Modeling of in-cylinder pressure oscillations under knocking conditions: A general approach based on the damped wave equation

The modeling of the in-cylinder pressure oscillations under knocking conditions is tackled in this work. High frequency pressure oscillations are modeled by the explicit integration of a partial differential wave equation augmented with a time-dependent dissipation term. The general solution of such equation is determined by the Fourier method of separation of variables whereas the integration constants are obtained from the boundary and initial conditions. The integration space is a cylindrical acoustic cavity whose volume is that of the combustion chamber evaluated at the knock onset. The domain of integration is assumed to be formed by a finite set of small volumes having the shape of annulus sectors. This approach involves that knock region can assume more realistic shape of the kernels where abnormal combustion initiates. The initial conditions are evaluated by means of a two-zone thermodynamic model applied to low-pass filtered experimental pressure cycles. The damping coefficient and the knock region are model parameters to be assigned or identified experimentally by means of a proper least-squares optimization process. Experimental data obtained on a direct injection spark ignition engine, operating under knocking conditions at different speeds, have been used to validate the model both in time and frequency domains.