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

Abstract Manifold-based combustion models have traditionally been limited to asymptotically nonpremixed or premixed combustion relying on one-dimensional manifold equations in terms of either mixture fraction or progress variable, respectively. More recently, multiple formulations of two-dimensional manifold equations have been proposed for multi-modal combustion in terms of both mixture fraction and some progress 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 homogeneous progress variable. In this work, a formulation for two-dimensional manifold equations in mixture fraction and generalized progress variable is developed that simultaneously satisfies all three of these desirable properties. Utilizing its flexibility, the generalized progress variable is constructed to be exactly spatially homogeneous for asymptotically nonpremixed combustion by explicitly relating its definition to solutions of one-dimensional manifold equations in mixture fraction. This definition of the generalized progress variable leads to trivial simplification of the two-dimensional manifold equations into (quasi-)one-dimensional equations in mixture fraction for asymptotically nonpremixed combustion. Within these two-dimensional manifold equations, three scalar dissipation rates (the mixture fraction dissipation rate, the generalized progress variable dissipation rate, and the cross-dissipation rate) appear as coefficients, and, to close the manifold equations, models are required for the dependence of these dissipation rates on the mixture fraction and generalized progress variable. Analytical model forms for the scalar dissipation rates, the development of which is aided by the new definition for the generalized progress variable, are compared against simulations of a laminar lifted coflow flame, and existing models 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 poor model, and an improved model is proposed in which the normalized cross-dissipation rate changes sign across 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 the sensitivity of the solutions of the one- and two-dimensional manifold equations to variations in these scalar dissipation profiles and indicates a relative insensitivity of the thermochemical state to the scalar dissipation rate profiles, especially compared to the sensitivity to the overall magnitude of the scalar dissipation rate or the inclusion of the often neglected cross-dissipation rate.