Experimental study on the flow separation and self-excited oscillation phenomenon in a rectangular duct

Acta Astronautica 133 (2017) 158–165

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Experimental study on the flow separation and self-excited oscillation phenomenon in a rectangular duct

MARK

⁎

Bing Xiong ,1, Zhen-Guo Wang1, Xiao-Qiang Fan, Yi Wang National University of Defense Technology, 410073 Changsha, People's Republic of China

A BS T RAC T To study the characteristics of flow separation and self-excited oscillation of a shock train in a rectangular duct, a simple test case has been conducted and analyzed. The high-speed Schlieren technique and high-frequency pressure measurements have been adopted to collect the data. The experimental results show that there are two separation modes in the duct under M3 incoming condition. The separation mode switch has great effects on the flow effects, such as the pressure distribution, the standard deviation distribution and so on. The separation mode switch can be judged by the history of pressure standard deviation. When it comes to the self-excited oscillation of a shock train, the frequency contents in the undisturbed region, the intermittent region, and the separated bubble have been compared. It was found that the low-frequency disturbance induced by the upstream shock foot motions can travel downstream and the frequency will be magnified by the separation bubble. The oscillation of the small shock foot and the oscillation of the large shock foot are associated with each other rather than oscillating independently.

1. Introduction The isolator is a key component for an air-breathing scramjet, which plays a significant role in preventing the interactions between the combustor and the inlet [1]. Under the high back-pressure condition induced by the combustion, the boundary layer in the isolator will be separated and the typical shock train (or pseudo shock) will form in the isolator [2]. Commonly, the isolator is simplified to be a rectangular duct in the research. In recent years, much research has been done to study the characteristics of the shock train and much has been learnt about it, such as the length [3], the fine structure [4,5], the self-excited oscillation of a shock train [6,7], and so on. Matsuo [8] and Gnani [9] have reviewed the previous research in 1999 and 2016. Flow separation induced by back-pressure is a prevalent phenomenon in a scramjet isolator. The asymmetric separation often occurs in the isolator under high back-pressure condition [10,11], which has significant effects on the quantity of the outflow of the isolator. The asymmetric separation phenomena have been found in some other devices, such as the convergent-divergent nozzle which works under the over-expanded condition [12]. Papamoschou et al. [12,13]made pressure measurements and found that there was a low-frequency, piston-like unsteady shock motion. After that, Johnson et al. [14] made

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1

some new studies based on the previous work and they found that the unsteadiness of the shock motion was coupled to enhanced shear layer instability. What is more, the flow separation mode may switch in the isolator, which has ever been observed in previous experiments [15]. When it comes to the separation mode switch, Yu et al. [16] studied the separation mode switch in an over-expanded single expansion ramp nozzle. They found that the separation patterns changed between the restricted shock separation and the free shock separation during the startup process. The separation mode switch has never been discussed in an isolator, especially the effects of the separation mode switch on the flow characteristics. Actually, it is very important to know and even predict the separation mode switch in an isolator. In addition, the self-excited oscillation of a shock train is also a main problem in an isolator [17–19]. The self-excited oscillation means that the shock train keeps moving back and forth without any external excitations. The motion of the shock train is coupled with the pressure fluctuations which may generate noise or fluctuated wall loads. In previous studies, many researchers have studied this phenomenon. Yamane et al. [20] measured the pressure at several locations to determine the correlation coefficient and the coherence between them and they found that the upstream turbulent disturbance is the source of the high-frequency oscillation. Sugiyama et al. [21]

Corresponding author. E-mail addresses: [email protected] (B. Xiong), [email protected] (Z.-G. Wang). Science and Technology on Scramjet Laboratory, National University of Defense Technology, Changsha, People's Republic of China.

http://dx.doi.org/10.1016/j.actaastro.2017.01.009 Received 27 July 2016; Received in revised form 5 January 2017; Accepted 6 January 2017 Available online 09 January 2017 0094-5765/ © 2017 IAA. Published by Elsevier Ltd. All rights reserved.

