# Aerodynamic Simulation of Helicopter Based on Polyhedron Nested Grid Technology
## 2 CFD Simulation Method  
### Governing Equation and Solution Method  
CFD simulation examples in this paper are all based on the Reynolds average Navier-Stokes (N-S) equation in the conservation form, as shown in Equation (1):  
\[\frac{\partial}{\partial t}\iint\limits_{V}\stackrel{{\rightarrow} }{{W}}dV+\iint\limits_{S}\left(\stackrel{{\rightarrow}}{{F}}_{c}- \stackrel{{\rightarrow}}{{F}}_{v}\right)\stackrel{{ \rightarrow}}{{n}}dS\ =\ 0 \tag{1}\]  
where, \(\stackrel{{\rightarrow}}{{W}}=\left[\rho,\rho u,\rho v,\rho w,\rho e \right]^{T}\) is the conservation variable, \(\stackrel{{\rightarrow}}{{F}}_{c}=\left(f,g,h\right)\) is the convection flux, and \(\stackrel{{\rightarrow}}{{F}}_{v}=\left(a,b,c\right)\) is the viscous flux. These parameters can be expressed as:  
\[f\ =\ \left[ \begin{array}{c}\rho u\\ \rho u^{2}+p\\ \rho uv\\ \rho uw\\ \rho pe+up\end{array} \right]\,\ g\ =\ \left[ \begin{array}{c}\rho v\\ \rho uv\\ \rho v^{2}+p\\ \rho vw\\ \rho ve+vp\end{array} \right]\,\ h\ =\ \left[ \begin{array}{c}\rho w\\ \rho uw\\ \rho vw\\ \rho w^{2}+p\\ \rho ve+wp\end{array} \right] \tag{2}\]  
\[a\ =\ \left[ \begin{array}{c}0\\ \tau_{xx}\\ \tau_{yx}\\ \tau_{zx}\\ \tau_{zx}\\ u\tau_{xy}+v\tau_{yx}+w\tau_{zx}-q_{x}\end{array} \right]\,b\ =\ \left[ \begin{array}{c} 0\\ \tau_{xy}\\ \tau_{zy}\\ \tau_{zy}\\ \tau_{zy}\\ u\tau_{xy}+v\tau_{yy}+w\tau_{zy}-q_{y}\end{array} \right]\ c\ =\ \left[ \begin{array}{c}0\\ \tau_{xz}\\ \tau_{yz}\\ \tau_{zz}\\ u\tau_{xz}+v\tau_{yz}+w\tau_{zz}-q_{z}\end{array} \right] \tag{3}\]In the viscous flux, the stress can be expressed as:  
\[\tau_{xx} = \frac{2}{3}\mu\bigg{(}2\frac{\partial u}{\partial x}-\frac{ \partial w}{\partial y}-\frac{\partial w}{\partial z}\bigg{)},\ \tau_{xy}\ =\ \tau_{yx}\ =\ \mu\bigg{(}\frac{\partial w}{\partial x}+\frac{ \partial u}{\partial y}\bigg{)},\ q_{x}\ =\ -\text{K}\frac{\partial T}{\partial x}\] \[\tau_{yy} = \frac{2}{3}\mu\bigg{(}2\frac{\partial w}{\partial y}-\frac{ \partial u}{\partial x}-\frac{\partial w}{\partial z}\bigg{)},\ \tau_{yz}\ =\ \tau_{zy}\ =\ \mu\bigg{(}\frac{\partial w}{\partial y}+\frac{ \partial w}{\partial z}\bigg{)},\ q_{y}\ =\ -\text{K}\frac{\partial T}{ \partial y} \tag{4}\] \[\tau_{zz} = \frac{2}{3}\mu\bigg{(}2\frac{\partial w}{\partial z}-\frac{ \partial w}{\partial x}\bigg{)},\ \tau_{xz}\ =\ \tau_{zx}\ =\ \mu\bigg{(}\frac{\partial w}{\partial x}+\frac{ \partial u}{\partial z}\bigg{)},\ q_{z}\ =\ -\text{K}\frac{\partial T}{ \partial z}\]  
By introducing the relationship between pressure and temperature, the closed equations can be obtained:  
\[\text{p}\ =\ \rho(\gamma-1)\bigg{[}e-\frac{1}{2}\Big{(}u^{2}+v^{2}+w^{2} \Big{)}\bigg{]} \tag{5}\]  
\[\text{p}\ =\ \rho\text{RT} \tag{6}\]  
Among them, \(\rho\) represents the density of air, p represents the pressure, \(e\) represents the total energy per unit mass of the gas, \(\mu\) represents the viscosity coefficient, K represents the thermal conductivity coefficient, \(T\) represents the temperature, \(q_{x}\), \(q_{y}\), and \(q_{z}\) represent the components of the heat flow along three directions, \(\gamma\) represents the specific heat capacity, for ideal gas, \(\gamma\ =\ 1.4,\ \text{R}\ =\ 287.3\). The Spalart-Allmaras (S-A) turbulence model [18] is adopted in the simulation of this paper. This model is a single-equation turbulence model, which does not derive the precise transport equation of the vortex-viscosity coefficient, but a simplified equation obtained by referring to experimental data. Thanks to Spalart's convenience in aviation calculation, rich experience, and large amount of test data, the S-A model has been widely applied in aviation, especially in shock wave capture. Different interpolation schemes are formed because of different discrete strategies for inviscid flux on the interface of grid cells. The Roe format [19] of flux difference splitting (FDS) in the third-order upwind format is adopted in this paper. This format has strong shock capture capability and is a method to calculate inviscid flux by approximating the Riemann solution.