Characterizing Lean Spark Ignition Combustion Instability in Terms of a Low-Order Map

We investigate lean-fueling cyclic dispersion in spark ignition engines in terms of experimental nonlinear mapping functions representing the connection between past and future combustion events. Nonlinear mapping functions provide a relatively easy method for identifying the deterministic dynamics associated with lean combustion instability, even in the presence of very high levels of noise. Observed experimental maps appear to have strong similarities to those predicted by an existing nonlinear spark ignition engine model. Differences between the observed map and model predictions become more pronounced at very lean fueling and high residual fraction. Map function details are shown to be useful in model validation, identifying model deficiencies, and comparing the characteristics of different engines. We expect that such maps will also be useful for developing real-time control strategies. * Corresponding author: [email protected] Proceedings of the 2 Joint Meeting of the U.S. Sections of the Combustion Institute Introduction In previous publications [1-5], we report strong evidence for the presence of noisy nonlinear determinism in the combustion dynamics of spark-ignition engines operated under very lean fueling conditions. These observations indicate that cycle-to-cycle combustion variations can be described in terms of simple (i.e., low-order) nonlinear maps that correlate combustion quality in one cycle with combustion quality in succeeding cycles. The dominant physical mechanism involved appears to be the impact of residual fuel and air on ignition in subsequent cycles. In the investigation described here, our objective has been to explore how low-order deterministic features can be separated from the high dimensional (i.e., noisy) part of the combustion process. Our objective is to develop quantitative descriptors of the deterministic component that can be used for comparing the combustion performance of different engines, the combustion performance of different cylinders within the same engine, or the combustion behavior predicted by detailed CFD models with experimental results. In addition, we anticipate that better definition of deterministic components in combustion instability will improve the chances for implementing active control strategies that can extend engine operation into regions where fuel efficiency and emissions are improved. We begin our investigation with the working hypothesis that if a low-order deterministic relationship exists between successive cycles, then it should be possible to reconstruct the basic features of this relationship even when the observations are heavily contaminated with parametric and measurement noise. The basis for such a hypothesis can be illustrated by considering numerical experiments with a simple nonlinear map, the so-called logistic equation, which has been widely investigated [6]. The usual form of the logistic map is: [ ] ) i ( x 1 ) i ( x k ) 1 i ( x − = + (1) where x(i) represents the scalar value of the system state on iterate i , x(i+1) is the value on the succeeding iterate, and k is a feedback parameter. Depending on the value of k, the output of repeated iterations of equation (1) can be a single value, a repeating sequence of two or more values (i.e., periodic), or a non-repeating but bounded series of infinite length (i.e., deterministic chaos). Changing k from lower to higher values (e.g., from k=2 to k=3.8) produces a series of dynamical state transitions referred to as a period doubling bifurcation sequence. It is now well understood that the source of this complex behavior in the logistic map is the nonlinear feedback from prior to future iterates. The parameter k directly contributes to this feedback, larger values of k producing greater instability. As we explain more fully elsewhere [1-5], it appears that cyclic combustion variations in spark-ignition engines undergo the same basic type of bifurcation sequence with increasingly lean fueling. Specifically, residual cylinder gas creates the same type of dynamic instability as the numerical feedback in the logistic equation. An important complication, however, is that parametric noise in the engine blurs the distinctness of the observed dynamics. Although the engine map is more complex than the logistic map, there still appear to be qualitative similarities. Further, by adding parametric noise to the logistic map, we expect to see other useful similarities. Figure 1(a) illustrates a return plot for the logistic map when k=3.2. The solid line represents the locus of the entire map for all initial x(i) values between 0 and 1. The open square on the diagonal represents the period-1 fixed