1. INTRODUCTION

Optical flow field measurements are indispensable for combustion diagnostics and enable an efficient design of technical combustions with regard to a reduction of fuel consumption and pollutant emission [1,2]. Optical flow measurements, however, are disturbed by light refraction caused by inhomogeneous refractive index fields of combustion flows. Particle image velocimetry (PIV) is a widely used optical measuring method that evaluates the displacement of so-called seeding particles between two successively recorded camera images. The velocity fields are approximated by the difference quotient between the particle displacements and the time difference, where the time difference is adjusted using pulsed illumination. Typically, particle diameters of about 1 µm are used, which allows the approximation that the repercussions on the flow are negligible and that the seed particles have a negligible slip. In standard PIV measurements, the particle displacement is observed in a measurement plane with one camera while the measurement plane is illuminated by a light sheet. The observation with a camera allows the determination of two velocity components parallel to the measurement plane. The third velocity component perpendicular to the measurement plane can be measured with a second camera observing the particle displacement, which is called stereoscopic PIV [3]. In a three-dimensional measurement volume, all three components of the velocity field can be measured with typically at least four cameras, which is known as tomographic PIV [4].

PIV measurements are generally subject to fundamental limits of measurability. In combustion flows, PIV measurements are challenging because of additional contributions to the measurement uncertainty. An overview of fundamental limits of imaging flow field measurement techniques is summarized by Fischer 2017 [5]. The discrete detection of the pixel (px) intensities leads to a measurement error of the particle positions [6]. Furthermore, the intensity of the light sheet is approximately Gaussian distributed. Therefore, intensity variations between the consecutive particle images occur due to the particle movement. An error of the particle displacement detection by the PIV evaluation algorithm of 0.1 px results [7], which limits the minimum achievable measurement uncertainty of the velocity field. In combustion flows, light emissions from the flame produce additional intensity variations in the particle images, which can be reduced by using optical bandpass filters on the camera with a center wavelength of the laser light illumination [8]. Further, the thermophoresis effect leads to a contribution to the PIV measurement uncertainty budget [9].

Light refractions due to inhomogeneous refractive index fields, such as those occurring in flames, also cause fundamental PIV measurement errors [10]. The measurement error due to inhomogeneous refractive index fields was recently theoretically studied for standard, stereoscopic, and tomographic PIV [11]. The resulting measurement uncertainty was estimated based on the measured inhomogeneous refractive index field of a hot jet flow. Increased measurement errors for stereoscopic and tomographic PIV compared to standard PIV were predicted. However, an experimental quantification of resulting measurement errors for other applications influenced by inhomogeneous refractive index fields was not performed. Therefore, a short summary of studies analyzing the influence of light refraction for supersonic flows, two phase flows in micro channels, and reacting flows is given.

In a supersonic flow, the particle position errors were quantified using the background-oriented Schlieren (BOS) technique [12]. Based on the measured particle position errors under the assumption that the gradients of the refractive index field remain constant in the camera viewing direction, the velocity errors were calculated. Additionally, the influence of the particle position errors in planar supersonic shock waves was analyzed regarding the angle between the observation direction and the planar shock wave [13]. The resulting position errors and velocity errors were quantified by ray-tracing simulations and experiments. However, the results for supersonic flows cannot be transferred to PIV measurements within combustion flows, because the shape of the inhomogeneous refractive index field that has a major impact on the resulting light refraction is different.

In a two-phase flow in a microchannel, the PIV measurement error due to light refraction was corrected by adaptive optics [14]. The technique is based on a phase transition that causes a jump in the refractive index. Since there is a continuous change in the refractive index in combustion flows, the technique cannot be applied here.

In flame flows, a qualitative estimation of the resulting PIV measurement errors was performed [9]. The laser light sheet is deflected, but this is negligible in flame flows on a laboratory scale. Furthermore, measurement errors of the particle positions are caused by distorted and blurred particle images. The influence of particle position errors can be neglected if the time between two successive laser pulses is shorter than the characteristic time for a significant flame front variation. However, the influence of particle motion within the curved refractive index field of the flame was not considered. The influence of fluctuating refractive index fields of a turbulent flame and a window contaminated with oil droplets was analyzed for the PIV measurement of a jet flow [15]. The disturbance was introduced in the optical path from the camera to the jet flow. Therefore, the results cannot be transferred quantitatively to measurements, where the light sheet is located inside the flame. A quantification of standard PIV measurement errors due to light refraction in combustion flows was recently performed with a direct measurement approach [16] based on the insertion of a particle position reference object. The estimated maximum systematic measurement error inside a flame flow with an equivalence ratio of 2.4 amounts to 4% and the random error to 6%. However, the quantified measurement errors are specific to the flame configuration under investigation. Thus, the influence of the Reynolds number or the flame diameter on standard PIV measurement errors inside flame flows is still an open question. Furthermore, the measurement errors for stereoscopic PIV in combustion flows are still unknown.

