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ABSTRACT
Three-dimensional direct numerical simulations (DNS) data of statistically planar turbulent spray flames propagating into monodisperse droplets for different values of droplet diameter ad and droplet equivalence ratio ϕd have been used to analyze the statistical behavior of the fuel mass fraction variance and its transport in the context of Reynolds-averaged Navier–Stokes (RANS) simulations. The algebraic closure, which was previously derived for high Damköhler number turbulent stratified mixture combustion, has been shown not to capture statistical behavior of
for turbulent spray flames, because the underlying assumptions behind the original modeling are invalid for the cases considered in this analysis. The modeling of the unclosed terms of the variance
transport equation (i.e., the turbulent transport term T1, the reaction rate contribution T3, the evaporation contribution T4, and the dissipation rate term –D2) has been analyzed in the context of RANS simulations. The models previously proposed in the context of turbulent gaseous stratified flames have been considered here to assess their suitability for turbulent spray flames. Model expressions have been identified for
and −D2 which have been shown to perform satisfactorily in all cases considered in the current study. However, the model previously proposed for T3 in the context of turbulent gaseous stratified flames has been found to be inadequate for turbulent spray flames and further consideration of the modeling of this unclosed term is therefore necessary.
Nomenclature
| Arabic | = | |
| ad | = | droplet diameter |
| A | = | coefficient which determines fuel mass fraction distribution on Burke–Schumann diagram |
| Bd | = | Spalding mass transfer number |
| c | = | reaction progress variable |
| CP | = | specific heat at constant pressure |
| Cu | = | correction for drag coefficient |
| Cv | = | specific heat at constant volume |
| = | model parameters | |
| D | = | mass diffusivity |
| D0 | = | diffusivity in unburned gas |
| D1 | = | molecular diffusion term in the variance transport equation |
| D2 | = | dissipation term in the variance transport equation |
| Da | = | Damköhler number |
| = | turbulent kinetic energy | |
| L11 | = | integral length scale for turbulent velocity fluctuation |
| LV | = | latent heat of droplet evaporation |
| m | = | model parameter |
| Nuc | = | corrected Nusselt number for droplets |
| p | = | pressure |
| = | partial pressure at the droplet surface | |
| P(YF) | = | PDF of fuel mass fraction YF |
| P(ξ|YF) | = | PDF of mixture fraction ξ conditional on fuel mass fraction YF |
| P(YF, ξ) | = | joint PDF between fuel mass fraction YF and mixture fraction ξ |
| = | Favre joint PDF between fuel mass fraction YF and mixture fraction ξ | |
| Pr | = | Prandtl number |
| = | Reynolds-averaged value of a general quantity | |
| = | Favre-averaged value of a general quantity | |
| q″ | = | Favre fluctuation of a general quantity |
| Red | = | droplet Reynolds number |
| s | = | ratio of oxidizer to fuel by mass under stoichiometric condition |
| S | = | segregation factor |
| Smod | = | modified segregation factor |
| Sc | = | Schmidt number |
| Shc | = | corrected Sherwood number |
| = | unstrained laminar burning velocity at equivalence ratio ϕg | |
| t | = | time |
| tchem | = | chemical timescale |
| te | = | initial turbulent eddy turnover time |
| T | = | nondimensional temperature |
| = | dimensional temperature | |
| = | adiabatic flame temperature | |
| Td | = | dimensional droplet temperature |
| T0 | = | unburned gas temperature |
| T1 | = | turbulent transport term in the variance transport equation |
| T2 | = | generation/destruction term in the variance transport equation due to scalar flux |
| T3 | = | reaction rate contribution to the variance transport equation |
| ui | = | ith component of nondimensional fluid velocity |
| u′ | = | root mean square fluctuation velocity |
| = | droplet velocity vector | |
| WF, WO | = | molecular weight of fuel and oxidizer |
| = | droplet position vector | |
| xi | = | ith Cartesian coordinate |
| YF | = | fuel mass fraction |
| YF∞ | = | fuel mass fraction in pure fuel stream |
| YFst | = | fuel mass fraction under stoichiometric condition |
| Ymax and Ymin | = | maximum and minimum values of fuel mass fraction according to the Burke–Schumann relation |
| YO | = | oxidizer mass fraction |
| YO∞ | = | oxidizer mass fraction in pure air stream |
| Greek | = | |
| α | = | heat release parameter |
| αT | = | thermal diffusivity |
| αW | = | parameter in the presumed joint PDF |
| α1, α2, α4 | = | model parameters |
| β1, β2, β4, βϵ | = | model parameters |
| γ | = | ratio of specific heats of constant pressure to constant volume in gaseous phase |
| γ4 | = | model parameter |
| δth | = | thermal laminar premixed flame thickness for the stoichiometric mixture |
| = | dissipation rate of turbulent kinetic energy | |
| = | dissipation rate of fuel mass fraction variance | |
| = | dissipation rate of mixture fraction variance | |
| η | = | Kolmogorov length scale |
| λ | = | thermal conductivity of the gaseous phase |
| λW | = | parameter in the presumed joint PDF |
| μ | = | dynamic viscosity |
| μt | = | eddy viscosity |
| ξ | = | mixture fraction |
| ξmax and ξmin | = | maximum and minimum values of mixture fraction within the domain of definition |
| ξst | = | mixture fraction under stoichiometric condition |
| ψ, ψ1 | = | general primitive variable |
| ρ | = | gas density |
