1 Introduction

In hydrocarbon flames electronically excited radicals, e.g. CH(A2Δ, B2 Σ −), OH(A2 Σ +), C2(d 3 Π g ), HCO(A2A″, B2A′), CO(A1 Π,d 3Δ,a 3 Π) and CO2(A1B2), are formed by very exothermic reactions. The concentrations of these electronic excited species in the reaction zone of air-fuel flames exceed by far their thermal equilibrium levels [2]. Therefore, the chemiluminescence of the species represents a natural inherent flame indicator of the progress of the combustion process. Because of simplicity compared to advanced laser-techniques, e.g. LIF (Laser-induced Fluorescence), Rayleigh- and Raman-scattering, CARS (Coherent Anti-Stokes Raman-Scattering) or DFWM (Degenerate Four-Wave Mixing) etc., the utilization of chemiluminescence in combustion research has gained increasing interest during the last two decades: E.g. the chemiluminescence of OH∗ in premixed laminar and moderate turbulent hydrocarbon flames [3–9] was found to be correlated with heat release rates. The latter represent the dominating physical quantity which considerably determines the stability of combustion processes [7]. Furthermore, in various investigations it was shown that ratios of chemiluminescence intensities \(I_{\mathrm{CH}^{*}}/I_{\mathrm{OH}^{*}}\) [5, 8] and \(I_{\mathrm{C2}^{*}}/I_{\mathrm{CH}^{*}}\) [9, 10] are measures for the equivalence ratio which represents an ultimate parameter for diffusion flames. Therefore, chemiluminescence measurements are a prospective tool for non-intrusive diagnostic of combustion phenomena.

Unfortunately, chemiluminescence signals suffer from a lack of spatial information compared to laser techniques. E.g., in references [11, 12] heat release rate values were obtained in turbulent premixed ethylene-air flames using 3 different methods: (1) simultaneous CH2O–OH- 2D-LIF, (2) flame surface density method and (3) OH∗-chemiluminescence. The authors showed line of sight integrated images of the OH∗-chemiluminescence giving some qualitative information about the heat release rate, but compared with other methods the latter does not allow to measure local values.

For the detection of spatially resolved chemiluminescence distributions advanced optical techniques have to be applied. For this purpose, in reference [13] a Cassegrain optical probe was introduced. The technique was improved with respect to the spatial resolution by the insertion of specific masks placed in front of the probe face [14, 15]. These probes have been used frequently to investigate the behavior of chemiluminescence signals under various combustion conditions [5, 15–21]. This technique was applied to determine the equivalence ratio in the reaction zone of laminar methane-air premixed flames by measuring the ratio of \(I_{\mathrm{CH}^{*}}/I_{\mathrm{OH}^{*}}\) [15]. As mentioned above, a monotonic dependency of the latter quantity and the equivalence ratio was found. Similar results were found in counter-flow turbulent natural gas-air flames in reference [5]. Additionally, the authors recommend that CH∗- and OH∗-emission intensities as markers for the heat release rate in the flames. On the contrary to other investigations, both the intensity \(I_{\mathrm{C2}^{*}}\) and the ratio \(I_{\mathrm{C2}^{*}}/I_{\mathrm{OH}^{*}}\) were found to be inappropriate for measurements of the heat release rate and the equivalence ratio, respectively. In reference [19] a Cassegrain probe was used to measure the air-fuel ratio in a stratified-charge single-cylinder spark-ignition engine. In reference [21] the effect of fuel type on equivalence ratio in premixed flames was investigated.

Besides spurious contamination of the measuring signal from the probe volume by chemiluminescence light emitted from excited species located along the propagation direction of the wanted signal, the limitation to single point measurements is the major drawback of Cassegrain techniques. The latter restriction can be eliminated, if optical emission tomography is used, which allows the detection of 2 or even 3-D distributions of chemiluminescence emission. Tomographic measurements carried out at a single detection angle can successfully be applied, if the chemiluminescent object under investigation possesses rotational symmetry. In this case Abel-inversion- or onion-peeling-methods can be used to estimate the three-dimensional distribution of OH∗ [22, 23]. Assuming rotational symmetry, time-averaged emission distributions of OH∗ and CH∗ in various cryogenic turbulent flames were reconstructed in references [24–27].

