Interior noise prediction of inter-coach space of high-speed maglev trains based on wavenumber decomposition on aerodynamic excitation

Abstract

Current research on noise and vibration control of high-speed maglev trains pays more attention to far-field noise, while the level of interior noise has a direct impact on the ride comfort and should be placed equal weight on. In this paper, inter-coach space, one of the main pressure fluctuation sources of a specific type of high-speed maglev train with a design speed of 600 km·h⁻¹, is taken as the research object. The turbulent and acoustic components of wall pressure fluctuations (WPF) are separated based on a wavenumber-frequency analysis approach, and then each component is applied as different forms of source input to the vibroacoustic model, namely, finite element method-boundary element method (FEM-BEM) and statistical energy analysis (SEA) for low- and high-frequency ranges respectively, to investigate the contribution of both components to interior acoustic cavity in all frequency range. It can be seen quantitatively from the results that the amplitude of turbulent component is generally much higher than that of the acoustic one, but it can be vice versa when it comes to the interior response. The conclusion drawn in this paper are able to provide guidance for future researches on more targeted interior noise control of high-speed maglev trains.

Keywords

High-speed maglev train interior noise inter-coach space wavenumber-frequency analysis finite element method-boundary element method statistical energy analysis

Introduction

As high-speed EMS maglev system and superconducting technology improve by leaps and bounds, high-speed maglev trains have become the fastest means of ground transportation. However, aerodynamic noise is still one of the bottlenecks restricting its further speed increase. The intensity of aerodynamic noise is proportional to the sixth to eighth power of the running speed, so that aerodynamic noise will become the dominant fraction of running noise as the speed increases further.¹ Compared with other ground vehicles with more mature applications, such as automobiles and high-speed trains, the aerodynamic noise of high-speed maglev trains is investigated relatively late. Due to the absence of pantograph and the installation of bogie skirtboards, the geometry of high-speed maglev trains is more regular than that of traditional wheel-rail trains. Nevertheless, most of the existing numerical investigations into high-speed maglev trains adopt the simplified entire train model including the streamlines of head and tail car,² while few discussions are made on noise genesis mechanism and transmission characteristics at inter-coach space. Researches on far-field aerodynamic noise indicate that the wake region is the main quadrupole noise source, while the head car and mid cars mainly dipole.^3,4 Conclusions are also drawn that inter-coach space is also a disturbance source and generates quadrupole noise to a non-negligible extent.⁵ The Ffowc Williams-Hawkings (FW-H) acoustic analogy method⁶ is generally used for far-field noise prediction, which efficiently separates near field turbulence flow and far field sound propagation located in the same physical medium and thus greatly saving computational resources compared to direct numerical simulation (DNS). The method is based on the assumption that there is no obstacle between the sound source and the receiver, but the mechanism and calculation of interior noise are far more complicated than such a case, which are rarely investigated in the domain of maglev transportation systems.

The noise issues of multiple means of ground transportation share quite a lot of similarities. Unlike the hazards of far-field aerodynamic noise, the level of interior wind noise can have a direct impact on the ride comfort. A large number of experimental studies on automobiles and high-speed trains have shown that leakage noise and transmission noise are the main components of wind noise inside vehicles, while the latter can be further divided into structure-borne noise generated by turbulent wall pressure fluctuation (TWPF, also referred to in some literature as hydrodynamic wall pressure fluctuation) and air-borne noise generated by acoustic wall pressure fluctuation (AWPF).^7,8 Depending on their nature, TWPF and AWPF are also referred to in some literature as the incompressible and compressible components of wall pressure fluctuations (WPF), or pseudo-sound and sound, respectively. The transmission efficiency of TWPF and AWPF is quite different. In general, the power level of AWPF is much lower than that of TWPF in vast frequency ranges, but it can be vice versa when it comes to the interior response,⁹ which results from the different resonant frequency of vorticity waves and acoustic waves with the bending wave of vehicle body materials. At low Mach numbers, wavenumber decomposition (WND) has been proved to be a feasible approach to separate the two fractions, which take advantage of the difference in phase speed between the freestream and the sound wave. Once the two fractions are separated, they can be applied to the vehicle interior noise prediction model as different forms of power input, and their corresponding response in the interior acoustic cavity can then be obtained. And, in terms of the vehicle interior noise prediction, suitable models should be adopted as per the frequency range: finite element method-boundary element method (FEM-BEM) for low frequency, which solves the structural vibration formula and the acoustic radiation integral formula simultaneously¹⁰; finite element-statistical energy analysis (FE-SEA) for medium frequency, which is suitable for the case in which the modal density of substructure or subsystem differs greatly¹¹; and statistical energy analysis (SEA) for high frequency, which divides complex systems into several modal groups, thus modularizing large systems into several independent subsystems that are convenient for analysis in a statistical sense.¹²

In the earlier studies regarding the interior noise of high-speed trains, the influence of sound and pseudo-sound on the interior acoustic cavity was more often than not investigated separately. The source input of the former were derived from field experiment data and SEA parameters were determined through measurement in the acoustic laboratory,¹³ while input of the latter from incompressible simulation^14,15 or track tests¹⁶ without taking into account the air-borne path. In recent years, with the development of computational fluid dynamics (CFD) and computational aeroacoustics (CAA), many attempts have been made to model transmission path and to predict interior noise. Attention has been already paid to the influence of acoustic pressure components on interior sound pressure level (SPL) in several studies regarding automobiles and high-speed trains, and the main areas of concern included front side window and rearview mirror of cars,^8,9,17–20 pantograph platform²¹ and side wall^22,23 of high-speed trains, etc. However, interior SPL prediction of high-speed maglev trains on the basis of different transmission paths, to the best of our knowledge, has not yet been published. High-speed maglev trains differ from traditional wheel-rail trains in terms of geometry, running speed level and running environment. The far-field aerodynamic noise of high-speed maglev trains has already been investigated as an independent topic, while as for interior noise, maglev also deserves special attention since individual problems may occur, for instance, the difficulty of wavenumber filtering in the low frequency range as the Mach number increases, the re-determination of contribution rate of the two components as the running environment changes, etc.

