# Vortex-Induced Vibration of Deep-Sea Mining Pipes: Analysis Using the Slicing Method
## 2 Theoretical Analysis
### Governing Equations and Boundary Conditions  
Derived from fundamental principles such as the conservation of mass, [PERSON]'s second law of motion, and the conservation of energy, the foundational equations of fluid mechanics encompass the equations for mass conservation, momentum, and energy. In scenarios where thermal gradients do not play a significant role, such as with the flow of incompressible viscous fluids, temperature variations are generally negligible. Consequently, for the analysis within the fluid dynamics domain under these conditions, the equations primarily considered are those pertaining to the conservation of momentum and mass.  
(1) Mass conservation equation  
When the mass of a particle remains the same over a certain period of time, and the mass of the fluid in it remains stable, then the motion of the particle follows the principle of mass conservation. The specific mass conservation formula can be expressed as Formula (3):  
\[\frac{\partial\rho}{\partial t}+\
abla\cdot\left(\rho\overset{\rightarrow}{ \mathrm{U}}\right)=0, \tag{3}\]  
where \(t\) represents time, \(\
abla\) is the Hamiltonian operator, and \(\overset{\rightarrow}{\mathrm{U}}\) represents the velocity vector of the fluid.  
For incompressible fluid, the density is constant, and we can simplify the above formula:  
\[\
abla\cdot\overset{\rightarrow}{\mathrm{U}}=\frac{\partial\mathrm{u}}{ \partial\mathrm{x}}+\frac{\partial v}{\partial\mathrm{y}}+\frac{\partial \mathrm{w}}{\partial\mathrm{z}}=0, \tag{4}\]where the components of \(\overset{\rightarrow}{\mathrm{U}}\) in the x, y, and z directions are represented by u, v, and w, respectively.  
(2) Momentum conservation equation  
The resultant force acting on a particle is equal to the increase in momentum on the particle, expressed as follows:  
\[\frac{\overset{\rightarrow}{\mathrm{DU}}}{\mathrm{Dt}}=\overset{ \rightarrow}{\mathrm{F}}-\frac{1}{\rho}\
abla\overset{\rightarrow}{\mathrm{P} }+\
u\
abla^{2}\overset{\rightarrow}{\mathrm{U}}, \tag{5}\]  
where \(\frac{\overset{\rightarrow}{\mathrm{DU}}}{\mathrm{Dt}}\) is the random derivative operator, \(\overset{\rightarrow}{\mathrm{F}}\) is the mass force, \(\
abla\) represents the Hamiltonian operator, and \(\
u\) is the viscosity coefficient of fluid motion.  
(3) Turbulence model and solution method  
[PERSON] [21] proposed the SST k-\(\omega\) model, which is particularly suitable for calculating fluids with reverse pressure differences [12], and which consists of two components, the dissipation rate (\(\omega\)) and the turbulent kinetic energy (k):  
\[\rho\frac{\partial\mathrm{k}}{\partial\mathrm{t}}+\mathrm{k}\rho\frac{ \partial\mathrm{u_{i}}}{\partial\mathrm{x_{i}}}=\frac{\partial}{\partial \mathrm{x_{i}}}\left[\Gamma_{\mathrm{k}}\frac{\partial\mathrm{k}}{\partial \mathrm{x_{j}}}\right]+\widetilde{\mathrm{G}}_{\mathrm{k}}-\mathrm{Y_{k}}+ \mathrm{S_{k}}, \tag{6}\]  
\[\rho\frac{\partial\omega}{\partial\mathrm{t}}+\rho\omega\frac{\partial \mathrm{u_{i}}}{\partial\mathrm{x_{i}}}=\frac{\partial}{\partial\mathrm{x_{j} }}[\Gamma_{\omega}\frac{\partial\omega}{\partial\mathrm{x_{i}}}]+\mathrm{G}_{ \omega}-\mathrm{Y_{\omega}}+\mathrm{D}_{\omega}+\mathrm{S_{\omega}}, \tag{7}\]  
The turbulence simulation method for numerical simulation analysis in this paper is based on the SST k-\(\omega\) model of the Reynolds mean method, coupled with the application of overlapping mesh and dynamic mesh.