For the analysis of the measurement errors for stereoscopic PIV caused by light refraction in combustion flows and the influence of the Reynolds number on the errors, a quantification of the resulting systematic velocity errors for standard and stereoscopic PIV measurements in reacting flows with different Reynolds numbers is performed. In Section 2, the measurement approach for the determination of the resulting measurement errors caused by inhomogeneous refractive index fields is described. The BOS technique is used to measure the mean refractive index field of the reacting flow, and the resulting particle position error is obtained from a ray-tracing simulation. Here, also the volumetric self-calibration method is analyzed regarding remaining position errors after tomographic reconstruction. Based on the quantified position errors within the flame flows, the resulting errors of the velocity fields measured with stereoscopic or standard PIV are calculated. Section 3 describes the design of the burner and the measurement setup for BOS and stereoscopic PIV measurements. The determined position and velocity errors are analyzed in Section 4 for different volume flows and thus for different Reynolds numbers. Finally, a conclusion and outlook are given in Section 5.

2. MEASUREMENT APPROACH

A. Position Error

The position errors of PIV images are determined measuring the refractive index fields of the flame flows by the BOS technique described in Section 2.A.1 and ray-tracing simulations described in Section 2.B.2.

 

Fig. 1. Principle of BOS. The light rays reflected from a background pattern to a camera with viewing direction $\vec k$ are deflected by the refractive index field $n(\vec r,t)$. The resulting deflection angle $\vec \varepsilon$ is calculated by the distance between the background and the center of the refractive index field and the image distortion determined by a correlation based evaluation algorithm.

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1. Background-Oriented Schlieren Technique

The BOS technique measures the deflection of light rays propagating through the inhomogeneous refractive index field of the measurement object. Since it is a line-of-sight measurement, the reconstruction of the three-dimensional refractive index field with one camera viewing direction is possible only with symmetrical assumptions. Therefore, an axisymmetric burner is used to fulfill an axisymmetric distribution of the average refractive index fields. The Abel inversion is then applied to reconstruct the three- dimensional refractive index fields of the flame flows.

In Fig. 1, field and gas mixture distribution of turbulent flame flows causes the inhomogeneous refractive index fields $n(\vec r,t)$ with $\vec r = (x,y,z{)^{\rm{T}}}$, which are generally time dependent. The light rays reflected from a background pattern towards the camera with viewing direction $\vec k$ in $z$ direction are deflected on their way through the inhomogeneous refractive index field $n(\vec r,t)$ of the flame above the burner. A deflection angle $\vec \varepsilon (x,y,t)$ between the deflected light ray and the nondeflected light ray is determined by $\tan \vec \varepsilon (x,y,t) = \frac{{{{\vec \xi}_{{\rm{BOS}}}}(x,y,t)}}{{{z_{{\rm{BG}}}}}}$ with the assumptions that the nondistorted light ray from the background pattern to the camera is approximately perpendicular to the image distortion ${\vec \xi _{{\rm{BOS}}}}$ and the intersection between the deflected and nondeflected light ray is at $z \approx 0$. The assumptions are justified for long distances ${z_{{\rm{BG}}}}$ between the center of the refractive index field and the background pattern compared to the diameter of the inhomogeneous refractive index field. The image distortion ${\vec \xi _{{\rm{BOS}}}}(x,y,t)$ is calculated by a PIV evaluation algorithm from a nondistorted and distorted camera image of the background pattern. The background pattern usually consists of a randomly distributed point pattern to get high contrast between different image areas. The measured deflection angles $\vec \varepsilon (x,y,t)$ depend on the light paths through the inhomogeneous refractive index field and are described by [17]

2. Ray-Tracing Simulations

Based on the reconstructed refractive index fields of the examined flame flows, ray-tracing simulations are performed to determine the systematic particle position errors of the distorted PIV images.

