The three-dimensional reconstruction of a flame in the present work uses the schlieren optical technique to data acquisition in form of refractive index gradient and it also uses the assumption of cylindrical geometry to process the results.

### Schlieren technique theory

Schlieren technique was used to acquire the visualization of gradient of refractive index in two-dimensions caused by the hot gases in a combustion flame. The light ray propagation through a transparent media was used to describe how schlieren technique works, as shown in Fig. 1. When a light ray travels through a transparent media with a thickness *W*=*ζ*_{2}−*ζ*_{1} and a refractive index *n*=*n*(*x*,*y*,*z*) it experiences certain deflection angle forming a projection of the object in the image plane. The ray propagation in an inhomogeneous medium is well described by the eikonal equation given by [1, 2, 6]:

$$ \frac{d}{ds}\left(n \frac{d{\textbf{r}}}{ds} \right)= \nabla n\ . $$

(1)

Where *ds* is the arc length defined as *d**s*^{2}=*d**x*^{2}+*d**y*^{2}+*d**z*^{2} and **r** is a position vector. The light ray passes through the inhomogeneous medium in the *z*-direction from *ζ*_{1} to *ζ*_{2} changing *ds* into *dz* each displacement of *dz* generates a small angle deflection. Assuming the configuration depicted in Fig. 1 the Eq. (1) changes into the form:

$$ \frac{\partial}{\partial z}\left(n\frac{\partial x}{\partial z}\right) =\frac{\partial n}{\partial x}\ , $$

(2)

$$ \frac{\partial}{\partial z}\left(n\frac{\partial y}{\partial z}\right) =\frac{\partial n}{\partial y} \ , $$

(3)

integrating Eqs. (2) and (3) in both sides from *ζ*_{1} to *ζ*_{2}, it is obtained:

$$ n\left(\frac{dx}{dz}\right)_{\zeta_{2}}-n\left(\frac{dx}{dz}\right)_{\zeta_{1}} =\int_{\zeta_{1}}^{\zeta_{2}}\frac{\partial n}{\partial x}dz $$

(4)

and

$$ n\left(\frac{dy}{dz}\right)_{\zeta_{2}}-n\left(\frac{dy}{dz}\right)_{\zeta_{1}} =\int_{\zeta_{1}}^{\zeta_{2}}\frac{\partial n}{\partial y}dz \ , $$

(5)

However, when the ray propagates through *ζ*_{1} there is not deflection angle and then:

$$ \left(\frac{dx}{dz}\right)_{\zeta_{1}}=0= \left(\frac{dy}{dz}\right)_{\zeta_{1}} \ , $$

(6)

as shown in Fig. 1. Substituting now the condition expressed in Eq. (6) into Eqs. (4) and (5) respectively, these change into:

$$ \left(\frac{dx}{dz}\right)_{\zeta_{2}}=\frac{\Delta x}{f}\ , $$

(7)

$$ \left(\frac{dy}{dz}\right)_{\zeta_{2}}=\frac{\Delta y}{f}\ . $$

(8)

where *Δ**x* and *Δ**y* are the infinitesimal displacements subtended by the angles *ε*_{x} and *ε*_{y} in the *Z*−*X* and *Z*−*Y* planes, respectively.

Considering hot gases from combustion in the flame as the object of study, it can be assumed that the information obtained in *x* direction is more significant than the corresponding information in the *y*-direction due to the cylindrical shape of flame. For this reason in this work, in Eq. (7) the *x* direction was used. Afterward, taking the approximation of small angle deviations tan(*ε*_{x}) can be changed by *ε*_{x}. As shown in Fig. 1, tan(*ε*_{x})=*Δ**x*/*f*, with *f* the focal length of the optical system which projects the image of the test volume, so using the approximation *ε*_{x} = *Δ**x*/*f* as in [6] and employing Eqs. (4) and (7) it is obtained:

$$ \epsilon_{x} = \int_{\zeta_{1}}^{\zeta_{2}}\frac{1}{n}\frac{\partial n}{\partial x}dz. $$

(9)

