Shanguang GUO, Yun WU, Hua LIANG

Science and Technology on Plasma Dynamics Laboratory, Air Force Engineering University, Xi’an 710038, China

KEYWORDS Shock wave;Boundary layer;Interaction;Flow mechanism;Instability

Abstract The coherent structure and instability of the interaction of incident shock wave with boundary layer developing on a compression corner are experimentally studied. The experiments are carried out in a supersonic wind tunnel of Mach number 2.Particular attention is paid to shock patterns and unsteady shock motions induced by the separation bubble.The high-speed schlieren is used to visualize the flowfield evolution and to characterize the instability. The snapshot proper orthogonal decomposition of schlieren sequences is applied to investigate the primary coherent structure in the flowfield. Fast Fourier transform and continuous wavelet transformation are applied to characterize the instability. The results show that there are large-scale low-frequency oscillations of the shock waves and small-scale high-frequency pulsations in the separation region.The peak frequency of shock oscillation is mainly concentrated in the range of 100-1000 Hz. The pulsation of the small flow structure in the separation bubble is mainly concentrated above 12.5 kHz. Based on the results of experimental analysis, the preliminary mechanism of the largescale instability of such interaction is obtained.

1. Introduction

The interaction of shock wave/boundary layer developing on the compression corner is particular important at supersonic speeds due to their occurrence in supersonic intakes, at offdesign operation, when the ramp shock wave impinges on the bent cowl at some distance downstream of the leading edge,1which can damage the intake structure or, at least,severely limit its performance.2,3For comprehensive reviews on computational and experimental investigations of the Shock Wave/Boundary Layer Interaction (SWBLI), please refer to the studies of Panaras,4Gaitonde,5and Knight and Mortazavi.6In most studies, SWBLI is caused by a single adverse pressure source, such as incident shock wave7,8and compression corner.9,10

The interaction of the oblique shock wave impinging on the boundary layer is intensively investigated to obtain the spacetime organization and unsteadiness in a shock induced separated boundary layer.11,12Polivanov et al.13presented that two-point correlation in the separation bubble was obtained and low-frequency oscillation of the reflected shock waves was revealed to be related to pulsations in the inflow turbulent boundary layer.Tan et al.14,15performed the microcosmic evolution of coherent vortical structures in interactions with the ice-cluster planar laser scattering technique. Sriram and Jagadeesh1studied the correlation for length of impinging shock-induced large separation bubble and conclude that the separation length follows a similarity law independent of the Reynolds number.

Flow mechanisms of interactions between oblique shock wave induced by the compression corner and the boundary layer have been extensively explored.16,17Yi et al.18revealed high space-time resolution three-dimensional instantaneous structure of the incident SWBLI. Huang et al.19,20experimentally assessed the low frequency unsteadiness of a canonical swept separation caused by a slanted 90°-step discontinuity over an axisymmetric turbulent boundary layer. Tumuklu et al.21performed the unsteadiness of shock-laminar boundary layer interactions of hypersonic flow over a double cone and demonstrated the strong coupling between the separated region with the entire shock system.

The small-scale pulsation and large-scale fluctuations in the SWBLI are effected by many factors,such as upstream boundary layer,22,23shock incidence angles,24Mach number25and Reynolds number.26There are many mathematical analytical tools to quantify these scales and to deepen our understanding of the SWBLI physical properties, such as snapshot Proper Orthogonal Decomposition (POD)27and Fast Fourier Transform(FFT).28Mustafa et al.29,30applied snapshot POD to the Krypton tagging velocimetry results to investigate the separated flow structure and evaluate the effect of the compression-corner angle on the turbulent kinetic energy.Sartor et al.31applied a Fourier analysis on a series of schlieren snapshots to characterize the structure of perturbations in the transonic SWBLI.

Interactions of boundary layer with incident shock and corner compression are compared by Alessandro et al.32and the results show that the two case exhibits almost identical flow characteristics with the same inflow parameters and separation-bubble length. However, there have been relatively few investigations of SWBLI induced simultaneously by incident shock wave and compression corners. In this paper, we present experimental results obtained in the boundary layer along a compression corner at a Mach number of 2.0 impinged by an oblique shock wave. Attention is focused on the shock patterns and shock motions induced by the separation bubble.

