Appl. Math. Mech. -Engl. Ed., 40(2), 283 292 (2019) Applied Mathematics and Mechanics (English Edition) https://doi.org/10.1007/s10483-019-2423-8 Transition control of Mach 6.5 hypersonic flat plate boundary layer Yunchi ZHANG 1,2,, Chi LI 1,2 1. State Key Laboratory for Turbulence and Complex Systems, Peking University, Beijing 100871, China; 2. Department of Aeronautics and Astronautics, College of Engineering, Peking University, Beijing 100871, China (Received Sept. 5, 2018 / Revised Oct. 15, 2018) Abstract An artificial disturbance is introduced into the boundary layer over a flat plate to investigate the effect on the transition process in the Mach 6.5 wind tunnel at Peking University. A linear stability theory (LST) is utilized to predict the evolution of the eigenmodes, and the frequency of the artificial disturbance is chosen according to the LST results. The artificial disturbance is generated by glowing discharge on the surface of the plate close to the leading edge. The Rayleigh-scattering visualization and particle image velocimetry (PIV) measurements are performed. By comparing the experimental results with artificial disturbances with those under the natural condition (without artificial disturbances), the present paper shows that the second-mode instability waves are significantly stimulated by the artificial disturbances, and the boundary layer transition is effectively triggered. Key words hypersonic, boundary layer, transition control, glowing discharge Chinese Library Classification O354.4 2010 Mathematics Subject Classification 76K05 Nomenclature T 0, total temperature, K; p 0, total pressure, Pa; U, free-stream velocity, m/s; Re, unit Reynolds number, m 1 ; x, streamwise location from the leading edge, mm; y, wall normal coordinate, mm; f, frequency, Hz; α, normalized streamwise wavenumber; β, normalized spanwise wavenumber; ω, normalized frequency; ψ, propagation angle of the eigenmode, ( ); U, velocity amplitude of the eigenmode wave, m/s; c r, phase velocity of the eigenmode, m/s; δ, thickness of the boundary layer, mm. Citation: ZHANG, Y. C. and LI, C. Transition control of Mach 6.5 hypersonic flat plate boundary layer. Applied Mathematics and Mechanics (English Edition), 40(2), 283 292 (2019) https://doi.org/ 10.1007/s10483-019-2423-8 Corresponding author, E-mail: zhangyunchi@pku.edu.cn Project supported by the National Natural Science Foundation of China (Nos. 10921202, 11221061, 11632002, 11521091, 91752000, and 11602005) c Shanghai University and Springer-Verlag GmbH Germany, part of Springer Nature 2019
284 Yunchi ZHANG and Chi LI 1 Introduction The subject of laminar-turbulent boundary layer transition in hypersonic flows is attracting increasing interest from researchers as the transition process significantly affects aerodynamic quantities such as drag and heat transfer, which are of great importance to the aerodynamic control and thermal protection systems of a hypersonic vehicle. Among several possible transition paths for hypersonic flows [1 2], the typical path of low environmental disturbance levels has received the most attention, as a hypersonic vehicle typically travels in near space with small-amplitude environmental fluctuations. The first stage of this scenario is the boundary layer receptivity by which the disturbances in the free-stream are sensed and trapped in the boundary layer to form the initial unstable waves. In the second stage, the initial unstable waves experience the linear amplification, which can be well interpreted by a linear stability theory (LST). When the eigenmodes increase to a certain threshold, the nonlinear evolution comes into effect, which can be partially explained by a parabolized stability equation method. The boundary layer eventually breaks down to the turbulent flow. As far as the feasibility and effectiveness in experiments are concerned, the eigenmode growth stage is the most appropriate stage to take measures to control the transition process, considering that the receptivity process is complex [3] to control in experiments without introducing additional disturbances and that, when nonlinear effects become prominent, artificial methods are not likely to control the transition effectively. For low-speed flows, both the transition scenario and important flow structures before the onset of transition, such as Tollmien-Schlichting (T-S) waves and cavity solitons, have been investigated in depth [4 9]. However, because of compressibility, the case is more complex for a hypersonic boundary layer. According to the LST, different eigenmodes exist for a hypersonic boundary layer, among which the first- and second-mode instability waves are basic eigenmodes for a typical geometry model without a pressure gradient, for example, a flat plate or cone. The first mode is of vortical instability, which is the corresponding part of a T-S wave in a hypersonic boundary layer. On the other hand, the second-mode instability is of acoustic nature. The LST predicts that, in a hypersonic boundary layer, the two-dimensional (2D) secondmode instability wave is more amplified than the three-dimensional (3D) first-mode instability wave [10], which indicates that the second-mode instability plays a dominant role in the boundary layer transition. However, because of the complex receptivity process and different wind tunnel conditions, experimental results are not always coincident with the predictions [11 13].Inorder to study receptivity and wave processes in supersonic and hypersonic flows, experiments have been conducted with different methods to introduce artificial disturbances into the boundary layer [14 19]. It is proposed that, during the breakdown stage of a Mach 4.5 flat plate boundary layer, no matter which mode is prominent at first, the eigenmode modifies the profile of the mean flow, which significantly changes the stability characteristics of the first mode, leading to the significant growth of the first mode and further modification of the mean flow profile towards turbulence [20]. In this study, LST calculations are performed to determine the frequency range of the firstand second-mode instabilities, and then experiments are conducted, utilizing glowing discharge of certain frequencies to introduce artificial disturbances into a Mach 6.5 flat plate boundary layer. The effects on the transition process are studied with Rayleigh-scattering visualization and particle image velocimetry (PIV) measurements. 2 Experiment setup 2.1 Facility and test conditions The experiment was conducted on a flat plate with a sharp leading edge for Mach 6.5 at the Peking University Φ300 mm hypersonic quiet wind tunnel [21 22]. The experimental conditions are presented in Table 1.
