**Analysis of Aerodynamic Noise Characteristics of High-Speed Train Pantograph with Di**ff**erent Installation Bases**

## **Yongfang Yao 1,2, Zhenxu Sun 1,\*, Guowei Yang 1,2, Wen Liu <sup>3</sup> and Prasert Prapamonthon 1,4**


Received: 22 April 2019; Accepted: 4 June 2019; Published: 6 June 2019

**Abstract:** The high-speed-train pantograph is a complex structure that consists of different rod-shaped and rectangular surfaces. Flow phenomena around the pantograph are complicated and can cause a large proportion of aerodynamic noise, which is one of the main aerodynamic noise sources of a high-speed train. Therefore, better understanding of aerodynamic noise characteristics is needed. In this study, the large eddy simulation (LES) coupled with the acoustic finite element method (FEM) is applied to analyze aerodynamic noise characteristics of a high-speed train with a pantograph installed on different configurations of the roof base, i.e. flush and sunken surfaces. Numerical results are presented in terms of acoustic pressure spectra and distributions of aerodynamic noise in near-field and far-field regions under up- and down-pantograph as well as flushed and sunken pantograph base conditions. The results show that the pantograph with the sunken base configuration provides better aerodynamic noise performances when compared to that with the flush base configuration. The noise induced by the down-pantograph is higher than that by the up-pantograph under the same condition under the pantograph shape and opening direction selected in this paper. The results also indicate that, in general, the directivity of the noise induced by the down-pantograph with sunken base configuration is slighter than that with the flush configuration. However, for the up-pantograph, the directivity is close to each other in Y-Z or X-Z plane whether it is under flush or sunken roof base condition. However, the sunken installation is still conducive to the noise environment on both sides of the track.

**Keywords:** high-speed train; pantograph; aerodynamic noise; large eddy simulation; acoustic finite element method

## **1. Introduction**

Over the last twenty years, high-speed trains have played a major role in community and urban development. It is well known that, as the speed of a high-speed train increases, aerodynamic problems that can be neglected at low speeds become more serious. For such cases, the problem of aerodynamic noise cannot be avoided and must be addressed. Therefore, better understanding of the aerodynamic noise characteristics is needed. To alleviate the aerodynamic noise problems, the shape of complex parts is optimized and appropriate sound barriers are commonly used. Presently, a large number of high-speed trains can reach an operating speed of about 300–350 km/h. With these speeds, the trains cause serious aerodynamic noise pollution to passengers and surroundings. Zhang [1] indicated that aerodynamic noise becomes a main pollution source when train speed exceeds 250–300 km/h. His results indicated that, with the increase of train speed, the noise increases rapidly and the aerodynamic noise almost increases with the sixth power of the train speed. To date, the aerodynamic noise characteristics of high-speed trains have been investigated numerically and experimentally. Based on numerical approaches, aero-acoustic analysis is applied to simplified and real-oriented high-speed train geometries to evaluate aerodynamic noise. The numerical approaches could be divided into direct and hybrid methods. The direct methods commonly use the direct numerical simulation (DNS), large eddy simulation (LES), or detached eddy simulation (DES) to simulate the flow field and aerodynamic noise simultaneously. In this kind of methods, the grid scale and energy capture requirements differ between the analysis of the flow field and the analysis of the acoustic propagation. It is found that the magnitude of sound pressure is smaller when compared to that of the dynamic pressure in the flow field. The direct methods are required to adopt high-order, low-dissipation, low-dispersion discretization schemes to produce appropriately accurate solutions [2]. Moreover, since the shape of high-speed trains is rather complex, the use of the direct methods can suffer from high computational costs. Consequently, the hybrid methods seem at present more attractive for practical engineering problems. In the hybrid methods, sound sources in the near field and sound propagation in the far field are solved separately. The pressure fluctuations are solved using a high-order technique of LES, DES or are modeled from turbulence statistics obtained from Reynolds Average Navier-Stokes (RANS) simulations. Then, the pressure fluctuations are analyzed by means of acoustic methods to solve the acoustic propagation. Presently, the trend of the research on the aerodynamic noise of high-speed trains is mostly based on the hybrid methods.

On simplified train geometries, it is found that effects of flow separation and vortices from the head, the tail, and the car connections are the main sources of aerodynamic noise. Sun et al. [3] studied the near-field and far-field aerodynamic noise characteristics and comprehensively assessed the noise level of the key parts in a simplified CRH3 high-speed train model using the Non-Linear Acoustic method (NLAS) and the Ffowcs Williams–Hawkings (FW-H) acoustic analogy approach. Their results indicated that the head and the tail are the main noise sources. In addition, the results indicated that the rough areas with cavities or hump faces on the train structure contribute significantly to the aerodynamic noise. Moreover, it was found that the car connection area is also a noise source. Liu et al. [4] used unsteady incompressible flow analysis to obtain the fluctuation pressure on the train surface, and the FW-H method was adopted to predict the noise propagation to the far field. They studied the spectral characteristics of the head surface of a simplified high-speed train, and the aerodynamic noise sources and the aerodynamic noise distribution in the far-field region. Their results revealed that aerodynamic noise could be greatly reduced as long as the shapes of train head and body are optimized. Aerodynamic noise of more realistic high-speed geometries has been investigated by including more complicated components, such as pantographs and bogies. It was observed by Zhu and Jing [5], and Sun et al. [6] that the aerodynamic noise sources come from the pantographs, the bogies, the car head, and the rear of the car. Previously, King et al. [7] showed that the correlation between the logarithm of the aerodynamic noise and the running speed of the pantograph is approximately linear. Takaishi et al. [8,9] used LES and the compact Green's function to simulate the distribution of the dipole noise source on the bogie and the surface of the pantograph. Their results showed that periodic vortices induced by unstable shear layer separation at the leading edge of the bogie section in the flow provide the major part of sound generation [8]. Furthermore, Takaishi et al. [9] indicated that the dipole sound sources around the pantograph make a strong aerodynamic noise source due to the fact that the dipole sound sources are formed strongly in the shear layer close to the model surface. Yoshiki et al. [10] used the Lattice Boltzmann Method (LBM) to calculate the aerodynamic noise of the pantograph, and numerical results gave good agreement with the experimental results obtained by wind tunnel test. Liu [11] used LES to simulate the unsteady incompressible flow on the pantograph, and the far-field aerodynamic noise of pantograph was calculated by Lighthill's acoustic analogy. The characteristics of sound pressure level, frequency spectrum and the relationship between

sound pressure level and speed was investigated. It was found that the sound pressure level increases significantly with the increase of vehicle speed and is approximately linear with the logarithm of vehicle speed. Tan and Xie [12] used LES, the scale adaptive simulation(SAS), the improved delayed detached eddy simulation with shear-stress transport k-ω (IDDES SST k-ω), the delayed detached eddy simulation with shear-stress transport k-ω (DDES SST k-ω), and the delayed detached eddy simulation with realizable k-ε(DDES Realizable k-ε) models to investigate the flow-field structures, the aeroacoustic sources, and the aeroacoustics of pantographs. By means of a hybrid method of NLAS and FW–H acoustic analogy, Yu et al. [13] studied the aerodynamic noise of the pantograph system, specifically to predict the influence of the pantograph covers on noise in the speed range. Besides the methods used to conduct acoustic propagation as mentioned above, there are still other methods for universal acoustic propagation calculations such as the acoustic boundary element method (BEM) and the acoustic finite element method (FEM). CFD transient simulation is used to obtain the fluctuating pressure in the time domain on the boundary of the pantograph, and is transformed into the frequency domain to form the boundary conditions of the acoustic finite element to further solve the far-field noise. By means of the combined CFD/FW-H the acoustic analogy with BEM, aerodynamic noise from a pantograph was predicted by Sun et al. [14] and by Zhang et al. [15]. According to Zhang et al. [15], they used LES with high-order finite difference schemes to analyze the near-field unsteady flow around the pantograph, while the far-field aerodynamic noise was predicted using the CFD/FW-H acoustic analogy. The results of the surface pressure fluctuations were used in BEM to predict aerodynamic noise sources of the pantograph and the far-field sound radiation. The results showed that the aerodynamic noise originates mainly from the top regions of the pantograph rather than from the bottom regions. Also, the results indicated that the noise generated from the pantograph oriented opposite to the direction to the motion is lower than that oriented in the same direction to the motion, by as much as 3.4 dB(A). Sun et al. [14] used DES to analyze the flow field. They also used BEM to predict the aerodynamic noise from a pantograph, giving the spatial and spectral characteristics of the noise around the pantograph. Several previous experiments also showed evidence that the pantograph is an important component that generates high intensity of noise. For example, Kitagawa and Nagakura [16,17] showed that the main road-side noise is the aerodynamic noise produced by the pantograph and bogie. Noger et al. [18] tested a 1/7th scaled train model with and without pantograph in a wind tunnel and showed that the space near the rear vertical face of the cavity is the most complex and turbulent region. This is the origin of the most important noise generation. Their research suggested that the modification of the cavity geometry with passive devices or active control can be an effective method for reducing the radiated noise. Lawson and Barakos [19] indicated that the length/height ratio of the cavity has a greater impact on its internal flow field, which then also affects pantograph noise. However, the spatial distribution and directivity of far-field noise in the experiment was not discussed deeply in their work. Hence, the influence on noise from the cavity needs to be further explored.

Overall, few studies on the effects of the installation base configuration of the pantograph on the aerodynamic noise characteristics have been done so far. In fact, the flow field of a high-speed train can be seriously disturbed by non-flush surfaces of the train body and then complex flow separation and vortex shedding are caused. This disturbance leads to fluctuating surface pressure that becomes a significant aerodynamic noise source, which radiates to far field regions. The mechanism of aero-acoustic propagation and aerodynamic noise physical characteristics caused by complex structures such as the pantograph and the bogies remain challenging and need to be further discussed. The objective of the present work is to numerically investigate the effects of the pantograph with different installation bases of a high-speed train on the aerodynamic and on the acoustic characteristics using LES with the acoustic FEM. The spectral characteristics and directivity of the aerodynamic noise are discussed. This work can provide high-speed train designers and investigators with useful knowledge of the aerodynamic noise characteristics under different configurations of the installation base of the pantograph.

## **2. Computational Models**

The pantograph used in the present study consists of insulators, chassis, upper arm, lower arm rod and double slide, as depicted in Figure 1. The upper arm and lower arm rod can move vertically, and are controlled by the base frame mechanism. On the roof of the train body, the pantograph is installed, either flush or sunken. Namely, the flush surface configuration of the roof is used as the baseline for a comparison with the sunken surface configuration. Scenarios of the up and down pantograph with the different configurations of the pantograph installation on the surface roof are shown in Figure 2.

**Figure 1.** Model of pantograph.

**Figure 2.** Scenarios of pantograph with different configurations of installation base: (**a**) model of down-pantograph with flush installation; (**b**) model of up-pantograph with flush installation; (**c**) model of down-pantograph with sunken installation; and (**d**) model of up-pantograph with sunken installation.

## **3. Computational Methods**

The commercial CFD program STAR-CCM+, which is based on the finite-volume method, is used to simulate the flow field around the pantograph and the installation base. The LES method is used to conduct the dynamic flow characteristics, including the fluctuating pressure on the pantograph and the base. Then, the fluctuating pressure in the time domain is transformed into the frequency domain by the Fast Fourier Transform (FFT) method and the acoustic FEM with commercial program LMS Virtual lab is used to evaluate the acoustic noise of radiation from the pantograph.

## *3.1. Large Eddy Simulation*

According to LES, the large-scale eddies in a fully turbulent flow are computed directly and the influence of the small eddies on the large-scale eddies is modeled. The continuity equation and the Favre averaged compressible Navier–Stokes equations, as expressed mathematically in Equations (1) and (2), respectively, are solved implicitly.

$$\frac{\partial \rho}{\partial t} + \frac{\partial}{\partial \mathbf{x}\_i} (\rho \overline{u}\_i) = 0 \tag{1}$$

$$\frac{\partial}{\partial t}(\rho \overline{u}\_i) + \frac{\partial}{\partial \mathbf{x}\_i}(\rho \overline{u}\_i \overline{u}\_j) = \frac{\partial}{\partial \mathbf{x}\_j}(\sigma\_{i,j}) - \frac{\partial \overline{p}}{\partial \mathbf{x}\_i} - \frac{\partial \tau\_{i,j}}{\partial \mathbf{x}\_j} \tag{2}$$

where σ*i*,*<sup>j</sup>* and τ*i*,*<sup>j</sup>* are defined as Equations (3) and (4), respectively.

$$
\sigma\_{i,j} = \left[ \mu (\frac{\partial \overline{u}\_i}{\partial \mathbf{x}\_j} + \frac{\partial \overline{u}\_j}{\partial \mathbf{x}\_i}) \right] - \frac{2}{3} \mu \frac{\partial \overline{u}\_i}{\partial \mathbf{x}\_j} \delta\_{i,j} \tag{3}
$$

$$
\pi\_{\vec{\imath},\vec{\jmath}} = \rho \overline{u\_{\vec{\imath}} \underline{u}\_{\vec{\jmath}}} - \rho \overline{\overline{\overline{u}}\_{\vec{\imath}}} \overline{u}\_{\vec{\jmath}} \tag{4}
$$

where *ui* is the resolved filtered velocity vector, σi,*<sup>j</sup>* is the stress tensor caused by molecular viscosity, and τ*i*,*<sup>j</sup>* is the sub-grid scale (SGS) stress tensor, representing the diffusive effect of the sub-grid scale eddies on the resolved ones. The Smagorinsky model [20] is used to model the SGS stresses because of its simplicity.

