# High-Speed Vehicles in Low-Pressure Tunnels--Influence of Choked Flows
## 3 Sensitivity to Key Parameters
### Vehicle Parameters  
#### 3.1.1 Cross-Sectional Area  
Figure 7a,b shows pressures along vehicles with cross-sectional areas of 2.5 m\({}^{2}\) and 7.5 m\({}^{2}\), respectively. By inspection, this has a huge influence on the pressures, especially on the maximum values ahead of the vehicle. There are two key reasons for this. The more obvious of the two is simply that increases in vehicle area correspond to decreases in the area available for flow along the annulus. This reduces the rate of backflow and hence increases the rate of flow driven ahead of the vehicle, thereby increasing the rate of compression in that region. The less obvious of the two key reasons is that the period during which choked flow exists increases with increasing vehicle area. For the three cases presented (areas of 2.5, 5.0 and 7.5 m\({}^{2}\)) the durations are approximately 37, 80 and 104 s, respectively. During these periods, the backflow along a vehicle scales approximately with density. In this description, changes in the vehicle:tunnel blockage ratio correspond to differing vehicle areas in a given tunnel. They could alternatively arise for a given vehicle in tunnels with differing areas. From such comparisons, it has been found that the correlations with blockage ratio are close, but not exact. Small differences exist as a consequence of Reynolds number effects that depend upon size as well as shape. Nevertheless, for initial design purposes in any study, it will be sufficient to focus on blockage ratios instead of assessing the vehicle and tunnel areas independently. It is possible to compare data from these simulations with specific predictions from published 3-D CFD analyses when the latter are presented in aggregate forms devoid of 3-D details. Table 2 compares drag coefficients given by [PERSON] et al. (2021) [3] with corresponding values obtained from a series of simulations such as those presented in the base case and in the above figure. The chosen blockage ratios are those listed by [PERSON] et al., but the comparisons require interpolation from their data to give the vehicle Mach number used herein, namely 0.5825. This is the ratio of the vehicle speed (200 m/s) and the speed of sound in the ambient conditions (343.3 m/s). Table 2 by [PERSON] et al. includes data for vehicle Mach numbers of 0.5 and 0.6 so errors due to the interpolation will be quite small. For this purpose, the drag coefficient is evaluated as  
\[C_{d}=\frac{Drag\ force}{1/2\rho_{amb}A_{z}{U_{z}}^{2}} \tag{1}\]  
where the suffices \"_amb_\" and \"z\" denote ambient conditions and the vehicle, respectively. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Blockage**} & \multicolumn{3}{c}{**[PERSON] et al. (2021) [3]**} & \multicolumn{3}{c}{**Present 1-D Analysis**} \\ \cline{2-6}  & **Ma = 0.5** & **Ma = 0.6** & **Ma = 0.5825** & **at t = 50 s** & **at t = 75 s** \\ \hline
**0.2** & 0.21 & 1.29 & 1.20 & 1.21 & 1.36 \\
**0.3** & 1.50 & 3.01 & 2.89  
Some warnings are necessary regarding any conclusions drawn from these data. First, although the comparisons are based on vehicles with the same maximum cross-sectional areas, their longitudinal profiles differ strongly, and this will necessarily influence conditions in the wakes behind them. It is to be expected that drag forces obtained by 3-D CFD for a vehicle of uniform cross-section would also differ from those for the shaped vehicle reported in Table 2. Second, data from [PERSON] et al. [3] are presented for an asymptotic state, whereas the 1-D data correspond to conditions that are evolving in time. The importance of this can be inferred from the proportional difference between conditions at the beginning and end of the period of constant speed. This proportion increases with the blockage ratio, thereby implying that the time required to approach asymptotic conditions also increases. As a consequence of these differences, it is not safe to make quantitative comparisons between the two sets of data.