Schlieren measurements of shock train flow fields in a supersonic cylindrical isolator at Mach 2

Abstract

In a supersonic cylindrical isolator at Mach 2, the structures and frequency characteristics of shock train flow fields were experimentally studied by the schlieren measurement method. According to the design principle of parallel light through schlieren windows in a cylindrical duct, a high-precision conformal optical window pair was designed and integratively processed before. Based on a self-built pipeline structure with conformal windows in a direct-connect wind tunnel under adjustable back-pressure conditions, the shock surfaces in a cylindrical isolator at Mach 2 were first captured by the schlieren method. Then, the schlieren photographs were corrected by a nonlinear image transformation algorithm for the restoration of real shock train structures, and the experimental results were compared with numerical simulation results quantitatively. Finally, the shock train positions were calculated by an image recognition algorithm to analyze the self-excited oscillation frequency characteristics of shock train structures. The methods and experiments in this study enriched optical observation methods of supersonic flows through non-rectangular cross-section isolators in scramjet.

Graphical abstract

1 Introduction

Hypersonic flight is an inevitable trend in the aerospace development and one of the military and strategic goals of most countries in the world. As the most ideal power source and key technology of hypersonic vehicles, the scramjet engine has been widely focused on by many international researchers in recent decades (Tanno and Tanno 2021; Zhao et al. 2022). The main structure of a scramjet includes four parts: an inlet, an isolator, a combustion chamber, and an exhaust nozzle (Kim et al. 2022; Sam et al. 2023). Different from the other air-breathing engines, the isolator is a core component unique to the scramjet, which is between the inlet and the combustion chamber. On the one hand, the scramjet isolator can isolate the inlet from the high back-pressure of the combustion chamber. On the other hand, the scramjet isolator can stabilize the pre-combustion shock train for the normal operation of the inlet and highly efficient combustion in the combustion chamber (Vanstone et al. 2018; Hou et al. 2020). In the actual operation of hypersonic flight vehicles flying at Mach number ranging from 5 to 6, the inflow velocity of scramjet is reduced to Mach 2–2.5 by the compression inlet. Actual wind tunnel tests can reproduce the complex flow field structures in isolators by controlling upstream and downstream flow field conditions.

The conventional structures of the isolator include a constant area or micro-expansion pipe with a rectangular or circular cross-section. Though the configurations of the isolator are relatively simple, the flow field structures in them are very complex. When the supersonic incoming flow conditions are fixed and the high back-pressure is generated in the combustion chamber, flow separation occurs in the isolator and a shock wave/boundary layer interference phenomenon occurs, which forms the typical "shock train" structure (Deng et al. 2017). The typical shapes of shock train (Matsu et al. 1999) can be classified as x-type (also known as oblique shock train) and λ-type (also known as normal shock train). The shock train structures are mainly affected by the upstream Mach number and the boundary layer thickness based on the fixed duct configuration. Meanwhile, the positions of shock train can be detected by monitoring the turbulent nature and boundary layer unsteadiness (Li et al. 2018), and need to be controlled to avoid the unstart phenomenon. Meanwhile, the self-excited oscillation of shock train can be detected by monitoring the changes of overall shock train structures in position, which is one of the most important unsteady characteristics of shock wave–boundary layer interactions (Xiong et al. 2017).

Currently, for the research of complex flow fields in scramjet isolators by ground wind tunnel tests, the corresponding flow field conditions need to be realized by experimental facilities. At the entrance of an isolator, the designed Laval nozzle can provide incoming flow conditions. While, comparing to the actual thermal choke caused by the high back-pressure from the combustion chamber, the back-pressure conditions can be provided by the mechanical choke from adjustable throttling devices downstream of the isolator.

Based on the target flow field generated in isolators, appropriate detection methods should adapt to specific isolator configurations. Compared with contact detection methods using sensors to measure flow field parameters in discrete measuring point locations, non-contact optical detection methods can record spatiotemporal variation information of complex flow fields by optical measurement methods. Among numerous non-contact optical detection methods, the schlieren measurement technique (Vaisakh and Muruganandam 2020; Shigeta et al. 2022; KONG et al. 2023) is the most popular technique. The schlieren method uses a schlieren apparatus to record the deflection angles of the parallel light caused by the inhomogeneity of flow field along the light path at set cutoff rates. Then, the wave structures and high-frequency characteristics of shock train in the isolator can be extracted from large numbers of schlieren images taken by a high-speed camera. As a qualitative to quasi-quantitative measurement technique (Settle 2001), during schlieren measurements of complex flow fields inside wind tunnel ducts, high-quality schlieren optical windows become core optical components within the measuring light path.

Up to now, based on the manufacturing technique and application convenience of schlieren observation windows in isolators, the schlieren measurement technique has been only limited in the detection of constant area or micro-expansion rectangular isolators (Weiss et al. 2010; Wang et al. 2021), simply using plane windows on both sides of the duct. Plane windows not only ensure that the windows’ internal faces fit the flow field configuration well, but also ensure that the parallel rays pass through the bilateral optical windows without deflection. This greatly reduces the design and manufacturing difficulty of high-precision schlieren optical windows.

However, as the conventional configuration in scramjets, the cylindrical isolator plays an important role in engineering application, which has better connectivity to the cylindrical combustion chamber, greater ability to carry back-pressure and lighter weight than the rectangular cross-section isolator (Lin et al. 1993). Currently, as one of the earliest studied structures, the flow field characteristics in the cylindrical isolator could only been researched by numerical simulation methods (Lin et al. 2006) and contact detection methods (Wang et al. 2022).

