Oblique shock train motion based on schlieren image processing

Longsheng XUE, Chuan CHENG, Chengpeng WANG,*,Lantian ZHANG, Kang LI, Keming CHENG

a College of Aerospace Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China

b Key Laboratory of Unsteady Aerodynamics and Flow Control, Ministry of Industry and Information Technology, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China

c Beijing Institute of Tracking and Telecommunication Technology, Beijing 100094, China

d Aerospace System Engineering Shanghai, Shanghai 201109, China

KEYWORDS Frequency components;Hysteresis loop;Maximum Correlation Detection (MCD);Schlieren image processing;Shock train oscillation

Abstract In this paper, shock train motion in a Mach number 2.7 duct is studied experimentally,and large numbers of schlieren images are obtained by a high-speed camera.An image processing method based on Maximum Correlation Detection(MCD)is proposed to detect shock train motion from the schlieren images, based on which the key structures, e.g., separation positions and separation shock angles on the top and bottom walls, can be analysed in detail.The oscillations of the shock train are generated by rhombus and ellipse shafts at various excitation frequencies.According to the analysis of MCD results,the distributions of the frequency components of shock train oscillation generated by the two shafts are distinctly different, in which the motion generated by the ellipse shaft is much smoother;shock train motion is mainly characterized by the oscillation of separation position while the separation shock strength is not so sensitive to downstream disturbance;there is a hysteresis loop relation between the downstream pressure and separation position.?2022 Chinese Society of Aeronautics and Astronautics.Production and hosting by Elsevier Ltd.This is an open access article under the CC BY-NC-ND license(http://creativecommons.org/licenses/by-nc-nd/4.0/).

1.Introduction

Shock train is a common phenomenon existing in most supersonic duct flows, e.g., the supersonic/hypersonic wind tunnel,and inlet/isolator of hypersonic vehicle.The shock train configuration is usually characterized by a series of shocks/pseudo-shocks interacting with the boundary layer and separation flow,1resulting in complex structures, including normal shock train2and oblique shock train,3which depend on incoming flow properties.4,5Sufficient understanding of the structure mechanism and motion behaviour of shock trains is significant for novel design methods and efficient flow control.

Oscillation is one of the most representative features of shock train motion, including self-excited oscillation6,7and forced oscillation.8,9The former is closely related to the unsteadiness of shock wave-boundary layer interactions, and the latter is mainly induced by upstream or downstream flow disturbances,possibly leading to propagation of the separation region and complex interactions with background waves.10,11For an air-breathing scramjet,12,13it is difficult to conduct a completely stable combustion, i.e., the influence of the downstream unsteady flow on the shock train in the isolator is inevitable.14,15In fact, combustion could strongly affect the shock train and cause a high-frequency(>100 Hz)pressure disturbance.16The upstream propagation or large amplitude oscillation of the shock train may threaten the performance of the inlet and isolator,greatly reducing the total pressure recovery,or even worse, causing an unstart.17,18In recent years, much attention has been given by researchers to the response of shock train to pressure disturbances.Xiong et al.3proposed that the mechanism by which the moving shock train responds to downstream excitations is to keep moving so that the relative Mach number ahead of the shock train changes to match the varying imposed back pressure.In addition,they studied both the symmetrical19and asymmetrical7oblique shock train structures,and it was observed that the asymmetrical modes may switch to each other during one case.Cheng et al.9also studied an asymmetrical oblique shock train and found that the shock train always holds a same asymmetrical mode when moves upstream and downstream.The work of Klomparens et al.6demonstrated that there exists a linear relationship between back pressure and the average location of the shock train.Wang et al.4found that the location of shock train changed by back pressure may have a sudden movement when shock train passes through a local separation region, and in their another study,10it was found that there is a hysteretic effect and phase lag between the shock position and pressure distribution when the shock train travels along a different path for upstream and downstream movements.Similarly,Klomparens et al.8also observed that there is a hysteresis effect during the shock train’s cyclic motion that causes the leading shock to travel along different paths during the upstream and downstream moving portion of the cycle.Both Tan et al.11and Shi et al.16proved that the shock train structure may be strongly interacted with background waves, resulting in complex motions and various flow patterns.Literature indicate that the complex characteristics of shock train are caused by coupling results of multiple factors, among which downstream pressure disturbance is the main inducement to shock train motion.

