseini, U. Rist, A. Hanifi, and D. S. Henningson, “ Optimal wavepackets in streamwise corner flow,” J. Fluid Mech. 766, 405–435 (2015). https://doi.org/10.1017/jfm.2015.18 Balachandar and Malik11. S. Balachandar and M. R. Malik, “ Inviscid instability of streamwise corner flow,” J. Fluid Mech. 282, 187–201 (1995). https://doi.org/10.1017/S0022112095000097 performed a three-dimensional inviscid stability analysis to consider the interaction between two sidewall boundary layers and confirmed the existence of an unstable corner mode. Parker and Balachandar1414. S. J. Parker and S. Balachandar, “ Viscous and inviscid instabilities of flow along a streamwise corner,” Theor. Comput. Fluid Dyn. 13, 231–270 (1999). https://doi.org/10.1007/s001620050117 additionally considered the effect of viscosity and found a symmetric corner mode. Schmidt and Rist22. O. T. Schmidt and U. Rist, “ Linear stability of compressible flow in a streamwise corner,” J. Fluid Mech. 688, 569–590 (2011). https://doi.org/10.1017/jfm.2011.405 extended the theoretical analysis to a compressible flow regime and discovered an odd-symmetric corner mode with relatively large disturbance amplification to accelerate the transition. Galionis and Hall,1616. I. Galionis and P. Hall, “ Spatial stability of the incompressible corner flow,” Theor. Comput. Fluid Dyn. 19, 77–113 (2005). https://doi.org/10.1007/s00162-004-0153-1 Schmidt and Rist,1717. O. T. Schmidt and U. Rist, “ Viscid-inviscid pseudo-resonance in streamwise corner flow,” J. Fluid Mech. 743, 327–357 (2014). https://doi.org/10.1017/jfm.2014.31 and Schmidt et al.1818. O. T. Schmidt, S. M. Hosseini, U. Rist, A. Hanifi, and D. S. Henningson, “ Optimal wavepackets in streamwise corner flow,” J. Fluid Mech. 766, 405–435 (2015). https://doi.org/10.1017/jfm.2015.18 also performed nonlinear stability analysis to show the preference for transition near the corner region.The above-mentioned viewpoints or conclusions are mainly drawn from numerical and theoretical perspectives, while a few experiments about the corner boundary layer transition are limited to low-speed and incompressible configuration.99. M. Zamir, “ Similarity and stability of the laminar boundary layer in a streamwise corner,” Proc. R. Soc. London A 377, 269–288 (1981). https://doi.org/10.1098/rspa.1981.0124 As for supersonic experiments, the available data for corner flow are currently concentrated on the turbulent status.19,2019. Z. Wang, J. Chang, J. Zhang, and D. Yu, “ Evolution of subsonic and supersonic corner vortices in a supersonic cascade,” Aerosp. Sci. Technol. 95, 105509 (2019). https://doi.org/10.1016/j.ast.2019.10550920. K. Sabnis and H. Babinsky, “ Nozzle geometry effects on corner boundary layers in supersonic wind tunnels,” AIAA J. 57, 3620–3623 (2019). https://doi.org/10.2514/1.J058310 Therefore, an experimental study about the natural transition in a streamwise corner boundary layer is presented herein.The experiments were conducted in the KD-02 supersonic wind tunnel of the National University of Defense Technology. To get a uniform and relatively quiet exhausted stream, this suction-type wind tunnel has been updated four times in past decades and currently has equipped with multiple honeycomb screens, a well-designed nozzle with a large contraction section, and a wave-attenuation expansion section.2121. D. P. Wang, Z. X. Xia, Y. X. Zhao, Q. H. Wang, and B. Liu, “ Vortical structures of supersonic flow over a delta-wing on a flat plate,” Appl. Phys. Lett. 102, 061911 (2013). https://doi.org/10.1063/1.4790286 The wind tunnel supports detailed investigations about transition and turbulence in boundary layers.21,2221. D. P. Wang, Z. X. Xia, Y. X. Zhao, Q. H. Wang, and B. Liu, “ Vortical structures of supersonic flow over a delta-wing on a flat plate,” Appl. Phys. Lett. 102, 061911 (2013). https://doi.org/10.1063/1.479028622. Q. C. Wang and Z. G. Wang, “ Structural characteristics of the supersonic turbulent boundary layer subjected to concave curvature,” Appl. Phys. Lett. 108, 114102 (2016). https://doi.org/10.1063/1.4944536 The free-stream Mach number is M=2.95 and the actual inflow parameters of the present study are listed in Table I. Figure 1 shows the corner model formed by two perpendicular flat plates, where the length and width of each flat plate are 500 and 150 mm, respectively. Ahead of experiments, the model surface is specifically treated with a black paint to weaken light reflection, and the sharp wedge angle of each plate leading edge is chosen as 10° to avoid the detached shock wave angle of 33.7° at a Mach number of 2.95. The adiabatic wall temperature is estimated by Taw=T∞(1+rγ−12M∞2)=267.9 K with a turbulent recovery factor r=Pr1/3=0.89 and Prandtl number Pr=0.71.2323. N. Murray, R. Hillier, and S. Williams, “ Experimental investigation of axisymmetric hypersonic shock-wave/turbulent-boundary-layer interactions,” J. Fluid Mech. 714, 152–189 (2013). https://doi.org/10.1017/jfm.2012.464 The wall temperature variation during experiments is basically constant in short intervals.2424. Q. Wang, Z. Wang, and Y. Zhao, “ On the impact of adverse pressure gradient on the supersonic turbulent boundary layer,” Phys. Fluids 28, 116101 (2016). https://doi.org/10.1063/1.4968527