Over the past several decades, the circular cross-section ducts have attracted a lot of research on internal flow fields by optical measurement methods based on the transmitted parallel light. In the field of engine visualization, Kaiser et al. (2013) set a pair of cylindrical positive meniscus lenses outside of a concentric cylindrical pipe to perform schlieren measurements through the combustion engine. Fujikawa et al. (1994) and Kozuka et al. (2003) used a transparent collimating cylinder (TCC) for schlieren measurements through an internal combustion engine. Comparing to the scheme of discrete elements with corrected meniscus cylindrical lenses (Kaiser et al. 2013), the scheme with integrated windows (Fujikawa et al. 1994; Kozuka et al. 2003) improved the compactness of the schlieren measurement light path and reduced the number of optical components for assembling, but increased the processing difficulty. In the field of supersonic flow visualization, Hosseini and Takayama (2010) used a fully transparent tube at the end of a diaphragmless shock tube for the study of toroidal shock wave by the double-exposure holographic interferometry. Takayama (1983) conducted experimental studies of the shock wave propagation and diffraction past a circular cross-sectional 90° bend by a double-exposure holographic interferometer. While, based on the measurement principle of the double-exposure holographic interferometry, the valid information can be extracted from the adjacent two exposures, and the image distortions caused by the inaccuracy in manufacturing of optical windows could be avoided.

Hence, for schlieren measurements through circular cross-section ducts, high-precision schlieren windows are the basis of a measurement system, which can not only keep the inner duct shape but also provide a clean measurement background. Meanwhile, as consumables in wind tunnel tests and core optical components in the measurement light path, conformal optical windows should also have characteristics of rapid manufacturing and convenient application.

Until now, no experimental study on the shock train structure in cylindrical isolators by schlieren measurements for a scramjet background has been conducted. The macroscopic characteristics of shock train in supersonic circular ducts have been studied by many researchers (Om and Childs 1985), and the characteristics of shock train were clarified (Waltrup and Billig 1973). However, the spatiotemporal variation image information consisting of macroscopic structures and characteristics of the shock train in supersonic cylindrical isolators have not been observed and fully researched.

Based on the research gaps above, the main contributions of this study were as follows. First, based on the design principles of schlieren windows in non-rectangular cross-section ducts, a high-precision conformal optical window pair of a cylindrical isolator was integratively designed and processed before. Second, based on the experimental platform of a direct-connect wind tunnel, the pipeline configuration was built, and the function of each component was displayed. Third, the flow fields in the cylindrical isolator at Mach 2 were numerically calculated. Fourth, the schlieren photographs were corrected by a nonlinear transformation algorithm based on the geometric optical structure of conformal optical windows for the restoration of real shock train structures inside the cylindrical isolator. Fifth, the shock train structures in a cylindrical isolator at Mach 2 were photographed by the schlieren measurement technique for the first time, and the quantitative comparison analyses were conducted. Lastly, the positions of shock train were calculated by the image recognition algorithm for the self-excited oscillation frequency analysis.

2 Experimental setup and numerical method

2.1 Design and manufacture of conformal optical windows

As shown in Fig. 1, during schlieren measurements through ducts with non-rectangular cross-sections, the inner flow channel S0 is a complex surface. The core design and manufacturing principles of conformal optical windows can be divided into three parts:

Fig. 1

a Measured parallel optical path through the complex flow field S0 and b the ray passing through the micro-surface around the emergent point A₃

2.1.1 Flow field conformability of optical windows

During the schlieren measurements through ducts, the primary goal is to keep the inner surfaces S3 conformal with the actual flow channel shape S0, which can retain the actual flow field characteristics to the greatest extent. Hence, the designed inner surface should adopt the same surface shape as the flow channel. While, after the ultra-precision machining, the manufacturing errors and installation errors will make the inner surfaces and the target flow channel fit imperfectly, and the system errors should be strictly controlled within the tolerance range.

2.1.2 Transmitted waverfront error of zero

In the measuring parallel light path, the actual schlieren measurement effect would be disturbed by conformal optical windows. Based on the high-precision machined optical window, the transmitted wavefront error becomes the core optical index through the measuring light path which determines the initial light intensity distribution without the flow field interference on a high-speed camera. When the Ray₁ passes through the inner surface S3 and outer surface S4 in turn, there exists two refraction processes.

The first refraction process takes place at the point A₂. Based on the designed inner surface S3, the normal vector $\overline{n}_{1} = (n_{1x} ,n_{1y} ,n_{1z} )$ at the incident point A₂ can be obtained easily. Then, the refracted ray travels the distance of A₂A₃ in the conformal optical window. The second refraction process takes place at the point A₃. When the micro-surface P₃ around the point A₃ has been designed, the normal vector $\overline{n}_{3} = (n_{3x} ,n_{3y} ,n_{3z} )$ decides the direction of emergent rays. Then, based on the equal optical path principle, in order to achieve the target of zero transmitted wavefront error, the optical path value of different rays from the wavefront O₁A₁ to wavefront O₄A₄ should remain the same in Eq. (1).

$$A_{1} A_{2} + A_{2} A_{3} \cdot n_{P} + A_{3} A_{4} = O_{1} O_{2} + O_{2} O_{3} \cdot n_{P} + O_{3} O_{4}$$

(1)

By enumerating rays within the incident light region, the corresponding micro-surfaces around the emergent points can construct the whole outer surface.

