**Qiaorui Si 1, Biaobiao Wang 1, Jianping Yuan 1, Kaile Huang 1, Gang Lin <sup>1</sup> and Chuan Wang 1,2,\***


Received: 4 September 2019; Accepted: 25 October 2019; Published: 2 November 2019

**Abstract:** The radiated noise of the centrifugal pump acts as a disturbance in many applications. The radiated noise is closely related to the hydraulic design. The hydraulic parameters in the multistage pump are complex and the flow interaction among different stages is very strong, which in turn causes vibration and noise problems because of the strong hydraulic excitation. Hence, the mechanism of radiated noise and its relationship with hydraulics must be studied clearly. In order to find the regular pattern of the radiated noise at different operational conditions, a hybrid numerical method was proposed to obtain the flow-induced noise source based on Lighthill acoustic analogy theory, which divided the computational process into two parts: computational fluid dynamics (CFD) and computational acoustics (CA). The unsteady flow field was solved by detached eddy simulation using the commercial CFD code. The detailed flow information near the surface of the vane diffusers and the calculated flow-induced noise source was extracted as the hydraulic exciting force, both of which were used as acoustic sources for radiated noise simulation. The acoustic simulation employed the finite element method code to get the sound pressure level (SPL), frequency response, directivity, et al. results. The experiment was performed inside a semi-anechoic room with a closed type pump test rig. The pump performance and acoustic parameters of the multistage pump at different flow rates were gathered to verify the numerical methods. The computational and experimental results both reveal that the radiated noise exhibits a typical dipole characteristic behavior and its directivity varies with the flowrate. In addition, the sound pressure level (SPL) of the radiated noise fluctuates with the increment of the flow rate and the lowest SPL is generated at 0.8*Q*d, which corresponds to the maximum efficiency working conditions. Furthermore, the experiment detects that the sound pressure level of the radiated noise in the multistage pump rises linearly with the increase of the rotational speed. Finally, an example of a low noise pump design is processed based on the obtained noise characteristics.

**Keywords:** multistage centrifugal pump; flow-induced noise; numerical calculation; acoustic analogy

#### **1. Introduction**

Multistage pumps with high pressure are widely used in water supply facilities [1–4]. The flow in the multistage pump is highly unsteady and greatly influenced by the interference between the different stages [5,6]. These give rise to the pressure pulsations, mechanical vibrations, and the radiated noise in various pump components [7–10]. Especially, the intensive pressure pulsations are an important source of hydrodynamic excitation force, which in turn produces fluid borne noise and structure-borne noise. Further, fluid-borne noise is a major contributor to radiated noise as well as leading to increased fatigue in the system components. Additionally, the radiated noise is felt as a disturbance in some applications and many conditions are subject to stringent requirements concerning the limitation of noise. Therefore, the investigation of the radiated noises from the centrifugal pump is significant to pump designers.

Computational fluid dynamics (CFD) has been used widely in many engineering fields with the rapid development of computer technology [11–16]. Previous experimental and computational study of the noise generation in the centrifugal pump assured that the noise source in the centrifugal pump is composed of the mechanical noise and flow-induced noise [17–19]. The radiation noise of the pump system can be significantly reduced by increasing the machining accuracy and using components such as low noise motors. However, the low-frequency noise induced by the fluid flow is very difficult to be eliminated. Recently, computational fluid dynamics were widely used to predict the flow-induced noise and analyze its corresponding sound source [20,21]. By employing the Lighthill's acoustic analogy theory, Howe [22] concluded that the dipoles source is the main source of the flow-induced noise in the centrifugal pump. In addition, Kato [23] proposed a one-way coupled simulation method that combines CFD and structure analysis. Ding [24] and Si [25] used the combined CFD/ computational acoustics (CA) method to simulate the hydraulic noise of centrifugal pumps. Keller [26] theoretically and experimentally investigated the effects of the pump-circuit acoustic coupling on the blade-passing frequency perturbation induced by fluid-dynamic interaction between the rotor and the stator, which provides some theoretical supports for the above numerical works. By employing the large eddy simulation (LES) and the Ffowcs Williams and Hawkings (FW-H ) acoustic method, Gao [27] concluded that the design operation generates the lowest total sound pressure level. Jiang [28] reported the fluid-induced noise by using the Fluid-solid coupling method and regarded it as a reliable way to predict the fluid-induced noise in the rotating machinery. Sergey [29] simulated the flow-induced noise based on the acoustic-vertex method, which divided the fluid mechanics equation into the sound item and the vortex item. Rual [30] studied the unsteady flow near the tongue and found that the size of the mesh, the time step, and the turbulence model had great influence on the accuracy of the sound simulation. Furthermore, a hybrid method that transforms the information of the unsteady flow into the source of the sound is widely used [31–33]. This method could improve simulation accuracy with lower computational resources [34]. To be specific, the detached eddy simulation (DES) model, which uses the RANS model to simulate the flow near the wall and employs the LES model to simulate the flow far away from the wall, could improve the computational efficiency and produce equal accurate unsteady results [35]. For the acoustic simulation, the finite element method (FEM) method could predict the acoustic distribution at the low frequency well and show more tolerance to the geometric irregularity.