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x T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11 T12

Ma3

56

optical access

C1

Ø200 B1 B2 B3 B4 B5 B6 B7 B8 B9 B10 B11 B12

203

x

L= 400 Fig. 1. Schematic diagram of the wind tunnel.

images are captured using a Photron Fastcam SA5 high-speed camera fitted with a 105 mm lens at a frame rate of 1000 Hz and a resolution of 768×512 pixels sensitivity. The light source used in our experiments is a constant one and the exposure time of the camera is chosen as 1/ 1000 s. Along with the Schlieren visualization, the high-frequency pressure measurements were adopted. The pressure measurements were made by 24 high-frequency transducers labeled with T1–T12 for the top wall and B1–B12 for the bottom wall. The 12 top transducers are located equidistantly along a line displaced by 10 mm from the median plane and the 12 bottom transducers are located equidistantly along the center line. These pressure transducers are all piezo-electric type. The measuring range of each transducer is 0–100 kpa, and the overload pressure is 200% F.S (full scale). The temperature shift below 0.05% F.S/K, and the comprehensive accuracy is ± 0.8% F.S. For the data acquisition, the anti-aliasing filter has been adopted to cut off the frequencies which are higher than 1/2 the sample frequency. The locations of the 24 transducers are presented in Table1. The pressure signals were all sampled at a rate of 5 kHz.

concluded that the source of the self-excited oscillation is in the shock train region. However, the mechanism behind the shock train oscillation remains still unclear. The main aim of this paper is to examine the separation mode switch in a rectangular duct and the effects it brings to the flow fields, such as the pressure distribution, the self-excited oscillation of shock train, and so on. In addition, the characteristics of shock train oscillation have been analyzed for two separation modes. It is hoped that the experimental phenomenon will have inspiration on the future research. 2. Experimental facility and data acquisition 2.1. Experimental facility The experiments presented in this paper are conducted in a continuous supersonic wind tunnel of NUDT (National University of Defense Technology) Scramjet Laboratory. The Schematic diagram of the wind tunnel is illustrated in Fig. 1. The atmospheric air passes through a 2-dimensional Mach 3 nozzle designed with viscous correction. The test section is a rectangular duct with the cross section 120 mm wide and 56 mm high. The total length of the test section is 400 mm. The reference location x=0 is defined as the inlet of the test section and x is the distance between the local position and the inlet of the duct, as shown in Fig. 1. In the side of the rectangular duct, the optical window is opened. It is shown in Fig. 1 as the dashed circle. In the downstream of the test section, a throttling device is equipped and the throttling effects are generated by a throttling valve, as presented C1 in Fig. 1. The valve is controlled by an actuating motor, thus leading that the throttling ratio and the turning speed is adjustable. The throttling ratio (TR) is defined as following:

TR = Ath / Aduct

3. Experimental results 3.1. The switch of flow separation mode In the internal duct flow, the structure of the shock train will become asymmetric when the incoming Mach number is high. That is to say, the large scale separation of the boundary layer occurs on one side of the duct (the top wall or the bottom wall), and the small scale separation occurs on the other side. The asymmetric flow separation phenomena in a symmetric duct have been observed by many researchers in experiments or CFD. In our experiments, the switch of flow separation mode has been observed and the effects bought by the separation mode switch will be analyzed. In one test case, the TR is set to change as the Fig. 2 shown and the pressure histories of transducer T10 and B10 are also presented in Fig. 2. The wall pressure is non-dimensioned by the reference pressure pref which is 3 kpa. The theoretical static pressure (isentropic flow) at the outlet of the Ma3 nozzle is 2.76 kpa for total pressure 1 atm. So we choose 3 kpa, which is close to 2.76 kpa, as the reference pressure. At about t=2 s, the downstream valve steps to an angle and keeps constant, thus leading the shock train forms in the duct. After t=2 s, the downstream throttling ratio keeps constant. However the shock train begins to self-excited oscillated in the duct, and the wall pressure thus fluctuates violently. It can be obviously seen that the mode of the pressure fluctuations changes at about t=2.850 s. During the phase 1, the pressure at the top wall T10 fluctuates more violently than the

