Sabelnikov and Lipatnikov (Fluids 5:109, 2020) derived exact but unclosed transport equations for the conditioned (to the reactant mixing zone) moments and probability density function (PDF) of the reaction progress variable for a reaction wave with an infinitely thin reaction zone propagating in constant-density turbulence at low Damköhler number. The unclosed terms are conditioned quantities such as the conditioned Laplacian or molecular flux γu⟨Δc⟩u,c\gamma_u\langle\Delta c\rangle_{u,c} and γuD⟨∇c⟩u,c\gamma_u D\langle\nabla c\rangle_{u,c}, the conditioned scalar dissipation γu⟨N⟩u,c\gamma_u\langle N\rangle_{u,c}, and the conditioned velocity ⟨u⟩u,c\langle\mathbf{u}\rangle_{u,c}, together with reaction-sheet surface-density terms, all embedded in an equation system whose boundary conditions are Dirac-delta-supported generalized functions at the reaction sheet. The authors explicitly left the closure problem outside the scope of that work, and the problem — to close the derived equation system in its stated regime — remains open.

Background

In premixed turbulent combustion a reaction wave converts reactants to products; its structure is commonly described by a reaction progress variable c(x,t)c(x,t), equal to 0 in fresh reactants and 1 in equilibrium products. Two dimensionless groups characterize the interaction between chemistry and turbulence: the Damköhler number DaDa, the ratio of a turbulent time scale to the chemical time scale, and the Karlovitz number Ka=Ret1/2/DaKa=Re_t^{1/2}/Da, which grows when turbulence intensifies relative to the reaction. The classical Bray–Moss–Libby (BML) statistical model of premixed turbulent flames (Bray and Moss 1977; Bray, Libby and Moss 1985) addresses the flamelet regime of large DaDa: the turbulent flame brush — the spatial zone swept by the wrinkled flame — is an intermittent mixture of fresh and burned pockets separated by thin flamelets, the PDF of cc is bimodal with Dirac delta functions at both c=0c=0 and c=1c=1, and closures are built on zone-conditioned (to unburned or burned gas) averaging; zone-conditional closure models in this line exist for the high-DaDa case (e.g., Lee and Huh 2004). In the thin-reaction-zone regime at low DaDa and high KaKa, turbulent eddies penetrate and thicken the preheat zone while molecular mixing smooths small-scale wrinkles of the reaction-zone surface, and this flamelet picture fails. Sabelnikov, Yu and Lipatnikov (2019) developed a theory, supported by direct numerical simulation (DNS), in which an infinitely thin reaction sheet (where c=1c=1 and ∣∇c∣|\nabla c| is fixed so that the molecular flux into the sheet yields the laminar consumption speed SLS_L) is adjacent to a thick mixing zone where 0≤c<10\le c<1; consequently the PDF of cc contains a single Dirac delta function at c=1c=1 and is formally analogous to the PDF of a passive scalar mixed in a turbulent jet. Sabelnikov and Lipatnikov (2020) turned this physical scenario into a mathematical framework: splitting the field into the reactant mixing zone and equilibrium products by an indicator function χu\chi_u (following Kataoka's approach to conditioning, which avoids ill-defined products of discontinuous Heaviside functions with Dirac deltas), they derived exact local field equations and, after ensemble averaging, exact transport equations for the conditioned moments of cc (their Equations (26)–(29)) and for the conditioned and unconditioned PDFs of cc (their Equations (62)–(84)), including boundary conditions for the PDF at the reaction sheet; the reaction rate appears as a singular term SLΣS_L\Sigma, where Σ=∣∇χu∣\Sigma=|\nabla\chi_u| is the reaction-sheet surface density, a Dirac-delta-supported generalized function. Here γu\gamma_u is the probability of the reactant zone, ⟨q⟩u,c\langle q\rangle_{u,c} denotes a quantity qq conditioned both on being in the reactant zone and on a particular value of cc there, N=D(∇cu)2N=D(\nabla c_u)^2 is the scalar dissipation, and DD is the molecular diffusivity (the setting assumes a unit Lewis number, i.e. equal thermal and molecular diffusivities, single-step chemistry, and constant density). The derived equations are exact but unclosed: they contain the unknown terms γu⟨Δc⟩u,c\gamma_u\langle\Delta c\rangle_{u,c} (equivalently the conditioned flux γuD⟨∇c⟩u,c\gamma_u D\langle\nabla c\rangle_{u,c}), the conditioned scalar dissipation γu⟨N⟩u,c\gamma_u\langle N\rangle_{u,c}, the conditioned velocity ⟨u⟩u,c\langle\mathbf{u}\rangle_{u,c}, and the conditioned displacement-speed–surface-density product ⟨Sd⟩rΣ\langle S_d\rangle_r\Sigma (where the subscript rr denotes conditioning at the reaction sheet), none of which is expressible through the solved-for moments or PDF. The paper states that 'the closure problem for the derived equations is beyond the scope of the present study.' Whereas closure of zone-conditioned equations is classical in the high-DaDa flamelet limit, no closure exists for this low-DaDa regime in which the mixing zone is thick, the PDF is unimodal with a single delta function, and the boundary conditions are distributional. Later work in this line has produced a partial asymptotic result — the variance scaling c′2‾∝Da\overline{c'^2}\propto Da at low DaDa — and a review of the adjacent zone-conditioned-averaging formalism, while other studies pursue independent closure strategies.

Problem Statement

Close the exact conditioned-moment and probability-density-function transport equations derived by Sabelnikov and Lipatnikov (2020) for propagation of a single reaction wave with an infinitely thin reaction zone in constant-density homogeneous isotropic turbulence at high turbulent Reynolds number and Damköhler number Da<1Da<1 (single-step chemistry, unit Lewis number, statistically planar configuration). Specifically, construct closure relations for — or exact/asymptotic evaluations of — every unclosed conditioned quantity in that equation system: the conditioned Laplacian term γu⟨Δc⟩u,c\gamma_u\langle\Delta c\rangle_{u,c} or, equivalently, the conditioned molecular flux γuD⟨∇c⟩u,c\gamma_u D\langle\nabla c\rangle_{u,c}; the conditioned scalar dissipation γu⟨N⟩u,c\gamma_u\langle N\rangle_{u,c} with N=D(∇cu)2N=D(\nabla c_u)^2; the conditioned velocity ⟨u⟩u,c\langle\mathbf{u}\rangle_{u,c}; and the conditioned displacement-speed–surface-density term ⟨Sd⟩rΣ\langle S_d\rangle_r\Sigma. The target is one coherent, self-contained closed system in which the moment-level and PDF-level closures are mutually consistent (moments computed from the closed PDF solution satisfy the closed moment equations), the exact generalized-function structure is preserved (the derived boundary conditions for the PDF at the reaction sheet and the δ(c−1)\delta(c-1)-supported terms are represented consistently), realizability holds (nonnegative PDF, normalization γu+γb=1\gamma_u+\gamma_b=1), and the correct known limits are recovered (the passive-scalar mixing limit, the variance scaling c′2‾∝Da\overline{c'^2}\propto Da at low DaDa). The task is restricted to the stated setting, and extensions beyond it (variable density, non-unity Lewis number, multi-step chemistry, heat losses) are out of scope.

References

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