However, instantaneous (non-symmetric) distributions of chemiluminescence in turbulent flames can only be achieved using computed tomography. In this case the reconstruction of the chemiluminescence distribution is achieved solving the inverse problem of tomography based upon Radon transform.

The Radon integral transform R(α,s) [28] is given by

(1)

with δ[(xcosα+ysinα)−s] the one-dimensional Dirac delta function. As illustrated in Fig. 1, the OH∗-chemilumines-cence emission intensity f(x,y) only contributes to the integrated signal R(α,s) along the path of the ray s=xcosα+ysinα.

Fig. 1
figure 1

Schematic illustration of two-dimensional Radon transform

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For practical applications of tomography the Radon transform R(α,s) has to be measured experimentally across the total projected size of the object under investigation (e.g. flame) for various projection angles α κ , ranging from 0∘ to 180∘. The graphical representations of R(α κ ,s) are also called sinograms (see e.g. Fig. 6). For steady-state cases, R(α κ ,s) is measured at different angles α κ by a single linear (one-dimensional) detector, which is rotated around the object under investigation [29]. However, for e.g. (unsteady) turbulent flames this strategy fails, because temporal fluctuations of f(x,y) are fast compared to the timescale necessary to rotate the detector around the object under investigation. Consequently, a pre-selected number of detectors (k=1,…,N) has to be positioned at all detection angles α κ in order to detect R(α κ ,s) simultaneously.

Until now, only few articles are devoted to instantaneous tomographic techniques. First were Hertz and Faris who applied computed tomography for chemiluminescence measurements [30]. They investigated emission distribution of CH∗ in atmospheric methane-oxygen flame using only three simultaneous viewing angles. Later on, Ishino and Ohiwa measured a broadband chemiluminescence emitted from a fuel rich turbulent premixed propane/air flame (ϕ=1.43) [31]. In the latter work, a custom-made multi-lens (40 detection angles) camera in combination with very high-speed black and white negative film allowed the detection of generally \(\mbox{C}_{2}^{*}\)-emission (between 400 and 600 nm) originated in reaction zones of this very rich flame. Floyd et al. used five commodity CCD-cameras and system of mirrors to create a 3D tomographic setup for 10 simultaneous views [29, 32]. The cameras, equipped with standard photographic lens systems, collected the whole spectrum of visible light. In reference [29] possibilities of the tomographic setup were demonstrated for a methane-oxygen flame consisting of 21 separate laminar diffusion jet flames. In reference [32] the computed tomography apparatus was applied for investigation of visible chemiluminescence of premixed turbulent opposed jet flames of 3.6 m/s bulk velocity and equivalence ratios of 0.7, 0.8 and 0.9. With an exposure time of 250 μs the chemiluminescence distributions integrated over the complete visible spectrum were achieved in these flames.

In reference [1] we developed and tested a cost-effective optical-emission tomographic setup possessing 10 simultaneous views in various specially shaped laminar premixed flat flames as well as in a diffusion flame. The use of an arrangement consisting of only a single intensified CCD and 10 Kepler-telescopes surrounding the object under investigation in combination with fiber cables for optical signal transmission (description see below and [1]) led to a very essential increase in the sensitivity of the apparatus. Therefore, OH∗-chemiluminescence distributions in the flames mentioned above could be reconstructed in 2D planes with spatial resolution of ∼1 mm at exposure times down to 100 μs. The high sensitivity and the multiplicity of different views allow for the detection of two-dimensional maps of instantaneous OH∗-chemiluminescence also in turbulent flames. As a consequence, this approach also clears the way for instantaneous 2D measurements of chemiluminescence of different species offering prospective possibilities of 2D detection of the heat release rate and the equivalence ratio. Furthermore, if the object under investigation possesses isotropic turbulence Taylor-hypothesis can be utilized. In this case, full 3D-informations can in principle be obtained, if 2D measurements are carried out at a fixed height above the burner with high repetition rates (>1 kHz) using an appropriate high-speed camera. Unfortunately, isotropic turbulence may not be found in flames with strong recirculation zones, like our bluff-body flame.

