^{1}

^{1}

^{2}

^{1}

^{2}

The shape of the modal duct of an acoustic wave propagating in a muffling system varies with the internal geometry. This shape can be either as a result of plane wave propagation or three-dimensional wave propagation. These shapes depict the distribution of acoustic pressure that may be used in the design or modification of mufflers to create resonance at cut-off frequencies and hence achieve noise attenuation or special effects on the output of the noise. This research compares the shapes of acoustic duct modes of two sets of four pitch configurations of a helicoid in a simple expansion chamber with and without a central tube. Models are generated using Autodesk Inventor modeling software and imported into ANSYS 18.2, where a fluid volume from the complex computer-aided-design (CAD) geometry is extracted for three-dimensional (3D) analysis. Mesh is generated to capture the details of the fluid cavity for frequency range between 0 and 2000Hz. After defining acoustic properties, acoustic boundary conditions and loads were defined at inlet and outlet ports before computation. Postprocessed acoustic results of the modal shapes and transmission loss (TL) characteristics of the two configurations were obtained and compared for geometries of the same helical pitch. It was established that whereas plane wave propagation in a simple expansion chamber (SEC) resulted in a clearly defined acoustic pressure pattern across the propagation path, the distribution in the configurations with and without the central tube depicted three-dimensional acoustic wave propagation characteristics, with patterns scattering or consolidating to regions of either very low or very high acoustic pressure differentials. A difference of about 80 decibels between the highest and lowest acoustic pressure levels was observed for the modal duct of the geometry with four turns and with a central tube. On the other hand, the shape of the TL curve shifts from a sinusoidal-shaped profile with well-defined peaks and valleys in definite multiples of

In vortex motion, the curvature of the streamlines introduces the action of centrifugal force which must be counterbalanced by a pressure gradient in the fluid [

Effects of vortex in acoustics have also been studied by Oosterhuis et al., [

The configuration of the helicoid and the central tube in a simple expansion chamber which is the subject of this investigation is a modified version of the Herschel-Quincke tube [

The wave equation describing sound in one dimension at a position x is given by [

Two arrangements (Model A and Model B) each with four variations of the helicoid within a simple expansion chamber of overall dimensions as indicated in Figure

Overall dimensions and orientation of the SEC

Nomenclature of the Helicoid geometry

For the first arrangement, acoustic propagation is such that flow is possible through the mid-section and around the spaces after sudden expansion at the entry to the chamber defined by the helicoid within the SEC and later joining the main stream due to contraction at the end of the expansion chamber. Effects of back-pressure are minimized since there is no barrier in the direct path of propagation of the acoustic wave, but possibility of generation of aerodynamic sound exists as a result of vorticity created by the helicoid. In the second arrangement, a concentric tube of 50mm diameter is added at the expansion chamber for all the variations shown in the second column in Table

Configurations of test specimens with and without a central tube.

| | |
---|---|---|

(1) | | |

| ||

(2) | | |

| ||

(3) | | |

| ||

(4) | | |

The geometries were modeled in 3D using Autodesk Inventor software and imported into Ansys, where fluid volume from the complex CAD geometry was extracted for 3D analysis. The geometry was meshed based on the maximum frequency of 2000Hz and sonic speed of 343 m/s, giving a mesh size of approximately 0.02m. Transient Computational Fluid Dynamics (CFD) analysis was carried out after definition of acoustic properties and application of acoustic boundary conditions and loads. Details of boundary conditions defined are included in the Appendix at the end of this paper.

Postprocessed results included a map of acoustic sound pressure field computed at each element node, acoustic time frequency plot, and acoustic power result plot. These are summarized in Figures

Acoustic Pressure maps for models with and without central tube.

| | |
---|---|---|

(1) | | |

| ||

(2) | | |

| ||

(3) | | |

| ||

(4) | | |

Acoustic pressure map for the Simple Expansion Chamber.

Transmission Loss of a Simple Expansion Chamber.

Transmission loss of helicoid without central tube.

Transmission Loss of helicoid with central tube.

