1 Introduction

Gas flows have been theoretically, numerically, and experimentally studied. Visualization is an important experimental technique used to investigate or measure gas flows because gas is generally invisible due to its transparency. Flow visualization techniques yield flow patterns, which provide a simplified and intuitive understanding for users (Mueller 1985). Various gas flow visualization methods have been established. The schlieren and shadowgraph methods are commonly used. However, the information obtained from these two measurement methods is qualitative, and they do not yield quantitative data. Therefore, quantitative visualization methods, such as Mach–Zehnder interferometry, particle image velocimetry (PIV), planar laser-induced fluorescence (PLIF), and molecular tagging velocimetry (MTV) are used. Mach–Zehnder interferometry is a quantitative technique for measuring the density distributions (Zhang et al. 2015). In this technique, split laser beams are passed through the test and reference sections, and the beams interfere with each other and generate interference fringes. The density field is obtained by analyzing the fringes. This technique requires complex optics and its spatial resolution is limited by the fringe size. PIV is a quantitative technique that obtains the velocity field by analyzing the displacement of particles in the flow (Brend et al. 2020). However, disadvantageously, particles that serve as tracers must be mixed in the flow field to be measured. PLIF is a quantitative technique utilized for visualizing density, temperature, and other species-specific distributions through laser excitation. It is typically employed in flows containing excited species, such as reactive flows (Chaib et al. 2023). In the context of low-temperature flows, excitation from the ground state is essential, demanding high-power lasers for multiphoton excitation (Kinefuchi et al. 2019) or seeding agents like acetone (Thurber and Hanson 2001). MTV is a velocimetry technique that necessitates large-scale implementation for laser excitation of premixed tracers. As a technique similar to MTV, femtosecond laser electronic excitation tagging (FLEET) is also used to obtain velocity distributions as a means of tagging nitrogen-based flow fields (Kinefuchi et al. 2017).

Background-oriented schlieren (BOS) and laser Rayleigh scattering (LRS) are quantitative flow measurement techniques. The principle of BOS is the same as that of the schlieren method, but generally, an image sensor is placed on one side of the observation target flow field and a background image with a certain pattern is placed on the other side (Raffel 2015). A random dot pattern is usually used for the background. The light rays from the background image refract due to the refractive index difference caused by the density change in the flow, and the image sensor captures them with the displacement of the background pattern. By comparing the images with and without the density change, the density fields can be quantitatively obtained from the displacement to integrate the Poisson equation. BOS can be applied to a large flow area, and it is an inexpensive method as it requires simple optics. In BOS, the solution is markedly influenced by the selection of boundary conditions for integrating the Poisson equation (Komuro et al. 2019). Therefore, in BOS, the requirement to know the boundary density values for maintaining high accuracy limits its applicability. Extrapolation is also employed for the boundary condition because of its ease of implementation, but it has been reported to result in reduced accuracy (Gong et al. 2018). Furthermore, the density field of a flow with a strong density gradient, such as shock waves, cannot be directly obtained using BOS. Thus, additional procedures, such as computed tomography processing on multiple images taken from various directions, are required to reconstruct the density field (Ota et al. 2011).

The LRS method is a nonintrusive and unseeded optical measurement method used to acquire the density field. In LRS, the flow is illuminated by a laser beam, which is scattered by the molecular particles of the flow. The scattered light intensity is proportional to the flow density; therefore, the density field can be obtained by capturing the intensity distribution of the scattered light (Miles et al. 2001). LRS is a powerful tool for studying complex fluid flows, including supersonic flows (Ground et al. 2018), combustion, and flames (Mielke and Elam 2009; Park et al. 2022), because the flow density can be directly obtained from the scattering intensity and short pulse lasers can freeze a flow field in time, allowing instantaneous density measurements. However, when measuring a large area, a wide laser sheet is required, which consequently necessitates a high-power laser to obtain sufficient the scattering signal and ensure measurement accuracy. This sometimes limits the measurement area size, which is a disadvantage of LRS. Also, LRS is inherently difficult to conduct near walls due to the reflection, which has limited its use.

