Veried Computations of Laminar Premixed Flames

The required spatial discretization to capture all detailed continuum physics in the reaction zone for one-dimensional steady laminar premixed hydrogen-air ames described by detailed kinetics and multi-component transport is accurately estimated a priori by a simple mean free path calculation. To verify this, a robust method has been developed to rigorously calculate the nest length scale a posteriori. The method reveals that the nest length scale is at the micron-level. This result is consistent with an estimate from the underlying molecular collision theory, and orders of magnitude smaller than the discretization scales employed in nearly all multi-dimensional and/or unsteady laminar premixed ame simulations in the literature. It is well understood that in any mathematically based scientic theory, associated computations should have delit y with the underlying mathematics, and the underlying mathematical model has to represent the observed physics. The rst issue is demonstrated by comparing computational results with another known solution and/or performing a formal grid convergence study, while the second issue is demonstrated by comparing the computational predictions with experimental data. Addressing these two issues, in this order, is a necessity in any computational study to build condence in both the simulation strategy and the underlying mathematical model. The exercise of demonstrating the harmony of the discrete solution with the foundational mathematics is known as verication. 1 For multi-scale problems, verication is dicult due to the range of the spatiotemporal scales, which may span many orders of magnitude. In this kind of problem, usually modeled by highly nonlinear equations, signican t coupling across the scales can occur, so that errors at small scales can rapidly cascade to the large scales. Moreover, the strength of the coupling across the scales is not known a priori. So, all the physical scales of the mathematical model, temporal and spatial, have to be captured in order to have full condence that predictions are repeatable, grid-independent, and thus veriable. Subsequently, in the validation step one can choose what physical phenomena and to what accuracy one wants to reproduce experiments. The main aim of this paper is to rigorously determine the required spatial resolution to capture all physical scales in a standard multi-scale problem: the steady one-dimensional laminar premixed ame propagating freely at atmospheric pressure in a stoichiometric mixture of hydrogen-air described by detailed kinetics and multi-component transport. Here, the robust method to calculate the length scales employed in Powers and Paolucci 2,3 for gas phase detonation is implemented with modication for deagration. The method is robust in that it has little dependence on the details of the underlying numerical method used to calculate the laminar ame. It simply requires a local determination of the state of the system, which is followed by a Jacobian formulation, and a generalized eigenvalue analysis. As such, it is able to estimate with great accuracy the length scales on a fundamental mathematical, non-numerical, basis. The minimum length scale which must be resolved in order for the mathematical model to be veried is thus determined. In the rst section, the governing partial dieren tial equations (PDEs) for unsteady reactive o w are presented. This is followed by a reduction of the PDEs into a system of dieren tial algebraic equations (DAEs) which describes the spatial evolution of the state variables. Following a short description of the generalized