Statistical Analysis of Scalar Dissipation Rate Transport in Turbulent Partially Premixed Flames: A Direct Numerical Simulation Study

Abstract

Statistically planar turbulent premixed and partially premixed flames for different initial turbulence intensities are simulated for global equivalence ratios < φ >= 0.7 and < φ >= 1.0 using three-dimensional Direct Numerical Simulations (DNS) with simplified chemistry. For the simulations of partially premixed flames, a random distribution of equivalence ratio following a bimodal distribution of equivalence ratio is introduced in the unburned reactants ahead of the flame. The simulation parameters in all of the cases were chosen such that the combustion situation belongs to the thin reaction zones regime. The DNS data has been used to analyse the behaviour of the dissipation rate transports of both active and passive scalars (i.e. the fuel mass fraction Y F and the mixture fraction ξ ) in the context of Reynolds Averaged Navier-Stokes (RANS) simulations. The behaviours of the unclosed terms of the Favre averaged scalar dissipation rates of fuel mass fraction and mixture fraction (i.e. ε Y = ρ D∇Y F · ∇Y F /ρ and ε ξ = ρ D∇ξ · ∇ξ /ρ) transport equations have been analysed in detail. In the case of the ε Y transport, it has been observed that the turbulent transport term of scalar dissipation rate remains small throughout the flame brush whereas the terms due to density variation, scalar-turbulence interaction, reaction rate and molecular dissipation remain the leading order contributors. The term arising due to density variation remains positive throughout the flame brush and the combined contribution of the reaction and molecular dissipation to the ε Y transport remains negative throughout the flame brush in all cases. However, the behaviour of scalar-turbulence interaction term of the ε Y transport equation is significantly affected by the relative strengths of turbulent straining and the straining due to chemical heat release. In the case of the ε ξ transport, the turbulent transport term remains small throughout the flame brush and the density variation term is 2 Flow Turbulence Combust (2011) 86:1-44 found to be negligible in all cases, whilst the reaction rate term is exactly zero. The scalar-turbulence interaction term and molecular dissipation term remain the leading order contributors to the ε ξ transport throughout the flame brush in all cases that have been analysed in the present study. Performances of existing models for the unclosed terms of the transport equations of ε Y and ε ξ are assessed with respect to the corresponding quantities obtained from DNS data. Based on this exercise either suitable models have been identified or new models have been proposed for the accurate closure of the unclosed terms of both ε Y and ε ξ transport equations in the context of Reynolds Averaged Navier-Stokes (RANS) simulations.

Statistical Analysis of Cross Scalar Dissipation Rate Transport in Turbulent Partially Premixed Flames: A Direct Numerical Simulation Study

“…In the current study, transport quantities including viscosity μ, thermal conductivity λ and the density-weighted mass diffusivity ρ D are taken to be equal for all species and independent of temperature. Similar assumptions have been made in several DNS studies [13][14][15][16][17][24][25][26][27][28][29][30][31][32][33][34][35][36][37][38][39] in the past. In reality, μ, λ and ρ D vary non-linearly with temperature at constant pressure (e.g.…”

“…By contrast, fuel mass fraction gradient ∇Y F alignment with local principal strain rate is dependent on the relative strengths of turbulent straining and the strain rate induced by heat release [14,17,35,39]. The fuel mass fraction gradient ∇Y F is shown to align with the most extensive principal strain rate when strain rate induced by heat release overwhelms the effects of turbulent straining and vice versa [14,17,35,39]. The quantity 2ρ D∂ξ /∂ x i .∂u i /∂ x j is modelled by C p1 × ρu i ξ ×ε/k according to Mantel and Borghi [9] where C P1 is a model constant.…”

“…The statistical behaviours of T (i) 31 and T (ii) 31 are dependent on the alignment of local principal strain rates with fuel mass fraction gradient ∇Y F and mixture fraction gradient ∇ξ respectively. It has been found that mixture fraction gradient aligns preferentially with the most compressive principal strain rate [39]. By contrast, fuel mass fraction gradient ∇Y F alignment with local principal strain rate is dependent on the relative strengths of turbulent straining and the strain rate induced by heat release [14,17,35,39].…”

“…By contrast in case B, the strain rate induced by heat release tends to overcome the effects of turbulent straining which gives rise to preferential alignment of ∇Y F with the most extensive principal strain rate [14-17, 35, 39] whereas ∇ξ predominantly aligns with the most compressive principal strain rate. The alignment of ∇Y F and ∇ξ with the local principal strain rates for the flames in the current study has been studied in elsewhere and interested readers are referred to [39] for the relevant discussion. This misalignment between ∇Y F and ∇ξ in case B leads to a nonnegligible probability of finding intermediate values of cos θ Yξ (i.e.…”

“…The quantity 2ρ D∂ξ /∂ x i .∂u i /∂ x j is modelled by C p1 × ρu i ξ ×ε/k according to Mantel and Borghi [9] where C P1 is a model constant. In order to account for the competition between the straining due to turbulence and heat release induced strain rate the quantity 2ρ D∂Y F /∂ x i .∂u i /∂ x j can be modelled by an expression given by: [39] where Da * L = S b ρ 0k /(δ bρε ) is the local density-weighted Damköhler number and N B = (ρ 0 −ρ b )S b /ρ b 2k/3 is a modified Bray number [30] and is a model parameter. Under the condition where the effects of flame normal acceleration overcome the effects of turbulent velocity .…”