Direct aeroacoustic modelling of grazing flow over porous duct liner with CE/SE method

Abstract

There is increasing need for reducing aeroacoustic noise in various industrial applications carrying unsteady flows such as power plants, heating ventilation and air-conditioning (HVAC) and exhaust systems of vehicles. Adoption of bulk reacting porous liners in dissipative lined ducts to achieve noise reduction is one of popular acoustic treatment. In most applications of porous absorbers, the liner is placed on hard walls of a duct over which sound waves propagate. The presence of a grazing flow has significant influence on the performance of a porous liner. It was firstly reported in the experiments of Aurégan and Singh (2014) with a bulk porous liner made of metallic foam that the presence of grazing flow induces loss of transmission loss with a certain range of frequencies in which unusual oscillatory increase in transmission coefficients are observed. Because of the difficulty in measuring the magnitudes of flow physical properties such as velocity and pressure inside the complex microstructure of porous liner, the details of the underlying mechanism causing the phenomenon can only be further studied through numerical modelling. The objective of the present study is to develop a numerical model that can capture the interaction between flow dynamics and acoustic behavior of porous liner under grazing flow. The success of the development of the numerical model can help better understand the mechanism responsible for transmission instability as well as allow optimization of the design and manufacturing of porous liner in future applications. Impedance boundary condition is widely used in computational aeroacoustics to simulate lined surfaces. Unlike locally reacting liners (point reacting liners) whose surface impedance does not depend on the angle of incidence and its normal impedance can be applied at the boundary directly, it is not always appropriate to use the measured or modelled normal impedance to represent a bulk reacting liner. So far there is no time domain impedance modelling for bulk reacting porous liners under grazing acoustic incidence and grazing flow problems existing in literature. In fact, since the acoustic wave is allowed to propagate along the direction parallel to the wall of the porous liner, the acoustic field inside and outside the porous liner are coupled and the liner cannot be classified with a single point number impedance. In this regard time domain impedance modelling may not be an appropriate approach to study aeroacoustic problems of porous liner exposed to grazing acoustic incidence and flow. On the other hand, direct modelling via aeroacoustic simulation (DAS) approach, that solves the compressible flow equations rather than the simple wave equation for nonlinear pressure propagations, has been proven able to capture the coupling of acoustic fields inside and outside porous liner and nonlinear interaction between flow dynamic and acoustics intrinsically. Thus, DAS is adopted in the current study to model the grazing flow over porous liner problems. Direct aeroacoustic modelling calculates the flow dynamics and acoustic fields simultaneously by solving the inhomogeneous unsteady compressible Navier-Stokes (N-S) equations and the perfect gas equation of state with source terms which model the porous effect to the aeroacoustical flow. Since the computational cost to model the microscopic behavior of flow and acoustics through the pores is very huge, an approach of modelling of flow and acoustics inside porous material in a macroscopic view by using volume averaging over a representative elementary volume is adopted in the present study. In the prescription of source terms, Brinkman penalization method (BPM) and Brinkman-Forchheimer­extended Darcy model (BFDM) are attempted to model the flow in porous medium. The proposed single domain formulation comprising both fluid and porous material regions of computational domain eliminates the need to specify boundary conditions at fluid and porous interface explicitly. An in-house space-time conservation element and solution element (CE/SE) based time-domain DAS method is modified for the calculation of duct aeroacoustic problems with porous liner. A formulation of governing equations with BFDM for two dimensional compressible flow for solving aeroacoustic problem of porous liner is proposed in the present study based on approaches of modelling low speed incompressible flow transport in hydrogeophysics and convective heat transfer in thermodynamics. The energy equation in the governing equations is modified for aeroacoustic applications by describing the rate of work done of Darcy, Forchheimer and Brinkman terms. Newton's iterative method is applied to treat the stiff source term and the change of the flux at the clear fluid-porous interface is solved by using fictive cell method. The capabilities of the numerical modelling are examined by comparison of numerical results of benchmark duct aeroacoustic problems. The comparison results show that the proposed single domain formulation using BFDM can accurately calculate the flow dynamics and acoustics in both fluid and porous region as well as at their interface. The numerical calculation of the acoustic behavior of porous liner under low Mach number grazing flow validates the modelling has capability in capturing the interactions between the flow dynamics and acoustics. In the present study, two dimensional aeroacoustic problems in porous liner having a rigid frame (metallic foam) under low Mach number flow (M = 0.2) is investigated. The unusual oscillation of transmission coefficient of the porous liner over the same range of frequencies can also be captured using this numerical model and the mechanism responsible for it is uncovered with the numerical results.