## *3.2. Acoustic Finite Element Method*

The acoustic FEM is used to compute the sound pressure level (SPL). This method can take the reflection effect of the car body structure surface on the noise [21] into consideration. The acoustic FEM procedure starts from calculating the noise propagation from the Hemholtz equation as defined in Equation (5):

$$
\overline{\nabla}^2 \cdot p(\mathbf{x}, y, z) + k^2 p(\mathbf{x}, y, z) = \mathbf{f}(\mathbf{x}, y, z) \tag{5}
$$

where *p*(*x*, *y*, *z*) is the acoustic pressure, *k* = 2π*f*/*c* is the wave-number, and *f* is the frequency. The corresponding wavelength is computed from Equation (6):

$$
\lambda = 2\pi/k = 2\pi c/\omega = \text{c/}f\tag{6}
$$

where c is the free-stream speed of sound. Then, the FFT method is adopted for spectral analysis. The square of the amplitude of pressure wave is expressed in terms of the summation of sine and cosine functions, as shown in Equation (7):

$$A^2(\omega\_k) = 2\left[\frac{1}{N}\sum\_{n=1}^{N-1} (p\_n - \overline{p})\cos[\frac{2\pi nk}{N}]\right]^2 + 2\left[\frac{1}{N}\sum\_{n=1}^{N-1} (p\_n - \overline{p})\sin[\frac{2\pi nk}{N}]\right]^2\tag{7}$$

where *P*<sup>n</sup> is a data set and represents the fluctuating pressure of the *Nth* step; *n* = 0, ... , *N* − 1. Meanwhile, ω<sup>k</sup> = k/*N*δ*t*, where δ*t* is the time step. Finally, the SPL is computed by Equation (8).

$$SPL(\omega\_k) = 10 \log(A^2/P\_{ref}^2) \tag{8}$$

where *<sup>P</sup>*ref is constant and equal to 2 <sup>×</sup> <sup>10</sup>−<sup>5</sup> Pa.

## *3.3. Computational Domain and Boundary Conditions*

The computation domain is depicted in Figure 3. The flow field is tested to confirm that it is large enough to eliminate boundary effects. The computational domain size is 96.5*H* × 45*H* × 20*H*, where *H* is the height of the train and *H* = 3.55 m. The origin of the Cartesian reference system locates at the distance x = 29 m from the train geometry. The inflow velocity is 350 km/h. The Reynolds number, *Re,* is 2.25 <sup>×</sup> 107 based on inflow velocity and the train's height *H.* A three-car train model with the pantograph is used to study the effects of the pantograph with different configurations of the installation base on the aerodynamic noise characteristics. The effects of other complicated components, such as bogies and windshields, are neglected to save computational cost.

**Figure 3.** Computational domain for flow field and aerodynamic noise.

The physical time step of the calculation is 10−<sup>4</sup> s and 20 sub-steps are used within each time step. The total physical calculation time is 0.4 s. The Courant–Friedrichs–Lewy number (CFL = uΔt/Δx) is less than 1 so that the calculation is converged within each time step. Boundary conditions are set as follows: (1) The surface of the train is a fixed wall with non-slip and penetration conditions. (2) Both lateral sides and the top of the computational domain are given as the far field condition with the characteristic line method, and the air flows in the positive x-direction. (3) The ground is set to be a moving wall, and its velocity is the same as the incoming velocity of the air. Figure 4 shows the observer locations used to evaluate the acoustic pressure spectrum and directivity in the near and far field of the pantograph, namely, two positions at y = 5 and 25 m for the spectrum characteristics, and planes y = 0, 10 and 20 m for the distributions of aerodynamic noise, respectively. Furthermore, five positions in the x-direction i.e. x = 0, 5, 10, 15 and 20 m are monitored to investigate the overall sound pressure level (OASPL) caused by the pantograph with the different base configurations. It should be noted that these five positions are 3.5 m above the ground.

**Figure 4.** Locations of aerodynamic noise observers.

#### *3.4. Mesh Strategy*

The STAR-CCM+ mesher is used to generate the computational grids. On the grid arrangement of flow field computation domain, the first layer thickness is 0.05 mm, the geometric grid growth rate is 1.1, and the number of the grid layers close to the train surface is 10 to resolve flow in the boundary layer. The grid size on the pantograph surface and train body surface is 10 mm and 60 mm, respectively, and mesh refinement rear the train body and pantograph is employed to make the computational grid fine enough to capture flow field physics. An isotropic mesh is used in the whole flow field except near wall boundaries. Three sets of grids, i.e., coarse, medium and fine grids, are used to test the grid sensitivity of the simulations. The total number of grid cells in the three grid configurations are 15, 30 and 60 million cells, respectively. Four mesh refinement zones including small, middle, big and wake region are set up, as shown in Figure 5. The mesh size of each refinement zone is listed in Table 1.

**Figure 5.** The distribution of refinement zone.

**Table 1.** The mesh size of the refinement zone.


Figure 6 shows the distribution of time-averaged slipstream velocity along a sampling line at a distance of 2 m from the center of the train, 1.3 m above the ground predicted by the coarse, medium and fine meshes. The non-dimensional results of slipstream velocity *U*non is defined as:

$$dJ\_{\rm non} = \sqrt{(V\_{\rm x} - V\_{\infty})^2 + V\_{\rm y}^2 + V\_{\rm z}^2} / V\_{\infty} \tag{9}$$

where *V*x, *V*<sup>y</sup> and *Vz* are the velocity components along the x-direction, y-direction and z-direction, respectively, as shown in Figure 3.

**Figure 6.** The distribution non-dimensional time-averaged slipstream velocity.

Figure 6 shows that the time-averaged non-dimensional slipstream velocity shows larger difference between the coarse and the fine meshes near the train in Region 2 and in the wake propagation Region 3, indicating that the coarse mesh is inadequate analytical accuracy, while the medium mesh exhibits good consistence with the fine mesh. Thus, the medium grid configuration is adopted to calculate characteristics of flow field, as it gives a satisfactory balance between accuracy and computational costs. Figure 7 shows the medium grids around the train surfaces. Because pantograph noise is the main research content, the area near the pantograph is refined. The total number of the computational cells used is about 32 million.

**Figure 7.** Medium sized computational mesh: (**a**) grid around pantograph; and (**b**) enlarged view of pantograph grid.

## *3.5. Validation of Computational Methods*

Many structures of the pantograph could be treated as cylinders. The acoustic finite element method is first validated by a cylindrical flow calculation, as shown in Figure 8. The cylinder has a diameter of 10 mm, the length of the cylinder is πD, and the radius of the flow field is 15D. The trimmer meshing method is employed, and the first layer thickness is 0.0035 mm with a growth ratio of 1.1 in the wall-normal direction. The value of normalized wall-normal distance y+ is defined as:

$$\mathbf{y}^{+} = \frac{\boldsymbol{u} \cdot \mathbf{y}}{\nu} \tag{10}$$

where y is the distance from the wall, <sup>u</sup><sup>∗</sup> is the friction velocity (u<sup>∗</sup> <sup>=</sup> τw ρ 1/2), <sup>τ</sup><sup>w</sup> is wall shear stress, <sup>ρ</sup> is the air density and ν is the air kinetic viscosity. The values of y+ are below 1 in downstream of the front facing stagnation line on the cylinder surface, which is adequate for the LES model. The cylindrical surface grid size is 0.25 mm. The total number of cells is about 6 million. Figure 8 shows the boundary conditions, namely, the pressure far-field boundary with a uniform inlet velocity of 72 m/s and a gauge pressure of 0 Pa, and the parallel plane (on the two ends of a cylinder) is set as periodic.

**Figure 8.** Circular cylinder model domain and discretization.

The LES method is used to calculate the flow field. The pressure–velocity coupling method is addressed by the Semi-Implicit Method for Pressure Linked Equations (SIMPLE) algorithm. The discrete algebraic equation is solved by the Gauss–Seidel iteration technique. Figure 9 shows the generation, development, and detachment of the leeward vortices downstream of the cylinder. The vortices alternate axially in the downstream direction of the flow, forming a von Karman vortex street. It can be seen that the leeward side vortices are mainly generated when the airflow flowing from the cylindrical wall surface leaves the leeward side wall surface. The generation, development and shedding of the vortices lead to noise generation. The acoustic FEM is used to solve the far-field noise at an inflow Mach number is 0.21. In this study, Kato equation [22] is used to correct the SPL obtained from numerical simulations because the spanwise length of the LES computational domain is limited by the available computational resources. As a result, the simulation spanwise length, Lsim, of the geometry is shorter than the experimental spanwise length, Lexp, reported in Jacob et al. [23]. To compare the predicted SPL with experimental data, the value obtained from numerical simulations must be corrected. The SPL is monitored 185D above directly the cylinder. Figure 10 shows the validations of the aerodynamic results and SPL results of 185D. It can be seen that the pressure coefficient (Cp) and the corresponding Strouhal number (St) of the vortex shedding predicted by the present study agrees with the experimental results and other numerical simulation solutions obtained in [23–26]. Therefore, it is reasonable to use the FEM for the far-field noise characteristic prediction in the following analysis.

**Figure 9.** Vortex shedding on plane at 50% span(s<sup>−</sup>1).

**Figure 10.** Validations of the Cp and SPL results at 185D (where f0. = 1508Hz).

This paper seeks to show that the numerical method is able to model the evolution of the structures in the gap between the solid bodies. A comparison between downstream flow characteristics of the cylinder obtained by measurements and simulation using the same mesh level changes in the cylinder wake as used for the pantograph simulation is performed. As shown in Figure 11, the cylinder has a diameter of 0.05715 m, the separation distance, L, between the cylinders is 3.7D. The length is 3D. The trimmer meshing method is employed, the first layer thickness is 0.001 mm with a growth ratio of 1.2 in the wall-normal direction. The cylindrical surface grid size is 1 mm. The total number of cells is about 8.5 million. The pressure outlet is set as a gauge pressure of 0 Pa, the inlet condition is set as a uniform velocity of 44 m/s, and the other parallel planes are given as symmetry.

**Figure 11.** Geometry and computational domain for flow field of two cylinders in tandem.

Figure 12a shows the generation, development and detachment of the leeward vortices downstream of the two cylinders. The predicted root-mean-square (rms) of the pressure coefficient (Cp) on the cylinder downstream cylinder surfaces is compared with the previous work [27], in Figure 12b. The angle "Theta" is measured from the upstream stagnation point and is positive in the clockwise direction. The predicted results show that this method is suitable to obtain the flow characteristics in the gap and downstream between the solid bodies.

**Figure 12.** (**a**) Instantaneous iso-surface normalized Q-criterion (Q = 100); and (**b**) the root-mean-square (rms) of the perturbations in Cp.

