Numerical and Experimental Analysis of the Effect of Acoustic Loads on the Source Characterization for an Internal Combustion Engine Exhaust System

The identification of source characteristics, commonly characterized as the source strength and source impedance, is essential for predicting the acoustic performance of an internal combustion (IC) engine exhaust system. This study contributes to a theoretical analysis of the effect of the acoustic parameters of loads (i.e., four-pole parameters, load impedance, and radiation impedance of the tailpipe) on the identification error of the source characteristics for an IC engine. A model based on the linear time-invariant hypothesis was constructed. A dispersion estimation function of the source strength and a deviation estimation function of the source impedance were established as indicators to test the identification accuracy. A three-dimensional (3D) multifield coupling numerical simulation method, which can thoroughly consider the influences of airflow and temperature, was applied to obtain the acoustic parameters of loads and compare them with those obtained by a one-dimensional (1D) analytical method. Then, the acoustic parameters of loads and the radiated sound pressure level of the tailpipe obtained in the measurement were substituted into the source characteristics identification model. Based on the analysis and calculation, the estimated error of the source characteristics acquired by using the 3D numerical simulation method is significantly lower than that by using the 1D analytical method. Moreover, the experiment and the 3D multifield coupling numerical simulation method were used to verify the identification results of the engine source characteristics. The results show that the far-field sound pressure level of the tailpipe predicted via the 3D numerical simulation method agrees well with the experimental results, indicating that accurate acoustic parameters of loads can effectively improve the source identification accuracy for an IC engine.


I. INTRODUCTION
An exhaust system is an essential component of an internal combustion (IC) engine; the exhaust system controls the noise of exhaust, thereby reducing the environmental noise pollution of motor vehicles, and adjusts the exhaust noise quality to meet human hearing requirements. Moreover, the The associate editor coordinating the review of this manuscript and approving it for publication was Jenny Mahoney. noise control performance of the exhaust system is significantly related to the source characteristics of the IC engine. To design a muffler system capable of accommodating an IC engine, the identification of engine source characteristics is of great importance [1].
Many research studies have been performed in the field of source characteristics identification for IC engines. Source characteristics are associated with a source model, which aims to provide a load-independent and quantitative VOLUME 8, 2020 This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ description of the engine source. A linear time-invariant model, which represents the engine as a black box, satisfies the requirements of this principle and plays a key role in the prediction and design of an exhaust system [2]. In a linear time-invariant model, engine source characteristics are represented by the source strength and impedance, considerably simplifying the engine modeling process. The engine source characteristics (i.e., source strength and impedance) can be identified via an engine bench test. Many methods, which can be classified as either direct or indirect, have been proposed for the experimental identification of source characteristics. Direct methods, based on standing waves and random excitation, require an external source [3]. Hence, measuring the source characteristics of IC engines in exhaust pipes is sometimes difficult owing to the high gas temperature and acidic turbulent gas flow [4]. Due to these severe conditions, indirect methods, such as the two-load [5], [6], three-load [7], four-load [8]- [10], and multiload [11]- [13] methods, are more widely used in practical applications. These indirect methods consider the effects of the number of loads and the optimization of the identification algorithm on the source characteristics identification accuracy. Generally, increasing the number of acoustic loads and choosing an appropriate identification algorithm, such as the multiload method, can effectively improve the calculation accuracy of engine source characteristics [14]. However, with an increasing number of loads, the computational load of the experiment for identifying the engine source characteristics will increase exponentially.
To render the selection of loads more efficient, several scholars studied the effects of the load combination and load structure on the identification of source characteristics. Zheng et al. [15] analyzed the load selection effect on the identification error of engine source characteristics based on a theoretical error analysis and proposed practical rules for selecting loads that can improve the source characteristics accuracy for an IC engine and reduce the required number of loads used in the measurement. Hynninen et al. [16] developed a novel ''capsule pipe method'' for the experimental determination of acoustic sources for large IC engines, in which it is difficult to vary the load in such a harsh environment. Capsule pipes were used for the acoustic load impedance variation, and engine source data were derived with success.
The above research on the identification accuracy of engine source characteristics mainly focused on optimizing the identification algorithm, selecting the number of loads, and determining the load impedance variation. However, no scholars have considered the influence of the accuracy of the acoustic parameters of loads (i.e., four-pole parameters, load impedance, and radiation impedance of the tailpipe) on the source identification error for an IC engine. Among all the linear time-invariant models, the one-dimensional (1D) analytical method is generally used to calculate the acoustic parameters of loads. However, the 1D analytical method can consider only a uniform flow field; that is, the sound propagation medium in the pipe is treated as a uniform flow field [17]. For the identification of loudspeaker, compressor, and other sound source characteristics, the calculation of the acoustic parameters of loads by using the 1D analytical method generally does not cause large errors. However, in the exhaust system of an IC engine, the medium in which sound propagates in the acoustic load pipe is a hot nonuniform flow field that contains large temperature and velocity gradients. For the 1D analytical method, adequately describing the characteristics of the sound field in the acoustic load pipe is impossible, and the results of the acoustic parameters of loads obtained by the 1D analytical method will contain enormous errors, which may considerably affect the accuracy of the source characteristics identification for an IC engine.
In this work, the effects of acoustic parameters of loads on the source characteristics identification error for an IC engine are theoretically analyzed. The three-dimensional (3D) multifield coupling simulation method is applied rather than the 1D analytical method to obtain more accurate acoustic parameters of loads. The influences of the airflow velocity and temperature gradient on the sound field in exhaust load pipes are fully considered. Furthermore, an engine bench experiment and the multifield coupling simulation method are used to test the reliability of the source characteristics identification results.
The remainder of this paper is organized as follows. In Section II, the theoretical model for source characteristics identification is constructed. Section III introduces the solution for the acoustic parameters of loads. The identification and analysis of source characteristics are presented in Section IV. The identification results are verified and discussed in Section V, followed by conclusions in Section VI.

