Jun Wu， Haitao Xu， Fengcheng Song， Jun Xu， Yanling Li and Tao Wang,✉

（1. Aeronautical Engineering Institute, Civil Aviation University of China, Tianjin 300300, China；2. College of Electronic Information and Automation, Civil Aviation University of China, Tianjin 300300, China；3. State Key Laboratory of Precision Measuring Technology and Instruments,Tianjin University, Tianjin 300072, China）

Abstract: A 3D temperature field reconstruction method using the colored background oriented schlieren (CBOS) method is proposed to address image blurring due to the different refractive index of the multi-wavelength light and significant errors produced when the traditional background oriented schlieren (BOS) method is applied to high-temperature gas. First, the traditional method is employed to reconstruct the non-uniform 3D temperature field. Second, the CBOS method is applied to correct the distortion. Then, by analyzing the correlation coefficient among different color points of the colored background pattern, the non-uniform temperature field is reconstructed much more accurately. Finally, the experimental results are verified by applying the Runge-Kutta ray-tracing method and the thermocouple contact measurement method. The maximum average temperature error of the CBOS-reconstructed temperature field is 12.92°C, compared with the thermocouples. Therefore, an accurate three-dimensional reconstruction of the temperature field can be achieved by the proposed method effectively.

Key words: 3D temperature field reconstruction；colored background oriented schlieren method；ABEL inverse transform；image processing

The visual measurement method, as a noncontact detection method, is widely implemented in flow and temperature field measurement because of its high speed and accuracy. For this purpose, the schlieren method was proposed by A. Toepler in 1864[1]. Gary et al. introduced the development of the recent schlieren method since 2015[2]. In the traditional schlieren method, a large number of precise optical instruments such as laser emitters are needed. C. Alvarez-Herrera used the traditional schlieren method to reconstruct the temperature field in two dimensions[3].However, this method requires not only a large number of optical instruments but also complex optical path design and environmental conditions. Moreover, it is hard to obtain multi-directional temperature field data simultaneously, so a three-dimensional temperature field is difficult to reconstruct by the traditional schlieren method.

The BOS method proposed by Meier et al.[4],which combines the traditional schlieren method and the particle image velocimetry method(PIV), eliminates the need for traditional schlieren method equipment such as precision lenses and cutting diaphragms, which reduces the condition requirements of the measurement. This method obtains the density and the temperature field reconstruction by measuring the offset of the background image, which has already been applied to wind tunnel measurement, engine flow field measurement, etc. Micheal et al. compared three different schlieren methods, including the traditional schlieren method and the BOS method, by taking pictures of the light source[5]. Jenna et al. compared the advantages and disadvantages of the traditional schlieren method and the BOS method by measuring the flow field[6]. Sayan et al. explored the application of particle image processing algorithms in the displacement measurement of background striated images[7].Lalit et al. investigated the experimental design of the BOS method based on PIV image processing[8]. Michalski et al. examined the application of the BOS method in the measurement of nozzle formation flow field reconstruction[9]. Lilly,Elsinga et al. studied the application of the BOS method in density field reconstruction[10-11]. Gannavarpu and Alan et al. analyzed the heat transfer process during the reconstruction of the temperature field with the traditional BOS method[12-13]. H. Zhou et al. employed traditional BOS technology to reconstruct a two-dimensional temperature field[14].

However, the BOS method is prone to errors in high-temperature areas, which reduces the accuracy of the temperature field reconstruction.Sangyoon et al. investigated the use of the BOS method to reconstruct an engine’s tail flow field in a high-temperature environment formed by an aero-engine[15]. When the BOS method is applied to the reconstruction of a high-temperature field,it is easily affected by the high-temperature environment, which causes low accuracy. For example, when the background image is blurred due to the high-temperature environment, it makes it difficult for BOS to capture scattered specks on the background image that reduce accuracy.

To improve the precision of the BOS method, the colored BOS method has been proposed in this paper which replaces the black and white points in the BOS method with three-color points, which are red, blue, and green, to reconstruct the 3D temperature field. To analyze the accuracy of the three-dimensional temperature field reconstructed with the CBOS method, the BOS method is also used to reconstruct the temperature field. The temperature field reconstructed by the CBOS method is compared with the actual temperature field by using the point contact measurement method and the Runge-Kutta ray tracing algorithm method.

1 Principle of the CBOS Method

The BOS method is based on the measurement of the deviation of light passing through a gaseous medium, as expressed by the Gladstone-Dale formula

where ρ is the gas density and G(λ) is the Gladstone-Dale constant[16].

According to Eq. (1) and Eq. (2), the refractive index of light is affected by the wavelength λ. According to the relationship between the refractive index and the deflection angle of the light in the Eq. (3), the relationship between the deflection angle of the light and the wavelength of the light can be obtained.

