Modelling of Variance and Co-variance in Turbulent Flame–Droplet Interaction: A Direct Numerical Simulation Analysis

Abstract

The advancement of high-performance computing has made computational fluid dynamics (CFD) simulations a viable alternative to expensive experimentation. Most industrial flows are innately turbulent, and the modelling of turbulent flow remains a challenging task. This complexity is augmented in turbulent droplet combustion simulations by the complex interaction of heat and mass transfer associated with evaporation, fluid dynamics, combustion and heat release. As a result of this, the fidelity of CFD simulations of turbulent reacting flows remains sensitive to the accuracy of the combustion modelling, which principally focuses on the prediction of mean chemical reaction and heat release rates. The closure of the mean reaction rate in the context of Reynolds-averaged Navier–Stokes (RANS) simulations in the combustion of turbulent droplet-laden mixtures often requires the knowledge of the variances of the fuel mass fraction \( {\text{Y}}_{\text{F}} \) fluctuations \( \left( {\widetilde{{{\text{Y}}_{\text{F}}^{{{\prime \prime }\,2}} }}} \right) \), the mixture fraction \( \upxi \) fluctuations \( \left( {\widetilde{{\upxi^{{{\prime \prime }\,2}} }}} \right) \) and co-variance \( \left( {\widetilde{{{\text{Y}}_{\text{F}}^{{\prime \prime }}\upxi^{{\prime \prime }} }}} \right) \), where \( {\bar{\text{q}}} \), \( {\tilde{\text{q}}} = \overline{{\uprho{\text{q}}}} /{\bar{\uprho }} \) and \( {\text{q}}^{{\prime \prime }} = {\text{q}} - {\tilde{\text{q}}} \) are Reynolds average, Favre mean and Favre fluctuation of a general quantity q and \( \uprho \) is the gas density. Algebraic and transport equation-based closures of \( \widetilde{{{\text{Y}}_{\text{F}}^{{{\prime \prime }\,2}} }} \), \( \widetilde{{\upxi^{{{\prime \prime }\,2}} }} \) and \( \widetilde{{{\text{Y}}_{\text{F}}^{{\prime \prime }}\upxi^{{\prime \prime }} }} \) have previously been considered in the context of purely gaseous phase combustion where variations in equivalence ratio exist. Whilst limited effort has been directed to the modelling of the fuel mass fraction variance \( \widetilde{{{\text{Y}}_{\text{F}}^{{{\prime \prime }\,2}} }} \) and mixture fraction variance \( \widetilde{{\upxi^{{{\prime \prime }\,2}} }} \) for turbulent combustion in droplet-laden mixtures, the statistical behaviour of \( \widetilde{{{\text{Y}}_{\text{F}}^{{\prime \prime }}\upxi^{{\prime \prime }} }} \) and its transport in turbulent spray flames are yet to be considered in detail. Furthermore, the validity of existing closures for \( \widetilde{{{\text{Y}}_{\text{F}}^{{\prime \prime }}\upxi^{{\prime \prime }} }} \) and the unclosed terms of its transport equation, which were originally proposed for purely gaseous phase combustion, are yet to be assessed for turbulent spray flames. These gaps in the existing literature are addressed by analysing the statistical behaviours of \( \widetilde{{{\text{Y}}_{\text{F}}^{{{\prime \prime }\,2}} }} \), \( \widetilde{{\upxi^{{{\prime \prime }\,2}} }} \) and \( \widetilde{{{\text{Y}}_{\text{F}}^{{\prime \prime }}\upxi^{{\prime \prime }} }} \) as well as the terms of their transport equations using a three-dimensional compressible Direct Numerical Simulation (DNS) database of statistically planar turbulent flames propagating into droplet-laden mixtures where the fuel is supplied in the form of monodisperse droplets ahead of the flame. This chapter focuses on the effects of droplet diameter \( {\text{a}}_{\text{d}} \) and droplet equivalence ratio \( \phi_{\text{d}} \) (i.e. fuel in liquid droplets to air ratio by mass, normalized by fuel-to-air ratio by mass under stoichiometric conditions) on the statistical behaviours of \( \widetilde{{{\text{Y}}_{\text{F}}^{{{\prime \prime }\,2}} }} \), \( \widetilde{{\upxi^{{{\prime \prime }\,2}} }} \) and \( \widetilde{{{\text{Y}}_{\text{F}}^{{\prime \prime }}\upxi^{{\prime \prime }} }} \) and their transport in detail. Furthermore, the validity of the existing models for the unclosed terms of \( \widetilde{{{\text{Y}}_{\text{F}}^{{{\prime \prime }\,2}} }} \), \( \widetilde{{\upxi^{{{\prime \prime }\,2}} }} \) and \( \widetilde{{{\text{Y}}_{\text{F}}^{{\prime \prime }}\upxi^{{\prime \prime }} }} \) transport equations, which were originally proposed for gaseous stratified mixture combustion, has been assessed for turbulent combustion in droplet-laden mixtures. Based on this exercise, either the modification of existing models has been suggested or new models are proposed, wherever necessary, based on the physical insights extracted from DNS data.