(1)

where Ath means the throttling area caused by the Butterfly Valve and Aduct means the cross area of the test section. During each test, the throttling ratio (TR) can be set to any value between 0% (Butterfly Valve is parallel to the flow direction) and 52.14% (Butterfly Valve is perpendicular to the flow direction). The Butterfly Valve is controlled by a type of servo motor, which has a feedback system. Thank to the feedback system, the movement of the Butterfly Valve can be controlled precisely. Thus, the real-time throttling ratio is known. By keeping the constant throttling ratio, a steady back-pressure condition will be generated in the downstream of the test section and then the shock train can form in the duct. A huge vacuum container is equipped in the outlet of the wind tunnel. In our experiment, the stagnation pressure of the incoming flow is 1 atm ± 0.5 kpa, and the stagnation temperature is 288 K ± 1.5 K.

Table1 The location of 24 transducers at top wall and bottom wall.

2.2. Data collection Transducer x(mm) Transducer x(mm)

The structure of the shock train in the duct is captured by the highspeed Schlieren images. The window shown in Fig. 1 provides the optical access for Schlieren visualization. During the test, high-speed 159

T1,B1 44 T7,B7 224

T2,B2 74 T8,B8 254

T3,B3 104 T9,B9 284

T4,B4 134 T10,B10 314

T5,B5 164 T11,B11 344

T6,B6 194 T12,B12 374

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7

pT10 /pref

the separation bubble, the free interaction theory [22] and the shock polar [24,25] have been used. Zhukoski et al. [26] analyzed the turbulent boundary layer separation in front of a forward-facing step, and concluded the relation between pressure ratio p2/p1 and the incoming Mach number M1. The relation is shown as follows:

wind tunnel strats

6 5

separation mode switches

4 3

p2 =1 + 0. 5M1 p1

wind tunnel strats throttling ratio steps

2 1 7

wind tunnel strats

pB10 /pref

6 5

p2 2γ =1 + (M12sin2 β −1) p1 γ +1

separation mode switches

4 3

throttling ratio steps

2 0

1

2

3

t(s)

4

5

6

1/2 ⎛ γ +1 1 1 ⎞ sin β =⎜ + 2⎟ M1 ⎠ ⎝ 4γ M1

7

0

1

TR=0%

2

3

t(s)

4

5

6

7

Fig. 2. The non-dimensioned pressure histories at transducer T10 and B10.

bottom wall B10. During the phase 2, the phenomenon reversed, which results from the switch of the separation mode. Fig. 3 illustrates two instantaneous Schlieren images for the phase 1and phase 2 respectively. As shown in Fig. 3, the first bifurcated shock and a successive shock are captured clearly, and the bifurcated shock is formed in the asymmetric “X” type. The core flow goes more upwards in phase 1 and goes more downwards in phase 2, which results from the different scale of the separation bubbles at both sides of the duct. It can be seen that a large separation bubble Sl forms at the bottom wall and a small separation bubble Ss forms at the top wall during the phase 1. During the phase 2, this phenomenon reversed. Actually, the shock train keeps self-excited oscillating in the type of the structure shown in (a) during phase 1 and oscillating in the type of the structure shown in (b) during phase 2. In this paper, the separation mode shown in (a) is defined as “Bottom-Large-Separation (BLS)” and the separation mode shown in (b) is defined as “Top-Large-Separation (TLS)”. It should be made clear that the TLS mode (like Fig. 3(b)) lasted to the end of this experiment without returning to the BLS mode (like Fig. 3(a)). In our experiments, we have found three experimental phenomena, the separation mode switches from BLS to TLS (like as on Fig. 2), the separation mode switches from TLS to BLS (like as on Figs. 2 and 3) an d the separation mode does not switch. It seems that they are random phenomena. To theoretically analyze the geometry of the separated shock and

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354

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(3)

Therefore, the theoretical separated shock angle under Ma3 incoming condition is 30. 3°. The degree of the separated shock for the Schlieren images can be measured, and the results are presented in Table 2. It can be seen from Table 2 that, the separated shock at the top wall and the bottom wall has almost the same strength regardless of the separation mode. Moreover, the separated shock angle can be well predicted with the free interaction theory. In addition, the style of the shock reflection can be predicted using the shock polar. The deflected flow angle θ can be known using the following equation. Thus, the theoretical deflected angle is 13. 1°