In order to further verify the relationship between the wavelength and the refractive index,this paper applied a simulation to reveal the relationship between the deflection angle of the light and the wavelength of the light. It is assumed that the refractive index field is a circular temperature field with a radius of 50 mm. The temperature field is conformed to a Gaussian distribution, and the highest temperature is 1 000 K.The relationship between the wavelength and the deflection angle can be obtained, as shown in the Fig.1. Therefore, when BOS is used in temperature field reconstruction, the refractive index field caused by different wavelengths is different,which may cause the background image to be blurred.

Fig.1 Wavelength and deflection angle simulation results

For the ideal constant pressure under constant pressure conditions, according to the Gladstone-Dale’s equation formula, the relationship between density and temperature can be obtained by the simplification expressed as

As shown in Fig.2, a deflection occurs when the light generated by the background image passes through the measured temperature field.The dotted line represents the projection of the half-point of the background on the CCD when there is no temperature field, while the solid line is the deflection of the light passing through the temperature field. In the case of spot projection,ΔH′is the speckle offset corresponding to the actual background image, and ΔH is the actual offset pixel corresponding to the speckle offset after adding the temperature field. Based on this process, the offset corresponding to all the spots can be obtained.

Fig.2 Background oriented schlieren method schematic

Assume that all of the records are paraxial rays, and the deflection angle is extremely small so that the deflection angles in the x and y directions can be calculated by Eq. (3) and

Meanwhile, the relationship between the horizontal and the vertical deflection angle of the light and the horizontal gradient of the refractive index and the vertical gradient of the refractive index can be known from the formulas of the light deflection angles from Eq. (8) and Eq. (9).

The CBOS method proposed in this paper is mainly used to solve the problem of reduced accuracy, such as with image blurring, when the BOS method is applied to temperature field reconstruction. The CBOS method uses a computer to generate a large number of disordered scattered spots instead of the black and white points in the BOS method. These background points are placed in the background pattern, and these background points should cover 35%–70%of the background image[18], as shown in Fig.3.

As illustrated, all points are composed of red, green, and blue dots, because the primary colors are red, green, and blue, which are easily captured by color CCD cameras. Various pure color points on the background image are evenly distributed at the same volume and proportion.For color CCD cameras, red, green, and blue are stored directly by the sensor in the original format without processing and compression. As displayed in Fig.3, the processing process of the schlieren image in the smear method of the colored background is as follows: the first step is to take a picture of the background image; the second step is to divide the background image spot into the following parts: ①pure red spots;② pure blue spots; ③ pure green spots; ④ all spots[19]; The third step is to apply the PIV particle image processing method to calculate the offset of all spots on the picture; the fourth step is to obtain the offsets of the three pure color points of red, blue and green separately; the fifth step is to calculate whether the offset of the color point satisfies the standard deviation range δ of the corresponding black and white point offsets. According to the reference, the standard deviation range is 0.36 in Ref.[20]; the sixth step is to calculate the displacement of all points. The shift and the offset of the three primary colors are used to obtain the average of the cross-correlation coefficients; in the seventh step, the average offset of all positions can be determined; the eighth step, the density field can be reconstructed by the ABEL inverse transform through the calculated average offset. The ABEL transformation formula is given in Eq.(10), where I(x) is the density gradient under the condition of column symmetry and ε(r)ε(r) is the radial distribution[21]. The distribution radius is normalized so that the ABEL inverse transformation can be obtained, as shown in Eq. (11).

Fig.3 Illustration for CBOS

Then, applying the formulas to the density field reconstruction yields the results in Eqs.(12)(13).

Fig.4 CBOS method experiment setup

Finally, through the achieved density field,according to Eq.(4), the relationship between the density field and the temperature field is used to obtain the temperature field reconstruction.

2 Experiments

A circular symmetrical electric furnace with a maximum power of 2 000 W is used in the experiment. The CCD camera resolution is modified to be 1 920×1 080 pixels and f is 1.9 mm. The distance between the camera and the center of the electric furnace is 478 mm, and the distance from the center of the electric furnace to the background grain image is 246 mm.

The size of the background image is 210mm×297 mm, and the number of red, green,and blue dots (size 5 pixels) on the background image is 300 each, or 900 in total, which are randomly generated by the computer. The laboratory temperature condition is 27 °C, and the calculated environmental refractive index is 1.000 260. The pixel offset value of the background image can be measured by taking photos.The experimental site is shown in Fig.4.

2.1 Temperature field reconstruction experiment by using the BOS method

To reconstruct the temperature field, the ABEL inverse transformation is used to perform an iterative reconstruction of the section at the flame height z=0 mm in the x-direction, and this process is repeated every 90 pixels in the z-direction. The iteration results are as follows in Fig.5.

Fig.5 Refractive index of BOS

According to the relationship between the density field and temperature in Eq. (4), the corresponding temperature gradient can be obtained. The three-dimensional temperature field is reconstructed by the BOS method, shown in Fig.6.

Fig.6 3D temperature field (BOS)

2.2 Temperature field reconstruction experiment by using the CBOS method

Next, the offset of the red, green, and blue color points in the x-direction is conducted separately, and the correlation coefficient between the offset of these three solid colors and all of the spots is calculated. By applying the inverse ABEL transformation, the change in the refractive index gradient is obtained. The corrected refractive index can be obtained as shown in Fig.7.