This study focuses on the density measurement because density is an important flow parameter along with velocity, and under certain conditions, it can also be used to estimate pressure and temperature. We propose and demonstrate a new density field measurement method that overcomes the weaknesses of BOS and LRS. Specifically, this study aims to address the boundary condition setting issue in BOS and the limited measurement area issue in LRS, thus achieving mutual complementarity between the two methods. BOS captures the displacement image in the wide area of a flow field, while LRS measures the only boundaries of the BOS measuring area. Then, the flow density obtained by LRS is applied as the boundary condition in the integration process of the displacement image for BOS. This allows the removal of the BOS limitation of the boundary condition and enables flow measurements over a wide range. Furthermore, the LRS only needs to be applied to a narrow area near the BOS boundary, which can enhance the energy density of the laser sheet and eliminate the need for high-power lasers. The proposed method is applied to a heated turbulent air jet, and its usefulness and applicability are discussed by comparing it with results from Computational Fluid Dynamics (CFD). Furthermore, based on the comparison, this study highlights the importance of boundary conditions in BOS and provides the setting guidelines.

2 Experimental method and data analysis

2.1 Heated air jet

The target flow for validating the methodology is an axisymmetric heated air jet under atmospheric conditions. An electric inline air heater (Fintech SAH200V-3KW/29PH) was used to heat the air. The heater had no feedback control, but the outlet temperature was monitored using Yokogawa TX10 digital thermometer and the temperature fluctuation during the experiment was less than 1 K. The rated power and voltage of the heater were 3 kW and 200 V, respectively. The air temperature was controlled by changing the input voltage to the heater. A straight cylindrical nozzle whose inner diameter Dn was 17 mm was attached downstream of the heater. The temperature of the air supplied from the heater was measured using a K-type thermocouple installed between the heater and nozzle. It was installed 50 mm upstream from the nozzle exit. The air flow rate was controlled by a mass flow controller (Horiba Stec, SEC-E431X, ± 1 standard litter per minutes accuracy) and was fixed at 50 standard litter per minutes (1.1 g/s) in the experiment. To measure the temperature of the hot exhaust air, six thermocouples were placed on the center of axis of the jet at 5 mm intervals in the flow direction. A thermocouple with a wire diameter of 0.1 mm was employed to minimize as much disturbance to the flow as possible.

2.2 Laser Rayleigh scattering

LRS is a flow diagnostic method for measuring the density of gases. When an air is illuminated by a laser light, the total Rayleigh scattering signal \(S_{{\text{R}}}\) from the air is expressed as follows:

$$\begin{array}{*{20}c} {S_{{\text{R}}} = \varepsilon I_{{\text{L}}} NV\mathop \int \limits_{{\Delta {\Omega }}}^{{}} \frac{{\partial \sigma_{{\text{R}}} }}{{\partial {\Omega }}}d{\Omega },} \\ \end{array}$$
(1)

where \(\varepsilon\) is the optical system constant, \(I_{{\text{L}}}\) is the incident laser intensity, \(N\) is the number density of the air, \(V\) is the scattering volume, \(\sigma_{{\text{R}}}\) is the Rayleigh cross section of air, and \({\Omega }\) is the solid angle for detection. Normalization of Eq. (1) to air at standard ambient conditions yields

$$\begin{array}{*{20}c} {\frac{{S_{{\text{R}}} }}{{S_{{{\text{R}}0}} }} = \frac{N}{{N_{0} }},} \\ \end{array}$$
(2)

where \(S_{{{\text{R}}0}}\) is the Rayleigh scattering under standard ambient conditions with the laser intensity \(I_{{\text{L}}}\). \(N_{0}\) is the number density under the standard ambient conditions. The standard ambient condition in this study was the static atmospheric air condition. Therefore, from Eq. (2), the flow number density \(N\) can be obtained based on the Rayleigh scattering signals with and without flow, \(S_{{\text{R}}}\) and\(S_{{{\text{R}}0}}\), respectively.