In Figure

From the geometry of the basic structure, the area expansion ratio (m =D/d) is 3 while the ratio L/D = 7 implying that the chamber used is axially longer. The speed of sound^{−1}. In Figure ^{th} trough_{m} is given by

Equation (

The results of the maps in Table

For Model 2a, a sudden spike in the value of TL sets in at about 1150Hz while, for Model 2b, this occurs much early at 600Hz. For Model 3a, the sudden spike in the value of TL is observed at 700Hz while for Model 3b this occurs at about 450Hz. For these three sets of models, effects of 3D wave propagation are experienced much earlier with the central tube in place as compared with when the tube is not included. Both Models 4a and 4b exhibit the same characteristics where effects of 3D wave propagation set in at about 1400Hz with a maximum TL of 92 decibels occurring at 1900Hz. A broadband attenuation is observed for Model 4 for frequency range between 1400Hz and 1900Hz. However, the similarities in TL characteristics do not reflect in the acoustic pressure maps, but the geometries with and without the central tube appear to produce the same TL characteristics.

With the introduction of the helicoid within the SEC, regions of fairly low acoustic pressure are observed throughout this section. The occurrence of these low-pressure regions is likely to affect the overall acoustic pressure in the respective configurations. As the wave propagates downstream, there is a general reduction in acoustic pressure levels with patterns indicating noise attenuation. As the number of turns increases from one to four, the pattern also modifies to give an overall low acoustic pressure downstream as seen in Model 4b in Table

In this paper, Ansys 18.2 software has been used to simulate acoustic energy distribution in a simple expansion chamber with helicoid. Four variations of the helicoid have been used to predict the behaviours of acoustic energy as the pitch is varied. Results present the transformation of acoustic energy due to plane wave propagation as observed for a simple expansion chamber to a complex combination of plane wave and 3-dimensional wave propagation induced by the helicoidal geometry. Consolidation and reconfiguration of regions of high and low acoustic pressures are observed. The inclusion of the central tube provides two distinct paths for the propagation of acoustic waves. This leads to acoustic wave cancellation and hence noise attenuation if there is a phase difference between the two propagating waves as they meet downstream. The introduction of the central tube also contributed to the change in acoustic energy distribution.

Transmission loss characteristics of four variations of a helicoid in a simple expansion chamber have also been investigated. The TL curves show “peaks” for which the muffler is efficient and “valleys” where TL approaches or tends to zero indicating that all the acoustic energy radiates through the outlet of the muffler geometry. At the maximum TL, most of the acoustic energy injected is reflected by the cavity and comes back through the inlet section.

At lower frequencies, results of the models show plane wave propagation characterized by uniform and periodic shape of the transmission loss graph. This can also be seen in the pressure field map of respective models downstream and upstream of the expansion chamber. At higher frequencies, however, complex pressure fields are formed indicating that wave propagation is not planer anymore and thus will not favour acoustic propagation. This is indicated in the pressure field map in the simple expansion volume by the different pressure levels with the different colour shades. An interpretation of this on the transmission loss graph is the high transmission loss values that follow the plane wave propagation profile on the graphs.

From the analysis of the acoustic maps around the helicoid fitted in a simple expansion chamber, the geometry can be further developed to improve transmission loss characteristics of the geometry and hence higher noise attenuation.

Analysis settings

Frequency spacing: Linear

Range (min and max): 0 to 2000

Solution interval: 50

Solution method: Full

Output controls: General Miscellaneous

Acoustic Body

Mass density: 1.204 kgm^{−3}

Sound speed: 343.24 ms^{−1}

Acoustic-structural coupled body: Uncoupled with symmetric Algorithm

Behaviour: Compressible

Other boundary conditions:

Acoustic normal surface velocity: -1ms^{−1} at inlet, and 0 phase angle

Transparent port setting specified for inlet and outlet

Acoustic radiation boundary defined for the inlet and outlet.

The data used to support the findings of this study are available from the corresponding author upon request.

The authors declare that they have no conflicts of interest.

The authors acknowledge the financial support offered by the Africa-ai-Japan Project towards this research.