TABLE I. Main flow parameters of the free stream. M∞, U∞, c∞, p∞, T∞, ρ∞, μ∞, and Re denote the Mach number, the streamwise velocity, the sound speed, the static pressure, the ambient temperature, the adiabatic wall temperature, the density, the dynamic viscosity coefficient calculated by Sutherland equation, and the unit Reynolds number, respectively.

M∞(U∞/c∞)U∞(m/s)T∞(K)p∞ (Pa)ρ∞ (kg/m3)μ∞ (Pa s)Re/m2.95619105.129359.73×10−27.29×10−68.095×106The nanoparticle-based planar laser scattering (NPLS) system is employed to accurately resolve the flow structure in a high signal-to-noise ratio.2525. Y. X. Zhao, S. H. Yi, L. F. Tian, and Z. Y. Cheng, “ Supersonic flow imaging via nanoparticles,” Sci. China Ser. E 52, 3640–3648 (2009). https://doi.org/10.1007/s11431-009-0281-3 This system includes a nanoparticle generator to uniformly distribute seeds, a double-pulsed Nd:YAG laser to generate a light sheet with a wavelength of 532 nm and a stable energy of 520 mJ per 6 ns, and an IMPERX B 4020M CCD camera with a resolution of 4000 × 2672 pixels22. O. T. Schmidt and U. Rist, “ Linear stability of compressible flow in a streamwise corner,” J. Fluid Mech. 688, 569–590 (2011). https://doi.org/10.1017/jfm.2011.405 and a repetition rate of 2 Hz. During experiments, the seeded flow is illuminated by a thin laser sheet and captured by the camera, as shown in Fig. 1. The interaction between laser and nanoparticles leads to Rayleigh scattering that shows bright edges around fluid structures. If nanoparticles are concentrated at these edges, then the resulting strong Rayleigh scattering will create a nearly white interface to distinguish different physical structures. The NPLS system incorporated with the aforementioned wind tunnel enables us to observe the evolution of the boundary layer and judge the transition position.As shown in Fig. 1, two laser sheets are considered for experimental observation. The laser plane in Fig. 1(a) keeps a distance of z/δ=70.4 toward the side flat plate, whereas in Fig. 1(b) the distance to the bottom flat plate is y/δ=1.1, slightly larger than the following measured thickness of laminar boundary layer. In Fig. 1(a), the laser plane stays far away enough from the sidewall surface so that the local boundary layer can be treated as the kind evolving along a flat plate. Figure 2 shows the transition process of the flat-plate boundary layer. This laminar boundary layer begins to be unstable at around Rex=2.395×106, where hairpin structures are generated, and then rapidly break down, forming a complex turbulence field. The thickness of the compressible laminar boundary layer is about δ=1.42 mm in Fig. 2(b). The space-time evolution of vortical structures inside the boundary layer can be obtained by comparing Figs. 2(a) and 2(b). The movement speed of each vortical structure reaches 0.95 U∞, but its shape has not changed significantly in a present short time interval, which is featured by fast movement and slow change.

