Nomenclature
Ma,	Mach number;	μ,	dynamic viscosity coefficient, kg/(m·s);
T,	temperature, K;	x,	coordinate along the flow, mm;
T0,	total temperature, K;	y,	coordinate normal to the surface, mm;
P0,	total pressure, MPa;	τp,	particle relaxation time, s;
t,	time, s;	τf,	flow time scale, s;
U∞,	freestream velocity, m/s;	δ,	boundary layer thickness, mm;
Reunit,	unit Reynolds number, m-1;	f,	glow discharge frequency, kHz;
ρ∞,	freestream density, kg/m3;	F,	dimensionless frequency.

1 Introduction

Laminal-turbulence transition and the instability of boundary layers have been studied for more than a century. Reshotko^[1], Fedorov^[2], and many others have provided useful reviews and discussions. It is generally accepted that transition is a complicated process involving numerous parameters and mechanisms. Lee and Wu^[3], Lee and Li^[4], Lee and Fu^[5], Lee et al.^[6], Lee^[7-8], and Meng et al.^[9] experimentally observed three-dimensional nonlinear wave packets called soliton-like coherent structures, which are considered to be of fundamental importance in understanding the transition process and may reconstruct the possible universal scenario for wall-bounded flow transition. Yang et al.^[10] and Yang and Pullin^[11-12] improved the method for identifying turbulence structures and developed a topologically preserved tracking method to perform multiscale geometric analysis of isotropic turbulence.

Transition in hypersonic boundary layers has drawn much attention over the past decades for its key influence on aerodynamic and aerothermal problems. For hypersonic flat-plate boundary layers with small free-stream disturbances, two major instability modes were first identified by Mack in 1969^[13], namely, the first and second modes. First-mode instability at high speeds is analogous to Tollmien-Schlichting waves in incompressible flows, while second-mode instability belongs to trapped acoustic waves reflected in a waveguide between the solid wall and sonic line^[14].

Numerous studies have been performed on the whole process of transition at low disturbance level, involving the generation of unstable modes (receptivity problem)^[15-16], the downstream evolution of these modes (stability problem)^[17], and the nonlinear breakdown to turbulence^[18]. Many of these have focused on the evolution of the first and second modes. Bountin et al.^[19] and Dong and Luo^[20] carried out hot-wire experiments and numerical simulations, respectively, on sharp cones at Mach 6, and concluded that first-mode instability was the leading effect in transition. Kendall^[21], Kimmel et al.^[22], and especially Stetson et al.^[23] systematically studied the development of unstable modes in the boundary layer in hypersonic wind tunnels. They believed that second-mode instability played a dominant role in the transition process in high Mach number hypersonic flows. Recently, Chen et al.^[24] analyzed in detail the stability of a hypersonic boundary layer using the linear stability theory, the nonlinear parabolized stability equation, and other theoretical tools. The interaction between the second mode and low-frequency waves (including the first mode and the Görtler mode) and the parametric resonance involved were revealed. The results agreed well with the experiments. Zhu et al.^[25] studied aerodynamic heating in a cone boundary layer at Mach 6 by experiments and numerical simulations. They found that the second-mode wave generated intense aerodynamic heating in the early stage of transition, whose value may exceed the heat flux in the developed turbulent state. This process had a direct relationship with the high-frequency fluid compression and expansion caused by the second-mode wave, and a new principle of aerodynamic heating was constructed. Lee and Chen^[26] reviewed recent progresses in the study of transition in the hypersonic boundary layer.

Herein, an experimental study of instability modes in a hypersonic flat-plate boundary layer is reported. Glow discharge is introduced as an artificial disturbance to investigate the evolution of first- and second-mode instabilities in the boundary layer.

2 Experimental setup 2.1 Wind tunnel and test model

The experiments were performed in the Mach 6.5 quiet wind tunnel at Peking University. The tunnel has an open-jet configuration with nozzle exit diameter of 300 mm, and is currently the largest operational hypersonic quiet wind tunnel in the world. Continuous operation time of the wind tunnel is up to 30 s. The experimental parameters are listed in Table 1.