## **4. Results and Discussion**

## *4.1. Flow Characteristics of Pantograph Area*

In this section, the flow characteristics around the pantograph and the installation bases are discussed. As shown in Figures 13 and 14, the vorticity and the instantaneous iso-surface vorticity as shown by the normalized Q-criterion, and the velocity streamlines around the pantograph are presented, respectively. In Figure 13, the flow separates over the pantograph surface. A series of vortices detaches from the leeward side of each rod and they interact with each other downstream. The hairpin vortices of different size are observed. In addition to the common characteristics of the flow field mentioned above, Figure 13a,b also shows that the flow field characteristics of the pantograph are different due to the opening directions of the pantograph. For the down-pantograph with flush installation base, the incoming flow first impacts on the corner of the upper and lower arm rods, and then strikes the rear of the carbon skateboard and the base in the downstream area. The turbulent upstream flow field hits downstream complex structure like the carbon skateboard and base again, which makes the flow field in the wake region more complex. Because the contact area between the pantograph and the incoming flow is concentrated, the eddy size under the larger contact area is also larger. For the up-pantograph, the distance between different structures on the pantograph is larger, and the inflow flows directly to the downstream area after impact with the carbon skateboard. Compared with the down-pantograph, the rear eddy scale is smaller. For Figure 13c, when the pantograph is placed in the cavity, the flow field above the cavity shielding part is almost the same as when it is installed on the flush surface. However, for the case that the pantograph is installed in the sunken base, the flow field characteristics have something in common with the general flow field in a cavity, as previously studied in [28–30]. However, the presence of the pantograph affects the pressure fluctuation on the cavity surface, and then it further affects the noise induced by the pantograph. For a pantograph that is installed in the sunken cavity, the base and insulator are basically located in the cavity, avoiding direct collision with the incoming flow. Generally, the flow past a cavity exhibits a strong relationship with the cavity length (L) to height (D) ratio as studied by Lawson and Barakos [19]. For the present study, the ratio of L/D of the cavity with pantograph is about 4.7. Therefore, according to the criterion of Lawson and Barakos [19], it is considered as the open cavity. In addition, the pressure field in the cavity is coupled with the shear layer shed from the cavity upstream edge, and the pressure fluctuation intensity in the cavity depends on the characteristics of the shear layer and on the evolution of momentum and vorticity during the impact of the shed shear layer with the cavity rear bulkhead as studied by Ouyang et al. [31]. Because the presence of the pantograph, the typical open cavity flow field structure is destroyed. As shown in Figure 14, one can observe that the air-flow acts on the pantograph first, and then vortices shed by the pantograph impinge on the surface downstream. A part of the air flow returns to the front of the cavity and continues to act on the pantograph. This differs from an empty cavity flow in which the shear layer shed at the cavity upstream edge impacts the cavity downstream edge, unobstructed. The pressure wave (feedback pressure wave) propagating upstream after acting on the surface of the cavity collides with the pressure wave reflected from the front wall when it arrives upstream at the last time. The air circulation inside and outside the cavity exchanges, resulting in pressure fluctuation in the cavity. The pressure fluctuation on the base and insulator surface even as well as the overall pressure fluctuation on the surface of the pantograph is affected by the flow in the cavity.

**Figure 13.** Instantaneous iso-surface normalized Q-criterion (Q = 100/s2) and vorticity magnitude(1/s) on: (**a**) down-pantograph with flush base; (**b**) up-pantograph with flush base; (**c**) down-pantograph with sunken base; and (**d**) up-pantograph with sunken base.

**Figure 14.** Streamlines superimposed on color iso-levels of the time-averaged velocity vector magnitude: (**a**) down-pantograph with sunken base; and (**b**) up-pantograph with sunken base.

## *4.2. Aerodynamic Noise Characteristics*

## 4.2.1. Aerodynamic Noise Source

In this section, sources of aerodynamic noise are discussed as it is expected that the flow field of the different base configurations of the pantograph may have significant effects on noise spectrum characteristics and on the noise directivity, which could lead to a change in the perceived noise along the train track. Therefore, the aerodynamic pressure amplitude distributions on the pantograph at 200 Hz and 1000 Hz are compared. As shown in Figure 15, it can be observed that along the direction of the flow velocity, the aerodynamic pressure amplitude in the front regions of the pantograph is lower than that in the rear regions of the pantograph. This may be explained by the fact that the front parts always collide with the incoming flow sharply, forming a windward stagnation area. As a result, the fluctuating pressure is relatively small. Flow with higher levels of vorticity, as shown in Figure 13, passes over the rear parts of the pantograph. As a result, the amplitude of aerodynamic pressure fluctuation in the rear region is larger. Meanwhile, the amplitude of the aerodynamic pressure decreases when the frequency increases from 200 Hz to 1000 Hz. Where the flush configuration is used under the down-pantograph condition, the maximum aerodynamic pressure fluctuation at 1000 Hz is smaller than that at 200 Hz by as much as 18 dB. The phenomenon that the maximum aerodynamic pressure fluctuation decreases with the increasing frequency also can be observed by Zhang et al. [15]. At the low frequency, the SPL distribution of the aerodynamic pressure on the surface of the pantograph is non-uniform. The largest amplitude aerodynamic pressure mainly features on the supporting slide, the rotating shaft, the lower arm, and the leeward surface of the installation base. At 1000 Hz, the aerodynamic pressure amplitude distribution is more uniform than that at 200 Hz. It seems that, although the aerodynamic pressure amplitude distribution over the pantograph surfaces are different at these two frequencies, the largest amplitudes are always located on the downstream base frame, the connection between the upper and lower arms and the downstream position of the double skateboard bow. These aerodynamic pressure fluctuations act as dipole noise sources.

200 Hz

**Figure 15.** *Cont.*

**Figure 15.** Normalized aerodynamic pressure fluctuation amplitude at 200 and 1000 Hz on: (**a**) down-pantograph with flush base; (**b**) up-pantograph with flush base; (**c**) down-pantograph with sunken base; and (**d**) up-pantograph with sunken base.

#### 4.2.2. Aerodynamic Noise Radiation and Attenuation Characteristics

To study aerodynamic noise radiation and attenuation characteristics induced by the up- and down-pantograph with the different base configurations, the contours of spatial distribution in near-field and far-field noise at three planes in the y-direction i.e. y = 0 (near-field), 10 and 20 m (in far-field) are compared, as depicted in Figure 16. In this analysis, the acoustic finite element method is directly used to deal with the whole field noise, including near-field y = 0 plane noise, without considering the effect of quadrupole noise and corresponding convection effect. The main reason is that Mach number of the incoming flow is only 0.278 in this calculation. As it is known that the ratio of quadrupole noise to dipole noise is proportional to the square of Mach number [32], the influence of the quadrupole noise at this speed is relatively small, and it is reasonable to ignore it in the present study. As the pantograph is the noise source, the sound pressure decreases with the y distance. The contours also indicate that the regions with higher sound levels for both frequencies are in the vicinity of the pantograph. These distributions are more uniform at the high frequency. It is shown that, as frequency increases, the sound pressure amplitude decreases.

**Figure 16.** Spatial distribution of near-field (Y = 0 m) and far-field (Y = 10 m, Y = 20 m) noise at 200 Hz and 1000 Hz on: (**a**) down-pantograph with flush base; (**b**) up-pantograph with flush base; (**c**) down-pantograph with sunken base; and (**d**) up-pantograph with sunken base.

To obtain the noise characteristics in the far field, two positions in the y-direction, i.e. at y = 5 m and y = 25 m from the center line of the train, as shown previously in Figure 4, are evaluated, as plotted in Figure 17. Some phenomena are observed. Firstly, the noise amplitude at y = 25 m is lower than that at y = 5 m at all frequencies. Next, it is shown that the noise caused by the down-pantograph is higher than that by the up-pantograph. Finally, the noise from the pantograph with the flush base configuration is of higher amplitude than that with the sunken base configuration irrespective of whether the pantograph is under the up or down condition. These phenomena could be explained by the fact that, for the sunken configuration, the vortices generated in the rear of the insulator and the base move inside the cavity and the movement to the surroundings is blocked by the cavity surface. This may explain the predicted weakening of the far-field noise. Additionally, one can see that, for frequencies below 200 Hz, the noise caused by the sunken type configuration is almost the same as that from the flush. However, at higher frequencies, the noise caused by the sunken configuration of the pantograph is lower than that by the flush configuration in general. This is because the large vortices produced by the sunken insulator are blocked by the cavity, therefore their streamwise growth and spread are constrained. As a result, large eddies are not easily broken into small ones.

**Figure 17.** Noise predictions at the far field monitoring points. (**a**) Noise of down-pantograph with flush base; (**b**) Noise of up-pantograph with flush base; (**c**) Noise of down-pantograph with sunken base; and (**d**) Noise of up-pantograph with sunken base.

To further study the trend of the overall sound pressure levels (OASPL) at different distances along x (parallel to the train) and y (normal to the train) directions, Figure 18a,b shows the OASPL in the far field along the x and y directions, respectively. Figure 18a indicates that the OASPL is inversely proportional to the distance of sound source for all cases. To predict the OASPL in the region from the pantograph to the far field, it is found that the logarithm function, OASPL = OASPL0 − b × ln(x + c), is fitted appropriately using the values of the OASPL under the relation between the OASPL and the distance in the range of 0 ≤ y ≤ 30 m. The four correlations obtained by the fitted curves are mathematically expressed as follows: (1) OASPL = 107.3 − 8.9 × ln(x − 0.50) for the down-pantograph with the flush base; (2) OASPL = 105.5 − 8.8 × ln(x − 0.03) for the up-pantograph with the flush base; (3) OASPL = 102.9 − 9.1 × ln(x − 0.12) for the down-pantograph with the sunken base; and (4) OASPL = 97.1 − 7.9 × ln(x − 0.46) for the up-pantograph with the sunken base. Another observation is that the differences between the OASPL obtained by the up- and down-pantograph with the flush configuration base at each position are about 1–2 dB, whereas the differences between the OASPL obtained by the up- and down-pantograph with the sunken base at each position are about 4–5 dB. Besides, under the same configuration base, the OASPL obtained by the down-pantograph is higher

than that by the up-pantograph. To evaluate the OASPL caused by the pantograph in the far-field region, it is acceptable if the distance from the sound source is more than 25 m, the sound source is assumed as a point source [13]. Therefore, the sound pressure propagation can be considered as the spherical surface from the point source center and the attenuation value ΔL of the OASPL from the point source in the far-field regions is computed by the equation: ΔL = 20log(r/r0). This can be used to the predict the geometric attenuation of noise. Figure 18b shows the OASPL along the x direction at y = 25 m. The down-pantograph always produces higher noise than the up-pantograph for both configuration bases. Besides, Figure 18a,b indicates that the OASPL in the far field along the x and y directions are always found that for all x and y positions the down-pantograph with the flush base provides the highest OASPL. The lowest OASPL is obtained by the up-pantograph with the sunken base. In addition, the OASPL caused by the up-pantograph with the flush base is higher than that by the down-pantograph with the sunken base.

**Figure 18.** Comparison of OASPL at different distances of pantographs with different configuration bases: (**a**) with pantograph as reference position, OASPL at different vertical distances from center line of track in horizontal plane; and (**b**) OASPL of up- and down-pantograph with different configuration bases at different monitoring points along moving direction of train.

## 4.2.3. Aerodynamic Noise Directivity

In this section, the aerodynamic noise directivity around the pantograph with the different configurations is presented. According to the ISO3095-2013 standard [33], the far-field noise receivers should be located 7.5 or 25 m away from the center line of the track and 1.2 or 3.5 m above the ground. The present study investigates the far-field noise characteristics around the pantograph with a distance of 7.5 m in the radial direction and 3.5 m above the ground. The OASPL in the X-Y, Y-Z and X-Z planes is discussed, as shown in Figure 19, and the OASPL is considered at every five degrees. It is found that in all the planes the noise directivity induced by the sunken pantograph is slightly smaller than the flush pantograph in general. Only at some angles, the noise induced by the sunken pantograph is equal to or larger than that by the flush pantograph. Besides, some interesting phenomena are observed in each plane. First, in the X-Y plane, when the pantograph is used with the flush surface, a relatively high OASPL is found in the rear arc of the pantograph (−5◦ ≤ θ ≤ 5◦), and along the sideline over the ranges 85◦ ≤ θ ≤ 125◦ and 235◦ ≤ θ ≤ 275◦. When the pantograph is installed in the sunken surface, the rear arc still has the higher OASPL. However, the high OASPL values are also predicted at other angles. In the down-pantograph, a large part of the structure is located inside the cavity. The corresponding amplitude of the four corners of the cavity is slightly higher than that of other positions. This may be because of the effects of the sunken configuration. Specifically, during the process of noise propagation, the sound wave can reflect within the cavity of the sunken surface, four corners can easily become the convergence area of reflected sound waves. One can see that the directivity of noise is not completely symmetrical the X-Y plane. Similar instances of incomplete symmetry are also reported in the literature [13]. The main reason is that the far-field noise monitors are usually placed at

every 5–10 degrees around the pantograph in different planes. This may omit part of the amplitude between the monitors, resulting a certain impact on the results. In addition, in the calculation of the sound field, the frequency resolution has a slight impact on the results. However, further detailed analysis of the up- and down-pantograph in the cavity has shown that there is almost no difference when the frequency resolution is increased from 10 Hz to 5 Hz, namely, the overall noise difference is less than 1 dB. In the Y-Z plane, the noise distributions obtained by the two installation bases are nearly symmetric and the higher sound level area is found in in the range of 60◦ ≤ θ ≤ 120◦. It also shows that, in this range of theta with the pantograph under the down configuration, the OASPL obtained by the flush installation base is higher than that by the sunken base configuration. However, under the up-pantograph condition, the OASPL values obtained by the two configuration bases are close to each other. In the X-Z plane, the down-pantograph with the flush base produces higher noise than the sunken base in all directions. For the up-pantograph, the noise amplitudes of the flush and sunken bases are very close to each other in general. Nonetheless, it is seen that at some angles, the noise induced by the sunken base is higher than the flush base. In both Y-Z and X-Z plane, for the up-pantograph with the flush base or sunken base, it can be seen that the noise amplitudes are very close to each other in general, especially in the Z direction directly above the body. This may be due to the effect of sound wave reflection from the cavity floor and that the up-pantograph itself is a large source of noise in this direction.