II. THEORETICAL MODEL FOR SOURCE CHARACTERISTICS IDENTIFICATION
A linear time-invariant model assumes that engine exhaust noise is produced in a steady state as a linear system and simplifies the entire engine into a black box with stable source characteristic parameters.

A. LINEAR TIME-INVARIANT MODEL FOR AN IC ENGINE
When an engine exhaust system is assumed to have linear and time-invariant characteristics, the characteristics of the engine can be simplified into an equivalent acoustic source, which can be represented by the source strength P E and source impedance Z E . Based on the acoustoelectric analogy method, an electrical circuit representation can be established for an acoustic source-load mode [8]. According to the circuit principle, the equation for representing the source-load interaction can be expressed as follows: where p L is the acoustic pressure between the source and load and Z L is the load impedance for the whole exhaust system. For convenience, straight pipes with various lengths are used as acoustic loads for the identification of engine source characteristics. Then, Eq. (1) can be rewritten as where N is the number of loads. In Eq. (2), the number of acoustic loads determines the number of equations. The ratios of these equations can be used to solve for the source impedance and strength, which forms the theoretical basis of the indirect method [8]. However, the impedances of the acoustic loads Z L n in Eq. (2) are still unknown variables that need to be solved. According to a previous study [17], the impedance of an acoustic load pipe can be derived by the transfer matrix method.
According to the theory of plane wave propagation in ducts with flow [17],the transfer matrix for the exhaust pipes can be expressed as where p and U are the pressure and volume velocities, respectively, and A, B, C, and D are the four-pole parameters of the straight pipes. In previous studies, the 1D analytical method was extensively used to calculate the acoustic parameters of loads for source characteristics identification. However, the 1D analytical method can consider only a uniform flow field, thus neglecting the influences of complex temperature and velocity gradient fields in the load pipes of an IC engine. This study aims to investigate the effects of the acoustic parameters of loads on the identification of source characteristics. Thus, the solution for the four-pole parameters of load pipes is a key topic and will be discussed in Section 3.
With the four-pole parameters of acoustic loads, the engine and the exhaust system adopted herein can be diagrammed as an electrical circuit representation, as shown in Fig. 1. Then, the entire model of the exhaust system can be expressed as where U E is the volume velocity of the engine source, U 0 is the volume velocity at the end of the load pipes, and Z R is the radiation impedance at the end of the tailpipe.