To prevent air liquefaction, the upstream flow is preheated by a large electric heater to a nominal stagnation temperature

The stagnation pressure P₀ is set to be 0.61 MPa in the experiments. The flow is investigated from t = 8 s after the wind tunnel starts to work normally and the flow parameters become stable. The total temperature T₀ and the total pressure P₀ are monitored by a thermocouple and Kulite pressure transducer, respectively. The unit Reynolds number Re_unit is calculated as

(1)

where ρ_∞ and U_∞ are the density and the velocity of the freestream, respectively. μ is the dynamic viscosity evaluated from Sutherland's law as

(2)

where

and S=110.4 K for air.

The model used in the present study is a flat plate with a sharp leading edge and the zero angle of attack, as shown in Fig. 1. The main dimensions of the plate are as follows: the length 650 mm, the width 200 mm, and the thickness 25 mm. It is long enough to witness the occurrence of transition but short enough to avoid the interference of shock waves reflected by the wind tunnel wall. It is a little thicker than ordinary flat plate models to put in the wire of the glow discharge device and the particle channel of the particle image velocimetry (PIV) system.

Glow discharge is a phenomenon of gas discharge that glows with a colored light at low pressure. In the environment of a hypersonic wind tunnel (about 600 Pa), a stable glow discharge can be generated between two electrodes if loaded with a suitably high voltage and high-frequency current. In the present experiments, two 50 mm long, 2 mm wide strip electrodes made of tungsten are inlaid on the surface of the plate, located parallel to and 100 mm away from the leading edge. An insulation support is designed to mount the electrodes and separate them by 1 mm. An AC power is custom-made to drive the electrodes. The voltage is approximately 3 kV, and the frequency can be tuned continuously from 10 kHz to 180 kHz. Under these conditions, we can obtain a stable linear glow discharge uniformly distributed over the entire length of the electrodes with the same frequency as the power. The ability to generate controlled perturbation with adjustable frequency makes glow discharge a convenient way to add an artificial disturbance^[27].

2.2 Measurement methods

The flow field is observed by the Rayleigh scattering visualization method and quantitatively investigated by a modified PIV method.

2.2.1 Rayleigh scattering visualization

The Rayleigh scattering visualization technique can obtain clear information on the tiny structures in the boundary layer. In this method, carbon dioxide (CO₂) of 99.99% purity is added into the upstream flow. The mass ratio of CO₂ is controlled to be no more than 5% of the whole freestream flow to minimize the influence of gas composition change on disturbance evolution and transition. A laser sheet is projected vertically from the top window of the experimental cabin to the test model, and aligned with the centerline of the plate (see Fig. 2). When the experiment begins, the static temperature in the main flow is less than 50 K, and the CO₂ gas condenses into dry ice particles. The diameter of the particles (several nanometers) is much smaller than the wavelength of the laser (532 nm), under which condition Rayleigh scattering occurs and brightens the main flow region. While in the boundary layer, the dry ice particles sublimate to gas owing to the relatively high temperature near the wall, this region remains dark. The difference in brightness of these two regions will visualize the temperature disturbances around the boundary layer and reveal the flow structures. This technique has been used by Smits' and Lee's group^[28-29] to investigate various hypersonic flows.

In our experiment, the flow field is illuminated by the ZK Laser double-cavity Nd:YAG laser, generating light pulses with a wavelength of 532 nm and maximum energy of 400 mJ per pulse. The sampling rate is 10 Hz. The thickness of the light sheet is approximately 1 mm, and the width is over 300 mm. The flow field is viewed by two PCO edge 5.5 CCD cameras from a side window of the experimental cabin. Each camera is equipped with a Nikkor Micro 200 mm lens, and the resolution is 2 560× 2 160 pixels. They are located side by side to observe the adjacent position of the plate. The DG645 digital delay device is used to ensure that the shooting times of the two cameras are consistent. Pictures from the two cameras are calibrated and spliced together to obtain a total view of 280 mm along the flow direction.

2.2.2 PIV

Reliable flow measurement near the wall is a core requirement for the research on wall-bounded flow transition. Here, PIV is used to measure the instantaneous boundary layer velocity field on the flat plate. The photographic apparatus and related settings remain the same as those used in the flow visualization. TiO₂ is chosen as the tracer particle because of its stability and high refractive index. For PIV experiments at low-speed flows or hypersonic turbulent boundary layers, particles can be injected from the upstream and mix well with the flow. However, the particle is quite difficult to enter a hypersonic laminar boundary layer because of the Saffman force caused by the large shear stress near the wall. To solve this problem, we make a 2 mm wide slot on the surface of the plate to introduce the particles directly into the boundary layer. It is parallel to and 130 mm away from the leading edge. A hollow chamber inside the model filled with particle/air mixture is designed under the slot, and linked to a stirrer and flow meter outside the experimental cabin. The flow rate is restricted to 20 mL/min to minimize the influence of the jet. When the experiment begins, the hypersonic flow sucks particles automatically from the slot.