The aim of this article is to test and to validate our tomographic apparatus in turbulent methane-air diffusion flames by reconstructing the 2D-distributions of the OH∗-chemiluminescence.

2 Tomographic setup

The tomographic system is extensively described in reference [1]. Ten angular equally separated Kepler-telescopes surround the flame in a semi-circle configuration. These telescopes serve for two major purposes: (1) to provide selective transmission of well defined (parallel and coplanar) rays, which is a prerequisite for the proper tomographic reconstruction of f(x,y). (2) To adopt the size of the object under investigation—the flame—to the detector (image) size.

As shown in Fig. 2, the OH∗-emission (λ cw=307 nm) originating from the flame is deflected into the vertical direction by UV-enhanced 45o-mirrors mounted on top of each telescope. The telescopes consist of two anti-reflection coated fused silica lenses, respectively: the objective lens (focal length \(F_{\mathrm{1@310nm}} =474 \pm 2.3~\mbox{mm}\); diameter ϕ 1=75 mm) and the ocular lens (\(F_{\mathrm{2@310nm}}= 71 \pm 0.5~\mbox{mm}\); ϕ 2=30 mm). A circular aperture placed in the focal plane of both lenses of the telescope restricts the transmission of divergent rays emitted by the flame and consequently controls the resolution of the device (see [1]). In our arrangement the aperture diameter ϕ p =15 mm is chosen because this value represents a good compromise between the signal strength on the one hand and the optical resolution of the system (∼1 mm) on the other hand. It is necessary to note that the total resolution of the tomographic reconstruction is defined by many factors: the optical resolution of the apparatus itself, the grid of the reconstruction, the experimental data quality (signal/noise ratio) and complexity of the emission distribution. Therefore, the total resolution of the apparatus can be worse than the optical one. Nevertheless, for comparably simple OH∗-distributions and rather high signal/noise-ratios the total resolution of the reconstruction should be close to the optical resolution of the device.

Fig. 2
figure 2

Schematic illustration of the tomographic reconstruction setup. Left figure: top view, right figure: side view of the arrangement consisting of 10 Kepler telescopes (optional 19). The OH∗-chemiluminescence signals propagating in the horizontal plane are deflected into the vertical direction by UV-enhanced 45∘ mirrors

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Each telescope projects the image of the flame onto a face of an optical cable consisting of 90 single fibers (core diameter: 100 μm; total diameter including cladding and polyamide jacket: 125 μm), respectively. These fibers are arranged in a single row side by side (centre to centre separation: 125 μm) forming a linear array. The magnification factor of the telescope is (\(F_{1}/F_{2})_{\mathrm{@310nm}} = 6.7\). The maximal size of the object under investigation is approximately 70 mm and is mainly restricted by the clear aperture of the first lens.