B. SOLUTION OF THE SOURCE STRENGTH
The far-field radiation sound pressure of an acoustic load tailpipe is measured in an anechoic room. The radius of the acoustic load pipes is much smaller than the distance between the microphone and the nozzle. Therefore, the tailpipe radiation process can be simplified as radiation from a point source. According to the radiation theory of a monopole source and the definition of the sound intensity [18], the relationship between the far-field radiated sound pressure and the volume velocity at the end of the load pipes can be expressed as where I R n is the sound intensity, ρ 0 and c 0 are the density and sound velocity in the anechoic room, respectively, k 0 is the wavenumber, and P R is the radiated sound pressure of the response point far from the end of the tailpipe with radius R. By combining Eq. (4) and Eq. (5), the relationship between the source strength P E and the sound pressure at the far-field response point P R is expressed in Eq. (6).
It can be seen from Eq. (6) that each acoustic load can provide a source strength. According to the hypothesis of a linear source model, the source strength should be independent of the number of acoustic loads. Therefore, the average value of the sound strength derived from each load is regarded as the ultimate engine source strength, as expressed in Eq. (7).

C. SOLUTION OF THE SOURCE IMPEDANCE
Based on the linear time-invariant model for an IC engine, the nonlinear equations for solving the source impedance can be obtained using the Prasad formula [8]: where The engine source impedance can be acquired by solving Eq. 8. The detailed derivation process is described in Ref. [8].

D. DISPERSION AND DEVIATION ESTIMATION FUNCTIONS
To estimate the source identification result accuracy, a dispersion estimation function for the source strength level and a deviation estimation function for the source impedance were established. VOLUME 8, 2020 As presented in Eq. (6), each acoustic load can provide a source strength. The value of the source strength can be affected by errors from the source impedance Z E , the radiation impedance Z R , and the four-pole parameters of acoustic loads. However, the source strength P E should be invariable due to the linear time-invariant hypothesis. Therefore, a function is built to estimate the dispersion of the source strength level, which can reflect the magnitude of the error in the solution of the source strength. The dispersion estimation function is expressed as follows: where D(P E ) is the value of the dispersion estimation function.
From Eq. (8), the solution of the source impedance Z E is determined by the acoustic parameters of loads (i.e., the load impedance Z L , four-pole parameters, and radiation impedance Z R ) and the spectrum of the response point pressure p R . The error of p R can be well controlled via an engine bench test through a reasonable arrangement of the test device. However, the error of the acoustic parameters of loads is related to the solution method. The influences of the acoustic parameters of loads on the estimation error of the source impedance can be analyzed by using a deviation estimation function, which is defined as the following equation: where Z E = R E +jX E is the solution of the source impedance obtained by using Eq. (8). E m represents the deviation generated when the solution of the source impedance is substituted back into Eq. (8).
As presented in Eq. (10), (N − 1) deviations exist for the N loads. For convenience, the total deviation can be rewritten as where E t indicates the estimated source impedance deviation, which can be used to quantitatively estimate the error of the source impedance solution.

III. SOLUTION OF THE ACOUSTIC PARAMETERS OF LOADS
Based on the above theoretical analysis, the acquisition of acoustic parameters of loads is the key to determining the engine source characteristics. The acoustic parameters of loads are obtained in two ways in this article. The first is the 1D analytical method based on the theory of plane wave propagation (1D-AM), and the second is the 3D multifield coupling numerical simulation method (3D-MCNSM).

A. 1D-AM
In previous studies, the 1D-AM was extensively used to calculate the acoustic parameters of loads. Based on the theory of plane wave propagation with flow [17], the four-pole parameters of straight ducts can be represented as A L n B L n C L n D L n = e −jkMl en coskl e n jZ 0 sinkl e n jZ −1 0 sinkl e n coskl e n , (n = 1, 2, . . . , N ) (12) where M is the Mach number, M = v/c, and v and c are the flow velocity and sound velocity, respectively. In this work, additional variables were used to simplify the equation, such as k = ω/c(1 − M 2 ), Z 0 = ρc/(πr 2 ), and l e n = L n + l, where ρ is the density of the flow in the pipe, L is the pipe length, l is the acoustic correction length, l = 0.6r/ (1 − M 2 ), and r is the pipe radius. According to the transfer matrix method [17], the impedance of the loads is written as The radiation impedance Z R can be obtained using the following equation [17]: where R c is the convective reflection coefficient. R c can be solved using the theory deduced by Munt [19] and has been proven to be applicable for pipe termination with hot and subsonic flow [20]. The expression can be written as follows: (15) where θ is the phase of the reflection coefficient and F + (u) is the transform of the vortex layer displacement. Based on the above equations, the acoustic parameters of loads can be obtained using the 1D method. Then, by substituting the acoustic parameters of loads into Eq. (6) and Eq. (8), the source strength and source impedance can be obtained using the 1D-AM.
Eq. (12) shows that the 1D-AM can consider only the effect of a uniform flow field on sound propagation, whereas the large temperature and velocity gradient fields in the axial and lateral regions of the load pipe are ignored. Many studies have shown that a velocity discontinuity in a moving medium will change the transmission and reflection characteristics of acoustic waves [20], [21], and a temperature discontinuity has a major effect on the propagation directivity of the transmitted waves [22], [23]. Therefore, ignoring the temperature and velocity gradient fields in the load pipe will generate significant errors in the solution of the acoustic parameters of loads.