A camera observing a particle at the position ${\vec r_{\rm{P}}} = ({x_{\rm{P}}},{y_{\rm{P}}},{z_{\rm{P}}}{)^T}$ inside the flame illuminated by a light sheet at ${z_{\rm{P}}}$ will record distorted particle images due to light refraction, and therefore, an error of the measured particle position $\vec \xi$ will result; see Fig. 2. The amount of light refraction affecting a light ray propagating from the particle to the camera depends on the optical path inside the refractive index field and thus on the camera viewing direction $\vec k$. This is quantitatively described by the fundamental equation for the propagation of light:

 

Fig. 2. Schematic of occurring light ray deflection of a particle located inside a refractive index field. The reflected light rays from a particle illuminated by a light sheet are deflected. The amount of light refraction and the resulting measurement error of the particle position depend on the observation light path of the camera.

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B. Velocity Error

The systematic particle position errors determined by ray-tracing cause measurement errors of the velocity fields measured with PIV. Error propagation is used to determine the resulting velocity errors for standard and stereoscopic PIV measurements inside the examined flame flows. The velocity errors $\Delta {v_i}$ in the direction of $i = x,y$ caused by particle position errors are determined by

The stereoscopic self-calibration method [22] is also affected by light refraction, which is not considered here. The stereoscopic self-calibration method uses a set of simultaneously acquired particle images. Disparity maps are computed using the cross-correlation between the particle images projected onto the measurement plane from each camera. The disparity maps are interpreted as an offset between the light sheet plane and the calibrated measurement plane. Based on triangulation, a plane is approximated to the disparity maps, which is interpreted as the light sheet plane. The discrepancy between the approximated and calibrated plane is corrected by adjusting the parameters of the calibration model. Since the disparities also map the particle position errors caused by light refraction, the correction of the measurement plane is disturbed inside flame flows. However, the particle position errors occur only locally in the measurement plane, so an averaging over the entire measurement plane reduces this effect. Nevertheless, a remaining offset of the corrected measurement plane to the light sheet plane is to be expected after self-calibration. Therefore, it is advantageous to perform the stereoscopic self-calibration method without a flame to avoid a misalignment between the light sheet plane and the calibrated measurement plane.

As a conclusion, the systematic position errors of the PIV images are simulated by ray-tracing and the velocity field is measured by PIV. Using Eq. (6), the velocity errors for the standard PIV camera or rather for each of the stereoscopic PIV cameras are calculated. For the stereoscopic PIV evaluation, the individual velocity errors of each camera are inserted into Eq. (7) to Eq. (9) to determine the velocity errors for the triangulated velocity components.

3. EXPERIMENTAL SETUP

The BOS and stereoscopic PIV measurements are applied to different premixed propane flames. A description of the used burner and flame configurations is given in Section 3.A. The BOS and PIV measurement setups are described in Section 3.B and Section 3.C, respectively.

A. Measurement Object

An axisymmetric burner is used to generate axisymmetric flames on average. A schematic of the burner is shown in Fig. 3. The volume flow of the air and propane inflow is set by mass flow controllers. The mixture chamber of the burner is filled with spheres to get a homogeneous mixture and prevent unintentional vortex structures or asymmetric flow conditions in the burner outlet. The outlet is formed by a pipe with a diameter of 41.8 mm and a length of 300 mm.

 

Fig. 3. Schematic of the burner.

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The measurement errors of the particle positions caused by the inhomogeneous refractive index fields of the flame flows depend on the distance of the observation light rays inside the refractive index field and on the amount of occurring refractive index gradients. So, the PIV measurement error depends on the temperature field and the distribution of the gas mixture, and therefore, on the flame size. In the experiments, the flame shape is varied using different volume flows with an equivalence ratio of $\phi = 2.4$. The used volume flows and the resulting Reynolds numbers are listed in Table 1.

Table 1. Air and Propane Volume Flows in ${\rm{L}}\;{\rm{mi}}{{\rm{n}}^{- 1}}$ and Reynolds Numbers for the Examined Flame Configurations

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Fig. 4. PIV measurement setup. The light sheet is located at the center of the burner outlet.

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B. BOS Measurement

The BOS measurement is performed with a sCMOS camera (5.5 Mpx) and a 50 mm focal length objective with an $f$-number of $f/{{16}}$. The distance between the background and the burner ${z_{{\rm{BG}}}}$ is 75.1 cm, and the distance between the camera and the burner is 98.6 cm. The spatial resolution of the background image is 280 µm. The background pattern is generated by the particle image generator of the software PIVlab and is illuminated by eight high power LEDs with a center wavelength of 532 nm. The camera images were taken with an exposure time of 10 ms and a global shutter.

