Multi-field coupling prediction for improving aeroacoustic performance of muffler based on LES and FW-H acoustic analogy methods

Abstract

Based on the large eddy simulation (LES) and Ffowcs Williams and Hawkings (FW-H) equation, a multi-field coupling method is presented for aeroacoustic prediction of a muffler with high-speed and high-temperature exhaust gasflow. A three-dimensional finite-volume model of the muffler is established by using the LES and FW-H acoustic analogy (FW-H-AA) methods. Experimental validations of the simulated results suggest a good accuracy of the combined LES and FW-H-AA approach. Some factors influencing on noise attenuation, such as the gasflow velocity, temperature and the structural parameters of the muffler are analyzed. The results show that the aerodynamic noise and turbulent kinetic energy (TKE) are mainly attributed to the structural mutations in the muffler. The outlet sound pressure level (SPL) increases with the inlet gasflow velocity and decreases with temperature. According to the factor analysis results, the target muffler is modified by adding a fillet transition to the end of inserted tube and redesigning the structures where the TKE concentrated for improving the aerodynamic performance. In terms of the outlet SPL, the inner TKE and the backpressure of the muffler, the modified muffler is significantly improved by the maximum reductions of 3-5dB in SPL, 10–20% in TKE and 0.5–2.5 kPa in backpressure. The presented method might be extended to other kinds of muffler for aeroacoustic calculation and improvement design.

Keywords

Resonant muffler Aeroacoustic performance Multi-field coupling Large eddy simulation FW-H acoustic analogy

Introduction

A large number of previous studies show that traffic accidents are related to vehicle noise.¹,² As an important part of engine exhaust system, muffler has been widely applied and continuously improved in recent decades. In the early designs of a vehicle muffler, only the sound attenuation performance (SAP) without airflow was considered. In an actual working condition of the muffler, however, the SAP was affected by the high-speed and high-temperature exhaust gasflow. In view of the generating mechanism of sound, it is a coupling problem of acoustics, fluid mechanics and thermodynamics. With the emergence of new techniques such as turbocharging and tail gas treatments, the mechanism of exhaust noise generation has become more complex. Thus, it is still of great theoretical and practical significances to study the generating mechanism and improvement scheme of the vehicle mufflers.

The mufflers can be classified into three types: resistance, reactive and resistance-reactive mufflers. The reactive resonant muffler is suitable for eliminating noise at low and medium frequencies, and is often used in vehicle exhaust systems. The resonant mufflers without gasflow have been widely studied. The plane wave theory was applied to study the performance of a single-cavity muffler.³ The transmission losses of one-dimensional mufflers were calculated by using the transfer matrix and sound propagation methods.⁴ In term of the transmission loss, the structural parameters of a muffler, such as the length of inserted tube and the thickness of perforated plate, etc., is investigated by using the transfer matrix method.⁵ The effects of structural parameters on the noise reduction and its frequency range of some single-cavity mufflers were studied by using the two- and three-dimensional computation methods, respectively.⁶,⁷ Selamet⁸,⁹ compared the one-, two- and three-dimensional models for sound attenuation analysis of a resonator, and deduced the corresponding computation formula of transmission loss. The finite element method (FEM), which was introduced by Crocker¹⁰ into the transmission loss calculation of a muffler, has been used for structural improvement of the vehicle mufflers.¹¹ A comparison with theoretical results showed that the FEM has sufficient accuracy in the calculations of resonance frequency and transmission loss of mufflers.¹² Accordingly, a large number of FEM-related researches on noise attenuation prediction of different types of mufflers with no mean flow, such as perforated-tube and/or expansion mufflers, were extensively conducted in the past few decades.^13–16 The influences of structural parameters on muffler performance have been widely discussed, which is helpful for design of vehicle mufflers.

Due to that the high-speed and high-temperature gasflow exists in the exhaust mufflers, the research on the mufflers with gasflow has attracted more attention in recent years. The aerodynamic noises in a muffler may be produced by the interactions among the gasflow, rigid wall and the fluid elements. The unstable free-shear layers of gasflow are formed at some sudden structure changes in the cavity, which leads to the formation and breakup of eddies, and thereby the aerodynamic noise sources. The three-dimensional (3 D) boundary element analysis of a simple expansion muffler with mean airflow showed that the airflow may cause a slight shift of transmission loss curve moving toward low frequencies.¹⁷ Ashcroft¹⁸ studied the aerodynamic noise of a resonator under low Mach number, which suggested that the noise radiations can be mainly attributed to the dramatic changes of air pressure and density in the shear layers close to the cavity openings. The effects of structural parameters, such as the diameter and length of chamber, the diameters of inlet and outlet pipes and the distance of connecting pipes, on the transmission loss of an expansion muffler with mean airflow were experimentally investigated.¹⁹ By using the modal matching method, the flow and sound fields of a perforated tube muffler with different structure units were studied.²⁰ The results showed that the transmission loss of a reactive muffler increases with the increase of the gas-flow velocity, which is contrary to the conclusion from a resistance muffler. The effects of the structural parameters on the pressure loss of an inserted tube muffler and the aerodynamic noise of a simple perforated tube muffler were further studied.²¹,²² The experimental results of Emmet²³ showed that, for a simple expansion chamber, the higher the gasflow velocity is, the higher the sound pressure level (SPL) of induced aerodynamic noise. Based on the simulation results of temperature, velocity and pressure fields in a vehicle muffler, a structural optimization for reducing back pressure of the exhaust system was conducted.²⁴ According to the simulated results from the flow and sound fields, a fluid impact coefficient, which can be used to predict aerodynamic noise of the expansion mufflers, is proposed.²⁵ Kang²⁶ studied the relationship between aerodynamic noise and airflow velocity of a perforated tube muffler. The results showed that transmission loss of the muffler increases with Mach number, and the frequency band of noise elimination extends to relative high frequencies. In the recent years, the large eddy simulation (LES) turbulence model has attracted wide attention in the field of aerodynamic noise prediction. Based on a variable cross-section pipe, the simulated aerodynamic noise from the LES approach was validated by experiment.²⁷ It has been found that, for predicting the transmission loss of an expansion muffler with airflow, the LES turbulence model is more accurate than the Reynolds time-averaged equation model.²⁸ Dan²⁹ built a 3 D LES model and simulated the jet flow noises of the straight and expansion tailpipes. More recently, the LES-FEM coupled method and the LES and FW-H hybrid method were applied for analyzing the aerodynamic noise of a vehicle intake system and the flow-induced noise of natural gas manifolds.³⁰,³¹ The results suggested that the maximum aerodynamic noise always appears in a mixing region with large velocity potential and violent flow turbulence. It can be seen that most of the previous studies are based on different types of simple muffler. For the complex mufflers used on real vehicles, the effectiveness of LES method needs to be further validated.

