A posteriori assessment of the flamelet regime diagram for non-premixed turbulent combustion LES: self-consistency of flamelet-model solutions and explicit-versus-implicit filtering

Chan and Ihme (2017) introduced a flamelet regime diagram for non-premixed turbulent combustion that classifies, pointwise in a flame, whether a classical one-dimensional flamelet closure is adequate, whether multi-dimensional flamelet effects must be retained, and whether the local numerical grid resolves the reaction zone. The diagram is parameterized by a flamelet regime parameter $\psi_\alpha$ — the ratio of flame-aligned (tangential) to flame-orthogonal species diffusion evaluated in a locally rotated frame — together with the ratio of grid spacing to reaction-zone thickness, and it was validated only through a priori filtering of direct numerical simulation (DNS) data. The authors explicitly posed two open questions that gate the diagram's deployment as a model-selection tool in production large-eddy simulation (LES): (1) self-consistency — whether an LES closed with a flamelet model can produce a solution whose own resolved fields, fed back into the regime diagram, invalidate the flamelet formulation in regions where that closure is being used; and (2) filtering — how the difference between the explicit top-hat filtering employed in the a priori analysis and the implicit grid filtering of practical LES affects the evaluation of $\psi_\alpha$ and the resulting regime classification. Both questions remain open: no a posteriori LES test of the regime diagram and no quantification of explicit-versus-implicit filtering effects on $\psi_\alpha$ have been published.

Background

In non-premixed (diffusion) combustion, fuel and oxidizer are supplied separately and burn across a mixing layer. The local mixing state is described by the mixture fraction $Z$ , a conserved scalar equal to 0 in the oxidizer stream and 1 in the fuel stream. The flamelet concept, in the formulation of Pitsch and Peters (1998), treats the thin reaction zone of such flames as a quasi-one-dimensional laminar structure (a "flamelet") embedded in the turbulent flow and parameterized by $Z$ , so that the thermochemistry can be solved once in mixture-fraction space and tabulated — the basis of most combustion closures used in large-eddy simulation (LES) of non-premixed flames. Classical theory expects this picture to hold when the reaction zone, of thickness $\delta_R$ , is thinner than the Kolmogorov length scale $\eta_K$ (the size of the smallest turbulent eddies); the ratio $Ka_K=(\delta_R/\eta_K)^2$ is a Karlovitz number for non-premixed flames, with $Ka_K\le 1$ the conventional flamelet-validity condition and $Ka_K=100$ an extreme theoretical limit. Later work by Scholtissek et al. (2015) showed that diffusion of species and heat along the flame surface (tangential, or multi-dimensional, diffusion), which the classical one-dimensional flamelet equation omits, can become comparable to flame-normal diffusion; they derived scaling criteria for when these multi-dimensional flamelet effects matter. LES resolves only the large turbulent scales: the resolved fields are filtered versions of the true fields, and the unresolved (subgrid-scale) contributions must be modeled. In principle the filter is an explicit convolution with a chosen kernel of width $\Delta$ (e.g., a homogeneous top-hat filter applied to DNS data); in practice, the numerical grid and discretization scheme act as an implicit, only loosely characterized filter. An a priori test evaluates a model or diagnostic on explicitly filtered DNS data, whereas an a posteriori test runs the actual simulation with the model active and examines the solution that the model itself produced. A standard resolution-quality measure is Pope's criterion $P_k=k_{\mathrm{res}}/(k_{\mathrm{res}}+k_{\mathrm{sgs}})$ , the fraction of turbulent kinetic energy contained in the resolved motions; values near unity indicate that the grid resolves the energy-containing scales. Chan and Ihme (2017) constructed a flamelet regime diagram for non-premixed combustion as follows. Working in a locally rotated coordinate frame whose first axis $X_1$ aligns with the mixture-fraction gradient (so $X_2,X_3$ are tangential to the mixture-fraction isosurface), they defined the flamelet regime parameter $\psi_\alpha$ of species $\alpha$ as the ratio of the flame-aligned diffusion terms to the flame-orthogonal diffusion terms of the species transport equation: $\psi_\alpha\le 1$ marks the regime where the classical one-dimensional flamelet formulation applies, $\psi_\alpha>1$ the multi-dimensional flamelet regime identified by Scholtissek et al. A length-scale argument — using the inner reaction-zone thickness concept and the dissipation-element notion (the notional turbulent structure whose width scales with $\eta_K$ ) — yields $\psi/(2Z_{\mathrm{st}})=(\delta_R/\eta_K)^2=Ka_K$ : the classical Karlovitz criterion $Ka_K\le 1$ thus places the boundary of the isolated flamelet (Peters) regime at $\psi=2Z_{\mathrm{st}}$ , distinct from the diffusion-ratio boundary $\psi=1$ at which the multi-dimensional flamelet regime is taken to begin, and $2Z_{\mathrm{st}}\le\psi\le 1$ is the transition zone between them. The diagram, drawn in the plane of grid spacing $\Delta$ relative to $\delta_R$ (abscissa) and $\psi$ (ordinate), therefore connects directly to the classical Karlovitz-number criterion while additionally encoding the local numerical resolution. Chan and Ihme verified the construction through a priori analysis of two DNS datasets: a turbulent lifted hydrogen jet flame in heated coflow at jet Reynolds number 8000 (Yoo et al., 2009) and a turbulent dimethyl-ether slot-jet flame at Reynolds number 13,050 (Bhagatwala et al., 2015). They found that $\psi_\alpha$ is approximately species-independent, that the regime classification is largely preserved when the DNS data are filtered with explicit top-hat filters of width $\Delta/\Delta_{\mathrm{DNS}}\in\{8,32,128\}$ , and that combining the diagram with Pope's criterion gives a joint flamelet-model/resolution diagnostic that could guide model selection — including the choice between classical and multi-dimensional flamelet closures and adaptive mesh refinement — in non-premixed LES. The entire demonstration, however, is a priori: the regime diagram has never been exercised on an actual LES solution.

Problem Statement

Carry out the a posteriori assessment of the Chan–Ihme flamelet regime diagram that the source paper explicitly identifies as further work. The problem has two linked parts that share one scientific objective — establishing whether the regime diagram can be trusted as a grid-aware, self-auditing model-selection tool in production non-premixed LES. (1) Self-consistency: perform large-eddy simulations of a turbulent non-premixed flame closed with a flamelet model, evaluate $\psi_\alpha$ and the grid-resolution ratio $\Delta/\delta_R$ from the LES-resolved fields exactly as the regime diagram prescribes, and determine whether the diagram invalidates the flamelet formulation in regions where that closure is applied — that is, whether the diagnostic is self-consistent with the model that generated the solution, and what this feedback implies for the reliability of the diagram as a model-selection criterion. (2) Explicit versus implicit filtering: quantify the differences between the explicit top-hat filtering used in the a priori analysis and the implicit grid filtering of typical LES, in the evaluation of $\psi_\alpha$ and in the resulting regime classification.

References

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