Analytical and Numerical Analysis of the Acoustics of Shallow Flow Reversal Chambers

Abstract

Flow reversal chambers are mainly used to accomplish a compact silencer design needed on a vehicle. Generally in this configuration the inlet and outlet ports are on the same face and the flow direction is reversed. During many years different authors have tried to develop 1D and 3D models for evaluating the acoustic performance of circular and rectangular reversing chambers. Ih [1] categorizes four methods for evaluating the acoustic performance of the reversing chamber. The first involves utilizing analysis techniques for other types of muffler elements having similar acoustic performances [2]. Analysis techniques for extended inlet/outlet expansion chambers may be used to approximate the behavior of a reversing chamber in which the length-to-diameter ratio is large. When the length-to-diameter ratio is small, the reversing chamber approximates the behavior of a short expansion chamber. In this case, exact predictions of the acoustic performances cannot be made and, moreover, the method itself is a trial-and-error one. The second is a mode-matching method at the discontinuities [3-5], but this is tedious to formulate and the transmission matrix for this type of muffler has not been obtained. A simplified version (third method) of this method has been developed for plane wave propagation, in which the sound pressures and particle velocities at the area discontinuities are matched [6, 7]. However, this method is restricted to a very small frequency range below the cut-off frequency of the first asymmetric mode, i.e., the (1, 0) mode, and the peaks of the transmission loss curves are not correctly predicted due to the disregard of the higher order modes. Furthermore, when the length-to-diameter ratio is small, the actual acoustic performance deviates appreciably from the theoretical transmission loss predicted by this one-dimensional analysis method. The fourth method involves using numerical methods such as finite element analysis [8] and the finite difference method [9], or possibly, the boundary element method. These numerical techniques have some merits in the treatment of more complicated geometries, such as that of an elliptic cross-section and/or a chamber with a pass tube [10], but a great many mesh points or mesh elements are required to deal with the high frequency range, so that the execution time for computation is long and the costs are high. It is also difficult to describe the total exhaust system by incorporating the transmission matrix of each silencer element.Lindborg et al. [11] modeled the flow reversal chamber by two port method. The system under study is broken down into a set of linear subcomponents that are described individually and then assembled in a network. Each component is treated as a black box that is defined at the inlet and outlet ports where plane waves are assumed. This is an efficient tool, but for complicated geometries such as the flow reversal chamber the decomposition into subcomponents is not obvious. Three different approaches are used for the two port modeling of a flow reversal [11]; 1- Large quarter wave resonator 2- More detailed representation consisting of cones and quarter wave resonators 3- A simplification of the second approach into a simple Pipe 6 From the results of this study, it can be concluded that the acoustic characteristics of shallow flow reversal chambers can be modeled, with engineering precision, up to cut on frequency of the first higher order mode using simple two-port elements. Good results were achieved modeling the flow reversal chamber as a simple straight duct connecting the inlet and the outlet. Munjal [12] devised a numerical collocation method. This method is easily applicable to rectangular as well as circular expansion chambers, but is limited to integer multiple area expansion ratios due to its inherent concept of discrete geometrical partitioning. Analytical methods have been introduced over the years. These methods fall into two main groups, one-dimensional and three-dimensional models. However as Ih [13] has mentioned, if the length of the chamber is much shorter than its width, then a large number of modes should be counted for calculating transmission loss even for the very low frequency range and this fact, arising mainly from the higher order acoustic modes generated at area discontinuities which do not fully decay before they reach the counterpart port, because the inlet and outlet are very close to each other. This leakage phenomenon means that the one-dimensional models are quite far from the actual performances even in the low frequency region. Three-dimensional models provide a very simple and exact approach to theoretical prediction of acoustical performance of plenum and reversing chambers. A three-dimensional mathematical formulation for mufflers with circular or rectangular cross-section with arbitrary location of inlet/outlet is derived by using the Eigen function expansion technique by Ih [13, 14]. The same problem is solved by the use of Green's function by Kim and Kang [15] for circular chambers and by Venkatesham et al. [16] for rectangular chambers. These methods take into account the effect of higher order modes which is necessary for successful analysis of a flow reversal chamber. The basic idea for these models stems from the fact that these chambers are in general regular in shape, which permits the use of series of orthogonal eigenfunctions. However, mufflers used in industry are not exactly rectangles or cylinders. Usually they are a bit curved at the edges to increase the stiffness. It is of interest for industry to know how this difference can alter the TL curve. This problem can be solved by FEM, however this method would be expensive and time consuming. One purpose of this thesis work is to investigate other methods for predicting TL of such chambers. One method could be to approximate the chamber which is curved at the edges with one which has sharp edges and then use the available theoretical models like the Green's function method to get TL curve. In the present study we want to find out how to do this approximation. The other possible method can be Neural Network. However this method needs some training data to train the neural network. Data for training can be obtained either through experiment or FEM. The effect of mean flow velocity is not studied here; However it has been found to be of negligible effect when Mach number is smaller than about 0.03 [17]. Besides, when the mean flow velocity is smaller than about M = 0.1, the convective contributions can be considered as negligible second order quantities and flow-generated noise may often be neglected. Further, if the mean flow velocity is small, the flow-generated 7 noise as well as pressure losses can be greatly reduced without degradation of the acoustic performance by streamline guidance: i.e., by using special l/O connecting geometries such as bell mouths and perforated bridges with high perforation ratios over 20% [14]