Current experiments for the transition over a flat-plate boundary layer have been performed multiple times. In these experiments, the transition position is found to fluctuate randomly. This randomness might be caused by the unsteady incoming flow that consequently changes the sensitivity of the boundary layer to inflow disturbances, thereby affecting the natural transition position. As the fluctuated transition position for a flat-plate boundary layer is close to Rex=2.478×106, subsequent experiments for the corner boundary layer transition will also focus on this area.

Figure 3 shows the random transition behavior of the streamwise corner boundary layer acquired at the spanwise plane in Fig. 1(b). Three typical types of transition corresponding to boundary layer stability are experimentally identified with the NPLS system. The transition process itself is highly unsteady, while in present measurements, different kinds of transition types cannot be maintained throughout the streamwise evolution and are prone to irregularly switch with each other.In Fig. 3(a), there are remarkably elongated streaks distributed both in the corner and plate sides. Andersson et al.2626. P. Andersson, L. Brandt, A. Bottaro, and D. S. Henningson, “ On the breakdown of boundary layer streaks,” J. Fluid Mech. 428, 29–60 (2001). https://doi.org/10.1017/S0022112000002421 observed similar streak structures in an incompressible planar Poiseuille flow, which is an important flow element to induce turbulence.2727. W. Schoppa and F. Hussain, “ Coherent structure generation in near-wall turbulence,” J. Fluid Mech. 453, 57–108 (2002). https://doi.org/10.1017/S002211200100667X An apparent consensus views that the streak instead of the instability wave is a precursor for the bypass transition caused by external disturbances like free-stream noise in the wind tunnel. If the streaky flow is stable, then it will attenuate and terminate under the influence of viscosity, or on the contrary, it bends along the streamwise direction and results in the generation of streamwise vorticity. A majority of streaks are finite in length and one representative length in Fig. 3(a) is 28δ. These streaks usually cannot be sustained all the time due to a lack of enough excitation. One representative exception is illustrated inside a white dotted box, where the streaky flow after evolving a certain distance is deformed into streamwise vorticity and a related transition to turbulence is expected to occur at the downstream position.In Fig. 3(b), the natural transition to turbulence takes place in the near-corner region, while the flow over the plate area far away from the sidewall remains in a laminar state. In general, the transition occurring in the corner side is consistent with the results of theoretical analysis.2,16,172. O. T. Schmidt and U. Rist, “ Linear stability of compressible flow in a streamwise corner,” J. Fluid Mech. 688, 569–590 (2011). https://doi.org/10.1017/jfm.2011.40516. I. Galionis and P. Hall, “ Spatial stability of the incompressible corner flow,” Theor. Comput. Fluid Dyn. 19, 77–113 (2005). https://doi.org/10.1007/s00162-004-0153-117. O. T. Schmidt and U. Rist, “ Viscid-inviscid pseudo-resonance in streamwise corner flow,” J. Fluid Mech. 743, 327–357 (2014). https://doi.org/10.1017/jfm.2014.31 The red dotted line illustrates the spatial development trend of transitional wake flow, somewhat similar to the transition behavior caused by a vortex generator.2828. Q. Q. Ye, F. F. J. Schrijer, and F. Scarano, “ On Reynolds number dependence of micro-ramp-induced transition,” J. Fluid Mech. 837, 597–626 (2018). https://doi.org/10.1017/jfm.2017.840 The beginning of transition is usually linked with wave-like perturbations. Once the perturbation is generated at the corner, it will move with the mainstream and propagate along the spanwise direction probably in the form of a Mach wave. The propagating perturbation triggers the laminar boundary layer nearby and finally forms a wedge-shaped transition zone under such a drive. The turbulent wedge situated around the corner region is also ascribed in a direct numerical simulation performed by Schmidt et al.2929. O. Schmidt, B. Selent, and U. Rist, “ Direct numerical simulation of boundary layer transition in streamwise corner-flow,” High Perform. Comput. Sci. Eng. 13, 337–348 (2013). under conditions of M = 0.8. Ahead of the wedge region, the transition to turbulence caused by the streak can be found in the white dotted box. Meanwhile, a transitional-turbulent spot is formed with recognizable boundaries and gradually evolves into a streak and streamwise vorticity as the energy cannot be maintained.3030. X. H. Wu, “ New insights into turbulent spots,” Annu. Rev. Fluid Mech. 55, 45–75 (2023). https://doi.org/10.1146/annurev-fluid-120720-021813 Since the transitional-turbulent spot is in the infancy stage of fully turbulent flow, it further proves that the instant flow on the flat-plate side stays in a state of laminar or re-laminar instead of transition.In contrast to the phenomena of Fig. 3(b), Fig. 3(c) shows that the transition starts from the flat-plate area, while the corner area still maintains a laminar state