# Demand Law of Fabric Weight on the Airflow Velocity of a Gas-Assisted Model
## 2 Model Establishment and Simulation
### Turbulent Kinetic Energy and the Turbulent Dissipation Rate  
There are a large number of turbulent flows in nature. This kind of flow is very complex, and it has significant practical applications. The laminar flow is stable and forms turbulent flow. The most obvious feature of turbulence is its randomness [15]. The gas-assisted model is not only affected by many factors, such as the Reynolds number and incoming turbulence intensity, but these influencing factors are also interrelated and restrict each other [16]. In the process whereby the air enters the outside through the pipe and then exerts pressure on the conveyed fabric to prevent it from falling, it must first be considered whether the flow state of the air is turbulent or laminar [17]. Through the criterion of the flow state--the critical Reynolds number--the flow Reynolds number Re can be calculated  
Figure 2: Inner and outer domain meshing: (**a**) simulation of the internal and external domains; (**b**) mesh division.  
and compared with the critical Reynolds number to judge the flow state [18]. The formula for the flow Reynolds number is  
\[Re=\frac{\rho vL}{\mu} \tag{7}\]  
where \(\rho\) is the density of the fluid, \(v\) is the velocity of the fluid, \(L\) is the characteristic length, and \(\mu\) is the viscosity coefficient. \(\rho\) is 1.29 kg/m\({}^{3}\), \(L\) is 220 mm, and \(\mu\) is 1.46 \(\times\) 10\({}^{-5}\) Pa\(\cdot\)s. [19].  
\[v=\frac{1.43}{\sin 40^{\circ}}=2.22\ \text{m/s} \tag{8}\]  
The above data can be substituted into the formula to obtain the Reynolds number:  
\[Re=\frac{\rho vL}{\mu}=\frac{1.29\times 2.22\times 0.22}{1.46\times 10^{-5}}=43,2 44.22 \tag{9}\]  
The critical Reynolds number is 4300, judgment criteria by fluid state, when the Reynolds number exceeds 4300, it can be determined that the fluid state is turbulent at this time. Turbulence intensity describes the degree of wind speed change with time and space; it reflects the relative intensity of the fluctuating wind speed and is the most important characteristic quantity for describing the characteristics of atmospheric turbulence [20].  
There are two main reasons for turbulence. One is that, when the air is flowing, it will be rubbed or blocked by the ground's roughness. The other is the vertical movement of the airflow caused by the difference in the air weight and atmospheric temperature [21]. According to experience, turbulence intensity can be solved using the following formula:  
\[I=0.16 Re^{-\frac{1}{8}}=0.16\times 53,261.10^{-\frac{1}{8}}=4.11\times 10^{-2} \tag{10}\]  
Studying the laws of turbulent motion plays an increasingly important role in solving practical turbulence problems [22]. Accurately solving the turbulent flow field has always been one of the main goals of fluid mechanics. The turbulent kinetic energy is estimated by using the turbulence intensity above as  
\[K=\frac{3}{2}(UI)^{2}=\frac{3}{2}\times\ (1.14\times 4.18\times 10^{-2})^{2}=3.41 \times 10^{-3} \tag{11}\]  
When the turbulent kinetic energy is transferred from the outer scale to the inner scale and reaches a dynamic equilibrium, the turbulent kinetic energy dissipation rate can be estimated by the flux of the turbulent kinetic energy in the inertial subregion [23]. The turbulence scale is estimated as  
\[\varepsilon=C_{\mu}^{\frac{3}{4}}\frac{k^{2}}{l}=0.09^{\frac{3}{4}}\times \frac{\left(3.41\times 10^{-3}\right)^{\frac{3}{2}}}{0.4\times 0.008}=1.02\times 10^{-2} \tag{12}\]  
See Table 1 for the first simulation input data of the first pipeline.  
The pressure distribution obtained from the simulation is shown in Figure 3.  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Simulation Parameters** & **Value** \\ \hline Pipe outlet speed in \(x\)-direction \(V_{x}\) (m/s) & 1.43 \\ Pipe outlet \(y\)-direction speed \(V_{y}\) (m/s) & 1.70 \\ Turbulent kinetic energy (m\({}^{3}\)/s\({}^{2}\)) & 0.00341 \\ Turbulent kinetic energy dissipation rate (m\({}^{2}\)/s\({}^{2}\)) & 0.0102 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Simulation parameters.