2.1.3 Transmitted light's deflection angle of zero

The zero transmitted wavefront error can only decide the wavefront value change between the incident wavefront and the emergent wavefront, but the direction of the emergent beam cannot be determined. During the actual schlieren measurement process, the parallel light passes from the collimating mirror to the convergent mirror along the straight light path. When the conformal optical windows are set in the middle of the schlieren apparatus, the transmitted beam region may be distorted, but the direction of the emergent rays should be parallel to the incident rays in Eq. (2).

$$\begin{gathered} A_{1} A_{2} \parallel A_{3} A_{4} \hfill \\ O_{1} O_{2} \parallel O_{3} O_{4} \hfill \\ \end{gathered}$$

(2)

When the designed flow channel S0 changes from the complex flow channel to the cylinder which has the same cross-section shape along the flow channel, the three-dimensional design method of the conformal optical window can be simplified to two-dimensional design method in Fig. 2. According to the work of conformal optical windows before (Ou et al. 2022), based on the cylindrical isolator with the radius (R₁) of 50 mm, the thickness (t₀) of 30 mm, and the observation width (L₁) of the parallel rays inside the internal pipe of 50 mm, the corresponding conformal optical window is designed according to the three principles above in Table 1, whose transmitted wavefront error (peak-to-valley (PV)) was 0.0126λ, and the transmitted wavefront error (root mean square (RMS)) was 0.0027λ when the diameter of the incident light on the surface S3 is 50 mm, which meets the requirements of the Maréchal criterion (Maréchal, 1947).

Fig. 2

Structure diagram of conformal optical windows of a circular cross-section duct

Table 1 Parameters of unilateral conformal optical window

Based on the design parameters of acrylic conformal optical windows, for the sake of rapid processing and free alignment, an integrative manufacturing scheme of the conformal optical window pair was creatively proposed before. After the ultra-precision machining of the inner and outer surfaces of the conformal optical windows, the finished model is shown in Fig. 3a. The diameter of inner surfaces in the optical window region is 99.995 mm and the inner diameter of the aluminum alloy pipe is 100.028 mm measured by the coordinate measuring machine (CMM), and the difference between unilateral radius values is 16.5 μm meeting the high-precision conformal requirements. Within the interferometric measurement range of 84.89 mm × 46.24 mm, the transmitted wavefront error (PV value and RMS value) of bilateral optical windows was 12.189λ and 2.658λ in Fig. 3b, respectively.

Fig. 3

a Finished model of the cylindrical isolator with conformal optical windows and b interferometric measurement results

The integrated processing cylindrical isolator with conformal optical windows not only omits the assembling process of conformal optical windows, but also provides the interface for mounting from the ultra-precision machine tool to the wind tunnel pipes, which meets the requirements for rapid manufacturing and ease of use. Meanwhile, the high-precision conformal optical windows with transmitted wavefront error at wavelength level can meet the requirements of high accuracy schlieren measurements of wave structures in the cylindrical isolator. While, based on the fixed internal cylindrical pipe configuration, as the observation area through the internal duct increases, the optical region of the conformal optical window's outer surface enlarges and the surface shape deviation of the outer aspheric cylinder from the best-fitting cylinder increases. Then, the large surface shape deviation causes the loss of machining accuracy and may result in the workpiece finishing exceeding the working stroke of the fast tool servo. By considering the processing difficulty and observation area through the internal duct comprehensively, the width L₁ of the parallel beam through the internal duct is limited in 50 mm (half of the duct diameter), which can already provide the visualization area for researches of shock train structures and positons in subsequent Sects. 3.2 and 3.3.

2.2 Experimental setup

The tests were conducted in a supersonic air-breathing wind tunnel of the National University of Defense Technology’s Scramjet Laboratory. Based on the designed pipe configuration, the shock train structures in the cylindrical isolator were first recorded by the schlieren measurement technique. As shown in the schematic diagram in Fig. 4a, the main components of the experiment were the nozzle section, test section, divergent section, throttling section, and chamber. Figure 4b shows the experimental photograph of the wind tunnel setup. The incoming flow passed through the ①airflow collection section, ②stabilization section, ③connecting section, ④nozzle section, ⑤test section, ⑥divergent section, and ⑦throttling section in turn, and it finally reached the ⑨vacuum chamber through the ⑧connecting pipes. The ⑩schlieren measurement system was set on both sides of the isolator, and all the pipeline configurations were fixed on the ⑪experimental platform.

Fig. 4

a Schematic diagram and b experimental photograph of the wind tunnel setup

The upstream component of the test section was a supersonic nozzle with a Mach number of 2. The diameter and length of the test section were 100 mm and 140 mm, respectively. The observation areas on the outer surface and inner surface were 60 mm × 96 mm and 50 mm × 96 mm, respectively. In consideration of the sensitivity of the window surface accuracy to the punched holes of the transducers, the high-frequency pressure detectors (sampling frequency: 10 kHz, range: 0 ~ 200 kPa, and accuracy: 0.25% F.S) were installed on the upper wall of the divergent section to monitor the wind tunnel starting state and wall pressure variation characteristics. The stagnation pressure of the incoming flow is 101.3 kPa ± 0.5 kPa, and the stagnation temperature is 300 K ± 2 K.

