A posteriori Assessment of the flamelet regime diagram for non-premixed turbulent combustion LES: self-consistency under a flamelet closure and explicit-versus-implicit filtering effects on the regime parameter

Chan and Ihme (Combustion and Flame 186:220-235, 2017) developed a flamelet regime diagram, based on the regime parameter $\psi_\alpha$ (the ratio of flame-aligned, tangential diffusion to flame-orthogonal, classical one-dimensional flamelet diffusion for species $\alpha$ ), that identifies where conventional one-dimensional flamelet models are applicable with respect to the local grid resolution and flow/flame structure. The diagram was verified only a priori, by applying explicit top-hat filters to a direct numerical simulation (DNS) of a turbulent lifted hydrogen jet flame. Two questions must be answered before the diagram can serve as a model-selection tool in production large-eddy simulation (LES): (1) self-consistency - whether an LES generated with a flamelet closure produces resolved fields whose own regime diagram invalidates the flamelet formulation in parts of the domain; and (2) how the implicit filtering of the numerical grid and discretization, in contrast to the explicit filter used a priori, affects the evaluation of $\psi_\alpha$ and the resulting regime classification. All subsequent uses of the regime diagram remain a priori analyses of DNS data; no a posteriori LES assessment of the diagram is known to date, so both questions remain open.

Background

In non-premixed (diffusion) combustion, fuel and oxidizer mix and react in thin sheets whose structure is often modeled by the flamelet concept: the reactive layer is treated as an ensemble of locally one-dimensional, laminar diffusion flames embedded in the turbulent flow, with species mass fractions and temperature governed by flamelet equations parameterized by the mixture fraction $Z$ (a conserved scalar varying from 0 in the oxidizer stream to 1 in the fuel stream) and its scalar dissipation rate (the rate at which molecular mixing erodes mixture-fraction gradients, which sets the local flamelet time scale). The classical Pitsch-Peters flamelet formulation retains only diffusion in the direction normal to the mixture-fraction iso-surfaces, whereas the multi-dimensional flamelet equations also contain tangential diffusion, i.e. transport along the stoichiometric $Z$ surface caused by flame propagation and mixture-fraction curvature. Chan and Ihme (2017) quantified the competition between these two types of diffusion through a regime parameter $\psi_\alpha$ for each species $\alpha$ : the ratio of the flame-aligned (tangential, curvature and multi-dimensional) diffusion terms to the flame-orthogonal (classical flamelet) diffusion terms, evaluated pointwise in a reference frame rotated so that one axis aligns with the local mixture-fraction gradient, an arrangement that avoids noisy second derivatives in mixture-fraction space. A length-scale argument gives $\psi_\alpha \sim (l^Z/l^\perp)^2$ , where $l^Z$ scales with the reaction-zone thickness $\delta_R$ and $l^\perp$ with the Kolmogorov length scale $\eta_K$ (the size of the smallest turbulent eddies), so that $\psi_\alpha/(2Z_{st}) \approx (\delta_R/\eta_K)^2 = Ka_K$ , a Karlovitz number defined for non-premixed flames, with $Z_{st}$ the stoichiometric mixture fraction. Following Peters' criteria, $Ka_K \le 1$ marks the regime in which the reaction zone is thinner than the smallest turbulent eddy and isolated one-dimensional flamelets are valid, and $Ka_K = 100$ is the upper validity limit of flamelet theory. These criteria define a regime diagram in the plane spanned by the normalized grid spacing $\Delta/\delta_R$ (abscissa) and $\psi_\alpha$ (ordinate): the Peters regime below the $Ka_K = 1$ diagonal, a transition region within $2Z_{st} \le \psi_\alpha \le 1$ in which the one-dimensional flamelet representation degrades, and the multi-dimensional flamelet regime above $\psi_\alpha = 1$ , where interaction between flamelets invalidates isolated one-dimensional models. In the source diagram, the $Ka_K = 1$ and $Ka_K = 100$ conditions are two distinct diagonals (since $\psi_\alpha/(2Z_{st}) = Ka_K$ ), and the line $\psi_\alpha = 1$ is a separate boundary: the transition region lies between the $Ka_K = 1$ diagonal and the $\psi_\alpha = 1$ line, while $Ka_K = 100$ serves as Peters' separate upper validity limit for flamelet theory, used to assess the applicability of the flamelet formulation rather than bounding the transition region. Because a mere change of grid resolution produces only a horizontal shift in the diagram, the construction relates model validity to the resolution actually used in a simulation. Chan and Ihme verified the diagram entirely a priori: they applied explicit top-hat filters (a box filter that averages over a cubic region of a given width) of varying widths to a DNS of a turbulent lifted hydrogen jet flame in a heated coflow, and found that the prevalence of one-dimensional flamelets is largely unchanged with filter width, supporting the approximation of $\psi_\alpha$ from the resolved (filtered) fields of an LES through an estimator that uses only resolved terms. An a priori analysis filters known reference data, whereas an a posteriori analysis evaluates the diagnostic inside a simulation whose solution is itself produced by the model under test. Practical LES also never applies an explicit mathematical filter: the computational grid and discretization act as an implicit filter whose effective transfer function is generally unknown and differs from a top-hat filter. The source paper therefore left open whether the regime diagram survives deployment in actual LES practice.

Problem Statement

Perform the a posteriori assessment of the Chan-Ihme flamelet regime diagram that the 2017 source explicitly identifies as the required next step, with the overarching objective of establishing whether the diagram can be used as a reliable model-selection tool in production LES of turbulent non-premixed combustion. Concretely: (1) carry out large-eddy simulations of at least one well-documented turbulent non-premixed flame (for which DNS or experimental reference data exist) using a flamelet-model closure as the combustion model; evaluate the regime parameter $\psi_\alpha$ and the resulting regime classification from the resolved LES fields using the estimator of Chan and Ihme (2017); and determine the self-consistency of the construction - whether the flamelet-model-generated solution contains regions in which the regime diagram, evaluated from that same solution, indicates that the flamelet formulation is invalid or inapplicable, and what this feedback implies for the reliability of the diagram as a diagnostic. (2) Quantify the effect of the implicit filtering performed by the numerical grid and discretization of the LES, in contrast to the explicit top-hat filtering used in the a priori analysis, on the evaluation of $\psi_\alpha$ and the resulting regime classification. Determine whether, and under what conditions, the regime diagram can be deployed as a standard model-selection diagnostic in LES practice, and quantify the residual bias, defined as the discrepancy between the regime classification obtained from the implicitly filtered LES fields and that obtained from explicitly filtered reference data at matched effective filter widths. A single well-documented flame, as specified above, is sufficient scope, provided the answer states explicitly the conditions to which its verdict is limited.

References

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