Development of High-Fidelity Numerical Methodology for Prediction of Vehicle Interior Noise Due to External Flow Disturbances Using LES and Vibroacoustic Techniques

Abstract

A pleasant and quiet cabin in driving a car is one of the most critical factors affecting a customer’s choice in a market. As the traditional noise sources such as power trains become less, the relative contribution of aerodynamic noise to the interior noise of a road vehicle becomes even more critical. In this study, a high-fidelity numerical methodology is developed for the reliable prediction and analysis of the interior transmitted noise due to external flow disturbance. The developed numerical methodology is based on the sequential application of the high-resolution LES technique, wavenumber–frequency transform, and vibroacoustic model. First, the compressible LES techniques with high-resolution grids are employed to accurately predict the external turbulent flow and aeroacoustic fields due to the turbulent flow, at the same time, of a vehicle running at a speed of 110 km/h. Second, surface pressure fluctuations on the front windshield and side windows, obtained from the LES simulation, are decomposed into incompressible and compressible ones using the wavenumber–frequency transform. Lastly, the interior sound pressure levels are predicted using the vibroacoustic model, which consists of the finite element (FE) and statistical energy analysis (SEA) methods. For the efficient computation of the vibroacoustic interaction between the vibration of the vehicle windows and the acoustic field inside the cabin room, the FE and SEA methods are applied in low- and high-frequency ranges, respectively. The predicted interior sound pressure spectral levels agree well with the measured ones. In addition, although the magnitudes of the compressible pressure components are generally lower than those of the incompressible ones, the compressible field is found to contribute more to the interior noise in high-frequency bands. The physical mechanism of the higher transmission is shown to be related to the coincident effect between the compressible pressure field and the structural vibration of the vehicle window.

1. Introduction

The vehicle interior noise caused by high-speed external flow is one of the critical issues for product developers to consider in a design state. The improvement of the NVH (noise, vibration, and harshness) technologies has led to a significant reduction in vehicle noise caused by vibration from an engine, the interaction of tire/road, and transmission system [1]. The sound power of aerodynamic noise is known to increase in proportion with the sixth to eighth power of the vehicle speed. On the other hand, the structure-borne noise from the engine, tire/road interaction, and transmission system is only proportional to the first to third power of the speed [2]. This fact implies that the aerodynamic noise should be more dominant as the running speed of the vehicle increases. Aerodynamic noise becomes even more critical due to the rapid commercialization of electric vehicles without traditional combustion engine noise. The physical process of the interior transmission of external flow disturbances can be understood by three sequential mechanisms: external flow excitation on the windshields of a vehicle, the vibrational response of the vehicle windshields to the external flow disturbances, and the acoustic response of a cabin room to the windshield vibration.

The external flow fluctuations can be categorized into two kinds of pressure fields. One is an incompressible pressure field due to convected turbulent boundary-layer flow, and the other is compressible pressure fluctuation due to aerodynamic noise. The former is called hydrodynamic pressure or pseudo-sound, and the latter is called acoustic pressure. A few studies focused on developing an effective algorithm to separate incompressible and compressible pressure fields to investigate the transmission characteristics of those through a cabin’s wall panel. Herpe et al. [3] applied a direct noise computation code of the compressible three-dimensional Navier–Stokes equations to accurately predict external aerodynamic and acoustic wall pressure fluctuations due to a car side mirror over a flat plate. They separated the aerodynamic and acoustic components using the wavenumber–frequency diagram of the surface pressure field. However, there were significant differences in the power spectral density of surface pressure fluctuations between the predicted and measured ones. Lee et al. [4] proposed an effective signal processing technique based on wavenumber–frequency transform. They also successfully decomposed the surface pressure field on the side window of an automobile into two distinct hydrodynamic and acoustics surface pressure fields by using the wavenumber–frequency diagram of the surface pressure field. Lee et al. [5] applied the wavenumber–frequency transform to the external surface flow fields of a high-speed train running at a speed of 300 km/h in an open space to assess the relative power levels due to each pressure field. However, these studies did not compare with measured data to validate the predicted surface pressure fields.

An appropriate test model is essential to quantitatively assess the indoor cabin noise of an automotive due to external flow disturbances in an actual situation. Hartmann et al. [6] reported the numerical and experimental results on the interior noise due to the airflow around the A-pillar and the side mirror of a generic vehicle model developed by a consortium of German automotive manufacturers. The model is built in such a way that only the noise transmission through the front side window is relevant. This study used different CFD solvers to predict the transient pressure loading. Cho et al. [7] also compared numerous numerical solvers of computational fluid dynamic and vibroacoustic methods to accurately predict interior noise transmission excited by a pillar vortex of a simple land vehicle model, the Hyundai Simplified Model (HSM). These studies focused on benchmarking different numerical methods without a detailed analysis of related physical and numerical issues. He et al. [8] combined the Spalart–Allmaras detached eddy simulation (DES) method with a statistical energy analysis (SEA) model to investigate the relative contribution of the convective and acoustic pressure fluctuations around a car side window to the interior of a full-sized DrivAer clay model. They revealed that the contribution of the acoustic fraction was more dominant than the convective one. However, the DES model uses the RANS solver in the boundary layer flow, which captures only the coherent structure related to the tonal aerodynamic noise source. Furthermore, the SEA model was used as a vibroacoustic solver, for which the validity is limited in the middle- and high-frequency range and only in terms of the averaged band spectrum.

The reliable vibroacoustic solver and computational aeroacoustic methods are critical for the accurate prediction of interior wind noise of a land vehicle. In this respect, some studies paid close attention to figuring out the transfer characteristics of the wall pressure fluctuations to an interior cabin through the front side glass of the vehicle. Van Herpe et al. [9] developed an analytical solution for the sound transmitted into a rigid rectangular cavity by a flexible plate excited by the turbulent boundary layer of an external grazing flow. The averaged sound power was calculated to estimate the contribution of the compressible and incompressible pressure components to the flow-induced noise inside the cavity. However, this study was limited to the problem of simple geometry. He et al. [10] studied the transmission characteristics and mechanism of the exterior wall pressure fluctuation on the side window of a full-size DrivAer model to the interior wind noise by using a simplified rectangular window–cabin equivalent model. They investigated the fundamental cause of different transmission efficiency between two pressure components. Although the analytic model provided a clear theoretical interpretation of different transmission characteristics between the compressible and incompressible pressure fields, it is difficult to extend the model to practical problems involving complex geometries for which analytic models are generally not available.

The structural components of high-speed vehicles are inevitably subject to and, thus, excited by external surface pressure fields within turbulent boundary layers due to the external flow disturbances induced by their motions [11,12,13,14,15]. From a fluid–structure interaction (FSI) perspective, the accurate computation of structural vibration induced by turbulent boundary layers is still a very challenging problem. The input power for the structural vibration must consider not only the random excitation force but also the vibrational characteristics of the structure itself [16,17]. Although the injected power formula is applied through various assumptions, most input power formulas do not reflect the vibration characteristics of the structure [18,19,20].

In this study, a reliable, systematic numerical methodology is proposed for the accurate and efficient prediction of the interior noise of a high-speed vehicle due to external flow disturbances. The developed numerical methods consist of three sequential methods suitable to each physical step causing the indoor transmission of external compressible and incompressible surface pressure fields. In the first step, the large eddy simulation (LES) technique with high-resolution grids is used to accurately compute external aerodynamic and aeroacoustic fields around a vehicle. In the second step, the wavenumber–frequency transform is used to decompose the surface pressure fluctuations on the windows of a vehicle into incompressible and compressible ones. The separation is based on the physical fact that the phase speeds of incompressible and compressible pressure waves differ. Lastly, the interior sound pressure levels are predicted using the vibroacoustic model based on the coupled finite element (FE) and SEA methods with the input of each incompressible and compressible pressure field. The developed numerical methods are applied for the benchmark problem of the HSM [7].

