# Helicopter Blade-Vortex Interaction Airload and Noise Prediction Using Coupling CFD/VWM Method
## 2 Computational Method
### Aerodynamic Model
#### 2.1.2 Viscous Wake Model  
As the wake movement is independent of the initial conditions after leaving the rotor at a distance, the Lagrangian-based viscous wake method [25] is adopted. In the VWM, the wake vorticity is discretized into a set of regular distributed vortex particles, and the process of wake rollup, distortion, and dissipation is numerically simulated with a Lagrangian description. The discrete vorticity field can be described by the vorticity kinematic and dynamic equations [21]:  
\[\frac{d\mathbf{x}_{p}(t)}{dt}=\mathbf{u}\big{(}\mathbf{x}_{p},t\big{)} \tag{2}\]  
\[\frac{d\mathbf{\Gamma}_{p}}{dt}=\mathbf{\mathrm{ff}}_{p}\cdot\
abla\mathbf{u}\big{(}\mathbf{x} _{p},t\big{)}+\
u\
abla^{2}\mathbf{\mathrm{ff}}_{p} \tag{3}\]  
where \(\mathbf{x}_{p}(t)\) is the position of the particle, \(\mathbf{\alpha}_{p}\) is the vector-valued total vorticity inside the particle \(p\), \(\mathbf{u}(\mathbf{x}_{p},t)=\mathbf{u}_{\infty}+\mathbf{u}_{p(ind)}\), \(\mathbf{u}_{\infty}\) is the free stream velocity, and \(\mathbf{u}_{p(ind)}\) is the local induced velocity. In the right-hand side of Equation (3), the first term is the stretching-effect term and the velocity field is analytically differentiated to solve the velocity gradient. The second term is the viscous diffusion term, and the Particle Strength Exchange (PSE) method [26] is utilized for the solution.  
For the viscous wake simulation, the blade surface is treated as a vorticity source that sheds the vorticity into the rotor wake. At one time-step, between the \(n\)-th blade azimuth and the \((n+1)\)-th blade azimuth, new vortex particles are generated behind each rotor blade for azimuthal and radial variation of the bound circulation. The radial and time derivatives of the bound vortex denote the circulation of the trailed and shed vortices, respectively. Therefore, the vorticity of new vortex particles is expressed as follows:  
\[\mathbf{\omega}=-\frac{d\mathbf{\Gamma}_{b}}{\mathrm{dt}}+\mathbf{v}_{b}\
abla\cdot\mathbf{ \Gamma}_{b} \tag{4}\]  
where \(\mathbf{\Gamma}_{b}\) is the blade-bound vortex circulation at the \((n+1)\)-th azimuth, and \(\mathbf{v}_{b}\) is the relative velocity of the blade section at the \((n+1)\)-th azimuth due to the free stream velocity, rotor rotation velocity, and blade flapping velocity. The total vorticity source \(\mathbf{\omega}\) is distributed onto an interpolation surface which represents the convected trajectory of the blade during that time-step. An equal annulus area rule is adopted for the blade spanwise discretization for aerodynamics. The vorticity variation of the new vortex particles shed into the rotor wake is governed by Equation (3) and the vorticity strength \(\mathbf{\alpha}_{p}\) in it is obtained by the following formulation:  
\[\mathbf{\alpha}_{p}=\mathbf{\omega}V_{p} \tag{5}\]  
where \(V_{p}\) is the volume of particle \(p\), which takes the product of the cross-sectional area and the length of the filament segment \(p\), as the new vorticity is discreted into a set of filament segments with the vortex core radius during the time-step.