2. Experimental apparatus and method

The facility used for the present experiments is Mach number 2 wind tunnel at Science and Technology on Plasma Dynamics Laboratory in China,as shown in Fig.1.Wind tunnel parameters are the same as described by Gan et al.33The wind tunnel consists of an inlet section,a rectification section,a Laval nozzle, a test section, a diffuser, and a vacuum tank. An optical observation window with a diameter of 250 mm is installed on the left and right sides of the test section for flowfield diagnosis. The diameter of the nozzle is 300 mm and the nominal Mach number of the nozzle is 2. The air source of the wind tunnel is the ambient atmosphere and its operating time is more than 2 s. The total temperature and total pressure of the incoming flow are 296 K and 96 kPa,respectively.The unit Reynolds number of the test section is 1.1×107.

The experimental model is clearly visible in Fig. 2, composed of a compression ramp with a compression angle of 24° and an incident shock wave generator with a deflection angle of 19°.The length and width of the shock wave generator are 150 mm and 75 mm, respectively. The compression corner consists of a ramp mounted on the plate.The length and width of the flat plate are 400 mm and 110 mm,respectively,and that of the ramp are 100 mm and 80 mm, respectively. The leading edge of the ramp is 300 mm from the leading edge of the plate.In order to facilitate the discussion of the experimental results,the origin of the coordinate system employed herein is located at the leading edge of the flat plate with y-axis oriented normal to the flat-plate surface and the x-axis oriented in the direction of the flow.The leading edge of the shock generator is located at (209 mm, 126 mm).

Combs et al.3found that the technologies based on schlieren system and fast-response surface-pressure measurements had a good agreement in the of motion analysis of the forward lambda-shock foot in the separation region. Therefore, Highspeed schlieren is selected as a main measurement technique in this paper. Schlieren images are obtained using a classical Toepler Z-type schlieren apparatus with mirrors of focal length 3000 mm. The light source is Xenon lamp. High-speed image sequences were obtained using the Phantom v2512 with a resolution of 512 pixel×400 pixel, operated at a frame rate of 50 kHz.The entire recording time for each operation is 100 ms.

3. Results and discussion

3.1. Schlieren image analysis

The density-gradient formula of the flowfield obtained from the schlieren is following,34,35

where, ρ represents density of the air, x represents streamwise coordinate, D represents deflection determined by the gray level of the image at a pixel,K represents Gladstone-Dale constant, L represents the width of the flowfield along the optical path and F represents the focal length of the decollimating lens. According to the above formula, the local gray level of the image is proportional to the density gradient of the air.Therefore, the following analysis based on the gray level of the schlieren image is actually an analysis for the density gradient.

The typical instantaneous schlieren image of streamwise density gradient is presented in Fig. 3. A sinuous pattern of bright and dark bands is evident in the figure, indicating all the flow structure of the flowfield clearly. The shock waves are indicated by bright lines on the image and the expansion waves are visualized as a dark region. The incident shock is formed below the shock wave generator. The boundary layer develops on the flat plate and interacts with a shock wave induced by the compression ramp, referred to as a separation shock. The incident shock and separation shock meet to produce Mach reflection, where Mach stem, reflected shock I,reflected shock II and slip layer are present . The large-scale separation bubble is present in the figure, which is resulted of combination of the separation shock/boundary layer interaction and incident shock/boundary layer interaction.The terminal shock wave is formed downstream and intersects with the slip layer. The expansion fan is shown in the image downstream of the flowfield.

Fig. 1 Supersonic wind tunnel.

Fig. 2 Experimental model.

Fig. 3 Characteristic structure at t=0 ms.

Fig. 4 Shock polar.

To better understand the complicated flow pattern in the present study, the pressure-deflection shock polar is used to describe the shock wave reflection, as shown in Fig. 4. In the figure,the abscissa is the flow deflection angle and the ordinate is the pressure obtained behind the oblique shock wave. The incident shock angle and separation shock angle are 52° and 47°, respectively. The incident shock wave and the separation shock wave are on the I-polar. The reflected shock I is the intersection point of I-polar and R1-polar and the reflected shock II is the intersection point of I-polar and R2-polar. As shown in figure, the direct Mach reflection is generated。The flow direction behind the reflected shock I and II is different,which leads to the Mach stem becoming curved. Through the limiting effect of the slip layer and the wall surface, the pressure behind the reflected shock I and the reflected shock II gradually become identical.

