# Application of a Coupled Eulerian-Lagrangian Approach to the Shape and Force of Scientific Balloons
## 3 Quasi-Static Shape Solving During the Ascent Process
\begin{cases}101325\big{(}(288.15-0.0065z)/288.15\big{)}^{5.25577}&0\; \text{m}<z\leqslant 11000\;\text{m}\\ 22632\text{exp}(-(z-11000)/6431.62)&11000\;\text{m}<z\leqslant 20000\;\text{m}\\ 5474.87\big{(}(216.65+0.0010(z-20000))/216.65\big{)}^{-34.163}&20000\;\text{m}< z\leqslant 32000\;\text{m}\end{cases}  \tag{3}\]  
When the balloon reaches a certain height (\(z\)), the corresponding atmospheric pressure (\(P_{air}\)) at that height can be obtained. The mass of the helium inside the zero-pressure balloon (\(m_{he}\)) remains constant, and the pressure (\(\rho_{he}\)) is equal to the atmospheric pressure. According to the ideal gas law,  
\[P_{he}=\rho_{he}R_{g}T,V=m_{he}/\rho_{he}, \tag{4}\]  
we calculate the volume of helium inside the balloon at this height [25]. In the Eulerian domain that wraps the balloon, we select the corresponding volume of helium and apply an upward load to simulate buoyancy, as shown in Figure 6b. After it stabilizes, we calculate the balloon's shape at this height. By selecting different heights and repeating this process, the balloon's shape at various heights during its ascent can be calculated. After rotating the shapes and setting them as shell elements, we create the zero-pressure balloon. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Parameter** & **Value** & **Parameter** & **Value** \\ \hline Volume & 4185.33 m\({}^{3}\) & Surface area & 1256 m\({}^{2}\) \\ Arc length & 18.86 m & Maximum section diameter & 21.13 m \\ Altitude of balloon & 31.4 m & Thickness of balloon membrane & 23 m \\ Elastic modulus of membrane & 182 MPa & Poisson’s ratio & 0.3 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Design parameters of the zero-pressure balloon for simulation. As shown in Figure 7, from left to right, the volume of helium gas inside the membrane increases from small to large. After applying boundary conditions and waiting for stabilization, seven figures are captured. Each figure corresponds to different gas pressures inside the zero-pressure balloon as the helium gas expands. Comparing this with the changes in the external air pressure shown in Figure 5b, which lead to shape changes in the balloon, it can be seen that the simulation results obtained using the CEL method show that, as the internal helium volume increases, the volume of the balloon head expands, and the zero-pressure point along the balloon's axis gradually moves downward. This matches well with the numerical results. Therefore, the CEL method can be used to simulate the fluid-structure coupling problem in scientific balloon ascent, where there are dual internal and external flow fields. Additionally, the general contact established between the balloon membrane and the internal/external flow fields not only accounts for the coupling between the fluid and the membrane, but also calculates the self-contact of the membrane. When the balloon membrane is not fully expanded, it folds and contacts itself. The use of a 2D line-based numerical method is not suitable for the calculation and presentation of such results. The self-contact setting prevents convergence issues during the simulation due to membrane contact and clearly shows large deformations, folds, and wrinkles in the computed results.