B. 3D-MCNSM
For an IC engine, the airflow in the exhaust pipe is a complex multiphysics coupling field. The heat dissipation effect of the pipe wall leads to a temperature gradient inside the pipe, and the friction near the pipe wall causes the flow field to become nonuniform within the pipe. In addition, the propagation of sound around the nozzle is also affected by the complex jet flow around the tailpipe. All these factors affect the 1D-AM solution accuracy for the acoustic parameters of loads. Hence, this study adopts the 3D-MCNSM to calculate the propagation of sound under the conditions of the complex, multiphysics coupling flow field to address these problems.
The hot nonuniform flow field in a load pipe was previously simulated via the commercial CFD code Fluent [24]. A meshed model with a structured grid was established for the flow field calculation, as shown in Fig. 2.
As presented in Fig. 2, the straight pipe is the acoustic load pipe, the diameter of which is 0.048 m. The cube at the end of the acoustic load pipe represents the external air domain with dimensions of 0.8 × 0.6 × 0.6 m. The meshed models are all structured grids for simpler discretization and formulation, which can reduce the computational load and improve efficiency. The number of grids is 430,000. The meshes of the acoustic load pipe and the outer nozzle of the tailpipe, where the velocity varies considerably, are densely refined, and the average mesh size is approximately 2 mm.
The governing equation of the three-dimensional flow field is where is the Hamiltonian operator, ⊗ is the tensor product, E is the total energy, τ ν is the viscous stress, q T is the heat flux, ρ is the flow density, p mt is the flow pressure, and v is the flow velocity in the three-dimensional space.
In the fluid simulation model, the realized κ − ε turbulence model was selected. The κ−ε turbulence model equations can