**Figure 19.** Noise directivity induced by pantograph in different planes (the left side figures show the down condition and the right side figures show the up condition): (**a**) directivity of X-Y plane; (**b**) directivity of Y-Z plane; and (**c**) directivity of X-Z plane.

## **5. Conclusions**

This work presents the numerical study of the aerodynamic noise characteristics induced by a pantograph over a simplified train geometry with the different configurations of the installation base using the large eddy simulation (LES) with finite element method (FEM). The numerical results are carried out in terms of the flow field around the pantograph and its installation base, the spatial distribution and spectrum characteristics, and the noise directivity in near and far fields under the up and down conditions at the train speed of 350 km/h. Through the discussion, the following interesting phenomena are drawn:

(1) The complex shape of the pantograph is the main reason for its induced aerodynamic noise. A series of vortices detaches from the leeward side of each rod and they interact with each other and with downstream solid structures. The aerodynamic pressure fluctuation is generated on the rods induces aerodynamic noise. The main aerodynamic noise sources are located on the downstream base frame and on the double skateboard bow, at the connection between the upper and lower arms. The aerodynamic noise sources over the pantograph surfaces are different at 200 Hz and at 1 kHz.

(2) For the spatial distributions, along the direction of the flow velocity, the aerodynamic pressure fluctuation amplitude in the front regions of the pantograph is lower than that in the rear regions of the pantograph. These distributions are more uniform at 1 kHz than at 200 Hz. In addition, the amplitude of the noise is lower at the higher frequency.

(3) In the far-field regions along the x and y directions, for all x and y positions under the pantograph shape and opening direction selected in this paper, the down-pantograph with the flush configuration provides the highest OASPL. The up-pantograph with the sunken configuration produces the lowest OASPL. In addition, the OASPL caused by the up-pantograph with the flush base is higher than that by the down-pantograph with the sunken base. Thus, the pantograph with sunken configuration is a better choice in a practical engineering application.

(4) On the directivity, the noise induced by the pantograph with the sunken surface is of lower amplitude than that with the flush surface, especially in the down-pantograph. For up-pantograph, the amplitude of the noise from the sunken surface configuration is close to that from the flush surface configuration in Y-Z plane and in the X-Z plane. In the X-Z plane, the results even show noise amplitude from the sunken surface configuration is predicted to be higher than that from the flush configurations at some angles in some angles. However, this still does not affect the fact that the sunken installation is conducive to a quieter noise environment on both sides of the road.

It should be noted that effects of noise transmission to the exterior and interior of high-speed train need to be further studied, especially in the case of the pantograph with the sunken configuration. The present study considers only the aerodynamic noise generated by the wall fluctuating pressure of the pantograph in the exterior noise transmission to the environment. In fact, there are strong vortices in the regions near the pantograph and spatial distribution of near-field, such as Y = 0 m, may obtain a more accurate result. These are the quadrupole noise sources that contribute to aerodynamic noise. Since the operating speed is not too high (Ma = 0.278), the proportion of noise generated by the quadrupole source, which scales by u8 [34], is smaller than that from dipole sources, which scales as u<sup>6</sup> [35], where u is the train velocity. Thus, the induced noise by the quadrupole source is not included in this study.

**Author Contributions:** Conceptualization, Z.S. and G.Y.; Data curation, Y.Y.; Formal analysis, Y.Y.; Funding acquisition, Z.S. and G.Y.; Investigation, Y.Y. and Z.S.; Methodology, Y.Y. and W.L.; Project administration, Z.S. and G.Y.; Resources, Z.S. and G.Y.; Software, Y.Y. and W.L.; Supervision, Z.S. and G.Y.; Validation, Y.Y. and Z.S.; Visualization, Y.Y.; Writing—original draft, Y.Y.; and Writing—review and editing, Y.Y., P.P., and Z.S.

**Funding:** This research is funded by Advanced Rail Transportation Special Plan in National Key Research and Development Project, grant number 2016YFB1200601-B13 and 2016YFB1200602-09.

**Acknowledgments:** The author would like to gratefully acknowledge the Computing Facility for the "Era" petascale supercomputer of Computer Network Information Center of Chinese Academy of Sciences. The first author would like to thank University of Chinese Academy of Sciences and Institute of Mechanics under Chinese Academy of Sciences for all support.

**Conflicts of Interest:** The authors declare no conflict of interest.

## **References**


© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## *Article* **Vibroacoustic Optimization Study for the Volute Casing of a Centrifugal Fan**

**Jianhua Zhang 1,\*, Wuli Chu 2,3, Jinghui Zhang <sup>1</sup> and Yi Lv <sup>1</sup>**


Received: 4 February 2019; Accepted: 21 February 2019; Published: 27 February 2019

## **Featured Application: Authors are encouraged to provide a concise description of the specific application or a potential application of the work. This section is not mandatory.**

**Abstract:** A numerical optimization is presented to reduce the vibrational noise of a centrifugal fan volute. Minimal vibrational radiated sound power was considered as the aim of the optimization. Three separate parts of volute panel thickness (ST: the side panel thickness; BT: the back panel thickness; FT: the front panel thickness) were taken as the design variables. Then, a vibrational noise optimization control method for the volute casing was proposed that considered the influence of vibroacoustic coupling. The optimization method was mainly divided into three main parts. The first was based on the simulation of unsteady flow to the fan to obtain the vibrational noise source. The second used the design of experiments (DoE) method and a weighted-average surrogate model (radial basis function, or RBF) with three design variables related to the geometries of the three-part volute panel thickness, which was used to provide the basic mathematical model for the optimization of the next part. The third part, implementing the low vibrational noise optimization for the fan volute, applied single-objective (taking volute radiated acoustical power as the objective function) and multi-objective (taking the volute radiated acoustical power and volute total mass as the objective function) methods. In addition, the fan aerodynamic performance, volute casing surface fluctuations, and vibration response were validated by experiments, showing good agreement. The optimization results showed that the vibrational noise optimization method proposed in this study can effectively reduce the vibration noise of the fan, obtaining a maximum value of noise reduction of 7.3 dB. The optimization in this study provides an important technical reference for the design of low vibroacoustic volute centrifugal compressors and fans whose fluids should be strictly kept in the system without any leakage.

**Keywords:** centrifugal fan; unsteady flow; vibroacoustics; fluid-structure-acoustic coupling; optimization

## **1. Introduction**

The centrifugal fan is considered a common turbomachinery that is widely used in the ventilation systems of the ship cabin and other sites, bringing comfortable working and living environments for people. However, the noise and vibrations generated with the fan running troubled researchers; thus, the study of the mechanisms of noise and vibration generation and propagation became more and more important. Most of the current studies on fan noise have dealt with aeroacoustic problems. However, the noise is induced not only by internal turbulent flow, but also by flow-induced structure

vibration. In some particular application environments, the fluid should be strictly kept within the fan's systems (e.g., petrochemical compressors and large fans with fan system inlets and outlets are entirely connected to the extended pipe) without any leakage, and the aerodynamic noise-induced unsteady flow of a fan cannot directly spread to the outside. At this moment, the fan casing and the inlet and outlet pipe vibration noise caused by the vibrations of the volute surface are predominant. Therefore, an intensive study of the generation mechanism of the vibrational noise and the noise reduction method is necessary.

In fact, the fan noise induced by unsteady flow belongs to fluid–structure coupling noise, and the impeller and volute can be classified as an elastomer; in particular, the volute vibration cannot be neglected in large fans [1]. In addition, the aeroacoustic and vibroacoustic calculations usually require high computational resources; in order to reduce the computational cost and have an accurate response, hybrid methods are applied. With respect to the vibroacoustics of casings, such as the vibrational noise of car body and compressor casings, the hybrid finite-element method/boundary-element method (FEM/BEM) approach and the hybrid finite-element method/statistical energy analysis (FEM/SEA) approach are often used. It could be also appropriate to cite Citarella and Federico [2], who have made a comprehensive literature review of both the structural and acoustic modeling methods that are used nowadays to predict the vibroacoustics' performance. They pointed out that lower frequencies, where the tonal resonances are significant, are calculated applying finite element methods (FEM), whereas for higher frequencies, a statistic energy approach can be chosen. Besides, the FEM/BEM method is usually used to perform the free-field sound radiation analysis of open domains. Armentani [3,4] carried out a vibroacoustic analysis for the chain cover of a four-stroke four-cylinder diesel engine through an FEM–BEM coupled approach, while Bianco [5] described an innovative integrated design verification process, based on the bridging between a new semiempirical jet noise model and a hybrid finite-element method/statistical energy analysis (FEM/SEA) approach for calculating the acceleration produced at the payload and equipment level within the structure, vibrating under the external acoustic forcing field. However, there are few studies on the vibration noise induced by the vibration of the casing in a centrifugal fan. This type of noise is prominent in large-scale fan systems and fans with closed pipelines.

At present, research on the vibration noise induced by casing vibration resulting from impeller outlet unsteady flow is usually conducted using simulation methods. A prediction method based on a method of combining boundary element method (BEM) calculations with experimental measurement was proposed by Koopmann [6]. In this method, the aerodynamic noise is isolated, the volute vibrations induced by the unsteady flow are calculated separately, and the pressure fluctuations required for noise and vibration calculations are obtained experimentally. On this basis, some scholars such as Hwang [7], Cai [8,9], and Lu [10] have used the same method to calculate the vibrational sound radiation of a compressor and the T9-19 No.4 industrial centrifugal fan. Cai Jiancheng [11] calculated the vibrational sound radiation of a volute casing of the same centrifugal fan using a fluid–structure–acoustic coupling method. Indeed, this BEM method discretizes the Lighthill equations by applying a free-field Green function integral. At present, the Green function integral method can only solve problems with simple geometric boundaries, and those complex boundaries must be simplified in the free field. Without doubt, this simplification does not consider reflection and scattering effects in the noise propagation. Based on the above advantages, the finite element method (FEM) for solving noise radiation has been recognized by scholars. Durand [12] predicted the structural acoustics of automotive vehicles through using the FEM model. However, there is a computational disadvantage when the finite element method solves the structural acoustic problem of a closed domain. To overcome this disadvantage, automatically matched boundary layers (AMLs) were introduced to simulate the unbound boundary of the exterior computational domain. The outermost layer exposed to the AML surface that satisfied the Sommerfeld radiation condition was defined as a non-reflecting boundary. Based on the FEM method, Zhang [13] performed the aerodynamic noise of the centrifugal fan using the FEM method, and achieved higher prediction accuracy while using less computing resources. To reveal and reduce

the noise primarily generated by the freezer fan unit, Onur [14] investigated the vibration and acoustic interactions between the structure and the cavity inside the freezer cabinet, and the FEM method was also used. Zhou [15] performed a vibroacoustic analysis of a centrifugal compressor with connecting piping systems, in which the sound was induced by the unsteady flow in the centrifugal compressor and pipes, and the same FEM method was used. These studies have been of benefit in promoting the development of the research of vibrational sound radiation on structural casing, allowing for a deeper understanding of vibrational noise during the fan operation, and have provided a useful reference for the noise reduction of such machinery.

The purpose of vibrational noise research is to explore the generation mechanism of vibrational noise, and then propose targeted methods of vibration and noise reduction. Concerning vibration and noise control, there are certain means: controlling the vibrational source, such as vibration absorption and vibration isolation [16]; dynamic vibration absorption; damping vibration control [17]; and structural vibration control [18–27]. At present, a structural vibration control method that meets specific requirements by modifying the dynamic characteristics of the controlled object without adding any subsystem is a research hotspot. Moreover, current structural vibration control is focused on structural optimization. However, a centrifugal fan casing belongs to a thin-casing structure, and the vibrational sound power of the thin casing is a quadratic function of the structural vibration velocity [18,19]. As optimization reduces the structural vibration speed of a fan casing, it can be concluded that the sound power radiation must be reduced within a specific range. The optimal design of a thin-casing structure usually uses the panel thickness as the design variable and the square sum of the vibration velocities of the nodes on the wall as the optimization target function [20,21]. Adopting the mentioned method, Zhou et al. [22] and Lu et al. [23] implemented the optimization study on structure vibration control and noise reduction for the T9-19 No.4A centrifugal fan.