In view of the limitations of existing researches, this paper takes inter-coach space, one of the main pressure fluctuation sources of a specific type of high-speed maglev train with a design speed of 600 km·h⁻¹ as the research object, analyses the wavenumber-frequency domain characteristics of simulated surface pressure fluctuations and then estimates the contribution of each component to the interior SPL. Firstly, the numerical simulation model is established with a detailed description of the viscous model, the geometry, the computational domain as well as boundary conditions, and the simulation results of three-dimensional, unsteady, compressible CFD are displayed, which is mainly reflected in External flow field simulation. Secondly, the principle of wavenumber-frequency analysis is introduced, three-dimensional discrete Fourier transform (3D-DFT) is performed on simulated data in the spatial-temporal domain to obtain the three-dimensional PSD spectrum in the wavenumber-frequency domain, and the two components of WPF are separated through integrating different regions of the PSD, which is mainly reflected in Wavenumber analysis of surface pressure fluctuations. Thirdly, each component is applied as different forms of power input to suitable interior noise prediction models as per the frequency range, and their respective contribution to interior SPL are calculated in all frequency range, which is reflected in Interior noise prediction. The conclusions drawn in this paper may provide guidance for future researches on more targeted noise and vibration control of high-speed maglev train.

External flow field simulation

Target model and details on simulation

In this paper, a full-scale local numerical model of inter-coach space of a specific type of high-speed maglev train is established. In order to reconcile the precision with computational efficiency, necessary simplification of geometry has been made. The views of computational domain and the detailed features of inter-coach space are shown in Figure 1, where the length of a single coach L = 24.5 m is taken as the reference length. From the side view, the rear of coach 1 with a length of l₁ = l_n,1 + l_w = 0.3000 L and the front of coach 2 with l₂ = l_n,2 + l_w = 0.5000 L are connected through a full outer windshield with 2l_w = 0.0335 L, where l_n,1 and l_n,2 are the net lengths of coach 1 and coach 2 included in the local model, respectively. The total length of model is l = l₁ + l₂ = 0.8000 L. From the front view, the maglev train runs on a T-shaped guideway. The distance between the top of the train and the guideway surface is h = h_c + h_sus = 0.1298 L, including the height of the coach h_c = 0.1294 L and the suspension gap h_sus = 0.0004 L. The width w_c = 0.1510 L, while the width of the upper surface of the guideway w_g = 0.0857 L. From the perspective of the whole computational domain, since maglev trains usually run on the elevated line, the train and the guideway are set in the middle of the domain, and the coaches extend to the boundaries in the front as well as in the back, so that the length of the computational domain is equal to the length of the local model, namely, L_D = l = 0.8000 L. Sufficient distances should be left in the four directions of up, down, left, and right to ensure the full development of the outer flow field around the train, with the height of the computational domain set to H_D = 0.8163 L, and the width W_D = 0.8163 L. The cross section ABCD that truncates coach 1 is set as the velocity inlet with the velocity 166.6667 m·s⁻¹ (equal to the running speed of 600 km·h⁻¹), while KLMN that truncates coach 2 the pressure outlet with 1 standard atmospheric pressure. The top, bottom, left and right sides (BLMC, AKND, DCMN, ABLK) are defined as symmetric boundaries so that the normal velocity of which is 0. From a detailed perspective, the outer windshield is made of hard rubber and adopts smooth curve transition. The surface of the coach is defined as fixed boundary with no-slip condition, where the roof wall and the floor are defined as homogeneous thick aluminum plate, and also does the side wall since the side windows has no significant convex or concave structure and their disturbance to the flow field is negligible. The guideway is set as concrete material, and is defined as moving wall with a velocity equal to the inlet velocity so as to simulate the ground effect. The full outer windshield is directly connected with the interior acoustic cavity, regardless of the influences of functional structures of the interior, such as seats, handrails, HVAC equipment, etc. on interior SPL distribution, which is often adopted as a way of simplification in other researches on high-speed trains. Additionally, sound reflections may significantly increase the magnitude of AWPF.²³ As high-speed maglev lines are generally routed in areas where buildings are sparsely spaced, sound reflections from sound barriers and surrounding buildings are not considered in this paper.

Figure 1.

Computational model. (a) Computational domain. (b) Detailed features of inter-coach space.

Then, the commercial pre-processing software ANSYS ICEM CFD 19.0 is used to conduct structural mesh generation on the surface and surroundings of the maglev train. Two refinement zones Ref1, Ref2 are set around the inter-coach space with the dimensions of both are defined in Figure 2. The surface mesh size on the outer windshield and the spatial grid scale in Ref1 are yet to be determined. The spatial grids in Ref2 and outside the refinement zones are set with the maximum size of 10.0 mm, 200 mm, respectively. A boundary layer with 10 layers is set in the near wall area, where the thickness of the first later is 0.01 mm and the stretching ratio is 1.2. The non-dimensional wall distance y⁺ is less than 1, and the total number of the spatial grids outside the refinement zone Ref1 is about 17.7 million.

Figure 2.

Illustration of setting of refinement zones.

Finally, the flow field simulation is conducted with commercial CFD software ANSYS Fluent 19.0. Prior to collecting WPF data on the outer windshield, steady simulation should be carried out to convergence. Then, the steady simulation results are applied to initialize the unsteady simulation. For solving the external flow field, the fluid must be set as compressible since AWPF is a compressible wave. Hence, for the simulation of steady flow field, a density-based implicit solver is adopted, and Realizable k-ε is selected as the viscous model. Then, detached eddy simulation (DES) model with Spalart-Allmaras model for the near wall boundary layer, which is validated by wind tunnel tests as the most suitable near wall model for simulating WPF,¹⁸ is adopted for unsteady flow field calculation. The time step of unsteady simulation is set to 5.0 × 10⁻⁵ s with a maximum of 30 iterations in each time step, and 6.0 × 10³ time steps are calculated in total, in which WPF data are collected in the last 5.0 × 10³ time steps. The main modelling schemes adopted for the CFD simulations are summarized in Table 1. The mathematical expression of the turbulence model is described by Wilcox²⁴.

Table 1.

Main modelling schemes adopted for the CFD simulations.

