Example: Acoustics in a Muffler

Introduction

• The damping effectiveness of a muffler is studied in the frequency range 100─1000 Hz

• In the low-frequency range, reactive damping prevails. Therefore, resistive damping is not included in the model.

• The purpose of the model is to show how 3D acoustics can be treated in a fairly complex geometry consisting of several separate sections and pipes divided by thin, but perfectly rigid, walls.

Acoustics in a Muffler

Geometry

Acoustics in a Muffler – Problem Definition

Inlet

OutletResonator chambers

Domain Equations

• Frequency domain acoustics is goverened by a slightly modified Helmholtz’s equation for the acoustic pressure, p.

• Material properties are density, 0, and speed of sound, c.

• Note that the density cannot be eliminated from the equation unless it is a global constant

Acoustics in a Muffler – Problem Definition

2

20 0

0 2p

p , fc

Boundary Conditions

• The natural boundary condition corresponds to inward normal acceleration. Three types are used:

Acoustics in a Muffler – Problem Definition

00

p

n

pc

ipn

00

0000

2p

c

ip

c

ipn

Hard wall

Incoming plus outgoing plane wave

Outgoing plane wave

Implementation

• Different dependent variables are used in the pipes and in the resonator chambers in order to decouple the pressure on the inside of the pipe walls from the pressure on the outside.

• The continuity of the pressure where the pipes open into the resonator champers, and at the openings between the chambers, is enforced through boundary conditions.

Acoustics in a Muffler – Implementation

Results

• The plot shows the pressure amplitude and energy flow at 470 Hz

• Energy is conserved as no dissipation has been included in the model

• The frequency is close to one of the resonance frequencies of the middle section.

Acoustics in a Muffler – Results

Results

• The damping coefficent shows typical ”dips” at the resonance frequencies of the separate chambers.

Acoustics in a Muffler – Results