By reducing fan casing vibration, an effect of casing noise reduction is achieved using the aforementioned optimization method, which sets the vibration (node vibrational speed) as the target function. However, this method does not consider the propagation of sound waves and sound boundary influences on the calculation results; thus, deviation is nearly inevitable. The integration of structural–acoustic optimization correctly eliminates these drawbacks. This method has been used in the automotive field, and shows that the sound radiation generated by the excitation of body surface vibrations on the engine is substantially reduced [24–27] after optimization. Based on the aforementioned advantages of optimization, the authors proposed a vibration–acoustic integrated optimization design method that is suitable for turbomachinery volute. In investigating the vibrational noise of the studied marine centrifugal fans, the following three aspects were the focus:

(1) In this study, we proposed a numerical method for one-way fluid–solid–acoustic coupling. The rationality of this one-way coupling is verified by a volute wall vibration test.

(2) To analyze the influence factors of vibrational noise and reduce the vibrational sound radiation induced by unsteady flow in the fan, a detailed theoretical derivation of vibration noise is put forward.

(3) To control the vibrational noise of a certain type of marine centrifugal fan volute, an optimization method considering the influence of vibroacoustic coupling is proposed. Under the premise of whether volute total mass constraints, accordingly taking the panel thickness of the volute casing (FT: the front panel thickness, ST: the side panel thickness, BT: the back panel thickness) as the design variable, this study conducted low vibrational noise single target (taking the volute vibrational radiated sound power as the target function) and multi-target optimization (taking the volute vibrational radiated sound power and total mass as the target function).

## **2. Centrifugal Fan Description**

The studied machine was a ventilating centrifugal fan with four main components (conical bell mouth, shrouded impeller, volute casing with conventional tongue, and conical flow rates throttle) driven by an AC inverter motor with adjustable angular speed between ~0–3600 r/min; the design rotational speed was specified as 2920 r/min, as shown in Figure 1. The main dimensions and

characteristics of the investigated fan for this study are presented in Table 1. The ambient air was intake from the inlet pipe. The tests for characterizing the aerodynamic and acoustic behaviors of the fan system were made in a resilient installation to fulfill ship system noise and vibration requirements (according to standards GJB4058-2000 China [28] and GB-T1236-2000 China [29]). Figure 2a shows the details of the test system installation and data collection procedure. The following maximum measurement errors were obtained for the different magnitudes: total pressure ±2% (±10 Pa), flow rate: ±2% (±0.05 m3/s), and shaft power ±2% (±50 W).

**Figure 1.** Component representation for the test fan.

**Figure 2.** Mesh details of fan. (**a**) Section workspaces; (**b**) Mesh details of Section 1; (**c**) Mesh details of Section 2.


**Table 1.** Fan dimensions.

## **3. Vibroacoustic Aerodynamics**

## *3.1. Numerical Methods*

CFD (Computational fluid dynamics) technology has been proven to be a very useful tool in the analysis of the internal flow of turbomachinery both in design and performance prediction. It has been widely applied to simulate the fully three-dimensional (3D) unsteady flow in centrifugal turbomachinery. In this part, the unsteady aerodynamic forces that excite vibrational noise are calculated by CFD. The whole unsteady flow for the entire impeller-volute configuration was conducted using the computational fluid dynamics (CFD) code ANSYS CFX. The numerical simulation is based on a finite-volume numerical method that employs an incompressible flow model to solve the Unsteady Reynolds Averaged Navier–Stokes equations (URANS). The characteristic Mach number of the simulated fan described by the blade tip circumferential velocity was *u*2/*c* = 0.18 (<0.3); therefore, the flow was guaranteed to be incompressible. The continuity equation and momentum equations were solved independently of the energy equation because of the isothermal flow. The standard *k-ε* turbulence model, which Ballesteros-Tajadura [30] and Cai [31] applied to capture wall pressure fluctuations, was used in the present simulation of the unsteady flow field. A coupled solver, which uses a fully implicit discretization scheme to solve all of the equations (corresponding to the velocity and pressure), was used. However, a second-order high-resolution discretization scheme was used for the convection terms, and a second-order backward Euler scheme was used for the transient terms.

For the three-dimensional calculations, a couple of high-quality hexahedral structural grids were employed to define the flow domains. Details of the grid features and meridian grid cross-section, including the radical gap and cavity around the volute, are shown in Figure 2. More details about the grids of this fan have been reported in the previous work (Jianhua Zhang et al.) [13]. The numerical deviations result from the grid number needed to be removed, and a grid-independent validation of the fan total pressure coefficient and efficiency was performed. Figure 3 shows the influence of the grid size on the fan total pressure coefficient and efficiency. From Figure 3, we can see that the grid was independent when the total number of grid points exceeded 2.8 million. In addition, by increasing the grid size to 5.7 million, the total pressure coefficient with respect to the flow rate was nearly unchanged, compared with the smaller total pressure coefficient over a small flow rate range that was close to the best efficiency point (BEP).

**Figure 3.** Gridsize independent: (**a**) fan total pressure and efficiency with grid size; (**b**) total pressure coefficient with flow rate.

The modeled boundary conditions were considered to have greater physical meaning for turbomachinery flow simulations. In this case, the CFD simulation process began with a steady flow calculation using the frozen-rotor approach, and nonslip conditions were specified at the solid walls. In addition, for the near-wall flow region, a scalable wall function treatment based on the logarithmic law [32] was applied to cause the mean value of *y*<sup>+</sup> (*y*+ is the dimensionless distance from the wall; it is used to check the location of the first node away from a wall) to vary between 30–300 (the recommended values by ANSYS CFD code to ensure a high calculation accuracy), as shown in Figure 4. Therefore, by applying this method, the number of the grid points was greatly reduced without reducing the calculation accuracy. For the unsteady flow, a transient rotor/stator grid interface based on the sliding grid technique was applied, which allowed unsteady interactions between the impeller and volute casing. A time step of 5.7089e−<sup>5</sup> s was used for the calculation of the unsteady interactions, which was sufficient for the dynamic analysis. The time step was related to the rotational speed of the impeller, and the time step was specified such that the impeller rotated once in 360 steps (a blade passage defined 30 steps). The number of iterations was adjusted to reduce the residual below an acceptable value at each time step. At each time step, a reduction of 10−<sup>5</sup> (five orders of magnitude) in the residuals for the given variables in the cells was required. The unsteady simulation was initialized using a steady solution, and over 15 revolutions (approximately 5400 time steps) were required to converge on a periodic unsteady solution.

**Figure 4.** *y*<sup>+</sup> of the simulated fan.

#### *3.2. CFD Validation*

#### 3.2.1. Experiments Description

In this part, the Dynamic Pressure Testing System (DPTS) was employed to obtain the information of the casing pressure fluctuations. This system contained XCQ-080-5G Kulite highfrequency dynamic pressure sensors (Kulite, Leonia, NJ, America), standard power supplies, 8300 AU amplifiers (Econ, Hangzhou, China), and AVANT MI-7016 signal acquisition (Econ, Hangzhou, China). The test system installation is shown in Figure 5. There are 44 measurement locations on the casing surface, which were at five locations (p01–p05) in the vicinity of the volute tongue, and another six monitoring locations (p06–p11) were evenly distributed along the circumferential direction of the volute. Table 2 summarizes the angular coordinates of the measurement positions over the volute surface. The locations of monitoring points are shown in Figure 6. The origin of the angle is the volute tongue. At each angular position, four axial measurements were made at the following *Z*/*B* coordinates: 0.17, 0.27, 0.34, and 0.75 (*B* is the volute axial width, and *Z* is the axial measurement position from the volute rear casing). *Z*/*B* = 0.07 corresponds to the impeller hub, and *Z*/*B* = 0.36 corresponds to the impeller shroud. More details about the DPTS installation and test have been reported in the previous work (Jianhua Zhang et al.) [13].

**Figure 5.** Installation of test system. (**a**) Test equipment; (**b**) The flow chart of test system installation.


**Table 2.** Angular coordinate of the measurement points over the volute.

**Figure 6.** Sketches of the fan with the measurement points.

#### 3.2.2. Aeodynamic Characteristics Validation

Using the defined test bench, the overall performance of this fan as predicted by the CFD calculations was compared for different flow rates. The non-dimensional flow rate and the total pressure rise were transformed by:

$$\varphi = \frac{4Q}{D\_2^2 \mu\_2 \pi} \tag{1}$$

$$
\psi = \frac{P\_T}{\rho u\_2^2} \tag{2}
$$

The best efficiency point (BEP) at a rotational speed 2920 rpm corresponded to a flow rate Q = 3.3614 kg/s (ϕ = 0.1659) and a total pressure rise PT = 3182 Pa (ψ = 0.41954). Figure 7 indicates that the measured total pressure coefficient and efficiency agreed well with the three-dimensional steady-state calculations. The expected trend of decreasing total pressure coefficients with increasing flow rates can be observed in the experimental and numerical curves. In addition, the performance curves between the numerical simulation and experimental test results were perfectly consistent regards of larger grids (nearly double the size of the small grids).

**Figure 7.** Comparison results between numerical and experimental curves.

#### 3.2.3. Unsteady Pressure Fluctuation Validation

The pressure fluctuations on the volute casing induced by the unsteady flow at the impeller outlet are the significant vibration noise sources; thus, the validation of the unsteady aerodynamics is very important. In this study, to confirm the simulation accuracy, a comparison analysis between the dynamic pressure measurement results of volute casing and the numerical calculation was conducted. Due to the similar spectrum signals of the different measurement points on the volute surface, only a few typical points were selected for analysis and confirmation. Figure 8 shows the power spectrum of the volute pressure fluctuations at the BEP at the typical selected measurement points: point 01, point 06, and point 09. The left figure shows the experimental measurement results, and the right figure shows the numerical results. It can be found that the numerical and experimental amplitude of the blade-passing frequency (BPF), which presented the predominant frequency component, were in good agreement at the three most important axial positions (*Z*/*B* = 0.17, 0.27, and 0.34). More details about the results discussion have been reported in the previous work (Jianhua Zhang et al.) [13].

**Figure 8.** Power spectrum of volute pressure fluctuation at three measurement points with the flow rate best efficiency point (BEP); (**a**) Point 01 (exactly at the tongue); (**b**) Point 06 (52◦ from the tongue); (**c**) Point 09 (187◦ degree from the tongue).

#### **4. Volute Vibroacoustic Model, FEM Validation, and Simulation**

For the simulated fan, the displacement of the volute vibration was very small, and the flow was incompressible. Furthermore, the characteristic Mach number was smaller than 0.3. Therefore, the volute's vibration influence on the internal flow was neglected. Therefore, one-way fluid–solid coupling was applied in the simulation. Jiang et al. [33] applied a one-way coupling technique that validated the rationality of an unsteady flow-induced vibration of a centrifugal pump. The validation of one-way coupling is also presented in this study. For details, please refer to Section 4.2.

## *4.1. Vibroacoustic Mathematical Model*

For a continuous system of an actual structure, which was dispersed by FEM, the dynamic balance equation is as follows:

$$\mathbf{M}\ddot{\mathbf{x}}(t) + \mathbf{C}\dot{\mathbf{x}}(t) + \mathbf{k}x(t) = \mathbf{F}(t) \tag{3}$$

As the structure is subjected to external harmonic force, the external force can be expressed as follows:

$$F(t) = \mathbf{F} \cdot e^{jwt} \tag{4}$$

The modal vectors are linearly independent of each other. Therefore, the response of the dynamic under any excitation can be regarded as the coupling of the systematic modes and the modal participation factors (MPFs) of each order. At this point, the displacement response can be expressed as follows:

$$\mathbf{x}(t) = \sum\_{i=1}^{n} \mathbf{q}\_{i} y\_{i}(t) = \boldsymbol{\Phi} \mathbf{Y} \tag{5}$$

In the formula, *ϕ<sup>i</sup>* represents the ith mode shape of the structure, and *yi*(*t*) represents the ith mode coordinate, which is called the ith MPFs; **Φ** = [ϕ1ϕ<sup>2</sup> ··· ϕ*n*]; **Y** = [*y*1*y*<sup>2</sup> ··· *yn*]. Substituting Equation (5) into Equation (3) and multiplying **Φ***<sup>T</sup>* on the two laterals yields the following:

$$
\mathbf{\dot{Q}} \cdot \mathbf{\dot{M}} \mathbf{\dot{Q}} \cdot \mathbf{\ddot{Y}} + \mathbf{\dot{Q}}^T \mathbf{C} \mathbf{\dot{Q}} \mathbf{\dot{Y}} + \mathbf{\dot{Q}}^T \mathbf{K} \mathbf{\dot{Q}} \mathbf{Y} = \mathbf{\dot{Q}}^T \mathbf{F} \mathbf{e}^{j\omega t} \tag{6}
$$

Using the orthogonality of the modal vectors for the mass, damping, and stiffness matrixes, we obtain independent coefficients of the single degrees of freedom of *n* items. Therefore, the original system can be regarded as linear superposition independent coefficients of single degrees of freedom of *n* items.

