# Study on Influencing Factors of Helicopter Brownout Evolution Based on CFD-DEM
## 2 Numerical Simulation Theory  
### Governing Equation  
In computational fluid dynamics, the computer obtains the physical parameters of the flow field by solving the governing equations, which are generally considered as the mass conservation equation, momentum conservation equation and energy conservation equation [16].  
The mass conservation equation is also called a continuity equation. According to the law of mass conservation, the mass of fluid passing through the surface of volume in the flow field is consistent with the variable controlling the mass of fluid in the volume.  
\[\dot{\
ot{P}}_{CS}\rho v_{n}dA+\frac{\partial}{\partial t}\iint_{CV}\rho dV=0 \tag{1}\]where \(\rho\) is the density of the fluid, \(v_{n}\) is the normal velocity, \(A\) is the surface area of the control volume, and \(V\) is the volume The continuity equation of any point in the flow field can be deduced by using the above integral form of mass conservation equation:  
\[\frac{\partial\rho}{\partial t}+div\Big{(}\rho\overrightarrow{v}\Big{)}=0 \tag{2}\]  
This is the continuous differential equation of unsteady compressible flow.  
For steady flow:  
\[\frac{\partial\rho}{\partial t}=0 \tag{3}\]  
For incompressible flow, \(\rho\) is a constant  
\[div\left(\overrightarrow{v}\right)=\frac{\partial v_{x}}{\partial x}+\frac{ \partial v_{y}}{y}+\frac{\partial v_{z}}{z}=0 \tag{4}\]  
In the following cases, the rotor tip Mach number is much less than 1, and it could be approximately considered as an unsteady incompressible flow.  
A momentum conservation equation, known as a Navier-Stokes (N-S) equation, refers to the change rate of fluid momentum relative to time in the fluid micro cluster, which is the sum of external forces on the fluid micro cluster. The formula is as follows:  
\[\frac{\partial\rho\overrightarrow{U}}{\partial t}+div\left(\rho \overrightarrow{UU}\right)=\rho\overrightarrow{f}-grad(p)+div\left( \overrightarrow{\tau}\right) \tag{5}\]  
where \(\overrightarrow{f}\) is the volume force, \(\rho\overrightarrow{U}\) is the mass flow rate, \(p\) is the pressure, and \(\overrightarrow{\tau}\) is the shear stress tensor.  
For incompressible fluid, the above formula can be simplified and expressed as follows:  
\[\frac{dv_{x}}{dt}=f_{x}-\frac{1}{\rho}\frac{\partial p}{\partial x}+\
u\left( \frac{\partial^{2}v_{x}}{\partial x^{2}}+\frac{\partial^{2}v_{x}}{\partial y ^{2}}+\frac{\partial^{2}v_{x}}{\partial z^{2}}\right) \tag{6}\]  
\[\frac{dv_{y}}{dt}=f_{y}-\frac{1}{\rho}\frac{\partial p}{\partial y}+\
u\left( \frac{\partial^{2}v_{y}}{\partial x^{2}}+\frac{\partial^{2}v_{y}}{\partial y ^{2}}+\frac{\partial^{2}v_{y}}{\partial z^{2}}\right) \tag{7}\]  
\[\frac{dv_{z}}{dt}=f_{z}-\frac{1}{\rho}\frac{\partial p}{\partial z}+\
u\left( \frac{\partial^{2}v_{z}}{\partial x^{2}}+\frac{\partial^{2}v_{z}}{\partial y ^{2}}+\frac{\partial^{2}v_{z}}{\partial z^{2}}\right) \tag{8}\]  
where the kinematic viscosity \(\
u=\mu/\rho\) is the ratio of dynamic viscosity to density.  
The above are the governing equations of continuous medium hydrodynamics. The computer divides the flow field into a large number of grids based on the finite volume method and solves them. In some cases, it is necessary to add other equations for closure.