In order to change the positions of shock train in the whole duct, the drive motor and the computer outside of the throttling section controlled the different rotation angles of the throttling plate for adjusting throttling ratios. The schematic diagram and physical picture of the throttling plate are shown in Fig. 5a and b. By extracting the high-frequency pressure data in the divergent section, the locations of shock train can be detected first. Then, the subsequent angle adjustment of the throttling plate can be performed for generating the back-pressure conditions by the mechanical choke.

Fig. 5

a Schematic diagram and b physical picture of the throttling plate

When the integratively processed cylindrical isolator with conformal optical windows is installed in the wind tunnel pipeline, the physical picture and cross-section diagram of the test section are shown in Fig. 6a and b, respectively. The integrated cylindrical isolator can be connected to the upstream and downstream pipes by the flange structures on both ends of the pipe, and bilateral conformal windows need to be assembled correctly with the measured parallel light passing through the schlieren apparatus.

Fig. 6

a Physical picture and b cross-section diagram of the test section

Based on the constructed pipeline in the wind tunnel, a z-type schlieren apparatus, whose schematic diagram is shown in Fig. 7, can take photos of shock surfaces in the cylindrical isolator. The mirror M1 and M2 had same parameters, whose focal length and diameter were 2 m and 200 mm. The halogen light source is placed at the position S. The model of the high-speed camera was FASTCAM SA5 (Company: Photron) whose sensor is 12-bit ADC (Bayer system color, single sensor) with 20-μm pixel. During the course of photographing, the focal length of the camera lens was set to 200 mm. The frame rate of the camera, whose resolution was 1024 × 1024, was set to 1000 fps with an exposure time of 1/3000 s. The rays emanating from the light source S were reflected by the plane mirror P1 and collimating spherical mirror M1. Then, the rays were incident into the cylindrical isolator in the state of parallel light. After the rays entered the internal flow channel, the density change of the flow field in the test region can be recorded by parallel rays. Then, the rays passed through the opposite conformal optical window and were reflected by the converging mirror M2 and plane mirror P2 successively. Then, the rays were cut by the vertical knife-edge at a cutoff rate of 50% and finally entered the high-speed camera.

Fig. 7

Schematic diagram of z-type schlieren apparatus

2.3 Numerical method and code validation

Computational fluid dynamics (CFD) simulations of the integral structures of the nozzle section, cylindrical isolator, and divergent section were conducted using Fluent. The three-dimensional Reynolds-averaged Navier–Stokes (RANS) equations were discretized by the finite volume method using a density-based methodology and were adopted to numerically calculate the flow field structures in the cylindrical isolator. The k-ω SST turbulence model was used in the simulation. The flux type was set as Roe-averaged flux difference splitting (Roe-FDS), and the two-order upwind scheme was adopted for the spatial discretization. To ensure the convergence of the simulation data, each residual and the mass flow ratio between the inlet and the outlet surfaces were monitored during the computational process. The incoming flow was set to the ideal gas with the pressure-far-field boundary condition at the inlet of the nozzle. At the outlet of the divergent section, the pressure-outlet condition was adopted with the back-pressure, which was similar to the actual wind tunnel conditions. The three-dimensional structured grid was used in the simulation with mesh refinement on the wall, whose bottom grid height was 0.01 mm with equal-ratio encryption. Correspondingly, the wall y + value of the bottom grid near wall was less than 5. Adiabatic and no-slip boundary conditions were enforced at the walls.

In order to confirm the accuracy of the CFD method used in this article, the mesh independency was analyzed first. The basic flow field geometry is generated, and three different mesh scales are compared which include a coarse mesh (grid length: 4 mm, number of grids: 522,240), a moderate mesh (grid length: 2 mm, number of grids: 1,044,480), and a refined mesh (grid length: 1 mm, number of grids: 2,085,120) in Fig. 8a. Then, three meshes were calculated with the same CFD method, and the pressure distributions along the bottom wall in different mesh scales are shown in Fig. 8b. From the simulation results, it can be seen that there is a slight difference between the moderate mesh and refined mesh. In order to improve computational efficiency, the moderate mesh is chosen to calculate the flow field in this work.

Fig. 8

a Three mesh scales for display and b pressure distribution along the bottom wall

To verify the numerical method, the study of structures of shock waves in straight rectangular ducts (Kawatsu et al. 2005) was chose for the comparative analysis, whose height H of the test model is 30 mm, and stagnation pressure P₀ of the inlet is 100 kPa. The comparison results of wall pressure distributions between the numerical and experimental results are shown in Fig. 9. As the results show, the wall pressure distribution calculated by the computational method was almost equal to the experimental results.

Fig. 9

Comparison results of wall pressure distributions between the numerical and experimental results

3 Experimental result and analysis

3.1 Nonlinear image distortion correction

Starting from the schlieren measurement theory and finished high-precision conformal optical windows presented in Sect. 2.1, during the schlieren imaging process shown in Fig. 10, the high-speed camera focuses on the surface S which is in the middle of the cylindrical isolator. The surface S is placed at the location outside of one focal length $f_{2}$, and the distance between the surface S and the mirror M2 is $d + f_{2}$.