TR=22.85% TR=0%

(2)

where β is the theoretical degree of the separated oblique shock. By solving the Eqs. (1′) and (2), the degree of the separated oblique shock is:

wind tunnel strats

phase 2

phase 1

throttling ratio

1

(1')

where p2 is the pressure in the separation bubble and p1 is the pressure upstream of the separated shock. The Rankine-Hugoniot relation is presented as follows:

tanθ =2cotβ

M12sin2 β −1 2 M1 (γ + cos2β )+2

(4)

Fig. 4 presents the shock polar for the incoming Mach 3 condition, the first deflected angle is 13. 1° as shown in Fig. 4. According to the shock polar and the previous research [24], the shock reflection is a regular reflection (RR), which is the same as the experimental results. However, the asymmetric shock structure cannot be predicted theoretically because that separation points at top wall and bottom wall locate at different streamwise position. Actually, the flow separation mode switched from BLS to TLS without any changes of the boundary conditions. In a rectangular duct, the separation mode switch induced by the changes of the backpressure condition has been observed in experiments and CFD. In the over-expanded supersonic nozzles, Papamoschou et al. [12] found that the large scale separation bubble may occur at the top wall or the bottom wall under the same back-pressure condition, which is very similar to the phenomenon observed in this test case. Essentially, the separation mode switch may come from the intrinsic instability of the separated flow. Actually, the separation mode switch may result in many other changes in the flow fields, such as the pressure fluctuations and the wall pressure distributions. Fig. 5 shows several instantaneous pressure distributions along the top wall and the bottom wall for both BLS and TLS. At t=1.5 s, the pressures along the top wall and the

394

214

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x(mm)

294

Su

314

Su

bifurcated shock Sl

(a) phase 1

successive shock

bifurcated shock

Sl

(b) phase 2 Fig. 3. the instantaneous shock train structures captured during phase 1 and phase 2.

160

334

354

374

394

successive shock

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smoothly along the top wall. Fig. 6 presents the pressure standard deviation (std) distribution along the top wall and the bottom wall for both BLS and TLS. In the BLS mode (Fig. 6(a)), the pressure standard deviation at the bottom wall is much smaller than the top wall in the downstream of the shock foot, which implies that the pressure fluctuates more violently at the small separation side. In the BLS mode, the transducer T7 and T8 lies in the top wall intermittent region (the separation shock foot oscillates back and forth over this region) and the transducer B6 and B7 lies in the bottom wall intermittent region. It can also be seen from Fig. 6(a) that the σT7 ,σT8 almost equals to σB6 ,σB7 respectively, which implies that the top separated shock foot and the bottom separated shock foot has almost the same strength. This phenomenon corresponds to the ‘free interaction theory’ [22]. In addition, the bottom pressure std keeps decreasing along the wall, and the top pressure std decreases firstly and then increases to a bigger value at transducer T12. According to the Schlieren results, the bigger value at T12 is induced by the second shock wave. In the TLS mode, the characteristics of the pressure standard deviation distribution reverse at the top and bottom wall. The separation mode switch is a common phenomenon in the inletisolator flow. It is important to predict the separation mode switch. According to Fig. 6, the difference between the pressure std at the top wall and the bottom wall is a possible method. Fig. 7 presents the history of the pressure std at transducer T10 and B10, and the std is computed with a moving time window 50 ms. When top wall pressure std is larger than the bottom wall, the separation mode is judged to be the BLS, and the reversed condition is judged to be the TLS. It can be judged from Fig. 7 that the flow separation switches at t=2.860 s, and the separation mode is BLS before 2.860 s and is TLS after that. In addition, there indeed exists the reverse transition from the TLS mode to the BLS mode. Because the changes brought by the reverse transition are very similar to the other transition, and the changes are just reverse. Thus, the reverse transition should not be discussed any more in this paper.