Time dependency	Steady	Unsteady
Turbulence model	Realizable k-ε	Spalart-Allmaras DES
Solver	Density based	Density based
Flux type	Roe-FDS	Roe-FDS
Gradient discretization	Least squares cell based	Least squares cell based
Flow discretization	(To be determined)	(To be determined)
Turbulent kinetic energy discretization	QUICK
Turbulent dissipation rate discretization	Second-order upwind
Modified turbulent viscosity discretization		Second-order upwind
Transient formulation		Second-order implicit

In order to capture the acoustic waves propagating in the flow field whose amplitudes are significantly lower than the turbulent pressure waves, theoretically, the smaller the maximum spatial grid scale of Ref1 is set, or the higher the order in the discretization procedure is adopted, the more accuracy would be gained in solving for compressible flow field. However, such an improvement will inevitably result in an excessively large number of grids as well as poorer convergence of calculation, leading to the consumption of enormous computational resources. In an effort to achieve the optimal balance between accuracy and cost, three different simulation schemes SchA, SchB and SchC are considered, and the objective is to determine appropriately the aforementioned yet-to-be-determined grid parameters or numerical methods by comparing the values of the independence indicators, the convergence time of steady simulation and the efficiency of unsteady simulation. The different schemes for the independence validation are specified in Table 2. For the grid scales adopted in schemes SchA and SchC, the mesh distribution is shown in Figure 3. Then, the total A-weighted SPL (abbreviated as A-SPL in tables) at sensors V1, V2 and V3 artificially set at coordinates (0.0959 L, 0.3061 L, 0.1429 L), (0.3000 L, 0.3061 L, 0.1429 L) and (0.5041 L, 0.3061 L, 0.1429 L), respectively, are selected as independence indicators. The total A-weighted SPL at each sensor is obtained by integrating the sound source surfaces using the FW-H acoustic analogy method. So as to take the quadrupole effect into account, the interface of Ref1 and Ref2, which is labeled intf1 in Figure 2, is defined as penetrable integration surface, while the train body that is not wrapped in intf1 is defined as rigid sound source surface. In order to save computational resources in the stage of independence validation, the default convergence criteria are adjusted in steady simulation that once the residual of continuity is less than 0.001 and the residual of other physical quantities is less than 0.0033, the calculation is judged to converge, while in unsteady simulation, WPF data collection begins immediately after the steady simulation reaches convergence, and the number of time steps in which WPF data is collected is reduced to 5.0 × 10². The time taken for steady simulation to reach convergence and the average time taken for each time step in unsteady simulation are recorded, then time indices are de-dimensioned by taking the calculation result of SchA as 1.0, and finally the results are shown in Table 3.

Table 2.

Simulation schemes for the grid/numerical method independence validation.

Scheme	Maximum grid/mesh size in Ref1 (mm)	Total number of spatial grids (million)	Flow discretization method
SchA	7.0	97.4	First-order upwind
SchB	6.0	146.3	First-order upwind
SchC	7.0	97.4	Second-order upwind

Figure 3.

Mesh generation with the surface mesh size on the outer windshield and the spatial grid scale in Ref1 controlled within 7.0 mm.

Table 3.

Independence indicators and non-dimensional computing time indices under simulation schemes SchA, SchB and SchC.

Scheme	A-SPL at V1 (dBA)	A-SPL at V2 (dBA)	A-SPL at V3 (dBA)	Steady conv. time	Unsteady time per step
SchA	107.258	115.223	120.841	1.0	1.0
SchB	107.826	115.041	121.640	2.29	11.97
SchC	109.955	118.182	123.712	14.89	9.95

It can be seen from the results that with SchA as the control group, when the maximum grid scale in Ref1 in SchB is adjusted from 7.0 mm to 6.0 mm, the total A-weighted sound pressure at sensors V1, V2 and V3 are +6.76%, −2.07% and +9.64% higher (the minus sign indicates lower) than that in SchA, respectively, while the efficiency of unsteady simulation was found to decrease sharply, which is mainly due to more iterations per time step on average and the longer time spent on each iteration itself. SchC adopts second-order upwind in the flow discretization procedure while the maximum grid scale in Ref1 remains unchanged. Theoretically, higher order numerical method (e.g., second-order upwind, third-order MUSCL and other higher order methods) can avoid dissipating acoustic waves of minimal amplitude than the first-order upwind scheme, as evidence, the total A-weighted sound pressure at sensors V1, V2 and V3 are +27.8%, +43.6%, +26.9% higher than that in SchA. However, the time consumption in steady simulation increases drastically, indicating that the simulation that adopts higher order method converges far more slowly. Under the current hardware configuration constrained by our budget, the total time spent of SchB and SchC when the total number of time steps in unsteady simulation reaches 6.0 × 10³ could be on the order of magnitude of tens of days. For economic and efficiency considerations, SchA is adopted as the scheme for numerical simulation in this paper, and further work is recommended be done to establish the accuracy of the CFD method for capturing these waves.

Simulation results and analysis

In this section, the numerical simulation results of external flow field are presented with the genesis mechanism of aerodynamic noise qualitatively explained. Figure 4 shows a snapshot of velocity magnitudes distribution around the train on an xy plane at z = 0.0600 L. It can be seen from the result that significant change in the velocity magnitude is induced at inter-coach space and affects the flow pattern downstream widely, which can be attributed to flow instability due to the open cavity structure. More specifically, a fluid-acoustic mechanism characterized by recirculation and impingement on the shear layer on the downstream edge of the windshield is formed, thus inducing significant aerodynamic noise.

Figure 4.

Instantaneous iso-contour of velocity magnitude on an xy plane (z = 0.0600 L and t = 0.0755 s).

According to the aerodynamic noise theory, dipole sound sources are mainly caused by WPF and wall shear stress. Figures 5 and 6 show the instantaneous iso-contours of WPF and wall shear stress on the surface of the outer windshield, respectively. Figure 5 shows the distribution of fluctuating pressure magnitudes at three transients t = 0.0755 s, t = 0.1600 s and t = 0.2830 s (corresponding to time steps 1510, 3200 and 5660 respectively), and it can be seen that the positive and negative fluctuating pressure exhibit a characteristic of interval distribution, especially on the downstream edge of the windshield. The positive and negative pressure regions evolve over time and exhibit periodic characteristics, which is consistent with the previous conclusions that inter-coach space is the main dipole source. Figure 6 shows the distribution of wall shear stress at the same transients, and the area with high wall shear stress is mainly distributed on the rear of the downstream edge, which is related to the phenomenon that the shear layer or vortices hit the area.

Figure 5.

Instantaneous iso-contour of fluctuating pressure of windshield surface. (a) t = 0.0755 s. (b) t = 0.1600 s. (c) t = 0.2830 s.

Figure 6.