Transition control of Mach 6.5 hypersonic flat plate boundary layer 285 2.2 Test model A 200 mm 650 mm flat plate model with a sharp leading edge was used for both the Rayleigh-scattering visualization and the PIV experiments. As shown in Fig. 1, a 2 mm 100mm slot was grooved 130 mm downstream from the leading edge to seed particles with 20 ml/min air flows in the PIV experiments, while for the visualization experiments no air was injected. Property Table 1 Experiment conditions Value Free-stream Mach number 6.5 Total temperature T 0 /K 415 Total pressure p 0 /kpa 650 Unit Reynolds number Re /m 1 6.0 10 6 Free-stream velocity U /(m s 1 ) 875 Operating run time/s 12 Fig. 1 Flat plate model (color online) Two 2 mm 50 mm electrodes parallel to the leading edge were mounted on the surface of the plate. A dielectric barrier was designed to separate the electrodes by 1 mm. The electrodes were driven by the high-voltage AC power to generate glowing discharge as the artificial disturbance. The output of the high-voltage power was approximately sinusoidal. The frequency of glowing discharge was controlled through tuning the AC power continuously from 10 khz to 140 khz, and the peak to peak voltage was approximately 2.8 kv. The output of AC power is shown in Fig. 2. Fig. 2 Output signal of AC power, which is the output snapshot of the AC power captured by the oscilloscope (color online)
286 Yunchi ZHANG and Chi LI 2.3 Rayleigh-scattering visualization and PIV technique The setup of the Rayleigh-scattering and PIV experiments is shown in Fig. 3. The Rayleighscattering technique was first used at Princeton to visualize various flow structures in hypersonic flows [23 24]. The authors injected CO 2 gas upstream of the test section with a mass flow no greater than 5% that of the free stream. Because of the low static temperature in the test section (44 K in this case), the CO 2 condenses to minute solid particles of which the diameters are significantly smaller than the laser wavelength (532 nm). Close to the model, where the temperature is higher than the sublimation point of CO 2, it remains gaseous. When the laser illuminates, the solid particles scatter and brighten the region in the grayscale charge-coupled device (CCD) images. On the other hand, as the gaseous CO 2 does not exhibit the scattering effect, the region close to the test model is dark on the grayscale images. Therefore, the boundary between the dark and bright areas denotes the sublimation line, according to which certain flow structures can be visualized. - Fig. 3 Setup of PIV experiments (side-view) (color online) The PIV is the most powerful technique to obtain the instantaneous flow field in a boundary layer. However, for a high-speed flow, given the significant velocity gradient in the near-wall region, it is likely that the interrogation will fail to converge to a correct result. To overcome this problem, Zhu et al. [25] developed an image-preprocessing method by adding stationary artificial particles in the wall region and succeeded in resolving the second-mode waves in a hypersonic boundary layer. This method has been widely used in the measurement of highspeed flows [26 29]. With the help of stationary synthetic particles, Jia et al. [30] developed a PIV image-preprocessing method to obtain high spatial resolution velocity profiles close to a moving interface. In this PIV experiment, the TiO 2 particles and air mixture were fed into the boundary layer through a 100 mm 2 mm slot on the flat plate 150 mm from the leading edge. To minimize the disturbance introduced by the jet, the air mass flow rate was controlled by a flowmeter to no greater than 20 ml/min. 3 Results 3.1 LST results Artificial disturbances in the frequency range of second-mode instability were introduced in the experiments as opposed to natural transition experiments. Calculations based on the LST were performed to determine the frequency range of the first- and second-mode instabilities.