The downstream pressure disturbance could travel in the counterflow direction through the subsonic boundary layer to induce shock train motions, and thus an important step to experimentally study the forced oscillation of a shock train in a duct is the generation of the pressure disturbance.However,because of the safety problem and high cost,combustion is not easy to perform in general supersonic wind tunnels;instead, downstream flow throttling is commonly used to simulate the pressure disturbance, e.g., in the study of Xiong et al.,3the oscillation of shock train was generated by downstream periodic throttling, or in the study of wang et al.,4the upstream propagation of shock train was induced by continuously closing outlet.According to the research focus of experiments in the literature, there are two main ways to design a device of a downstream disturbance generator: one is to generate linearly or near-linearly continuous throttling,e.g., a moving plug11,17,20,21or rotary flap,4,22–24which drives gradual flow choking and can be applied to simulate the combustor pressure rise;the other is to generate periodic throttling,e.g.,a noncircular shaft8–10or a throttling fan,3,19which drives oscillatory flow choking and can be applied to simulate the pressure fluctuation of unsteady combustion.Therefore, the first method is suitable for investigating the propagation of shock train and unstart phenomena, and the second method is more often used for learning forced oscillation phenomena.

In a duct with a rectangular cross section,the shock train is generally treated as a two-dimensional configuration,3,6–10,19and thus shock train oscillation could be simplified as the motion of the leading structure, including the changes of separation positions and separation shock angles on the top and bottom walls.Schlieren is one of the most widely employed techniques for monitoring shock train oscillation, which demands an efficient image-processing method.For an oblique shock train, the leading structure is quite similar to an‘‘X”,and thus the edge detection method based on the Hough transform is usually employed to detect the shock location automatically from large quantities of schlieren images.In the works of Xiong3and Klomparens6,8et al., shock locations were extracted from the schlieren images by edge detection method,which provided valuable data for motion analyses.However,edge detection requires the images to maintain high visual quality, and background waves might affect the accuracy.Thus, the application of edge detection may be restricted on the flow pattern that shock train is interacted with background waves.For motion analysis, Dynamic Mode Decomposition(DMD),25which requires less image quality, could be employed to analyse the motion modes and frequency components of flow structures.DMD is helpful for unsteady flow analyses, e.g., Grilli26and Priebe27et al.studied Shock Wave-Boundary Layer Interaction (SWBLI) phenomena near a compression ramp based on DMD, which provided the evidence for supporting the hypothesis that SWBLI phenomena for the low-frequency motion of the shock are a consequence of the inherent dynamics between separation bubble and shock and are not driven by upstream coherent structures.Therefore,image processing method plays an important role in the analysis of dynamic flow pattern.

The goal of the current work is to examine the forced oscillation characteristics of shock trains driven by different disturbance generators.On one hand, an image-processing method based on Maximum Correlation Detection(MCD)is proposed to detect the shock train structures, and the DMD method is also discussed in detail to compare with MCD.On the other hand, two shafts with different profiles, rhombus and ellipse,are employed to generate downstream excitation frequencies,and the response of the shock train to both disturbances is analysed based on the MCD and DMD methods.

2.Image processing methods

2.1.Dynamic Mode Decomposition (DMD)

A shock train in Mach number 2.7 flow is usually characterized by asymmetrical separation shock-shock interaction,9,10and the separation positions on the top and bottom walls are not on the same vertical plane, as shown in Fig.1.The shock train in the duct is unsteady due to downstream pressure disturbance, in which separation position oscillates between upstream and downstream, as shown in Fig.2.To research the relationship between shock train motion and downstream disturbance,the dynamic characteristics of shock trains should be described quantitatively; hence, image processing is necessary.