with only some streak structures. Furthermore, Fig. 3(d) shows that the transition occurs both in the corner and the flat-plate areas almost at the same time. These transition scenarios are strikingly distinct from previous theoretical conclusions that the corner region is conventionally considered to be more unstable in facilitating the transition. Through a large number of repeated experiments, it is found that the corner boundary layer transition does not always start at the corner but has strong randomness. This randomness, on the one hand, is partly caused by the differences between the disturbance-free environment assumed by theory and the relatively noisy condition existing in a realistic wind tunnel even in a low-noise level.3131. C. Hader and H. F. Fasel, “ Towards simulating natural transition in hypersonic boundary layers via random inflow disturbances,” J. Fluid Mech. 847, R3 (2018). https://doi.org/10.1017/jfm.2018.386 On the other hand, the stability theory struggles to well-resolve the nonparallel wall-bounded flow in a fully three-dimensional and highly transient feature.1414. S. J. Parker and S. Balachandar, “ Viscous and inviscid instabilities of flow along a streamwise corner,” Theor. Comput. Fluid Dyn. 13, 231–270 (1999). https://doi.org/10.1007/s001620050117As the transition of the corner boundary layer owns three typical forms in a random manner, it is remarkably interesting to question whether the randomness of the transition location has statistical dominance to some extent. The intermittent factor is a recognized indicator to characterize the occurrence frequency of turbulence. The intermittent value is defined as 0 and 1 for the laminar flow with streaks or uniform background and for the turbulent flow with streamwise vortical structures, respectively.32,3332. R. A. Humble, S. J. Peltier, and R. D. W. Bowersox, “ Visualization of the structural response of a hypersonic turbulent boundary layer to convex curvature,” Phys. Fluids 24, 106103 (2012). https://doi.org/10.1063/1.476183333. N. Reuther and C. J. Kähler, “ Effect of the intermittency dynamics on single and multipoint statistics of turbulent boundary layers,” J. Fluid Mech. 897, A11 (2020). https://doi.org/10.1017/jfm.2020.384 To extract the vortices and exclude the streaks from a NPLS image, an open-source machine learning toolkit3636. Anna et al., ilastik, version 1.4.0rc5, https://www.ilastik.org/ (2010–2022). developed in the European Molecular Biology Laboratory is applied to interactively training a pixel classification model and thereby distinguishing the region with vortices.34,3534. S. Berg, D. Kutra, T. Kroeger, C. N. Straehle, B. X. Kausler, C. Haubold, M. Schiegg, J. Ales, T. Beier, M. Rudy, K. Eren, J. I. Cervantes, B. Xu, F. Beuttenmueller, A. Wolny, C. Zhang, U. Koethe, F. A. Hamprecht, and A. Kreshuk, “ ilastik: Interactive machine learning for (bio)image analysis,” Nat. Methods 16, 1226–1232 (2019). https://doi.org/10.1038/s41592-019-0582-935. Q. C. Wang, Z. G. Wang, and Y. X. Zhao, “ Structural responses of the supersonic turbulent boundary layer to expansions,” Appl. Phys. Lett. 109, 124104 (2016). https://doi.org/10.1063/1.4963382 As shown in Fig. 4, the instant flow field and its identified elements are depicted clearly, where the turbulent vortices and their edges are basically captured, while the streaks with conspicuously long shapes are eliminated. The captured features are converted into a binary image, processed with a 4 × 4 median filter, and then adopted to account for the intermittent factor. In this work, the gross number of experimental images reaches 600, a sufficient number for statistically independent results.Figure 5 shows a statistical mean value of the intermittence factor in a region of interest, where natural transition possibly occurs in the streamwise corner boundary layer. Noise points marked in Fig. 4(a) are polluted with an intermittence factor of 1 due to light reflection and bad camera spots. Beyond these noise points, the magnitudes of the intermittence factor are less than 1, demonstrating directly the randomness of the boundary layer transition. The sidewall region (z/δ < 1.4) is cropped to eliminate the boundary layer effects. Seen from the statistical map of the intermittence factor, the flat-plate region occupies a smaller range colored in brightness, whereas comparatively this range becomes relatively greater in the region adjacent to the corner and the flat plate. Moreover, the bright range of the intermittence factor in the pretty near-corner region keeps larger than that in the flat-plate side, which evidently indicates that the transition in the near-corner boundary layer is more frequent than that over the flat-plate one. The random transition over a corner boundary layer is also proved to be three-dimensional as the brightness distribution of the intermittence factor varies along the spanwise direction in Fig. 5. The Reynolds number for corner transition is approximated to be 2.230×106. This Reynolds number is smaller than the value of flat-plate transition Rex=2.478×106 and seems to be much smaller because the present laser sheet stays a certain distance from the bottom wall.

