# Study of Solid Particle Erosion on Helicopter Rotor Blades Surfaces
## 2 Erosion Equations and CFD Setup  
### Erosion Equations  
The erosion of helicopter blade models was examined numerically using ANSYS 15.0 Fluent (Ansys Inc.). The Euler-Lagrangian method, fluid flow equation, and particle motion equation were examined using finite volume and discrete phase methods.  
To calculate changes in pressure, velocity, and kinetic energy in simulated models, conservation equations for mass and momentum are required. The Reynolds transformation form of the Navier-Stokes equation is necessary to solve the momentum and energy conservation equations. The generalized equation of state for the Reynolds-average Navier-Stokes equation is given by Equations (1) and (2):  
\[\frac{\partial\rho}{\partial t}+\frac{\partial}{\partial x_{i}}(\rho U_{i})=0 \tag{1}\]  
\[\frac{\partial}{\partial t}(\rho U_{i})+\frac{\partial}{\partial x_{i}}(\rho U _{i}U_{j})=-\frac{\partial P}{\partial x_{i}}+\frac{\partial}{\partial x_{j}} \bigg{[}\mu(\frac{\partial U_{i}}{\partial x_{j}}+\frac{\partial U_{j}}{ \partial x_{i}}-\frac{2}{3}\delta_{ij}\frac{\partial U_{k}}{\partial x_{k}}) \bigg{]}+\frac{\partial}{\partial x_{i}}(\overline{-\rho u_{k}^{\prime}u_{j} ^{\prime}}) \tag{2}\]  
Equations (3) and (4) describe the \(k\)-\(\omega\) SST turbulence model in CFD erosion analysis. In the \(k\)-\(\omega\) SST turbulence model, there are two important parameters: turbulent kinetic energy \(k\) and turbulent vortex frequency \(\omega\):  
\[\frac{\partial(\rho k)}{\partial t}+\frac{\partial(\rho U_{i}k)}{\partial x_ {i}}=\overline{P_{k}}-\beta^{\prime}\rho k\omega+\frac{\partial}{\partial x_{ i}}\bigg{[}(\mu+\alpha_{k}\mu_{t})\frac{\partial k}{\partial x_{i}}\bigg{]} \tag{3}\]  
\[\begin{array}{ll}\frac{\partial(\rho\omega)}{\partial t}+\frac{\partial(\rho U _{i}\omega)}{\partial x_{i}}&=\alpha\rho S^{2}-\beta\rho\omega^{2}+\frac{ \partial}{\partial x_{i}}\Big{[}(\mu+\alpha_{\omega}\mu_{t})\frac{\partial \omega}{\partial x_{i}}\Big{]}\\ &\qquad\qquad\qquad+2(1-F_{1})\rho\sigma_{\omega^{2}}\frac{1}{\alpha}\frac{ \partial k}{\partial x_{i}}\frac{\partial\omega}{\partial x_{i}}\end{array} \tag{4}\]  
where \(\rho\) is the fluid density, \(\mu_{t}\) is the turbulent viscosity, \(\mu\) is the dynamic viscosity, \(F_{1}\) is the first blending function, and \(F_{2}\) is the second blending function. The values of these constants are as follows [35]:  
\(\beta^{*}=0.09\), \(\alpha_{1}=5/9\), \(\beta_{1}=3/40\), \(\alpha_{k1}=0.85\). \(\alpha_{\omega 1}=0.5\), \(\alpha_{2}=0.44\), \(\beta_{2}=0.0828\), \(\sigma_{k2}=1\), \(\sigma_{\omega 2}=0.856\).  
The discrete phase method and the Euler-Lagrangian method were used to solve Equation (5) for particle motion [36]:  
\[\frac{dV_{P}}{dt}=F_{D}(U-V_{P})+\frac{g(\rho_{P}-\rho)}{\rho_{P}}+F_{others} \tag{5}\]  
In the above formula,  
\[F_{D}=\frac{18\mu C_{D}\text{Re}}{\rho_{P}d_{P}^{2}\cdot 24} \tag{6}\]  
where \(F_{D}\) is the drag force per unit mass of the particle; \(U\) is fluid velocity; \(V_{P}\) is velocity; \(\rho_{P}\) is particle density; \(g\) is gravitational acceleration; \(d_{p}\) is particle diameter; \(C_{D}\) is drag coefficient; \(F_{\text{others}}\) is other forces per unit mass, and \(R_{e}\) is relative Reynolds number.  
Erosion wear refers to the phenomenon that the solid particles or the fluid are corroded, worn, etc. caused by the surface of the material. The wear rate is usually used to measure the amount of wear of the material. The expression of wear rate is Equation (7) [37]:  
\[R_{erosion}=\sum_{p=1}^{N_{particles}}\frac{m_{p}C(d_{p})f(\alpha)v^{b(v)}}{A_{ face}} \tag{7}\]  
where \(m_{p}\) is particle quality; \(C(d_{p})\) is particle diameter function; \(\alpha\) is particle impact material surface angle; \(f(\alpha)\) is impact angle function; \(v\) is particle impact relative speed; \(b(v)\) is relative speed function;\(A_{fac}\) is particles on the surface of the material erosion wear area; and \(N_{particles}\) is the number of particles hitting the material at the wear. The unit of \(R_{erosion}\) is \(kg/(m^{2}/s)\).