The shock train motion illustrated in Fig.2 is driven by a periodic disturbance, of which the frequency components are indicated in Fig.3.Fig.2(a)and Fig.3(a)are original schlieren images and the Fast Fourier Transform(FFT)result of downstream disturbance pressure,respectively.The DMD method is well suitable for restructuring the dominant dynamic modes of motion, and the Power Spectral Density (PSD) can be clearly presented by a frequency-mode energy map(the description of DMD is given in Appendix A).Accordingly, Fig.2(b) is restructured by 31 of the largest energy modes that are computed from 2000 schlieren images, and the corresponding frequency-mode energy map is shown in Fig.3(b).Both downstream pressure and shock train motion indicate the same dominant frequency of 104 Hz and the same harmonic frequency of 208 Hz.The instance demonstrates that the major energy of shock train motion resides in several dominant modes, which work at the same dominant frequencies as downstream disturbance.The restructured images show the main flow interaction structure, while some details that oscillate at relatively weak energy are neglected, e.g., the leading shock-shock interaction configuration, and as a result, the influences of downstream pressure disturbance on the separation position or separation shock strength cannot be identified by the DMD method.

Fig.1 Sketch for illustrating motion of an asymmetrical shock train in a duct.

Fig.2 Schlieren images for illustrating shock train motion of Mach number 2.7 flow.

Fig.3 Frequency components of shock train motion based on various methods.

2.2.Maximum Correlation Detection (MCD)

Complex background waves exist in original schlieren images,as shown in Fig.2(a), resulting in difficulty in detecting the leading separation shocks based on the edge detection method,of which the accuracy is less than 50%;hence,a novel method for shock train location is necessary.The oblique shock train oscillation in a Mach number 2.7 duct usually holds a relatively stable structure moving upstream and downstream,9,10which demonstrates that the leading interaction configurations are similar at different positions;in other words,all the leading shock structures in the successive schlieren images present a high correlation.Therefore, the current study proposes a method for detecting shock train motion based on MCD.

The first step of the MCD method is to create a template of the leading shock train structure and then obtain the positions that present the maximum correlation with the template from all the input images (the template can be one of the restructured images, which will be introduced in the following part).An example is given in Fig.4, where matrix M0(w0× h0) is a template (Fig.4(a)) for the leading shock train structure,and the grey values are expressed by a vector m0.One of the input images (Fig.4(b)) that is w (w > w0) in width and h0in height can be divided into finite matrices, and each matrix expressed by mihas the same size as m0.The correlation coefficient ribetween m0and mican be written as

Similarly, some other details, e.g., separation shock angle,can also be obtained by the MCD method.As shown in Fig.4,for angle detection,the matrix Mβ0(wβ×hβ)is created as a template, which is expressed by a vector mβ0containing the grey values of the top separation shock.Assuming that the separation position Oiin an input image has been determined first,then the sector around Oican be divided into finite matrices expressed by Mβi, and the corresponding vectors are mβi.The correlation coefficient rβibetween mβ0and mβican be computed by Eq.(1).The maximum value of rβidetermines the angle of the top separation shock, and the detection accuracy depends on the matrix size and dβ.Fig.5(c)shows the distribution of the correlation coefficient of the instance in Fig.4,where dβ ranges from 0.01°to 2°(the lines of dβ>0.01°move downwards in turn for differentiation,i.e.,rβi-0.05,rβi-0.10,rβi- 0.15, and rβi- 0.20).Therefore, to save calculating resources, dβ should be reduced gradually from a relatively high value to a small value.Fig.5(d) shows the time history of the top separation shock angle of the instance in Fig.2,and it can be seen that the shock strength oscillates between β = 34° and β = 36°.

Fig.4 An instance for explaining MCD method.

Fig.5 Instances for illustrating MCD results.

Consequently, the main configuration of the shock train,e.g.,separation positions on the top and bottom walls,leading separation shocks, and reflected shocks, can be identified by the MCD method, based on which the shock train configuration can be restructured by the weighted average of several successive images at the same position of the shock train.If N images are used to restructure the shock train, the vector vi(grey values of the ith image) of the shock train at time tican be expressed as follows:

where Ω≥0 is a no-dimensional parameter for controlling the transient state.A small Ω makes the result tend to be an average image while a large Ω makes the result tend to be an instantaneous image.Fig.6(a) illustrates the weighting curves of fH(k)with various Ω ranging from 0 to 1000,indicating that a larger Ω corresponds to a sharper curved line, which spends higher weightings on nearby images while lower weightings on farther images.Fig.6(b) shows several restructured images with various Ω ranging from 0 to 1000, in which the first one is an original image for comparison.It can be observed that the restructured image with Ω = 1000 is quite similar to the original one.With the decrease of Ω,the details of the flow pattern become fewer and fewer,and some of the details including the background waves almost disappear when Ω reduces to 0.In current study, the Ω = 20 is employed to restructure the images shown in Fig.2(a), and the results are shown in Fig.2(c), which clearly indicate that the leading shock train configuration is intensified while background waves are denoised, greatly facilitating shock train motion analysis.Additionally, vi+k,j+Oi+k-Oidenotes grey value in local position of the leading shock determined by MCD, and the position can also be determined by artificial measurement, e.g., ten original images where the shock locations were measured artificially were used to restructure a template with Ω = 0 shown in Fig.4(a),which performed better in accuracy of MCD than any of the original ones.

In summary, shock train oscillation in a duct is characterized by complex interaction configurations.Both the DMD and MCD methods are suitable for analysing the frequency components of shock train motion.The frequency components and dynamic modes of the overall shock train characteristics can be presented by the DMD method, while the frequency components of several detailed features, e.g., positions and angles, can be identified by the MCD method.DMD could decompose the dynamic modes of shock train motion and restructure the dominant oscillation characteristics, while MCD could restructure the key structures of the shock train and denoise background waves.

3.Shock train motion analysis based on image processing

3.1.Experimental setup

Fig.6 Restructured images with various Ω.

Fig.7 Sketch for illustrating experimental setup.

Experiments were performed in a blowdown-type supersonic wind tunnel, as shown in Fig.7.The pressure air source is supplied by three connected spherical air tanks.Each tank is 640 m3in volume at a pressure of 660 to 680 kPa.In the test section, the incoming flow is from a two-dimensional nozzle,of which the rectangular outlet is 40 mm in width and 45.1 mm in height.A Mach number 2.7 nozzle was employed for the current experiments.The focused flow configuration exists in a two-dimensional duct 580 mm in length downstream of the nozzle, and there are 0.3° expansion angles on both the top and bottom walls to eliminate the effect of a continuously thickened boundary layer.Two sidewalls of the duct are embedded with two optical glass windows for schlieren.

The schlieren window of 200 mm in diameter is located downstream of the nozzle, as shown in Fig.1, and an NAC(NAC Image Technology)Hotshot high-speed camera operating at a frame rate of 1 kHz with a 10 s sampling time and a resolution of 720 pixel × 720 pixel was employed to take schlieren images.The schlieren images for each test were taken during the effective running time (tr) of the wind tunnel when the incoming flow total pressure was stable,as shown in Fig.8(a).The downstream disturbance generator is a noncircular shaft driven by a motor, which generates two disturbance cycles by one rotation in the middle of the duct, as shown in Fig.1.The shaft is located 537 mm downstream of the nozzle outlet.Two alternative shafts with different profiles were used in the tests: rhombus and ellipse, as shown in Figs.8(b) and 8(c), respectively.The throttling level generated by the two shafts can be expressed as follows:

where TR,a,b and n are the throttling level,major axis,minor axis and rotation rate of the shaft, respectively; the subscripts‘‘r”and‘‘e”denote rhombus and ellipse shafts,respectively;hsis the height of the duct at the position of the shaft.Herein ar=14 mm,br=10 mm,ae=12 mm,and be=9 mm,then.

where ω is phase angle, thus the integral values of throttling levels by the two shafts during one rotation (0 to 2π) are similar.Figs.8(b) and 8(c) show the curved lines of the throttling level during one rotation, indicating that the rhombus shaft could generate both dominant and secondary excitation frequencies, while the ellipse shaft only generates the dominant excitation frequency.Below the shaft, there is a Kulite XTEL-190 M fast-response transducer mounted on the central line of the bottom wall, which measures pressure at a rate of 10 kHz with a 20 s sampling time using data acquisition cards.The conditions of incoming flow(Mach number Ma and Reynolds number Re) and downstream disturbance for each test are summarized in Table 1, where fddenotes the dominant excitation frequency of a downstream disturbance.

