Mathematical Modelling of Linear/Non-Linear Wave Reflective Boundaries in Automotive Exhaust Systems

Time-domain numerical modelling of performance for complete engine systems is now routine in most automotive design offices. To achieve this with a minimal computer run-time overhead, drastic simplification of the exhaust silencer box models is required. Complete exhaust systems are often modelled as simple capacitances bounded by orifices. This is justified on the grounds that the influence of exhaust system geometric detail on the engine torque-speed characteristic is less significant than that of the backpressure of the ‘lumped’ system. In contrast to engine performance, acoustic modelling of silencer boxes is usually carried out in the frequency-domain, since this offers a computationally cost-effective means of including very complex internal geometries with minimal computer time overhead. One limitation of these models is the difficulty of characterising the engine as an acoustic source, though empirical methods are often employed.

This paper describes the development of a ‘hybrid’ model in which a frequency-domain silencer box boundary condition is integrated into the MERLIN time-domain engine code. The silencer box is modelled by its frequency dependent reflection and transmission coefficients at each boundary. These are calculated using the LAMPS frequency-domain acoustic model. An engine order-based scheme is used to map the time-domain silencer box boundary into the frequency-domain, and back again, on a once per cycle basis. The model is limited to steady engine speeds only. Predictions of time-pressure histories up and downstream of a silencer box are compared with time-domain measurements. Good agreement in both trend and magnitude is demonstrated.