Based on the above discussions, the studies on sound attenuation of mufflers have been developed from the one-dimensional theoretical calculations to the three-dimensional (3 D) simulations and experiment. However, the vehicle muffler with high-temperature and high-speed gasflow is still difficult to establish an accurate theoretical model for predicting its acoustic property, gas-flow field and sound energy distribution. Therefore, aiming at an exhaust muffler with gasflow, in this paper, a simulation model is established by combining the LES and FW-H acoustic analogy (FW-H-AA). The effects of flow, temperature and structural parameters on the performance of muffler are analyzed. Based on the obtained results, the structure of a muffler is further optimized, which may provide a reference for design of the resonant mufflers.

Theoretical background

Les numerical method for gasflow simulation

Aeroacoustics is a branch of acoustics that studies noise generation via fluid mechanics, in which the motion of fluid follows the governing equations of continuity, momentum and energy. When the gasflows passing through a muffler chamber, the internal turbulences will be generated, thus affecting the muffler performance. The turbulences consist of some vortices in various scales, and the large eddies contain most of the energy in the turbulences. In the movement process, the large eddies are gradually broken into small-scale eddies, which leads to energy dissipations due to the viscous fluid. All parameters in turbulent motion are closely related to time and space. Compared to the experimental ones, the simulation methods for turbulent flow calculation with the advantages of low cost and high efficiency can be classified into direct numerical simulation (DNS), Reynolds-averaged N-S (RANS) and large eddy simulation (LES).³² Theoretically, the DNS can directly solve the N-S equations and obtain the turbulences at all scales and all the information of flow field. Because of a huge amount of computation caused by the very detailed grids, the DNS is only suitable for simple fluid-flow models. Compared to the DNS, the RANS has less computational complexity, but its mathematical model is based on the isotropic assumptions, which inevitably results in greater errors at the positions of turbulence concentration. In order to obtain the complex turbulence information as that in exhaust mufflers, a method that considers both the accuracy and complexity of computation is needed. The LES method can achieve this goal. Consider the LES of turbulent compressible flows, the flow variables are density weighted or Favre averaged.³³ Based on the basic governing equations of fluid, the LES divides the eddy scales by using a filter function, and solves the large-scale eddies directly. And then a subgrid model is used to simulate the small-scale eddies. The relationship between the small and large eddies is established by using the subgrid stress.³⁴ The pulsating component $\bar{ϕ}$ of a large-scale eddy of a turbulent parameter $ϕ$ can be defined as,

\bar{ϕ} (x, t) = \int_{- \infty}^{+ \infty} ϕ (ξ, t) G (x - ξ, Δ) d ξ

(1)

where,

G (x - ξ, Δ)

is the Gauss filter function,

d ξ

is the volume element,

Δ

is the filter scale. The low-pass filtering function in the spectral space can be expressed as,

\hat{G} (X) = u (X_{c} - | X |)

(2)

where, u(x) is the step function, X_c is the upper cut-off frequency of the filter. According to Gauss theorem, the filtering equation can be obtained as,

G (x - ξ, Δ) = {(\frac{6}{π Δ^{2}})}^{1 / 2} e^{- 6 {| x - ξ |}^{2} / Δ^{2}}

(3)

The LES equations can be derived by filtering the N-S equation of fluid. The filtered governing equations are obtained by applying above filter function as follows,

\frac{\partial}{\partial t} (ρ \bar{ϕ}) + \frac{\partial}{\partial x_{i}} (ρ \bar{ϕ} {\bar{u}}_{j}) = - \frac{\partial \bar{p}}{\partial x_{i}} + \frac{\partial}{\partial x_{j}} (μ \frac{\partial \bar{ϕ}}{\partial x_{j}}) - \frac{\partial τ_{i j}}{\partial x_{j}}

(4)

