Reaction-modified turbulent diffusion beyond the constant-density, single-step regime: variable-density coupling, multi-step chemistry, and a predictive turbulent chemical time scale

Theory and three-dimensional direct numerical simulations (Elperin, Kleeorin, Liberman, Lipatnikov, Rogachevskii, and Yu, Phys. Rev. E 96, 053111 (2017)) established that a chemical reaction suppresses turbulent diffusion, yielding a closed turbulent diffusion coefficient $D_T(\mathrm{Da},\mathrm{Re},\mathrm{Pr})$, but only under constant density, one-way coupling, a single first-order-like reaction, and an empirically measured mapping factor $\theta$ between laminar and effective turbulent chemical time scales, with validation up to $\mathrm{Re}=200$. The open problem is to extend this validated theory to realistic reacting flows: variable-density flows with two-way heat-release coupling, multi-step chemistry with intermediate species and other reaction orders, the full Damköhler-number range at finite Reynolds number including crossovers, and a predictive (non-fitted) theory of $\theta$, validated at combustion-relevant parameters.

Background

When a passive substance is advected by turbulence, its mean concentration spreads as if diffused with a turbulent diffusion coefficient $D_T$ that is vastly larger than the molecular diffusivity. Elperin, Kleeorin, and Rogachevskii (Phys. Rev. Lett. 80, 69 (1998)) showed that if the substance undergoes a chemical reaction, the reaction modifies $D_T$ itself, and Elperin, Kleeorin, Liberman, and Rogachevskii (Phys. Rev. E 90, 053001 (2014)) derived the dependence of $D_T$ on the Damköhler number $\mathrm{Da}=\tau_0/\tau_c$ (the ratio of the turbulent turnover time $\tau_0$ to the chemical time $\tau_c$), using the spectral $\tau$ approximation — a closure that expresses deviations of third-order turbulent moments from their background-turbulence values through deviations of second-order moments with a scale-dependent relaxation time. This 2014 theory, valid at very large Reynolds ($\mathrm{Re}$) and Péclet numbers, predicts strong suppression of turbulent diffusion with increasing $\mathrm{Da}$, and was validated against mean-field simulations of a reactive front obeying the Kolmogorov–Petrovskii–Piskunov (KPP) or Fisher equation — a reaction–diffusion equation with a nonlinear source term whose planar front propagates at a speed set by the diffusivity and reaction rate (Brandenburg, Haugen, and Babkovskaia, Phys. Rev. E 83, 016304 (2011)).

The 2017 source paper generalized that theory to large yet finite $\mathrm{Re}$, obtaining $D_T(\mathrm{Da},\mathrm{Re},\mathrm{Pr})$ in closed form ($\mathrm{Pr}$ the Prandtl number, the ratio of momentum to species diffusivity), and validated it with three-dimensional direct numerical simulations (DNS, i.e. numerical solution of the full Navier–Stokes and reaction–advection–diffusion equations without turbulence modeling) of a finite-thickness reaction wave in forced, homogeneous, isotropic, incompressible turbulence at $\mathrm{Re}=50$, $100$, and $200$. The validation required a mapping factor $\theta=\max\langle W\rangle/\max W$ — the ratio of the maximum mean reaction rate to the maximum laminar reaction rate — measured empirically from the DNS to relate the DNS Damköhler number to the theoretical one.

The theory's validity rests on restrictive assumptions, all explicit in the source: (1) constant fluid density and one-way coupling (small mass-loading, negligible heat release), adopted deliberately so that the direct effect of reactions on turbulent transport could be isolated from density-change effects such as counter-gradient transport in flames; (2) a single reaction source of first-order-like form $W(c)$, mimicking an Arrhenius rate through model counterparts of the Zeldovich number $\mathrm{Ze}=E_a(T_b-T_u)/(RT_b^2)$ — the dimensionless activation energy, with $T_u$ and $T_b$ the unburned and burned-gas temperatures, $E_a$ the activation energy, and $R$ the gas constant — and of the heat-release factor $\tau=(\rho_u-\rho_b)/\rho_b$, the relative density decrease across the wave, with both set to $6$ even though the density is held fixed; (3) asymptotic finite-$\mathrm{Re}$ results emphasizing large $\mathrm{Da}$; and (4) an empirically fitted $\theta$. Adjacent later work by Sabelnikov, Yu, and Lipatnikov (Phys. Fluids 31, 055104 (2019), DOI 10.1063/1.5090192; and Fluids 5, 109 (2020), DOI 10.3390/fluids5030109) characterized constant-density reaction waves at low Damköhler numbers $0.01<\mathrm{Da}<1$ with a different, reaction-sheet framework, finding $\sqrt{\mathrm{Da}}$ scaling of consumption velocity — but did not extend the spectral-$\tau$ $D_T$ theory, treat heat release, or supply a predictive $\theta$. A predictive, validated theory of reaction-modified turbulent transport under realistic combustion conditions therefore remains lacking.