Table 2 The comparison between the theoretical and experimental results for the separated degree.

Phase 1 Phase 2

Separated shock angle

Schlieren images

Theoretical analysis

Errors

30. 3°

7.3%

βu

32. 7°

βd

34. 2°

11.4%

βu

34. 7°

12.6%

βd

32. 6°

7.1%

p2 / p1

18 16 14

top wall

bottom wall

12 10 8 6 4 2

13.1º -45

-30

0

-15

13.1º 0

15

θ

30

45

Fig. 4. The shock polar for the incoming Mach 3 condition.

bottom wall increase naturally thanks to the absence of shock train and there is no difference between the top wall pressure distribution and the bottom wall pressure distribution. Thanks to the second shock wave (see Fig. 3), the pressure fluctuates along the top wall in the downstream of the shock foot at t=2.4 s and 2.6 s. Conversely, the pressure increases smoothly along the bottom wall owing to the large separation bubble. The shock foot (the pressure rise point) locates at different positions at t=2.4 s and 2.6 s because of the self-excited oscillation. At t=3 s and 3.5 s, the separation mode has switched from BLS to TLS. As a result, the pressure fluctuates along the bottom wall and increases

3.2. Characteristics of shock train self-excited oscillation As what has been mentioned above, the shock train keep moving back and forth in the duct rather than keeping stable in a position. The effects of the asymmetric flow separation and the separation mode switch on the characteristics of self-excited oscillation will be analyzed. The power spectrum density for the wall pressure transducer can be obtained with Fast Fourier Transform (FFT) analysis method. Fig. 8

6

6

t=1.500s t=2.400s t=2.600s t=3.000s t=3.500s

5

4

ps/ppref

ps/pref

5

3

4

3

2

2

1

t=1.500s t=2.400s t=2.600s t=3.000s t=3.500s

0

0.2

0.4

x/L

0.6

0.8

1

1

0

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x/L

(b) Bottom wall

(a) Top wall

Fig. 5. The pressure distribution along the top wall and the bottom wall for both TLS and BLS.

161

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1

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0.8

top wall bottom wall

0.6

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0

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0.7

σp/pref

σp/pref

0.7

0.6

0.8

0

1

(a) phase 1

0

0.2

0.4

x/L

0.6

0.8

1

(b) phase 2 Fig. 6. The pressure standard deviation distribution at the top wall and the bottom wall for BLS and TLS.

upstream of the shock foot is not affected by the shock train oscillation, and it is induced by the incoming turbulent flow. The transducer T8 and B7 locates at the top wall and the bottom wall intermittent region respectively. The 80% percent accumulative frequency for T8 is 170.1 Hz and for B7 is 173.6 Hz, which are also very close. It implies that the oscillation of the small shock foot (at the top wall) and the oscillation of the large shock foot (at the bottom wall) are associated with each other rather than oscillating independently. However, the previous research on the unsteadiness of the shock/boundary layer interaction [23] show that the intermittent-region frequencies decrease as the magnitude of separation region increases. This conclusion is not applicable for the shock train oscillation. The transducer B10, B11 and B12 locates underneath the large separation bubble at the bottom wall. The FFT results show that the 80% percent accumulative frequency for B10, B11 and B12 are 342.2 Hz, 521.6 Hz and 547.4 Hz respectively. It shows that the more downstream the transducer locates in the large separation bubble, the more high frequency contents it has. The mechanism may be that the low-frequency disturbance induced by the upstream shock foot motions travels downstream and the frequency of the upstream disturbance is magnified by the large scale vortex in the separation bubble. At the top wall, the 80% percent accumulative frequency for T10 is 202.6 Hz, which is a little higher than the frequency of the shock foot oscillation. The upstream disturbance is magnified a little by the small separation bubble. At transducer T12, the 80% percent accumulative frequency is 231.0 Hz, which is much smaller than the frequency at B12. The reason is that the pressure fluctuation at transducer T12 is decided by the small separation bubble, so the frequency contents at T12 is very close to the frequency contents at T10. Similarly, Fig. 9 illustrates the FFT results for five top transducers and four bottom transducers for the TLS mode. The transducers T5 and B5 locate underneath the upstream turbulent boundary layer. Their pressure fluctuations are decided by the incoming flow and the pressure spectrums are very close to each other. The transducer T8-T10 locate underneath the large separation bubble in the top wall and the transducer B10 locate underneath the small separation bubble in the bottom wall. The transducer B12 locates in the reattaching flow downstream of the small separation bubble. The frequency at B12 (locates in the reattaching flow) is much smaller than the frequency at transducer T12 (locates in the large separation bubble), which is due to the absence of magnified effects of the large scale vortex. It can be seen from Fig. 3(b) and Fig. 5(b) that the transducer B12 locates near the second successive shock. Thus, the oscillation of the second shock may be affected by the