\frac{\partial \bar{ρ}}{\partial t} + \frac{\partial}{\partial x_{i}} (ρ \bar{ϕ} {\bar{u}}_{j}) = 0

(5)

where x_i and x_j (i, j = 1,2,3) are the coordinates X, Y and Z; u_j denotes the filtered velocity components

\bar{u}

\bar{v}

and

\bar{w}

along with X, Y and Z, respectively;

\bar{p}

is the filtered pressure;

μ

is the molecular viscosity;

ρ

is the Favre-averaged fluid density;

τ_{i j}

is the subgrid scale stress,

τ_{i j} = ρ \bar{u_{i} u_{j}} - ρ \bar{u_{i}} \bar{u_{j}}

, which indicates that the small-scale variables acting on the large-scale ones. After filtering, the large-scale eddies can be directly solved, while the small-scale eddies need to be further solved by a subgrid model. The key point is to obtain the turbulent eddy viscosity in the subgrid expressions. A subgrid-scale component can be obtained by a subgrid model that is defined as,³⁵

τ_{i j} - \frac{1}{3} τ_{k k} δ_{i j} = - 2 μ_{t} {\bar{S}}_{i j}

(6)

where

μ_{t}

is the turbulent eddy viscosity coefficient;

k = 1, 2, 3

;

δ_{i j}

is the Kronecker delta.

δ_{i j} = {\begin{cases} 1, \overset{}{} i = j \\ 0, \overset{}{} i \neq j \end{cases}

(7)

{\bar{S}}_{i j} = \frac{1}{2} (\frac{\partial {\bar{u}}_{i}}{\partial x_{j}} + \frac{\partial {\bar{u}}_{j}}{\partial x_{i}})

(8)

Four subgrid models are commonly used in the LES, i.e., Smagorinsky, WALE, KET and WMLES models. As an improved version of the Smagorinsky model, the Smagorinsky-Lilly subgrid model³⁶ still involves in large dissipations and is not universally applicable.³⁷ The WMLES model can provide good transitions for the boundary layer interfaces through the RANS equations, but cannot accurately express the turbulent flow fluctuations. The WALE subgrid model takes the wall effect of turbulence into account. Comparing with the KET model, it is more suitable for describing the laminar characteristics near the walls, which can ensure the accuracy in simulations. Thus, the WALE subgrid turbulence model is adopted and applied to simulate the muffler performance in this paper. $μ_{t}$ can be determined by the WALE model as,

μ_{t} = ρ L_{s}^{2} \frac{{(S_{i j}^{d} S_{i j}^{d})}^{3 / 2}}{{({\bar{S}}_{i j} {\bar{S}}_{i j})}^{5 / 2} + {(S_{i j}^{d} S_{i j}^{d})}^{5 / 4}}

(9)

L_{s} = \min [k' d, C_{w} {(Δ_{x} Δ_{y} Δ_{z})}^{1 / 3}]

(10)

S_{i j}^{d} = \frac{1}{2} ({\bar{g}}_{i j}^{2} + {\bar{g}}_{j i}^{2}) - \frac{1}{3} δ_{i j} {\bar{g}}_{k k}^{2}

(11)

{\bar{g}}_{i j} = \frac{\partial {\bar{u}}_{i}}{\partial x_{j}}

(12)

where

k' = 0.42

is the Von Karman constant, d is distance of the closest wall, C_w, the WALE constant, is set to 0.325,

Δ_{x}

Δ_{y}

and

Δ_{z}

are the dimensions of the governed volume in three directions.

FW-H acoustic analogy method for acoustical simulation

The aeroacoustics is a cross discipline between the fluid dynamics and the acoustics. An aerodynamic noise is caused by the flow pulsations, which may be generated by the interactions between fluid and solid wall and/or fluid and fluid. Mathematically, Ffowcs Williams and Hawkings deduced the FW-H equations for the control surface of arbitrary motion in stationary fluid by using the generalized function theory.³⁸,³⁹ The FW-H equations may extract the aerodynamic sound sources by means of the acoustic analogy method.⁴⁰ Supposing that there is a moving control surface f(x_i,t)=0, if $\nabla f = n_{i}$ , $\partial f / \partial t = - v_{n} = v_{i} \circ n_{i}$ , n _i and v_i are the outward unit normal vector and moving velocity, the FW-H acoustic analogy (FW-H-AA) equation can be written as,

\frac{1}{c_{0}^{2}} \frac{\partial^{2} p'}{\partial t^{2}} - \nabla^{2} p' = \frac{\partial}{\partial t} {[ρ_{0} v_{n} + ρ (u_{n} - v_{n})] δ (f)} - \frac{\partial}{\partial x_{i}} {[P_{ij} n_{j} + ρ u_{i} (u_{n} - v_{n})] δ (f)} + \frac{\partial^{2}}{\partial x_{i} \partial x_{j}} [T_{ij} H (f)]

(13)

where,

ρ

, x_i and x_j are the same parameters as those in the LES,

ρ_{0}

is the density of quiescent medium, c₀ is the sound speed in quiescent medium, p’ is the observed sound pressure at t moment, u_i is the component of flow velocity along direction x_i, u_n and v_n are the flow velocity component perpendicular to the integral plane and the moving velocity component along the integral plane,