The main contributions of the present study are fivefold. Firstly, a highly accurate LES simulation with high-resolution grids is carried out with a focus on the accurate computation of external flow, including both incompressible and compressible flow fields. The predicted surface pressure spectra show excellent agreement with the measured ones. Secondly, a detailed signal processing algorithm based on the wavenumber–frequency transform is developed to accurately separate incompressible and compressible surface pressure fields on the windows of the HSM as input for the pressure loading of the vibroacoustic model. The successful separation of these two fields enables the assessment of the relative contribution of the incompressible and compressible pressure fields to the interior noise of the vehicle and, thus, an effective measure for suppression of interior wind noise. Thirdly, a vibroacoustic solver based on the combination of the FE and SEA methods is developed for efficient computation of the interior acoustic field due to the vibration of windows. Specifically, the FE and SEA methods are applied in low- and high-frequency bands, respectively. The modal overlap factor is used to determine the low- and high-frequency bands. Fourthly, the injected power formula that reflects the vibration characteristics of the structure is defined, and a method that can be applied to practical problems is presented, showing excellent agreement between the predicted and measured sound pressure spectral levels. Lastly, the physical mechanism leading to more contribution of the compressible surface pressure field to the interior sound in high-frequency bands is made clear.

2. Numerical Methods and Target Model

2.1. Large Eddy Simulation

Compressible LES techniques were used to accurately predict external flow and acoustic disturbances around a vehicle running at a speed of 110 km/h. The governing equations of LES were obtained by applying a low-pass spatial filter to the Navier–Stokes equations, thus computing only large eddies directly. The three-dimensional compressible unsteady filtered Navier–Stokes equations with the Smagorinski–Lilly model to predict subgrid-scale turbulent viscosity can be written in the following form:

$$\frac{\partial \rho }{\partial t}+\frac{\partial \left(\rho {\overline{u}}_i\right)}{\partial x_i}=0,$$

(1)

$$\frac{\partial \left(\rho {\overline{u}}_i\right)}{\partial t}+\frac{\partial \left(\rho {\overline{u}}_i{\overline{u}}_j\right)}{\partial x_j}=\frac{\partial {\sigma }_{ij}}{\partial x_j}-\frac{\partial \overline{p}}{\partial x_j}-\frac{\partial {\tau }_{ij}}{\partial x_j},$$

(2)

$$\frac{\partial \left(\rho {\overline{h}}_s\right)}{\partial t}+\frac{\partial \left(\rho {\overline{u}}_i{\overline{h}}_s\right)}{\partial x_j}-\frac{\partial \overline{p}}{\partial t}-{\overline{u}}_j\frac{\partial \overline{p}}{\partial x_i}-\frac{\partial }{\partial x_i}\left(\lambda \frac{\partial \overline{T}}{\partial x_i}\right)=-\frac{\partial }{\partial x_j}\left[\rho \left(\overline{u_i{h}_s}-{\overline{u}}_i{\overline{h}}_s\right)\right],$$

(3)

where ${\sigma }_{ij}$ is the stress tensor due to molecular viscosity, which is defined as

$${\sigma }_{ij}=\left[\mu \left(\frac{\partial {\overline{u}}_i}{\partial x_j}+\frac{\partial {\overline{u}}_j}{\partial x_i}\right)\right]-\frac{2}{3}\mu \frac{\partial {\overline{u}}_l}{\partial x_l}{\delta }_{ij},$$

(4)

and ${\tau }_{ij}$ is the subgrid-scale stress tensor, defined using the Boussinesq hypothesis in the form

$${\tau }_{ij}=-2{\mu }_t\left({\overline{S}}_{ij}-\frac{1}{3}{\overline{S}}_{kk}{\delta }_{ij}\right),$$

(5)

were ${\mu }_t$ is the subgrid-scale turbulent viscosity, and ${\overline{S}}_{ij}$ is the strain-rate tensor for the resolved scale, which represents the length scale of turbulence components less than the grid size, in the form

$${\overline{S}}_{ij}=\frac{1}{2}\left(\frac{\partial {\overline{u}}_i}{\partial {\overline{x}}_j}+\frac{\partial {\overline{u}}_j}{\partial {\overline{x}}_i}\right).$$

(6)

For the LES Smagorinsky model, the subgrid-scale turbulent viscosity is defined by

$${\mu }_t=\rho L_s^2\left|\overline{S}\right|,$$

(7)

where $\left|\overline{S}\right|=\sqrt{{\overline{S}}_{ij}{\overline{S}}_{ij}}$, and $L_s$ is the mixing length for the subgrid-scale defined as

$$L_s=\min \left(\kappa d,C_s\Delta \right),$$

(8)

were, $\kappa$ is the von Karman constant, d is the distance to the nearest wall, $C_s$ is the Smagorinsky constant, and $∆$ is the local grid size defined as $∆=V^{1/3}$. In Equation (3), $h_s$ and $\lambda$ are the sensible enthalpy and thermal conductivity, respectively. The subgrid enthalpy flux term on the right side of Equation (3) is approximated using the gradient hypothesis in the form

$$\left[\rho \left(\overline{u_i{h}_s}-{\overline{u}}_i{\overline{h}}_s\right)\right]=-\frac{{\mu }_t}{P{r}_t}{C}_p\frac{\partial \overline{T}}{\partial x_j},$$

(9)

where $C_p$ is the specific heat for the fluid, and $P{r}_t$ is the subgrid Prandtl number. The numerical discretization methods are listed in

Table 1. Discretization schemes of governing equations.

Equations	Discretization Method
Pressure	Second-order
Momentum	Bounded central differencing
Energy	Second-order upwind
Transient formulation	Second-order implicit

. In the normal direction of the wall, the following equation is imposed as the wall boundary for the LES:

$$u_w=0,$$

(10)

where $u_w$ is the fluid velocity in the normal direction to the wall. The pressure far-field condition (also known as the characteristic boundary condition) was applied as the non-reflecting boundary condition for acoustic non-reflection in the domain’s boundary. The pressure far-field boundary condition is based on the introduction of Riemann invariants. Under the assumption of an ideal gas flow, the entropy, pressure, and density are related as follows [21,22]:

$$s_b=\frac{c^2}{\gamma {\rho }^{\gamma -1}},$$

(11)

$${\rho }_b={\left(\frac{c^2}{\gamma s_b}\right)}^{1/\left(\gamma -1\right)},$$

(12)

$$p_b=\frac{{\rho }_b{c}^2}{\gamma },$$

(13)

where $\gamma$ is the specific heat ratio. The LES was numerically realized using the commercial software ANSYS Fluent (Ver. 19.1) [22].

2.2. Wavenumber–Frequency Transform

The wavenumber–frequency diagram of fluctuating surface pressure can be obtained by taking the three-dimensional Fourier transform of the surface pressure in the space–time domains, thereby revealing the distribution of the magnitude of the specific pressure wave component in terms of the wavenumber k and the frequency ω. One of the advantages of using the wavenumber–frequency spectrum over the space–time diagram is that the phase speed of the specific wave component constituting the wave-like surface pressure field in the space–time domain can be determined. The phase speed $v_p=\omega /k$ can be used as a criterion to decompose the surface pressure fluctuation into the incompressible and compressible components. The former is convected at the velocity proportional to the mean flow velocity $U_0$, while the latter propagates at the velocity equal to the vector sum of mean flow velocity and the sound speed $c_0$. The detailed algorithm for the wavenumber–frequency analysis is described below.

