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Enstrophy is an intrinsic feature of turbulent flows, and its transport properties are essential for the understanding of premixed flame-turbulence interaction. The interrelation between the enstrophy transport and flow topologies, which can be assigned to eight categories based on the three invariants of the velocity-gradient tensor, has been analysed here. The enstrophy transport conditional on flow topologies in turbulent premixed flames has been analysed using a Direct Numerical Simulation database representing the corrugated flamelets (CF), thin reaction zones (TRZ) and broken reaction zones (BRZ) combustion regimes. The flame in the CF regime exhibits considerable flame-generated enstrophy, and the dilatation rate and baroclinic torque contributions to the enstrophy transport act as leading order sink and source terms, respectively. Consequently, flow topologies associated with positive dilatation rate values, contribute significantly to the enstrophy transport in the CF regime. By contrast, enstrophy decreases from the unburned to the burned gas side for the cases representing the TRZ and BRZ regimes, with diminishing influences of dilatation rate and baroclinic torque. The enstrophy transport in the TRZ and BRZ regimes is governed by the vortex-stretching and viscous dissipation contributions, similar to non-reacting flows, and topologies existing for all values of dilatation rate remain significant contributors.
Turbulent flows are inherently rotational in nature and the extent of this rotation is often quantified in terms of vorticity
Turbulent flows exhibit eight generic canonical local flow configurations underneath an apparently random fluid motion2, 3, and these distinct flow topologies are categorised depending on the values of first, second and third invariants (i.e. P, Q and R, respectively) of the velocity gradient tensor, A ij=u i/x j=S ij+W ij, where the symmetric strain-rate tensor is S ij=0.5(A ij+A ji) and the anti-symmetric rotation rate tensor is W ij=0.5(A ij−A ji). Three eigenvalues of A ij, λ 1, λ 2, and λ 3, are the solutions of the characteristics equation λ 3+Pλ 2+Qλ+R=0 with its invariants P, Q and R as specified below2:
The discriminant, D=[27R
2]/108, of the characteristic equation λ
2+Qλ+R=0 divides the P−Q−R phase-space into two regions: the focal (D>0) and nodal (D<0) topologies2, 3. The A
ij tensor exhibits one real eigenvalue and two complex conjugate eigenvalues for focal topologies, whereas A
ij shows three real eigenvalues for nodal topologies. The surface D = 0 leads to two subsets r
1a and r
1b in P−Q−R phase space which are given by Refs. 2 and 3: r
2)3/2/27 and r
1b = P(Q−2P
2)3/2/27. In the region D>0, A
ij has purely imaginary eigenvalues on the surface r
2, which are given by R=PQ. The surfaces r
1b and r
2, where r
2 is described by PQ−R=0, divide the P−Q−R phase space into eight generic flow topologies, referred to as S1-S8, as shown schematically in Fig. 1
2, 3. To date, a large body of literature4–11 concentrated on the local flow topology distribution from various viewpoints for incompressible fluids where
In turbulent flow, the enstrophy (i.e. Ω=ω i ω i/2 where ω i is the i th component of vorticity) and vorticity fields affect the statistical behaviours of the second and third invariants, Q and R, which in turn affect the distributions of local flow topologies S1–S8. The instantaneous transport equation of Ω=ω i ω i/2 is given by23–30:
The term T I is the vortex-stretching contribution to the enstrophy transport, whereas term T II arises due to misalignment of gradients of density and viscous stresses. The term T III is responsible for molecular diffusion and dissipation of enstrophy due to viscous action, whereas the term T IV signifies the dissipation of enstrophy due to dilatation rate. The term T V is the baroclinic torque term which arises due to misalignment between pressure and density gradients.
Hamlington et al.23 concentrated on different mechanisms of vorticity generation and the alignment of vorticity with local principal strain rates in the flames representing the thin reaction zones regime of premixed turbulent combustion. Chakraborty25 showed that the alignment of vorticity with local principal strain rates is significantly affected by the regime of combustion and the characteristic Lewis number (i.e. ratio of thermal diffusivity to mass diffusivity). Both studies23, 25 reported a predominant alignment of vorticity with the intermediate principal strain rate in turbulent premixed flames, which is similar to non-reacting turbulent flows31. However, the relative alignment of vorticity with the most extensive and most compressive principal strain rates is significantly affected by the strength of dilatation rate25, which is influenced by the regime of combustion and the characteristic Lewis number. Hamlington et al.23 showed that enstrophy decays significantly in the burned gas across the flame brush, whereas Treurniet et al.24 demonstrated an opposite trend for the flames with high density ratio between the unburned and burned gases. This behaviour has been explained by Lipatnikov et al.26 by analysing the terms of enstrophy and vorticity transport equation for weakly turbulent premixed flames representing the corrugated flamelets regime.
