Automotive Muffler Simulation
|Introduction||Model 2: Muffler with Perforation|
|Frequency Domain Acoustic Model||Model 3: Muffler with Sound absorptive Pads|
|Domain||Post-Processing and Visualization|
|Model 1: Classical Muffler||References|
This study is to simulate the sound propagation within three different kinds of mufflers, and compare their performance varying from the frequency to . The mufflers analyzed here are a classical muffler, a perforated muffler and a padded muffler. The sound pressure distribution and the transmission loss spectrum will be calculated to show the damping behavior of these types of mufflers.
The symbols and corresponding units used throughout this tutorial are summarized in the Nomenclature section.
Please refer to the information provided in "Acoustics in the Frequency Domain" for more general theoretical background for acoustics.
To study the performance of a muffler over a range of frequencies, an acoustic model is built in the frequency domain based on the source-free Helmholtz partial differential equation (PDE):
The classical acoustic muffler consists of three parts: an inlet pipe, an outlet pipe and expansion chambers. Note that the inlet and outlet pipes are assumed to have infinite extension, and are modeled with a radiation boundary condition and an absorbing boundary condition, respectively.
There are two common ways to further improve the muffler performance. The first method is to use perforated pipes within the muffler, and the second method is to cover the walls of the muffler with sound absorptive pads. Each technique has its advantages and disadvantages, and will be explained in the later section.
There are three types of the boundary conditions involved in this example. To model the incoming sound wave at the inlet a radiation boundary condition is used.
This radiation boundary condition is very general. However, in the 3D case it may use up more memory that is available. To reduce the memory requirement, we observe that the inlet is at . Therefor the boundary normal is in the direction . This boundary unit normal can be specified as a parameter and will simplify the boundary condition.
To model the outgoing wave at the outlet an absorbing boundary condition is specified.
On the remaining walls of the muffler the default sound hard boundary conditions are used. Since the sound hard boundary condition is the default boundary condition nothing needs to be further specified.
In acoustics simulations the wavelength of a sound wave needs to be resolved by a sufficiently fine mesh in order to get an accurate numerical solution. Here we set the max edge length to 12 nodes per , which means that there will be at least twelve elements per wavelength in each direction of the wave propagation.
To study the behavior of the muffler over the frequency range , the PDE model is solved repetitively with ParametricNDSolveValue.
Last, we consider a muffler with sound absorption materials padded on the inner walls. Common sound absorptive materials are made of fiberglass or steel wool. Depending on the density and the mean radius of the material , the flow resistivity values typically lie in range of to .
A technique to model the sound propagation within absorptive materials is called "Equivalent Fluid Method" . The idea is to characterize the sound attenuation by inserting the effective density and the effective speed of sound in the original Helmholtz equation (3). and are complex valued and can be calculated with an empirical formula derived by Delany and Bazley . For the low-frequency range where the non-dimensional parameter , Kirby and Cummings  presented the following more accurate approximations:
To see the general damping behavior of acoustic waves, the sound propagation is visualized within the classical muffler. The harmonic wave relation (7) is applied to transform the solution into the time domain:
See this note about improving the visual quality of the animation.
Within the inlet pipe and the first expansion chamber, the sound wave appears to be fixed in space but it simply oscillates in time. This type of wave is known as the standing wave, and often forms several local extrema of the sound pressure.
This excess sound pressure is called "Back-pressure" , which will restrict the gas flow within the muffler and reduce the power output of the car engine. To see how the perforated and padded mufflers improve this issue, it is better to visualize the amplitude distribution of the sound pressure waves.
Both the perforated and padded mufflers improve the effect of back-pressure that is seen in the classical muffler. Because the sound pressure wave is allowed to dissipate within the muffler pipes, the perforated muffler created the lowest back-pressure among the three cases.
Another important quantity to measure the performance of an acoustic muffler is the transmission loss (TL), which is defined as the ratio between the incident and the transmitted sound power. The formulas for the transmission loss, sound power and sound intensity are given by:
For the classical muffler, the transmission loss curve appears in a wavelike pattern with few dips. These dips correspond to the resonance frequencies of the muffler, where the sound transmission is enhanced and the muffler becomes less effective. The padded muffler, however, not only smooths out these dips but also increases the overall transmission loss values, especially for the higher frequencies, making it a more effective muffler in that range. Note that although the perforated muffler has the best effect on the back-pressure issue, it has generally smaller transmission loss values over the spectrum except for the frequency around .
More advanced mufflers can be constructed by combining the features of absorptive pads and perforation . If there is a targeting frequency range where the noise is to be controlled, the dimension of each expansion chamber can also be customized to meet the functional requirements.
As explained in a separate tutorial, the discrete Fourier transform generates Fourier coefficients with the symmetric property that . For each pair of the Fourier coefficients and , the corresponding frequency component is .
Based on the output audio and the amplitude spectrum, the engine noise has been significantly damped by all of the mufflers. For the low frequency range which the noise is mostly composed of, the padded muffler shows the best sound attenuation.
|c||speed of sound||[m/s]|
|ω||sound wave angular frequency||[rad/s]|
|f||sound wave frequency||[Hz]|
|R||radius of the expansion chamber||[m]|
|L||length of the expansion chamber||[m]|
|r||radius of the inlet/outlet pipe||[m]|
|l||length of the inlet/outlet pipe||[m]|