The three-dimensional temperature field reconstructed by the CBOS method is shown in Fig.8.

Fig.7 Refractive index of CBOS

2.3 Verification experiments by using Runge-Kutta ray-tracing method and thermocouple

The three-dimensional temperature field reconstructed by the CBOS method is shown in Fig.8.

Fig.8 3D temperature field (CBOS)

With a known temperature field, based on the principle of light propagation, the propagation path is a straight line when it is in a uniform refractive index medium and the propagation path in the medium of variable refractive index is biased. Meanwhile, the isotherm is established in the temperature field by the differential method, so that a temperature gradient field is built, and the refractive index distribution gradient corresponding to the temperature gradient can be achieved by using

Moreover, through the refractive index corresponding to each temperature gradient interval,an accurate light propagation trajectory can be achieved because the ray trajectory of each corresponding interval is obtained by the Runge-Kutta tracing algorithm principle.

According to Eq. (14) and the Runge-Kutta formula, we can calculate the ray tracing formula by using the Runge-Kutta method, as follows.

Here ∇t is the step size, indicating the increment of t. R is the position matrix in Eq.(16)and T is the ray vector, in Eq. (17), where α, β are the angles between the ray vector and the x,y, and z coordinate axes. D is the variable, in Eq.(18).

Utilizing the Runge-Kutta ray tracing algorithm of Eq. (15), the direction and coordinates of the light can be obtained[22]. The iterations are used to compute the coordinate points of the light during the refraction process and to reconstruct the trajectory of the light in the temperature gradient.

Fig.9 shows a schematic diagram of the reconstruction of the optical path by using the ray tracing algorithm, and the experimental setup is shown in Fig.10. The industrial high-precision platinum-iridium thermocouple applied is model type WRP-130 S. The temperature measurement accuracy of the equipment is in range of 0 – 1 600 ℃,with an error of ±1.5 ℃. The accuracy of the checkerboard calibration plate is 0.01 mm.

Fig.9 Ray-tracing algorithm

Fig.10 Runge-Kutta ray-tracing method experiment site

This paper examined multiple sets of distortion errors at different heights and radii. Fig.11 displays a part of the comparison of the corner detection result and the distortion errors, which are caused by both the BOS reconstruction temperature field and the CBOS reconstruction temperature field.

Fig.11 Deviation value

According to the results obtained by the ray tracing algorithm, the average error of the BOS method is 0.025 9 mm when radius=0 mm, the average error of the CBOS method is 0.003 28 mm;when radius=20 mm the average error of the traditional schlieren method is 0.033 5 mm, and the average error of the CBOS method is 0.020 2 mm.Through the calculation of different radius errors at a different height, the following conclusions can be drawn: ① The distortion caused by the temperature field reconstructed by the CBOS method is closer to the distortion caused by the actual temperature field than that caused by the BOS method. ② The CBOS method produces distortion in the high-temperature region with better effect, which is closer to the distortion in the actual temperature field.

After using the corner detection and ray reconstruction algorithms to verify the temperature field, this paper uses the point contact measurement method to measure the temperature of the temperature field at 0 mm, 27 mm, 54 mm,81 mm, 108 mm, and 135 mm, using the S-type thermocouple and comparing the experimental results. Fig.12 is a comparison of the temperature reconstructed by the BOS method, the CBOS method, and the measured temperature with the thermocouple.

Fig.12 Temperature comparison

The maximum temperature of the actual temperature field at different heights measured by the thermocouple is compared with the highest temperature of the reconstructed BOS temperature field and the reconstructed CBOS temperature field at different heights. The maximum average temperature error of the BOS reconstructed temperature field is 54.27°C, and the maximum average temperature error of the CBOS reconstructed temperature field is 12.92°C. By analyzing the results, it can be concluded that the temperature fitting effect of the reconstructed CBOS temperature field is better than that of the reconstructed BOS temperature field.

3 Conclusion

In this paper, the CBOS method is used to analyze the shift of the color points of the background image under the influence of a high-temperature temperature field to achieve the reconstruction of the non-uniform temperature field.The Runge-Kutta ray-tracing algorithm and the corner detection method are applied to verify the two temperature fields. After applying the ray tracing algorithm, the experimenters use the point contact measurement method to measure the temperature field with the thermocouple and to verify the experimental results. In conclusion,the verification proves that the three-dimensional temperature field can be reconstructed by the CBOS method. Moreover, because the CBOS method solves the blurred background image problem found with the BOS method, the reconstruction accuracy is significantly higher than the BOS method. Especially in high-temperature areas, the temperature field effect of CBOS reconstruction is better than that of BOS reconstruction. Compared with the accuracy of the traditional background shading method, the accuracy of the colored background shading method is improved by 13.8% in the reconstruction of the non-uniform temperature field. The method in this paper provides a new method for the measurement of high-temperature temperature fields, such as the tail spray temperature field of aircraft engines and rocket engines.