Recent analyses by Chakraborty et al.27 and Dopazo et al.30 demonstrated that the characteristic Lewis number significantly affects the vorticity generation within the flame, and a combination of augmented flame normal acceleration and high extent of flame wrinkling for small values of Lewis number may give rise to a significant amount of vorticity generation within the flame brush due to baroclinic torque. Bobbit and coworkers28, 29 demonstrated that the enstrophy transport in statistically planar flames propagating in homogeneous isotropic turbulence for large values of Karlovitz number is governed by the relative balance between the vortex-stretching and viscous dissipation similar to its non-reacting counterpart, and also revealed that the choice of chemical reaction mechanism does not affect the qualitative nature of the enstrophy transport. The enstrophy field in turbulent premixed flames using cinema- stereoscopic particle image velocimetry (PIV) measurements of rim-stabilised turbulent premixed flames has been investigated32–34 and confirmed some of the observations based on DNS data.
The aforementioned analyses23–29, 31–34 provided invaluable insight into the flame-turbulence interaction, but the nature of the enstrophy transport conditional on generic local flow topologies S1–S8 in different premixed combustion regimes is yet to be analysed in detail. Such an analysis is expected to reveal the canonical flow configurations which make dominant contributions to the enstrophy transport for different combustion regimes.
In order to address the aforementioned gap in the existing literature and to meet the above objectives an existing detailed chemistry DNS database21, 35 has been considered consisting of three H 2-air flames with an equivalence ratio of ϕ=0.7, representative of the combustion processes in the corrugated flamelets, thin reaction zones, and broken reaction zones regimes of combustion. A detailed chemical mechanism36 involving 9 steps and 19 chemical reactions is considered for this analysis. The unburned gas temperature T 0 is taken to be 300 K, which gives rise to an unstrained laminar burning velocity S L=135.62 cm/s under atmospheric pressure. The inlet values of normalised root-mean-square turbulent velocity fluctuation u′/S L, turbulent length scale to flame thickness ratio l T/δ th, Damköhler number Da=l T S L/u′δ th, Karlovitz number Ka=(ρ 0 S L δ th/μ 0)0.5(u′/S L)1.5(l T/δ th)−0.5 and turbulent Reynolds number Re t=ρ 0 u′l T/μ 0 for all cases are presented in Table 1, where ρ 0 is the unburned gas density, μ 0 is the unburned gas viscosity, l T is the most energetic length scale, δ th = (T ad−T 0)/max|T|L is the thermal flame thickness with T, T ad and T 0 being the instantaneous dimensional temperature, adiabatic flame temperature and unburned gas temperature, respectively, and the subscript ‘L’ is used to refer to unstrained laminar flame quantities.
The cases investigated in this study are representative of three regimes of combustion: case A: corrugated flamelets (Ka < 1), case B: thin reaction zones (1<Ka<100) and case C: broken reaction zones regime (Ka>100)22. The Karlovitz number can be scaled as
The distributions of natural logarithm (to cover the wide dynamic range) of normalised enstrophy field (
Figure 2 (top row) further shows that in case A the enstrophy value increases locally from the unburned to the burned gas side of the flame due to flame-induced vorticity generation. In contrast, the magnitude of enstrophy decreases from unburned to burned gas side of the flame for cases B and C. Furthermore, Fig. 2 indicates that S1, S2, S3 and S4 topologies are predominantly present in the burned gas in all cases, but the topology distribution within the flame is significantly different between cases A–C. This can be substantiated from Fig. 3 which shows the probability of finding each flow topology for different values of Favre-averaged reaction progress variable
Figure 3 indicates that S1–S4 topologies remain major contributors within the flame front for all three cases. As shown in Fig. 1, S1–S4 appear for all values of P, and thus they remain dominant contributors, which is consistent with previous findings by Cifuentes et al.19. The dilatation rate
The variation of the normalised Reynolds-averaged enstrophy
In order to identify the physical mechanisms responsible for the difference in the enstrophy statistics between cases A-C, which are representative of combustion situations in three different combustion regimes, the variation of
The vortex-stretching term
The term-by-term contributions of each individual topology to the enstrophy transport (i.e.