Transition control of Mach 6.5 hypersonic flat plate boundary layer 287 To perform a linear stability analysis for the Mach 6.5 flat plate boundary layer, namely, to track the evolution of an eigenmode of a certain frequency in the boundary layer, three steps are needed [13]. Firstly, the undisturbed mean flow is calculated with the self-similarity assumption. Secondly, for a given streamwise wavenumber α (which is approximately proportional to the frequency f) and a spanwise wavenumber β, the single-domain spectral method is used to obtain the temporal growth rate ω at the initial location x = 110 mm. Thirdly, the evolution of the eigenmode downstream from the initial location is calculated using the fourth-order finite difference method for sequential locations downstream at x = 110mm. It can be seen in Tables 2 and 3 that the phase velocity is approximately 90% of the main stream velocity for both the first- and second-mode instabilities. According to the LST, the 3D first-mode wave has the largest amplification rate, while the 2D second-mode wave with β =0is the most unstable. As seen in Fig. 4, the first mode lies in a significantly lower frequency range than the second mode, and the N-factor is not sensitive to the frequency. A wide frequency range of the second mode with β = 0 experiences growth from the location where the artificial disturbance is introduced (x = 110mm) and the N-factor is more sensitive to the frequency compared with the first mode. The phase velocity is approximately 0.9 times the main stream velocity for both the first- and second-mode instabilities. Table 2 LST data for the first mode of different frequencies at the initial location x = 110 mm f/khz α β ω c r/u 15 0.017 3 0.041 7 0.014 79+0.000 536 9i 0.854 8 20 0.022 7 0.048 3 0.019 71+0.000 610 6i 0.867 8 25 0.028 0 0.053 9 0.024 62+0.000 651 4i 0.878 1 30 0.033 4 0.058 6 0.029 58+0.000 677 2i 0.886 4 Table 3 LST data for the second mode of different frequencies at the initial location x = 110 mm f/khz α β ω c r/u 70 0.744 0 0 0.068 930+0.000 218 9i 0.926 6 75 0.079 4 0 0.073 960+0.000 334 7i 0.931 5 80 0.084 2 0 0.078 830+0.000 629 1i 0.936 3 85 0.089 3 0 0.083 750+0.001 176 0i 0.938 4 90 0.094 7 0 0.088 706+0.001 694 0i 0.936 7 95 0.100 3 0 0.093 630+0.001 978 0i 0.933 5 - - Fig. 4 N-factor predicted by LST, (a) first-mode instability and (b) second-mode instability
288 Yunchi ZHANG and Chi LI 3.2 Rayleigh-scattering visualization Figure 5 shows the Rayleigh-scattering visualization along the centerline of the plate model without and with the artificial disturbance. Fig. 5 Side-views of Rayleigh-scattering visualization Figure 5(a) shows the long-wave structures, with a wavelength of approximately 45 mm, before the boundary layer breaks down to turbulence at x = 600 mm. No short waves are observed. By contrast, when glowing discharge of 105 khz is on, as shown in Fig. 5(b), periodic short-wave structures can be observed between x = 380 mm and x = 440 mm with a wavelength of approximately 9.3 mm. This is twice the laminar boundary layer thickness δ ( 20x/ Re ) and is the characteristic of the second-mode instability. The second-mode waves disappear downstream, and the boundary layer breakdown shifts to x = 500mm, which is significantly closer to the leading edge than the case without the artificial disturbance (natural transition). For the case with glowing discharge of 90 khz, as shown in Fig. 5(c), the second-mode waves appear at x = 360 mm, and there is coupling downstream between the short and long waves. The boundary layer breakdown occurs farther downstream than the case with glowing discharge of 105 khz at x = 580 mm, but still upstream of the case without glowing discharge. Therefore, it can be concluded that the second-mode waves are stimulated, and the boundary layer transition is triggered effectively by glowing discharge with different frequencies in the band predicted by the LST. Additionally, according to the LST results, the phase velocity of the second mode is approximately 0.9 times the main stream velocity. Therefore, the frequency of the second-mode wave captured by the visualization can be estimated as f 85 khz. The frequency differs from the most-amplified frequency predicted by the LST, which could be explained by the non-parallel effect and nonlinear interactions between different modes. 3.3 PIV measurements Two 1 376 pixels 1 024 pixels CCD cameras were placed side by side to obtain the particle images. Each camera covered a streamwise field of 82 mm, and the overlay field of the two camera views was 2 mm. The center view of the two cameras covered the field between x = 300 mm and x = 462 mm. The time delay of the laser pulses was set to 1 μs, and the sample rate was 10 Hz. The evaluation was started with 64 pixels 64 pixels coarse samples and then 32 pixels 32 pixels medium samples to capture the mean displacement fields of the time series. With the mean results as reference, instantaneous recordings were then evaluated with 16 pixels 16 pixels samples to resolve the fine structures in the boundary layer. Smaller samples were unavailable because of the low particle density close to the wall.
