Modelling Sub-Grid Passive Scalar Statistics in Moderately Dense Evaporating Sprays

Abstract

Spray evaporation in spatially decaying turbulence is simulated using carrier-phase direct numerical simulations (CP-DNS). The CP-DNS cover a much wider parameter range than earlier fully resolved DNS of regular droplet arrays that were used to calibrate scaling laws for sub-grid closures of the distribution of mixture fraction and its conditionally averaged scalar dissipation. The scaling laws include the effects of sub-grid interactions between droplet evaporation and turbulence, and here they are assessed by direct comparison with the statistics from the CP-DNS. Two issues can be observed: Firstly, care must be taken when interpreting the CP-DNS statistics as the lack of resolution of the point particle surface could impact on the values in DNS cells that are located within the (unresolved) quasi-laminar wake. Secondly, the scaling laws present similar agreement with CP-DNS data as had been observed before for the fully resolved DNS. The scaling law for the scalar dissipation approximates the DNS statistics well, independent of the droplet number density, Stokes number and turbulence intensity. The estimated mixture fraction distribution (the PDF) is good for the mixture fraction values larger than a suitable average value but deteriorates for smaller mixture fractions due to inherent model limitations. The data corroborate that the scaling laws for turbulent micro-mixing can potentially serve as sub-grid closures for mixture fraction based combustion models such as flamelet and conditional moment closure approaches in large eddy simulations and may provide better approximations than existing expressions derived from single-phase non-premixed combustion.

Appendix: Derivation of the Mixture Fraction PDF in Inter-droplet Space

Here, spherical symmetry and constant thermodynamic properties are assumed for simplification. The continuity and mixture fraction transport equation for the droplet evaporation in the spherical coordinate system can be written as

$$ \frac{1}{r^{2}}\frac{d}{dr}\left( \rho u r^{2} \right) = 0, $$

(A.1)

$$ \frac{1}{r^{2}}\frac{d}{dr}\left( \rho u r^{2} f - \rho r^{2} D \frac{df}{dr}\right) = 0. $$

(A.2)

We introduce the evaporation rate of the droplet, J_m, and the integral of Eq. A.1 is then given by

$$ \rho u r^{2} = \frac{J_{m}}{4\pi}. $$

(A.3)

Equation A.2 is rewritten by substitution of Eq. A.3 as

$$ \frac{df}{dr}=\frac{d}{dr}\left( \frac{4\pi \rho D r^{2}}{J_{m}}\frac{df}{dr}\right). $$

(A.4)

The integral of Eq. A.4 from the droplet surface at r_s to a distance r is

$$ f - f_{s} = \frac{4\pi \rho D r^{2}}{J_{m}}\frac{df}{dr} - \frac{4\pi \rho D {r_{s}^{2}}}{J_{m}}{\frac{df}{dr}} |_{r=r_{s}}, $$

(A.5)

where f_s is the mixture fraction at the droplet surface. The boundary condition of f at the droplet surface is given by

$$ f_{d}J_{m} = f_{s}J_{m} - 4\pi \rho D {r_{s}^{2}} {\frac{df}{dr}} |_{r=r_{s}}, $$

(A.6)

where f_d is the mass fraction of the evaporating fuel in the droplet and it equals 1 for a single component droplet. Substituting Eq. A.6 into Eq. A.5 yields

$$ \frac{df}{dr} = \frac{J_{m}\left( f-f_{d}\right)}{4\pi r^{2} \rho D}. $$

(A.7)

The scalar dissipation N_f can be derived using Eq. A.7 as

$$ N_{f} = D\left( \frac{df}{dr}\right)^{2} = \frac{J_{m}\left( f_{d}-f\right)}{4\pi r^{2} \rho} \left|\frac{df}{dr}\right|. $$

(A.8)

The PDF of mixture fraction is evaluated by an identity formula

$$ P_{f} df \equiv c4\pi r^{2} dr, $$

(A.9)

where c is the droplet number density. Equation A.9 is rewritten with the substitution of Eq. A.8, viz.

$$ P_{f} = \frac{4\pi r^{2} c}{\left|df/dr\right|} = \frac{4\pi r^{2} c}{N_{f} \frac{4\pi r^{2} \rho}{J_{m}\left( f_{d}-f\right)}}=\frac{cJ_{m}\left( f_{d}-f\right)}{\rho N_{f}} \approx \frac{cJ_{m}\left( f_{d}-f_{2}\right)}{\rho N_{f}}. $$

(A.10)

It is important to note that the identity in Eq. A.9 assumes lack of interaction between the mixing fields of the different droplets and accounts for multiple droplets by simple multiplication by c. This will be - as shown in this paper - a sufficiently accurate approximation for dilute sprays but will lead to larger inaccuracies for P_f with f smaller than the average mixture fraction value, $f < \tilde {f}$, if the droplet loading is high as for these values the mixing fields will interact.