If,

$$
\Phi^T \mathbf{M} \Phi = m\_{\mathrm{i}}, \quad \Phi^T \mathbf{C} \Phi = \mathbf{C}\_{\mathrm{i}}, \quad \Phi^T \mathbf{K} \Phi = \mathbf{K}\_{\mathrm{i}}, \quad \Phi^T \mathbf{F} = \mathbf{F}\_{\mathrm{i}} \tag{7}
$$

Then, substituting Equation (7) into Equation (6), the transformation is as follows:

$$m\_i \ddot{y}\_i(t) + \mathcal{C}\_i \dot{y}\_i(t) + K\_i y\_i(t) = F\_i e^{j\omega\_i t} \tag{8}$$

If,

$$
\omega\_n = \sqrt{\frac{k\_i}{m\_i}}, \quad \zeta\_i = \frac{c\_i}{2\sqrt{k\_i m\_i}} \tag{9}
$$

Then, substituting Equation (9) into Equation (8) results in the following:

$$
\ddot{y}\_i(t) + 2\mathcal{J}\_i \omega\_n \dot{y}\_i(t) + \omega\_n^2 y\_i(t) = F\_i e^{i\omega t} \tag{10}
$$

Using the theory of ordinary differential equations, we obtain the stable solution of Equation (4) as follows:

$$y\_i(t) = \frac{F\_i}{\omega\_{\text{fl}}^2 - \omega^2 + 2j\zeta\_i\omega\omega\_{\text{nl}}}e^{j\omega t} \tag{11}$$

*Appl. Sci.* **2019**, *9*, 859

Introducing the frequency ratio λ = *ωn*/*ω* and the dimensionless vibration mode amplification factor *β<sup>i</sup>* results in the following:

$$\beta\_i = \frac{1}{\sqrt{\left(\lambda^2 - 1\right)^2 + \left(2\zeta\_i \lambda\right)^2}}\tag{12}$$

Ordering:

$$\psi\_i = \arctan \frac{2\zeta\_i\lambda}{\lambda^2 - 1} \tag{13}$$

Substituting equations (12) and (13) into Equation (11) results in the following:

$$y\_i(t) = \frac{F\_i}{\omega^2} \beta\_i e^{j(\omega t - \psi)} \tag{14}$$

At this point, the vibrational displacement is as follows:

$$\mathbf{x}(t) = \sum\_{i=1}^{n} \boldsymbol{\mathfrak{p}}\_{i} y\_{i}(t) = \sum\_{i=1}^{n} \boldsymbol{\mathfrak{p}}\_{i} \frac{F\_{i}}{\omega^{2}} \boldsymbol{\beta}\_{i} e^{j(\omega t - \boldsymbol{\upvarphi})} \tag{15}$$

The vibrational velocity is as follows:

$$\dot{\mathbf{x}}(t) = \sum\_{i=1}^{n} \boldsymbol{\mathfrak{op}}\_{i} \boldsymbol{y}\_{i}(t) = \sum\_{i=1}^{n} \boldsymbol{\mathfrak{op}}\_{i} \frac{F\_{i}}{\omega} \beta\_{i} e^{j(\omega t - \boldsymbol{\uppsi} + \frac{\pi}{2})} \tag{16}$$

The active output power is as follows:

$$\mathcal{W}\_{o, \text{active}} = \int\_{S} \text{Re}(l\_n) ds = \frac{1}{2} \int\_{S} \text{Re}(pv\_n^\*) ds \tag{17}$$

The relationship between the plane wave sound pressure p and the surface velocity *v*∗ *<sup>n</sup>* is as follows:

$$p = \rho\_0 c\_0 v\_n^\* \tag{18}$$

Substituting Equation (18) into Equation (17) results in the following:

$$\mathcal{W}\_{\text{o,active}} = \int\_{S} \text{Re}(I\_n) ds = \frac{1}{2} \rho\_0 c\_0 \int\_{S} \text{Re}(\upsilon\_n^{\*2}) ds \tag{19}$$

Substituting Equation (16) into Equation (19) results in the following:

$$\mathcal{W}\_{\text{o,active}} = \int\_{S} \text{Re}(I\_{\text{n}}) ds = \frac{1}{2} \rho\_{0} c\_{0} \int\_{S} \text{Re}(\sum\_{i=1}^{n} \mathbf{q}\_{i} \frac{F\_{i}}{\omega} \beta\_{i} e^{j\left(\omega t - \mathbf{q} + \frac{\pi}{2}\right)}) \,\text{d}s \tag{20}$$

According to Equation (20), it can be concluded that the structural acoustic radiation power is mainly determined by the modal shape *ϕi*, the applied exciting force F*i*, and the frequency amplification factor βi. Therefore, the following methods can be used to control vibrational noise:

(1) With the structural model and material determined, the vibrational sound radiation can be weakened by attenuating the amplitude of the applied exciting force;

(2) With the determination of the exciting force, the geometric parameters of the structural model are modified to reduce the modal shape;

(3) To reduce the amplitude amplification factor, the natural frequency and the external excitation force frequency should be avoided.

Regarding the studied fan volute structure, the structural mode can be changed by controlling the thickness distribution of the structure if the geometry, the stiffness, and the constraint position are all fixed.

## *4.2. Volute Vibration Simulation and Validation*

The finite element analysis method is one of the important methods to obtain the vibrations of the structure surface. In this study, N × Nastran, the commercial software made by the Siemens Company, was used to calculate the modal and vibration response of the volute. The finite element model (FEM) of the volute was selected by using a high-quality surface quadrilateral mesh, as shown in Figure 9. The thickness of the volute panel is relatively small (up to six mm), and the shell63 element is selected for the FEM, as the shell63 element has both bending and membrane capabilities, and can suffer from both plane and normal loads. The volute FEM with a total of 46,182 shell63 element grids was divided into three main sections according to the different thickness properties. The front panel thickness (FT) and the back panel thickness (BF) were set to six mm, and the volute side panel thickness was set to five mm (ST). In addition, the model material was steel, the density ρ = 7800 kg/m3, the elastic modulus = 2.06e11 pa, and Poisson's ratio *υ* = 0.3. The volute casing was fixed to a supporting stand by 10 fastening bolts at the casing front. The volute panel rear (near the motor) was connected by four fixed bolts, and the three translational degrees of freedom of the nodes at the bolts were restricted to zero. The panel thickness distribution and the degree of freedom constraints on the volutes are shown in Figure 9.

**Figure 9.** The finite element model (FEM) of the volute.

To validate the one-way fluid–solid coupling, vibration analysis was performed, and the results were compared with those of experimental vibrational analysis. The LMS Test Lab test system was used to complete the vibrational test of the fan casing. To eliminate the vibrational disturbances on the volute originating from the imported pipe and outlet throttle flow, elastic connections were used in two positions: at the connection between the transition section of an inlet and the volute, and between the volute outlet and the throttle valve. The flexible installation should meet the requirements of GJB4058-2000 (The Noise and Vibration Measurement Method of Ship Equipment) [28]. There is some major equipment required for this test, such as an LMS SC310W signal analyzer, a B&K 4513 accelerometer, and a B&K 4514 accelerometer. The background noise is ignored because of the lower value compared to the actual value of the fan. One hand of the accelerometer is fixed on the volute by bonding, and the other hand is directly connected to the data processing and analysis notebook by the data line. The arrangement of the vibration sensor is shown in Figure 10. There are 16 vibration measurement locations on the casing surface. The first five measuring points are arranged near the border between the back panel of the volute and the side panel of the volute. The first measuring point is located near the tongue, and the second through fifth measuring points, #2–5, are respectively arranged at the positions of 0◦, 90◦, 180◦, and 270◦. The sixth to ninth measuring points, which are located at the edge of the plate between the back panel of the volute and the motor along with the

connecting plate in the circumferential direction, are arranged at an interval of 90 degrees. At the front panel of the volute, with a 90-degree interval in the direction of a counterclockwise rotation layout, are measuring points 10 to 12. The vibration measuring point of the volute side panel is in the middle of the axial width of the volute arranging measuring points 13 to 15. Measuring point 13 is defined as the starting point; measuring points 14 and 15 are also arranged in the side volute at an interval of 140 degrees; and the measuring point 16 is at the outlet of the volute side panel. The vibration test and the dynamic pressure test of the volute are carried out at the same time, and the data are, respectively, collected in different control computers. Then, the data are sequentially extracted to complete the post-processing.

47

**Figure 10.** The arrangement of the vibration sensor. (**a**) The distribution of measuring points of experiments; (**b**) The distribution of measuring points of simulation.

The definition of the total vibrational amplitude is shown as follows:

$$VAL = \sqrt{\sum\_{i=1}^{n} a\_{fei}^2} \tag{21}$$

where, *a f ei* represents the vibration acceleration at any frequency in the spectrum.

Figure 11 shows a comparison of the numerical and experimental results of the total vibration levels of various vibrational positions in the range of 20 to 3000 Hz. The calculated vibrational measurement positions are arranged according to the vibrational test. Most importantly, it should be stated that the volute casing vibration measurements, the vibration response calculation, and the vibrational noise production are all carried out on the fan design flow rate, the best efficiency point (BEP). As seen in the Figure 11, the calculations are in good agreement with the experiments; the detailed results and analysis refer to the reference [34]. Moreover, a comparison between the experimental and the numerical results shows that it is reasonable and effective to adopt the one-way fluid–structure–acoustic coupling method. Figure 12 presents the vibration acceleration spectrum of

the selected three measuring positions (corresponding to the volute rear panel [BT], the volute front panel [FT], and the volute side panel [ST]). It can be seen from Figure 12 that the spectrum waveforms at each measuring position are similar, and the maximum amplitude of vibration acceleration presents at the fundamental frequency, indicating that the fundamental frequency, the blade-passing frequency (BPF), is the major component for volute vibrations induced by unsteady flow.

**Figure 11.** The compassion of numerical and experimental amplitude of normal acceleration.

**Figure 12.** The spectrum of acceleration on the three different measuring points.

## *4.3. Volute Vibroacoustics Estimation Method*

The vibroacoustic simulation was performed using the LMS Virtual Acoustics commercial code, and the volute acoustical FEM model is shown in Figure 13. It was similar to the acoustical finite element mesh that was used for aerodynamic noise calculations [13]. Taking into account the characteristic of radiated vibrational noise, the volute's inlet and outlet were completely enclosed. More importantly, according to the requirements on element size driven by maximum frequency, the computational acoustic mesh had to satisfy each wavelength corresponding to six elements. An acoustical mesh with a maximum element size of 15 mm was applied in the sound computation, and guaranteed a spatial resolution at the maximum frequency of 3236 Hz of six points per wavelength. Atmospheric boundary layers (AMLs) were introduced to simulate the unbound boundary of the exterior fluid domain. The outermost layer exposed to the AML surface that satisfied the Sommerfeld radiation condition was defined as a non-reflecting boundary. Then, a field point mesh based on standard ISO3744 [35] that enclosed the entire calculation domain was established using an approximate free-field engineering method.

**Figure 13.** The volute acoustical FEM model.

Figure 14 shows the numerical evaluation method of the volute vibroacoustic coupling. It can be seen that the one-way fluid–structure–acoustic coupling method is divided into three main steps. The first involves the acquisition of the vibrational source of the volute based on the unsteady flow calculation on the centrifugal fan, and then transformation of the extracted time-domain fluctuation data into frequency-domain data through FFT, providing basic data for the next vibration response and vibroacoustic calculation. The second steps involves the interpolation of the frequency-domain node pressure of the fluid into the corresponding structural FEM nodes according to Equation (5) (where *Pi* (*i* = 1, 2, 3, 4) is the source node pressure load, *P*<sup>A</sup> is the target node pressure load, and *di* (*i* = 1, 2, 3, 4) is the distance from the source node to the target node; Figure 15 is a sketch of the geometric interpolation algorithm), assignment of the interpolated node pressure of the structure to the boundary loads of vibroacoustics, and then application of the structural FEM to obtain the modal participation factor of the volute. The third step involves loading the modal participation factor and vibroacoustic boundary loads that were obtained during the second step in order to calculate the volute vibrational sound radiation using the modal superposition vibroacoustic method.

$$P\_A = \frac{P\_1 \frac{1}{d\_1} + P\_2 \frac{1}{d\_2} + P\_3 \frac{1}{d\_3} + P\_4 \frac{1}{d\_4}}{\frac{1}{d\_1} + \frac{1}{d\_2} + \frac{1}{d\_3} + \frac{1}{d\_4}}\tag{22}$$

**Figure 14.** The flow chart of the numerical evaluation method of volute vibroacoustic coupling.

**Figure 15.** The diagram of the geometric interpolation algorithm.

Figure 16 presents the spectrum chart of the vibrational sound radiation of volute casing, and the vibrational noise at the fundamental frequency (BPF) is obvious. Besides, the distribution of the vibrational sound radiation and the normalized velocity of the volute casing surface at the fundamental frequency is presented in Figures 17 and 18. It can be observed that the distribution shape of the surface sound pressure and surface normal velocity on the volute have identical characteristics, and the outlet of the volute side panel near the volute tongue region and the volute back panel at 180◦ from the tongue presented very strong vibrational acoustic radiation values. In addition, the previous study [36] showed that the normal vibration velocity of the volute was the decisive factor that determined the volute surface acoustic radiation. Moreover, the theoretical derivation of Section 4.1 (according to Equation (19)) shows that the acoustical power that characterized the vibrational acoustic energy is also a quadratic function of the vibrational velocity (according to Equation (19), Zhou [22]) indirectly reducing the volute surface acoustical radiation through a decrease in the surface normal velocity of the volute casing.