Figure 1 displays the experimental setup for the LRS measurement of the heated air jet. The laser source for LRS was a Nd/YAG laser (Litron Nano). The second harmonic of Nd/YAG, 532 nm, was obtained using a second harmonic generator unit, and it was used to illuminate the air molecules and obtain their Rayleigh scattering. The pulse width and pulse energy of the second harmonic were 10 ns and 70 mJ, respectively. The laser beam was converted into a laser sheet through a beam expander comprising cylindrical concave and convex lenses. The 15-mm-wide laser sheet passed through the center axis of the hot air flow in the radial direction, as shown in Fig. 1. Filtered Rayleigh scattering (FRS) is an improved version of LRS (Limbach and Miles 2014). In FRS, molecular filters, such as iodine filter for 532 nm, are used to reduce the stray light, such as scattering form dust particles, optical components, and experimental equipment. However, in addition to the molecular filters, FRS requires high-power and wavelength tunable lasers to precisely match the laser wavelength to the transmitted wavelength of the molecular filter. In this study, LRS, which does not require wavelength tuning and molecular filters, was chosen for simplicity. The supply air system had a dust filter, but some obvious high-intensity points were observed in the images due to the scattering from the dust particles in the ambient air around the jet, as shown later. The scattering signals over the third quartile were replaced to the averaged intensity to filter the noise. The averaged intensity was calculated over the region where the laser sheet was emitted. In addition, a Gaussian filter with a standard deviation of 7 was applied to reduce pixel bleeding from the dust.

Fig. 1
figure 1

Setup for the laser Rayleigh scattering measurement

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Furthermore, the noise attributed to particles is primarily caused by Mie scattering. To mitigate this effect, the camera is tilted at an angle of approximately 45 degrees to capture the laser sheet, taking into account the angular characteristics of Mie and Rayleigh scatterings, as illustrated in the lower part of Fig. 1. The captured images were corrected for the tilt angle through projective transformation. For simplicity, a single-lens reflex camera (Nikon D7500, 5568 × 3712 pixel) with a zoom lens (Nikon AF-S NIKKOR 18–140 mm 1:3.5–5.6GED) was used as a two-dimensional image sensor to observe the Rayleigh scattering of the hot air. The exposure time was set to 1 s, and a Rayleigh scattering image was taken during a single laser shot with the pulse duration of 10 ns. The effect of the long exposure time had negligible effect on the image because the experiment was conducted in a darkroom.

2.3 Background-oriented schlieren

BOS is based on the schlieren method. However, compared to traditional schlieren methods that qualitatively visualize density gradients, BOS quantitatively acquires the density distribution in the flow field. Figure 2 presents the optical configuration of BOS. A transparent film with a random dot pattern printed was placed as the background across the optical path of the laser light source. Background patterns are usually placed on the opposite side of the camera across the flow field. However, in this study, the random dot pattern was directly added to the laser light, which enabled the acquisition of more light than conventional techniques, thereby making it particularly well suited for observing high-speed flows. As the flow density in the measurement region changed, the position of the dots in the obtained image moved in proportion to the density gradient based on the following Gladstone–Dale relation:

$$\begin{array}{*{20}c} {n - 1 = G\rho ,} \\ \end{array}$$
(3)

where \(n\) is the index of refraction, \(G\) is the Gladstone–Dale constant, and \(\rho\) is the flow density. The density distribution can be obtained by comparing the images with and without the flow. First, the displacement vector \(\left( {\Delta x,\Delta y} \right)\) of each dot was calculated by comparing the images taken with and without the hot flow using template matching by direct cross-correlation. The template matching was performed using an open-source code called OpenPIV (OpenPIV Group 2022). To reconstruct the density field from the displacement field, numerical integration is necessary. Two integration methods can be employed: either by solving the Poisson equation or through line integration. The latter method, which is the simplest to implement, has the disadvantage of introducing line noise (Richard and Raffe 2001). Therefore, in the present study, the Poisson equation was selected as the integration method.