C. PIV Measurement

The resulting flow velocity fields of the examined flames are measured by stereoscopic PIV measurements. A schematic of the measurement setup is shown in Fig. 4. The light sheet (thickness $\approx$ 1.5 mm) is generated with a pulsed laser with a pulse length of ${\lt}\!{{10}}\;{\rm{ns}}$ and a pulse energy of 200 mJ. The sCMOS cameras (5.5 Mpx) with observing directions ${\vec k_{1,2}}$ measure the particle movement inside the light sheet plane at $z = 0$ with observation angles of ${\alpha _{1,2}} = \pm {32^ \circ}$. Objectives with a focal length of 50 mm and an $f$-number of 16 are used for the cameras. The resulting projected pixel size is 48 µm. Imaging the used titan dioxide seeding particles with a diameter of 0.4 µm leads to a diffraction-limited particle image size of 3.5 px.

 

Fig. 5. Measured average deflection angles ${\varepsilon _x}$ in $x$ direction. The Reynolds numbers increase to the right direction.

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Fig. 6. Reconstructed refractive index fields of the flame configurations based on the measured average deflection angles ${\varepsilon _x}$ in $x$ direction using the Abel inversion.

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4. MEASUREMENT RESULTS

For the studied flames, the particle position errors are determined by BOS and ray-tracing, which is presented in Section 4.A. Here, also the performance of the volumetric self-calibration method is analyzed regarding the remaining position errors for tomographic reconstruction. Additionally, a verification of the assumed axisymmetry of the refractive index field and the ray-tracing algorithm is performed by comparing simulated and measured light ray deflections. The resulting velocity errors caused by the position errors of the stereoscopic PIV measurements are analyzed in Section 4.B.

A. Position Error

1. BOS Measurements

The measured average deflection angles are used to reconstruct the axisymmetric three-dimensional refractive index fields by the Abel inversion described in Section 2.A.1. The resulting particle position errors are quantified using ray-tracing simulations. The measured deflection angles are averaged over 300 single measurements captured with a repetition rate of 15 Hz. The distortion ${\xi _{{\rm{BOS}},x}}$ of the BOS images in $x$ direction is determined using a commercial iterative PIV evaluation algorithm, where the minimum interrogation window size of $16 \times 16\;{\rm{p}}{{\rm{x}}^2}$ and a grid step size of 12 px are applied. The resulting average deflection angles ${\varepsilon _x}$ in $x$ direction are shown in Fig. 5. For a confidence interval of 0.95, the resulting measurement uncertainty is less than 1%, except for a few outliers. The flames with Reynolds numbers of at least ${\rm{Re}} = 300$ show larger flame diameters, which results from the displacement of the flame front by the inner unburned core flow, as is also visible in the measured flow fields. The larger flame flows also cause larger deflection angles with up to ${\pm}1.3\;{\rm{mrad}}$. Furthermore, at the transition between the unburned core flow and the reaction zone, the temperature gradients also cause light deflections, which are opposed to the light deflections at the flame front.

The measured average deflection angles ${\varepsilon _x}$ are used for reconstruction of the refractive index fields by Abel inversion. The fields are shown in Fig. 6. Since the refractive index field is calculated for each side of the determined symmetry axis, a reconstruction artifact is visible at the center line by a small step of the refractive index values, which indicate small deviations from the axisymmetry of the flame. Thus, due to the refractive index step, the simulated light deflection is high at the center line compared to the measured light deflections and therefore, the resulting refractive index gradients at the center line are neglected for ray-tracing simulations. Despite the small deviations from axisymmetry, an accurate description of the reconstructed refractive index field is proven with verification simulations in Section 4.A.2. The refractive index fields for Reynolds numbers of at least 300 show an increased maximum refractive index in the center of the flame flows, which arises from the core flow of cold unburned propane–air mixture. The local minimum of the refractive index fields indicate the influence of the flame fronts, which are related to a smaller density due to increased temperatures.