3.2.Influence of downstream disturbance on shock train motion

Fig.8 Curved lines of upstream flow property and downstream throttling of wind tunnel.

Table 1 Conditions of incoming flow and downstream disturbance.

The shock train configuration is related to various influences,such as the Mach number,Reynolds number,and downstream pressure disturbance.For a relatively low Mach number, e.g.,experiments in Mach number 2 flow conducted by Xiong et al.,3,19the shock train was approximately symmetrical,while in current experiments at Mach number 2.7, a symmetrical configuration was never observed.There are two possible modes existing in asymmetrical shock trains:Top-Large Separation (TLS), shown in Fig.9(a), and Bottom-Large Separation (BLS), shown in Fig.9(b).In the work of Xiong et al.7at Mach number 3, both BLS and TLS were observed, and the two modes might switch to the other in one test.However,the switch phenomenon was not observed in any current case,although both BLS and TLS alternately appeared in different tests.Additionally, the current experiments cannot predict BLS or TLS; hence, regardless of which one was formed at first, the shock train oscillation would hold the mode during the effective running time of the wind tunnel.

Based on the MCD method mentioned above, several key structural parameters can be described in detail,including separation positions on the top and bottom walls and separation shock angles from the top and bottom walls, which are denoted as xt, xb, βtand βb, respectively, as shown in Fig.9.The time histories of the parameters for self-excited oscillation(Case 1, fd= 0 Hz) and forced oscillations (Case 3 and Case 10, fd= 21 Hz) are illustrated in Fig.10.It can be observed that the self-excited oscillation does not indicate a distinct dominant frequency, and the time history of each variable is similar to white noise according to the FFT results.When a 21 Hz excitation frequency was exerted, the shock train positions(xtand xb)fluctuated with the periodic change in disturbance pressure (pd).Although the curve shape of the time histories generated by a rhombus shaft was different from that generated by an ellipse shaft,the oscillation periods(48 ms)of the two motions were very close.The influence of downstream disturbance on the amplitudes of separation shock angles (βtand βb)was not apparent,and thus the values fluctuated without an obvious shape.For the TLS mode, βt≈34.8° and βb≈32.1°, while for the BLS mode, βt≈32.1° and βb≈34.8°(the standard deviations are approximately 1°),indicating that the separation shock angle on the top is distinctly different from that on the bottom, and the massive separation region corresponds to a relatively larger separation angle.

The statistical results of the oscillation cycles, including pd,xtand βt,are summarized in Fig.11,in which the curved lines of shock train motion during one rotation of the downstream shaft are average values computed by more than 200 successive oscillation cycles containing 10000 schlieren images.It can be observed from Figs.11(a) and 11(b) that there are two cycles during one rotation,and the curved lines of downstream pressures (pd) generated by rhombus and ellipse shafts are similar to the curved lines of the throttling levels shown in Figs.8(b)and 8(c), respectively.For the rhombus shaft, as shown in Fig.11(a), both dominant and secondary disturbances can be observed, while for the ellipse shaft shown in Fig.11(b), there is only a dominant disturbance.The shock train moves upstream with increasing pdand moves downstream with decreasing pd; hence, the peak of pdcorresponds to the valley of xt.However, the shock train motion is approximately 5 ms delayed behind the downstream disturbance pressure because of the distance between the separation position and shaft.The separation shock angle βtincreases with the shock train moving upstream and decreases with the shock train moving downstream, while it is noted that the change in separation shock angle βtis approximately 6 ms delayed behind separation position xt, indicating that the shock train motion senses the propagation of pressure disturbance earlier than the separation shock strength.The loop maps of hysteresis relation between shock location and downstream pressure generated by different shafts at fd= 21 Hz are illustrated in Fig.12,demonstrating different paths along which the leading shock moves upstream and downstream during oscillation cycles.It can be observed from Fig.12(a)that the upstream moving portion is distinctly different from the downstream moving portion generated by rhombus shaft (dominant and secondary excitation frequencies), while the upstream and downstream moving portions generated by ellipse shaft (only dominant excitation frequency) are nearly centrosymmetric, as shown in Fig.12(b).Therefore, the secondary excitation frequency may damage the central symmetry of hysteresis paths of shock train motion.