| ρd | = | droplet density |
| ρ0 | = | unburned gas density |
| σ | = | turbulent Schmidt number |
| τ | = | heat release parameter |
| = | relaxation/decay timescales for droplet velocity, diameter, and temperature | |
| ϕd | = | droplet equivalence ratio |
| ϕg | = | equivalence ratio in gaseous phase |
| = | reaction rate of fuel | |
| = | fuel reaction rates when the fuel mass fraction values are given by YF11 and YF12 (YF21 and YF22) respectively at a mixture fraction ξ41 (ξ42). | |
| ΩY | = | the term given by |
| Subscript | = | |
| d | = | droplet (i.e., in liquid phase) |
| g | = | gaseous phase |
| l | = | liquid phase |
| ref | = | reference value |
| Superscript | = | |
| g | = | gaseous phase |
| s | = | saturated state |
Nomenclature
| Arabic | = | |
| ad | = | droplet diameter |
| A | = | coefficient which determines fuel mass fraction distribution on Burke–Schumann diagram |
| Bd | = | Spalding mass transfer number |
| c | = | reaction progress variable |
| CP | = | specific heat at constant pressure |
| Cu | = | correction for drag coefficient |
| Cv | = | specific heat at constant volume |
| = | model parameters | |
| D | = | mass diffusivity |
| D0 | = | diffusivity in unburned gas |
| D1 | = | molecular diffusion term in the variance transport equation |
| D2 | = | dissipation term in the variance transport equation |
| Da | = | Damköhler number |
| = | turbulent kinetic energy | |
| L11 | = | integral length scale for turbulent velocity fluctuation |
| LV | = | latent heat of droplet evaporation |
| m | = | model parameter |
| Nuc | = | corrected Nusselt number for droplets |
| p | = | pressure |
| = | partial pressure at the droplet surface | |
| P(YF) | = | PDF of fuel mass fraction YF |
| P(ξ|YF) | = | PDF of mixture fraction ξ conditional on fuel mass fraction YF |
| P(YF, ξ) | = | joint PDF between fuel mass fraction YF and mixture fraction ξ |
| = | Favre joint PDF between fuel mass fraction YF and mixture fraction ξ | |
| Pr | = | Prandtl number |
| = | Reynolds-averaged value of a general quantity | |
| = | Favre-averaged value of a general quantity | |
| q″ | = | Favre fluctuation of a general quantity |
| Red | = | droplet Reynolds number |
| s | = | ratio of oxidizer to fuel by mass under stoichiometric condition |
| S | = | segregation factor |
| Smod | = | modified segregation factor |
| Sc | = | Schmidt number |
| Shc | = | corrected Sherwood number |
| = | unstrained laminar burning velocity at equivalence ratio ϕg | |
| t | = | time |
| tchem | = | chemical timescale |
| te | = | initial turbulent eddy turnover time |
| T | = | nondimensional temperature |
| = | dimensional temperature | |
| = | adiabatic flame temperature | |
| Td | = | dimensional droplet temperature |
| T0 | = | unburned gas temperature |
| T1 | = | turbulent transport term in the variance transport equation |
| T2 | = | generation/destruction term in the variance transport equation due to scalar flux |
| T3 | = | reaction rate contribution to the variance transport equation |
| ui | = | ith component of nondimensional fluid velocity |
| u′ | = | root mean square fluctuation velocity |
| = | droplet velocity vector | |
| WF, WO | = | molecular weight of fuel and oxidizer |
| = | droplet position vector | |
| xi | = | ith Cartesian coordinate |
| YF | = | fuel mass fraction |
| YF∞ | = | fuel mass fraction in pure fuel stream |
| YFst | = | fuel mass fraction under stoichiometric condition |
| Ymax and Ymin | = | maximum and minimum values of fuel mass fraction according to the Burke–Schumann relation |
| YO | = | oxidizer mass fraction |
| YO∞ | = | oxidizer mass fraction in pure air stream |
| Greek | = | |
| α | = | heat release parameter |
| αT | = | thermal diffusivity |
| αW | = | parameter in the presumed joint PDF |
| α1, α2, α4 | = | model parameters |
| β1, β2, β4, βϵ | = | model parameters |
| γ | = | ratio of specific heats of constant pressure to constant volume in gaseous phase |
| γ4 | = | model parameter |
| δth | = | thermal laminar premixed flame thickness for the stoichiometric mixture |
| = | dissipation rate of turbulent kinetic energy | |
| = | dissipation rate of fuel mass fraction variance | |
| = | dissipation rate of mixture fraction variance | |
| η | = | Kolmogorov length scale |
| λ | = | thermal conductivity of the gaseous phase |
| λW | = | parameter in the presumed joint PDF |
| μ | = | dynamic viscosity |
| μt | = | eddy viscosity |
| ξ | = | mixture fraction |
| ξmax and ξmin | = | maximum and minimum values of mixture fraction within the domain of definition |
| ξst | = | mixture fraction under stoichiometric condition |
| ψ, ψ1 | = | general primitive variable |
| ρ | = | gas density |
| ρd | = | droplet density |
| ρ0 | = | unburned gas density |
| σ | = | turbulent Schmidt number |
| τ | = | heat release parameter |
| = | relaxation/decay timescales for droplet velocity, diameter, and temperature | |
| ϕd | = | droplet equivalence ratio |
| ϕg | = | equivalence ratio in gaseous phase |
| = | reaction rate of fuel | |
| = | fuel reaction rates when the fuel mass fraction values are given by YF11 and YF12 (YF21 and YF22) respectively at a mixture fraction ξ41 (ξ42). | |
| ΩY | = | the term given by |
| Subscript | = | |
| d | = | droplet (i.e., in liquid phase) |
| g | = | gaseous phase |
| l | = | liquid phase |
| ref | = | reference value |
| Superscript | = | |
| g | = | gaseous phase |
| s | = | saturated state |
Acknowledgments
The authors are grateful to EPSRC UK and N8/ARCHER for financial and computational support, respectively.