**Figure 16.** The spectrum chart of vibrational sound radiation of the volute casing (numerical).

**Figure 17.** The distribution of the vibrational sound radiation of the volute casing surface at the fundamental blade-passing frequency (BPF). (**a**) Side; (**b**) Back; (**c**) Front.

**Figure 18.** The distribution of the vibrational velocity of the volute casing surface at the fundamental frequency (BPF). (**a**) Side; (**b**) Back; (**c**) Front.

## **5. Vibroacoustic Optimization Strategy**

## *5.1. Optimization Objective Selection and Volute Panel Thickness Parametrization*

Determining the appropriate optimization objective function is the key problem of optimizing calculations. Section 4.1 of this paper provides two means to control vibroacoustics. Control method (1) can be used to modify the geometrical shape of the volute tongue or wavy edge of the blade to weaken the pressure fluctuations on the volute casing. Control method (2) can be used to modify the model geometry, thickness distribution, and constrained position, and reduce the modal shapes. However, the shapes and natural frequency are determined by the structural thickness distribution, structural stiffness, and the constrained position. The modal shapes can be changed by controlling the thickness distribution, when the geometry, stiffness, and constrained position of the structure are fixed.

Concerning the studied centrifugal fan volute with several welding panels, this study developed an optimization method of vibroacoustics by changing different panel thicknesses (FT, BT, and ST) to attenuate the volute vibrational acoustic radiation. Regarding vibroacoustic optimization, we must first determine the optimization target function. From the literature review, it is generally considered that the uniform sound pressure at the received position or the radiated sound power at the structural surface should be set as the optimization function. The selection of the target function depends on the research problems, and the target function is divided into two categories according to different attributes. One is a noise optimization problem (internal noise control problem) applied in a closed domain. In this case, one or a few sound pressure levels of a specific measuring point are taken as the target function [24–27]. Concerning the noise control problem in the open domain, the external acoustic power on the structural surface was chosen as the optimal target function. This method has been proven by many scholars [24,25,37].

For the type of target function, the sound pressure at the arranged receiving position can be clearly determined, but it may result in judgments distortion, such that a low sound pressure value at an arranged position may be obtained, and a high sound pressure value at other points can be presented. Thus, it is very important to choose such target functions. Fortunately, Marburg [38] proposed an improved target function (F), which is defined as follows:

$$F = \widetilde{F}^{\frac{1}{n}} = \left(\frac{1}{\omega\_{\text{max}} - \omega\_{\text{min}}} \int\_{\omega\_{\text{min}}}^{\omega\_{\text{max}}} \Phi\{p\_l(\omega)\} d\omega\right)^{\frac{1}{n}}\tag{23}$$

In Equation (23), *φ*{*pl*(*ω*)} represents the given weighting function, which is replaced by the following equation:

$$\phi\{p\_l(\omega)\} = \begin{cases} \left(p\_l - p\_{ref}\right)^n, & p\_l > p\_{ref} \\ 0, & p\_l < p\_{ref} \end{cases} \tag{24}$$

Where *n* = 1, which is the average value of sound pressure in the frequency spectrum; and *n* = 2, which is the root mean square (RMS) value of the sound pressure.

For the second type of problem, Koopmann and Fahnline [6] proposed an optimization method that takes the external radiated sound power as the target function, and is suitable for the optimal noise control method in this study. They provided the root mean square (RMS) expression of sound pressure in an enclosed space as follows:

$$\mathcal{W} = \int\_{\mathcal{S}} \text{Re}(I\_n) ds = \frac{1}{2} \int\_{\mathcal{S}} \text{Re}(pv\_n^\*) ds \tag{25}$$

Where S refers to the structural surface, and the structural surface is discretized based on FEM. At this time, the structural radiated sound power can be regarded as the sum of the individual radiated sound power on the FEM mesh [39]. Thus, after organization, Equation (25) can be changed to the following:

$$\mathcal{W} = \sum\_{j=1}^{N\_\varepsilon} \mathcal{W}\_j = \frac{1}{2} \text{Re}(\sum\_{j=1}^{N\_\varepsilon} \frac{1}{2} \int\_{S\_j} p\_j v\_{nj}^\* dS) \tag{26}$$

In the formula:

N*e*—the number of finite elements on the structural casing;

*Sj*—the area of *j*th finite element, m2;

P*j*—the sound pressure of *j*th finite element, pa;

v∗ *nj*—the normal velocity of *j*th finite element, m/s.

The vibrational noise of the volute structure of the marine centrifugal fans that is studied in this paper belongs to the typical external opening noise radiation problem. Therefore, the second type of optimization target function should be adopted.

The previous analysis in Section 4 shows that the structure modal shape can be changed by controlling the structural thickness distribution. Therefore, the volute thicknesses of the three panels (FT, BT, and ST in mm) were specified as the design variables. Thus, the design objectives can be achieved by adjusting the combination of different volute panel thicknesses in the optimization design. The thickness of each volute panel is parameterized using FEM. Figure 19 shows the volute parameterization structural FEM. Since the panel thickness of each volute is generally less than 10 mm, we assigned the panel thickness of each volute to be from four mm to 10 mm.

**Figure 19.** The volute parameterization structure.

## *5.2. RBF Approximation Surrogate Models and Validation*

The high computational cost of each vibration and acoustic simulation to predict the vibroacoustics performance of each volute panel thickness design makes a direct optimization approach to find an optimum low-vibration noise volute unfeasible. Indeed, an optimized cycling takes 24–26 h. In addition, the optimization process terminates due to the failure of a single simulation program. Therefore, the long time of the optimal iterative procedure is the major bottleneck in the optimization of acoustics performance for all complex structures. Instead, the use of a surrogate model, which was constructed from and used in lieu of the actual simulation model, represents a valuable alternative to increase the speed of the optimization. For this reason, a metamodeling approach was chosen for vibroacoustic optimizations. Therefore, the combination of a design of experiment (DoE) and an approximation model (AM) was used to determine the relationship between the structural design variables and the target function, and provide the basic data and models for the subsequent optimization. According to the studied centrifugal fan volute, taking each volute thickness (FT, BT, and ST) as design variables, each design variable was given five levels. Table 3 provides the levels of distribution of the design variables. Due to the smaller number of design variables, the full-factor

method was adopted to collect sample points of the design space constructed by the three variables collected; thus, a total of 125 sample points was collected.

**Table 3.** The levels of distribution of each volute panel thickness. FT: the front panel thickness, ST: the side panel thickness, BT: the back panel thickness.


For each sample point, the aforementioned vibroacoustic coupling assessment method was used to calculate the radiated sound power and the total volute mass of the volute structure surface of each combination. The approximate model approach is generally divided into two broad categories: first, least squares fitting, also known as response surface methodology (RSM); and the second, interpolation. RSM uses polynomial functions to fit the design space. In addition, the RSM fitted the complex response relationships through regression models due to the simple algebraic expression. In addition, because of the advantages of being systematic and practical, the RSM had been used in a broad range [40–43], but the approaching effect is not as good as that of neural network and other methods for highly complex functional relationships (a complex relation of nonlinear function). The radial basis function (RBF) belongs to the interpolation algorithms, and is the second approximate model method. Since the neural network model has a strong ability to approximate complex nonlinear functions, the learning speed is fast, has excellent generalization ability, and is highly fault-tolerant. It is used by many scholars and engineers in aircraft wingtips [44], highway traffics [45], turbomachinery [46], engineering optimization [47], compressors [48], gas cyclone separator [49], MMES controllers [50], and many other applications. The relationships between the vibrational sound radiation of the volute casing surface and the volute panel thickness are typically nonlinear. Thus, this study used the RBF model to approximate and fit the design space.

Figure 20 shows the fitting procedure of the vibroacoustic optimization of volute casing. It can be seen from Figure 20 that this procedure is mainly divided into two parts. The first is the creation of a design space for the collected sample points using the assessment method of vibroacoustic coupling. The FEM model of fan volute was established by UG; then, ANSA constructed FEM mesh, and the Nastran code solved the volute modal participation factor. At last, the vibrational sound radiation of volute was used by LMS Virtual Acoustics. In addition, all the calculation codes were integrated into the multi-disciplinary optimization platform, Isight. The second built an RBF approximation model (RBF surrogate model) instead of a simulation loop, as mentioned for the first part. Thus, the RBF method with Isight code was used to establish an approximate alternative model.

In fact, the approximate model can be considered an approximate approach for the physical model, in which the precision of the approximate model is affected by the number of sampling points. At present, statistical theory with analysis of variance is usually used to verify the effectiveness of an approximate model. However, the approximate model is usually tested by the complex correlation coefficient *R*<sup>2</sup> (0 < *R*<sup>2</sup> < 1) in engineering, and Shi [51] provided its mathematical expression. The closer the value of *R*<sup>2</sup> is to one, the more precise the approximation model. Figure 21 shows the error schematic diagram of each response surface model. In Figure 21, the Kirchhoff SPW (dB) represents the radiated sound power of the volute surface, and the mass (kg) represents the total mass of the volute. It can be seen that all the response surface models are infinitely close to the value one; thus, the approximate model that was established using the mentioned method can completely replace the real simulation loop. The parametric analysis related to the radiated sound power of the volute surface is performed, which will be discussed in the following sections.

**Figure 20.** Design space and radial basis function (RBF) approximation model processing for optimization.

**Figure 21.** The error schematic diagram of each response surface model. (**a**) The precision of sound power on the volute surface; (**b**) The precision of total mass.

## *5.3. Single-Objective Optimization Procedure*

The single-objective optimization was conducted to improve the radiated sound power of the fan volute by using a weighted-average surrogate model with three design variables related to the geometries of the three-part volute panel thickness. The collection of sample points that was described in the previous text, and based on the RBF approximation model, provides the basic database and the analysis model of the target function for the optimization of the vibroacoustic coupling of the volute structure. In this part, the single-objective optimization took three panel thicknesses (FT, BT, and ST in mm) as design variables, and took the vibrational sound radiated power of the volute surface (Kirchhoff SPW (dB)) as the target function. In addition to these, the single-objective optimization was divided into two parts: the first part maintained the volute total mass as invariable, and for the second part, there was no mass constraint on the volute. Since the thickness of the volute panel is generally less than 10 mm, the author assigned the volute panel thickness as from four mm to 10 mm. Therefore, the mathematical model optimized in this section is as follows:

The objective function, *Ws* (minimum):

$$\mathcal{W}\_{\mathbf{s}} = \sum\_{\mathbf{j}=1}^{N\_{\mathbf{c}}} \mathcal{W}\_{\mathbf{j}} = \frac{1}{2} \text{Re}(\sum\_{\mathbf{j}=1}^{N\_{\mathbf{c}}} \frac{1}{2} \int\_{S\_{\mathbf{j}}} p\_{\mathbf{j}} v\_{\mathbf{nj}}^{\*} dS)\_{\mathbf{s}} \text{ (vibrational sound radiated power of solute surface } \boldsymbol{\beta} \text{W) (27)}$$

Variables:

$$\{\text{FT}, \text{ST}, \text{BT}\} \tag{28}$$

Constrained conditions:

 ≤ FT ≤ 10 ≤ ST ≤ 10 ≤ BT ≤ 10 (29)

Constrained variable:

$$M\_T = S\_{FT} \cdot FT + S\_{ST} \cdot ST + S\_{BT} \cdot BT,\text{ (total mass, kg)}\tag{30}$$

Defining the sound power level as follows:

$$10 \times \lg^{(\mathcal{W}\_s / \mathcal{W}\_{ref})} \tag{31}$$

In Equation (31), *Wref* represents the reference value of sound power, *Wref* = <sup>1</sup> × <sup>10</sup>−<sup>12</sup> <sup>W</sup>

Figure 22 shows the flow chart of the single-objective optimization. The single objective optimization uses the simulated annealing algorithm (ASA) to implement a global search. The optimal result of the approximate model in the previous text is assigned as the initial value, and the global optimization iterates 10,000 steps and takes 12 minutes. Then, the value is locally optimized using mixed integer sequential quadratic programming (MISQP), which iterates more than 12 steps in several seconds. The results of the single-objective optimization of the vibroacoustics of a volute surface will be discussed in the following sections.

