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Flamelet models for premixed combustion, which are based on equations formulated and solved in progress variable space, have been proposed in the past, but have not been adopted for chemistry reduction methods. This is due to one limitation of these models: they need a closure for both the magnitude and the shape of the gradient (or scalar dissipation rate) of the progress variable, which is essential for an accurate prediction of the flame displacement speed. So far, solution methods for the aforementioned models require gradient information as an input, which is either modelled and non-generic, or extracted from a previous physical space flame solution for the analogous problem. The objective of this work is to provide a self-contained solution method for freely-propagating premixed flamelets in progress variable space, by solving an additional flamelet equation for the gradient of the progress variable. With this, the novel method provides both magnitude and shape of the gradient. Studying hydrogen-air and methane-air configurations, it is demonstrated that an accurate prediction of the laminar flame speed without the necessity for further input parameters can be obtained.

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Interactions between premixed and non-premixed reaction zones can lead to complex mixed combustion regimes, here denoted as multi-regime combustion, which pose challenges to many conventional combustion modeling approaches. Such conditions occur in most practical combustors and can originate from partial premixing, mixture inhomogeneities/stratification, hot product recirculation, or local flame extinction and re-ignition. Therefore, novel equations are derived for modeling multi-regime combustion which are formulated with respect to a twodimensional composition space spanned by mixture fraction and reaction progress variable. Contrary to previous works, the dependency of the progress variable on the mixture fraction is considered in the new model. This is achieved by splitting the progress variable gradient into an aligned and an orthogonal component with respect to the mixture fraction gradient and the latter is used to define the second coordinate. In the theory that follows, a balance equation for the progress variable on mixture fraction iso-surfaces is formulated. Using this balance equation together with the orthogonal coordinate system, the transformation of species and temperature equations to the 2D composition space yields a novel set of equation without so-called cross-terms. This is advantageous since cross-terms obtained with previous approaches lack a general closure and it is uncertain if it exists at all. Furthermore, the approach allows to naturally distinguish between non-premixed and premixed combustion regimes, auto ignition, and it covers multi-regime combustion characteristics. The theory is validated and discussed by means of a fully resolved solution of a laminar triple flame using detailed chemistry. At first, regions which exhibit premixed, non-premixed or multi-regime combustion characteristics are identified. The triple flame solution then serves as a database from which all relevant theoretical relations are post-processed and validated. In comparison to budgets of conventional 1D flamelet equations for premixed and non-premixed combustion it is shown that only the full set of transport terms considered in the 2D equations accurately balances chemical source terms everywhere in the triple flame, especially in regions where 1 multi-regime combustion prevails.

Solving flames in scalar or composition space usually requires the modeling of scalar dissipation rates, or scalar gradients, which appear in the Jacobian of the transformation from the physical-space coordinates to the scalar space coordinates attached to the flame surface. Recently, Scholtissek et al. (2019) have discussed a self-contained solution for freely propagating premixed flames, in which the gradient distribution of the coordinate is also solved in scalar space. This approach is here extended to include curvature, strain and effects of unsteadiness. The resulting method is applied to a constant pressure and homogeneous ignition case, a stagnation flame, a tubular flame and a spherical expanding flame, capturing accurately effects of unsteadiness, strain and curvature. The results are systematically validated against physical-space solutions and experiments. In addition to the possibility of exploring these various canonical problems with a single set of equations, it is shown that another major outcome of this new formulation lies in the possibility of studying steady flames subjected to negative strain.

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