Instantaneous iso-contour of wall shear stress of windshield surface. (a) t = 0.0755 s. (b) t = 0.1600 s. (c) t = 0.2830 s.

To identify the three-dimensional vortex structures around the train, Figure 7 computes the instantaneous iso-surfaces of the Q-criterion. It can be seen that the vortex structure is mainly distributed near the downstream edge of the windshield, and its extension downstream is relatively limited, which is related to the small scale of inter-coach space and the large streamwise length-to-depth ratio of the full outer windshield. Return to Figure 5, it can be seen that the outline of the vortex structure in contact with the windshield surface is basically consistent with the areas with higher WPF, which further indicates that vortex shedding is the main cause of surface pressure fluctuations.

Figure 7.

Instantaneous iso-surface of Q-criterion (Q = 2 × 10⁵ and t = 0.0755 s).

The above work shows the instantaneous distribution of magnitudes of flow variables from a time domain perspective, while as for the frequency domain, Figure 8 selects three representative monitoring points and shows their respective frequency spectra of PFL. It can be seen that pressure fluctuations exhibit significant low-frequency characteristics. As the frequency increases, PFL on the windshield surface decreases rapidly, and more qualitatively, PFL at the same monitoring point at 5000 Hz decreases by about 17 dB compared to 200 Hz. Figure 9 shows the spectra of pressure fluctuation level (PFL) on the windshield surface at six different frequencies, which is computed through performing FFT on all surface mesh points.

Figure 8.

Spectra of pressure fluctuation level at monitoring points. (a) Location of three monitoring points P1, P2 and P3. (b) Spectra of pressure fluctuation level at P1, P2 and P3.

Figure 9.

Spectra of pressure fluctuation level on windshield surface. (a) f = 200 Hz. (b) f = 500 Hz. (c) f = 1500 Hz. (d) f = 2500 Hz. (e) f = 3500 Hz. (f) f = 5000 Hz.

Wavenumber analysis of surface pressure fluctuations

Wavenumber analysis theory

The unsteady flow field solved by Spalart-Allmaras DES model is a hybrid of turbulent component and acoustic component. The turbulent pressure fluctuations propagate in the convective direction relying on the motion of fluid particles, while the acoustic pressure fluctuations radiate and propagate in all directions relying on the compression and expansion of air. The relationship among the frequency f, the phase speed v_p and the wavelength λ of a pressure wave is as follows:

f = \frac{v_{p}}{λ}

(1)

The definition of wavenumber is the reciprocal of wavelength, as shown in equation (2):

k = \frac{1}{λ} = \frac{f}{v_{p}}

(2)

According to the definition of wavenumber, when two pressure waves propagate at distinct speeds, their wavenumbers at the same frequency are different. The propagation speed of turbulent pressure is determined by the convective velocity, which is equal to the running speed of the train and is much lower than the sound velocity in air, while the propagation speed of acoustic pressure is the vector sum of the sound velocity and the convective velocity, thus having smaller wavenumber than turbulent pressure at the same frequency due to much higher propagation speed. In other words, the phase speed v_p can be used as a criterion to decompose the pressure field into its turbulent component and its acoustic component. Figure 10 shows the frequency-wavenumber relationship curves of turbulent pressure, acoustic pressure, and the bending wave of hard rubber plate vibration at the convective velocity of 166.6667 m·s⁻¹, where the bending wave wavenumber of hard rubber plate with a thickness of d = 5 mm, a Young’s modulus of E = 6 MPa, a density of ρ = 930 kg/m³, and a Poisson’s ration of μ = 0.5 is estimated with the analytic equation for the bending wave phase velocity of an infinite plain plate.¹⁷ It can be seen that as frequency increases, the difference in wavenumber between turbulence pressure and acoustic pressure becomes larger. However, the coincidence frequency with turbulent component f_c = 33.1 kHz is far beyond the range of human hearing, and let alone that with the acoustic one. Unlike the case of either tempered glass or thick aluminum plate, f < f_c holds in all frequency range for the assumed infinite hard rubber plate, making it impossible for forced bending wave to propagate, but the edge effect still enables sound radiation for finite planes, which conforms to the real case of windshield area and is analyzed detailly in this paper.

Figure 10.

Frequency-wavenumber relationships of turbulent pressure, acoustic pressure and bending wave of a 5 mm thick hard rubber plate (u₀ = 166.6667 m·s⁻¹ and c₀ = 340 m·s⁻¹).

In order to obtain the PSD of WPF in the wavenumber-frequency domain, firstly, the spatial-temporal WPF data at inter-coach space collected from numerical simulation can be characterized using the following correlation function:

R (x, y, t; σ_{x}, σ_{y}, τ) = E 〈 p^{'} (x, y, t) \cdot p^{'} (x + σ_{x}, y + σ_{y}, t + τ) 〉

(3)

where R is the correlation function, x, y are the spatial coordinates, σ_x, σ_y are the distance components in the directions of x, y, τ is the time delay, and p’ is the fluctuating pressure. The correlation function of the spatial-temporal WPF data can be converted to the wavenumber-frequency domain using a three-dimensional Fourier transform:

S (k_{x}, k_{y}, f) = \int_{- \infty}^{+ \infty} \int_{- \infty}^{+ \infty} \int_{- \infty}^{+ \infty} R (x, y, t; σ_{x}, σ_{y}, τ) e^{- j (k_{x} σ_{x} + k_{y} σ_{y} + f τ)} d σ_{x} d σ_{y} d τ

(4)

where S( · ) represents the PSD function, k_x, k_y are the wavenumber components in the direction of x, y. The PSD of WPF can be estimated using the periodogram method, as follows:

S (k_{x}, k_{y}, f) = \lim_{L_{x}, L_{y}, T \to \infty} \frac{1}{L_{x} L_{y} T} E 〈 {\hat{p}}_{L, T} (k_{x}, k_{y}, f) \cdot {\hat{p}}_{L, T}^{*} (k_{x}, k_{y}, f) 〉

(5)

where,

{\hat{p}}_{L, T} (k_{x}, k_{y}, f) = \int_{- \frac{T}{2}}^{\frac{T}{2}} \int_{- \frac{L_{y}}{2}}^{\frac{L_{y}}{2}} \int_{- \frac{L_{x}}{2}}^{\frac{L_{x}}{2}} p^{'} (x, y, t) e^{- j (k_{x} x + k_{y} y + f t)} d x d y d t