Another challenge for the usage of PIV in hypersonic flows is the particle-tracing performance. The particle-tracing capability is assessed by the ratio

(3)

where τ_p is the particle relaxation time as it passes through a velocity step, for example, a shock wave, and τ_f is the slipping time of the flow defined as

(4)

in which du is the spatial variation of velocity, and dl is the distance for the variation.

In our case, the particle relaxation time is experimentally evaluated as

τ_f is usually calculated as

(5)

where δ is the boundary layer thickness. Therefore,

indicating satisfactory particle tracing performance.

The main difficulty in PIV measurement of the hypersonic near-wall flow is the large in-plane displacement and gradient. Zhu et al.^[30] developed a novel image preprocessing method to predict PIV images with very large displacement. Before starting the interrogation, stationary artificial particles are imposed in the solid wall region (see Fig. 3). By this process, the initial estimation of the near-wall flow can be corrected by the data from both the real particles and stationary artificial particles, which reduces the large random uncertainties encountered in traditional processing. Interrogations in subsequent iterations tend to converge to the correct results and provide an accurate prediction. This method has been widely used in hypersonic measurements^[31-33]. Jia et al.^[34-35] extended this method to the moving curved interface problem and obtained high-resolution results near a moving rotor blade.

3 Results and discussion

First, Rayleigh scattering visualization of the boundary layer is carried out to obtain a complete picture of the flow and identify the effect of glow discharge, as shown in Fig. 4. The field of view is along the flow direction, from 340 mm to 620 mm away from the leading edge. Figure 4(a) is applied with a glow discharge of 17 kHz, which belongs to the first-mode frequency band. We can easily see that the boundary layer starts with the laminar flow, then two regular first-mode waves along with a series of second-mode waves appear from 380 mm to 480 mm, and finally the transition to turbulence occurs. In contrast, no glow discharge is applied in Fig. 4(b). Hence, no regular first- or second-mode waves are seen in the boundary layer, and only several long waves with various wavelengths appear. This comparison indicates that the introduction of artificial disturbance in the first-frequency can enhance both the first- and second-mode waves in the boundary layer.

Next, PIV experiments are conducted to quantitatively investigate the flow field and verify our theory. The first step is to determine the discharge frequency. Thus, elementary linear stability analysis is performed to check the difference in disturbance growth rates under different artificial disturbance frequencies. The basic flow is set consistent with the experimental conditions. Then, three random frequency disturbances (18 kHz, 28 kHz, and 33 kHz; all within the first-mode frequency range) with their corresponding maximum growth rate angles are added to the basic flow separately at x=100 mm (the glow discharge location) to calculate the downstream growth rates using the fourth-order finite difference method. As shown in Fig. 5, the N-factor of different frequencies does not show much difference. Thus, we choose a series of frequencies f, including 12 kHz, 17 kHz, 25 kHz, 34 kHz, and 39 kHz, to determine the most effective one for PIV experiments. The corresponding dimensionless frequencies are calculated as follows:

(6)

The values are shown in Table 2.

After the experiments and calculation of the instantaneous velocity field, (0, 40) kHz and (60, 110) kHz bandpass filters, which belong to the first- and second-mode frequency ranges, respectively, are performed to analyze the corresponding modes. Specifically, space Gaussian filtering with window widths of 7 mm and 12.8 mm is performed separately. Thus, disturbances with wavelength less than 7 mm or 12.8 mm are filtered out. The linear stability analysis and double-frame Rayleigh scattering visualization results confirm that the phase velocity of the disturbance is approximately 90% of the main flow, or approximately 770 m/s. From the wavelength and velocity, we can easily figure out that the previous operation is equivalent to a 110 kHz and 60 kHz low-pass filter. Then, subtracting the latter by the former, we can obtain a (60, 110) kHz bandpass-filtered disturbance field. Similarly, we can obtain the 40 kHz low-pass-filtered velocity field (e.g., the upper image in Fig. 6(a)) and the (0, 40) kHz bandpass-filtered disturbance field (the lower image in Fig. 6(a)). The low-pass-filtered velocity field can intuitively display the disturbance waves, while the bandpass-filtered disturbance field can mark the wave peak and valley clearly.