Transition control of Mach 6.5 hypersonic flat plate boundary layer 289 After calculating the instantaneous flow field, space Gaussian filtering was performed with specified window widths to obtain the waves of interest. For example, in Fig. 6(a), waves with wavelengths smaller than 8.1 mm are filtered out. The coupling of long and short waves, namely, the second-mode waves, can be seen in Fig. 6(a). The second-mode waves are distinct in the region between x = 400 mm and x = 450 mm, which coincides with the visualization results in Fig. 5(b). In Fig. 6(b), waves with wavelengths smaller than 30 mm are filtered out, and long waves related to the first mode are observed. Fig. 6 Space Gaussian filtering for instantaneous flow field with glowing discharge of 105 khz with window widths of (a) 8.1 mm and (b) 30 mm (color online) To obtain the evolution of the amplitude of the first and second modes, band filtering was performed. For example, to obtain the second mode centered at a frequency of 85 khz, 71 khz and 96 khz were chosen as the cut-off frequencies for the filtering. Assuming that the phase velocities of both the first and second modes were equal to 0.9 times the main-stream velocity, this was achieved by filtering the flow field waves with windows 8.1 mm and 11 mm in the width, respectively, and then subtracting the 8.1 mm-window filtering result from the 11 mm-window filtering result. Figure 7(a) shows the disturbances in the flow field after band filtering between 71 khz and 96 khz, which is the range of most-amplified frequencies predicted by the LST, as shown in Fig. 4(b). Similarly, the evolution of the first mode is shown in Fig. 7(b) by band filtering between 30 mm and 15.5 mm. Similar results are obtained for the flow field without glowing discharge. The comparison of the eigenmode evolutions with and without glowing discharge is shown in Fig. 8. The result is obtained by averaging 20 instantaneous flow field results. Fig. 7 Space Gaussian band filtering for instantaneous flow field with glowing discharge of 105 khz, (a) second-mode evolution and (b) first-mode evolution (color online) It can be seen in Fig. 8 that both the first- and second-mode instabilities are more amplified in the case with glowing discharge of 105 khz. The first-mode waves in both cases increase
290 Yunchi ZHANG and Chi LI monotonically from x = 325mm to x = 420 mm. In the case with glowing discharge, the amplitude of the first mode increases from 1.7% to 3.3% of the main-stream velocity, while in the case without glowing discharge, the amplitude of the first mode increases from 1.7% to 2.7% of the main-stream velocity. The amplitude of the second-mode waves increases to a peak and then decreases monotonically. The peak is located at approximately x = 420 mm, and the peak values are 2.8% and 2.4% of the main-stream velocity for cases with and without glowing discharge, respectively. The evolution is significantly different from the LST results that the second-mode waves centered at 85kHz should increase monotonically in the range from x = 300 mm to x = 460 mm. Fig. 8 Comparison of amplitude evolution of eigenmode It is noteworthy that, because of the Gaussian filtering method, the amplitudes of both the first- and second-mode waves are damped. 4 Conclusions This study utilizes glowing discharge as an artificial disturbance to study its effects on the transition of the boundary layer for a Mach 6.5 flat plate model. Rayleigh-scattering visualization indicates that the second-mode instability waves are significantly stimulated by the glowing discharge within the frequency range of the second mode, and the transition is effectively triggered as opposed to the case without glowing discharge. In addition, PIV measurements are performed to obtain the evolution of the eigenmode instability before the location where the boundary becomes turbulent. The results indicate that the first-mode amplitude increases monotonically, while the amplitude of the second mode first increases and then is damped, which is significantly different from the results predicted by the LST. This behavior could be explained by the nonlinear effects, and more detailed PIV measurements and numerical simulations are required to clarify the dynamics. References [1] MORKOVIN, M. V., RESHOTKO, E., and HERBERT, T. Transition in open flow systems: a reassessment. Bulletin of American Physical Society, 39, 1 31 (1994) [2] FEDOROV, A. Transition and stability of high-speed boundary layers. Annual Review of Fluid Mechanics, 43, 79 95 (2011) [3] JIANG, X. Y. and LEE, C. B. Review of research on the receptivity of hypersonic boundary layer (in Chinese). Journal of Experiments in Fluid Mechanics, 31, 1 11 (2017) [4] LEE, C. B. New features of CS solitons and the formation of vortices. Physics Letters A, 247, 397 402 (1998)
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