Sabelnikov and Lipatnikov (Fluids 5(3):109, 2020) derived an exact mathematical framework for the thin-reaction-zone regime of turbulent reacting flows at low Damköhler number, in which an infinitely thin reaction sheet at c=1c=1 borders a thick mixing zone 0≤c<10\le c<1. The framework yields exact transport equations for the reaction-progress-variable moments conditioned to the mixing zone and for the conditioned probability density function, together with a generalized-function boundary condition at the reaction sheet and an integral constraint. The equations are exact but unclosed: the conditioned molecular-transport operator, the conditioned scalar dissipation, the joint conditioned velocity-scalar correlations, and the mean reaction-surface density (which carries the delta-function-supported reaction term) are all undetermined, and the authors state that the closure problem is beyond the scope of their paper. The open problem is to construct closure models for the unclosed conditioned terms, consistent with the exact boundary condition and integral constraint, that turn the framework into a predictive description of highly turbulent premixed combustion, with extension beyond the constant-density, single-step, unity-Lewis-number setting as acknowledged follow-on tasks.

Background

Turbulent premixed combustion, in which fuel and oxidizer are mixed before ignition and a reaction wave propagates through the mixture, is statistically described using a reaction progress variable cc, equal to 0 in fresh reactants and 1 in equilibrium products. The relative importance of chemistry and turbulence is measured by the Damköhler number Da=τT/τL\mathrm{Da}=\tau_T/\tau_L, the ratio of the turbulent eddy-turn-over time τT=L/u′\tau_T=L/u' (with u′u' the root-mean-square turbulent velocity and LL an integral length scale) to the laminar flame time τL=δL/SL\tau_L=\delta_L/S_L (with SLS_L the laminar flame speed, δL=D/SL\delta_L=D/S_L the laminar flame thickness, and DD the molecular diffusivity), and by the Karlovitz number Ka=Ret1/2/Da\mathrm{Ka}=\mathrm{Re}_t^{1/2}/\mathrm{Da}. The classical statistical theory, the Bray-Moss-Libby (BML) model, treats the turbulent flame as an ensemble of infinitely thin flamelets separating pure reactants from pure products and works with zone-conditioned averages, i.e. averages weighted by indicator (intermittency) functions that equal 1 inside a given zone and 0 outside, a device introduced for intermittent turbulent flows by Dopazo (1977). The BML hypothesis holds only in the limit of large Da\mathrm{Da} and small Ka\mathrm{Ka}, where turbulence wrinkles the flame but does not penetrate it.

