Modeling pressure oscillations under knocking conditions: a partial differential wave equation approach

In this work the authors present a model to simulate the in-cylinder pressure oscillations due to knock. Pressure oscillations are predicted by the explicit integration of a Partial Differential Wave Equation (PDWE) similar, in its structure, to the so-called “Equation of Telegraphy”. This equation differs mainly from the classical wave formulation for the presence of a loss term. The general solution of such equation is obtained by the Fourier method of variables separation. The integration space is a cylindrical acoustic cavity whose volume is evaluated at the knock onset. The integration constants are derived from the boundary and initial conditions. A novel approach is proposed to derive the initial condition for the derivative of the oscillating component of pressure. It descends, conceptually, from the integration of the linearized relation between the derivative of pressure versus time and the expansion velocity of burned gas. In practice, the required calculation parameters are evaluated by means of a two zone thermodynamic processing of single, low-pass filtered pressure cycles. The damping constant, the size and position of the knocking volume of unburned gases at knock onset are the model parameters to be assigned or identified. The model was validated using a set of experimental data obtained using a four cylinder Direct Injection SI engine at different operating conditions. The position of the unburned gases at knock onset was identified through a mean square optimization process based on hybrid genetic algorithms. Even if a simple cylindrical geometry was adopted to include the mass of unburned gases, simulations reproduced experimental measurements with fairly good accuracy.