# Numerical Simulation and Optimization of the Airflow Field of a Forage Drum Mower
## 2 Cutter Structure and Theory Analysis
### Theory Analysis of Airflow Field  
The airflow field around a single drum was first analyzed. The single drum is basically a cylinder so the flow field around is approximately regarded as pure circulating flow [33]. As shown in Figure 2, the magnitude of the velocity \(v\) of a fluid microcluster at any point around a rotating drum is close to being inversely proportional to its polar radius \(r\) to the rotating center of the drum. Therefore, it can be expressed as follows:  
\[v=\frac{c}{r} \tag{1}\]where \(c\) is a constant. The velocity \(v\) can be decomposed along the X- and Y-directions as follows:  
\[\left\{ \begin{array}{c}v_{x}=-\frac{c}{r}\cdot\sin\theta=-\frac{cy}{x^{2}+y^{2} }\\ v_{y}=\frac{c}{r}\cdot\cos\theta=\frac{cx}{x^{2}+y^{2}}\end{array} \right. \tag{2}\]  
where \(\theta\) represents the angle between the line from the fluid microcluster to the rotating center and the X-direction. By substituting the above equation into the fluid continuity equation, the following can be obtained:  
\[\frac{\partial v_{x}}{\partial x}+\frac{\partial v_{y}}{\partial y}=\frac{ \partial}{\partial x}(-\frac{cy}{x^{2}+y^{2}})+\frac{\partial}{\partial y}( \frac{cx}{x^{2}+y^{2}})=c\cdot\frac{0}{\left(x^{2}+y^{2}\right)^{2}}=0 \tag{3}\]  
According to the above equation, the airflow around a single drum meets the conditions required for continuous flow. For a single fluid microcluster, the angular velocity is expressed as follows:  
\[\omega=\frac{1}{2}(\frac{\partial v_{y}}{\partial x}-\frac{\partial v_{x}}{ \partial y})=\frac{1}{2}\bigg{[}\frac{\partial}{\partial x}(\frac{cx}{x^{2} +y^{2}})-\frac{\partial}{\partial y}(-\frac{cy}{x^{2}+y^{2}})\bigg{]}=\frac{ c}{2}\cdot\frac{0}{\left(x^{2}+y^{2}\right)^{2}}=0 \tag{4}\]  
Obviously, the airflow is an irrotational flow. In the context of a pure circulating flow, the velocity circulation of any curve enclosing the origin is constant and its magnitude is expressed as follows:  
\[\Gamma=\int_{L}v\cdot\mathrm{d}s=\int_{0}^{2\pi}\frac{c}{r}\cdot r\cdot d \theta=2\pi c \tag{5}\]  
Figure 1: Structure of double-drum cutter. 1. Ground 2. Twisted blade 3. Tilting disc 4. Conical drum 5. Guide plates. Figure 2: Airflow velocity around a rotating drum. The flow function \(\Psi(x,y)\), which can be derived from the above formula, is expressed as follows:  
\[\mathrm{d}\Psi=v_{x}\mathrm{d}y-v_{y}\mathrm{d}x=\frac{\Gamma}{2\pi}(-\frac{y}{x ^{2}+y^{2}})\mathrm{d}y-\frac{\Gamma}{2\pi}(\frac{x}{x^{2}+y^{2}})\mathrm{d}x=- \frac{\Gamma}{2\pi}\cdot\frac{\mathrm{d}y^{2}+\mathrm{d}x^{2}}{x^{2}+y^{2}} \tag{6}\]  
By integrating it, the flow function can be expressed as:  
\[\Psi=-\frac{\Gamma}{4\pi}\ln(x^{2}+y^{2})=-\frac{\Gamma}{2\pi}\ln\sqrt{x^{2}+ y^{2}}=-\frac{\Gamma}{2\pi}\ln r \tag{7}\]  
For the velocity potential function \(\varphi(x,y)\):  
\[\mathrm{d}\varphi=v_{x}\mathrm{d}x+v_{y}\mathrm{d}y=\frac{\Gamma}{2\pi}(- \frac{y}{x^{2}+y^{2}})\mathrm{d}x+\frac{\Gamma}{2\pi}(\frac{x}{x^{2}+y^{2}}) \mathrm{d}y=\frac{\Gamma}{2\pi}\cdot\frac{x\mathrm{d}y-y\mathrm{d}x}{x^{2}+y^ {2}}=\frac{\Gamma}{2\pi}\cdot\frac{d(\frac{y}{x})}{1+(\frac{y}{x})^{2}} \tag{8}\]  
By integrating it, the velocity potential function can be expressed as:  
\[\varphi=\frac{\Gamma}{2\pi}\mathrm{arctg}\frac{y}{x}=\frac{\Gamma}{2\pi}\theta \tag{9}\]  
From Equations (7) and (9), it can be seen that the lines of constant stream function of the airflow field around a single drum are a series of concentric circles centered on the rotational axis of the cutter and that the isopotential lines are half-rays from the center. The above theory provides a reference for the analysis of the airflow field around a single-drum cutter. However, the airflow field around the cutter is also affected by various factors, including the structure of the blades and guide plates in practice (e.g., the shape and the position of the cutter), while the distribution and parameters of the airflow field tend to be inconsistent with the results of theoretical analysis.