At the opposite limit of intense turbulence (Da<1\mathrm{Da}<1, Ka≫1\mathrm{Ka}\gg1), Direct Numerical Simulations (DNS) and experiments show broad preheat zones stirred by turbulent eddies while the reaction zone itself remains thin — the thin-reaction-zone regime. Sabelnikov, Yu and Lipatnikov (2019) developed and DNS-validated a phenomenological theory of this regime, modeled as an infinitely thin reaction sheet bordering a thick mixing layer, predicting the mean consumption rate scaling UT∝u′ Da1/2U_T\propto u'\,\mathrm{Da}^{1/2}. Sabelnikov and Lipatnikov (2020) then derived the exact statistical framework audited here. The instantaneous field is split into a mixing zone 0≤c<10\le c<1, where molecular transport acts, and equilibrium products c=1c=1; the two are separated by a reaction sheet on which the normal gradient of cc is fixed at SL/DS_L/D so that the molecular flux into the sheet sustains the laminar consumption speed — a free-boundary problem of the classical Stefan type, in which the moving boundary must be found as part of the solution. The derivation assumes constant density, single-step chemistry, and unity Lewis number Le=1Le=1 (the ratio of thermal to mass diffusivity, so that a single scalar fully characterizes the mixture state). Using indicator functions in the manner of Kataoka (1986) — chosen because multiplying the governing equations by indicator functions naively produces products of Heaviside and Dirac delta functions, which are undefined in the theory of generalized functions (distributions, such as the Dirac delta δ\delta) — the authors derive exact transport equations for the moments of cc conditioned to the mixing zone and for the reaction-progress-variable probability density function (PDF). The full PDF decomposes as P(c)=γuPu(c)+γbδ(c−1)P(c)=\gamma_u P_u(c)+\gamma_b\delta(c-1), where γu\gamma_u and γb\gamma_b are the zone probabilities (intermittencies) and the Dirac delta at c=1c=1 represents the equilibrium products. Averaging yields exact but unclosed equations: they contain unknown correlations such as the conditioned molecular-transport operator γu⟨Δc⟩u,c\gamma_u\langle\Delta c\rangle_{u,c} (the Laplacian Δc\Delta c conditioned jointly to the mixing zone and to a fixed value of cc, with ⟨⋅⟩u,c\langle\cdot\rangle_{u,c} denoting this double conditioning), the conditioned scalar dissipation ⟨N⟩u,c\langle N\rangle_{u,c} where the scalar dissipation rate Nu=D∣∇cu∣2N_u=D|\nabla c_u|^2 measures the smoothing of scalar gradients by molecular diffusion, the zone-conditioned fluxes γu⟨∇c⟩u\gamma_u\langle\nabla c\rangle_u, the joint conditioned velocity-scalar correlations ⟨u⟩u,c\langle\mathbf{u}\rangle_{u,c}, and the mean reaction-surface density Σ\Sigma (mean area of the reaction sheet per unit volume), which carries the entire reaction term as the δ(c−1)\delta(c-1)-supported product-creation rate W=SLΣW=S_L\Sigma. The paper also derives a boundary condition for the conditioned PDF PuP_u at the reaction sheet and an integral constraint on PuP_u, both obtained with generalized-function techniques developed by Kuznetsov (1972) for passive-scalar mixing. The authors state explicitly that the closure problem for the derived equations is beyond the scope of their paper; closing them, together with the acknowledged extensions beyond the constant-density, single-step, Le=1Le=1 simplification (thermal expansion, preferential diffusion, detailed chemistry, heat losses), is what would make the framework predictive.

Problem Statement

Close the exact conditioned-moment and PDF transport equations derived by Sabelnikov and Lipatnikov (2020) for the thin-reaction-zone regime of turbulent reacting flows at low Damköhler number, in the setting stated by the source (a statistically planar single-step reaction wave of laminar speed SLS_L in constant-density, unity-Lewis-number, homogeneous isotropic turbulence unaffected by the wave, with Da<1\mathrm{Da}<1 and Ka≫1\mathrm{Ka}\gg1). Specifically, construct closure models — expressed in the resolved dependent variables such as the intermittencies γu,γb\gamma_u,\gamma_b, the conditioned PDF Pu(c)P_u(c), and the conditioned moments — for the unclosed terms of the derived equations, at minimum: (i) the doubly-conditioned molecular-transport operator γu⟨Δc⟩u,c\gamma_u\langle\Delta c\rangle_{u,c}; (ii) the conditioned scalar dissipation ⟨N⟩u,c\langle N\rangle_{u,c}, with Nu=D∣∇cu∣2N_u=D|\nabla c_u|^2; (iii) the zone-conditioned molecular fluxes of the moment equations, e.g. γu⟨∇c⟩u\gamma_u\langle\nabla c\rangle_u and γu⟨∇c2⟩u\gamma_u\langle\nabla c^2\rangle_u; (iv) the joint conditioned velocity-scalar correlations ⟨u⟩u,c\langle\mathbf{u}\rangle_{u,c}, γu⟨uc⟩u\gamma_u\langle\mathbf{u}c\rangle_u, γu⟨uc2⟩u\gamma_u\langle\mathbf{u}c^2\rangle_u; (v) the mean reaction-surface density Σ\Sigma, equivalently the δ(c−1)\delta(c-1)-supported product-creation rate W=SLΣW=S_L\Sigma. The closed system, together with the generalized-function boundary condition for PuP_u at the reaction sheet (c=1c=1) and the integral constraint on PuP_u, must form a well-posed system determining PuP_u, the conditioned moments, and γu,γb\gamma_u,\gamma_b, and must be consistent with the exact constraints. Within the stated constant-density, single-step, Le=1Le=1 setting, the closure must satisfy these exact constraints, form a well-posed closed system in the resolved dependent variables, and reproduce the DNS-validated mean-consumption-rate scaling UT∝u′ Da1/2U_T\propto u'\,\mathrm{Da}^{1/2} of the companion theory; extension of the closed framework to variable density (thermal expansion), non-unity Lewis number, multi-step chemistry, and heat losses is acknowledged follow-on work that is not required to resolve this task.