The linear arrays of fibers transmit solely OH∗-chemiluminescence signals originating from flat narrow layers or slices of the flame. The opposite ends of the fiber cables are merged together to a rectangular arrangement consisting of 10 rows×90 fibers/row=900 fibers. To avoid a crosstalk of adjacent fibers, respectively, the arrangement is optically coupled to a single image intensified CCD-camera (Princeton Instruments, ICCD PIMAX 512 RB) by a 1:1-optical relay system. The latter was modified compared with the one used in reference [1]. Instead of a pair of small (ϕ=25 mm) and short focal length (\(F_{\mathrm{@310nm}}= 120~\mbox{mm}\)) aspheric lenses we used a pair of aspheric anti-reflection coated fused silica lenses with bigger diameter and longer focal length (ϕ=50 mm, \(F_{\mathrm{@310nm}} = 120~\mbox{mm}\), respectively). The modification allows one to increase the field of view of the apparatus from 50 mm in reference [1] to the size of field of view of the telescopes themselves (70 mm). Thus, the setup registers simultaneously linear images of flame planes obtained from parallel rays at 10 different angles, respectively. All detection planes of the device were aligned with very good accuracy (<0.4 mm) with respect to each other. The alignment procedure is described in detail in reference [1]. Between the lenses and the ICCD-camera entrance window a Schott glass-filter UG11 (thickness 2 mm), which transmits predominantly OH∗-chemiluminescence signal, is used.

The registered image of the ICCD-camera is converted into a (α×s)-matrix by a special computer program. The angle (α)- and the number of fiber (s)-dependant elements of the matrix represent the integral intensity over the image area of each particular (α,s)-fiber, respectively. Therefore, the signal strength-information of the matrix elements (α,s) is the Radon transform R(α,s) multiplied by a sensitivity function F s (a,s), which is different for each particular fiber. In order to obtain F s (a,s), the device was calibrated using a laminar premixed flat flame of rectangular shape (5 mm×70 mm). The latter was placed in the centre of the setup with respect to the vertical and horizontal position, respectively. Each detection channel (α,s), consisting of (1) mirror, (2) telescope, (3) the individual fiber (α,s) and (4) the corresponding area of the ICCD was calibrated as a solid unit. Having assumed the uniformity of the flat flame, the sensitivity function F s (a,s) of each particular channel (α,s) was obtained by measuring the OH∗-signal intensity of the flat flame orientated parallel to the face of the corresponding telescope.

3 Tomographic reconstruction procedure

Equation (1) is a Fredholm integral equation of first kind with respect to the unknown function f(x,y). Therefore, the reconstruction of f(x,y) from the measured Radon transform R(α,s), the so called inverse problem of tomography, is a mathematically ill-posed problem, which can only be solved approximately [33].

First we discretize f(x,y) and R(α,s) according to our detection scheme (10 angles, 90 fibers). The step size of the discretization is chosen according to the separation between two adjacent fibers multiplied by the demagnification factor of the telescope (125 μm×6.7=0.83 mm) and the angle separation of the telescopes (18∘). The discrete Radon transform is given by

(2)

with f i,j the emission intensity of the numerical cell (i,j). I×J=90×90=8100 is the total number of numerical grid points. R k,n is the Radon transformed intensity (projection) for ray-number k at angle n(k−n ray) with K=90 the number of rays and N=10 the number of angles. \(D_{k,n}^{i,j}\) represents the projection matrix.

As extensively discussed in reference [1], the method based on minimization of the regularized function (MRF) is applied, because the latter is especially effective in both, presence of essential level of experimental noise and lack of angles covered by appropriate detectors. Additionally, MRF is very efficient for reconstructions of discontinuous functions.

For the investigations of turbulent flames short exposure times are required. In this case the level of the experimental noise plays an essential role for the quality of the reconstructed distribution. Discontinuous emission distributions reflecting the structure of thin flame zones are very usual in flames. For these reasons the minimization of the regularized function Φ(f,λ) is the method of choice for our purposes in order to reconstruct the OH∗-distribution properly. In a discrete form the regularized function is given by

(3)

with the regularization parameter λ. This parameter serves as a weighting factor of the last term on the right hand side of the formula. The latter represent a filter, which smoothies the solution of f(x,y). The function Φ(f i,j ,λ) is least squared sum minimized by an iterative variation of the f i,j -distribution based on the steepest descent method. The detailed description of both, the proper choice of the value of λ and the mathematical procedure in general, is given in reference [1].