Fig. 2
figure 2

Optical configuration of BOS

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The Poisson equation for axisymmetric flow field can be derived as follows. The deflection angle in the y-direction \(\varepsilon_{y}\) was expressed as follows:

$$\begin{array}{*{20}c} {\varepsilon_{y} = \frac{1}{{n_{0} }}\int \frac{\partial n}{{\partial y}}dz = \int \frac{\partial \delta }{{\partial y}}dz = \frac{\partial }{\partial y}\int \delta dz = \frac{{\partial \overline{\delta }}}{\partial y},} \\ \end{array}$$
(4)

where \(n\) is index of refraction of the heated air flow and \(n_{0}\) is the index of refraction of the surrounding ambient air. \(\delta\) is modified refractive index and defined as follows (Xiong et al. 2020):

$$\begin{array}{*{20}c} {\delta = \frac{n}{{n_{0} }} - 1} \\ \end{array}$$
(5)

\(\overline{\delta }\) in Eq. (4) is the line integrated δ in the z-direction. Introducing radial coordinates due to the axisymmetric flow field, following Abel transform is employed assuming optical path is parallel to the z-axis:

$$\begin{array}{*{20}c} {\overline{\delta }\left( y \right) = 2\mathop \int \limits_{y}^{\infty } \frac{\delta \left( r \right)r}{{\sqrt {r^{2} - y^{2} } }}dr.} \\ \end{array}$$
(6)

The radial distribution of modified refractive index \(\delta \left( r \right)\) can be obtained through following Abel inversion:

$$\begin{array}{*{20}c} {\delta \left( r \right) = - \frac{1}{\pi }\mathop \int \limits_{r}^{\infty } \frac{{\partial \overline{\delta }}}{\partial y}\frac{dy}{{\sqrt {y^{2} - r^{2} } }}.} \\ \end{array}$$
(7)

The Poisson equation is obtained from Eq. (4) by taking the partial derivative in the y-direction and then considering the same equation in the x-direction, followed by summing the two equations:

$$\begin{array}{*{20}c} {\frac{{\partial^{2} \overline{\delta }}}{{\partial x^{2} }} + \frac{{\partial^{2} \overline{\delta }}}{{\partial y^{2} }} = \frac{{\partial \varepsilon_{x} }}{\partial x} + \frac{{\partial \varepsilon_{y} }}{\partial y}.} \\ \end{array}$$
(8)

\(\varepsilon_{y}\) is geometrically calculated from trigonometry in Fig. 2 assuming \(\varepsilon_{y}\) is small:

$$\varepsilon_{y} = \frac{{Z_{{\text{B}}} }}{{Z_{{\text{D}}} Z_{{\text{I}}} }}\Delta y.$$
(9)

where \(z_{{\text{B}}}\) is the distance between the image plane and the background plate, \(z_{{\text{D}}}\) is the distance between the background plate and the center of the target flow (the center of the nozzle outlet in the present study), and \(z_{{\text{I}}}\) is the distance between the image plane and the lens which is almost equal to the focal length of the lens. These lengths were directly measured using a ruler after alignment of the equipment using a cross-line laser (Tajima GT4R-Xi). From the displacement vector \(\left( {\Delta x,\Delta y} \right)\) obtained from the BOS image, \(\varepsilon_{x}\) and \(\varepsilon_{y}\) were calculated using Eq. (9). Consequently, the right-hand terms of Eq. (8) were known. The Poisson equation was solved using the successive over-relaxation method to obtain \(\overline{\delta }\). Boundary conditions for integrating the Poisson equation have a significant impact on the solution, which will be discussed later. Barr inversion method (Barr 1962) was used for the Abel inversion to obtain \(\delta\). Finally, the index diffraction \(n\) and then density \(\rho\) were obtained using Eqs. (5) and (3), respectively.