Fig. 10

Schematic diagram of the schlieren imaging process through unilateral conformal optical window

Under the assumption of no conformal window, the amplification factor of the image on the surface S through the mirror M2 can be calculated in Eq. (3) (Settle 2001).

$$m_{1} = - \frac{{f_{2} }}{d}$$

(3)

Due to the existence of the conformal optical window which has an axisymmetric structure, there is corresponding amplification factor through unilateral window at different Y-direction positions in Eq. (4).

$$m_{2} = f(y) = \frac{{L_{in} (y)}}{{L_{out} (y)}}$$

(4)

where $L_{in} (y)$ is the distance between the incident ray and the optical axis, and the $L_{out} (y)$ is the distance between the emergent ray and the optical axis.

In the actual schlieren apparatus, the distance $d + f_{2}$ is much larger than the sum of the radius of flow channel and the thickness of unilateral window. Therefore, the total magnification factor can be changed to $m$ in Eq. (5).

$$m = m_{1} \cdot m_{2} = - \frac{{f_{2} }}{d} \cdot \frac{{L_{{{\text{in}}}} (y)}}{{L_{{{\text{out}}}} (y)}}$$

(5)

Based on the known parameters of mirror M2 and measured distance between the surface S and the mirror M2, in order to get the total magnification value $m$, the key point is to derive the value of amplification factor $m_{2}$. As the parallel rays passing through unilateral conformal optical window in Fig. 11, the marginal ray Ray₁ from the incident wavefront W₁ passes through the points A₁, A₂, A₃, and A₄ in turn and finally reaches the emergent wavefront W₂. The Ray₁ refracts twice through the conformal optical window, which occurs at the points A₂ and A₃.

Fig. 11

Schematic diagram of parallel rays passing through unilateral conformal optical window

Depending on Snell’s law and the theory of equal optical path, Eq. (6) and Eq. (7) can be derived (Kozuka et al. 2003).

$$\sin \theta_{1} = n_{P} \cdot \sin (\theta_{1} - \theta_{2} )$$

(6)

$$t(n_{P} - \cos \theta_{2} ) = t_{0} (n_{P} - 1)$$

(7)

where $n_{P}$ is the refractive index of the acrylic material, $t$ is the distance of the marginal ray Ray₁ traveling in the conformal optical window, and $t_{0}$ is the thickness of the conformal optical window.

By numerical calculation of incident parallel rays with different widths L_in, as shown in Fig. 12a which reveals a nonlinear proportional relationship, the amplification factor $m_{2}$ with different incident light widths $L_{in}$ can be calculated according to Eqs. (6) and (7) within the maximum range of 25 mm. The nonlinear proportional relationship has a great influence on the parameter calculation of real flow field structures in schlieren photographs, especially during quantitative schlieren measurements. Hence, the actual flow field structures can be restored from the original schlieren photograph within the visualization region by inverse transformation of the amplification factor $f^{ - 1} (y)$ in Fig. 12b, which corresponds to a nonlinear image distortion correction process.

Fig. 12

a Amplification factor with different incident beam widths L_in and b the process diagram of the nonlinear distortion correction

To verify the nonlinear image distortion correction process through conformal optical windows, during the construction of the wind tunnel pipeline, a weak positive lens with ultra-long focal length (model: PLCX-25.4–5151.0-C, focal length: 10 m, diameter: 25.4 mm) was placed at the central section of the cylindrical isolator in Fig. 13a. The weak positive lens can be regarded as a "calibration lens" (Hargather et al. 2012), which can play a role of automatic calibration in schlieren apparatus along the parallel light for quantitative studies. When the parallel light passes through the calibration lens in Fig. 13b, the deflection angles $\varepsilon_{r}$ of incident rays with different heights $r$ can be derived in Eq. (8).

$$\varepsilon_{r} = \arctan (\frac{r}{f}) \approx \frac{r}{f}$$

(8)

Fig. 13

a Calibration lens placed at the central section of the cylindrical isolator and b parallel rays passing through the calibration lens

For 50% cutoff rate by the knife-edge, the schlieren photograph within the observation region through the cylindrical isolator is shown in Fig. 14a. Then, the grayscale photograph is shown in Fig. 14b by converting from the schlieren photograph. By applying the nonlinear distortion correction method above to the original photograph, the image corrected along the Y-direction is shown in Fig. 14c. The edge of the corrected calibration lens is extracted by image processing method, and the calculated roundness value is 0.91 in Fig. 14d, which confirms the correctness of the nonlinear distortion correction algorithm. While, the roundness difference between the calibration lens and a standard circle (roundness value of 1) is mainly caused by the laying angle of the calibration lens and the pixel density of schlieren images.

Fig. 14

a Schlieren photograph, b grayscale photograph, c nonlinear distortion correction grayscale photograph of the calibration lens, and d roundness calculation result of the calibration lens region

Meanwhile, to verify the quantitative imaging effect of the calibration lens between the conformal optical windows, the grayscale intensity through the center line of the lens (blue dotted line in Fig. 14c) is extracted. The distribution of the grayscale value is shown in Fig. 15. Within the region of the calibration lens, the grayscale value shows a linear growth trend. This fully illustrates the feasibility of conformal optical windows for high-precision schlieren measurements without interference.

Fig. 15

Grayscale value distribution along the center line of the calibration lens

3.2 Schlieren measurement result and comparison analysis

According to the working condition at Ma = 2 of the incoming flow, the instantaneous schlieren photograph is captured in Fig. 16 with a steady back-pressure. When the throttling plate rotates to a fixed angle, the back-pressure is produced with the throttling ratio $R$ of 19.65% defined as follows

$$R = S_{t} /S_{d}$$

(9)

where $S_{t}$ indicates the throttling area connected with the butterfly valve, and $S_{d}$ indicates the cross area of the test section.