The spatial–temporal pressure field on a land vehicle can be characterized using the following correlation function:

$$R\left(\mathit{x};\mathbf{\xi },\tau \right)=\langle p^{\prime }\left(\mathit{x},t\right)·p^{\prime }\left(\mathit{x}+\mathbf{\xi },t+\tau \right)\rangle ,$$

(14)

where $\mathbf{\xi }$ is the distance vector, $\tau$ is the time delay, and $p^{\prime }$ is the perturbed pressure. The correlation function of the spatial–temporal pressure field can be converted to the wavenumber–frequency domain using a Fourier transform. The Fourier transform from the two-dimensional space–time domain to the wavenumber–frequency domain is written as

$$S\left(\mathit{k},\omega \right)={{\displaystyle \int }}_{-\infty }^{\infty }{{\displaystyle \int }}_{-\infty }^{\infty }{{\displaystyle \int }}_{-\infty }^{\infty }R\left(\mathbf{\xi },\tau \right)e^{-i\left(-\mathit{k}\mathbf{\xi }+\omega \tau \right)}d{\xi }_{1}d{\xi }_{2}d\tau .$$

(15)

The power spectral density (PSD) of the wall pressure fluctuations can be estimated using the periodogram method in the following form:

$$S\left(\mathit{k},\omega \right)=\underset{T,L_1,L_2\to \infty }{\lim }\frac{1}{L_1}\frac{1}{L_2}\frac{1}{T} \langle {\widehat{p}}_{L,T}\left(\mathit{k},\omega \right)·{\widehat{p}}_{L,T}^{*}\left(\mathit{k},\omega \right)\rangle ,$$

(16)

where

$${\widehat{p}}_{L,T}\left(\mathit{k},\omega \right)={{\displaystyle \int }}_{-T/2}^{T/2}{{\displaystyle \int }}_{-L_1/2}^{L_1/2}{{\displaystyle \int }}_{-L_2/2}^{L_2/2}{p}^{\prime }\left(\mathit{x},t\right)e^{-i\left(-\mathit{k}\mathit{x}+\omega t\right)}d{x}_{2}d{x}_{1}dt,$$

(17)

and ${\widehat{p}}_{\mathit{L},T}^{*}$ is a complex conjugate of ${\widehat{p}}_{\mathit{L},T}$. In Equation (16), $L_1$, and $L_2$ are the length of the space domain in each direction, and $T$ is the period. Lastly, the three-dimensional discrete Fourier transform equation used to obtain the three-dimensional averaged modified periodogram can be written in the following form [3,4,5]:

$$\begin{array}{c}{S}_{m,n,o}=S\left(m\Delta f,n\Delta k_{x_1},o\Delta k_{x_2}\right)\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}=\frac{\Delta x_1\Delta x_2}{N_t{N}_{x_1}{N}_{x_2}{F}_s}{\left|{\displaystyle \sum _{k=0}^{N_t-1}}{\displaystyle \sum _{l=0}^{N_{x_1}-1}}{\displaystyle \sum _{j=0}^{N_{x_2}-1}}{w}_{klj}{P}_{klj}{e}^{-2i\pi \left(\frac{km}{N_t}-\frac{nl}{N_{x_1}}-\frac{oj}{N_{x_2}}\right)}\right|}^2/\frac{1}{N_t{N}_{x_1}{N}_{x_2}}{\displaystyle \sum _{k=0}^{N_t-1}}{\displaystyle \sum _{l=0}^{N_{x_1}-1}}{\displaystyle \sum _{j=0}^{N_{x_2}-1}}{\left|w_{klj}\right|}^2\end{array}$$

(18)

where $w_{klj}$ is a discretized Hanning window.

In the three-dimensional periodogram, the incompressible and compressible parts can be separated by using the slanted Dirac cone, defined as

$$\omega =c_0\sqrt{k_{x_1}^2+k_{x_2}^2}+k_{x_1}{U}_0,$$

(19)

where $U_0$ is the free stream velocity. The acoustic wavenumber $k_a$, which is used as the criterion to distinguish between the acoustic and convection components, can be obtained by dividing both sides of Equation (19) with the speed of sound, giving

$$k_a=\frac{\omega }{c_0}=\sqrt{k_{x_1}^2+k_{x_2}^2}+k_{x_1}M,$$

(20)

where $M$ is the mean flow Mach number. The components that satisfy the unequal equation of $\left|k\right| \le k_a$ are classified as compressible waves and, otherwise, as incompressible ones.

Figure 1 illustrates the slanted Dirac cone for the case where $U_0=110 \text{km}/\text{h}$ and $c_0=343 \text{m}/\text{s}$. The wave components located inside the slanted Dirac cone can be classified as compressible waves and, otherwise, as incompressible ones.

2.3. Vibro-Acoustics Model

2.3.1. Finite Element Method

The equation of motion of the structure excited by an external force $F$ can be written in the following form [23,24]:

$$\left[M\right]\left\{\ddot{x}\right\}+\left[C\right]\left\{\dot{x}\right\}+\left[K\right]\left\{x\right\}=\left\{F\right\},$$

(21)

where $\ddot{x},\dot{x},\mathrm{and}x$ are the acceleration, velocity, and displacement, respectively, and M, C, and K are the mass, damping, and stiffness matrices of the structure, respectively. The equation of motion shows the relationship between pressure and displacement. An acoustic wave equation can be derived by the equation of state, which indicates the relationship between density and pressure as follows:

$${\nabla }^{2}p-\frac{1}{c^2}\frac{{\partial }^{2}p}{\partial t^2}=0,$$

(22)

$$c^2=\frac{\kappa }{\rho },$$

(23)

where $p$ and $c$ are the sound pressure and the speed of sound in air, and $\rho$ and $\kappa$ are the fluid density and bulk modulus.

The FEM is the most appropriate numerical technique for the dynamic analysis of this kind of vibroacoustic system. The vibroacoustic problems are most commonly described in an Eulerian formulation, in which the fluid is described by a single scalar function, such as the acoustic pressure, and the structural components are described by a displacement vector. Using Galerkin weighted residuals, expressing $p=\left[N_f\right]\left\{p_i\right\}$, and using ${\left[N_f\right]}^T$ as the weighting function to minimize the residuals, we obtain

$${{\displaystyle \int }}_V^{}{N}_f^T\left(\frac{1}{\kappa }\ddot{p}-\frac{1}{{\rho }_0}\nabla ·\nabla p\right)dV={{\displaystyle \int }}_V^{}{\dot{Q}}_s\delta \left(\mathit{x}-{\mathit{x}}_s\right)dV,$$

(24)

where $Q_s$ and $\delta$ are the volume velocity per unit volume and Kronecker’s delta, respectively. Because of the Gauss divergence theorem, Equation (24) can be expressed in the form

$$\begin{gathered} \frac{1}{\kappa }{{\displaystyle \int }}_V^{}{N}_f^T\ddot{p}dV+\frac{1}{{\rho }_0}{{\displaystyle \int }}_V^{}\nabla N_f^T·\nabla pdV-\frac{1}{{\rho }_0}{{\displaystyle \int }}_S^{}{N}_f^T\nabla p·n_{f}dS \\ ={{\displaystyle \int }}_V^{}{N}_f^T{\dot{Q}}_s\delta \left(\mathit{x}-{\mathit{x}}_s\right)dV. \end{gathered}$$

(25)

