In Reynolds-averaged simulations of turbulent flows with non-equilibrium chemistry, the mean chemical formation rates cannot be evaluated at mean flow values because the rate functions are nonlinear in density, temperature, and species concentrations. Molchanov, Yanyshev and Bykov (J. Phys.: Conf. Ser. 891, 2017) built an assumed probability-density-function (PDF) closure that approximates the unknown joint PDF of temperature and composition as a product of statistically independent PDFs, with density fixed at its mean, and validated it on hydrogen–air combustion. The source paper states that the general form of the PDF for chemically reacting turbulent flows is yet unknown and leaves the consequences of the independence factorization unquantified. This problem asks for a quantified account of the error introduced by the independence assumption across turbulence–chemistry regimes, taking correlated-joint-PDF or transported-PDF solutions and experiment as references.

Background

In a turbulent reacting flow, the instantaneous formation rate of each chemical species is a nonlinear function of mixture density ρ\rho, temperature TT, and the species concentrations C1,…,CNcC_1,\dots,C_{N_c}. When the flow equations are Reynolds-averaged (RANS, the Reynolds-averaged Navier–Stokes approach, in which turbulent fluctuations are statistically averaged rather than resolved), the mean formation rate is the integral of the rate function against the one-point joint probability density function P(ρ,T,C1,…,CNc)P(\rho,T,C_1,\dots,C_{N_c}) of the fluctuating state variables — a PDF here is the function whose integral over a region of state space gives the probability of finding the local state in that region. Evaluating the rate function at the mean values (the quasi-laminar assumption) can therefore be badly wrong, because fluctuations can either enhance or suppress reaction.

The transported-PDF approach pioneered by Pope (Prog. Energy Combustion Sci. 11, 1985) solves a modeled transport equation for this joint PDF directly, so no presumed shape is needed; its closure burden shifts to micro-mixing models (models of the molecular-scale mixing that decays scalar fluctuations), as reviewed by Haworth (Prog. Energy Combustion Sci. 36, 2010). A cheaper engineering alternative is the assumed (presumed) PDF method: postulate analytic shapes for the PDFs, with parameters obtained from modeled mean and variance equations. In the model of Molchanov, Yanyshev and Bykov (J. Phys.: Conf. Ser. 891, 012106, 2017) — following Gaffney, White, Girimaji and Drummond (Comput. Syst. Eng. 5, 1994) and the joint-beta-PDF formulations of Baurle and co-workers — the joint PDF is written as a product of a Gaussian or beta-function PDF for TT (the beta function, a two-parameter distribution on a bounded interval, here fixed by the mean and variance of the fluctuating variable) and a multidimensional beta-function PDF for the concentrations, while the density PDF is collapsed to a Dirac delta at its mean. The fluctuation variances are carried by a scalar-variance transport equation closed with a semi-empirical ratio RR of the scalar-dissipation to turbulence-dissipation time scales, its source term modelled by generation and dissipation alone. The authors validated the model on subsonic and supersonic hydrogen–air jet flames using the hydrogen-oxidation mechanism of Connaire et al. (Int. J. Chem. Kinet. 36, 2004), finding a negligible effect on the developed mean flow field but a considerable upstream shift of the ignition point in supersonic non-equilibrium cases — the regime, characterized by a Damköhler number (the ratio of flow time to chemical time) near or below unity, where chemistry is too slow to track the mean flow.

The paper itself states that the general form of the PDF for chemical processes is yet unknown and adopts the factorized form as a simplification. Baurle and Girimaji (Combustion and Flame 134, 2003) had already shown, within the same assumed-PDF family, how arbitrary temperature–composition correlations can be retained by transposing the assumed joint PDF into a temperature–enthalpy plane, and that these correlations matter in high-speed reacting flows; the same group's later work (Zubko et al., Khim. Fiz. Mezoskopiya, 2026) reports strong correlations between temperature pulsations and the mass fractions of radicals such as OH. These results bound the question: the exact general joint PDF is not expected to have a closed analytic form (it is flow-dependent), and the field's canonical high-fidelity route is to compute it via transported-PDF methods; what remains genuinely open for the factorized assumed-PDF class used by the source is the magnitude and regime-dependence of the error committed by discarding the correlations.