Fig. 16

Schlieren photograph of the shock train structure in the cylindrical isolator

The shock train structures are formed within the optical observation region of the cylindrical isolator with the flow from left to right in Fig. 16. Each pixel corresponds to 0.2 mm in the schlieren photograph. The boundary of the visualization area is set as the reference plane (Z = 0) shown in Fig. 4a. The first shock on the left is a bifurcated shock with an axisymmetric structure. In the middle of the isolator, there is a distinct Mach disk structure, which means that the cylindrical isolator also exhibits Mach reflection with the incoming flow at Mach 2. The second shock is a successive shock which locates downstream of the first shock, which is similar to the shock train structure in the rectangular isolator with the typical form of a normal shock wave (Lin et al. 1991).

For subsequent comparative analysis of shock train structures quantitatively, the schlieren photograph is transformed into a grayscale image and processed by the nonlinear distortion correction algorithm shown in Fig. 17.

Fig. 17

Distortion-corrected grayscale picture of the shock train structure in the cylindrical isolator

Starting from the theoretical analysis, in a study on two-dimensional rectangular isolators, Chapman et al. applied the shock wave/boundary layer "free interaction theory" to predict the separation form of shock train (Chapman et al. 1958), which is also used to try to explain the flow field in the cylindrical isolator in this section. On the basis of the study on the step flow separation region of the turbulent boundary layer in the two-dimensional flow structure by Zukoski (1967) and the Rankine–Hugoniot relation, the relationship between the pressure of the flow field separation zone and the Mach number in the mainstream (Xiong 2016) can be described as Eq. (10).

$$\sin \beta = \sqrt {\frac{\gamma + 1}{{4\gamma }}\frac{1}{Ma} + \frac{1}{{Ma^{2} }}}$$

(10)

where $\beta$ is the angle of the separated shock, $Ma$ is the mainstream Mach number in front of shock train, and $\gamma$ is the ratio of the specific heats. When $Ma$ = 2 and $\gamma$ = 1.4, the separated shock angle calculated by the theoretical formula is 43°, which can provide a theoretical reference for subsequent simulation results in Table 2.

Table 2 Quantitative comparative analysis results

Meanwhile, according to the simulation method in Sect. 2.3, the numerical schlieren results of the YZ cross-section and the XZ cross-section through the central axis of the cylindrical isolator are shown in Fig. 18a and b, respectively. The simulation results of the two cross-sections are similar, which means that the shock train structures in the cylindrical isolator are centrosymmetric. Meanwhile, the regions of white dotted boxes in each image are actual observation areas through the cylindrical isolator.

Fig. 18

Numerical schlieren results of the a YZ cross-section (iX = 0 mm) and b XZ cross-section (Y = 0 mm)

In conclusion, the quantitative comparisons among the original schlieren photograph in Fig. 16, the distortion-corrected schlieren photograph in Fig. 17 and the numerical schlieren result of the middle cross-section in Fig. 18a is displayed in Table 2. Table 2 shows the separated shock angles (β_u, β_d) on the upper and the bottom walls, the widths of the Mach disk (P₁P₂), and the distances between the first and second shock waves (O₁O₂). The deviations between the numerical schlieren result and original schlieren photograph result are shown in column of Deviation#1, and the deviations between the numerical schlieren result and distortion-corrected schlieren photograph result are shown in column of Deviation#2.

By comparing the results in column of Deviation#1 to Deviation#2, there is a significant reduction in deviations of the wall shock angles between the numerical simulation results and schlieren photograph results by distortion correction, which means that the distortion caused by the geometric structures of conformal optical windows has great influence on the quantitative information extraction of shock train structures. Meanwhile, due to the axisymmetric structure of conformal optical windows in the cylindrical isolator, the distortion occurs only in the Y-axis direction in Fig. 16, which would not affect the structure parameters in the X-axis direction such as the length of O₁O₂.

In order to analyze the deviations in the column of Deviation#2 and the comparison results of the Mach disk width P₁P₂ in more detail, the three-dimensional characteristics of shock train structures were displayed in the numerical simulation results, setting the YZ cross-section (X = 0 mm) as the reference location. The numerical schlieren results of different YZ cross-sections are shown in Fig. 19a–(h) within the same observation region, which correspond to the cross-section locations of X = 5, − 5, 10, − 10, 15, − 15, 20, and − 20 mm.

Fig. 19

Numerical schlieren images of different YZ cross-sections

The primary reason for the deviations between the simulations and experiments is that the schlieren photograph result was the overall effect of the superposition of the three-dimensional flow field structure along the light path.

From the simulation results in Fig. 19, except for the flow field in the central area of the pipe, the Mach reflection structures in the other regions also have strong three-dimensional structural properties, which have a significant impact on the schlieren photographs. As the distance between the cross-section and the central axis increases, the shock wave structure shows the Mach reflection phenomenon. Meanwhile, the corresponding Mach disk structure gradually disappears, and the width of the Mask disk gradually decreases. The actual Mach disk structure is mainly contributed by the flow field between the cross-sections of X =  ± 10 mm and the widest Mach disk structure appears on the YZ cross-section (X = 0 mm). Therefore, as the width of the Mach disk structure gradually reduces along the radial direction in the numerical schlieren results, the overall Mach disk width in the schlieren photograph was smaller than the width value in the middle cross-section. Meanwhile, the length P₁P₂ measured in Fig. 16 is greater than the length value in Fig. 17 is mainly caused by the stretch distortion in the Y-axis direction leading to an illusion that the length P₁P₂ in Fig. 16 corresponds to a smaller deviation in Deviation#1 column.