(6)

In equation (5), ${\hat{p}}_{L, T}^{*}$ is the complex conjugate of ${\hat{p}}_{L, T}$ , and in equation (6), L_x, L_y are the lengths of spatial domain in the directions of x, y, T is the time period. Finally, 3D-DFT is performed to obtain the three-dimensional averaged modified periodogram, which can be expressed in equation (7):

\begin{array}{l} S (m Δ k_{x}, m Δ k_{y}, o Δ f) = \\ \frac{\frac{Δ x Δ y}{N_{x} N_{y} F_{s} N_{t}} {| \sum_{p = 0}^{N_{x} - 1} \sum_{q = 0}^{N_{y} - 1} \sum_{r = 0}^{N_{t} - 1} w_{p q r} p^{'} (p Δ x, q Δ y, r Δ t) e^{- j 2 π (\frac{m p}{N_{x}} + \frac{n q}{N_{y}} + \frac{o r}{N_{t}})} |}^{2}}{\frac{1}{N_{x} N_{y} N_{t}} \sum_{p = 0}^{N_{x} - 1} \sum_{q = 0}^{N_{y} - 1} \sum_{r = 0}^{N_{t} - 1} {| w_{p q r} |}^{2}} \end{array}

(7)

where m, n, o, N_x, N_y, N_t, p, q, r are positive integers. In the wavenumber-frequency domain, Δk_x, Δk_y are the wavenumber intervals in the directions of x, y, Δf is the frequency interval. In the spatial-temporal domain, Δx, Δy are the distance intervals in the directions of x, y, Δt is the time interval, N_x, N_y are the numbers of nodes in the directions of x, y, N_t is the number of sampling points in time domain, and w_pqr is a three-dimensional discretized Hanning window, which is the product of the Hanning window functions in each dimension. Having converted the WPF data to the wavenumber-frequency domain, the wavenumber k = (k_x, k_y) is used as the criterion to distinguish between TWPF and AWPF at a certain frequency, as shown in equations (8) ∼ (10):

\begin{array}{l} S_{i} (o Δ f) = \\ Δ k_{x} Δ k_{y} \sum (m Δ k_{x}, m Δ k_{y}) \in Ω_{i} S (m Δ k_{x}, m Δ k_{y}, o Δ f), i \in {t, a} \end{array}

(8)

Ω_{t} = {(k_{x}, k_{y}) | 0.5 u_{0} k_{x} \leq f \leq 1.2 u_{0} k_{x}}

(9)

Ω_{a} = {(k_{x}, k_{y}) | c_{0} \sqrt{{k_{x}}^{2} + {k_{y}}^{2}} + u_{0} k_{x} \leq f}

(10)

where S_a(oΔf) is the PSD of AWPF, S_t(oΔf) is the PSD of TWPF, and each is obtained from the integral of discrete PSD spectrum performed in different regions of integration Ω_a and Ω_t characterized by distinct range of wavenumbers, which is visually illustrated in Figure 11.

Figure 11.

Illustrative region of integration Ω_a and Ω_t separating AWPF and TWPF in three-dimensional PSD spectrum (u₀ = 166.6667 m·s⁻¹ and c₀ = 340 m·s⁻¹).

Decomposition of surface pressure fields

The WND must be performed on a uniform, two-dimensional grid. However, the train surface at inter-coach space is irregular, so that the CFD grid cannot be applied directly for WND calculation. To solve this problem, the train surface at inter-coach space is divided into 8 quasi-two-dimensional monitoring areas, each contains the corresponding part of outer windshield and its adjacent train surface included in the refinement zone, referred to as LU (Left, upstream), LD (Left, downstream), TLU (Top-left, upstream), TLD (Top-left, downstream), TRU (Top-right, upstream), TRD (Top-right, downstream), RU (Right, upstream) and RD (Right, downstream), as shown in Figure 12. As the pressure fluctuation on side walls and roof is very small compared with that on windshield surface, such division and extension increase the length of WND windows in the streamwise direction without much information loss. Then, rectangular WND windows are set up for each monitoring area with WND grid evenly distributed in the windows, and the excitation data from the CFD grid points are mapped to the WND grid by an interpolation approach.

Figure 12.

Illustration of division of quasi-two-dimensional monitoring areas.

The values of spatial and wavenumber parameters in application of 3D-DFT for each monitoring area are listed in Table 4, while for frequency parameters, Δf = 18.02 Hz and f_max = 10001.3 Hz holds for all monitoring areas.

Table 4.

Parameters of 3D-DFT of the WPF signal.

Monitoring area	Spatial domain (spatial-temporal)				Wavenumber domain (wavenumber-frequency)
Monitoring area	Δx (mm)	N _x	Δy (mm)	N _y	Δk_x (m⁻¹)	k_xmax (m⁻¹)	Δk_y (m⁻¹)	k_ymax (m⁻¹)
LU/RU	7.697	154	7.632	294	0.8547	64.959	0.4487	65.516
LD/RD	7.512	295	7.509	287	0.4528	66.559	0.4656	66.585
TLU/TRU	7.470	161	7.476	218	0.8367	66.935	0.6193	66.880
TLD/TRD	7.361	301	7.353	257	0.4529	67.930	0.5312	67.997

Figure 13 shows the wavenumber-frequency spectra of two monitoring areas RU and TLD at k_y = 0. It can be seen that the turbulent pressure is concentrated near the characteristic line of convective velocity, while the acoustic pressure is mainly distributed inside the slanted cone. The two components can be easily separated based on the bright band.

Figure 13.

Wavenumber-frequency spectra of different monitoring areas at k_y = 0. (a) RU. (b) TLD.

As for RU, Figure 14 further shows the wavenumber-wavenumber spectra at the different frequencies, where the middle region represents acoustic pressure and the rectangular bright band on the right represents turbulent pressure. As the frequency increases, the turbulent pressure region shifts to the right, gradually widens, and the energy weakens, indicating that it has significant low-frequency characteristics. Due to the approximate symmetry of the flow field, the spectra of the other monitoring areas have similar characteristics. Given space limitations, they will no longer be displayed in the main text, and only the calculation results will be used as source input for subsequent interior noise calculations.

Figure 14.

Wavenumber-wavenumber spectra of monitoring area RU at different frequencies. (a) 504.6 Hz. (b) 1495.7 Hz. (c) 2504.8 Hz. (d) 3495.9 Hz.

