## 2 Research Object and Methods
### Theory of the Proposed CFD/CFA  
Reasonable numerical simulation methods can make up for the shortcomings of experiments and assist theoretical analysis in revealing the nature of physical phenomena. The proposed experiment-assisted CFD/CFA method in this paper is based on the [PERSON] acoustic analogy theory. Both hydrodynamics and hydroacoustic of the flow field are governed by the compressible Navier-Stokes equations. A direct numerical simulation (DNS) of these equations is impractical on most engineering problems, which calls for an alternative approach. One alternative consists of using an acoustic analogy, as first proposed by [PERSON] in the following form [22]:  
\[ \begin{array}{l}\frac{\partial^{2}}{\partial t^{2}}(\rho-\rho_{0})-{c_{0}}^{ 2}\frac{\partial^{2}}{\partial x_{i}\partial x_{j}}(\rho-\rho_{0})=\frac{ \partial^{2}T_{ij}}{\partial x_{i}\partial x_{j}}\\ T_{ij}=\rho v_{i}v_{j}+\delta_{ij}[(p-p_{0})-{c_{0}}^{2}(\rho-\rho_{0})]-\tau_ {ij}\end{array}  \tag{1}\]  
where \(\rho\) represents the density of the fluid, \(\rho_{0}\) denotes density at rest, \(v\) is velocity, \(p\) is pressure, \(T\) denotes the [PERSON]'s tensor, \(c_{0}\) is the speed of sound at rest, and \(\tau\) is the viscous stress tensor. In practice, an acoustic analogy uses a two-step procedure. In the first step, an unsteady flow analysis is used to compute hydrodynamics sources. The second step consists in computing the propagation and radiation of the fluid noise sources. The implementation of the second step can be achieved by  
Figure 1: Three-dimensional structure of high-speed centrifugal pump model. using frequency-domain formulation. Harmonic perturbations are considered in such a way that any perturbed quantity \(p\) can be written as  
\[p(x,t)=Re(\overline{p}(x)e^{iwt}) \tag{2}\]  
Away from the source region, the density fluctuations, \(\rho-\rho_{0}\), correspond to acoustic density fluctuations \(\rho_{\rm c}\). Then, [PERSON]'s equation can be rewritten using the frequency domain transform. \[\frac{\partial^{2}\overline{\rho_{c}}}{\partial x_{i}\partial x_{i}}+k^{2} \overline{\rho_{c}}=-\frac{1}{c_{0}}\frac{\partial^{2}\overline{T}_{ij}}{ \partial x_{i}\partial x_{j}} \tag{3}\]  
where \(k=\omega/c_{0}\) denotes the acoustic wavenumber. Equivalently, [PERSON]'s analogy in a frequency domain computation requires the Fourier transform of \(T\) in the source term, which means every frequency component of \(T\) is a function of all frequency components of \(v\). The weak variational statement as follows. \[\int\limits_{\Omega}{(k^{2}\overline{\rho_{c}}\delta\rho-\frac{\partial \overline{\rho_{c}}}{\partial x_{i}}\frac{\partial\delta\rho}{\partial x_{i}}- \frac{1}{c_{0}^{2}}\frac{\partial\overline{T}_{ij}}{\partial x_{j}}\frac{ \partial\delta\rho}{\partial x_{i}})dx}=0 \tag{4}\]  
In general, CFD simulation is used to compute \(T\) instead of direct acoustic calculation because of using the finite volume method, which differs from FEM about the way to compute derivatives. Moreover, the CFD mesh is usually finer than the acoustic mesh, which leads to a better approximation of derivatives \(\partial\overline{T}_{ij}/\partial x_{j}\). Through interpolation, the frequency domain sound source information is transferred to the acoustic grid, and the second step of the acoustic calculation is performed using the finite element/infinite element method in the present work. The flow chart of the computational process is shown in Figure 2. The proposed CFD/CFA method consists of three parts, namely, flow field solving, structure domain solving, and experimental supporting. Firstly, the computational flow domain and corresponding meshes were prepared to process the steady simulation. Simultaneously, the test rig was built to simulate the condition of the engine-cooling system and give out the inlet and outlet boundary conditions for the CFD. The obtained pump performance data, such as the pump head and efficiency, are used to verify or modify the appropriate grid and the numerical calculation model.