2. Ray-Tracing Simulation

Based on the reconstructed refractive index fields, ray-tracing simulations are performed to quantify the particle position errors of the PIV measurements caused by image distortion. To prove an accurate reconstruction of the refractive index field and to verify the implemented ray-tracing algorithm, verification simulations are performed first. For verification, the resulting light ray displacement ${\xi _{{\rm{Sim}},x}}$ in $x$ direction is simulated by ray-tracing and compared to the measured displacement ${\xi _{{\rm{BOS}},x}}$. The resulting deviations are plotted in Fig. 7. In the background at a distance of $z = - 0.75\;{\rm{m}}$ behind the burner, maximum deviations of only 18 µm occur, which is about 2% of the maximum measured displacement. As a result, the description of an axisymmetric refractive index field for each side of the determined symmetry axis is assumed to be acceptable, and the ray-tracing algorithm achieves an accurate simulation of resulting light deflection in the measured refractive index field.

 

Fig. 7. Absolute deviations between the measured light ray displacements ${\xi _{{\rm{BOS}},x}}$ in the background at $z = - 0.75\;{\rm{m}}$ with BOS and the verification simulations ${\xi _{{\rm{Sim}},x}}$ by ray-tracing based on the reconstructed refractive index field applying Abel inversion.

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Fig. 8. Simulated maximum particle position errors ${\xi _x}$ in the $x$ direction for each height caused by image distortion of PIV particle images due to the inhomogeneous refractive index fields for (a) standard PIV with one vertically oriented camera to the light sheet and (b) stereoscopic PIV with a camera observation angle in the $x$, $z$ plane of ${-}{{32}}^\circ$ to the normal direction of the light sheet.

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Although only stereoscopic PIV measurements are performed, also the resulting position errors for a standard PIV setup are quantified, which allows to compare the position errors and consequently, the resulting PIV measurement errors. The simulated position errors for the standard and stereoscopic PIV setups are illustrated in Fig. 8. For the stereoscopic case, only the position errors of one camera viewing direction are shown, because the position errors for the opposite camera viewing direction are qualitatively point symmetric with respect to the $y$ axis. The maximum position errors are located in the area of the reaction zone and increase for larger Reynolds numbers. Moreover, for the stereoscopic setup, the errors reach 34 µm, which is more than four times higher than for the standard setup with a maximum position error of 9 µm. Furthermore, in contrast to the standard PIV setup, the tilted viewing direction of the stereoscopic setup results in asymmetric position errors due to the differences between the observation light paths inside the flames.

 

Fig. 9. Maximum particle position error (blue circles) due to image distortion is plotted over the height $x$ (blue circles) for (a) standard PIV and for (b) stereoscopic PIV. Additionally, the mean flame diameter ${d_{\rm{F}}}$ (red asterisks) is shown for each height $y$ in (a).

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The resulting position errors depend on various parameters of the flame flow, such as the mixture fraction, nozzle geometry, ambient conditions, and the Reynolds number. An analytical description of the position error as a function of all these parameters will be a complex task that is beyond the scope of this paper. For the investigated flames, however, a significant dependence of the position error on the mean flame diameter can be expected, since the optical path length within the refractive index field increases with larger flame diameters. Therefore, the mean flame diameter ${d_{\rm{F}}}$ is calculated by the distance between the maximum of the measured mean deflection angles for the positive and negative $x$-half axis of the flame. It should be noted that the detection of the mean flame diameter is disturbed by fluctuations of the flames above a certain height. Instantaneous flame front detection could therefore lead to smaller flame diameters. The detected mean flame diameter is a consequence of the Reynolds number and depends on the height $y$. The influence of the mean flame diameter on position error is quantified here as a first step to analyze the complex behavior of position errors within flame flows.

The maximum position errors for standard and stereoscopic PIV are plotted in Fig. 9(a) and in Fig. 9(b), respectively, over the flame height $y$ (blue circles). Additionally, the detected mean flame diameter (red asterisks) is plotted in Fig. 9(a). There is a general trend towards larger position errors for larger flame diameters. Since the Reynolds number correlates with the mean flame diameters for the investigated flames, the position error also rises for increased Reynolds numbers. Whether this trend holds beyond the investigated Reynolds number region is unclear. For each flame, the maximum position error can be expected near the burner exit. In this region, the flame is comparatively stable, causing stronger refractive index gradients on average. For flames with Reynolds numbers of at least 200, an increase in the determined mean flame diameter is apparent above a certain flame height, which can be explained by the flame fluctuations. The flickering flame smears and reduces the measured mean deflection angles, and the maximum deflection angle is shifted to a larger radial distance with respect to the symmetry axis. This increase in mean flame diameter correlates with an increase in the slope of the maximum position errors. This can be explained by increased optical path lengths in the region with increased measured deflection angles, which are accompanied by increased refractive index gradients. For the flame with a Reynolds number of 1000, the detected mean flame diameter appears scattered for flames heights larger than $y = 80\;{\rm{mm}}$. Here also a small increase of the slope of the position error is apparent. In summary, larger flames cause larger position errors. However, for any single flame, the maximum systematic position error is expected to be near the burner exit. The fluctuations of the flames cause an increase in the mean flame diameter above a certain height, which can lead to a local increase in systematic position error. This effect seems to decrease for larger Reynolds numbers. Compared with typical particle diameters of about 1 µm, the measurement errors of particle positions are of the same order of magnitude for standard PIV measurements and one order of magnitude higher for stereoscopic PIV measurements. Especially in tomographic PIV measurements, this can lead to serious problems in the tomographic reconstruction of spatial particle distribution, because the uncertainty of the measured particle positions should be smaller than the particle diameter [4].