4 Bluff-body burner and flames under investigation

Initially we tested the tomographic device by reconstructing the 2-D OH∗-chemiluminescence distribution emitted from well defined laminar premixed flat flames of different, pre-known shapes as well as from a laminar diffusion flames [1]. Even for signal exposure times down to 100 μs, these investigations proceeded successfully – the reconstructed distributions were in very good agreement with the pre-known shape of the structures. These promising results were an indispensable prerequisite in order to apply the method in turbulent flames. Therefore, in this paper we investigate the reconstruction of OH∗-chemiluminescence distributions in non-laminar methane-air diffusion flames.

For these investigations we built a burner similar to the one described in reference [12]. In Fig. 3 a sketch of the burner is given. In contradistinction to reference [12], the burner we used has a perforated plate to unify the flow field and a shortened plenum. The degree of porosity of the plate is 0.2 (or blockage ratio BR=0.8) and the diameter of the perforations is 1 mm. According to reference [34] the turbulence intensity on the edge of the burner can be estimated as 10 % of the flow velocity. Therefore, the perforated plate guarantied the turbulent character of the flow field.

Fig. 3
figure 3

Axial cross section of the turbulent burner

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The fuel supply tube positioned at the axis of the burner has six radial openings 2 mm below the bluff-body edge. The openings (0.7 mm in height) serve for the CH4-supply of the annular nozzle of the bluff-body.

For the investigations two different flames are used:

  1. (1)

    The first one was a low throughput flame, slightly lifted (∼3 mm) above the bluff-body. In the following this flame is called cylindrical flame, since the cross section of the flame changed downstream only slightly. The throughputs were 10 sl/min for air and 2 sl/min for CH4. The exit velocity of the cold air-methane flow was 0.43 m/s. Measurements were carried out at 13.2 and 27.4 mm height above the bluff-body edge, respectively.

  2. (2)

    The second flame was a high throughput flame attached to the bluff-body. Since the flame expands downstream with a nearly conical shape it is called conical flame in the following. The throughputs were 100 sl/min for air and 2 sl/min for CH4. The exit velocity of the cold air-methane flow was 4 m/s. Measurements were carried out at 2.0, 7.6, 13.2, 18.8 and 24.4 mm height above the bluff-body edge, respectively.

5 Results and discussion

The 2-D distributions of OH∗-chemiluminescence in the cylindrical and the conical flame were reconstructed from the experimentally obtained Radon transforms by means of the iterative procedure mentioned above. Regularization parameter was chosen same for all cases (λ=3.5) according to L-curve method extensively explained in [1].

5.1 Cylindrical flame

The reconstructed distributions of the OH∗-emission of the cylindrical flame are depicted in Fig. 4. For this flame the exit velocity of the air-fuel flow (0.43 m/s) is actually equal to the maximal value of the laminar burning velocity (0.42 m/s, see e.g. [35]) for methane-air mixtures. Therefore, the flame is nearly laminar and behaves similar to a usual laminar diffusion flame. For this reason it is interesting to compare the recent results with those obtained in a laminar diffusion flame reported in reference [1].

Fig. 4
figure 4

The reconstructed distributions of the OH∗-chemiluminescence emission of the nearly laminar (cylindrical) flame are represented in the horizontal planes at two heights above the bluff-body edge H=13.3 and 27.2 mm. Figures (a), (b), (d), and (e) represent single-shot measurements. Figures (c) and (f) represent the reconstruction averaged over 20 single-shot measurements. The exposure time is 1 ms. The approximately hexagonal shape of the flame is caused by six radial openings serving for the CH4-supply of the annular nozzle of the bluff-body

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In both cases the shape of the flame cross sections looks like a thin continuous ring-like structure. Thicknesses of both flames—2 points of the numerical grid or 1.7 mm for the diffusion flame in reference [1] and 2–3 points or 1.7–2.5 mm for the cylindrical flame—were close to the resolution limit. The latter is given by 2×125 μm×6.7=1.7 mm in analogy to the Nyquist–Shannon sampling theorem defined in a time frequency domain or 0.83 mm according to full-width at half-maximum resolution definition.

