Mathematical Modeling of Porous Medium for Sound Absorption Simulations: Application of Multi-Scales and Homogenization

The modeling of porous medium has many applications whose techniques can be used in the fields of automotive, aerospace, oil exploration, and biomedical. This work concentrates on the Noise and Vibration (NV) development of automotive interiors but the ideas can be translated to the aforementioned areas. The NV development requires the setting of NV targets at dif-ferent levels. These targets are then translated to TL (Transmission Loss), IL (Insertion Loss), and Alpha (absorption) performance. Therefore, the ability to manage an efficient product development cycle, that entails analyzing vibro-acoustic environments, hinges on the premise that accurate TL, IL, or Alpha values pertaining to the different multi-layered porous materials can be calculated. Thus, there is a need to have a thorough understanding of the physics behind the energy dissipating mechanism that includes the effects of the fluid meandering through the pores of the material. The goal of this series is to model the acoustic and dynamic coupling via multi-scale and homogenizations techniques, thus subsequently understand where to incorporate the concepts of dynamic tortuosity, viscous and thermal permeability, as well as viscous and thermal lengths. This study will allow the ability to get a better understanding of the underlying processes and also provides tools to create practical concepts for determining the coefficients of the macroscopic equations. This will assist in attaining novel ideas for NV absorption and insulation.

Model for Frequency Dependence of Thermal Permeability in Order to Quantify the Effects of Thermal Exchange on Wave Propagation in Multi Layered Porous Medium

The ability to quantify and predict the energy absorption/transmission characteristics of multi-layered porous medium is imperative if one is involved in the automotive, launch vehicle, commercial aircraft, architectural acoustics, petroleum exploration, or even in modeling human tissue. A case in point, the first four aforementioned fields rely on effective Noise and Vibration (NV) development for their commercial success. NV development requires the setting of NV targets at different system levels. The targets are then translated to Transmission Loss (TL), Insertion Loss (IL), and absorption (Alpha) performance for the multi-layered porous materials being utilized. Thus, it behooves to have a thorough understanding of the physics behind the energy dissipating mechanism of the material that entails the effects of the fluid meandering through the pores of the material and its interaction with the structural skeleton. In this section of the project the focus is on the thermal interchange that occurs within the porous medium. Via the acoustic modeling at the micro/macro level it is shown how this thermal exchange affects the acoustic compressibility within the porous material. In order to obtain a comprehensive approach the ensuing acoustic modeling includes the effects due to relaxation process, thus bulk viscosity and instantaneous entropy functions (effects due to vibration of diatomic molecules of air) are incorporated into the equation. The instantaneous entropy functions are explained by means of the Boltzmann’s distribution, partition function, and quantum states. The concept of thermal length and its connection to thermal permeability is clarified. Lastly, the results for TL calculations employing the aforementioned thermal exchange into the Transfer Matrix Method with finite size correction, (FTMM), pertaining to a simple multi-layered material is compared with experimentally obtained data.

Mathematical Modeling of Porous Medium for Sound Absorption Simulations II: Wave Propagation and Interface Conditions

The application of porous medium has a myriad of applications in different industries: automotive, aerospace, civil (commercial, residential), environmental noise control, and biomedical. In the past, design questions involving porous material were addressed with seat-of-the-pants decisions that led to multiple/iterative prototypes and experiments that were costly and time consuming. The objective, in this series of publications pertaining to porous medium, is to establish tools that will lead to effective and accurate simulations involving porous medium. In this third installment of this series the focus is on establishing the constitutive equations using tensors and then applying Transfer Matrix Method (TMM) to calculate diffuse field Transmission Loss (TL) across structures that comprises of layers of different porous medium. The constitutive equations are obtained by relating information regarding the micro-structure make-up to macro level properties. In order to apply the TMM, the equations for wave propagation across different mediums need to be developed and in turn represent these propagation properties in a matrix format. Additionally, the boundary condition between each layer type is defined in order to ensure numerical stability. The author’s current research effort is running simulations for the automotive industry to predict NVH environments. Therefore, TL calculations pertaining to the materials that are utilized in the interior of automobiles are used, in this paper, as a test bed for the developed analytical tools. Case in point, the TL for a multi-layered material consisting of one panel and two different layers of foam is calculated and compared to experimental data. Future publication goals will be to apply these tools in the biomedical field;an example will be to model and run simulations of different organs like the liver and lungs that are po-rous in nature.