The present paper focuses on the numerical and experimental investigation of the characteristics of the radiated noise of the multi-stage pump. The sound source was extracted from the unsteady computation based on the DES turbulence model, and the simulation code of acoustic finite element method inlaid in software Actran 14 was used to simulate the distribution of the radiated noise of the model pump. In addition, the change of the radiated noise with the variation of the flow rate and the rotation speed is also discussed.

#### **2. Method and Basic Theory**

#### *2.1. Theory of the Acoustic Simulation*

Acoustic analogies are derived from the Navier–Stokes equations, which governed both flow field and the corresponding acoustic filed. The Navier–Stokes equations expressed as Formula (1) and (2) are rearranged into various forms of the inhomogeneous acoustic wave equation.

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

*Processes* **2019**, *7*, 793

$$\frac{\partial(\rho u\_i)}{\partial t} + \frac{\partial(\rho u\_i u\_j)}{\partial \mathbf{x}\_j} = -\frac{\partial p}{\partial \mathbf{x}\_i} + \frac{\partial \mathbf{c}\_{ij}}{\partial \mathbf{x}\_j} \tag{2}$$

where *eij* is the viscous stress tensor, and the last Lighthill function is expressed as:

$$\frac{\partial^2(\rho - \rho\_0)}{\partial t^2} - c\_0^2 \frac{\partial^2}{\partial \mathbf{x}\_i \partial \mathbf{x}\_j} (\rho - \rho\_0) = \frac{\partial^2 T\_{ij}}{\partial \mathbf{x}\_i \partial \mathbf{x}\_j} \tag{3}$$

where *Tij* is the stress tensor, and *Tij* = ρ*uiuj* − *eij* + δ*ij*[(*p* − *p*0) − *c*<sup>0</sup> <sup>2</sup>(<sup>ρ</sup> <sup>−</sup> <sup>ρ</sup>0)]; <sup>δ</sup>*ij* is the Kronecker function, ρ is the fluid density, ρ<sup>0</sup> is the undisturbed density, *c*<sup>0</sup> is the sound velocity, *t* is the time, *x* is the space coordinates, while the *i*, *j* represent the direction of the coordinate axis.

Furthermore, by employing several mathematical manipulations, the above function could be transformed as:

$$\int\_{\Omega} \left( \frac{\partial^2}{\partial t^2} (\rho - \rho\_0) \delta \rho + c\_0^2 \frac{\partial}{\partial \mathbf{x}\_i} (\rho - \rho\_0) \frac{\partial (\delta \rho)}{\partial \mathbf{x}\_i} \right) d\mathbf{x} = - \int\_{\Omega} \frac{\partial T\_{ij}}{\partial \mathbf{x}\_j} \frac{\partial (\delta \rho)}{\partial \mathbf{x}\_i} d\mathbf{x} + \int\_{\Gamma} \frac{\partial \sum\_{ij}}{\partial \mathbf{x}\_j} n\_l \delta \rho d\Gamma(\mathbf{x}) \tag{4}$$

The first term of the right is the volume source, and the second term is the surface source. In the acoustic simulation, the solution of the Navier-Stokes equation firstly assumes the water as incompressible to calculate the flow-induced acoustic source. Then, the compressibility of the water is to be considered to solve the acoustic wave propagation.