The boundary conditions are set to

$$\nabla p·n_f=-{\rho }_0{\ddot{u}}_{f_n},$$

(26)

$${\ddot{u}}_{f_n}·n_f=\frac{\nabla p·n_f}{{\rho }_0}=\frac{\dot{p}}{Z},$$

(27)

where $Z$ is the specific acoustic impedance. Using $p=\left[N_f\right]\left\{p_i\right\}$ and substituting Equations (26) and (27) into Equation (25), we obtain

$$\left[M_f\right]\left\{\ddot{p}\right\}+\left[B_f\right]\left\{\dot{p}\right\}+\left[K_f\right]\left\{p\right\}+\left[A_f\right]\left\{{\ddot{u}}_f\right\}=\left\{P_f\right\},$$

(28)

where $\left[M_f\right]$ is the compressibility matrix, $\left[K_f\right]$ is the inverse mass or mobility matrix, $\left[A_f\right]$ is the area matrix, $\left[B_f\right]$ is the impedance matrix, and $\left\{P_f\right\}$ is the acoustic source vector. In the coupled vibroacoustic analysis, the fluid pressure on the structure boundary causes surface traction on the structure as follows:

$$\left\{F_s\right\}={{\displaystyle \int }}_S^{}{N}_s^T\left\{\phi \right\}dS,$$

(29)

where $N_s$ is a shape function for the structure, and substituting $\phi =-p{n}_s$ into Equation (29) gives

$$\left\{F_s\right\}=-{{\displaystyle \int }}_S^{}{N}_s^{T}p\left\{n_s\right\}dS={{\displaystyle \int }}_S^{}{N}_f^T{N}_f\left\{n_s\right\}dS\left\{p\right\}=-\left[A\right]\left\{p\right\}.$$

(30)

The structure equation of motion then becomes

$$\left[M_s\right]\left\{\ddot{u_s}\right\}+\left[B_s\right]\left\{\dot{u_s}\right\}+\left[K_s\right]\left\{u_s\right\}=\left\{P_s\right\}-\left[A\right]\left\{p\right\},$$

(31)

where

$$\left[A\right]={{\displaystyle \int }}_S^{}{N}_s^T{N}_f\left\{n_s\right\}dS\mathrm{or}\left[A\right]=-{{\displaystyle \int }}_S^{}{N}_s^T{N}_f\left\{n_f\right\}dS,$$

(32)

where $N_s$ and $N_f$ are structure and fluid shape functions, respectively, and $n_s=-n_f$.

At the fluid–structure interface, the following relation should be satisfied:

$$n_f·{\ddot{u}}_f+n_s·{\ddot{u}}_s=0.$$

(33)

Equation (24) then becomes

$$\left[M_f\right]\left\{\ddot{p}\right\}+\left[B_f\right]\left\{\dot{p}\right\}+\left[K_f\right]\left\{p\right\}-\left[A^T\right]\left\{{\ddot{u}}_s\right\}=\left\{P_f\right\}.$$

(34)

The combined fluid–structure equations then become

$$\left[\begin{array}{cc}{M}_s& 0\\ -A^T& M_f\end{array}\right]\left\{\begin{array}{c}{\ddot{u}}_s\\ \ddot{p}\end{array}\right\}+\left[\begin{array}{cc}{B}_s& 0\\ 0& B_f\end{array}\right]\left\{\begin{array}{c}{\dot{u}}_s\\ \dot{p}\end{array}\right\}+\left[\begin{array}{cc}{K}_s& 0\\ 0& K_f\end{array}\right]\left\{\begin{array}{c}{u}_s\\ p\end{array}\right\}=\left\{\begin{array}{c}{P}_s\\ P_f\end{array}\right\}.$$

(35)

The FEM was numerically realized using the commercial software Simcenter3D (version 20.0).

2.3.2. Statistical Energy Analysis

Statistical energy analysis (SEA) is a theory proposed by Lyon et al. [25] in the 1970s under the assumption that multiple resonance modes participate in the system response. The SEA is a response analysis method that calculates the average response of the modes existing within an arbitrary frequency band from the difference in the power balance between coupled subsystems. Figure 2 shows the power flow relationship between two coupled subsystems. Each subsystem can receive, dissipate, or exchange power with other subsystems to which it is connected, where ${\text{Π}}_{1, in}$ and ${\text{Π}}_{2, in}$ are the power entering each subsystem, ${\text{Π}}_{1, diss}$ and ${\text{Π}}_{2, diss}$ are the power dissipated in each subsystem, and ${\text{Π}}_{12}$ and ${\text{Π}}_{21}$ are the power moving between subsystems, as shown in Figure 2 [25].

In many practical problems, a structural system consists of a large number of multimodal subsystems, which are physically connected to one or more other subsystems. The entire system is modeled as the sum of the modes of each subsystem that stores power. In SEA, power transfer between subsystems is performed between resonant systems having a resonant frequency in the frequency band of interest and is expressed as follows:

$${\text{Π}}_{ij}=\omega \left({\eta }_{ij}{E}_i-{\eta }_{ji}{E}_j\right),$$

(36)

$$N_i{\eta }_{ij}=N_j{\eta }_{ji},$$

(37)

where $N_i$ is the number of vibration modes existing in the frequency band of interest of the subsystem, $E_i$ is the total energy of $N_i$ vibration modes in the subsystem, $\omega$ is the center frequency, and ${\eta }_{ij}$ is the coupling loss factor. The power dissipated in subsystem $i$ is given by

$${\text{Π}}_{i,diss}=\omega {\eta }_i{E}_i,$$

(38)

where ${\eta }_i$ is the damping loss factor. The power balance equation for the k subsystems can be derived as

$${\text{Π}}_{i,in}={\text{Π}}_{i,diss}+{\displaystyle \sum }_{i\ne j}^k{\text{Π}}_{ij}-{\displaystyle \sum }_{i\ne j}^k{\text{Π}}_{ji}.$$

(39)

Substituting Equations (36) and (38) into Equation (39) gives the following equation:

$${\text{Π}}_{i,in}=\omega {\eta }_i{E}_i+\omega {\displaystyle \sum }_{i\ne j}^k\left({\eta }_{ij}{E}_i-{\eta }_{ji}{E}_j\right).$$

(40)

The second term on the right side of Equation (40) is the net transmission loss energy and can be expressed with the number of resonance modes $N_i$ of the subsystem i and the reciprocal relationship in the form

$$\begin{gathered} {\text{Π}}_{i,in}=\omega {\eta }_i{E}_i+\omega {\displaystyle \sum }_{i\ne j}^k{N}_i\left({\eta }_{ij}\frac{E_i}{N_i}-{\eta }_{ji}\frac{E_j}{N_i}\right) \\ =\omega {\eta }_i{E}_i+\omega {\displaystyle \sum }_{i\ne j}^k{\eta }_{ij}{N}_i\left(\frac{E_i}{N_i}-\frac{E_j}{N_j}\right). \end{gathered}$$

(41)

