# High-Speed Vehicles in Low-Pressure Tunnels--Influence of Choked Flows
## 1 Introduction  
The aerodynamics of trains in tunnels have been studied extensively during the past century, with the result that most of the phenomena that need to be considered by designers are well understood. Whilst it remains the case that some cannot be predicted with high confidence (e.g., wavefront steepening in long tunnels and its consequences for the emission of micro-pressure waves), all important ones are well understood in principle (e.g., [PERSON] et al. 2020 [1]). The same cannot safely be said for vehicles travelling at high speeds in tunnels that have been evacuated to very low pressures, notwithstanding the existence of studies providing valuable guidance (e.g., [PERSON] et al. 2019 [2]). No operational systems of this type currently exist, although extensive effort continues to be devoted to assessing many aspects of them--e.g., [PERSON], et al. (2021) [3], [PERSON], et al. (2021) [4], [PERSON] and [PERSON] (2020) [5], [PERSON] and [PERSON] (2021) [6]. Also, whereas various authors have addressed the aerodynamic behaviour in detail, most focus on flows local to the vehicle itself, usually with the primary purpose of reducing \"form\" drag--e.g., [PERSON] et al. (2017) [7], [PERSON] et al. (2012) [8], [PERSON] and [PERSON] (2018) [9] and [PERSON] (2012) [10]. It is therefore considered that a generic assessment of the relevant aerodynamics could be of value, and this paper is intended as a contribution to that process.  
The author's first involvement in the design of this type of railway was in connection with the Swissmetro project in the 1990s (Swissmetro 1999 [11]). Early plans for this related solely to Switzerland, although the designers subsequently had ambitions for major extensions of the scheme through much of Europe--as is also envisaged today. Initially, the intention was to operate the tunnels at very low pressure, primarily to reduce aerodynamic drag--and hence also the required power. However, the target ambient pressure increased as the project developed and pressures in the order of 10 kPa (i.e., 10% of an atmosphere) became seen as preferable. This was the result of many factors, not all of which related to airflows generated by the vehicles. Herein, however, only the latter are considered. This is because the paper is not intended to be indicative of any particular system or even anyparticular conceptual design. Instead, its purpose is to provide background information for persons responsible for studying such systems.  
Differences from studies of high-speed trains in conventional railways can be illustrated by asking questions such as:  
* Will high vehicle speeds cause sonic airspeeds and hence choked flow conditions?
* Will the low ambient pressures ensure relatively low maximum pressures?
* Will the low ambient pressures ensure relatively low maximum temperatures?
* Will the low ambient pressures have a strong influence on heat flows?
* How will multiple vehicles interact aerodynamically?
* How beneficial will it be to link tunnels with vehicles travelling in opposite directions?  
Readers who are able to answer such questions with high confidence need not read further. Others should expect to find some useful pointers although it is re-emphasised that the aim is to provide generic guidance, not to suggest the relative importance of the various phenomena in specific design studies.