# Analysis of Acoustic Flow Field Characteristics of Key Components of High-Speed Train Bogies
## 2 Calculation Model
### Theoretical Basis of Acoustic Flow Field Calculation
#### 2.1.1 Basic Equations of Control for Fluid Dynamics  
Since the acoustic field is a form of the flow field, it is also governed by the equations of fluid dynamics, especially more so in the aspect of describing fluid noise. Therefore, in the study of the noise generated by the fluid, there is need for acoustic and hydrodynamic calculations to solve the problem. In hydrodynamics, through the law of conservation of mass, [PERSON]'s second law, and the law of conservation of energy of the three basic principles of physics obtained by the equation of continuity, the momentum equation and the energy equation is the controlling equations of hydrodynamics, in which the description of the motion of the field of motion is very important. In this section, a brief introduction to the motion of fluid will be introduced [27]. This section will briefly introduce the basic equations of fluid motion. 1. Continuity equation-derived from the law of conservation of mass  
The continuity equation represents a specific manifestation of the law of mass conservation in fluid dynamics, which is employed to describe the preservation of mass during fluid motion. It is based on the utilization of the continuum medium model for fluids [28], wherein the fluid is perceived as a continuous and uniformly distributed medium, with both velocity and density being the continuously differentiable functions of spatial coordinates and time. In all instances of fluid motion, adherence to the law of mass conservation is observed, and the continuity equations derived from this principle can be applied even in non-constant flows. The corresponding differential expressions are provided as the following equation [29]:  
\[\frac{\partial\rho}{\partial t}+\frac{\partial(\rho u)}{\partial x}+\frac{ \partial(\rho v)}{\partial y}+\frac{\partial(\rho w)}{\partial z}=0, \tag{1}\]  
where the corresponding fluid velocity components \(u\), \(v\), and \(w\) in the coordinate \(x\), \(y\), and \(z\) directions. 2. Momentum equation-derived from the law of conservation of momentum ([PERSON]'s second law) is shown in the following equations [30]: \[\frac{\partial u}{\partial t}+u\frac{\partial u}{\partial x}+v\frac{\partial u }{\partial y}+w\frac{\partial u}{\partial z}=f_{x}-\frac{1}{\rho}\frac{\partial p }{\partial x}+v\bigg{(}\frac{\partial^{2}u}{\partial x^{2}}+\frac{\partial^{ 2}u}{\partial y^{2}}+\frac{\partial^{2}u}{\partial z^{2}}\bigg{)},\] (2) \[\frac{\partial v}{\partial t}+u\frac{\partial v}{\partial x}+v\frac{ \partial v}{\partial y}+w\frac{\partial v}{\partial z}=f_{y}-\frac{1}{\rho} \frac{\partial p}{\partial y}+v\bigg{(}\frac{\partial^{2}v}{\partial x^{2}}+ \frac{\partial^{2}v}{\partial y^{2}}+\frac{\partial^{2}v}{\partial z^{2}} \bigg{)},\] (3) \[\frac{\partial v}{\partial t}+u\frac{\partial v}{\partial x}+v\frac{ \partial v}{\partial y}+w\frac{\partial v}{\partial z}=f_{y}-\frac{1}{\rho} \frac{\partial p}{\partial y}+v\bigg{(}\frac{\partial^{2}v}{\partial x^{2}}+ \frac{\partial^{2}v}{\partial y^{2}}+\frac{\partial^{2}v}{\partial z^{2}} \bigg{)},\] (4) where \(f_{x}\), \(f_{y}\), and \(f_{z}\) correspond to the force per unit mass of each axis, i.e., the p-surface force distribution function, and v is the fluid kinematic viscosity [31]. 3. Energy equation (also known as Bernoulli's equation)-derived from the law of conservation of energy  
The energy equation in fluid mechanics describes how energy transforms within a fluid by considering factors such as density, temperature, internal energy, and other relevant parameters. This equation reflects the fundamental principle of conserving energy by incorporating the internal energy associated with circulation processes. Consequently, changes in fluidic energetics are considered while still upholding the law of conservation of energy.