δ (f)

and H(f) are the Dirac function and the Heaviside function, which satisfy,

H (f) = {\begin{cases} \begin{matrix}  \end{matrix} 1, \begin{matrix}  \end{matrix} f (x_{i}, t) > 0 \\ \begin{matrix}  \end{matrix} 0, \begin{matrix}  \end{matrix} f (x_{i}, t) < 0 \end{cases}

(14a)

δ (f) = \frac{\partial H (f)}{\partial f}

(14b)

where, P _ij and T _ij denote the stress and Lighthill tensor, respectively. T _ij is the Lighthill turbulence stress tensor, which can be calculated by,

T_{i j} = ρ v_{i} v_{j} + δ_{i j} [(p - p_{0}) - c_{0}^{2} (ρ - ρ_{0})] - τ_{i j}

(15)

where,

τ_{i j}

is the eddy viscosity stress, v_i and v_j is the velocity components along with x_i and x_j, p and p₀ are the pressures of the fluid and the quiescent medium (constant), respectively.

The gasflow in an exhaust system is driven by combustion of the engine. To predict the sound field using the FW-H, the particle velocities and pressures of the sound sources must be given, which can be determined from the LES. According to the LES, the sound sources are included in a closed space and their surfaces can be integrated. In term of the exhaust noise of a vehicle, the aerodynamic sound sources can be classified as: the monopole source generated by volume changes of the exhaust system (pipes and muffler), the dipole and quadrupole sources induced by the positive pressure and the shear stress between the exhaust system and its inner flowing gas. Under normal cruise condition, the volume change of exhaust system is very small. Thus, the exhaust noise may be mainly attributed to the dipole and quadrupole sound sources. The FW-H-AA equations, which can calculate the sounds at any position in the space, are as follows:

p^{'} (x, t) = p_{T}^{'} (x, t) + p_{L}^{'} (x, t) + p_{Q}^{'} (x, t)

(16)

4 π p_{T}^{'} (x, t) = \int_{f = 0} [\frac{ρ_{0} ({\dot{U}}_{n} + {\dot{U}}_{\overset{•}{n}})}{r {(1 - M_{r})}^{2}}] d s + \int_{f = 0} [\frac{ρ_{0} U_{n} {r {\dot{M}}_{r} + c_{0} (M_{r} - M^{2})}}{r^{2} {(1 - M_{r})}^{3}}] d s

(17)

4 π p_{L}^{'} (x, t) = \frac{1}{c_{0}} \int_{f = 0} [\frac{{\dot{L}}_{r}}{r {(1 - M_{r})}^{2}}] d s + \int_{f = 0} [\frac{L_{r} - L_{M}}{r^{2} {(1 - M_{r})}^{3}}] d s

(18)

where, T is the monopole; L is the dipole; Q is the quadrupole; f is a function that defines the moving surface;

ρ_{0}

is the density of the quiescent medium; u_i is the component of the local fluid velocity; U_n is the normal velocity of the source surface;

{\dot{U}}_{n} = {\dot{u}}_{i} {\hat{n}}_{i}

;

U_{\dot{n}} = u_{i} {\dot{\hat{n}}}_{i}

;

M_{i}

is the Mach number of the sound in the radiation direction;

M_{r} = M_{i} {\hat{r}}_{i}

;

{\dot{M}}_{r} = {\dot{M}}_{i} {\hat{r}}_{i}

; r is the distance between the observer and the source; L_i is the component of local force intensity that acts on the fluid;

{\dot{L}}_{r} = {\dot{L}}_{i} {\hat{r}}_{i}

;

L_{r} = L_{i} {\hat{r}}_{i}

;

L_{M} = L_{i} {\hat{M}}_{i}

; and the dot over a symbol means a source-time differentiation.⁴¹,⁴² Comparing with the DNS and RANS models, the LES turbulence model, which has good characteristics in description of the turbulent fluctuations, can not only ensure computation accuracy, but also enhance computation efficiency. The FW-H-AA method has advantages in solving far-field aerodynamic noises. To obtain more comprehensive information in both the flow and sound fields, it can be assumed that a combination of the LES and FW-H-AA methods might be suitable for the aeroacoustic simulations of a muffler with gasflow.

Modeling and experimental validation

Muffler modeling for aeroacoustic calculation

The physical structure of a vehicle muffler is shown in Figure 1, in which the inner exhaust pipe opens to the cavities A and B. There are uniformly distributed 12 holes with diameters of 3 mm at the front end of the intake pipe. 20 and 120 holes with diameters of 3.5 mm are uniformly distributed at the front end and the middle section of the upper and lower exhaust pipes (outlets of the muffler), respectively. The thicknesses of the perforated plate and division plate are 1.2 mm, and there are 160 holes with diameters of 5 mm evenly distributed on the perforated plate. The muffler chamber is divided into cavities A and B by the division plate. The near-field SPL monitoring points 1 and 2 are set at the front end of the inlet pipe and the outlet end of the lower exhaust pipe, respectively. A far-field monitoring point is set at a point with 0.5 m distance from the outlet end of the lower exhaust pipe. The fluid regions in the muffler are extracted and fed to the software STAR-CCM+ for polyhedral meshing. Considering both the accuracy and the amount of calculation, the basic mesh size is set to 5 mm, and the local meshes in the areas of small holes and sharp deformations are refined manually. The surface reconstruction model is adopted for surface meshing, and an automatic surface repair is conducted by curvature refinement. Through the polyhedron and prismatic layer mesh generators, the mesh accuracy is improved by running the refinement function. Finally, 607,329 body grids are generated, and the mesh model and the boundary layer diagram are shown in Figure 2. In the following simulations, the LES turbulence model and the WALE subgrid model are used in transient near-field computations. The inlet and outlet parameters are set as “speed” and “pressure”, respectively. And the remaining shells are set to “wall”. The three-dimensional k-ε model and the Curle and Roudman broadband sound sources are used for steady-state calculation. The number of iterations is 1000. The FW-H-AA model is adopted in far-field computation. The time length and step size in iterations are set to 0.12 and 2.5e-5 seconds, respectively. The discrete order is set to 2. Due to limitation of the test conditions, the engine speed is set to 1000 r/min at room temperature, i.e., the muffler inlet flow velocity is 10 m/s with a temperature of 300 K. The boundary conditions of the simulations are listed in Table 1. After numerical simulations, the simulated SPL results at the muffler inlet (point 1) and outlet (point 2) are validated by comparing with the experimental results in the following section.