Figure 3 presents the experimental setup of BOS. A random dot pattern was placed across the light path. The laser source was Cavitar Cavilux Smart UHS (50 W). The laser pulse width was set to 50 ns. In synchronization with the laser, the high-speed camera Phantom v1211 was used for the BOS imaging with a recording speed of 19 kfps with an exposure time of 1 μs. The spatial resolution of the high-speed camera was 768 × 768 pixels. The spatial resolution of the displacement vector was 340 × 460 pixels and the integration window for the BOS was set to 16 × 16 pixels. The schlieren method was performed by installing a knife edge and removing the background pattern.

Fig. 3
figure 3

Experimental setup for BOS and schlieren

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3 Results and discussion

The experimental conditions are summarized in Table 1. The supply air temperature was measured 50 mm upstream of the nozzle outlet and was varied among 467, 512, and 607 K. The dynamic pressure of the heated air flow was estimated as a maximum of 20 Pa. Therefore, the pressure was almost constant in the entire measurement region. Assuming the constant pressure throughout the measurement region, the flow temperature \(T\) was obtained from the number density \(N\) using the equation of state.

$$\begin{array}{*{20}c} {\frac{N}{{N_{0} }} = \frac{T}{{T_{0} }},} \\ \end{array}$$
(10)

where \(T_{0}\) is the ambient temperature, 292 K. The quantity measured in this experiment is the density of the air. However, since there are no sensors capable of directly measuring density for verification purposes, it was converted to temperature before comparison. In addition, temperature is more intuitively understandable as a physical quantity compared to density. Hence, the densities obtained through BOS and LRS were converted to temperature using Eq. (10) for the purpose of comparison. In the experiment presented in Table 1, thermocouples were employed to measure the air supply temperature and temperature distribution within the jet.

Table 1 Experimental conditions
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3.1 Schlieren visualization

Figure 4 displays the schlieren image of the heated air jet for each air temperature. The critical Reynolds number, at which the free jet undergoes a turbulent transition is 1000–2000 (Revill 1992). In this study, the Reynolds number was around 3,000 (Table 1) and the shear layer of the exhaust flow was under turbulent condition. Figure 5 shows the schematic of the turbulent free jet. The turbulent entrainment, wherein fluid from the surrounding environment is drawn into the jet, occurred due to the motion of the eddies. The jet temperature decreased in the downstream region due to the entrainment. A core of essentially no turbulence, called potential core, was present close to the nozzle exit, which was hardly affected by the jet diffusion (Lee and Chu 2003). Generally, the axial length of the potential core increases with decreasing Reynolds number (Bogey and Bailly 2006). However, the length in the present experiment was almost the same in all cases as shown in Fig. 4 because the Reynolds number did not significantly change with the air supply temperature. Since the schlieren image is the integral of the optical path, a slight fluctuation was observed in the potential core region due to the turbulence region formed by entraining stationary surrounding air around the potential core. The density gradient and the contrast of the images increased with the air temperature.

