Two-dimensional manifold equations for multi-modal turbulent combustion: Nonpremixed combustion limit and scalar dissipation rates

摘要

Manifold-based combustion models have traditionally been limited to asymptotically nonpremixed orpremixed combustion relying on one-dimensional manifold equations in terms of either mixture fractionor progress variable, respectively. More recently, multiple formulations of two-dimensional manifoldequations have been proposed for multi-modal combustion in terms of both mixture fraction and someprogress variable. However, none of these formulations simultaneously satisfies three desirable properties:computationally solvable manifold equations, an explicit transport equation for the progress variable,and the recovery of (quasi-)one-dimensional manifold equations for all asymptotic modes of combustion,a notable challenge for asymptotically nonpremixed combustion that requires a spatially homogeneousprogress variable. In this work, a formulation for two-dimensional manifold equations in mixturefraction and generalized progress variable is developed that simultaneously satisfies all three of thesedesirable properties. Utilizing its flexibility, the generalized progress variable is constructed to be exactlyspatially homogeneous for asymptotically nonpremixed combustion by explicitly relating its definitionto solutions of one-dimensional manifold equations in mixture fraction. This definition of thegeneralized progress variable leads to trivial simplification of the two-dimensional manifold equationsinto (quasi-)one-dimensional equations in mixture fraction for asymptotically nonpremixed combustion.Within these two-dimensional manifold equations, three scalar dissipation rates (the mixture fractiondissipation rate, the generalized progress variable dissipation rate, and the cross-dissipation rate) appearas coefficients, and, to close the manifold equations, models are required for the dependence of thesedissipation rates on the mixture fraction and generalized progress variable. Analytical model forms forthe scalar dissipation rates, the development of which is aided by the new definition for the generalizedprogress variable, are compared against simulations of a laminar lifted coflow flame, and existingmodels for the mixture fraction and generalized progress variable dissipation rates are shown to be adequate.For the cross-dissipation rate, a constant normalized cross-dissipation rate is shown to be a poormodel, and an improved model is proposed in which the normalized cross-dissipation rate changes signacross stoichiometric mixture fraction such that both lean and rich combustion are either simultaneously“back-supported” or simultaneously “front-supported”. Additional analysis is conducted to determine thesensitivity of the solutions of the one- and two-dimensional manifold equations to variations in thesescalar dissipation profiles and indicates a relative insensitivity of the thermochemical state to the scalardissipation rate profiles, especially compared to the sensitivity to the overall magnitude of the scalardissipation rate or the inclusion of the often neglected cross-dissipation rate.