References

  1. Vladimir A. Sabelnikov and Andrei N. Lipatnikov, 'A New Mathematical Framework for Describing Thin-Reaction-Zone Regime of Turbulent Reacting Flows at Low Damköhler Number', Fluids 5(3):109 (2020), DOI 10.3390/fluids5030109, https://doi.org/10.3390/fluids5030109
  2. V. A. Sabelnikov, R. Yu, and A. N. Lipatnikov, 'Thin reaction zones in constant-density turbulent flows at low Damköhler numbers: Theory and simulations', Physics of Fluids 31:055104 (2019), DOI 10.1063/1.5090192, https://doi.org/10.1063/1.5090192
  3. V. A. Sabelnikov, R. Yu, and A. N. Lipatnikov, 'Thin reaction zones in highly turbulent medium', International Journal of Heat and Mass Transfer 128:1201-1205 (2019), DOI 10.1016/j.ijheatmasstransfer.2018.09.098, https://doi.org/10.1016/j.ijheatmasstransfer.2018.09.098
  4. V. A. Sabelnikov and A. N. Lipatnikov, 'Scaling of reaction progress variable variance in highly turbulent reaction waves', Physics of Fluids 33(8):085103 (2021), DOI 10.1063/5.0059938, https://doi.org/10.1063/5.0059938
  5. Cesar Dopazo, 'Paul Andrews Libby Contribution to Zone-Conditioned Average Transport Equations in Intermittent Turbulent Flows', Combustion Science and Technology 195:3853-3874 (2022), DOI 10.1080/00102202.2022.2041619, https://doi.org/10.1080/00102202.2022.2041619
  6. Stefanie Tomasch, Nedunchezhian Swaminathan, Christoph Spijker, and Ivar S. Ertesvåg, 'Development of a turbulence dissipation based reaction rate model for progress variable in turbulent premixed flames', Combustion Theory and Modelling 26:896-915 (2022), DOI 10.1080/13647830.2022.2083525, https://doi.org/10.1080/13647830.2022.2083525
  7. Ivan Yakovenko, Alexey Kiverin, and Ksenia Melnikova, 'Computational Fluid Dynamics Model for Analysis of the Turbulent Limits of Hydrogen Combustion', Fluids 7(11):343 (2022), DOI 10.3390/fluids7110343, https://doi.org/10.3390/fluids7110343
  8. K. N. C. Bray, Paul A. Libby, and J. B. Moss, 'Unified modeling approach for premixed turbulent combustion—Part I: General formulation', Combustion and Flame 61:87-102 (1985), DOI 10.1016/0010-2180(85)90075-6, https://doi.org/10.1016/0010-2180(85)90075-6
  9. A. Y. Klimenko and R. W. Bilger, 'Conditional moment closure for turbulent combustion', Progress in Energy and Combustion Science 25:595-687 (1999), DOI 10.1016/S0360-1285(99)00006-4, https://doi.org/10.1016/S0360-1285(99)00006-4
  10. César Dopazo, 'On conditioned averages for intermittent turbulent flows', Journal of Fluid Mechanics 81:433-438 (1977), DOI 10.1017/S0022112077002158, https://doi.org/10.1017/S0022112077002158
  11. V. R. Kuznetsov, 'Probability of the concentration of a passive impurity in turbulent flows with transverse shear', Fluid Dynamics 7:776-781 (1972), DOI 10.1007/BF01205755, https://doi.org/10.1007/BF01205755
  12. I. Kataoka, 'Local instant formulation of two-phase flow', International Journal of Multiphase Flow 12:745-758 (1986), DOI 10.1016/0301-9322(86)90049-2, https://doi.org/10.1016/0301-9322(86)90049-2
  13. James F. Driscoll, Jacqueline H. Chen, Aaron W. Skiba, Campbell D. Carter, Evatt R. Hawkes, and Haiou Wang, 'Premixed flames subjected to extreme turbulence: Some questions and recent answers', Progress in Energy and Combustion Science 76:100802 (2020), DOI 10.1016/j.pecs.2019.100802, https://doi.org/10.1016/j.pecs.2019.100802