Droplet spatial distribution in a spray under evaporating and reacting conditions


In order to study spray combustion, an experimental test rig was developed at ONERA to partially characterize the flow conditions inside the combustion chamber of a gas turbine. Experimental campaigns using laser-based diagnostics were performed to provide an experimental database under reacting and non-reacting conditions. The paper first describes the Mie scattering image-processing to detect the droplets in the spray, and to calculate 2D maps of droplet number density and mean inter-droplet distance. The method is subsequently used to investigate the spray behavior under both reacting and non-reacting conditions according to global-averaging and phase-averaging methods. Experimental findings on the spatial droplet distribution in the spray are compared to the simple regular grid distribution and the Hertz–Chandrasekhar distribution. Results show that, under both conditions, there is an affine relationship between the inverse square root of the mean droplet number density and the nearest-neighbor inter-droplet distance. Moreover, observations suggest that the droplet spatial distribution fits more closely to a Hertz–Chandrasekhar distribution than a simple regular grid distribution, which may bring new insight for spray modeling.

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Lola Rousseau is supported by a doctoral grant from ONERA and Région Occitanie.

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Assuming that droplets are uniformly randomly distributed in a two-dimensional spray with a droplet number density\(\overline{n}\), then, for a single droplet, the probability of having no droplet in a sphere (with a radius\(r<dnn\)) centered on this droplet can be estimated by the Poisson law:

$$P\left( {\lambda ,k} \right) = \frac{{\lambda^{k} }}{k!}\exp \left( { - \lambda } \right)$$

where the parameter \(k\) corresponds to the number of droplets enclosed in a surface and \(\lambda\) is the product of the mean density and the surface. Considering a droplet, the probability \(H\left(dn{n}_{i}\right)\) that its nearest-neighbor is located at a distance \(dn{n}_{i}\) is equivalent to the probability of having no droplet enclosed in the surface \(\pi *dn{n}_{i}^{2}\) times the probability of having no droplet between \(dn{n}_{i}\) and \(dn{n}_{i}+d(dn{n}_{i})\). Thus, the probability is defined as:

$$H\left( {dnn_{i} } \right) = 2\pi \overline{n} dnn_{i} *\exp \left( { - \pi \overline{n} dnn_{i}^{2} } \right)$$

By definition, the mean value of the nearest-neighbor is:

$$\overline{{dnn_{i} }} = \mathop \int \limits_{0}^{\infty } dnn_{i} 2\pi \overline{n} dnn_{i} \exp \left( { - \pi \overline{n} dnn_{i}^{2} } \right)d\left( {dnn_{i} } \right)$$

Using the intermediate variable \(u=\pi \overline{n} dn{n}_{i}^{2}\) and after some algebraic operations, the mean value expression is:

$$\overline{{dnn_{i} }} = \frac{1}{\sqrt \pi } \overline{n}^{{ - \frac{1}{2}}} {\Gamma }\left( \frac{3}{2} \right)$$

By definition, the variance of this variable is:

$$\begin{aligned} Var(dnn_{i} ) &= \int_{0}^{\infty } {(dnn_{i} - \overline{{dnn_{i} }} )^{2} 2\pi \overline{n} dnn_{i} {\text{exp}}( - \pi \overline{n} dnn_{i}^{2} )d(dnn_{i} )} \\ &= \frac{1}{{\pi \overline{n} }}\left[ {\Gamma (2) - \Gamma } \right.\left( \frac{3}{2} \right)^{2} \Gamma \left( 1 \right) \\ \end{aligned}$$

or, equivalently, using the standard deviation:

$$\sigma_{dnn} = \frac{1}{{\sqrt {\pi \overline{n}} }} \left[ {{\Gamma }\left( 2 \right) - {\Gamma }\left( \frac{3}{2} \right)^{2} {\Gamma }\left( 1 \right)} \right]^{\frac{1}{2}}$$

These formulas use different values of the Gamma function: \(\Gamma \left(1\right)=1\), \(\Gamma \left(\frac{3}{2}\right)=\frac{\sqrt{\pi }}{2}\) and \(\Gamma \left(2\right)=1\).

By replacing these values inside the relationships, we obtain Eq. 9.

For the three-dimensional case, the reader may refer to the article of Chandrasekhar (1943).

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Rousseau, L., Lempereur, C., Orain, M. et al. Droplet spatial distribution in a spray under evaporating and reacting conditions. Exp Fluids 62, 26 (2021). https://doi.org/10.1007/s00348-020-03129-9

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