In Equation (41), $E_i/N_i$ is the average modal energy of the subsystem. The power balance equation for a multiple-degrees-of-freedom vibration system with a large number of natural vibration modes, as looked at above, can be rewritten as a determinant as follows:

$$\left\{\text{Π}\right\}=\omega \left[C\right]\left\{E\right\},$$

(42)

$$\left[\begin{array}{c}\begin{array}{c}\begin{array}{c}\\ {\text{Π}}_1\\ \end{array}\\ {\text{Π}}_2\end{array}\\ \begin{array}{c}\\ \end{array}\\ \begin{array}{c}\begin{array}{c}⋮\\ \begin{array}{c}\\ {\text{Π}}_k\end{array}\end{array}\\ \end{array}\end{array}\right]=\omega \left[\begin{array}{ccc}\left({\eta }_1+{\displaystyle \sum }_{i\ne 1}^k{\eta }_{1i}\right)N_1& -{\eta }_{12}{N}_1& \begin{array}{cc}\begin{array}{cc}\cdots & \end{array}\begin{array}{cc}& -{\eta }_{1k}{N}_1\end{array}& \end{array}\\ -{\eta }_{21}{N}_2& \left({\eta }_2+{\displaystyle \sum }_{i\ne 1}^k{\eta }_{2i}\right)N_2& \begin{array}{cc}\begin{array}{cc}\cdots & \end{array}\begin{array}{cc}& -{\eta }_{2k}{N}_2\end{array}& \end{array}\\ \begin{array}{c}\begin{array}{c}⋮\\ \begin{array}{c}\\ \end{array}\end{array}\\ -{\eta }_{k1}{N}_k\end{array}& \begin{array}{c}\begin{array}{c}⋮\\ \begin{array}{c}\\ \end{array}\end{array}\\ \cdots \end{array}& \begin{array}{cc}\begin{array}{c}\begin{array}{c}⋮\\ \begin{array}{c}\\ \end{array}\end{array}\\ \cdots \end{array}& \begin{array}{c}\begin{array}{c}\\ \begin{array}{c}⋮\\ \end{array}\end{array}\\ \left({\eta }_k+{\displaystyle \sum }_{i\ne 1}^k{\eta }_k\right)N_k\end{array}\end{array}\end{array}\right]\left[\begin{array}{c}\begin{array}{c}\begin{array}{c}\frac{E_1}{N_1}\\ \end{array}\\ \frac{E_2}{N_2}\end{array}\\ \begin{array}{c}\\ ⋮\end{array}\\ \begin{array}{c}\\ \frac{E_2}{N_2}\end{array}\end{array}\right],$$

(43)

where $\left\{\text{Π}\right\}$ and $\left[C\right]$ are the input power vector and coupling loss coefficient matrices, respectively, and $\left\{E\right\}$ is the average modal energy vector of the subsystem that is unknown in the equation.

2.4. Target Model and Details of LES Simulation

The entire computational domain and the target vehicle model are shown in Figure 3. The applied boundary conditions are summarized in

Table 2. Details on applied boundary conditions.

Equations	Discretization Method	Remarks
Inlet boundary	Velocity inlet	30.56 m/s, nonreflecting
Outlet boundary	Pressure outlet	101,325 Pa, nonreflecting
Side and upper boundary	Pressure far field	101,325 Pa, Ma = 0.09
Ground boundary	No-slip wall
HSM wall	No-slip wall

. The dimensions of the entire computation domain were set to be large enough to allow any flow and acoustic disturbances generated around a car body to leave without any reflection from the boundary plane, which is of critical importance for accurate aeroacoustic simulation. The target model was the HSM, which is a simplified vehicle model designed to reproduce complex vortex structures due to the main structure of an actual automobile, such as an A-pillar.

Cho et al. [7] measured the exterior and interior noise due to the external flow of the HSM at the full-scale Hyundai Aero-Acoustic Wind Tunnel (HAWT) of Nam-Yang Technical Center of the Hyundai Motor Company [26]. Additionally, the modal test and the transfer path analysis were performed to measure the frequency response of window vibration and cabin acoustics of HSM. Noise transmission through the body was assumed to occur only through two left- and right-side windows and the front windshield. Figure 4 shows the grid distribution for the entire computational domain at the central vertical cross-sectional plane. The maximum grid size around the HSM surface did not exceed 5 mm, which allowed for resolving an acoustic wave of the maximum target frequency of 5000 Hz with a minimum of 13 points per one wavelength. The tetrahedral cells were used together with 13 layers of prismatic elements near the wall surface, as shown in the zoomed plot in the right-top corner of Figure 3. The time interval ∆t was set to 1.25 × 10⁻⁵ s, producing the unsteady time-history data with a sampling rate of 80,000 Hz. For this case, the car cruising wind was set to be 110 km/h with a yaw angle of 0°. The whole computational domain consisted of about 250 million cells, which were constructed using Fluent Meshing (version: 19.1).

3. Unsteady Flow Analysis and Decomposition of Surface Pressure

3.1. Unsteady Flow Analysis

To ensure the convergence of the unsteady simulation, the aerodynamic drag $C_d$ was used, defined in the following form:

$$C_d=\frac{D}{0.5\rho u^{2}S},$$

(44)

where D is the drag force, $\rho$ is the air density, u is the flow velocity, and S is the reference cross-section area. The air density and the reference area were 1.225 kg/m³ and 0.85 m², respectively. Figure 5 shows the predicted time history of the drag coefficient of HSM. It was found that the oscillation magnitude became slightly lower after t = 2.8 s. On the basis of this observation, the simulation result was determined to be converged after t = 2.8 s, and flow simulation data after t = 2.8 s were used for further analysis.

Figure 6 shows the instantaneous iso-contours of velocity magnitude and static pressure at the central vertical plane. The entire computation domain was seen to be sufficiently large such that the strong wake flow generated behind the HSM was convected and left the computation domain without any spurious reflection from the boundary, which is of critical importance for successful aeroacoustic simulation.

Figure 7 shows the instantaneous iso-surface of the Q-criterion. It is seen that the strong turbulent flow separations from the A-pillar produced complex vortex structures, which in turn characterized the flow field disturbance around the side window. Figure 8 shows the reattachment line of the A-pillar vortex on the left window. The reattachment angle was found to be between 25° and 30°, which is in good agreement with the PIV (particle image velocimetry) measurements [7]. It reveals that fluid flow around the A-pillar had a strong swirling structure and induced the most large-scale coherent surface pressure fields on the side window. Figure 9b–f compare the power spectral density (PSD) of surface pressure between the predicted and measured values at the five sensor positions shown in Figure 9a. The measured data were taken from a previous study [7]. The PSD of simulated surface pressure fluctuations was obtained at a sampling frequency of 80,000 Hz by averaging 21 blocks of 8192 samples (0.1024 s) weighted by a Hanning window, with 80% overlap. It can be seen that there was excellent agreement between the two groups of results at all monitoring points, although underestimations were observed in the high-frequency range higher than ~3.5 kHz, for which the current grid and time resolutions seemed to be insufficient to resolve small vortex eddies responsible for the flow and acoustic disturbances in the corresponding high-frequency band.

3.2. Decomposition of Surface Pressure

The wavenumber–frequency transform was applied to the unsteady surface pressure fields on the side windows and the front windshield of the HSM shown in Figure 2 to separate the compressible part from the incompressible one. The wavenumber–frequency analysis was conducted by applying Equation (14) to the surface pressure fields. The sampling rate $f_s$ and the frequency interval $∆f$ were 80,000 Hz and 20.0 Hz, respectively.

Wavenumber–frequency analysis is available only for a rectangular surface. However, the side window and the front windshield glass of the HSM were not precisely a rectangle. To resolve this issue, the targeted surface for the wavenumber–frequency analysis was extended to become a rectangle. Figure 10 illustrates the extended rectangular surface pressure field, including the side window surface. Furthermore, signal processing information is shown in

Table 3. Parameters of three-dimensional DFT for wall pressure fluctuation signal of side windows.