The laminar flame in reference [1] has circular shape and is stable in time. On the contrary, at the bottom cross section (13.3 mm) the cylindrical flame has almost regular hexagon shape, see Fig. 4a, b and c. This is caused by the six radial openings in the fuel supply tube. Further downstream of the flow, the shape is distorted essentially, compare Figs. 4a, b and c with d, e and f.

From the comparison of the size of the distributions in Figs. 4a and d with those of (b) and (e) one can see that the flame front pulsates, respectively, keeping a nearly permanent shape of the flame. In the upper cross section of the flame, the amplitude of the pulsations is increased compared to the one at lower height, since the averaged flame front is essentially wider: ∼4 mm for the upper cross section and ∼2.5 mm for the lower one in this case (compare Figs. 4c and f). Even so the flame front propagates relatively slow and the turbulence level is actually rather weak, the structure of the flame is relatively complex. This is a consequence of the special design of the burner, in which the perforated grid and the bluff-body lead to complex flow conditions of the air-fuel mixture. Nevertheless, the apparatus allows a successful reconstruction of the 2-D maps of the OH∗-chemiluminescence distributions of the complex flame structure.

5.2 Conical flame

In a next step we concentrate our attention to the tomographic reconstruction of the OH∗-chemiluminescence distributions from flames offering a much higher degree of turbulence. Because the exit velocity of the cold gas flow (∼4 m/s) is approximately one order of magnitude higher than the maximal value of the laminar burning velocity, the conical flame is strongly turbulent.

The reconstructed distributions of the OH∗-chemi-luminescence emission of the conical flame are given in Figs. 5 and 6 for different heights above the bluff-body edge: 2 mm (Figs. 5a and b), 7.6 mm (Figs. 5c and d), 13.3 mm (Figs. 5e and f), 18.8 mm (Figs. 5g and h), and 24.4 mm (Fig. 6). Figures 5a, c, e, g and 6a, c represent distributions reconstructed from the corresponding Radon transforms obtained by averaging 60 single-shot events, respectively. The other figures are distributions from single-shot measurements.

Fig. 5
figure 5

The reconstructed distributions of the OH∗-emission of the turbulent (conical) flame are represented in the horizontal planes at various heights above the bluff-body edge H. The exposure time is 500 μs. Figures (b), (d), (f) and (h) represent results from single-shot events and (a), (c), (e) and (g) ones are obtained by averaging of 60 single-shot measurements

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Fig. 6
figure 6

The reconstructed distributions of the OH∗-emission of the turbulent (conical) flame are represented in the horizontal plane at H=24.4 mm above the bluff-body edge at different exposure time t exp. Figure (d) represents the reconstruction of the model distribution with addition of 14 % percents equally distributed noise (explanation see in text). Figures (b), (d) and (e) represent results from single-shot events and (a), (c) ones are obtained by averaging of 60 single-shot measurements

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From the figure it becomes obvious that the flame is strongly turbulent possessing a ring-like cross-sectional shape. Since the diameter of the OH∗-distributions increases linearly downstream the reactive flow, the flame possess a nearly conical shape: At 2 mm height above the burner (Figs. 5a and b), the size of the flame is similar to the bluff-body diameter (25 mm). With increasing height the diameter of the flame increases up to approximately 55–58 mm at 24.4 mm (Fig. 6).

It can further be seen that the flame consists of several leafs. This is a consequence of the six radial openings in the fuel supply tube. With increasing detection height, adjacent leafs of the flame begin to overlap with each other. Simultaneously, the characteristic thickness of the flame zone rises from 2–3 mm at the lower height up to ∼8 mm at the higher height above the burner. Both the average as well as the single-shot distributions show a similar shape and values for the OH∗-chemiluminescence intensities. This indicates that the flame zone is actually thick otherwise the single-shot measurements should give an essentially thinner flame width.

