Turbulent diffusion of chemically reacting admixtures beyond first-order, constant-density assumptions: predictive theory for the effective turbulent chemical time and the turbulent diffusion coefficient at finite Reynolds and Damköhler numbers

The theory of turbulent diffusion of chemically reacting gaseous admixtures (Elperin et al., Phys. Rev. E 90, 053001 (2014); Phys. Rev. E 96, 053111 (2017)) predicts that a chemical reaction with laminar chemical time $\tau_F$ depletes the turbulent diffusion coefficient $D_T$ of the reacting species, with $D_T \approx D_T^0/\mathrm{Da}$ at large turbulent Damköhler number $\mathrm{Da}$ , and gives $D_T(\mathrm{Da},\mathrm{Re},\mathrm{Pr})$ at large but finite Reynolds number, validated by direct numerical simulations (DNS) at $\mathrm{Re} \le 200$ . The theory, however, rests on restrictive assumptions: constant density with one-way coupling, a large-Reynolds-number expansion, an empirical 'mapping factor' $\theta$ (the laminar chemical time relative to the effective turbulent chemical time) measured from DNS rather than predicted, and — in the 2017 paper's simplified model and its DNS validation — a single first-order-like reaction step in one progress variable (the 2014 theory itself treats multicomponent single-step reactions up to third order, with mutual turbulent diffusion and cross-effects, but not multi-step kinetics with intermediate species). The open problem is to build a predictive, validated theory of turbulent diffusion in reacting flows that removes these assumptions: predicting $\theta$ from turbulence and chemistry parameters, covering multi-step chemistry and intermediate species, covering variable-density two-way-coupled flows with significant heat release, and characterizing $D_T$ over the full finite-Re, moderate-to-small-Da plane including crossovers. Later literature has progressed adjacent aspects — low-Damköhler wave-propagation scaling (Sabelnikov, Yu, Lipatnov 2019), exact unclosed low-Da frameworks (Sabelnikov and Lipatnov 2020), reaction-dependent eddy-diffusivity closures from dispersion analysis (Shende and Mani 2022), and variable-density DNS diagnostics (Yu et al. 2021; Pignatelli et al. 2021) — but none extends this theory's diffusion coefficient or provides a predictive $\theta$ , so the core remains open.