Compression corner SWBLI and incident SWBLI together lead to shock wave oscillations and separation-bubble pulsations in the flowfield. Fig. 5 shows a series of schlieren snapshots, which provides insight into the evolution process of flow pattern. The first image is 0 ms moment. Compared with the 0 ms, the root of the shock wave is moved forward at 0.5 ms and the large-scale vortex structure upstream of the reflected shock II becomes smaller. At 1.0 ms, the intensity of the root of the separation shock is weakened and the thickness of the reflected shock I increases. At 1.5 ms, the root of the separation shock continues to move forward and the structure of the reflected shock I exhibits obvious three-dimensional features. Meanwhile, the Mach stem disappears and the slip layer becomes blurred. At 2.0 ms, the separation-shock intensity continues to weaken,the three-dimensional characteristics of the reflected shock wave II are more obvious,and the size of the separation bubble is significantly reduced. At 2.5 ms, the size of the separation bubble continues to decrease, which causes the separation shock to move back and separation shock angle to increase. Moreover, the Mach stem reappears and the slip layer structure becomes clear. Based on the above analysis, the large-scale oscillation of the shock wave has a strong correlation with the scale of the separation bubble and the large-scale vortical structures in the separation bubble.

Fig. 5 Schlieren image sequences.

It can be seen from Fig. 5 that the streamwise oscillationdistance of the separation-shock root is 20 mm, and the distance between the leading edge of the separation bubble and the compression ramp is 63 mm,which is far greater than that caused by compression corner SWBLI in Ref.36or incident SWBLI in Ref.18

3.2. Flow structure identification

To provide further insight into the dynamics of the coherent structures associated with the SWBLI, snapshot POD is applied. There are not many applications of POD based on the schlieren to the supersonic flow in the literature.One example is that of Berry et al.37where a POD analysis was performed on the time-resolved schlieren field from the supersonic Multi-stream rectangular jets. Generally, largescale coherent structures produce large regions of spatially correlated density fluctuation, and thus are generally well captured by the most energetic POD modes. The eigenvalues and eigenvectors of the autocorrelation matrix of the density gradient are following,38,39

where ρ’is the gradient of density,H is density-gradient matrix made up of all gray-value snap shots in the data set,tkis time.In the present study, five thousand instantaneous densitygradient fields are used to construct the matrix H.

The eigenvalues are determined from,

where An.is the eigenvector matrix built up from the POD coefficients an, λndenotes the nth eigenvalue and corresponds to the energy contained within the nth eigenmode, Ntis index of schlieren snapshots at t time.

The spatial POD modes,Φncan be then expressed as a linear combination of the fluctuating snapshots,

Fig.6 shows the cumulative energy of the POD mode where only the first 1000 modes are exhibited. The first-order modal energy ratio is 99.32%,indicating that the first-order mode has the largest energy contribution to the original flowfield.This is because the orthogonal decomposition is performed on the entire flowfield, while the pulsation area is relatively small compared with the entire flowfield. As the mode increases,the cumulative energy approaches 1.0.

Fig. 6 Cumulative energy of POD modes.

Fig. 7 Typical modes extracted by POD.

Typical spatial eigenvectors obtained from the POD are presented in Fig. 7. It can be seen from the figure that Mode 6, accounting for 17.99‰ of the fluctuations, is characterized by the shock waves, the slip layer and the separation-bubble profile. Mode 41, representing 2.10‰ of the fluctuations,shows that the large-scale structures are upstream of the separation bubble. Mode 150, accounting for 0.53‰ of the fluctuations, shows that the large-scale vortex structure in the separation bubble is broken into middle-scale vortex structures and multiple strip structures appear on the ramp. Mode 250,representing 0.30‰of the fluctuations, presents that the oscillation amplitude of the separation shock and the reflection shock I become larger while the oscillation energy is further reduced. Moreover, the vortex-structure size in the separation region is further reduced and the strips on the ramp are broken into small-scale vortex structures. Mode 500, accounting for 0.14‰ of the fluctuations, performs that the vortex structure in the separation bubble is further broken and the small-small vortex structure also appears in the upstream boundary layer of the separation zone.It can be seen from the Mode 1000 that the shock wave structure almost disappears and the entire separation bubble is covered with the small vortex structure.Based on the above analysis, shock waves and vortices are the main flow structures in the flow field. In addition, the large-scale oscillation of the shock wave and the small-scale pulsation of the vortex in the separation bubble are the motive force of the flow instability.

3.3. Fluctuation characteristic of shock waves

In order to analyze the motion characteristics of the shock wave,it is necessary to extract the shock position and perform spectrum analysis on the schlieren sequences. If the sampling frequency is too small, discrete Fourier transform errors, such as aliasing distortion and spectral leakage,can affect the analysis results. According to Nyquist frequency, it must satisfy

where, fsis the sampling frequency and fhis the characteristic frequency of the flow field of interest.The sampling frequency of high-speed schlieren is 5×104frame/s, so the frequency below 2.5×104frame/s is the effective frequency.

To characterize flow unsteadiness,post-processing methods that track unsteady shock motion in time-resolved high-speed schlieren images have been developed. Fig. 8 shows the interrogation zones for the location of the three shock waves.Fig. 9 presents gray values in the separation-shock interrogation zone at 0 ms.It can be seen from the figure,the gray value first drops and then rises where there are two peaks. The gray level of the pixel represents the gradient of the flow field density,so the width between the two peaks is the thickness of the shock wave. It can be seen from Fig. 9, the shock thickness occupies three-pixel width. In the following analysis, the middle position of the two pixel peaks is taken as the shock position.

Fig. 10 shows the streamwise-location history of the three shock waves. From the figure, we can see that the standard deviation of the location history of the separation shock,11.4 mm, is largest and that of the reflection shock I,7.9 mm,is smaller.The standard deviation of the location history of the reflected shock II is relatively large, because the interaction of the reflected shock II with the separation bubble amplifies the instability of the shock wave.It can be seen from the curve profile in the figure that there is a strong correlation between the separation shock wave and the reflection shock wave I.

In order to analyze the characteristics of the shock position in the frequency domain, a FFT is performed on the position history of the shock waves.Fig.11 displays the Power Spectral Density (PSD) content of shock wave fluctuation where each shock has multiple peak frequencies. The maximal peak frequency of the separation shock, the reflected shock I and the reflected shock II are 357, 221 and 354 Hz, respectively. The maximal peak frequency of separation shock and reflected shock II are very close,indicating that these are frequency corresponding to the separation bubble,because the unsteadiness of the separation shock and reflected shock are directly affected by the separation bubble.As can be seen from the figure, the peak frequencies of the three shock waves are mainly concentrated between 100 Hz and 1000 Hz.

Fig. 9 Gray values in separation-shock interrogation zone.

Fig. 10 Unsteady shock wave location history.

In order to directly compare the oscillation characteristics of the three shock waves, the PSD curve of the shockposition history is smoothed by the median filter function, as shown in Fig. 12. It can be seen from the figure that the dominant frequency of the three shock waves is around 400 Hz.The pulsating power of the separation shock is the largest.At high frequencies (>2000 Hz), the pulsating power of the reflected shock I and the reflected shock II tend to be identical.

Fig. 11 FFT applied to position history of shock waves.

To give access to the time evolution of the spectral signature of the flow, the Continuous Wavelet Transforms (CWT)are performed on the fluctuating fields. The continuous wavelet transform is used to decompose a signal into wavelets that are highly localized.40,41With wavelet transforms, the spectral contents of the flow unfold to reveal the differences between different shock waves even more strikingly than in Fourier spectra.42Space-time analysis of the shock wave position is achieved with the use of the CWT based on the Morlet wavelet.The shape of the Morlet wavelet is determined by modulating a complex sinusoidal function by a Gaussian envelop,

where η is the non-dimensional angular frequency,τ is the nondimensional time. In the present paper, η is equal to 8.