The PIV results show that when generating 17 kHz glow discharge, corresponding to

disturbances of the same frequency can easily be recognized in the boundary layer, and sometimes the doubled frequency of 34 kHz, corresponding to

is seen as well, as shown in Fig. 6(b). Analogous results are acquired when applying a 34 kHz glow discharge. If no discharge is introduced, there will be no fixed disturbance frequency. The wavelength of the disturbance becomes irregular in each instantaneous velocity field (see Fig. 7). So far, the results are natural and easily acceptable. One interesting phenomenon is that when generating the glow discharge of 12 kHz, 25 kHz, or 39 kHz, corresponding to

disturbances of the same frequency first appear, which are predictable (see Fig. 8(a)). At the same time, the 17 kHz/34 kHz signal arises again in the velocity field (see Fig. 8(b)). These results imply that the 17 kHz disturbance seems more active compared with other first-mode frequencies, at least in our particular experimental environment. Thus, the 17 kHz glow discharge, corresponding to

is selected as the artificial disturbance to investigate the evolution of first- and second-mode instabilities in the boundary layer.

The effect of glow discharge on first-mode waves has been discussed above. Afterwards, the disturbance level of second-mode waves is also investigated by applying (60, 110) kHz bandpass filters to the PIV instantaneous velocity field. The filtered results show that the disturbance of second-mode waves increases significantly when 17 kHz glow discharge is introduced into the flow (see Fig. 9).

Experiments under different operating conditions are conducted, and comparisons of the average disturbance amplitude in different cases are shown in Fig. 10. The first three bars represent the first-mode disturbance amplitude under high-voltage 17 kHz discharge, low-voltage 17 kHz discharge, and no discharge, respectively. The next three bars represent the amplitudes of second-mode waves under corresponding conditions. It can be seen that glow discharge that is not strong enough has little influence on the evolution of instability waves, as the red bars show. The high-voltage discharge causes considerable growth to the average disturbance amplitude. An unexpected result is that the growth of second-mode disturbance is even greater than the first mode, although the 17 kHz glow discharge is introduced into the flow, which belongs to the first-mode frequency band.

Direct numerical simulation (DNS) is underway to determine the possible mechanisms that can explain this phenomenon. A simplified two-dimensional perturbation evolution is first analyzed. Similar to the experiments, a high amplitude of two-dimensional 17 kHz disturbance, corresponding to

is introduced into the basic flow at the same position as in the experiments, and the growth rates of the 17 kHz disturbance and its each order harmonics are investigated. Preliminary results show that the 17 kHz signal and its second-order harmonic 34 kHz (F=4.15 × 10^-5) experience slow growth along the flow direction, while its high-order harmonics, e.g., 68 kHz, 85 kHz, and 102 kHz (all in the second-mode frequency band), corresponding to

grow much faster at the beginning and reach similar magnitudes as first-mode disturbances. Traditionally, the second-mode wave is considered the dominant instability in hypersonic flows because of its extraordinarily fast growth rate^[2]. Its leading role in transition has also been doubted for its low initial amplitude and the consequent low share of total energy of free stream oscillations compared with the first mode^[19]. In our particular experiments, it seems that the identity as the high-order harmonics of the artificially introduced disturbance makes the 68 kHz, 85 kHz, and 102 kHz signals experience easier initial rapid growth, and show no absence of the typical rapid growth as the second-mode waves. The combination of these two mechanisms may be the path that leads to the unexpectedly significant enhancement of second-mode waves, but solid and detailed analysis requires support from subsequent DNS results.

4 Conclusions

In summary, glow discharge is introduced as an effective artificial disturbance to investigate the evolution of first- and second-mode instabilities in a hypersonic flat-plate boundary layer. The artificially introduced 17 kHz disturbance, which belongs to the first-mode frequency band, can effectively enhance the first-mode waves. In addition, it can enhance second-mode waves even more intensely. The double identity as the high-order harmonics of the artificially introduced disturbance and as the second-mode waves may cause specific frequency disturbances to experience very rapid growth and lead to greater enhancement of the second-mode waves compared with the first-mode waves. However, detailed experiments and numerical simulations are still needed to figure out this phenomenon.