Spatial–Temporal Domain			Wavenumber–Frequency Domain
$∆x$ (mm)	$∆z$ (mm)	$∆t$ (s)	$∆k_x$ (m−1)	$∆k_z$ (m−1)	$∆f$ (Hz)
2	2	0.0000125	1.1003	1.6523	20.0
$N_x$	$N_x$	$N$	$k_{xmax}$ (m−1)	$k_{zmax}$ (m−1)	$f_{max}$ (Hz)
455	303	4000	500.6	500.6	80,000

.

The power spectral density obtained from the wavenumber–frequency transform is plotted in Figure 11a. The slanted Dirac cone ($-c_0+U_0\le \omega /k\le c_0+U_0$) could be clearly identified with the convection hydrodynamic components. To highlight the characteristics of the power spectral density diagram more clearly, the two-dimensional spectrum at the cross-sectional plane of $k_z=0$ is shown in Figure 11b. Significant compressible components were identified between the characteristic lines for which the slopes corresponded to the phase speeds of $\left|-c_0+U_0\right|=310.56 \mathrm{m}/\mathrm{s}$ and $\left|-c_0+U_0\right|=370.56 \mathrm{m}/\mathrm{s}$, while most of the incompressible components were located on the characteristic wave line with slope $U_c\cong 0.9U_0=27.5 \mathrm{m}/\mathrm{s}$. It is known that the turbulent vorticity within the boundary layer is convected at the speed of $0.9U_0$ or less. These distinctly separated regions manifest that the LES simulation effectively captured the acoustic wave propagation, as well as the aerodynamic noise generation, in the external flow of the HSM. The corresponding power spectral density levels were obtained by integrating the power spectral density over the entire wavenumber ranges in the following form:

$$S\left(\omega \right)=\int \int S\left(\mathit{k}, \omega \right)d{k}_{x_1}{k}_{x_2}.$$

(45)

Figure 11c shows the computed PSD of the fluctuating pressure field. Although the predicted PSD of surface pressure showed good agreement with the measured one in the frequency range below 3.5 kHz, the decomposed PSDs were compared in the frequency range up to 5 kHz to highlight the general characteristics of the relative contribution between the compressible and incompressible pressure fields. It can be seen that the incompressible pressure parts dominated in most of the frequency range, but the relative contribution of compressible ones became comparable in the frequency range higher than 5 kHz.

The separated incompressible and compressible surface pressure fields in the space–time domain were obtained from the inverse Fourier transform of the wavenumber–frequency spectrum. The decomposed incompressible and compressible pressure fields in the rectangular space–time domain were used as input data for the vibroacoustic analysis to predict the wind noise inside the HSM cabin.

4. Vibroacoustic Analysis for Cabin Interior Noise

4.1. Frequency Limit for Calculation of Vibroacoustic Model: Modal Overlap Factor

SEA assumes that the energy balance between each element is achieved by the resonance mode. The mode density is a factor indicating how many resonance modes participate in energy balance in the corresponding frequency section. The SEA equation is derived under the assumption that the value of mode density is significant. Therefore, SEA is suitable for analysis in the high-frequency range. The mode density is a value obtained by dividing the number of resonant modes by the corresponding frequency section and is defined as follows:

$$n\left(f\right)=\frac{dN}{d\omega }.$$

(46)

Estimation of mode density can be obtained experimentally, theoretically, and analytically. In this study, the modal frequencies of the front windshield, side windows, and cabin cavity were calculated using FEM, and the modal density was computed using Equation (19). The mode density of the front windshield, side windows, and cabin cavity is shown in Figure 12.

On the basis of the predicted modal density, the modal overlap factor (MOF) was computed. This parameter gives approximate frequency criteria according to which FEM or SEA is chosen as a vibroacoustic model, and it is defined as follows [27]:

$$MOF=n\left(f\right)·{∆}_n=n\left(f\right)·\frac{\pi }{2}f\eta ,$$

(47)

where $n\left(f\right)$ is the mode density, ${∆}_n$ is the bandwidth of the modal oscillators, $f$ is the center frequency, and $\eta$ is the damping loss factor of the subsystem in the corresponding frequency band. When MOF < 1, a deterministic method such as FEM can be used, whereas, when MOF > 1, statistical methods such as SEA are more qualified [27].

Figure 13 shows the MOF calculated using the modal density of the front windshield, side windows, and cavity.

It can be seen that the frequency criterion of MOF = 1 was around 400 Hz. Therefore, in this study, to predict vehicle interior noise, the FEM was used in the frequency range of less than 400 Hz, and the SEA was used elsewhere.

4.2. Interior Noise Calculation with Vibroacoustic FE Model

The HSM, as shown in Figure 14, consists of three glass window parts (one front windshield and two side window glasses of 4 mm thickness), aluminum panels, and sound absorption pads. The absorption layers were constructed to reproduce the effect of the actual interior material in an actual vehicle and ensure that the principal transmission pass to interior noise is only through the side window and front windshield glasses.

Figure 15 shows the grid configurations for window glasses and acoustic cabin cavity. Determining the mesh size is the first step in finite element modeling. In general, the wavenumber of the resonant $k_{mode}$ is calculated such that the resonant frequency mode, which is more than twice the maximum frequency of interest, can be expressed well. In addition, the maximum convective $k_{c, max}$ and acoustic wave number $k_{a,max}$ for modeling the wall fluctuating pressure in the turbulent boundary layer are defined as follows:

$$k_{c,max}=\frac{{\omega }_{max}}{U_c},$$

(48)

$$k_{a,max}=\frac{{\omega }_{max}}{c_0},$$

(49)

where $U_c$ and $c_0$ are the convection velocity and the speed of sound, respectively. The mesh size $∆x$ of the finite element model is obtained as follows:

$$∆x=\frac{\pi }{k_{max}},$$

(50)

where

$$k_{max}\ge \max \left(2k_{mode}, 2k_{c,max}, 2k_{a,max}\right).$$

(51)

When the frequency range of interest is 0–500 Hz, according to Equation (50), the mesh size should be smaller than 7.5 mm for the plate and 88 mm for the cavity. In this case, the grid types (size) of window glasses and acoustic cavity were CTRIA3 (4 mm) and CTETRA (17 mm), respectively.

The acceleration and force data were measured at the four points on the glass in a frequency range up to 1600 Hz. The glass damping loss factor was estimated using the half-power bandwidth method from the selected resonance peaks. The regression fit was applied to predict the damping loss factor in the frequency range up to 5000 Hz. For the acoustic cabin, the damping loss factor could be determined with the following formula [28]:

$${\eta }_P=\frac{13.8}{T_{60}\omega },$$

(52)

where $T_{60}$ stands for the reverberation time measured inside a cabin at the frequency $\omega$. The sound absorption coefficient of the absorption layers could be estimated using the damping loss factor with the following formula:

$$\alpha ={\eta }_P\frac{8\pi fV}{c_{0}A}.$$

(53)

Then, the inner-wall impedance was calculated using the following equation:

$$Z=Z_0\frac{1+\sqrt{1-\alpha }}{1-\sqrt{1-\alpha }}.$$

(54)

The impedance determined using Equations (52)–(54) was applied as the boundary condition on the cavity wall. Figure 16 shows the applied damping loss factor and inner-wall impedance in the acoustic cavity of HSM.

4.3. Interior Noise Calculation with Vibroacoustic SE Model

4.3.1. SEA Model of HSM

SEA is based on the power balance equations in which the input power to subsystems is either dissipated within the subsystems or transmitted to coupled subsystems. In order to apply SEA, it is necessary to divide the system into appropriate subsystems and evaluate model parameters that determine the dissipated and transmitted power in each subsystem. Once this is achieved, the response variables can be assessed.