To summarize, this paper offers experimental evidence to show that the natural transition of a supersonic streamwise corner boundary layer can be categorized into three transition types according to the location where the transition occurs. These transition types are found to randomly switch in a region of interest among (a) transition in the corner side, (b) transition at the flat-plate side, and (c) concurrent transition over corner and flat plate. The transitioning flow field acquired at a spanwise plane near the boundary layer is used to post-process the intermittent factor of turbulent structures. The map of intermittent factor shows a large area in the corner region and its surroundings to flat plate, which demonstrates that the transition is more prone to happen in the near-corner region. The Reynolds number of present corner transition is about Rex=2.230×106 in comparison to the value of flat plate of 2.478×106. Current findings are different from a majority of linear and nonlinear analyses via stability theory, where the corner side is generally considered to transition in an early stage.

This research received the financial support from the National Natural Science Foundation of China under Grant No. 12272405. The authors thank the developers of the ilastik software and Dr. Wei Feng for his suggestion during use.

Conflict of Interest

The authors have no conflicts to disclose.

Author Contributions

Rui Yang: Conceptualization (equal); Data curation (equal); Investigation (equal); Writing – original draft (equal); Writing – review & editing (equal). Yu-xin Zhao: Conceptualization (equal); Data curation (equal); Funding acquisition (equal); Investigation (equal); Methodology (equal). Lican Wang: Conceptualization (equal); Software (equal); Visualization (equal); Writing – original draft (equal); Writing – review & editing (equal).

The data that support the findings of this study are available from the corresponding authors upon reasonable request.

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