The topologies S2, S7, S8 and S4 contribute significantly to the viscous torque term
Figure 6 suggests that the S4, S1 and S2 topologies are the leading order contributors to the combined viscous diffusion and dissipation term
The S2, S7, S1, S4 and S8 (in decreasing order of significance) topologies contribute significantly to the dilatation rate
The topologies which are significant contributors to
An existing three-dimensional DNS database21, 35 containing three freely propagating statistically planar H 2-air flames representing typical flame-turbulence interaction in the corrugated flamelets, thin reaction zones and broken reaction zones regimes of turbulent premixed combustion has been used to analyse the enstrophy transport conditional on flow topologies within the flame. The flow topologies have been characterized in terms of three invariants of velocity gradient tensor (P, Q and R), where the first invariant is the negative of dilatation rate and second and third invariants can be linked to strain rate and enstrophy, and their generation mechanisms2, 3, 11. In this analysis the flow topologies have been categorized in 8 types (i.e. S1-S8) depending on the location of velocity gradient tensor in P−Q−R space2, 3. It has been found that the weakening of dilatation rate with increasing Karlovitz number plays a key role in the enstrophy transport in turbulent premixed flames. The contributions to the enstrophy transport conditional on topology have been analysed in detail and it has been found that the enstrophy generation due to baroclinic torque weakens from the corrugated flamelets regime case to the broken reaction zones regime case. Further, the flow topologies S1–S4, which can be obtained for all values of dilatation rate, contribute significantly to the enstrophy and its transport in the thin reaction zones and broken reaction zones regimes of premixed turbulent combustion. However, the topologies (i.e. S7 and S8), which are obtained only for positive values of dilatation rate, also contribute significantly to the enstrophy transport in the corrugated flamelets regime.
A three-dimensional complex chemistry (9 steps and 19 chemical reactions according to a detailed chemical mechanism36) compressible flow DNS database of freely-propagating statistically planar turbulent H 2-air premixed flames with ϕ=0.7 has been considered for this analysis. An equivalence ratio of 0.7 is chosen because an H 2-air mixture for this equivalence ratio is known to be thermo-diffusively neutral36, such that the additional effects of the preferential diffusion are eliminated. Interested readers are referred to Im et al.35 and Wacks et al.21 for detailed information on the DNS database used for the current analysis and here a brief description of the numerical methodology is provided. The spatial discretisation is carried out using an 8th order central difference scheme for internal grid points and the order of differentiation gradually decreases to a one-sided 4th order scheme at the non-periodic boundaries. A fourth order Runge-Kutta scheme is used for explicit time advancement. The flame is initialised by a steady 1D planar laminar flame profile, and a pre-computed auxiliary divergence free, homogeneous, isotropic turbulence field generated using a pseudo-spectral method37 following Passot-Pouquet spectrum38 is injected through the inlet. The mean inlet velocity has been gradually modified to match turbulent flame speed as the simulation progresses. The temporal evolution of flame area has been monitored and the flame is taken to be statistically stationary when the flame area no longer varies with time. Turbulent inflow and outflow boundaries are specified in the direction of mean flame propagation and the transverse boundaries are taken to be periodic. The non-periodic boundaries are specified using an improved Navier Stokes characteristic boundary conditions (NSCBC) technique39.
The domain size is taken to be 20 mm×10mm×10mm (8 mm×2mm×2mm) in cases A and B (case C) and the domain has been discretised by a uniform Cartesian grid of 512 × 256 × 256 (1280 × 320 × 320). The grid spacing was determined by the flame resolution, ensuring about 10 grid points across δ th, and in all cases the Kolmogorov length scale remains bigger than the grid spacing (i.e. η≥1.5Δx where η and Δx are the Kolmogorov length scale and DNS grid spacing, respectively). Simulations have been carried out for 1.0t e, 6.8t e and 6.7t e (i.e. t e=l T/u ′) for cases A-C respectively, and this simulation time remains comparable to several previous analyses40–42.
The datasets generated during and/or analysed during the current study are available from the corresponding author on reasonable request.
N.C. and D.H.W. are grateful to EPSRC and N8/ARCHER. HGI was sponsored by KAUST and made use of the resources of the KAUST Supercomputing Laboratory and computer clusters.
V.P., N.C. and D.H.W. did the analysis and developed the post-processing code. The analysis was conceptualised by N.C. and M.K., H.G.I. generated the DNS data and M.K. helped with data processing and transfer.
The authors declare that they have no competing interests.
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