 

Fig. 10. Schematic of the optimization process from the volumetric self-calibration method. The curved light rays cause individual position errors ${\vec \xi _{{\rm{1,2,3}}}}$ of the particle images. The optimal reconstructed particle position ${{\vec r_{{\rm{P}}^\prime}}={\vec r_{\rm{P}}} + {\vec \xi_{\rm{tomo}}}}$ is estimated by minimizing the sum of the individual position errors ${\sum^{3}_{i=1}|(\vec r_{\rm{P}}+\vec \xi_{i})-(\vec r_{\rm{P}}+{\vec \xi _{{\rm{tomo}}}})|}$. In general, a position error ${\vec \xi _{{\rm{tomo}}}}$ remains.

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Fig. 11. Remaining position errors (a) ${\xi _x}$ in $x$ direction and (b) ${\xi _z}$ in $z$ direction for tomographic PIV for the measurement plane at $y = 3.9\;{\rm{cm}}$ after applying the volumetric self-calibration method.

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For the tomographic reconstruction of the particle positions, an intersection of the lines of sight from the particle images is necessary. The occurring light refraction caused by inhomogeneous refractive index fields lead to curved lines of sight and therefore to position errors of the particle images. This results in uncertainties of the reconstructed particle positions or even a prevention of intersecting lines of sight. Disparities, which are the remaining deviations from non-exact intersecting lines of sight and thus, a measure for the reconstruction error, increases for measurements influenced by inhomogeneous refractive index fields [23]. The volumetric self-calibration method addresses this problem by optimizing the calibration parameters for accurate intersections [22]. Simultaneously acquired particle images are used for this purpose. Each particle at the position ${\vec{r}_{\rm{P}}}$ has individual position errors ${\vec{\xi}{_i}}$ of the particle images for the cameras ${i=1,2,...,N_{\rm{C}}}$, Fig. 10. In general, a position error ${\vec{\xi}_{\rm{tomo}}}$ remains. In order to quantify the remaining position errors for the investigated flame flows, four cameras with observation angles of ${-}{22.5}^{\circ}$, ${-}{7.5}^{\circ}$, 7.5°, and 22.5° and perfect calibration parameters for the non-disturbed case without refractive index field are assumed. For each camera perspective, the resulting position error due to light refraction is simulated by ray-tracing and the remaining position error of the minimization process of the self-calibration method is calculated. The remaining position error ${\xi_x}$ in $x$-direction and ${\xi_{z}}$ in $z$-direction of the tomographic reconstruction for the measurement plane at $y = 3.9\;{\rm{cm}}$ are shown in Fig. 11 for the investigated flame flows. The structure of the remaining position errors is quite complex, and the position errors in ${\xi_x}$ in $x$-direction are one order of magnitude larger than in the $z$-direction. In the main camera viewing direction, the remaining position errors generally increases. As a result, the volumetric self-calibration method improves the particle reconstruction by minimizing the disparities. Although generally a decrease of the remaining particle position errors can be expected, an optimal correction of light refraction caused by inhomogeneous refractive index fields can only be achieved by the complementary knowledge of the occurring light ray deflections.

B. Velocity Error

The determined particle position errors cause measurement errors of the velocity fields. The velocity errors as described in Section 2.B also depend on the measured velocity field. Thus, the results of the velocity fields measured with stereoscopic PIV and the quantified velocity errors are described.

 

Fig. 12. Stereoscopic PIV measurement results. The average velocity fields of the examined flame configurations are listed in Table 1. The blue arrow indicates a velocity of $2\;{{\rm{m}}^{- 2}}$.