Fig. 4
figure 4

Schlieren image at the nozzle outlet at each air supply temperature

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Fig. 5
figure 5

Structure of the free turbulent jet

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3.2 Rayleigh scattering

Figure 6 shows the raw images of the Rayleigh scattering from the laser sheet for each air temperature. These instantaneous images were taken during the laser illumination period, 10 ns. The images showed distinct high-intensity points due to the scattering from the dust particles. These effects were reduced by the image processing as described in 2.2. The intensity of the Rayleigh scattering from air molecular decreased with increasing air supply temperature because of the low flow density. Figure 7 displays the air temperature distribution calculated by Eq. (10) using the LRS results for each air supply temperature. Three raw images are used for time averaging to create the temperature distribution in Fig. 7. The high-temperature region was present near the jet core. The flow appeared to be turbulent, similar to the schlieren image in Fig. 4. The potential core also had some fluctuation. According to previous studies (Crow and Champagne 1971; Abdel-Rahman et al. 1996), even within the potential core at Re = 103 ∼ 105, small velocity fluctuations are observed. Along with velocity fluctuations, temperature fluctuations are also present (Locke et al. 2018; DeBonis 2018). Thus, it is possible that this phenomenon was also observed in the present experiment. On the other hand, the signal-to-noise ratio at room temperature was 30, which was similar to a previous study (Grib et al. 2019); however, it decreased to less than 10 in the high-temperature region around 550 to 600 K because the Rayleigh scattering intensity was weak due to the high air temperature or low density. Therefore, it is also possible that a measurement error was included in the temperature fluctuation within the potential core shown in Fig. 7. The exhaust air temperature increased with the supply air temperature, as expected; however, the 607 K result seemed unreliable because the flow temperature was almost the same as in the 512 K result. This was attributed to the low signal-to-noise ratio; therefore, the 607 K result was excluded.

Fig. 6
figure 6

Raw image of Rayleigh scattering from the laser sheet for each air supply temperature

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Fig. 7
figure 7

Air temperature distribution calculated by LRS for each air supply temperature

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We used a high-power laser in this study not only to obtain the boundary conditions but also to observe the flow field shown in Fig. 7. However, LRS plays a crucial role solely in obtaining boundary conditions in the proposed method. Therefore, it is possible to apply the laser beam itself without sheeting or reduce the width of the laser sheet to increase the local Rayleigh scattering intensity. Narrowing the beam width can enhance its suitability and accuracy for low-density and high-temperature flows similar to those observed in the 607 K case. In addition, lower-power lasers can be applied by narrowing the beam to enhance scattering intensity for the boundary condition. Using lower-power lasers could lead to cost savings in experimental setups.

3.3 BOS/LRS complementary measurement

3.3.1 Boundary conditions for BOS

To reconstruct the density field or temperature distribution from the displacement field obtained in BOS, the Poisson equation provided in Eq. (7) needs to be solved. The choice of boundary conditions to integrate the Poisson equation significantly affects the solution (Gong et al. 2018; Komuro et al. 2019). However, no clear guidelines for setting the boundary conditions for BOS have been provided. In general, Dirichlet or Neumann boundary conditions are employed. A Dirichlet boundary condition specifies the solution’s value on the boundary, which can be challenging when density values at the boundaries are indeterminate. Typically, a constant value is input along the boundary in Dirichlet condition. In particular, this boundary condition is difficult to apply to complex flow fields. On the other hand, a Neumann boundary condition prescribes derivative values, with extrapolation often used, making it applicable in many instances due to the absence of a strict requirement for precise density values at the boundary. However, it’s important to note that the Neumann boundary condition may not always hold true from a physical perspective.

A BOS was applied to the flow around a turbine cascade (Gong et al. 2018). Neumann boundary conditions are typically employed in such complex flow due to the difficulty in specifying accurate density values for Dirichlet boundary conditions. However, in this case, they opted for a different approach by using the density distribution obtained through CFD on the boundary as Dirichlet boundary conditions. This resulted in improved accuracy when compared to the conventional Neumann boundary conditions. This suggests that even in complex flow fields, achieving higher accuracy is possible as long as the density distribution on the boundary can be accurately set as the boundary condition. To employ the CFD result as a boundary condition, validation of the CFD is necessary, but it is not always possible. Hence, in this study, we consider the use of the density distribution obtained through LRS as the boundary condition for BOS.

The configuration of the boundary conditions for BOS is shown in Fig. 8. At the upper boundary outside of the jet, the Dirichlet boundary condition \(\rho = \rho_{0}\) was applied assuming that the flow is static and ambient there. The Neumann boundary condition or \(\partial \rho = 0\) was applied at the center axis because it is impossible to specify the density value. The boundary conditions of the left and right boundaries including heated jet need to be investigated. Typically, Neumann conditions would be applied here. Herein, four types of boundary conditions cases A–D were considered, as shown in Table 2. For the Rayleigh scattering boundary condition, the three images were averaged and the obtained density distribution along the boundary was used as the boundary condition. Regarding the alignment of the LRS and BOS images, the same calibration chart was used for both to align and identify the corresponding points.