be represented as
where κ is the turbulent kinetic energy; v j is the flow velocity in one direction, j = 1, 2, 3, which represent the lateral, longitudinal, and vertical directions, respectively; x j is the coordinate position in the j − th direction; λ is the laminar viscosity coefficient; σ κ is the Prandtl number corresponding to the turbulent kinetic energy and is generally taken as 1; G 1 is the turbulent kinetic energy term generated by the laminar velocity gradient; ε is the turbulent dissipation rate; C 1 and C 2 are empirical constants generally taken as 1.47 and 1.92, respectively; σ ε is the Prandtl number corresponding to the turbulent dissipation rate; and G 2 is the turbulent kinetic energy term generated by the buoyant force.
In the simulation, the second-order upwind difference scheme was used to discretize the spatial derivatives of the Navier-Stokes equations. Additionally, the second-order implicit discrete scheme was used to discretize the temporal term to improve the computational accuracy. The pressure implicit split operator (PISO) algorithm was used to calculate the flow field. The wall material of acoustic load pipes is set as stainless steel with out coated material, which is consistent with the experimental setup. The boundary conditions of the flow simulation model are listed in Table 1. Fig. 3 shows the velocity and temperature fields for one of the acoustic load pipes. As shown in Fig. 3, the flow field in the measured load pipe and the jet flow around the tailpipe are not uniform and instead display complex velocity and temperature gradients, which have significant effects on the acoustic parameters of loads.  To calculate the propagation of sound, a 3D finite element model of the acoustic field was established. A tetrahedral mesh was used in the acoustic field calculation, and the total number of grids was 340,000, as shown in Fig. 4(a). Fig. 4(b) shows that virtual microphones were arranged within the pipe and at the far-field response point beside the grid elements. Ten virtual microphones were arranged at an equal distance from the entrance to the outlet of the load pipe. The far-field response points were set according to the experimental arrangement; that is, virtual microphones were arranged 0.5 m from the end of the tailpipe.
The boundary conditions of the acoustic field simulation model are listed in Table 2.
Before calculating the propagation of sound, the flow-field simulation results were mapped onto the acoustic finite element model, which was used to calculate the propagation of sound in the complex, multiphysics coupling flow field. Usually, the generated grid for a flow field computation is much denser than that of the acoustic finite element model. When mapping the flow-field simulation results by linear interpolation, the grid node information of the flow field will   be lost during the interpolation process. Therefore, an integral interpolation method was implemented to map the flow field onto the acoustic field. All the grid node information of the flow field distributed among the finite elements of the acoustic model was extracted via a shape function to ensure that all flow field simulation results could be used reasonably.
The results of the integral interpolation from the flow field to the acoustic finite element model are shown in Fig. 5.
Then, the acoustic field with a hot nonuniform flow field was calculated using commercial software ACTRAN/TM [21]. The simulation results are shown in Fig. 6.
It can be seen from Fig. 6 that under the influence of the complex multiphysics coupling flow field, the sound wave in the pipe no longer propagates in the form of a plane wave (for the case in Fig. 6, the cutoff frequency of the plane wave is approximately 7520 Hz), which is closer to the real sound propagation characteristics in an engine exhaust system. Therefore, more accurate acoustic parameters can be obtained through the 3D-MCNSM, which can be used to analyze the influences of the acoustic parameters of loads on the identification error of the source characteristics for an IC engine.
To obtain the impedance of the loads in the nonuniform, multiphysics, coupling flow field, the four-pole parameters of the load pipes and the radiation impedance at the end of the tailpipe should first be addressed. In this study, a two-load method was applied to the acoustic finite element model to address the four-pole parameters. Two impedance boundary conditions were set at the entrance of the load pipe: a rigid boundary (A) and a non-reflecting boundary (B). Subsequently, the sound pressure and volume velocity signals of the virtual microphones at the inlet and outlet were extracted to calculate the four-pole parameters of the load pipe. The solution model is shown in Eqs. (18) and (19): where A L , B L , C L , and D L are the four-pole parameters of the load pipe in the 3D-MCNSM and p and U represent the sound pressure and volume velocity, respectively. The subscripts A and B represent the two types of inlet boundary conditions, and the subscript numbers 1 and 2 indicate the inlet and outlet of the load pipe, respectively. The radiation impedance at the end of the tailpipe can be solved using Eq. (20).
By substituting the four-pole parameters and radiation impedance of the acoustic load pipe into Eq. (13), the load impedance of the acoustic pipe can be obtained using the 3D-MCNSM and compared with that generated using the 1D-AM, as shown in Fig. 7.
As shown in Fig. 7, the resonance peak frequencies of the load impedance extracted using the 3D-MCNSM are practically consistent with those obtained by the 1D-AM. However, the resonance peak frequencies of the 3D-MCNSM in the medium-to high-frequency range shift slightly toward the low-frequency direction. The main reason for this phenomenon is that the sound wavelength at medium-high frequencies is much shorter and is easily affected by the nonuniform flow velocity and temperature fields. The resonance peak amplitude of the impedance based on the 3D-MCNSM is lower than that based on the 1D-AM, which is mainly because the 3D-MCNSM considers the effect of temperature-dependent viscosity on the hot nonuniform flow.

IV. IDENTIFICATION AND ANALYSIS OF SOURCE CHARACTERISTICS
The theoretical model and error estimation method (i.e., the dispersion and deviation estimation functions) for identifying the source characteristics of an IC engine were established above. To obtain the inlet parameters (e.g., temperature, static pressure and flow rate) for the engine source characterization, performing an IC engine bench test is necessary.