With the increase in the distance of the YZ cross-section from the central cross-section, the corresponding separated shock angles on the upper and bottom walls increases in Fig. 19. Due to the superposition effect of the three-dimensional flow field structure, the real separated shock angles in the distortion-corrected schlieren photograph are larger than the simulation results in the YZ cross-section. As for the deviation between the separated shock angles in the schlieren photograph in Fig. 17 and the theoretically calculated value of 43°, this mainly results from the error of theoretical prediction based on simplified formula of "free interaction theory" above.

Moreover, the length O₁O₂ between the bifurcated shock and successive shock was related to the actual downstream condition which cannot be accurately simulated, leading to the deviation between the CFD simulation result and schlieren photograph results.

3.3 Self-excited oscillation characteristic of shock train

Based on the same test condition in Sect. 3.2, the shock train oscillated in a small amplitude and the Mach stem of the bifurcated shock can be observed within the observation region of conformal optical windows.

By converting schlieren photographs to grayscale pictures and applying the nonlinear distortion correction to binary images, the obtained images can reflect actual flow field structures inside the cylindrical isolator. Then, through the self-compiled image recognition algorithm for the Mach stem structure of the bifurcated shock, the locations and self-excited oscillation frequencies of shock train can be calculated.

The main steps of the algorithm are as follows:

(1) By extracting the schlieren images as the background of calculation when the wind tunnel is not running, the corresponding average grayscale image matrix $\overline{P}[i,j]$ can be calculated as Eq. (11).

$$\overline{P}[i,j] = \frac{1}{N}\sum\limits_{x = 1}^{N} {P_{x} [i,j]}$$

(11)

where $P_{x} [i,j]$ is the grayscale value matrix of each image, and $N$, $i$, and $j$ are the number of background images, row number, and column number of each background image, respectively. The maximum numbers of rows and columns are $I$ and $J$.

Then, the matrix difference of absolute values between the average grayscale image matrix $\overline{P}[i,j]$ and each background grayscale image matrix $P_{x} [i,j]$ can be described as Eq. (12).

$${\text{d}}P_{x} [i,j] = \left| {P_{x} [i,j] - \overline{P}[i,j]} \right|$$

(12)

(2) Based on the difference matrix $dP_{x} [i,j]$, the mean value $\overline{x}[i,j]$, variance value $s^{2} [i,j]$, and fluctuation factor $F[i,j]$ of each point in $N$ difference matrixes can be calculated as Eq. (13), Eq. (14), and Eq. (15).

$$\overline{x}[i,j] = \frac{1}{N}\sum\limits_{x = 1}^{N} {dP_{x} [i,j]}$$

(13)

$$s^{2} [i,j] = \frac{1}{N}\sum\limits_{x = 1}^{N} {(dP_{x} [i,j]} - \overline{x}[i,j])^{2}$$

(14)

$$F[i,j] = \overline{x}[i,j] + {\text{\{ }}s^{2} [i,j]{\text{\} }}^{\frac{1}{2}}$$

(15)

3. Due to the actual structure of shock train, the Mach stem of bifurcated shock is perpendicular to the flow direction. Hence, the direction along the flow field becomes the feature recognition direction. Hence, the fluctuation factor $F[i]$ of each row in the matrix when the wind tunnel is not running can be derived as Eq. (16).

$$F[i] = \frac{1}{J}\sum\limits_{j = 1}^{J} {F[i,j]}$$

(16)

4. Based on the derivation results above, when the wind tunnel is running, the single frame image containing the shock train structure is regarded as the target grayscale value matrix $T[i,j]$. By setting the threshold coefficient $\alpha$ of all rows for comparison, the corresponding points that are beyond the threshold can be selected as the value 1 forming the 0–1 matrix $L[i,j]$ in Eq. (17).

$$L[i,j] = \left\{ \begin{gathered} 1,T[i,j] > \alpha F[i] \hfill \\ 0,T[i,j] \le \alpha F[i] \hfill \\ \end{gathered} \right.$$

(17)

(5) Finally, by equalizing the vertical coordinate values of selected points whose values are 1 in matrix $L[i,j]$, the bifurcated shock positions of each target image can be recognized.

For exploring the self-excited oscillation characteristic of shock train, 1000 schlieren photographs between t = 4000 ms and t = 5000 ms during the test were selected. Firstly, in order to verify the correctness of the image recognition algorithm, the schlieren photographs taken at t = 4000 ms, t = 4250 ms, t = 4500 ms, and t = 4750 ms are converted to grayscale images and processed by nonlinear distortion correction method in Fig. 20a–d. As the results show, the recognized bifurcated shock positions (red lines) of shock train structures are accurate in Fig. 20.

Fig. 20

Schlieren photographs and distortion-corrected grayscale images at a t = 4000 ms, b t = 4250 ms, c t = 4500 ms, and d t = 4750 ms

Then, the recognized positions (Z direction) of shock train against time between 4000 and 5000 ms are shown in Fig. 21. From the location data, the shock train keeps moving backwards and forwards in the cylindrical isolator, whose mean position (dotted line) of shock train is 80.0 mm and the standard deviation σ of positions is 6.4 mm. The calculated average self-excited oscillation amplitude is 12.8 mm which is equal to 2σ.