Figure 1.

Structural model of muffler.

Figure 2.

Grid model of the muffler and the boundary layer diagram.

Table 1.

The boundary conditions of simulations.

Items	Values
Transient near-field computations	LES turbulence model , the WALE subgrid model
Inlet parameters	Speed, 10 m/s
Outlet parameters	Pressure
Steady-state calculation	Three-dimensional k-ε model, Curle and Roudman broadband sound sources
Number of iterations	1000
Far-field computation	FW-H-AA model
Discrete order	2
Engine speed	1000 r/min
Engine temperature	Room temperature
Muffler inlet flow velocity	10 m/s
Muffler inlet flow temperature	300 K
The peak values of far-field SPLs	87dB

Experimental setting and validation

In order to accurately measure SPLs of the muffler, a testing system is built in semi-anechoic room according to the relevant standards,⁴³,⁴⁴ as shown in Figure 3. The testing system is mainly composed of three parts: the airflow generation system (air blower, anemometer, anechoic box and connecting tube), signal acquisition system (microphones, data acquisition, computer and signal processing software) and the muffler. The experimental results for checking the sound isolation performance of the anechoic box are shown in Figure 4. The blower noise is isolated by the well-designed anechoic box, the SPL difference between the inside and outside of the anechoic box is above 10 dB, and it is demonstrate that the anechoic box has good sound isolation performance. The muffler is placed in the center of the anechoic chamber, and the distance between the muffler and the wall is more than 2 m. The data acquisition system LMS-SCM05 is used. The 4189-A-21 microphones are mounted through the drilling holes on the inlet and outlet pipes. The microphones 1 and 2 are respectively positioned at the inlet end of the intake pipe and the outlet end of the lower exhaust pipe. The experimental conditions are exactly the same as those in the simulations, i.e., the inlet air velocity and temperature are 10 m/s and 300 K. During tests, the sampling rate is set to 10 kHz. The air blower in the anechoic box is turned on and the wind speed 10 m/s is confirmed by the anemometer. When the airflow and noise signal reached steady states, the SPLs at the monitoring points of the muffler are measured. To verify the hybrid simulation approach, the calculated and measured SPL results at the inlet (monitoring point 1) and outlet (monitoring point 2) of the muffler are compared in Figure 5. It can be seen that the measured results are in good agreements with the simulation results, which proved the feasibility of both the established model and the combined LES and FW-H-AA method in muffler simulations. In view of the results at point 2, a few simulated SPLs at the lower frequencies below 400 Hz is higher than the measured ones about 1–5 dB, which might be attributed to the experimental boundary conditions that are very difficult to be set exactly the same as those in the simulations, for example, the exhaust resistances may be caused by the microphones inserted into the pipes, etc. A more thorough validation should provide the spatial details of flow field inside the muffler. Unfortunately, we currently don’t have such experimental equipment and conditions.

Figure 3.

Experimental setting for noise attenuation measurement of a vehicle muffler.

Figure 4.

The SPLs measured at the assumed points inside and outside of the anechoic box.

Figure 5.

Comparisons of the calculated and measured SPLs at the (a) outlet and (b) inlet of the muffler.

Factor analysis of the muffler

Due to the multi-field coupling, the calculated results of a muffler without gasflow are different from its actual muffling performance. Based on the validated model, the effects of gasflow velocity, temperature and structural parameters on the noise attenuation performance of the muffler are discussed in the following text.

Effects of gasflow velocity and temperature

Figure 6 gives the calculated flow-field contours of turbulent kinetic energy (TKE), the flow velocity and the velocity vectors, which are taken from the middle symmetrical cross section of the muffler model. The working conditions are set to the inlet flow velocity 60 m/s with a temperature 900 K. From Figure 6(a), the TKEs are mainly distributed at the end of inlet pipe, near the orifices of perforated plate, and inside the upper and lower exhaust pipes. The maximum TKE (740.07 J/kg) appears at the front end of the lower exhaust pipe. Some strong vortexes are generated at the places where the gasflow concentrates. Due to the broken shear layers on the “wall” surfaces, the larger TKEs are formed at the points of structural mutations. From the velocity contours in Figure 6(b), the regions with larger velocities are mainly distributed at the end of the inlet pipe, near the small holes on the perforated plate and inside the upper and the lower exhaust pipes, and the maximum velocity is located in the lower exhaust pipe. The gasflow enters the cavity A through the inlet pipe, and encounters the division plate that causes gasflow diffusions, resulting in a larger velocity field at the end of inlet pipe. The gasflow needs to converge to the exhaust pipes and are finally emitted into the outside atmosphere, thus the large velocity fields are generated in the upper and lower exhaust pipes. Comparing Figure 6(a) with Figure 6(b), the velocity and TKE contours of the gasflow corresponding to each other have a consistent pattern in distributions. The calculated near-field SPLs at the end of the exhaust pipe are shown in Figure 7, where the inlet flow velocity is 40 m/s, 50 m/s, 60 m/s, 70 m/s and 80 m/s respectively, and the temperature is 500 K. It can be seen that the SPL variation patterns with respect to frequency under different working conditions are similar. The noise energies are mainly distributed below 3000 Hz, the SPLs with a peak value between 800 Hz and 1000 Hz and decrease gradually above 1000 Hz. The SPL values increase with increasing of the gas-flow velocity. The reason can be explained that, because the mass flow increases with flow velocity, larger impacts between the gasflow and the structural mutations result in higher SPLs of the aerodynamic noises. Figure 8 shows the calculated far-field SPLs at 0.5 m from the end of lower exhaust pipe under the corresponding working conditions, which are similar to the change rule of near-field SPLs in Figure 7. The peak SPLs appear between 400 Hz and 2000 Hz. It can be seen from Table 2 that the peak SPLs of far-field noise increase with the increase of gasflow velocity, but have been reduced more than those of near-field noise. Figure 9 gives the calculated SPLs at the near-field monitoring point in the cases of temperature 500 K, 600 K, 700 K, 800 K and 900 K. The inlet flow velocity is 70 m/s. The change patterns of SPL at different temperatures are similar, and the influence of temperature on noise reduction is small. The sound energies are distributed below 3000 Hz, and the peak SPLs lie in the frequency range of 600–1000Hz.

Figure 6.

The calculated contours under the conditions of inlet velocity 60 m/s and temperature 900 K: (a) turbulent kinetic energy, (b) flow velocity and (c) flow velocity with velocity vectors.

Figure 7.

The calculated SPLs in the near sound field at different inlet flow velocities.

Figure 8.

The calculated SPLs in the far sound field at different inlet flow velocities.

Table 2.

The peak values of SPL in the near and far sound fields at different inlet flow velocities.

Flow velocity (m/s)	40	50	60	70	80
The peak values of near-field SPLs (dB)	119	122	126	129	131
The peak values of far-field SPLs (dB)	77	78	84	87	88

Figure 9.

The calculated SPLs in the near sound field at different inlet flow temperatures.

From Figure 9 and Table 3, it can be seen that the SPL of exhaust noise decreases with the increase of gasflow temperature. The SPLs at the far-field monitoring point is shown in Figure 10. The SPL variations with respect to frequency of far-field noise are similar to those of near-field noise. Compared to the peak SPLs of near-field noise in Table 3, the SPLs of far-field noise reduced more than 40 dB, especially for those at the frequencies below 600 Hz. Theoretically, according to the gas equation of state, when the inlet flow velocity is constant, the increase of temperature leads to the decreases of fluid density and mass flow rate, which may be the reason why the SPLs decrease with the increase of temperature. The TKE distributions in the middle symmetrical cross section of the muffler are shown in Figure 11, in which the velocity and temperature of the inlet gas are set to (a) 50 m/s, 300 K; (b) 60 m/s, 400 K; and (c) 70 m/s, 500 K, respectively. The TKEs are mainly distributed at the end of inlet pipe, the orifices of the perforated plate and inside the upper and lower exhaust pipes. The maximum TKEs are 563.37 J/kg, 812.84 J/kg and 1115.3 J/kg, respectively, which are all distributed at the points of gas-flow convergence near the front end of the lower exhaust pipe, and the minimum TKEs are distributed in the cavity B. Although the inlet flow velocity and temperature are different in three conditions, the obtained distributions of TKE are similar. It can be concluded that the influence of inlet flow velocity and temperature on the TKE is mainly reflected in TKE values, not in its distribution characteristics.

Table 3.

The peak values of SPL in the near and far sound fields at different inlet flow temperatures.

Flow temperature (K)	500	600	700	800	900
The peak values of near-field SPLs (dB)	129	127	126	124	119
The peak values of far-field SPLs (dB)	87	86	85	82	80

Figure 10.

The calculated SPLs in the far sound field at different inlet flow temperatures.

Figure 11.

Comparisons of turbulent kinetic energy distributions under different working conditions: (a) 50 m/s, 300 K; (b) 60 m/s,400K; and (c) 70 m/s, 500 K.

Effects of structural parameters

The structural parameters of a muffler, such as the length and chamfering form of inserted tubes, the thickness of perforated plate and the expansion ratio of chamber, etc., have significant influences on noise reduction characteristics. Based on the typical simple mufflers in Figure 12, the structural factor analyses are performed in this paper. The initial parameters and boundary conditions of the muffler model are: the inner diameter of the inlet and outlet pipes are 30 mm, the length of expansion chamber is 250 mm, the expansion ratio is 9, and the inlet flow velocity and temperature are 60 m/s and 900 K. The structural improvement scheme of the muffler including the parameters of inserted tube, perforation plate and expansion chamber is shown in Table 4. From the above analysis, the greater TKEs are generated at the end of inserted pipe. Thus, to modify the transition structure of pipe orifice may be expected to reduce aerodynamic noise. Figure 12(b) and (c) show a 45^° bevel angle and a radius 10 mm rounded chamfering structures. The following simulations are based on the muffler model in Figure 12(a), keeping the initial boundary conditions unchanged. The calculated SPLs at the outlet ends of mufflers with different lengths and chamferings of the inserted inlet tube are shown in Figure 13. The peak values of SPL are listed in Table 4. It can be seen that, replacing the straight end by the bevel and rounded chamferings, and the outlet SPLs decreased up to 5 dB and 7 dB, respectively. The rounded chamfering is the best. In view of the results in Figure 13(b) and Table 4, ranking by the outlet SPLs from high to low, the order is the mufflers with inserted tube lengths of 80 mm, 50 mm and 20 mm. This implies that reducing the inserted length of inlet tube is helpful for flow noise reduction of the target muffler. By changing the parameters of expansion ratio and chamber length, as listed in Table 4, the simulated SPLs at the muffler outlets are shown in Figure 14. The SPL orders from high to low are expansion ratio 4, 9, 16 and chamber length 220 mm, 250 mm, 280 mm, respectively. Considering the peak SPLs in Table 4, the acoustical performance of the target muffler may be improved by increasing the expansion ratio and/or the chamber length. Furthermore, by mounting a perforated plate in the middle plane of chamber length, the relationships among the plate thickness, perforation diameter and the SPLs at the outlet end of the muffler are investigated in this paper. The structural parameters of the perforated plate can be seen in Table 4. The corresponding simulation results are shown in Figure 15. The SPLs ranking from high to low are the plate thickness 1 mm, 2 mm, 3 mm and the perforation diameter 9 mm, 7 mm, 5 mm, respectively. Increasing the thickness of perforated plate and/or reducing the perforation diameter can effectively reduce output SPLs of the muffler. The above conclusions can provide a theoretical reference in improvement design of the muffler in the following text.

Figure 12.

The designed simple resonant mufflers with different chamfering form of inserted tube.

Table 4.

Structural modifications of the muffler and the calculated SPLs at the outlet ends.

Modified part	Structural parameters		Peak SPLs (dB)
Inserted tube	Chamfering forms at the tube end	No chamfering	103
		Bevel chamfering (45^o)	98
		Round chamfering (R10)	96
	Inserted length of the tube (mm)	20	100
		50	103
		80	106
Expansion chamber	Expansion ratio	4	115
		9	110
		16	106
	Length of the chamber (mm)	220	113
		250	110
		280	106
Perforated plate	Thickness of the plate (mm)	1	108
		2	106
		3	104
	Diameter of the perforation (mm)	5	105
		7	108
		9	113

Figure 13.

The calculated SPLs of the mufflers with different (a) chamferings or (b) lengths of the inserted tube.

Figure 14.

The calculated SPLs of the mufflers with different (a) expansion ratios or (b) lengths of the chamber.

Figure 15.

The calculated SPLs of the mufflers with different (a) thicknesses or (b) drilling diameters of perforated plate.

Improvement design of the muffler

Structural modifications based on flow field

Using the above mentioned methods, some structural modifications based on flow field analyses are imposed on a resonance muffler for improving its acoustical performance. The original muffler model is shown in Figure 16(a), where the arrangements of inlet and inner pipes, upper and lower exhaust pipes, perforated and division plates are as same as those in Figure 1. The thickness of plates is 1.2 mm. There are 12 holes with diameter of 3 mm and 20 holes with diameter of 3.5 mm are uniformly distributed at the front ends of the inlet pipe and the upper and lower exhaust pipes, respectively. 160 holes with diameter of 5 mm are distributed on the perforated plate. The working conditions of a vehicle muffler are related to its engine speed. When the rotation speed of a four-cylinder 1.6 L engine reaches 6000r/min, the muffler working condition may be set as: gasflow velocity 70 m/s and temperature 800 K. The calculated TKEs are shown in Figure 16(b). The TKE distribution might be explained as follows: the maximum TKE (1014.4 J/kg) at the front end of the lower exhaust pipe is caused by the concentrated gas passing through the perforations of the pipe; the larger TKEs at the end of the inlet pipe are caused by the vortexes generated from the gas impacting on the division plate; and the larger TKEs near the perforated plate are caused by the drastic changes in speed and direction of the gasflow when the gas passes through the drilling holes. According to the conclusions drawn in the above factor analyses, keeping basic size of the muffler, structural improvements of the muffler model are conducted. Considering both the acoustical and the aerodynamic performances of the muffler, such as the transitional fillet, the perforation numbers and the maximum TKE, etc., the specific modifications are as follows: adding a fillet transition with a radius of 10 mm at the end of the inlet pipe, changing the rows of perforation at the front end of the inlet pipe from 3 to 4, increasing the number of perforations at the front end of the lower exhaust pipe (from 2 to 4 rows), adding 6 holes at the front end of the upper exhaust pipe and 4 holes on the perforated plate around the inlet pipe. The modified muffler model and its TKE distribution are shown in Figure 17. Compared with those in Figure 16(b), the amplitudes of TKE in Figure 17(b) are significantly reduced. The maximum value of TKE at the front end of the lower exhaust pipe becomes 845.49 J/kg, which is 168.91 J/kg lower than that before modifications. The structural improvement effectively alleviates the TKE concentration in the flow field, and is beneficial to aerodynamic noise reduction. The calculated near- and far-field SPLs are shown in Figure 18. Comparing the SPL results before and after structural improvement, the changing patterns of SPL with respect to the frequency are similar, but the SPLs after the improvement are generally decreased. The peak SPLs in the near and far fields decreased from 124 dB and 82 dB to 119 dB and 78 dB, i.e., 5 dB and 4 dB reduction, respectively. Considering that the modification might affect the back pressure of a muffler, this paper also calculates the inner pressure distributions of the mufflers before and after improvement. The results are compared in Figure 19. The maximum pressures in the muffler before and after the improvement are 108.56 kPa and 108.33 kPa, respectively. After the improvement, the engine back pressure (exhaust resistance) is reduced by 0.23 kPa, which is helpful for improving engine combustion.