Fig.9 Two possible modes of asymmetrical shock train structures.

Fig.10 Time history of key structural parameters for describing shock train oscillation at Ma=2.7(ht is the height of throat,and p∞is the static pressure ahead of shock train).

A nondimensional variable Amp, i.e., the standard deviation divided by the standard deviation of self-excited oscillation, is defined to measure the amplitude level of each case.For self-excited oscillation, Amp of each parameter is 1.Fig.11(c)shows the amplitude levels at various excitation frequencies.It can be observed that with increasing excitation frequency, Amp of each parameter shows a declining tendency towards 1, and the amplitude levels generated by the ellipse shaft are smaller than those generated by the rhombus shaft.When fdis larger than 60 Hz,the amplitude level of the downstream pressure is close to 1,which demonstrates that the fluctuation of the downstream pressure at a high excitation frequency shows an amplitude equivalent to that of self-excited oscillation.For the separation shock angle,the amplitude level is always close to 1, indicating that the separation shock strength is not so sensitive to the excitation frequency.

Fig.11 Average oscillation cycles and amplitudes of βt, xt and pd.

Fig.12 Loop maps for illustrating hysteresis relations between xt and pd generated by different shafts.

The shock train motions of Cases 2 to 15 are excited by several stable frequencies, including 10, 21, 41, 62, 83, 104 and 123 Hz.Based on the DMD and MCD methods, the mode energies and PSDs of βt, xtand pdgenerated by the rhombus and ellipse shafts are summarized in Fig.13 and Fig.14,respectively.For the rhombus shaft, as shown in Fig.13, the DMD results show complex modes containing various harmonic frequencies, and there are also various frequency components with large energies existing in relatively low excitation frequencies (fd= 10, 21, 41 Hz) observed in the PSDs of both xtand pd, demonstrating that shock train oscillations in these cases are complex motions with large amplitudes.However, for the ellipse shaft, as shown in Fig.14,the dynamic modes are less complex than those generated by the rhombus shaft, and the frequency components of xtand pdare very clean, in which the energies of harmonic frequencies are much weaker than the dominant ones, indicating that the oscillation excited by the ellipse shaft is much smoother than that excited by the rhombus shaft, which is the same as that observed from Figs.11(a)and 11(b).With increasing excitation frequency, the numbers of frequency components in xtand pddecrease gradually.In the cases of relatively high excitation frequencies (fd= 62, 83, 104, 123 Hz), the oscillation energy of the dominant frequency decreases distinctly, and the harmonic components are too weak to be distinguished from noise;hence,shock train oscillation seems to be mild with small amplitudes in these cases excited by both rhombus and ellipse shafts.It is noted that the energy of the dominant frequency of βtis always weak, and the harmonic components are not strong enough to be distinguished from noise in all cases, indicating that separation shock angles fluctuate with relatively small amplitudes,which is the same as that observed from Fig.11(c).Both Fig.13 and Fig.14 demonstrate that the frequency components of xtare much more similar to the frequency components of pdthan the mode analysis or frequency components of βt,and thus the influences of downstream pressure disturbance mostly exert on the change in the separation position rather than the separation strength.

Fig.13 Frequency components of shock train motion generated by rhombus shaft at various excitation frequencies.

Fig.14 Frequency components of shock train motion generated by ellipse shaft at various excitation frequencies.

Fig.15 Time-frequency analysis of shock train motion affected by unstable disturbance.

The shock train motions of Cases 16 and 17 excited by the ellipse shaft are affected by a continuous increase in the incoming flow Reynolds number and a continuous increase in the downstream excitation frequency, of which the time history of xtand the corresponding time–frequency analyses are illustrated in Fig.15.As shown in Fig.15(a), although the oscillation centre of xtmoves downstream gradually due to the linearly increasing Reynolds number (1.32 × 106to 1.61 × 106), the time–frequency map indicates a stable frequency of 21 Hz, which is equal to the downstream excitation frequency.Fig.15(b) indicates that the oscillation centre of xtis stable even when the excitation frequency increases continuously from 10 Hz to 123 Hz.Therefore,the linear change in the incoming Reynolds number mostly affects the oscillation centre of the shock train while exerting little influence on the frequency components,and the downstream excitation frequency determines the frequency components of the shock train oscillation while not changing the oscillation centre.