Excerpts of the wavelet spectra are plotted for the shock waves in Fig. 13. It can be seen from Fig. 13(a) that the oscillation-power band of the separated shock is wider between 20 ms and 50 ms and is much narrower in the remaining time.As can be seen from Fig. 13(b), the oscillation-peak numbers and oscillation-power of the reflected shock I are smaller as compared with the separation shock. However, the profiles of the PSD distribution of the separation shock and the reflected shock I are similar, which clearly indicates that there is a strong dependence between these two shock waves.As can be seen from Fig. 13(c), the oscillation power of the reflected shock II is highly intermittent where the oscillation power is stronger in the three time periods of 5-20 ms, 27-65 ms and 75-95 ms and is much weaker in the remaining time.

Fig. 12 Power spectrum density smoothed by median filter function.

Fig. 13 Wavelet transform applied to position history of shock waves.

3.4. Spatial spectra distributions

In order to compare and analyze the unsteady characteristics of the shock wave and the separation bubble, it is necessary to perform spatial Fourier transform on the schlieren sequence of the entire flow field and analyze the spatial distribution of the pulsating power at the specified frequency.43Fig.14 shows the typical frequencies extracted from the time-resolved schlieren. The five-thousand successive pictures were used to calculate the frequency content of the pictures at each pixel point.

It can be observed from the figure that most of the fluctuations occur around the primary flow structure. The low frequencies are more dominating in the shock wave structure and the edge of the separation bubble. The high frequencies are more dominating in the separation bubble. From the Fig.14 at the frequencies of 0.1,0.5 and 1 kHz,it is shown that the spectrum values appears to be higher around the Mach stem, the aft edge of the reflected shock wave I and the reflected shock wave II, which correspond extremely well with the peak-frequencies observed in the shock wave position oscillations.As the frequency increases,the spectrum values tend to continue to be smaller.At the frequency of 2.5 kHz,the pulsating power of the shock and the separation bubble are of the same order of magnitude. From Fig. 14 at the frequency of 5 kHz, the higher values of spectrum have shifted from the shock wave to the separation bubble. Based on the further analysis of the instantaneous schlieren images, it is likely that 5 kHz is linked to the motion of the large-scale vortical structure in the separation bubble.In the 12.5 kHz figure,the small flow structure in the separation bubble is the primary characteristic of the entire flowfiled, indicating that 12.5 kHz relates to the small vortical-structure pulsation in the separation bubble.

4. Conclusions

This paper investigates the flow structure and unsteadiness of interactions between the incident shock wave and the boundary layer developing on a compression ramp in a Mach number 2 flow to examine its potential mechanisms. We focus our attention on the shock patterns and unsteady shock motions induced by the separation bubble. Quantitative analysis of high-speed schlieren imagery for this interaction is carried out by three mathematical analytical tools (POD, FFT,CWT) to identify coherent structure and fluctuation intensity generated within the flow field.

This interaction is relatively complicated. From the perspective of flow organization,there are incident shock,separation shock, reflected shock, Mach stem, slip layer and separation bubble. Meanwhile there are complicated interactions between each coherent structure, including shock/shock interaction, shock/boundary layer interaction, Mach reflection, etc. From the perspective of flow instability, there are large-scale low-frequency oscillations of shock waves and small-scale high-frequency pulsations in the separation bubble.The peak frequency of the shock wave is mainly concentrated in the range of 100-1000 Hz, and the pulsation of the small vortices in separation bubble is mainly concentrated above 12.5 kHz.

Fig. 14 Spatial spectra distribution at different specified frequencies.

The instability and separation-bubble size of this interaction is much larger than the interaction imposed by a single inverse pressure gradient (incident shock or compression corner). The interaction mechanism is preliminarily analyzed as follows.Inverse pressure gradient imposed by the combination of the incident shock wave and compression corner greatly amplifies the boundary-layer instability which causes largescale motions of the separation bubble. The dependent domains of the separated shock and reflected shock I are in the separation bubble, so these two shock waves also exhibit large-scale motions. Meanwhile, the large-scale motions of the separation bubble cause pressure fluctuating behind the reflected shock II, which leads to large-scale movement of the reflection shock II.

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 research is supported by the National Natural Science Foundations of China (Nos. 51907205 and 51790511) and the Natural Science Basic Research Plan in Shannxi Province of China (No. 2018JQ1011).