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Fig. 13. Resulting average velocity errors for (a) standard PIV and for stereoscopic PIV in (b) $x$ direction and in (c) $z$ direction. The stereoscopic PIV errors are one order of magnitude larger than for standard PIV. The relative velocity error components are the dominant contribution to the measurement uncertainty in the area of the flame front.

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The three components of the average flow fields measured by stereoscopic PIV are shown in Fig. 12. The average flow fields are based on 500 single measurements, which are evaluated by a commercial iterative PIV evaluation algorithm with an adaptive interrogation window size. The minimal interrogation window size is set to $32 \times 32\;{\rm{p}}{{\rm{x}}^2}$ with a grid step size of 16 px. To reduce the influence of the light emissions of the flame, a sliding minimum is subtracted from the particle raw images. The velocity vector validation is performed by applying a minimum peak ratio between the two highest correlation peaks of at least 1.4 and a moving average filter on the determined velocity fields. The particle images are captured with 15 Hz repetition rate and a time between two consecutive laser pulses of 100 ms. The flows with small Reynolds numbers show a thin convective driven structure, where for flames with larger Reynolds numbers, an unburned, relatively slow, impulse driven gas flow dominates the core flow structure surrounded by the reaction zone. The out-of-plane velocity component ${v_z}$ is slower compared to the in-plane components. In theory, assuming a perfectly symmetric flame, the velocity in $z$ direction in the measurement plane at $z = 0$ should be zero. Thus, minor deviations from a symmetric flow condition exist.

In Fig. 13, the calculated systematic velocity errors are shown for the standard configuration with Eq. (6) and for the stereoscopic PIV configuration with Eqs. (7) and (9).

The systematic velocity errors in the $y$ direction are not shown since they are not significant. For calculation of the systematic standard PIV measurement error, the same measured velocity field is assumed as it is measured with the stereoscopic PIV measurements shown in Fig. 12. The PIV evaluation algorithm is based on interrogation windows with a size of $16 \times 16\;{\rm{p}}{{\rm{x}}^2}$, which results in an averaging effect of the influence of the determined particle position errors. However, this effect also occurs with the BOS evaluation algorithm. Furthermore, Abel inversion also has an averaging affect. Thus, the averaging effect of the interrogation windows in the PIV evaluation is over compensated by the determination of the particle position errors with the BOS technique. The maximum systematic velocity errors for the standard and stereoscopic PIV configurations are located in the area around the flame front. In this area, the position errors are also maximal. The maximum systematic velocity errors for standard PIV amount to $0.3\;{\rm{mm}}\;{{\rm{s}}^{- 1}}$, whereas the stereoscopic PIV measurement errors amount to $0.7\;{\rm{mm}}\;{{\rm{s}}^{- 1}}$ for the $x$ direction and $0.9\;{\rm{mm}}\;{{\rm{s}}^{- 1}}$ for the $z$ direction. The results show that the stereoscopic PIV measurement error is more strongly affected for the out-of-plane velocity component than for the in-plane component. This is caused by the different sensitivity coefficients of  ${\vec s_x} = (0.5,0.5)$ and ${\vec s_z} = (- 0.8,0.8)$. The difference between the sensitivity coefficients depends on the camera viewing direction, which is described in Section 2.B. Neglecting velocity values in $x$ direction below $0.5\;{\rm{c}}{{\rm{m}}^{- 1}}$, the maximum relative systematic velocity error is 2% for the standard PIV setup, which is of the same order of magnitude as determined by a direct measurement approach of the position error in a similar premixed propane flame flow [16]. The maximum relative systematic stereoscopic PIV measurement error amounts to 3% in the $x$ direction and 13% in the $z$ direction, where velocity values in $z$ direction below $0.5\;{\rm{c}}{{\rm{m}}^{- 1}}$ are also neglected. In the flames with Reynolds numbers of at least 300, an unburned core flow exists, which is visible due to the higher refractive index values of the unburned propane compared with the surrounding air. In this region, the resulting systematic velocity errors of $0.01\;{\rm{mm}}\;{{\rm{s}}^{- 1}}$ for standard PIV and $0,02\;{\rm{mm}}\;{{\rm{s}}^{- 1}}$ in the $x$ direction and $0.04\;{\rm{mm}}\;{{\rm{s}}^{- 1}}$ in the $z$ direction for stereoscopic PIV are comparably small. At the detected mean flame front, the systematic velocity errors are $0.05\;{\rm{mm}}\;{{\rm{s}}^{- 1}}$ for standard PIV and $0.3\;{\rm{mm}}\;{{\rm{s}}^{- 1}}$ in the $x$ direction and $0.6\;{\rm{mm}}\;{{\rm{s}}^{- 1}}$ in the $z$ direction for stereoscopic PIV. The systematic velocity errors are calculated with Eq. (6), which consists of two terms, and each of the terms contributes velocity errors of the same order of magnitude. Thus, the velocity errors depend on the gradient and on the curvature of the refractive index field since the position error $\vec \xi$ and the gradient of the position error ${\nabla _i}{\xi _i}$ depend on the gradient and the curvature of the refractive index field. The trend towards larger position errors for larger volume flows is confirmed for the velocity errors. Comparing these results to the typical measurement errors for standard PIV of about 1% to 2% [24] and stereoscopic PIV of about 1% to 2% for the in-plane and 3% to 4% for the out-of-plane component [25], the resulting systematic PIV measurement errors caused by light refraction inside flame flows can be the dominant contribution to the budget of measurement uncertainty.