1

σp/pref

0.8

0.6

0.4

top wall bottom wall 0.2

0

2

2.5

3

t(s)

3.5

4

Fig. 7. The history of the pressure standard deviation at the top wall and the bottom wall. The point marked with star is judged as the time when the separation mode switches.

illustrates the FFT results for four top transducers and five bottom transducers. In the center of Fig. 8, the BLS in phase 1 is illustrated schematically. At first, the accumulative frequency percent p(fac) has been defined as follows: f

p ( fac ) =

∑0ac PSD ( f ) ∞

∑0 PSD ( f )

(2')

where PSD(f) is the power spectrum density for the frequency f content. Therefore, the parameter p ( fac ) means the percentage of the PSD which frequency contents below fac contain to the PSD that all frequency contents contain. The parameter p ( fac ) shows the weight of frequency contents which below fac . For example, the T5 FFT results show that fac (80%) is 219.8 Hz, which implies that the frequency contents which below 219.8 Hz has 80% percentage of the total frequency contents. In the BLS mode shown in Fig. 8, the transducersT5 and B5 locate in the upstream turbulent boundary layer and they are not affected by the shock foot motions. The FFT results show that the 80% percent accumulative frequency for T5 is 219.8 Hz and for B5 is 213.5 Hz, which are very close. It implies that the pressure fluctuation in the 162

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accumulative f percent 2000

1

0.8

0.6

0.4

0

1

0.8

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1

0.8

T8

T5 1500

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fac(80%)=170.1Hz

0.6

0.4

0.2

1

0

0.8

0.6

0.4

0.2

0

T10

T12

fac(80%)=202.6Hz

fac(80%)=231.0Hz

f(Hz)

fac(80%)=219.8Hz

accumulative f percent

accumulative f percent

accumulative f percent

0.2

1000

500

0

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power spectrum density [×103 ] power spectrum density[×105 ] power spectrum density [×10 4 ] power spectrum density [×10

T8

T5

5

]

T12

T10

separation bubble boundary layer B5

B7

B10

B11

B12

intermittent region accumulative f percent 2000

1

0.8

0.6

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1

B5

0.8

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1

B7

fac(80%)=213.5Hz

fac(80%)=173.6Hz

0.8

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1

accumulative ff ppercent 0.8

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accumulative f percent 0

1

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0.2

B10

B11

B12

fac(80%)=342.2Hz

fac(80%)=526.1Hz

fac(80%)=547.4Hz

0

f(Hz)

1500

accumulative f percent

accumulative f percent 0

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500

0 0.25

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power spectrum density

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4 4 [×103 ] power spectrum density[×105 ] power spectrum density [×104 ] power spectrum density [×10 ] power spectrum density [×10 ]

Fig. 8. The FFT results along the top wall and the bottom wall for the BLS separation mode.

scale of the separated shock foots are different, but they have almost the same strength. Secondly, the separation mode switch has great effects on the flow fields. In the BLS mode, the wall pressure fluctuates along the top wall and increases smoothly along the bottom wall, and the pressure standard deviation keeps decreasing along the bottom wall but decreases firstly and then increases to a bigger value at the top wall. In the TLS mode, the characteristics show the reversed phenomenon. Thirdly, the pressure fluctuation in the upstream of the shock foot is not affected by the shock train oscillation, and it is induced by the incoming turbulent flow. The low-frequency disturbance induced by the upstream shock foot motions can travel downstream and the frequency will be magnified by the separation bubble. Fourthly, the frequency of the shock foot oscillation is independent of the magnitude of separation region, and it implies that the oscillations of the two shock foots of a shock train are associated with each other.