Background

Chemically reacting species transported by turbulence occur in premixed combustion, atmospheric chemistry, and chemical-process engineering. When a scalar field $c$ — a reaction progress variable, with $c=0$ in reactants and $c=1$ in products — is advected by turbulence and simultaneously consumed by a reaction with laminar chemical time $\tau_F$ , the mean turbulent flux of the scalar is described by a turbulent diffusion coefficient $D_T$ that the reaction can strongly modify. Elperin, Kleeorin, and Rogachevskii (1998) first showed, using a path-integral treatment of a velocity field random in time (delta-correlated), that chemical reactions and phase transitions can substantially deplete turbulent transport. Elperin et al. (2014) rederived and extended this result using the spectral $\tau$ -approximation — a closure in which each Fourier mode of the turbulent velocity field decorrelates over a scale-dependent time $\tau(k)$ , valid at large Reynolds and Péclet numbers — finding $D_T \approx D_T^0/\mathrm{Da}$ at large turbulent Damköhler number, where $\mathrm{Da} = \tau_0/\tau_c$ is the ratio of the turbulent turnover time $\tau_0$ to the effective turbulent chemical time $\tau_c$ and $D_T^0 = \tau_0 u_0^2/3$ is the reaction-free turbulent diffusivity ( $u_0$ is the r.m.s. turbulent velocity). That theory is formulated for multicomponent admixtures undergoing single-step reactions: it derives explicit turbulent diffusion coefficients for second- and third-order reactions, together with mutual turbulent-diffusion and turbulent Dufour-type cross-effect coefficients. Elperin et al. (2017) generalized the theory to large but finite Reynolds number $\mathrm{Re} = \ell_0 u_0/\nu$ ( $\ell_0$ the integral length scale, $\nu$ the kinematic viscosity), obtaining $D_T(\mathrm{Da},\mathrm{Re},\mathrm{Pr})$ — their Eq. (12) — as an expansion of an exact spectral integral in the small parameter $a(\mathrm{Re},\mathrm{Pr}) = 2(1+\mathrm{Pr}^{-1})/\mathrm{Re}$ , where $\mathrm{Pr} = \nu/D$ is the Prandtl (or Schmidt) number, with asymptotic limits for $\mathrm{Da} \ll 1$ , $1 \ll \mathrm{Da} \ll \mathrm{Re}^{1/2}$ , and $\mathrm{Re}^{1/2} \ll \mathrm{Da}$ . To validate the theory they performed three-dimensional direct numerical simulations (DNS — numerical solution of the governing equations without turbulence models) of a finite-thickness reaction wave with first-order-like kinetics propagating in forced, homogeneous, isotropic, incompressible turbulence at $\mathrm{Re} = 50$ , $100$ , and $200$ , using a reaction rate with Zeldovich number $\mathrm{Ze} = 6$ and heat-release parameter $\tau = 6$ (combustion-like dimensionless parameters controlling the temperature sensitivity and the density jump in a real flame, here imposed on a constant-density flow). Because the theory is formulated in terms of a Damköhler number based on an effective chemical time in turbulence, the comparison required a 'mapping factor' $\theta = \tau_F/\tau_c$ — the ratio of the laminar wave time scale $\tau_F$ associated with the laminar reaction rate $W(c)$ to the significantly larger effective chemical time $\tau_c$ characterizing the mean reaction rate $\overline{W}(\overline{c})$ in turbulence, so that $\theta \le 1$ — measured empirically from the DNS reaction-rate statistics as $\theta = \max\{\overline{W}(\overline{c})\}/\max\{W(c)\}$ rather than predicted. Theory and DNS agreed well within these assumptions, but the assumptions themselves — constant density, one-way coupling (the reaction does not modify the flow; small mass loading), a single first-order-like reaction step in one progress variable (the 2017 DNS validation setup), a large-Re expansion, and a DNS-calibrated $\theta$ — limit applicability to realistic reacting flows, and their removal constitutes the open task addressed by this statement.

Problem Statement

Develop a predictive, validated theory of the turbulent diffusion of chemically reacting admixtures that removes the simplifying assumptions of the Elperin et al. (2014, 2017) theory. The single overarching objective is a closed theory of turbulent species transport in reacting flows — explicit expressions for the turbulent diffusion coefficient(s) and the effective turbulent chemical time — valid under realistic reacting-flow conditions, with independent validation. Jointly supporting this one objective, the theory should: (i) predict, rather than measure a posteriori from DNS, the mapping factor $\theta = \tau_F/\tau_c$ (the ratio of the laminar chemical time $\tau_F$ to the effective chemical time $\tau_c$ in turbulence) as a function of turbulence and chemistry parameters; (ii) cover multi-step chemistry with intermediate species, going beyond both the single first-order-like reaction step in one progress variable used in the 2017 DNS validation and the multicomponent single-step reactions (up to third order) already treated by the 2014 theory; (iii) cover variable-density reacting flows with two-way coupling, in which significant heat release modifies the velocity and density fields; and (iv) characterize $D_T$ over the full finite-Reynolds-number, moderate-to-small-Damköhler-number plane — including crossover regimes and higher-order corrections beyond the large-Reynolds-number expansion of the 2017 Eq. (12) — with validation beyond $\mathrm{Re} \le 200$ and beyond the specific parameter sets ( $\mathrm{Ze}=6$ , $\tau=6$ , limited $S_L/u_0$ ranges, with $S_L$ the laminar reaction-wave speed) used there. These components are facets of one derivation: each removes a specific assumption of the same spectral- $\tau$ theory, and a theory covering only a proper subset of (i)–(iv) does not resolve the objective; a theory covering all of (i)–(iv) should agree with independent DNS or experiments across the regimes addressed by components (i)–(iv).

References

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