5.3 Analysis of discrepancies

In contradiction to the average distributions, in the single-shot reconstructions essential contributions of short-wave structures are observed. From the comparison of Figs. 6b (exposure time 500 μsec) and 6e (exposure time 200 μsec) it is evident that the number of these contributions is noticeably increased with decreasing exposure time. However, the average distributions are very similar in both cases, when comparing Figs. 6a with 6c. This behavior could in principle be explained by two different reasons:

  1. (1)

    The increase of the high frequency structures for the shorter exposure time is an artefact caused by noise in the reconstruction. In this case, the high frequency structures are caused by an insufficient SNR (signal to noise ratio) of the single-shot data. The SNR is improved with increasing exposure times for the single-shot measurements or by averaging the single-shot data over many events. Therefore, these artificial fine structures do not appear in the figure with the longer exposure time.

  2. (2)

    The proper short-wave structures of the turbulent flame are depicted in Fig. 6e obtained by the shorter exposure time. In this case, the fine structures of the flame cannot be resolved in Fig. 6b, because they have been moved significantly during the exposure of the image. Therefore, these structures are washed-out and consequently vanish in the figure with the longer exposure time.

In order to find the right explanation we analyzed the sinograms of the flame at 24.4 mm height above the bluff-body edge. Figures 7c, e, g, d, f and h depict the sinograms for single-shot measurements, whereas in Figs. 7a and b the average of 60 single-shot measurements are given. Figures 7a, and c represent measured sinograms for the exposure time 500 μs in contrast to 200 μs for 7g. The OH∗-distributions reconstructed from the sinograms 7a, c and g are depicted in Figs. 6a, b and e, respectively. Figures 7b, d, f and h represent sinograms, which are (re)calculated from the distributions 6a and b, d, e using formula (2), respectively. The average measured and recalculated sinograms show good agreement. Some residual uncertainty is explained by uncertainty of calibration. From the comparison of the average sinogram (Fig. 7a) with the single-shot ones (Figs. 7c and g) it becomes obvious that the latter have an essential number of short-wave structures. Additionally, they are markedly increased for the shorter exposure time (compare Figs. 7c and g).

Fig. 7
figure 7figure 7

Sinograms at 24.4 mm height above the bluff-body edge. Figures (a), (c), (e) and (g) are measured sinograms, (b), (d), (f) and (h) are (re)calculated ones with using of sinograms (a), (c) (e) and (g). Figures (c), (d), (e), (f), (g) and (h) represent results from single-shot events and (a), (b) ones are obtained by averaging of 60 single-shot measurements. Exposure times are: t exp=500 μs for (a), (b), (c), (d), (e) and (f); t exp=200 μs for (g) and (h). Sinogram (e) represent sinogram (c) with addition of 14 % amplitude equally distributed noise (explanation see in text)

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In order to elucidate the nature of the short-wave structures we calculated the Euclidean-norm ∥f∥2 as a function of the residual ∥R−D f∥2 for a set of regularization parameter λ. These graphs (∥R−D f∥2,∥f∥2) are so called L-curves (for further details see [1]), which characterize the impact of experimental errors on the quality of the reconstruction.

The L-curves in Fig. 8 are calculated for the sinograms depicted in Fig. 7. In contradiction to Ref. [1], the curves depicted in Fig. 8 are normalized by exposure time (∥R−D f∥2/t exp,∥f∥2/t exp), which allows direct comparison of events taken at different exposure times. Indeed, these normalized L-curves calculated for averaged sinograms are very close to each other (compare diamonds and squares in Fig. 8), although the exposure times are essentially different. Obviously, the stronger signals for the longer exposure time leads to an increase in f which is compensated in (∥R−D f∥2/t exp,∥f∥2/t exp) by the increase in t exp.