In addition, the description of the radiated noise is related to time and space. Mathematically, this relationship could be expressed as an acoustic wave equation. The process of the sound radiation meets the basic physical law, which is expressed as:

$$(\rho\_0 + \rho') \left(\frac{\partial}{\partial t} + \nu \nabla \right) \nu = -\nabla (p\_0 + p') \tag{5}$$

$$\frac{\partial(\rho\_0 + \rho')}{\partial t} = (\rho\_0 + \rho')q' - (\rho\_0 + \rho')\nabla\nu\tag{6}$$

$$p' = \frac{\gamma p\_0}{\rho\_0} \rho' + \frac{\gamma(\gamma - 1)}{2\rho\_0^2} (\rho')^2 \tag{7}$$

where, ρ0, *v*<sup>0</sup> and *p*<sup>0</sup> respectively represent the static density, static velocity, and static sound pressure. And ρ , *v* , *p* , and *q* , respectively represent the increment of the density, velocity, sound pressure, and the mass. With some mathematical treatment, the acoustic wave equation is expressed as:

$$
\nabla^2 p' = \frac{1}{c^2} \frac{\partial^2 p'}{\partial t^2} - \rho\_0 \frac{\partial q'}{\partial t} \tag{8}
$$

where <sup>∇</sup><sup>2</sup> <sup>=</sup> <sup>∂</sup><sup>2</sup> <sup>∂</sup>*x*<sup>2</sup> <sup>+</sup> <sup>∂</sup><sup>2</sup> <sup>∂</sup>*y*<sup>2</sup> <sup>+</sup> <sup>∂</sup><sup>2</sup> <sup>∂</sup>*z*<sup>2</sup> and *<sup>c</sup>* = γ*p*<sup>0</sup> ρ0 , which is the sound velocity in the fluid.

The vibration at any time could be decomposed as several simple harmonic vibrations expressed as:

$$p' = p(x, y, z)e^{\text{wt}} \tag{9}$$

$$q'=q(\mathbf{x},\mathbf{y},\mathbf{z})\varepsilon^{\text{jwt}}\tag{10}$$

A combination of the above three equations could deduct the acoustic wave equation expressed as;

$$
\nabla^2 p(\mathbf{x}, y, z) - k^2 p(\mathbf{x}, y, z) = -j\rho\_0 \omega \eta(\mathbf{x}, y, z) \tag{11}
$$

where *k* = <sup>2</sup>π*<sup>f</sup> <sup>c</sup>* and this value represents the number of the waves, ω is the angular velocity.

#### *2.2. DES Method*

The DES method is a modification of a RANS model in which the model switches to a subgrid-scale formulation in regions fine enough for LES calculations, which calculates the sound source information and simultaneously cuts down the cost of the computation. It is initially formulated for the Spalart–Allmaras model, which is expressed as:

$$\frac{D\vec{v}}{Dt} = \mathbf{c}\_{b1}\vec{\nabla}\vec{v} + \frac{1}{\sigma}[\nabla \cdot (\left(\upsilon + \vec{v}\right) \cdot \nabla \vec{v} + \mathbf{c}\_{b2}(\nabla \vec{v})]^2 - \mathbf{c}\_{w1}f\_w[\frac{\vec{v}}{d\_w}]^2\tag{12}$$

where the variables that represent turbulent motion are the quantities directly solved by the S-A equation. The relationship with the motion viscosity coefficient is defined as:

$$
v\_t = \frac{\mu\_t}{\rho} = \overline{v} f\_{v1} \tag{13}$$

where *f <sup>v</sup>*<sup>1</sup> is a dimensionless function defined as:

$$f\_{v1} = \frac{\chi^3}{\chi^3 + c\_{v1}^3}, \chi = \frac{\overline{v}}{v} \tag{14}$$

where *v* is an expression of the molecular viscosity generation term expressed as:

$$\bar{S} = f\_{\text{v3}} \mathcal{S} + \frac{\vec{v}}{\kappa^2 d\_w^2} f\_{\text{v2}} \tag{15}$$

where *S* is the absolute value of vorticity, and *f <sup>v</sup>*<sup>2</sup> and *f <sup>v</sup>*<sup>3</sup> express dimensionless functions, which is respectively expressed as:

$$f\_{\rm v2} = \left(1 + \frac{\chi}{c\_{\rm v2}}\right)^{-3}, f\_{\rm v3} = \frac{(1 + \chi f\_{\rm v1})(1 - f\_{\rm v2})}{\chi} \tag{16}$$