Problem Statement

Extend the validated theory of reaction-modified turbulent diffusion — the dependence of the turbulent diffusion coefficient $D_T$ of a reacting scalar on the Damköhler number $\mathrm{Da}$, Reynolds number $\mathrm{Re}$, and Prandtl number $\mathrm{Pr}$, obtained by Elperin et al. (Phys. Rev. E 96, 053111 (2017)) under constant density, one-way coupling, and a single first-order-like reaction — to realistic reacting flows by relaxing the restrictive assumptions of the source theory:

1. Variable-density, two-way-coupled flows: determine how significant heat release (reaction-induced density and velocity changes, i.e. two-way coupling) modifies the suppression of turbulent transport by reactions, deriving the analogue of $D_T(\mathrm{Da},\mathrm{Re},\mathrm{Pr})$ (e.g. in density-weighted form) for a reacting wave or admixture in turbulence.
2. Chemistry generality: generalize the source's single first-order-like source term to multi-step reaction mechanisms with intermediate species, and to reaction orders other than first order.
3. Turbulent chemical time scale: replace the empirically measured mapping factor $\theta=\max\langle W\rangle/\max W$ with a predictive theory giving $\theta$ (equivalently, the mean reaction rate $\langle W(c)\rangle$ or the effective turbulent chemical time) as a function of turbulence and chemistry parameters.
4. Regime completeness: cover the crossover between the large-$\mathrm{Da}$ asymptotics and the moderate/small-$\mathrm{Da}$ regime at finite $\mathrm{Re}$ (and, in the constant-density limit, consistent with the low-$\mathrm{Da}$ results of Sabelnikov et al. (2019)), extending the theory beyond the parameter range in which the source theory was validated ($\mathrm{Re}\le 200$, $\mathrm{Ze}=\tau=6$, the source Prandtl/Schmidt numbers) toward combustion-relevant thermochemical parameters.

A full extension of the theory to realistic reacting flows requires all four relaxations together. Each numbered item is nonetheless a well-posed sub-problem that may be attacked independently.

References

1. T. Elperin, N. Kleeorin, M. Liberman, A. N. Lipatnikov, I. Rogachevskii, and R. Yu, "Turbulent diffusion of chemically reacting flows: theory and numerical simulations," Phys. Rev. E 96, 053111 (2017). DOI: 10.1103/PhysRevE.96.053111; arXiv:1612.02585; https://arxiv.org/abs/1612.02585
2. T. Elperin, N. Kleeorin, M. A. Liberman, and I. Rogachevskii, "Turbulent diffusion of chemically reacting gaseous admixtures," Phys. Rev. E 90, 053001 (2014). DOI: 10.1103/PhysRevE.90.053001; arXiv:1406.0021; https://arxiv.org/abs/1406.0021
3. T. Elperin, N. Kleeorin, and I. Rogachevskii, "Effect of chemical reactions and phase transitions on turbulent transport of particles and gases," Phys. Rev. Lett. 80, 69 (1998). DOI: 10.1103/PhysRevLett.80.69; https://doi.org/10.1103/PhysRevLett.80.69
4. A. Brandenburg, N. E. L. Haugen, and N. Babkovskaia, "Turbulent front speed in the Fisher equation: dependence on Damköhler number," Phys. Rev. E 83, 016304 (2011). DOI: 10.1103/PhysRevE.83.016304; arXiv:1008.5145; https://arxiv.org/abs/1008.5145
5. V. A. Sabelnikov, R. Yu, and A. N. Lipatnikov, "Thin reaction zones in constant-density turbulent flows at low Damköhler numbers: Theory and simulations," Phys. Fluids 31, 055104 (2019). DOI: 10.1063/1.5090192; https://doi.org/10.1063/1.5090192
6. V. A. Sabelnikov and A. N. Lipatnikov, "A New Mathematical Framework for Describing Thin-Reaction-Zone Regime of Turbulent Reacting Flows at Low Damköhler Number," Fluids 5, 109 (2020). DOI: 10.3390/fluids5030109; https://doi.org/10.3390/fluids5030109
7. O. B. Shende and A. Mani, "Closures for multicomponent reacting flows based on dispersion analysis," Phys. Rev. Fluids 7, 093201 (2022). DOI: 10.1103/PhysRevFluids.7.093201; https://doi.org/10.1103/PhysRevFluids.7.093201