Fig. 21

Location history of the shock train structure between 4000 and 5000 ms

By applying the fast Fourier transform (FFT) method to analyzing the location data, the amplitude spectrum distribution and power spectrum density (PSD) distribution of shock train positions are shown in Fig. 22a and b, respectively. From the results of the amplitude spectrum distribution, the position fluctuations shows the typical characteristic of low-frequency oscillation below 150 Hz, where the dominant frequency is 23.0 Hz, and the calculated 80 percent of accumulative frequency in the PSD distribution is 159.8 Hz.

Fig. 22

a Amplitude spectrum distribution and b PSD distribution of shock train positions

Meanwhile, the study of the relationship between the self-excited oscillation characteristic of shock train and the pressure oscillation characteristic downstream of the isolator was conducted. During the same time period of schlieren measurements, the pressure measuring points (A₁, A₂, A₃, and A₄) located on the upper wall of the divergent section record the pressure change during the test, whose locations are shown in Table 3.

Table 3 Locations of pressure measuring points in the divergent section

Figure 23 shows pressure histories of each location, and the pressure is normalized by P_ref, which is 50 kPa. It can be seen that the pressure tends to rise further downstream with a phenomenon of static pressure recovery.

Fig. 23

The pressure histories for A₁, A₂, A₃, and A₄

The frequency characteristics for A₁, A₂, A₃, and A₄ are obtained with the fast Fourier transform (FFT) method in Fig. 24. It shows that the pressure of each measuring point fluctuates with low-frequency oscillation below 150 Hz including the same dominant frequency of 23 Hz and high temporal harmonics, which has the same dominant frequency of the shock train structures. Hence, the stiffness of the overall shock train structure is good. The fluctuations of the wall pressures downstream of the isolator are induced by the shock train oscillation. The essence of this phenomenon is that the fluctuations in the separation region propagates downstream, which is the same as the rectangular isolators (Sugiyama et al. 1988).

Fig. 24

Amplitude spectrum distribution for a A1, b A2, c A3, and d A4 where the dominant frequencies are 23 Hz

As the power spectrum density (PSD) results show in Fig. 25, and the corresponding calculated 80 percent accumulative frequencies for A₁, A₂, A₃, and A₄ are 724.9 Hz, 902.9 Hz, 992.9 Hz, and 1030.9 Hz, respectively. The self-excited oscillation of shock train with a small-amplitude downstream of the cylindrical isolator results in the wall pressure fluctuation. The low-frequency oscillation induced by the bifurcated shock leads to the enlargement of the vortical structures in the separation region along the duct. Hence, the calculated 80 percent accumulative frequency of the PSD distribution gradually increases at the measuring points along the duct, which means that the high-frequency components gradually increase.

Fig. 25

PSD distribution for a A1, b A2, c A3, and d A4

In fact, vortices generated by the shock and boundary layer interaction would develop along the streamline direction (Z direction), along which the pressure gradient is positive. Then, vortices tend to unwind and undergo convection instability until they dissipate in almost isentropic turbulence. Meanwhile, the process involves a turbulence cascade with the appearance of small spatial scales and subsequently high temporal harmonics.

4 Conclusions

In summary, the schlieren measurement of shock train structures in a cylindrical isolator was realized for the first time in this study. The shock train structures and self-excited frequency characteristics were first studied by schlieren methods. At the same time, this experiment verified the feasibility of high-precision conformal optical windows for schlieren measurements of non-rectangular cross-section isolators. The main contributions of this study were as follows:

• 1. Starting from the design of conformal optical windows for non-rectangular cross-section isolators, an integrated processing conformal optical window pair of a cylindrical isolator had good connectivity to the wind tunnel pipelines and could be applied easily without assembling. The transmitted wavefront error (RMS value) of bilateral conformal optical windows was 2.658λ which met the requirement of high-precision schlieren measurements. In the future, the integrated processing method of the conformal optical window pair can provide support for the low-cost manufacturing and quick test of supersonic wind tunnel ducts.

• 2. Based on the air-breathing wind tunnel and the self-built pipeline structures in our laboratory, the schlieren photographs of shock train structures in the cylindrical isolator at Mach 2 were taken for the first time. Based on the nonlinear distortion correction method, the corrected images of the shock train structure were quantitatively analyzed with the numerical results and the theoretical calculated results, which meant that the image distortion correction of schlieren images through conformal optical windows was a crucial step before data analysis.

• 3. According to the self-compiled image recognition algorithm, the bifurcated shock positions of shock train structures were recorded. The self-excited oscillation frequency of shock train in the cylindrical isolator was first researched by the schlieren observation method. From schlieren measurement results, the position fluctuations of shock train structures showed the typical characteristic of low-frequency oscillation below 150 Hz with the dominant frequency of 23.0 Hz, which corresponded to the pressure oscillation characteristic downstream of the isolator.

However, there were some limitations of this study. The following improvements will be made in the future work:

• 1. In the next step of the design and manufacturing of conformal optical windows, depending on the high-precision optical surfaces, the observation area needs to be further expanded for the schlieren measurement of the boundary layer region in the cylindrical isolator.

• 2. In the selection of the schlieren window materials, the acrylic can be replaced by other materials for the schlieren observation through the cylindrical isolator under more extreme test conditions, such as a wind tunnel at higher Mach number.