Through performing integration using equation (8), the spectra of total fluctuating pressure, TWPF and AWPF of 8 monitoring areas are obtained, as shown in Figure 15. Due to the finite grid resolution result from limited computational resources, there is not enough data points in the PSD spectra for TWPF in the low frequency range below 100 Hz, while some of the data points for TWPF may exceed the maximum range of wavenumber in the high frequency range above approximately 5000 Hz. Hence, only the frequency range from 100 Hz to 5000 Hz is taken for analysis and calculation. As shown in Figure 15, both TWPF and AWPF exhibit significant low-frequency characteristics. For all monitoring areas, TWPF is absolutely dominant in the frequency range below 2000 Hz. In the range of 2000 Hz to 5000 Hz, as the frequency increases, AWPF gradually decreases its difference from TWPF and even exceeds TWPF, thus becoming the dominant pressure component in certain monitoring areas. Figure 16 further compares the WPF characteristics in different monitoring areas. It can be seen that the mirror parts on the left and right sides (e.g., LU and RU) exhibit extremely similar spectral characteristics, proving that the pressure excitation has good approximate symmetry, while the spectra of WPF upstream and downstream on the same side (e.g., LU and LD) show slight difference that the upstream side may have higher root mean square (RMS) fluctuating pressure in most frequency bands. This is probably because the selection of downstream monitoring areas itself includes more “non-windshield” fractions. The area of connection between each shell and the acoustic cavity is taken into account in subsequent interior noise calculations.

Figure 15.

Spectra of total WPF, TWPF and AWPF at each monitoring area. (a) LU. (b) LD. (c) TLU. (d) TLD. (e) RU. (f) RD. (g) TRU. (h) TRD.

Figure 16.

Spectra of single type WPF in different monitoring areas. (a) Total WPF. (b) TWPF. (c) AWPF.

Interior noise prediction

Low-frequency interior noise prediction

The FEM-BEM technique is the most appropriate numerical method for the prediction of the interior noise caused by structural vibration in the low frequency range. Using acoustic wave equation and Galerkin weighting function, the fluid equation of motion of the interior acoustic cavity is derived:²⁰

[M_{f}] {\ddot{p}} + [C_{f}] {\dot{p}} + [K_{f}] {p} + ρ {[R_{f s}]}^{T} {{\ddot{U}}_{s}} = {P_{f}}

(11)

where [M_f] is the compressibility matrix, [C_f] is the sound impedance matrix, [K_f] is the sound mobility matrix, ρ[R_fs] is the fluid-structure coupling compressibility matrix and [P_f] is the acoustic source vector. Similarly, the structure equation of motion is expressed as:

[M_{s}] {{\ddot{U}}_{s}} + [C_{s}] {{\dot{U}}_{s}} + [K_{s}] {U_{s}} = {P_{s}} + {P_{f s}}

(12)

where [M_s] is the structural mass matrix, [C_s] is the structural damping matrix, [K_s] is the structural stiffness matrix, [P_s] is the external structural excitation, and [P_fs] is the surface traction on the structure caused by fluid pressure on the structure boundary. [P_fs] = ∫_S{N}^Tp{n}dS, where {N} is the shape function of the element, {n} is the unit normal vector of the interface, S is the area of interface, p = {N}^T{p}. The structure equation then becomes:

[M_{s}] {{\ddot{U}}_{s}} + [C_{s}] {{\dot{U}}_{s}} + [K_{s}] {U_{s}} - [R_{f s}] {p} = {P_{s}}

(13)

where {R_fs} = ∫_S{N}{N}^T{n}dS is the fluid-structure coupling matrix. Then, the fluid-structure coupling equation is obtained:

[\begin{array}{l} [M_{s}] & [0] \\ ρ {[R_{f s}]}^{T} & [M_{f}] \end{array}] {\begin{cases} {\ddot{U}}_{s} \\ \ddot{p} \end{cases}} + [\begin{array}{l} [C_{s}] & [0] \\ [0] & [C_{f}] \end{array}] {\begin{cases} {\dot{U}}_{s} \\ \dot{p} \end{cases}} + [\begin{array}{l} [K_{s}] & - [R_{f s}] \\ [0] & [K_{f}] \end{array}] {\begin{cases} U_{s} \\ p \end{cases}} = {\begin{cases} P_{s} \\ P_{f} \end{cases}}

(14)

By solving equation (14), the low-frequency interior response of the vibroacoustic model can be obtained. TWPF and AWPF can be applied as excitation to the coupling model to simulate the interior sound pressure distribution and the frequency response at specific sensors. For the interior noise prediction in this section and the next section, the only sensor is set at coordinates (0.3000 L, 0, 0.0490 L). This is because there is no identity difference among the personnel in this area (multiple sensors are often set up in research on cab or car side window noise to simulate the different sounds heard by drivers and passengers). According to the vibroacoustic theory, in order to capture structural vibrations up to 500 Hz, the maximum size of the train surface grid should be controlled within 7.5 mm. Therefore, the surface grid scale used in the finite element-boundary element model in this section follows the maximum size of 7.0 mm in the refinement zone Ref1 in Figure 3. The frequency range of low-frequency interior noise prediction is from 100 Hz to 500 Hz. (For constant bandwidth with Δf = 18.02 Hz, the lower limit frequency of the first frequency band is 90.102 Hz, while the upper limit frequency of the last frequency band is 504.5712 Hz).

Before solving the finite element-boundary element model, the structural mode is automatically solved by the intermediate solver integrated into the commercial acoustic software ESI VA One 2021.1. Then, according to the different monitoring areas, the corresponding excitation obtained from WND is loaded onto multiple finite element subsystems. Finally, the low-frequency response induced by two pressure components of each monitoring area as well as the entire inter-coach space model are obtained. The low-frequency interior response of each monitoring area is shown in Figure 17. As can be seen from the figure, although the amplitude of AWPF is much smaller than TWPF in the low frequency range in each monitoring area, its contribution to the low-frequency interior noise is generally higher than that of TWPF. The interior response of AWPF of each side wall monitoring area (LU,LD,RU,RD) ranges from 52.5 dB to 74.5 dB, while that of each roof monitoring area (TLU, TLD, TRU, TRD) ranges from 42.5 dB to 70.7 dB. In contrast, interior SPL induced by TWPF is not as high as that by AWPF, more specifically, about 10 dB to 17 dB lower in most frequency bands in each monitoring area, and the response of TWPF of each roof monitoring area rapidly attenuates with the increase of frequency.