small separation bubble. In the upstream of the shock foot, the pressure fluctuation is affected by the incoming turbulent flow. Therefore, it can be concluded that the self-excited oscillation of shock train is independent of the separation mode and it is may be a natural instability. 4. Conclusions In this paper, the separation mode switch in a rectangular duct and the self-excited oscillation of the shock train were experimentally studied. The effects of the separation mode switch on the flow fields have been analyzed. The conclusions can be summarized as follows: Firstly, the boundary layer has two separation modes in a rectangular duct under the incoming Mach 3.0 condition. That is the “Bottom-Large-Separation (BLS)” and the “Top-Large-Separation (TLS)”. The separation mode can switch in the unsteady process. The

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accumulative f percent 22000 1

0.8

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2000

1

T5

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2000

1

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power spectrum density [×10 ] 3

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]

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power spectrum density

T6

0

0 0.15

[×104 ]

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0

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power spectrum density [×105 ]

T10

T8

0.8

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0.4

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0

fac(80%)=547.6Hz

fac(80%)=403.6Hz

1000

0 0.25

1

T12

1500

fac(80%)=310.8Hz

fac(80%)=173.4Hz

2000

T10

1500

1500

fac(80%)=267.3Hz

accumulative f percent

accumulative f percent 0

T8

T6

1500

f(Hz)

accumulative f percent

accumulative f percent

0.2

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0

power spectrum density [×105 ]

T12

separation bubble

boundary layer B5

B7

B12

B10

intermittent region accumulative f percent 2000

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0.2

accumulative f percent 0

2000

B5

0.8

0.6

0.4

0.2

1500

fac(80%)=269.2Hz

1000

0.15

0.1

0.05

0

0.8

0.6

0.4

0.2

0

2000

1

0.8

0.6

0.4

0.2

0

B12 1500

fac(80%)=269.2Hz

fac(80%)=220.6Hz

1000

1000

f 500

0.2

1

1500

fac(80%)=190.3Hz

f 0 0.25

2000

B10

1000

500

accumulative f percent

accumulative f percent 0

B7

f(Hz)

1500

1

power spectrum density [×103 ]

500

500

0 0.7 0.6 0.5 0.4 0.3 0.2 0.1

0

0

power spectrum density [× 105 ]

1

0.8

0.6

0.4

0.2

0

0 0.25

power spectrum density [×10 4 ]

0.2

0.15

0.1

0.05

0

power spectrum density [×106 ]

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Acknowledgements The authors gratefully acknowledge the support of the National Natural Science Foundation of China (Grant nos. 11372347 and 11572347) for this work. References [1] W.H. Heiser, D.T. Pratt, Hypersonic Air Breathing Propulsion, AIAA, Washington, D.C, 1994. [2] P.J. Waltrup, F.S. Billig, Structure of shock waves in cylindrical ducts, AIAA J. 11 (10) (1973) 1404–1408. [3] H.J. Tan, S. Sun, Preliminary study of shock train in a curved variable-section diffuser, J. Propuls. Power 24 (2) (2008) 245–252. [4] W. Huang, Z.-G. Wang, M. Pourkashanian, et al., Numerical investigation on the shock wave transition in a three-dimensional scramjet isolator, Acta Astronaut. 68 (1) (2011) 1669–1675. [5] R. Kamali, S.M. Mousavi, A.R. Binesh, Three dimensional CFD investigation of shock train structure in a supersonic nozzle, Acta Astronaut. 116 (2015) 56–67. [6] W.Y. Su, K.Y. Zhang, Back-pressure effects on the hypersonic Inlet-isolator pseudo shock motions, J. Propuls. Power 29 (6) (2013) 1391–1399. [7] R.L. Klomparens, M. Gamba, J.F. Driscoll, Boundary layer separation in a 3D shock train, AIAA 2015–1519 (2015).

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