A. EXPERIMENTAL SETUP FOR THE IDENTIFICATION OF SOURCE CHARACTERISTICS
A gasoline engine with four cylinders is utilized to perform the test to identify the engine source characteristics. An electric eddy current dynamometer is used on an engine bench. The acoustic load pipes and microphones are placed in a semi-anechoic chamber, and anechoic wedges are placed on the ground under the tailpipe to reduce the sound reflection from the ground, thus rendering the test environment close to a free field. The acoustic load pipes are made of stainless steel without coated material. The schematic of the test device is shown in Fig. 8.
A photograph of the experimental setup is shown in Fig. 9.
The eddy current dynamometer is used to control the rotational speed n of the engine. A thermocouple, a static pressure sensor, and a pitot pipe are arranged at the connection between the load pipe and the engine to test the inlet flow parameters of the load pipe, namely, the temperature T , static pressureP ss , and flow rate ν, respectively. VOLUME 8, 2020  The microphones are installed at the far-field response points, approximately 500 mm from the nozzle of the acoustic load tailpipes, to measure the far-field radiated sound pressure level. To prevent the hot air of the exhaust flow from affecting the microphones, a wind hood is installed over the head of each microphone. 26CA-type free-field microphones produced by G.R.A.S. are used to measure the sound pressure. The electrical signals of the microphones are collected and analyzed using LMS's TypeSCM02 data analyzer.
The experiment for the source characteristics identification is performed at a constant engine speed (in 500 rpm increments from 1000 to 5000 rpm). During the test, the engine runs under full load conditions, and its speed is adjusted through the eddy current dynamometer. Six pipes of different lengths are used as acoustic loads to measure the engine source characteristics, which are summarized in Table 3.

B. CALCULATION AND ANALYSIS OF SOURCE CHARACTERISTICS
The test for the identification of source characteristics is performed under steady-state conditions, and the far-field radiated sound pressure level at the response point in the frequency domain is obtained by applying a Fourier transform to the test data. The load impedance, four-pole parameters, and radiation impedance of the six loads are calculated using the 1D-AM and 3D-MCNSM separately. The impedance values are reselected using the method proposed in a previous study [15] for the calculation of the source strength and impedance, which can effectively improve the accuracy of the source characteristics identification.
The solution method for the identification of source characteristics is the same under steady-state conditions for the IC engine bench test. Hence, the bench test data under the  operating condition of 2500 rpm are employed to identify the source characteristics. The temperature, static pressure, and flow rate at the inlet of the load pipe under the condition of 2500 rpm are used to calculate the acoustic parameters of the loads through the method introduced in Section 3. By substituting the acoustic parameters of the loads and the radiated sound pressure level of the far-field response points into Eqs. (6) and (8), the source strength and impedance of the engine can be obtained. By using Eqs. (9) and (11), the values of the dispersion and deviation estimation functions can also be obtained, which are shown in Figs. 10 and 11. Fig. 10 shows that the dispersion estimate of the source strength level by using the 3D-MCNSM is more stable and much lower than that using the 1D-AM. In some frequency ranges, the source strength dispersion values of the 3D-MCNSM are approximately 6 dB lower than those of the 1D-AM. As shown in Fig. 11, the normalized deviation estimate of the source impedance obtained using the 3D-MCNSM is much lower than that acquired by the 1D-AM. Figs. 10 and 11 illustrate that the 3D-MCNSM can obtain more accurate engine source characteristics than the 1D-AM.
The normalized resistance, normalized reactance, and strength of the engine source obtained by the two methods are shown in Figs. 12, 13, and 14, respectively. The comparisons in Figs. 12 and 13 show that the fluctuations in   the source impedance predicted using the 3D-MCNSM are much less than those predicted by the 1D-AM. In addition, Fig. 12 shows that the real parts of the source impedance obtained by utilizing the 1D-AM contain many negative values, for example, in the frequency range of 150 Hz∼200 Hz and 550 Hz∼600 Hz. Nevertheless, the real parts obtained by using the 3D-MCNSM are nearly all positive. The real parts of the source impedance represent the source resistance. According to acoustic theory, the sound resistance should be positive. The negative values of the source resistance obtained by the 1D-AM are mainly caused by inaccurate acoustic parameters of the loads. Therefore, Fig. 12 strongly demonstrates that the 3D-MCNSM can obtain more accurate acoustic parameters of loads than the 1D-AM and that the acoustic parameters of loads have a significant effect on the identification error of the source characteristics for an IC engine.
As shown in Fig. 14, the peak frequencies of the source strength for the two methods are in good agreement. However, the peak amplitudes of the source strength for the two methods deviate greatly. To analyze the identification accuracy of the two methods, the radiated sound pressure from the tailpipes with different numbers of acoustic load pipes should be predicted via the identified source impedance and source strength and compared with the experimental results. In the next section, the engine source identification results will be verified and discussed.