Fig. 8
figure 8

L-curves normalized by exposure time t exp. The point of the regularization parameter λ=3.5 used for all reconstructions is pointed out with the arrows

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In contrast to the averaged sinograms, the L-curve for 200 μs exposure time (circles) differs from the one taken at 500 μs (triangles). The first is shifted to the upper right compared to the latter, which does not contradict to one of both hypotheses. In order to judge the influence of noise with respect to the shape of recalculated sinograms, we increased the amplitude of equally distributed noise added to each point of the sinogram of 500 μs exposure time until the L-curve of the modified sinogram (crosses) coincided to the L-curve calculated for 200 μs exposure time. This was achieved by the noise peak to peak value of 14 % of the global maximum of the 500 μs sinogram. The latter sinogram modified by noise is depicted in Fig. 7e. Nevertheless, even so the added noise leads to an increase in the short-wave structures of Fig. 7e compared to 7c, the recalculated sinograms look very similar for cases Fig. 7d and f, respectively. Obviously, the reconstruction procedure eliminates high frequency, self-uncorrelated noise with respect to Radon transform with very high efficiency.

On the contrary, the comparison between the measured (Fig. 7c and g) and the corresponding recalculated sinograms (Figs. 7d and h) shows that the short-wave contributions in the recalculated sinograms confidently reflects the contributions in the respective measured ones. In other words: the high frequency contributions of Fig. 7g are eliminated to a much lesser extent in Fig. 7h compared to Figs. 7e and 7f. This is an indication that the short-wave structures at the shorter exposure time reflect an inner fine structure of the turbulent flame rather than being an artefact caused by noise.

Another interesting observation gives the comparison of the distributions (Figs. 6b and d) calculated from the sinograms depicted in Figs. 7c and e, respectively. Although the “noisy” distribution (Fig. 6d) contains an additional variety of small points with rather low intensity, the main structure of both 500 μs distributions are very similar. On the contrary, the main structure of the 200 μs distribution (Fig. 6e) differs from those of 500 μs (Figs. 6b and 6d) essentially. In spite of the amounts of rather big and intense spots are approximately the same for both exposure times, the characteristic size of the spots is essentially less in the case of 200 μs. These behaviors of the distributions are naturally explained by presence of short-wave structures in the flame.

Therefore, an instant emission distribution of OH∗-chemiluminescence of a turbulent flame can be reconstructed within the spatial resolution of the device (∼1 mm, see [1]).

6 Conclusions

Our recently developed fast optical tomographic device described in reference [1] has been successfully applied to reconstruct the two-dimensional distributions of the OH∗-chemiluminescence in two significantly different non-premixed flames at various heights above the burner.

The first flame is slightly lifted and possesses more or less a cylindrical shape. From the reconstructed distributions of the OH∗-chemiluminescence of this near laminar flame one could nicely see that the flame has a thin continuous ring-like structure with the characteristic thickness close to the limit of the device resolution. The flame pulsates more or less keeping its form. With the height above burner the amplitude of the pulsations slightly increases.

The second investigated—non-lifted—conical flame is strongly turbulent. From the reconstruction it becomes obvious that

  • the flame possesses a ring-like cross section for the OH∗-chemiluminescence distribution and expands downstream with nearly conical shape

  • the flame consists of several leafs caused by a corresponding number of radial openings in the fuel supply tube

  • adjacent leafs slightly overlap at higher heights above the burner

  • the thickness of reaction zone increases with increasing heights above the burner

  • short-wave structures exist

  • the number of these short-wave structures increase if a shorter exposure time is chosen

  • these fine structures are real features of the flame rather than being an artefact caused by noise.

The results presented above show clearly that the tomographic device has successfully been used for spatially resolved measurements of the OH∗-chemiluminescence emission in turbulent flames. The detailed discussion further reveals that information about the fine structure from turbulent flames can be obtained by the technique. Therefore, the method is a useful tool, to get deeper insight into the understanding of turbulent combustion.