Figure 17.

Spectra of low-frequency interior noise input from different monitoring areas. (a) LU. (b) LD. (c) TLU. (d) TLD. (e) RU. (f) RD. (g) TRU. (h) TRD.

Figure 18 shows the interior response characteristics of two pressure components of the entire inter-coach model, which is equivalent to the superposition of the response of excitations of each monitoring area at the monitoring point. It can be seen from the figure that AWPF plays a dominant role in the low-frequency interior noise, whose interior response ranges from 69.4 dB to 81.0 dB using standard one-third octave, wherein in the frequency band above 160 Hz, its amplitude decreases rapidly with frequency. The interior SPL induced by TWPF ranges from 58.0 dB to 70.1 dB, which has a difference between 8.5 dB and 13.9 dB from that by AWPF in different frequency bands. This further indicates that AWPF is the main contributor to low-frequency interior noise at inter-coach space.

Figure 18.

Spectra of low-frequency interior noise. (a) Constant bandwidth (Δf = 18.02 Hz). (b) Standard one-third octave band.

Considering that the amplitude of TWPF in the external flow field is generally higher than 100 dB, even more drastic transmission loss is found in each frequency band of the low frequency range compared with other similar studies. This is because the coincidence frequencies between the hard rubber plate vibration and either of the two pressure waves are both far beyond the frequency range of human hearing, resulting in the frequency band from 100 Hz to 500 Hz being below both of the two coincidence frequencies of the bending wave of the windshield vibration, while for glass or thick aluminum plate, such frequency band is only lower than the coincidence frequency of acoustic pressure wave but is higher than the coincidence frequency of turbulent pressure wave. Such property of hard rubber may prevent bending wave from propagating and lead to higher transmission loss of the two pressure components. Therefore, for inter-coach space installed with full outer windshield, the structure has a more significant inhibition effect on the transmission of the turbulent pressure.

High-frequency interior noise prediction

To investigate the contribution of two components of WPF to interior noise in the high frequency range, a simplified SEA model is established for inter-coach space. The basic idea of SEA is to divide a complex structural system into multiple subsystems with different vibration modes based on the principles of modal similarity, mode number greater than 5, natural boundary, etc. The subsystem can be a vibrating structure, such as flat shell, cylinder, singly curved shell or doubly curved shell, or an enclosed acoustic space, such as an acoustic cavity. Each subsystem is independent in a statistical sense, and the average modal energy of each subsystem can be obtained by solving the power flow balance equation. The power flow balance relationship in a simple two-subsystem SEA model is shown in Figure 19.

Figure 19.

Power flow in two-subsystem SEA model.

Where P₁, P₂ are the power inputs of subsystem 1 and subsystem 2, E₁, E₂ are the average modal energy, η₁, η₂ are the damping loss factors and η₁₂, η₂₁ are the coupling loss factors between subsystems. According to the law of conservation of energy, the power output of each subsystem is equal to the power input minus the energy dissipated by the subsystem and the power loss between coupling subsystems. The energy balance equation of the two-subsystem SEA model is expressed as:

{\begin{cases} P_{1} = f (η_{1} + η_{12}) E_{1} - f η_{21} \frac{n_{1}}{n_{2}} E_{2} \\ P_{2} = - f η_{12} \frac{n_{2}}{n_{1}} E_{1} + f (η_{2} + η_{21}) E_{2} \end{cases}

(15)

where f is the center frequency within the analysis bandwidth and n₁, n₂ are the modal density of subsystem 1 and subsystem 2, respectively. For the case of a k-subsystem SEA model, the generalized energy balance equation is obtained, as follows:

f [\begin{array}{l} n_{1} (η_{1} + \sum_{i \neq 1} η_{1 i}) & - n_{1} η_{12} & \dots & - n_{1} η_{1 k} \\ - n_{2} η_{21} & n_{2} (η_{2} + \sum_{i \neq 2} η_{2 i}) & ⋮ \\ ⋮ & ⋱ & ⋮ \\ - n_{k} η_{k 1} & \dots & \dots & n_{k} (η_{k} + \sum_{i \neq k} η_{k i}) \end{array}] [\begin{array}{l} \frac{E_{1}}{n_{1}} \\ \frac{E_{2}}{n_{2}} \\ ⋮ \\ \frac{E_{k}}{n_{k}} \end{array}] = [\begin{array}{l} P_{1} \\ P_{2} \\ ⋮ \\ P_{k} \end{array}]

(16)

and the general formula is:

f [N] [L] {[N]}^{- 1} {E} = {P}

(17)

where [N] = diag(n₁, n₂, …, n_k) is the modal density matrix, [L] is the loss factor matrix, {E} is the average modal energy vector of subsystems that is unknown in the equation, and {P} is the power input vector of subsystems. Then, inter-coach space is simplified as a structural system with 44 subsystems including 19 flat shells, 24 singly curved shells and an acoustic cavity. The material properties and dimensions of the SEA shells are set consistent with that of external flow field simulation model except that each full outer windshield is simplified with a singly curved shell due to saddles are not allowed in the fitting of a doubly curved shell. The part below the floor is not directly connected to the acoustic cavity, so it is not considered in SEA modeling. The damping loss factors of all SEA shells are set to be 0.01. The material inside the interior acoustic cavity is air and the damping loss factor is also set to be 0.01. The determination method of subsystem parameters such as coupling loss factor and modal density can be referred to the publication of Crocker and Price²⁵. The two components of the external excitations, TWPF and AWPF, are applied as different forms or power input, where TWPF is loaded as turbulent boundary layer (TBL) while AWPF as diffuse acoustic field (DAF). The shrunk view of the interior noise prediction model which illustrates the division of subsystems and the loading mode of different excitations is shown in Figure 20.

Figure 20.

Shrunk view of interior noise prediction model.