5. CONCLUSION

PIV is a standard technique for contactless measurement of the velocity fields of combustion flows. The optical measurements are disturbed by inhomogeneous refractive index fields, which cause distorted particle images due to light refraction. Therefore, measurement errors of the particle positions result. The resulting errors of the measured velocity fields were not known for stereoscopic PIV and only for a certain flame configuration for standard PIV. For this reason, the systematic velocity errors for standard and stereoscopic PIV measurements for premixed propane flames with Reynolds numbers between 100 and 1000 were quantified using the BOS technique for measuring the mean refractive index fields of the flame flows. The particle position errors caused by light refraction were simulated by ray-tracing. Note that possible image blurring caused by light refraction is neglected here. Based on position errors, the resulting systematic velocity errors were calculated. Furthermore, the mean flame diameter was determined to analyze its influence on position and velocity errors.

The particle position error is up to 9 µm for the standard PIV setup and up to 34 µm for a stereoscopic PIV setup with viewing angles of ${\pm}{32^ \circ}$, which are an order of magnitude larger than for the standard PIV setup. The maximum position error occurs in the area around the flame front and a trend towards larger position errors for larger Reynolds numbers is observed. In addition, the position error is correlated to the mean flame diameter. The volumetric self-calibration method reduces disparities for tomographic reconstruction of particle positions. The disparities are a measure for the remaining particle position errors. However, although an improvement in reconstruction is achieved, particle position errors in the two-digit micrometer range remain. The resulting maximum systematic velocity error for standard PIV is $0.3\;{\rm{mm}}\;{{\rm{s}}^{- 1}}$, and for stereoscopic PIV $0.7\;{\rm{mm}}\;{{\rm{s}}^{- 1}}$ for the in-plane velocity component and $0.9\;{\rm{mm}}\;{{\rm{s}}^{- 1}}$ for the out-of-plane component. For the examined flame configurations, the position errors and velocity errors increase for larger Reynolds numbers. The relative systematic velocity errors of 2% for standard PIV as well as 3% and 13% for in-plane and the out-of-plane velocity components for stereoscopic PIV indicate a dominant contribution to the measurement uncertainty budget. Furthermore, the measurement errors for stereoscopic PIV can be one order of magnitude larger than for standard PIV. Thus, for the first time, the resulting systematic measurement errors for stereoscopic PIV caused by inhomogeneous refractive index fields in reacting flows were quantified, which in principle allows an error correction in future experimental studies by combining PIV and BOS measurements.

As an outlook, the influence of the equivalence ratio and different combustion fuels on the resulting PIV measurement error has to be clarified in future studies. Furthermore, the resulting measurement error for different burner geometries, e.g.,  swirl burners and large combustion chamber domains, has to be quantified. In general, the resulting random velocity error for stereoscopic PIV measurements is not known. Since larger random measurement errors have been determined for standard PIV measurements in a reactive flow compared to the systematic measurement error [16], a larger random measurement error can also be assumed for stereoscopic PIV measurements. In particular, the influence of turbulence on the resulting measurement error is an important issue here. For example, the random velocity error can be determined by combined simultaneous PIV and tomographic BOS measurements and by calculating the time-dependent particle position error for the measured instantaneous refractive index field. In addition, the influence of the observation angles for stereoscopic flow measurements and the resulting velocity errors of tomographic PIV measurements within flame flows due to an inhomogeneous refractive index field must be investigated.