## 3 Numerical Modeling
### Friction between Piston and Wall  
The implementations in the L1d code assume that the friction force is simply proportional to the pressure exerted either at the back or the front of the piston, and that the piston is a perfectly elastic body [27]. However, this method is specified not from physical considerations, but to produce the best agreement between the calculated and experimental values of object velocity or gas pressure. An improved model with a physical basis has been implemented for the Longshot by accounting for the inertia loads within the piston [29]. Since different types of pistons are used in different facilities; modifications were made based on the Longshot method to obtain a better agreement in the present study. Details are described below. The longitudinal stresses \(\sigma_{x}\) of a moving piston are obtained by considering successive static stresses (due to equal pressures applied on both ends of the piston) and dynamic stresses (due to the remaining pressure difference and the corresponding acceleration of the piston). The longitudinal stresses within a piston in arbitrary motion are then obtained by summing the static contribution and the dynamic one, as expressed by Equation (9). \[\sigma_{x}~{}=~{}\min\Big{(}p_{b},p_{f}\Big{)}+\frac{4F_{inertia}}{\pi D^{2}} \tag{9}\]  
where \(D\) is the diameter of the piston, \(F_{inertia}=m_{p}|a_{p}|\) are the piston inertia forces during the acceleration. According to [PERSON]'s law generalized to cylindrical coordinates, the strain-stress relationship in an isotropic medium are:  
\[\varepsilon_{r}~{}=~{}\frac{1}{E}(\sigma_{r}-\
u(\sigma_{\theta}+\sigma_{x})) \tag{10}\]  
\[\varepsilon_{\theta}~{}=~{}\frac{1}{E}(\sigma_{\theta}-\
u(\sigma_{r}+\sigma_{ x})) \tag{11}\]  
where \(E\) is [PERSON]'s modulus of the material, \(\sigma_{x}\) and \(\varepsilon_{r}\) are the radial stresses and strain, \(\sigma_{\theta}\) and \(\varepsilon_{\theta}\) are the azimuthal stresses and strain. For the specific case where the piston is bounded by the tube, only axial and radial deformations are allowed. As long as the diameter of the piston is larger than that of the tube, differences in the diameter \(\Delta r\) exist between the piston and tube, and then the radial and angular strains are obtained as follows:  
\[\varepsilon_{r}~{}=~{}\frac{\partial u_{r}}{\partial r}\approx\frac{\Delta r} {R_{p}}~{}=~{}\frac{2\Delta r}{D_{p}} \tag{12}\]\[\varepsilon_{\theta}\ =\ \frac{u_{r}}{r}+\frac{1}{r}\frac{\partial u_{\theta}}{ \partial\theta}\approx\frac{\Delta r}{R_{p}}\ =\ \frac{2\Delta r}{D_{p}} \tag{13}\]  
where \(R_{p}\) is the radius of the piston, and \(D_{p}\) is the diameter. Thus, the radial stress can be obtained from Equations (9)-(13). Besides, an upper limit for the local stresses \(\sigma_{r}\) may be associated to the maximum stresses allowable before yield would take place. \[\sigma_{r}\ =\ \min\left(\sigma_{yield},\frac{2E\Delta r}{D_{p}(1-v)}+\frac{v}{ 1-v}\left(\min\left(p_{b},p_{f}\right)+\frac{4F_{inertia}}{\pi D_{t}^{2}}\right)\right) \tag{14}\]  
The normal force \(F_{N}\) corresponding to the radial stresses applied on all friction surfaces of the piston is defined as:  
\[F_{N}=\sigma_{r}\ \pi D_{p}L_{f} \tag{15}\]  
where \(L_{f}\) is the length of the piston in contact with the tube. The friction force \(F_{f}\) is then assumed to be proportional to the normal force \(F_{N}\):  
\[F_{f}=\mu_{f}\ F_{N} \tag{16}\]  
The piston consists of PTFE and its Poisson's ratio \(v\) is 0.4; the sliding friction coefficient \(\mu_{f}\) of steel-PTFE is in the range of 0.04-0.05. [PERSON]'s modulus is about 0.5 GPa, and the yield stress is about 20 MPa [30]. At room temperature, the diameter of the piston is manufactured to be larger than that of the tube. However, freezing of the piston at 0 \({}^{\circ}\)C for at least four hours before the experiments is required to be able to insert the piston into the tube.