## 1 Introduction  
The silencer is the main device used for suppressing automobile noise. The most important goal for a high-performance silencer is to simultaneously have good aerodynamic and acoustic attenuation characteristics. However, these often contradict each other. Perforated tube silencers are well-balanced in terms of both characteristics. Thus, they are applied extensively to the exhaust systems of automobiles. Pressure loss is the main indicator used for evaluating the aerodynamic performance of a silencer. It has a negative influence on the efficiency of an engine. If the loss exceeds the backpressure limit, it will result in reduced engine power and an increase in fuel consumption [1]. Currently, the pressure loss is predicted by performing a 3D steady computation using computational fluid dynamics (CFD). This method has been widely accepted among researchers because of its high level of accuracy and adaptability [2; 3; 4]. [PERSON] et al. [5] computed the pressure loss of a simple expansion chamber muffler using a CFD simulation. [PERSON] et al. [6] predicted the pressure loss of concentric tube silencers with five different patterns of perforated elements using CFD analysis. [PERSON] et al. [7] employed the CFD approach to predict the pressure loss of an exhaust muffler, which was influenced by the insert duct, the position of the baffle, and the inlet air velocity. Transmission loss (_TL_) is one of the most important indicators used for evaluating the acoustic attenuation performance of a silencer. The gas from an automotive engine has a high temperature and speed, which will have a strong influence on the acoustic attenuation performance. High temperature changes, gas density, sound velocity and acoustic impedance [8], and the effects of flow velocity embody two sides: one is to affect the sound wave propagation in a medium, and the other is to reduce aerodynamic noise resulting from turbulence [9]. Various theoretical and experimental studies were conducted in the past in order to investigate the acoustic attenuation performance of a silencer under the influence of different temperatures and air flow [10; 11; 12]. [PERSON] et al. [13] investigated the acoustic characteristics of an expansion chamber with a constant mass flow and a steady temperature gradient. [PERSON] et al. [14] applied finite element and boundary element methods to evaluate the acoustic wave transmission characteristics in a medium with uniform and steady mean flow. [PERSON] [15] compared two numerical methods and two analytical methods of modelling automotive dissipative silencers with a uniform mean gas flow of Mach number (M). The comparison indicated a close similarity to the _TL_ predictions that were obtained for the silencers that were examined. With the rapid development of high-performance computers, the 3D numerical simulation method plays an increasingly significant role in sound field analysis. [PERSON] et al. [16] proposed a 3D time-domain technique based on the CFD approach to calculate the _TL_ of exhaust mufflers with different chambers. [PERSON] et al. [17] proposed a hybrid finite element approach--combining an acoustic velocity potential formulation in the central airway with a pressure-based wave equation in the outer chamber--to study the _TL_ of perforated dissipative silencers with heterogeneous properties in the presence of mean flow. [PERSON] et al. [18] employed the CFD approach to perform a 3D steady computation and obtain a temperature distribution inside a two-pass perforated tube silencer, and then defined the elements with a 5 \({}^{\circ}\)C difference in temperature as a collection. The air parameters of the corresponding temperatures were assigned to the defined collections using SYSNOISE software, in order to calculate the _TL_. However, most of the present works associated with predicting the _TL_ of perforated tube silencers assume the mean flow to be uniform.