Advances in Acoustics and Vibration

Volume 2018, Article ID 3276548, 6 pages

https://doi.org/10.1155/2018/3276548

## Effects of Central Tube on Shape of Modal Duct of a Helicoid in a Simple Expansion Chamber

^{1}Jomo Kenyatta University of Agriculture and Technology, Kenya^{2}Multimedia University of Kenya, Kenya

Correspondence should be addressed to Daniel Omondi Onyango; ek.ca.taukj.gne@idnomod

Received 9 April 2018; Accepted 2 December 2018; Published 11 December 2018

Academic Editor: Benjamin Soenarko

Copyright © 2018 Daniel Omondi Onyango et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

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 *π* for the simple expansion chamber, while that of the other two configurations depended on the variation in wavelength that affects the location of occurrence of cut-on or cut-off frequency. The geometry with four turns and a central tube had a maximum value of TL of about 90 decibels at approximately 1900Hz.

#### 1. Introduction

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 [1]. Aerodynamic sound is generated as a result of the movement of vortices, or of vorticity, in an unsteady fluid flow. Vortex flow can be introduced in fluid systems by modification of the path of flow in order to achieve desired effects. Such modifications include introduction of holes with sharp edges [2], barriers in the path of flow [3], or sudden contraction or expansion as is the case with the SEC [4]. Applications of this phenomenon in vortex motion have been successfully studied by Smith et al., [5] who identified two scattering mechanisms that allow neighbouring modes to interact; scattering occurs at significantly lower frequencies when the mean flow is present; an exchange of energy between mean flow and acoustic field occurs during scattering.

Effects of vortex in acoustics have also been studied by Oosterhuis et al., [6] who predicted the relationship between pressure drop and acoustic power dissipation. Changes in circulation or area of a vortex ring gives rise to a dipole sound field. When this sound source couples with a resonator, large amplitudes may be generated. The nonlinear behaviour of resonators where change of the output is not proportional to the change of the input occurring with flow separation makes such devices efficient sound absorbers. An acoustic wave may be greatly affected by factors such as mean flow, convection, refraction in shear, coupling with vorticity, and scattering by turbulence.

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 [7]. In the present arrangement, the path bypassing the central tube is made longer by varying the pitch of the helicoid for a fixed length of the chamber. Similar modifications have been studied by Selamet et al. [8], Karlsson and Glav [9], and Selamet and Easwaran [10], though the difference here is the elongation of the path along and around the axis of propagation as opposed to that parallel to the axis of propagation only in the Herschel-Quincke version.

The wave equation describing sound in one dimension at a position x is given by [11] where** p** is the acoustic pressure and

**is the speed of sound. Provided that the speed is a constant and not dependent on frequency, then the general solution of the equation becomeswhere**

*c**f*and

*g*are any two twice differentiable functions. This may be pictured as the superposition of two waveforms of arbitrary profiles, one

**travelling up the x-axis and the other**

*(f)***down the x-axis at speed**

*(g)***. The particular case of a sinusoidal wave traveling in one direction is obtained by choosing either**

*c***or**

*f***to be a sinusoid and the other to be zero, giving where**

*g***is the angular frequency of the wave and**

*w***is its wave number.**

*k*#### 2. Materials and Methods

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 1(a) were used in this analysis. Model A had been investigated by [12]. The helicoid geometry (Figure 1(b)) is defined by its outer diameter which coincides with the inner diameter of the expansion chamber (D = 150mm), inner diameter corresponding to the inner diameter of the inlet and outlet pipes (d = 50mm), pitch that varies (350, 175, 117, and 87.5mm), and an overall length corresponding to the inner length of the expansion chamber (L = 350mm). The thickness of the model is 5mm throughout.