Figure 17 shows the SEA subsystem identified for HSM. According to its physical component, the subsystem could be divided into four subsystems, i.e., one acoustic cavity, one front windshield glass, and two side window glasses. The power balance equations could be assembled in the following matrix:

$$\left\{\begin{array}{c}{\text{Π}}_{1,in}\\ {\text{Π}}_{2,in}\\ {\text{Π}}_{3,in}\\ 0\end{array}\right\}=\omega \left[\begin{array}{c}{\eta }_{11}\\ 0\\ 0\\ -{\eta }_{14}\end{array}\begin{array}{c}0\\ {\eta }_{22}+{\eta }_{24}\\ 0\\ -{\eta }_{24}\end{array}\begin{array}{c}0\\ 0\\ {\eta }_{33}+{\eta }_{34}\\ -{\eta }_{34}\end{array}\begin{array}{c}-{\eta }_{41}\\ -{\eta }_{42}\\ -{\eta }_{43}\\ {\eta }_{44}+{\eta }_{41}+{\eta }_{42}+{\eta }_{43}\end{array}\right]\left\{\begin{array}{c}{E}_{1,tot}\\ E_{2,tot}\\ E_{3,tot}\\ E_{4,tot}\end{array}\right\}.$$

(55)

4.3.2. SEA Parameters

The parameters of SEA include the input power, modal parameter, damping loss factors (DLF), and coupling loss factor (CLF). Parameters are factors needed to describe the total energy of each subsystem and the energy exchange process. The theoretical, experimental, and numerical parameters used in this study are described below in detail.

The power supplied to the plate was defined as follows [29],

$$\langle P\rangle ={{\displaystyle \int }}_A^{}\langle pv\rangle dA,$$

(56)

where $p$ is random excitation pressure, $v$ is the normal velocity of the plate, and 〈〉denotes mean value or expectation. Introducing the cross-power spectral density for pressure and velocity at x, Equation (56) reads as follows:

$$\langle P\rangle =\frac{1}{2\pi }{{\displaystyle \int }}_A^{}{{\displaystyle \int }}_{-\infty }^{\infty }{{\displaystyle \int }}_A^{}Y\left(x,x-,\omega \right)S_{p{p}^{\prime }}\left(\omega \right)d{A}^{\prime }d\omega dA,$$

(57)

where $Y\left(x,x-,\omega \right)$ is the admittance between a force at x′ and velocity at x, and $S_{p{p}^{\prime }}\left(\omega \right)$ is the power spectral density of wall pressure at x and x′, respectively. According to Equation (57), the power applied to the plate is expressed by admittance and PSD. This means that, in the case of the distributed pressure field input, the vibration characteristics of the plate should also be considered. Since obtaining admittance for all positions is challenging, the following formula is generally used [30]:

$${\text{Π}}_{p,in}=\frac{A_p\langle p_{TBL}^2\rangle }{{\pi }^{2}f{\rho }_p{h}_p}\left(\frac{U_c}{c_p}\right), U_c>c_p,$$

(58)

$${\mathrm{\Pi }}_{p,in}=\frac{A_p\langle p_{TBL}^2\rangle }{2\pi f{\rho }_p{h}_p}{\left(\frac{U_c}{c_p}\right)}^3\left[\frac{a_1}{6}+a_2{\left(\frac{U_c}{2\pi f{L}_s}\right)}^2\right], U_c<c_p,$$

(59)

where ${\rho }_p{h}_p$ is the surface density of the plate, $A_p$ is the plate area, $\langle p_{TBL}^2\rangle$ is the spatial distribution of the turbulent boundary layer (TBL) pressure, $a_1$ and $a_2$ are constants depending on the details of the turbulent flow and plate mode shapes, and $L_s$ is the distance between the wavenumber spectra for the TBL pressure and the bending mode shape of the plate. For aerodynamically fast modes, the following formula can be used [19]:

$${\text{Π}}_{p,in}={\displaystyle \sum }_{{\omega }_M\in \left[\mathsf{\Omega }-∆,\mathsf{\Omega }+∆\right]}\frac{4{\pi }^3}{M_d}{S}_{pp}\left(k_M,{\omega }_M\right), k_M\ll \frac{\omega }{U_c},$$

(60)

where $M_d$ is the mass per unit area of the plate, and the subscript M indicates the number indices of the plate and the cavity, respectively. Then, damping loss factors need to be determined. The two most common methods for experimentally determining the damping loss factor of a subsystem are the decay rate method and the half-power bandwidth methods. The related detailed descriptions are also provided in Section 4.2. Lastly, the coupling loss factors need to be determined. Energy transmission in subsystem i to j can be expressed in the form

$${\text{Π}}_{ij}=\omega {\eta }_{ij}{E}_i=\omega {\eta }_{ij}{\rho }_i{e}_{i}A{\nu }_i^2.$$

(61)

The acoustic power radiated by the plate into the acoustic cavity is then

$${\text{Π}}_{ij}={\rho }_0{c}_{0}A{\sigma }_i{\nu }_i^2,$$

(62)

where ${\rho }_i, e_i,$ and A are the density, thickness, and area of the front windshield and side window glass, respectively. By rearranging Equations (61) and (62), the coupling loss factor becomes

$${\eta }_{ij}=\frac{{\rho }_0{c}_0{\sigma }_i}{\omega {\rho }_i{e}_i},$$

(63)

where ${\sigma }_i$ is the radiation efficiency of plate. The radiation efficiency is defined as follows:

$$\sigma ={\displaystyle \sum }_i\frac{W_i}{{\rho }_0{c}_0\langle v^2\rangle A},$$

(64)

where $W_i$ is the radiated sound power emitted by the plate, and $\langle v^2\rangle$ is the spatially averaged RMS value of velocity across the surface. Estimation of radiation efficiency can be obtained experimentally, theoretically, and analytically. In this study, the radiated sound power and averaged RMS value of the velocity of the front windshield and side windows were obtained from the FEM. The radiation efficiency was then calculated by applying Equation (64). Figure 18 shows the computed radiation efficiency of the front windshield and side windows.

Conversion from energy to acoustic pressure in the acoustic cabin is needed to compare predicted results with the measured ones in terms of acoustic pressure spectrum. The conversion of a subsystem energy response value to another variable is based on the following fundamental relationship [30]:

$$E=M\langle v^2\rangle .$$

(65)

where M and $\langle v^2\rangle$ are the uniformly distributed mass and the mean-square velocity averaged over the space and the frequency of a subsystem. Therefore, the sound pressure in the cavity can be obtained from the following equation:

$$E=\frac{V\langle p^2\rangle }{{\rho }_0{c}_0},$$

(66)

where V and ${\rho }_0{c}_0$ are the total volume of the subsystem and characteristic impedance of the subsystem (in this case, air), respectively.

5. Numerical Results

Figure 19 summarizes the application process of the fluid–vibroacoustic model presented in this study.