On the other hand, from the Gladstone-Dale relation (*n*−1)=*K**ρ*, it is possible to obtain *∂**ρ*/*∂**x* which can be substituted into Eq.(9) to get:

$$ \frac{\partial \rho}{\partial x} = \frac{\delta x}{KWf}; $$

(10)

where *K* is the Gladstone-Dale constant which depends on the medium and the wavelength used in the measurement, *ρ* is the medium density, *W* is the object width, *δ**x* is the displacement in x direction when a light ray passes through an inhomogeneous medium, which is related linearly with the intensity obtained in the image plane according with Figs. 2 and 3, *f* represents the focal length of the second mirror. Integrating now the density field as described in the following equation:

$$ \rho(x) = \rho_{0}+ \frac{1}{KWf}\int_{x_{1}}^{x_{2}}\delta x dx, $$

(11)

where *ρ*(*x*) is the density of hot combustion gases along the *x* axis and *ρ*_{0} is the nominal density of air at room temperature. This result is relevant because it can be used to get the temperature *T*(*x*) along the *x* axis using the following relation where *T*_{0} stands for the reference temperature:

$$ T\left(x\right) = \frac{\rho_{0}}{\rho(x)}T_{0}\ . $$

(12)

As can be observed in Eq. (12) it is possible to obtain the temperature from a projection of the volumetric object, as was mentioned above. That is, in the two-dimensional projection of hot gases of flame there is intrinsic 3D information of the combustion process and its recovery is precisely the main objective of this work.

### 3D reconstruction of flame

The temperature function depicted in Eq. (12) was obtained from the schlieren technique, depicted in Figs. 2 and 3, applied to the volumetric object with cylindrical symmetry using the Abel transform described by the relation:

$$ g\left(x\right) =A\left[ \widehat{f} \left(r \right) \right] =\int_{x}^{\infty}\frac{\widehat{f} \left(r\right)rdr}{\sqrt{r^{2}-x^{2}}}\ . $$

(13)

Where *g*(*x*) is the Abel transform of *f*(*r*). Such that, to recover the 3D information, the inverse Abel transform should be used [7, 8]; several authors usually apply the inverse Abel transform to reconstruct a volume in three dimensions [7, 9–12]. The procedure proposed in this research for the 3D reconstruction is an alternative to the use of the Abel transform and additionally it is easy to implement, to achieve that purpose the outer product of two coordinate vectors *x* and *y* was applied using the next relation:

$$ A = A + xy^{t}, A \in \mathbb{R}^{m\times n}, x \in \mathbb{R}^{n}, y \in \mathbb{R}^{m}\ , $$

(14)

being *x* and *y* skinny matrices, such that the number of columns of *x* is equal to the number of rows in *y*, as shown in the following example:

$$ xy^{t}=(1,2,1) \left(\begin{array}{cc} &1\ \\ &2\ \\ &1\ \\ \end{array}\right)\ . $$

(15)

In this work, the *x* coordinate represents a row of temperature field associated to a height of flame, and *y*^{t} represents the transpose of *x* coordinate matrix [10]. Then using Eq. (15) the temperature slices at a specific height were obtained. Finally, doing a concatenation of all the temperatures slices, the flame volume was reconstructed.

So, under the cylindrical symmetry hypothesis, the shape of flame *h* at a specific *x* coordinate was obtained making a Gaussian fitting to the curve given by:

$$ h(x)=a e^{-(x-b)^{2}/2c^{2}}\ , $$

(16)

where *a*, *b* and *c* were calculated for each *x* coordinate, by means of minimizing the mean squared error between the estimated curve and the experimental data as shown in Fig. 8a, b, c and d.

On the other hand, it is well known that a fitting process introduces an absolute error [8], for this reason in order to validate the approximation used, the mean square error between the fitted curve and the experimental data was computed. The functions *h*(*x*) and Abel transform *g*(*x*) theoretically give the same shape when they are applied in one dimension, that means, both give the projection of light sheet that passes through a cylindrical transparent media cutting a circular slice. These functions describe the temperature profile in *x* direction obtained with the schlieren technique.