4.Conclusions

The current study proposes an image processing method for detecting shock train motion from large quantities of schlieren images based on MCD,which can analyse shock train motion by describing several key structures with time histories, e.g.,separation positions and separation shock angles on the top and bottom walls.The oscillations of the shock train generated by rhombus and ellipse shafts at various excitation frequencies were studied experimentally, and the schlieren images were processed by both DMD and MCD methods, based on which the following conclusions can be obtained.

(1)For processing schlieren images of shock train motion in a duct, both the DMD and MCD methods are suitable for analysing the frequency components of shock train oscillation,and the motion can be restructured by both methods.The difference is that DMD could analyse the dynamic modes of the whole flow field, while MCD could separately analyse several key structures of the motion.Additionally, the MCD method performs well in denoising background waves.

(2) The rhombus shaft generates both dominant and secondary disturbances, which contribute to a number of harmonic frequency components of shock train oscillation, while the ellipse shaft only generates a dominant disturbance, and the energies of the harmonic components of shock train oscillation are much weaker.The amplitude of the shock train position decreases distinctly with increasing excitation frequency,and at relatively high excitation frequencies (>100 Hz), the amplitude of forced oscillation shows a tendency towards the amplitude of self-excited oscillation.

(3) The shock train in a Mach number 2.7 duct has an asymmetrical configuration, and both TLS and BLS modes were observed in the current tests,of which the motion mainly contains the changes of separation positions and separation shock angles on the top and bottom walls.The separation shock strength was not so sensitive to downstream disturbance, and the fluctuation of the separation shock angle was delayed behind the separation position.There is a hysteresis loop relation between the downstream disturbance and separation position, and the hysteresis loop paths when shock train moves upstream and downstream which is excited by one dominant frequency are nearly centrosymmetric, while the central symmetry may be damaged if downstream pressure disturbance contains multiple frequency components.In addition,the oscillation centre of the shock train position could be affected by a linear change in the Reynolds number, while the oscillation centre does not react to the downstream excitation frequency.

Declaration of Competing Interest

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

Acknowledgements

This work was supported by the National Numerical Wind Tunnel Project of China, the National Natural Science Foundation of China (Nos.12002163 and 12072157), the Natural Science Foundation of Jiangsu Province, China (No.BK20200408), the China Postdoctoral Science Foundation(No.2022T150321) and the Key Laboratory of Hypersonic Aerodynamic Force and Heat Technology,AVIC Aerodynamics Research Institute, China.

Appendix A.Dynamic Mode Decomposition (DMD) is a technique for linear motion analysis, which shows a powerful advantage in detecting the dynamic characteristics of unsteady flow.It is assumed that shock train motion in a twodimensional duct is approximately linear, which is recorded continuously by finite schlieren images.As shown in Fig.1,pixel matrices with w denoting width and h height, which contain the same region of shock train motion, are extracted from the schlieren images.Each pixel matrix can be expressed by a vector vi, which includes all the w × h grey values.If n images are taken into consideration, the input data can be summarized into a larger matrix that is n in width and w×h in height as follows:

where X and Λ are the eigenvector matrix and eigenvalue matrix of A, respectively, and X-1is the inverse matrix of X.Thus, the dynamic modes of flow motion can be analysed from the eigenvalues and corresponding eigenvectors.The key step of the DMD method is to solve the eigenvalues of A.Because A is a very large matrix that is w × h in width and w × h in height, it is difficult to solve the eigenvalues directly.Refs.25–27demonstrated that the eigenvalues decay very fast, and several largest eigenvalues can well describe the linear motion.Therefore, a much smaller matrix Arcontaining the former r (r ≤n - 1) with the same eigenvalues as A can efficiently be used to analyse the motion.Accordingly,the Singular Value Decomposition (SVD) of V1,n-1is helpful for DMD.The SVD of matrix V1,n-1can be expressed as follows: