Modelling and Simulation in Engineering

Volume 2018 (2018), Article ID 9170317, 9 pages

https://doi.org/10.1155/2018/9170317

## Numerical Simulation of Thermoacoustic Wave Induced by Thermal Effects by Using Discontinuous Galerkin Method

^{1}Department of Informatics Engineering, Universitas Atma Jaya Yogyakarta, Yogyakarta, Indonesia^{2}Department of Mechanical Engineering, Universitas Gadjah Mada, Yogyakarta, Indonesia

Correspondence should be addressed to Pranowo Pranowo

Received 12 October 2017; Revised 26 December 2017; Accepted 15 January 2018; Published 8 March 2018

Academic Editor: Salim Belouettar

Copyright © 2018 Pranowo Pranowo and Adhika Widyaparaga. 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

This paper describes a numerical model based on discontinuous Galerkin method for thermoacoustic investigation. Numerical investigation was conducted to study the behaviour of thermoacoustic wave propagations induced by thermal effects in 2-dimensional enclosure. The compressible Navier-Stokes equations are used as the governing equations. The spatial domain was discretized by using unstructured discontinuous Galerkin method and the explicit fourth-order Runge-Kutta was used to integrate the temporal domain. The accuracy of the numerical results was assessed against other numerical methods and linear solutions. The comparisons with linear solutions show excellent agreement and comparisons with flux corrected transport (FCT) method show fair agreement.

#### 1. Introduction

In addition to being commonly caused by mechanical vibration, the initiation of acoustic waves can also be triggered by sudden temperature change. When a compressible fluid experiences rapid heating, the fluid will expand. This expansion results in a local pressure disturbance which then propagates. Due to viscous and thermal effects, the amplitude of the thermoacoustic induced wave will eventually be dampened. The oscillating motion of the particles due to wave propagation also results in convective heat transfer, thus resulting in heat transfer enhancement [1]. To better understand the mechanism of this phenomenon, numerical studies have been conducted to investigate the thermoacoustic wave.

Previous numerical studies with regard to thermoacoustics have mostly adopted control volume and finite volume methods. An early study was conducted by Worlikar et al. using a finite difference low Mach-number model for simulation of unsteady adiabatic and stratified flow in a thermoacoustic stack [2, 3]. By using the computed results, they analyzed the configuration stacks and the nonlinear response of the flow due to different acoustic driving amplitudes and frequencies. Their results also showed that the computed results have good agreement with experimental results and linear theory. Hantschk and Vortmeyer [4] used the fluent finite volume model to simulate a Rijke tube. In this model, the thermoacoustic wave was generated with one heated wire screen. They showed that the obtained results are in good agreement with experiments. Farouk et al. [5, 6] studied the behaviour of thermoacoustic waves in a nitrogen-filled two-dimensional cavity. The thermoacoustic waves were generated by heating or cooling the vertical walls of the cavity. The wall temperatures were altered both impulsively and gradually. A fully correct transport (FCT) method was used to solve the full compressible Navier-Stokes equation. Flow patterns were shown to strongly correlate to the rapidity of the wall heating process. Lin [1] expanded the work of Farouk et al. [5] investigating acoustic wave induced convection and transport in gases under normal and microgravity conditions. He concluded that the acoustic streaming can enhance the heat transfer and the temperature along a stack and cooling effect can be predicted. Nijeholt et al. [7] used the CFX finite volume model to study a traveling-wave thermoacoustic engine. They concluded that CFD codes could be used in the future to predict and optimize thermoacoustic systems. Ke et al. [8] carried out 2-dimensional numerical simulations of a thermoacoustic refrigerator driven at large amplitude. Their computation was based on a pressure-correction (SIMPLE) algorithm for compressible flows. It was revealed that the optimal length of heat exchanger should be proportional to the amplitude of gas in it. In addition, the optimal heat exchanger length should be close to the peak-to-peak displacement amplitude of the gas medium. Lourier et al. [9] used a semi-implicit pressure-based fractional step method and characteristic boundary conditions (NSCBC). They showed that the acoustic CFL limitation can be removed by using implicit NSBC.

The discontinuous Galerkin method is a combination of the finite element method with the finite volume method. As such, this method can handle a complex spatial domain with a good level of accuracy. In 1972, Reed and Hill [10] introduced a DG method to solve the neutron transport equation. Bassi and Rebay developed the DG method for compressible gas dynamics in which the DG method was used to solve the nonlinear Euler equation [11] and then extended to solve the compressible Navier-Stokes equation [12]. Their study showed that the DG method had a high level of accuracy and was robust for supersonic flows. Liu et al. [13] developed a reconstructed DG method for a solution of compressible Navier-Stokes equations on 3D hybrid grids. Viscous and inviscid flux used formulations developed by Bassi and Rebay [11, 12]. The study was capable of increasing the accuracy of the DG method by one order higher than the existing DG methods for compressible turbulent flows with low Mach numbers and high Reynolds numbers.

However, studies on thermoacoustic waves using the discontinuous Galerkin (DG) method are very scarce in literature. Gineste simulated a one-dimensional thermoacoustic system by using the discontinuous Galerkin [DG] method with linearized Euler equations [14]. His results showed that the DG method is effective and quite flexible to be applied to thermoacoustic problems. Based on the study carried out by Gineste, we have extended the solution approach to a two-dimensional problem by using full compressible Navier-Stokes equation as the governing equations.

#### 2. Mathematical Model and Numerical Scheme

The thermoacoustic wave propagation is modelled by the compressible Navier-Stokes equations. These equations can be expressed in a vector conservation form as follows:where is the vector of the conservative variables, are the nonlinear convective fluxes, and are the diffusive fluxes.where is the density of the fluid, and are the components of the momentum, is the pressure, is the total energy, is thermal conductivity, is dynamic viscosity, and is the stress tensor. The stress tensor is defined byIt is assumed that the fluid is ideal gas and the properties are constant.where is specific gas constant.

The diffusive fluxes contain second-order derivatives which are then converted to a system of first-order equations by adding auxiliary variables as conducted by Bassi and Rebay [12]. Thus, (1) can be written as [15]Auxiliary variables have 2 components, that is,By adding convective fluxes and diffusive fluxes into one partthen (5) becomes simpler:Because the discontinuous Galerkin methods are a combination of finite element methods and finite volume methods, the numerical steps are similar to those two methods. The 2-dimensional physical domain is composed of a number of triangular elements.The numerical solution in each element is approximated by nodal high order polynomial basis function .where are spatial coordinates, is the number of nodal points in each triangular element, and is the order of polynomial basis function.

The DG methods allow the solution to be noncontinuous across the element interfaces. The surface integral at the element interfaces, which is known as the numerical flux, is not neglected as it is in finite volume methods. The weak formulation can be obtained by applying the Galerkin integration by parts of the flux terms of the mathematical model as follows:The local Lax-Friedrich flux is used to approximate the inviscid fluxes and the central flux without penalty for diffusive fluxes [16]. Here denotes inner product, for example, .

The integration of the temporal domain was conducted using an explicit fourth-order Runge-Kutta [17]. We have implemented the numerical scheme into computer program based on the NUDG framework (https://github.com/tcew/nodal-dg) of which the details are described in [16].

#### 3. Results and Discussion

##### 3.1. Case 1: Point Heating Induced Wave

Case 1 investigates the effect of a temperature disturbance in initiating pressure wave propagation. The dimensions of domain are 5 m × 5 m (Figure 1) which consisted of 650 triangular elements. The gas medium was CO_{2} and the fields were initialized with