Fig. 8
figure 8

Configuration of boundary conditions for BOS

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Table 2 Boundary conditions for BOS
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3.3.2 Result of LRS/BOS complementary measurement

The BOS result of the air temperature distribution with each boundary condition for the air supply temperatures of 467 and 512 K are shown in Figs. 9 and 10, respectively. Initially, time averaging was applied to 400 frames of BOS images obtained with the frame rate of 19 kfps, i.e., the turbulent effect is averaged, for later comparison with the Reynolds-averaged numerical flow analysis and time-averaged thermocouple measurement results. Subsequently, the time-averaged Rayleigh scattering data were applied to these averaged BOS images as the boundary conditions, and the density distribution image was obtained by solving the Poisson equation. The density distribution was then recalibrated to temperature distribution using Eq. (10). The boundary conditions affected the temperature distributions in both the 467 and 512 K cases. In cases A and B, the right boundary condition was Neumann or\(\partial \rho = 0\), and evidently unnatural results are observed in both Figs. 9 and 10: the downstream temperature increased and exceeded the air supply temperature. The Neumann boundary conditions or \(\partial \rho = 0\) deviated from the observed phenomenon, indicating that they were physically inappropriate in these cases. The results of both Cases C and D were reasonable: the temperature decreased toward the downstream.

Fig. 9
figure 9

Air temperature distribution using BOS with various boundary conditions. Air supply temperature = 467 K

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Fig. 10
figure 10

Air temperature distribution using BOS with various boundary conditions. Air supply temperature = 512 K

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3.4 Evaluation of the proposed method

3.4.1 CFD implementation

CFD analysis was conducted for comparison with the experimental results. Ansys Fluent 2022 was used for the computation. The axisymmetric coordinate with a pressure-based solver was employed, and the calculation domain was 250 mm in the axial direction and 100 mm in the radial direction. The structural grid system comprised 30,000 and the grid convergence was confirmed. The turbulence was treated with Reynolds-averaged Navier–Stokes (RANS) equations, and k-ε turbulence model with the enhanced-wall treatment was used. Thus, the turbulent flow field was time averaged. Regarding the inlet boundary condition of the nozzle, the air supply temperature and flow rate presented in Table 1 were used. The temperature dependence of the viscosity of air was estimated using the Sutherland equation. The conjugate heat transfer was solved considering the nozzle wall.

The temperature distribution for the air supply temperature 512 K is shown in Fig. 11. A potential core, where the velocity and temperature fields were nearly uniform was observed just downstream of the nozzle exit, as shown in Fig. 5. The temperature decreased toward the downstream due to the surrounding air entrainment. Figure 12 shows the turbulent kinetic energy distribution for the air supply temperature 512 K. Almost no turbulence was observed in the potential core, but the turbulent kinetic energy showed the highest value around the potential core due to the turbulent entrainment.

Fig. 11
figure 11

CFD results of the temperature distribution for the air supply temperature of 512 K

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Fig. 12
figure 12

CFD results of the turbulent kinetic energy distribution for the air supply temperature of 512 K

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3.4.2 Comparison with CFD and thermocouples

Figures 13 and 14 present the comparison of the temperature at the center of axis among the BOS results, thermocouple measurements, and CFD results for the air supply temperatures of 467 and 512 K. The CFD results and thermocouple measurements exhibited good agreement, implying that the CFD results can be used to evaluate the time-averaged experimental results. The temperatures from the CFD and thermocouple measurements were almost constant along the axis because the measurement was conducted in the potential core region.