Figure 16.

The original muffler structure and its turbulent kinetic energy distribution (70 m/s,800K).

Figure 17.

The modified muffler structure and its turbulent kinetic energy distribution (70 m/s,800K).

Figure 18.

SPL comparisons of the mufflers before and after improvement in (a) near field and (b) far field (70 m/s, 800 K).

Figure 19.

Comparison of the pressure distributions in the mufflers (a) before and (b) after improvement (70 m/s, 800 K).

Discussions of the muffler performance before and after improvement

To investigate the performance of the improved muffler, further simulations are carried out under other two common working conditions: inlet flow velocity 70 m/s, temperatures 700 K and 900 K. The improvement results are checked by using the SPL at the end of exhaust pipe, the TKE in the chamber and the back pressure of the muffler before and after improvement. Figures 20 and 21 show the near- and far-field SPLs at the end of the exhaust pipe under the conditions of (70 m/s,700K) and (70 m/s,900K), respectively. It can be seen that, in both the near and far fields, the patterns of SPL varying with frequency are similar to those of (70 m/s,800K). Comparing with the original muffler under the condition of (70 m/s,700K), the near- and far-field peak SPLs of the improved muffler are reduced by 3 dB and 5 dB, respectively. The reductions of the maximum TKE at the front end of the lower exhaust pipe and the pressure difference of the muffler are 201.2 J/kg and 2.52 kPa. The corresponding reductions of the above parameters under the condition of (70 m/s,900K) are 4 dB, 5 dB, 154.14 J/kg and 0.907 kPa. As a summary, the calculated results under the three working conditions are listed in Table 5. From the results of different working conditions, it can be seen that the outlet SPL is reduced by 3-5dB, the TKE is reduced by 10–20% and the exhaust pressure difference is reduced by 6–20%. Generally speaking, the improved design has effectively enhanced the aerodynamic performance, and reduced the internal flow noise of the muffler.

Figure 20.

SPL comparisons of the mufflers before and after improvement in (a) near field and (b) far field (70 m/s,700K).

Figure 21.

SPL comparisons of the mufflers before and after improvement in (a) near field and (b) far field (70 m/s,900K).

Table 5

Comparison of the simulated results under three working conditions.

Conditions	Peak SPL reduction (dB)	TKE reduction (J/kg)	Pressure difference reduction (kPa)
70m/s, 700 K	3(near), 5(far)	201.2	2.52
70m/s, 800 K	5(near), 4(far)	168.91	0.582
70m/s, 900 K	4(near), 5(far)	154.14	0.907

Conclusions

This paper presents a multi-field coupling prediction method for improving the aeroacoustic performance of a muffler. The prediction method is combined by the large eddy simulation (LES) and FW-H acoustic analogy (FW-H-AA) approaches, which have higher accuracies in the near and far sound fields, respectively. Based on a vehicle-used muffler with gasflow, a finite volume model considering the coupling of flow, temperature and sound fields is established. The noise reduction characteristics of the muffler are simulated by using the LES and FW-H-AA methods and validated by experiment. The coupling influences of inlet gas velocity, temperature and structural parameters on the noise reductions of the muffler are further analyzed. The results suggest that the flow and sound fields have similar energy distributions under different working conditions. The inlet flow velocity and temperature have a slight influence on the energy contours, but the value of SPL at the muffler outlet increases with the increase of flow velocity and decreases with the increase of temperature. The exhaust noise can be effectively reduced by adding the chamferings and/or reducing the length of the inserted tube, increasing the thickness and/or reducing the perforation diameter of the perforated plate, reducing the expansion ratio and/or increasing the length of the expansion chamber. After structural improvement, the simulation results show that the turbulent kinetic energy (TKE) in the muffler is reduced by 10%–20%, the SPL at the end of the exhaust pipe is reduced by 3-5dB, and the exhaust pressure difference is reduced by 6%–20%. The noise reduction results and the aerodynamic performance of the muffler have been significantly improved. In applications, the simulation method proposed in this paper can be directly used in improvement design of vehicle mufflers, and can also be extended to other types of muffler for design and development.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the Project of National Natural Science Foundation of China (no. 51675324), and partly supported by the Program for Professor of Special Appointment (Eastern Scholar) at Shanghai Institutions of Higher Learning, China.

ORCID iDs

H Guo

YS Wang

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