Problem Statement

For Reynolds-averaged simulations of turbulent flows with non-equilibrium chemistry using the assumed-PDF closure family of Molchanov, Yanyshev and Bykov (2017) — in which the joint PDF of the thermochemical state is factorized into statistically independent PDFs for temperature (Gaussian or beta) and species concentrations (multidimensional beta), with the density PDF collapsed to a Dirac delta at its mean and the scalar variances supplied by a semi-empirically closed transport equation — establish quantitatively:

  1. the error in predicted mean formation rates and in the resulting mean flow field (including ignition location in non-equilibrium regimes) that is introduced by the statistical-independence factorization (that is, by writing the joint PDF as the product of these independent PDFs, with the density PDF collapsed to a Dirac delta at its mean);
  2. that error relative to closures that retain temperature–composition correlations (such as the transposed-PDF treatment of Baurle and Girimaji) and to transported-PDF solutions or experimental data, across turbulence–chemistry interaction regimes from near-equilibrium to strongly non-equilibrium;
  3. identify where the neglected correlations (or the semi-empirical scalar-variance closure) become the dominant error source.

References

  1. A. M. Molchanov, D. S. Yanyshev, L. V. Bykov, 'Influence of turbulent fluctuations on non-equilibrium chemical reactions in the flow', Journal of Physics: Conference Series 891, 012106 (2017), DOI 10.1088/1742-6596/891/1/012106, https://doi.org/10.1088/1742-6596/891/1/012106
  2. S. B. Pope, 'PDF methods for turbulent reactive flows', Progress in Energy and Combustion Science 11, 119–192 (1985), DOI 10.1016/0360-1285(85)90002-4, https://doi.org/10.1016/0360-1285(85)90002-4
  3. D. C. Haworth, 'Progress in probability density function methods for turbulent reacting flows', Progress in Energy and Combustion Science 36, 168–259 (2010), DOI 10.1016/j.pecs.2009.09.003, https://doi.org/10.1016/j.pecs.2009.09.003
  4. R. A. Baurle, S. S. Girimaji, 'Assumed PDF turbulence-chemistry closure with temperature-composition correlations', Combustion and Flame 134, 131–148 (2003), DOI 10.1016/S0010-2180(03)00056-7, https://doi.org/10.1016/S0010-2180(03)00056-7
  5. R. L. Gaffney, J. A. White, S. S. Girimaji, J. P. Drummond, 'Modeling temperature and species fluctuations in turbulent, reacting flow', Computing Systems in Engineering 5, 117–133 (1994), DOI 10.1016/0956-0521(94)90044-2, https://doi.org/10.1016/0956-0521(94)90044-2
  6. M. Ó Conaire, H. J. Curran, J. M. Simmie, W. J. Pitz, C. K. Westbrook, 'A comprehensive modeling study of hydrogen oxidation', International Journal of Chemical Kinetics 36, 603–622 (2004), DOI 10.1002/kin.20036, https://doi.org/10.1002/kin.20036
  7. V. P. Zhukov, I. N. Borovik, E. A. Strokach, 'Numerical Study of the Influence of Turbulent Diffusion Coefficients and Turbulent Prandtl Number on the Reactive Flow Simulation in a Combustor', Russian Aeronautics 63, 713–720 (2020), DOI 10.3103/S1068799820040200, https://doi.org/10.3103/S1068799820040200
  8. K. V. Tyul'kov, E. A. Strokach, I. N. Borovik, S. S. Blyakharskii, A. Sternin, 'Numerical Study of the Effect of Turbulent Schmidt Number on the Numerical Simulation of Processes in the Multi-Element GCH4-GO2 Combustor', Russian Aeronautics 64, 283–290 (2021), DOI 10.3103/S1068799821020161, https://doi.org/10.3103/S1068799821020161
  9. A. A. Zubko, A. S. Kozhemyako, A. M. Molchanov, A. O. Poletaev, M. S. Shilov, 'The effect of turbulent pulsations on the rates of chemical reactions', Khimicheskaya Fizika i Mezoskopiya 28 (2), 185–192 (2026), DOI 10.62669/17270227.2026.2.17, https://doi.org/10.62669/17270227.2026.2.17