Fig. 13
figure 13

Comparison of axial temperature distribution among BOS results, thermocouple measurements, and CFD results. Air supply temperature = 467 K

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Fig. 14
figure 14

Comparison of axial temperature distribution among BOS results, thermocouple measurements, and CFD results. Air supply temperature = 512 K

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For the 467 K air supply temperature case shown in Fig. 13, case D, i.e., boundary condition with the Rayleigh scattering for both the left and right boundaries, was the most accurate. The CFD results were based on the RANS analysis, wherein turbulence was treated by time averaging. The temperatures recorded by the thermocouples were the average of each one-second interval. The wavy characteristics in the BOS results stemmed from the flow turbulence, which did not disappear even after the averaging of 400 images. Additionally, the BOS results showed the integral along the optical path; thus, the turbulence around the potential core as shown in Fig. 12 influenced the BOS results. The 467 K case showed more turbulent than the 512 K case, which was because of the Reynolds number, as shown in Table 1. In the 512 K case, where the Reynolds number and turbulence were small, no significant difference were observed between cases C and D. This indicates that for cases with low turbulence or fluctuation, one laser sheet is sufficient for measurement. However, in the 467 K case, where the turbulence was large, case D exhibited a significant advantage over case C. The results indicate that two laser sheets are preferable for measurements. The turbulence intensity was higher downstream, as shown in the schlieren images in Fig. 4. Both cases B and C used Neumann boundary condition for one boundary and Rayleigh scattering for the other. However, case C was better than case B because LRS technique was better suited for measuring in more turbulent region. The results showed that Rayleigh scattering is especially effective in specifying boundary conditions in highly turbulent, unsteady, or non-uniform flow fields.

Figure 15 exhibits the temperature distribution near the nozzle exit for BOS with case D and CFD results with the air supply temperatures of 467 and 512 K. The potential core with a constant temperature was observed for all cases, and the BOS results accurately depict the flow field’s characteristics. Even in turbulent flows, it is possible to have valuable discussions about the mean flow characteristics. In the BOS results in Fig. 15, the influence of the turbulence region generated by entraining stationary surrounding air around the potential core was observed as the fluctuation because BOS has the inherent characteristic of providing line-of-sight integrated images. In this study, we primarily focused on axisymmetric flow; however, for two-dimensional flows, where BOS excels, it enables us to engage in more accurate discussions. Furthermore, when dealing flows without strong turbulence, the simultaneous measurement of BOS/LRS has the potential to capture the temporal evolution of the flow field.

Fig. 15
figure 15

Comparison of temperature distribution between BOS with Rayleigh scattering boundary condition (Case D in Table 2) and CFD results

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4 Conclusions

This study presented a novel method to quantitatively determine the density distribution of a wide area of a flow field with high accuracy. BOS has limitations in setting boundary conditions, while LRS disadvantageously requires a high-power laser for wide-area measurements. Therefore, a method that complements both BOS and LRS was proposed, i.e., BOS for wide-area imaging and LRS for setting the boundary condition for BOS as a Dirichlet boundary condition considering density distribution. Four types of boundary conditions for BOS were investigated for an axisymmetric heated turbulent jet. The time averaging of each image was performed to eliminate the effect of turbulence for comparison with the RANS analysis and thermocouple measurement results. The comparison demonstrated that the boundary condition defined by LRS yielded the most accurate result, indicating the applicability and usefulness of the proposed method. The results also emphasized the significance of boundary conditions in BOS and illustrated the setup guidelines.

It was found that the proposed method is useful to discuss the mean flow filed when the flow at the boundary of BOS is highly turbulent, unsteady, or non-uniform. In addition, for flows without strong turbulence, even with large spatial variation, the density of the instantaneous flow field can be obtained by simultaneously employing the BOS/LRS method. In addition, BOS could be utilized for flows with density discontinuities, such as shock waves, by applying LRS across the discontinuity and integrating the displacement using the LRS results as the anchor. However, in general, BOS is a line-of-sight integrated technique, and time variations and spatial non-uniformities along the integration path affect its accuracy. This limitation is also present in the proposed method, similar to BOS.