5.1. Validation of FEM and SEA Models

In this section, a vibroacoustic coupling analytical model is established for the validation of the numerical vibroacoustic models described in Section 4. The simplified plate–cavity geometry shown in Figure 20 is considered, featuring a homogeneous rectangular plate with all four sides simply supported. A cuboid cavity shape was used to represent the window–cabin system of the HSM. The modal approach was strongly penalized by the high modal density of the cavity. About 290,000 cavity normal modes were found below 10 kHz, leading to a massive size of the coupled matrix system to be solved. Van Herpe et al. [9] reported that there was no significant difference between the solution obtained using the full coupled system and the two-step approach above 100 Hz. The modal equation governing the plate and the acoustic pressure inside the cavity could be written as follows:

$$m_s\left({\omega }_M^2-{\omega }^2-i{\eta }_M{\omega }_M\omega \right)a_M\left(\omega \right)=-P_M\left(\omega \right),$$

(67)

$${\mathsf{\Lambda }}_N\left({\omega }_N^2-{\omega }^2-i{\eta }_N{\omega }_N\omega \right)b_M\left(\omega \right)={\omega }^2{\rho }_0{c}_0^2{\displaystyle \sum }_{M=1}^{\infty }{a}_M\left(\omega \right)C_{MN}.$$

(68)

Figure 21 compares the numerical results obtained using the current vibroacoustic FE and SEA models with those obtained from the analytic model for the simplified rectangular cavity model shown in Figure 20. The excitation surface pressure obtained from the LES result of HSM was used for the current validation computation. It can be found that there was good agreement between the two results. Note that the comparison for the SEA results is available only in the larger frequency bands, while that for the FEM results is available only in the narrower frequency bands.

5.2. Flow–Vibroacoustic Simulation Result of HSM

Finally, the numerical result obtained by applying the flow–vibroacoustic methods to the HSM was compared with the measured data. Figure 22 compares the calculated interior noise spectrum with the measured one for the HSM using the injected power formula of Equation (60). Over-prediction can be seen in the frequency range between 300 Ha and ~2500 Hz, whereas under-prediction can be seen in the frequency range higher than 2500 Hz. This result follows the trend of the wall pressure power spectrum in Figure 9a. The discrepancies between the experimental and numerical results were considered to be due to the error caused by Equation (60), which could not account for the vibration characteristics of the window. To improve the accuracy of numerical result, the power supplied to a structure needs to be determined considering the vibration characteristics of the side window and front windshield. Vibration displacement of a plate can be written as a general superposition of normal mode as follows [29]:

$$w\left(\mathit{x},\omega \right)={{\displaystyle \sum }}_{M=1}^{\infty }{a}_M\left(\omega \right)w_M\left(\mathit{x}\right).$$

(69)

Modal pressure expresses the degree to which the excitation field p spatially coupled to the $M$-th mode $w_M$ of the plate and can be defined as

$$P_M\left(\omega \right)=\frac{1}{A}{{\displaystyle \int }}_A^{}p\left(\mathit{x},\omega \right)w_M\left(\mathit{x},\omega \right)d\mathit{x}.$$

(70)

Under the assumption that $p\left(\mathit{x},t\right)$ is a random pressure field, the total pressure supplied to the plate is defined as follows:

$$P\left(\omega \right)=\frac{1}{2}Re\left(-{{\displaystyle \int }}_A^{}p\left(\mathit{x},\omega \right)v^{*}\left(\mathit{x},\omega \right)d\mathit{x}\right).$$

(71)

Substituting Equations (69) and (70) into Equation (71), followed by rearranging, we obtain the following equation:

$$P\left(\omega \right)=\frac{A{m}_s}{2}{{\displaystyle \sum }}_{M=1}^{\infty }{\eta }_M{\omega }_M{\left|p_M\right|}^2{\left|Y_M\left(\omega \right)\right|}^2.$$

(72)

Lastly, the power input ${\text{Π}}_{k,in}$ in the frequency band is defined as follows:

$$P\left(\omega \right)=\frac{A{m}_s}{2}{{\displaystyle \sum }}_{M=1}^{\infty }{\eta }_M{\omega }_M{{\displaystyle \int }}_{{\omega }_1}^{{\omega }_2}{\left|p_M\right|}^2{\left|Y_M\left(\omega \right)\right|}^{2}d\omega .$$

(73)

Assuming the admittance has a pronounced peak when $\omega ={\omega }_M$, and that the mode bandwidth ${\eta }_M{\omega }_M$ is small compared to ${\omega }_2-{\omega }_1$, the integral of the power admittance can be read as follows:

$${{\displaystyle \int }}_{{\omega }_1}^{{\omega }_2}{\left|Y_M\left(\omega \right)\right|}^{2}d\omega \approx \frac{\pi }{2m_s^2{\eta }_M{\omega }_M}.$$

(74)

The injected power by the surface pressure fluctuations to the window glass can be written in the form

$$P\left(\omega \right)=\frac{\pi A}{4m_s}{\displaystyle \sum _{M_1}^{M_2}}{\left|P_M\left(\omega \right)\right|}^2,$$

(75)

where $P_M\left(\omega \right)$ is the modal pressure. The input power of Equation (75) includes the modal pressure related to the mode shape of the plate.

In order to reflect the shape of the actual glass window, some data processing was required. The surface pressure was zero-padded in the regions other than the window. The modal shape obtained from the FEM analysis was regenerated in the rectangular domain. The modal pressure was calculated using the surface pressure and the mode shape generated in the rectangular domain. Figure 23 shows the regenerated pressure and mode shape.

Figure 24 compares the calculated interior noise spectrum with the measured one for the HSM using the injected power formula of Equation (56). First, it can be found that the numerical result obtained by combining the total surface pressure field from the high-resolution LES simulation result with the FE–SEA methods showed excellent agreement with the measured data. Figure 19 also shows the numerical results obtained using the individual contribution of hydrodynamic and acoustic surface pressure fields. It can be seen that the interior wind noise was dominated by the acoustic surface pressure field as the frequency increased, although the exterior acoustic energy was less than that of the hydrodynamic one in the corresponding frequency bands. This result reveals that the transmission efficiency of acoustic excitation was much higher than that of hydrodynamic excitation. In particular, a sharp increase in the relative contribution of compressible acoustic pressure was observed to begin in the frequency band around 4 kHz.

The vibration of the plate is closely related to the wavelength of the excitation pressure. The frequency at which the vibration wavelength of the plate becomes the same as the wavelength of the excitation pressure is called the coincidence frequency [31]. At the coincidence frequency, the vibration of a plate excited by sound waves acts as if it is transparent and has zero impedance. Under this condition, the sound waves incident on the plate are freely transmitted into the interior cabin. The coincidence frequency ($f_c$) of the front windshield and side window of the HSM was 3820 Hz and 3620 Hz, respectively, obtained from the maximum value of radiation efficiency. A greater contribution of compressible pressure components observed in the frequency band around 4 kHz can be considered due to this coincidence effect.

6. Conclusions

In this study, the vehicle interior noise caused by exterior flow disturbances was predicted and analyzed numerically. Firstly, the compressible large eddy simulation (LES) techniques were employed with a focus on the accurate prediction of the external flow field, including the acoustic field, around a vehicle running at a speed of 110 km/h. The predicted surface pressure spectrum was compared with the measured spectra, which confirmed the excellent agreement between the two results. Secondly, surface pressure fluctuations on the vehicle’s front windshield and side windows were decomposed into the incompressible and compressible ones using the wavenumber–frequency transform. It was shown that the wavenumber–frequency diagram successfully separated the compressible and incompressible parts. The wavenumber–frequency diagram also confirmed that the present LES method with high resolution in time and space successfully captured the compressible flow field, as well as the incompressible one. Thirdly, the injected power formula reflecting the vibration characteristics of the plate was presented and applied. Lastly, the interior sound pressure levels were predicted by combining the vibroacoustic FE and SEA models for the structure and cabin of the HSM with the surface pressure fields obtained from the LES simulation result. There was again excellent agreement between the predicted and measured sound pressure spectra. In addition, the numerical results confirmed that the contribution of compressible surface pressure field to interior noise level became more significant as the frequency increased. In particular, a sharp increase in the relative contribution of compressible pressure field to the interior sound was found from the frequency band of 3000 Hz, and the physical mechanism causing the higher transmission of acoustic disturbance in this frequency band was found to be related to the coincidence effect.