**Figure 22.** The flow chart of the single-objective optimization.

## *5.4. Multi-Objective Optimization Procedure*

In fact, the parameters of the vibrational sound power and total mass on the volute surface are somewhat contradictory. Thus, a multi-objective optimization method is needed to obtain the optimal volute thickness combination. The definition of the multi-objective optimization mathematical model is presented as follows:

The objective function, *Ws*, MT (minimize):

$$\mathcal{W}\_s = \sum\_{j=1}^{N\_\varepsilon} \mathcal{W}\_j = \frac{1}{2} \text{Re}(\sum\_{j=1}^{N\_\varepsilon} \frac{1}{2} \int\_{S\_j} p\_j v\_{\eta\_j^\ast}^\* dS), \text{ vibrational sound radiated power of volume surface, } \mathcal{W} \text{ : (32)}$$

$$M\_T = S\_{FT} \cdot FT + S\_{ST} \cdot ST + S\_{BT} \cdot BT \text{ (total mass, kg)} \tag{33}$$

Variables:

$$\text{[FT, ST, BT]}\tag{34}$$

Constrained conditions:

$$\begin{array}{l} 4 \le \text{FT} \le 10\\ 4 \le \text{ST} \le 10\\ 4 \le \text{BT} \le 10 \end{array} \tag{35}$$

Figure 23 presents a flow chart of multi-objective optimization. The multi-objective genetic algorithm termed NSGA\_2 was adopted to solve the multi-objective optimization with the optimal value obtained by taking the single-objective optimization of specific mass constraints as the initial values. The global search iterates 4800 steps, and the Pareto frontier solutions are marked. A solution satisfying the requirement is selected as the initial value of MISQP for mixed integer quadratic programming; then, the final solution satisfying the engineering requirement is obtained by iterating 15 steps again. Finally, the vibroacoustic estimation method mentioned in Section 4.3 is used to verify the precision of the optimal solution. The multi-objective optimization results will be discussed in the following sections.

**Figure 23.** The flow chart of multi-objective optimization.

#### **6. Results and Discussion**

#### *6.1. Results of Sensitivity and Parametric Analysis*

In order to investigate the influence of each design variable on the optimization objectives, it is necessary to perform the sensitivity analysis for the design variables. Sensitivity analysis is performed by studying the correlations between the design variables and the objectives. Figure 24 shows the correlation distribution of design variables and optimization objectives (Kirchhoff SPW, volute mass on the volute structure surface); the correlation theory and definition referred to the reference [52]. The positive value indicates that the optimization objective is proportional to the design variable, and otherwise is an inverse relationship. The closer the absolute value of the coefficient is to one, the higher the degree of correlation. It can be seen from Figure 24 that ST had the largest influence on all the objectives, followed by FT, then BT. Concerning radiated sound power, ST and FT present an inverse proportionality to the radiated sound power, which means that the larger the volute panel thickness (ST and FT), the smaller the radiated sound power.

**Figure 24.** The correlation distribution of design variables and responses (Kirchhoff SPW and volute mass on the volute structure surface).

However, the relationship diagram between each volute panel (ST, BT, FT) and objective functions (the Kirchhoff SPW or the total mass of the volute) in Figure 25 shows that BT and FT present nonlinearity characteristics for the acoustic power of the volute surface. With an increase of BT value, the vibrational radiated sound power of the volute surface increases first, and then decreases, and FT value presents the opposite change. However, the volute radiated sound power decreases with ST value if the ST does not exceed nine mm. This means that each panel thickness (BT, ST, FT) has an optimal value that satisfies the radiated sound power minimum. However, Figure 25 only shows the influence of each single thickness variable on the radiated sound power (the other two variables remain unchanged; the relationship between the other two variables and the noise radiation is determined). To determine the relationship between each variable combination (BT–ST; FT–ST; FT–BT) and the acoustic radiation power, Figure 26 gives the influence of each two-panel thickness variation on the radiated sound power. It can be observed that the FT–ST and BT–ST combinations present similar distribution characteristics. Therefore, the Kirchhoff SPW basically remained a constant smaller value when the ST was greater than 7.0 mm, and the BT was less than 8.5 mm or the FT was less than 7.0 mm. In addition, as the FT is larger and the BT is smaller or the FT and BT are both smaller (less than 5.5), the Kirchhoff SPW is observed to have a smaller value as shown in Figure 26c. Besides, linear relationships could be seen in Figure 25b, which means that the thicker the thickness of each volute panel, the greater the total mass of the casing.

**Figure 25.** The relationship profile between each panel thickness and objective function (the Kirchhoff SPW or the total mass of the volute). (**a**) The relationship of volute panel thickness and Kirchhoff SPW. (**b**) The relationship of volute panel thickness and total mass.

**Figure 26.** The relationship between each panel thickness (BT–ST; FT–ST; FT–BT) and radiated sound power of volute surface. (**a**) BT-ST; (**b**) FT-ST; (**c**) FT-BT.

In conclusion, the variation in ST is the most sensitive to the sound radiation power of the volute structure surface, while FT is the second-most sensitive, and BT is the least. Therefore, it is possible to obtain the smallest value of radiated sound power of the volute surface by optimizing the panel thickness combination (increasing the ST value and decreasing the FT and BT values).

## *6.2. Results of Optimization.*

After optimization, the vibrational radiated sound power of the volute surface is greatly reduced. Table 4 shows the massless constrained optimization results (the "optimized" represents the calculation results of the approximate model, and the "numerical validated" represents the vibroacoustic analysis with the optimum values). It can be seen that the radiated sound power on the volute structure surface decreases by 9.4 dB when ST increases to a maximum, while BT and ST locate at minimum values; unfortunately, at the same time, the volute total mass increases by 18.13%. Table 5 shows the optimization results with the volute total mass invariable. Even though the volute total mass remains the same, the radiated sound power on the surface of the volute will also be weakened by an average of 6.3 dB. In addition, the radiated sound power spectrum of the volute structural surface shown in Figure 27 also shows that the single-objective optimization significantly improves the radiated sound power on the volute surface at the fundamental frequency (BPF).



**Table 5.** The optimization results with the volute total mass invariable.

**Figure 27.** The radiated sound power spectrum of the volute structural surface.

In order to better understand the improvement effect of optimum panel thickness combinations on vibroacoustic performance before and after optimization, the comparison of distribution of vibrational sound radiation before and after the total mass invariable optimization at the fundamental frequency are given in Figures 28 and 29. From the comparison, it can be concluded that the sound pressure of the vibrational radiation close to the tongue on the side volute is greatly reduced by optimization, and

the other sound pressure of the vibrational radiation on the other area is also weakened to varying degrees. In addition to these results, the optimization also changed the directivity distribution of acoustical radiation, and very strong directivity was produced on the back panel side of the volute.

**Figure 28.** The distribution of vibrational sound radiation of original volute at the fundamental frequency. (**a**) The whole volute body; (**b**) XY section; (**c**) ZY section.

**Figure 29.** Distribution of vibrational sound radiation after the total mass invariable optimization at the fundamental frequency. (**a**) The whole volute body; (**b**) XY section; (**c**) ZY section.

The multi-objective optimization results, which took the volute surface vibrational radiated sound power and the total mass of the volute as the objective function of the optimization model, conformed to the actual needs. Multi-objective optimization solutions are not unique, and are usually presented in the form of solutions (Pareto front solutions). According to the distribution of Pareto front solutions in Figure 30, it can be seen that the radiated sound power on the volute surface and the total mass of the volute show an approximately linear inverse proportional relationship, and a smaller vibrational sound power can be obtained, when the volute mass (mass increase control: ± three kg) changes within a small range. Moreover, if the volute mass increased by two kg, the total acoustic power of the volute surface and the radiated sound power at BPF decreased by 7.3 dB and 6.9 dB, respectively, as shown in Table 6 and Figure 31. In addition to these, Figure 32 shows a comparison of the radiated sound power of the volute structure surface after multi-objective optimization. It can be seen that after optimization, the multi-objective optimization presents the same noise reduction effect on the volute structure surface as the single-objective optimization. In addition to these, the radiation sound pressure close to the tongue on the volute front panel is greatly reduced, the sound pressure is lower (under 10 dB) than at the other parts of the volute, and the sound pressure of the other volute parts is also greatly weakened. In summary, the radiated sound power of the volute structure surface obtained using the multi-objective optimization method is further reduced by nearly one dB compared to that of the single-objective optimization with mass constraints. It can be concluded that the optimization effect of the multi-objective method is obviously better than the single-objective optimization for this studied fan. Even considering the two conflicting objective functions, the multi-objective optimization can achieve a more balanced effect.

**Figure 30.** The distribution of Pareto front solutions.


**Table 6.** The multi-objective optimization results.

**Figure 31.** The frequency spectrum of total acoustic power of the volute surface.

**Figure 32.** The radiated sound power distribution of the volute structure surface after multi-objective optimization. (**a**) The whole volute body; (**b**) XY section; (**c**) ZY section.

Generally, the noise-reduction mechanism of single and multi-objective optimization is attributed to the major factor: the normal vibration velocity of the three-part volute surface, which was changed to be smaller by controlling the thickness combination with the unsteady aerodynamics determined and volute geometry fixed. The previous study [36] showed that the normal vibration velocity of the volute surface was the decisive factor that determined the radiated sound power. To comment on this major noise-reduction mechanism, the comparison of the normal vibration velocity of the volute surface before and after optimization were presented in Figures 18 and 33. It indicates that the normal vibration velocity of the volute surface was greatly diminished (especially the tongue region) after optimization, which inevitably leads to a significant reduction for the radiated sound power of the volute surface, as shown in Figure 32. In addition, the parametric analysis indicates that ST had the largest influence on sound radiated power of volute surface, followed by FT, and then BT. The sound radiated power of the volute surface could be sharply reduced if ST was much larger (larger than seven mm for this fan), while BT and FT were designed as lower values (the BT was less than 8.5 mm, and the FT was less than 7.0 mm for this fan). Less noise radiation could be achieved if the ST increased over a certain limit (the setting extremum), but this can lead to unpredictable mass gain and increased costs.

**Figure 33.** The distribution of the vibrational velocity of the volute casing surface at the fundamental frequency (BPF) after multi-optimization. (**a**) Side; (**b**) Back; (**c**) Front.

## **7. Conclusions**

To reduce this type of vibrational sound radiation, a vibrational noise control method of multi-disciplinary optimization that considered the influence of vibroacoustic coupling was proposed. The strategies employed in the vibroacoustic optimizations based on DoE and RBF optimization techniques were proved to be highly successful, and various optimal solutions were analyzed. Some preliminary conclusions are obtained in this paper as follows:

(1) The optimization results indicate that the three-part volute structure has an optimal thickness combination maintaining the volute mass constant, and the optimal design of the volute radiated sound power can be greatly reduced without any increase in material cost. Besides, the sensitivity analysis showed that ST is the most sensitive to the volute radiated sound power, followed by BT, and then FT, which is the smallest.

(2) The optimization process achieves the purpose of reducing the radiated sound power of the centrifugal fan volute. The radiated sound power on the volute casing surface decreased by 6.3 dB with mass constraint. Without a strict constraint of the volute mass, the optimization can be further applied to get a better thickness combination of the volute, thereby achieving better optimized vibrational noise results. The multi-objective optimization was more advantageous. It was found that the volute acoustical radiated power on the volute surface decreased by 7.3 dB when the total mass of the volute slightly increased (±3 kg). The optimization in this study provides an important technical reference for the design of low vibroacoustic volute centrifugal compressors and fans whose fluids should be strictly kept within the system without any leakage.

(3) In addition, the optimized thickness combination effectively reduces the normal vibration velocity of the volute surface, especially the volute tongue region, and thus significantly reduces the volute vibration radiation, which is also the noise reduction mechanism of this optimization method.

**Author Contributions:** J.Z. (Jianhua Zhang) and W.C. conceived the whole numerical simulations; J.Z. (Jinghui Zhang) contributed the optimization tools; Y.L. carried out the experimental validation and J.Z. (Jianhua Zhang) wrote the paper.

**Acknowledgments:** This research was supported by a grant from National Natural Science Foundation of China (No.51236006) and supported by Natural Science Basic Research Plan in Shaanxi Province of China (Program No. 2016JQ1043 and No. 2018JK0410).

**Conflicts of Interest:** The authors declare no conflict of interest.

## **Abbreviations**


## **References**


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