Then, as for loading pressure excitations, the power input of TWPF is calculated by:

P_{t} (x, y, x^{'}, y^{'}, f) = S_{t} (f) e^{[- d_{x} k_{c} | x^{'} - x | - d_{y} k_{c} | y^{'} - y | - j k_{c} \sqrt{{(x^{'} - x)}^{2} + {(y^{'} - y)}^{2}}]}

(18)

and the power input of AWPF:

P_{a} (x, y, x^{'}, y^{'}, f) = S_{a} (f) \frac{\sin [k_{c} \sqrt{{(x^{'} - x)}^{2} + {(y^{'} - y)}^{2}}]}{k_{c} \sqrt{{(x^{'} - x)}^{2} + {(y^{'} - y)}^{2}}}

(19)

where P_t and P_a are the power input of turbulent pressure and acoustic pressure, S_t(f) and S_a(f) are the PSD with respect to f, (x, y) and (x’, y’) are the spatial coordinated of any two points on the panel, d_x and d_y are the spatial attenuation coefficient in the direction of x, y, and k_c is the wavenumber of acoustic wave. By using the generalized energy balance equation, the energy response of each subsystem is solved and is further transformed into corresponding dynamic indices such as plate vibration level and SPL:

E = {\begin{cases} M_{i} 〈 {v_{i}}^{2} 〉 \\ \frac{〈 P^{2} 〉}{ρ_{0} {c_{0}}^{2}} V \end{cases}}, i = 1, 2, \dots, k

(20)

For the ith structural vibration subsystem, M_i is the mass and <v_i²> is the mean square vibration velocity in both time and space, and for the acoustic cavity subsystem, <P²> is the mean square sound pressure, V is the volume, and ρ₀ is the air density.

Similar to the prediction of low-frequency noise, Figures 21 and 22 show and compare the high-frequency interior noise induced by the two pressure components respectively. It can be seen from Figure 21 that the high-frequency interior noise induced by the two pressure components from different monitoring areas decreases with the increase of frequency. In the side wall monitoring areas, the sound pressure response of TWPF and AWPF is generally close in the frequency band below 2000 Hz, and the main difference frequency band is above 2500 Hz, where the amplitude of response of AWPF is about 4 to 10 dB larger than that of TWPF. In the roof monitoring area, the amplitude difference between the two response components in the frequency band below 2000 Hz is more obvious, where the amplitude of response of AWPF is about 5 dB to 20 dB higher than that of TWPF in the one-third octave band between 500 Hz and 2000 Hz, while in the frequency range between 2500 Hz and 5000 Hz, the amplitude of response of AWPF is still about 7 dB to 15 dB higher than that of TWPF in most frequency bands.

Figure 21.

Spectra of high-frequency interior noise input from different monitoring areas (at standard one-third octave band). (a) LU. (b) LD. (c) TLU. (d) TLD. (e) RU. (f) RD. (g) TRU. (h) TRD.

Figure 22.

Characteristics of interior noise in all frequency range (at standard one-third octave band). (a) SPL spectra. (b) Energy contribution.

Figure 22 shows the high-frequency noise spectra induced by the two pressure components of the inter-coach model. It can be seen that the amplitude of response of AWPF is also generally higher than that of TWPF in the high frequency range by a margin less than 9.6 dB in the frequency band below 2000 Hz and a margin ranging from 10.3 dB to 11.4 dB in the frequency band above 2500 Hz. The energy contribution of TWPF to the interior acoustic cavity (viz. converted decibels into absolute values for comparison) is generally lower than 40%, ranging from 16.7% to 35.9% except for the one-third octave band with the center frequency of 800 Hz (49.2%), 1000 Hz (61.0%) and 1250 Hz (46.6%) in spite of its enormous dominance in the external flow field. According to Langley and Shorter²⁶, when the frequency is lower than the coincidence frequency of the structure, the non-resonant modes of the structure are excited. Since these modes can also effectively radiate sound, the main sound transmission path of the structure is dominated by the mass control mode below the coincidence frequency. Although the high frequency range of 500 Hz to 5000 Hz is still below the two coincidence frequencies, the wavenumber range of AWPF is closer to that of bending wave of the windshield than that of TWPF, and the spatial correlation length of the acoustic pressure is larger than that of the turbulent pressure, which makes AWPF have higher energy distribution near the wavenumber of bending wave of the structural vibration, stimulate more modes of the windshield and radiate more noise into the acoustic cavity. This explains why the amplitude of TWPF is generally higher than that of AWPF in the full frequency range, but the interior noise is mainly induced by AWPF rather than TWPF, that is, AWPF has higher transmission efficiency.

The calculation results of SEA are further compared with the low-frequency interior noise spectra calculated by FEM-BEM. At the frequency division point of 500 Hz, the response of TWPF calculated by SEA is 2.94 dB lower than that by FEM-BEM, while the response of AWPF is 1.72 dB higher, and the total interior response is 1.18 dB higher. The error is within the tolerance range, indicating a good alignment between the two calculation models at the same frequency band. In the follow-up work, the FEM-BEM mesh scale can be optimized and more fitting SEA modeling can be adopted to try to minimize this error.

Conclusion

In this paper, inter-coach space of a specific type of high-speed maglev train with a design speed of 600 km·h⁻¹ is selected as the research object. Three-dimensional, compressible Spalart-Allmaras DES model is used to conduct high-precision numerical simulation of the unsteady flow field in this area. Wavenumber decomposition is performed on wall pressure fluctuation (WPF) data, and then both components of WPF are loaded into the corresponding vibroacoustic model as per the frequency range to predict their respective interior response quantitatively. It can be seen from the results that interior noise at inter-coach space is induced mainly by acoustic pressure rather than turbulent pressure, in spite of the fact that turbulent pressure has higher energy in the external flow field. Both components has high transmission loss since the frequency range is below the coincidence frequency of bending wave of structural vibration and resonant mode cannot be excited. Although there are differences in the specification of different types of maglev trains, and inter-coach space is only one of the excitation sources of the interior noise of maglev trains, it is still necessary to attempt interior noise prediction and transmission path decomposition at such a speed level. Also, due to the limited computational resources, the paper has made a certain compromise between accuracy and efficiency in the external flow field simulation, and it is suggested that further work should be done to establish CFD methodologies more adaptable to such problems so as to reduce the dissipation of minimal acoustic waves propagating in the flow field. The research conclusions of this paper can provide guidance for the future investigations into vibration and noise control of high-speed maglev trains, and more targeted active and passive control measures can henceforward be taken based on different transmission paths.

Footnotes

Declaration of conflicting interests

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

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This study was supported by the National 13th Five-Year Science and Technology Support Program of China (2016YFB1200602).

ORCID iD

Rusi Wang

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