*f w* is expressed as:

$$f\_{\rm av} = g(\frac{1 + c\_{\rm w3}^6}{g^6 + c\_{\rm w3}^6}) \stackrel{\frac{1}{\delta}}{\text{ }} g = r + c\_{\rm w2}(r^6 - r), r = \frac{\tilde{v}}{\tilde{S}\kappa^2 d\_{\rm w}^2} \tag{17}$$

Constant value is evaluated as:

$$\begin{aligned} c\_{b1} &= 0.1355, c\_{b2} = 0.622, \sigma = 2/3, \kappa = 0.41, \\ c\_{w1} &= \frac{c\_{b1}}{\kappa^2} + \frac{1 + c\_{b2}}{\sigma}, c\_{w1} = 0.3, c\_{w3} = 2, c\_{v1} = 71 \end{aligned} \tag{18}$$

DES method is based on the S-A model which replaces the feature-length *dw* as:

$$d = \min(d\_{\text{w}}, \mathbb{C}\_{\text{DES}}\Delta) \tag{19}$$

where Δ = max(Δ*x*, Δ*y*, Δ*z*) represents the maximum grid distance, *CDES* = 0.6.

The default value of the software is adopted in the article. The size of the characteristic length is related to the scale of the grid, and the DES method is an S-A model of the LES simulation.

#### *2.3. Simulation Procedure*

Considering that the flow field inside the multistage pump is complex and the acoustic simulation domain is irregular, the finite element method is used in this acoustic simulation which could avoid the inaccuracy during the transformation of the sound source. Firstly, the computational flow domain and corresponding structured grids were prepared to process the steady simulation of the pump flow field based on the experimental inlet and outlet boundary conditions. Simultaneously, the test rig

was built to use the obtained pump performance data such as the pump head to verify or modify the numerical calculation model. The shear stress transport (SST) turbulence model embedded in ANSYS CFX 14.1 software (ANSYS, Inc., Commonwealth of Pennsylvania, USA) was repeatedly used at this step. Afterwards, the correct fluid calculation domain and the appropriate grid were obtained. Secondly, the unsteady simulation of the pump flow field was obtained using the DES method and the detailed flow information, such as flow velocity, pressure, density etc, was extracted and transformed as the sound source in the acoustic simulation. Meanwhile, the acoustic calculation domains were built and the acoustic meshes were generated containing an interface setting. Finally, the radiated noise calculation was completed by the acoustic finite element method using Actran12.0 software (Free Field Technologies MSC Software Company, Mont-Saint-Guibert, Belgium). The acoustic characteristics of the model pump were analyzed after experimental verification. The flow chart of the simulation work is shown in Figure 1.

**Figure 1.** The flow chart of the radiated noise calculation of the multi-stage centrifugal pump.

#### **3. Numerical Method**

#### *3.1. Study Object*

The study object is focused on a self-priming multistage pump type, which is widely used to feed water to the high floors of a building. The field measurement revealed that this type of multistage pump generated a higher sound pressure level than the technical requirements. The basic analysis of this problem showed the noise is caused by the hydraulic design. The geometry features of the example multistage pump with five stages in this study is shown in Figure 2a. Its inflow is axial and its outflow is along the radial direction. Each stage of this pump is composed of one impeller and one vane diffuser. The impeller constitutes seven backswept blades, and the vane diffuser is two-dimensional with 12 guiding vanes and 12 returning vanes. The design impeller rotation speed of the pump is 2800 r/min and the design flow rate of the pump is 8 m3/h. The single-stage head of the pump is 10 m and the total head is a superposition of the stages. The detailed hydraulic parameters of each stage are presented in Table 1. In order to simplify the model, this study only takes the three-stages hydraulic part of the pump and makes a prototype as shown in Figure 2b for the simulation and experimental test.

(**a**) The cutaway view of the multi-stage centrifugal pump prototype.

(**b**) Test prototype of the simplified model pump.