V. VERIFICATION OF THE IDENTIFICATION RESULTS
To verify the identification results acquired by the two methods, the derived engine source strength and source impedance are used to predict the far-field radiated sound pressure level of the tailpipe for an expansion chamber muffler.
An expansion chamber with an insert pipe is used for the verification on the engine test bench. The far-field radiated sound pressure and the inlet flow parameters of the expansion chamber (i.e., temperature, static pressure, and flow rate) are measured through the experimental setup introduced in Section IV.A. The geometric dimensions of the expansion chamber with an insert pipe and a photograph of the test are shown in Fig. 15.
The 3D-MCNSM is used to predict the far-field radiated sound pressure of the expansion chamber with an insert pipe. The hot nonuniform flow field in the expansion chamber is simulated by using Fluent software with the inlet flow parameters obtained from the engine bench test. The flow field simulation results are mapped onto the acoustic grid model via integral interpolation. The propagation of sound within the hot nonuniform flow is calculated using ACTRAN. The engine source impedance and strength of the two methods   are applied to the entrance boundary of the acoustic finite element model to obtain the far-field radiated sound pressures at the response points. The acoustic simulation results are shown in Fig. 16.
The far-field radiated sound pressure levels simulated by the two methods are compared with the experimental results in Fig. 17.
As shown in Fig. 17, the radiated sound pressure level obtained using the 3D-MCNSM indicates better agreement with the experimental results than that obtained using the 1D-AM, especially at the exhaust order frequencies of the test engine. To perform a more accurate analysis of the source characteristics obtained using these two methods, the amplitudes of the sound pressure level at the main exhaust order frequencies are extracted to analyze the error relative to the test results, as shown in Fig. 18. Fig. 18 shows the relative errors of the radiated sound pressure at the main exhaust order frequencies from the two methods. As shown in Fig. 18, the error deviation from the test results of the 3D-MCNSM is much lower than that of the 1D-AM, especially from the 2nd order to the 8th order, which are the main order components of interest in controlling the exhaust noise emitted by an IC engine.

VI. CONCLUSION
This study aims to analyze the influences of the acoustic parameters of loads on the identification error of the source characteristics for an IC engine. The 1D-AM and 3D-MCNSM are used to calculate the acoustic parameters of the loads (i.e., four-pole parameters, load impedance, and radiation impedance of the tailpipe). The resonance peak frequencies of the load impedance extracted by using the 3D-MCNSM slightly shift toward the low-frequency direction compared with that obtained with the 1D-AM. Additionally, the resonance peak amplitude of the load impedance from the 3D-MCNSM is lower than that from the 1D-AM. The reason is that the 3D-MCNSM considers the effects of the multiphysics coupling flow field and the temperature-dependent viscosity in the hot nonuniform flow. Therefore, the 3D-MCNSM can give more accurate acoustic parameters of loads than the 1D-AM, and the former can be used to precisely consider the influences of the acoustic parameters of loads on the identification error of the source characteristics for an IC engine.
The source strength and impedance of the IC engine identified by the two methods are compared, as are the results of the dispersion and deviation estimation functions. Through these comparisons, the deviation estimate of the source impedance and the dispersion estimate of the source strength level by using the 3D-MCNSM are much lower than those predicted by using the 1D-AM. Moreover, the source impedance predicted by using the 3D-MCNSM is much more stable than that produced by using the 1D-AM. To verify the identification results of the engine source characteristics, an expansion chamber with an insert pipe is used for the engine test bench, and the derived engine source strength and source impedance are used to predict the radiated sound pressure level. The comparison shows that the radiated sound pressure level obtained using the 3D-MCNSM agrees better with the experimental results than that obtained by the 1D-AM.
In summary, the acoustic parameters of loads have a significant effect on the accuracy of the source characteristics identification for an IC engine. The 3D-MCNSM applied in this study can obtain more accurate acoustic parameters of loads considering a hot nonuniform flow field in an engine bench test, effectively reducing the identification error of the source characteristics for an IC engine.