# Investigation on Numerical Simulation of VIV of Deep-Sea Flexible Risers
## 2 Numerical Simulation Method of VIV of Riser  
### Fluid-Domain Governing Equations  
Viscous fluid flow is the macroscopic incompressible motion of fluid that abides by the laws of mass conservation, momentum conservation and energy conservation and is subject to the governing equations of the fluid. In this paper, we did not consider the energy conversion caused by temperature change during simulation; hence, we did not use the energy equation. Viscous fluid flow satisfies the Navier-Stokes (N-S) equation, and we employed the Reynolds-averaged numerical simulation (RANS) method. In this method, the Reynolds-averaged N-S equation is used to describe the fluid flow instead of the N-S equation. In Cartesian coordinates:  
Continuity equation:  
\[\frac{\partial\overline{u_{i}}}{\partial x_{i}}=0 \tag{1}\]  
Momentum equation:  
\[\rho\frac{\partial\overline{u_{i}}}{\partial t}+\rho\frac{\partial\overline{u_ {i}u_{j}}}{\partial x_{j}}=-\frac{\partial\overline{p}}{\partial x_{i}}+\mu \
abla^{2}\overline{u_{i}}-\rho\frac{\partial\overline{u^{\prime}_{i}u^{ \prime}_{j}}}{\partial x_{j}} \tag{2}\]  
where \(i\) and \(j\) take values of 1, 2 or 3, which, respectively, represent the components of variables in the X, Y and Z directions; \(P\) denotes the pressure; \(\
abla\) is the Hamiltonian operator; \(\overline{u}_{i}\) and \(\overline{u}_{j}\) stand for the time-average flow rate; and \(\overline{u^{\prime}}_{i}\) and \(\overline{u^{\prime}}_{j}\) signify the time-average value of the pulsating flow rate.  
Compared with the N-S equation, the RANS equation adds a group of unknown Reynolds stress terms which makes the RANS equation an unclosed equation. To solve the RANS equation, we must add a new supplementary equation known as the turbulence model.  
The boundary layer gradient around a flow is large, and an obvious pressure gradient and shearing action exist. Therefore, in this paper, we use the shearing pressure transmission turbulence model. This model was proposed by [PERSON] [25] in 1993 and can better capture the flow around and wake vortex near the wall. The model includes two equations: one for turbulent kinetic energy \(k\) and one for dissipation rate \(\omega\).  
\[\rho\frac{\partial k}{\partial t}+k\rho\frac{\partial u_{i}}{\partial x_{i}}= \frac{\partial}{\partial x_{j}}\left(\Gamma_{k}\frac{\partial k}{\partial x_ {j}}\right)+\widetilde{G}_{k}-Y_{k}+S_{k} \tag{3}\]  
\[\rho\frac{\partial\omega}{\partial t}+\rho\omega\frac{\partial u_{i}}{ \partial x_{i}}=\frac{\partial}{\partial x_{j}}\left(\Gamma_{\omega}\frac{ \partial\omega}{\partial x_{j}}\right)+G_{\omega}-Y_{\omega}+D_{\omega}+S_{\omega} \tag{4}\]  
where \(\widetilde{G}_{k}\) denotes the turbulence kinetic energy produced by the average velocity gradient, \(G_{\omega}\) signifies the turbulent kinetic energy generated by the equation, \(\Gamma_{k}\) and \(\Gamma_{\omega}\) represent the effective diffusivities of \(k\) and \(\omega\), \(Y_{k}\) and \(Y_{\omega}\) stand for turbulence terms due to diffusion, \(D_{\omega}\) is the orthogonal divergence term, and \(S_{k}\) and \(S_{\omega}\) are the source phases of the equation.  
To solve the governing equation of the fluid, we used the finite volume method in discrete form. We used the Fluent solver, which employs the coupled algorithm and overlapping grid technology. The second-order upwind scheme was used for the transient equation, and the fully implicit scheme was used for time dispersion.