# High-Speed Vehicles in Low-Pressure Tunnels--Influence of Choked Flows
## 3 Sensitivity to Key Parameters
### Thermodynamic Influences
#### 3.2.1 Heat Transfers at Tunnel and Vehicle Surfaces  
Perhaps the most important item in this category is heat transfer with the surfaces of the vehicle and tunnel. Two phenomena contribute, namely (i) convective heat transfer between the air and the surfaces and (ii) heat conduction within the surrounding material. The evolving surface temperatures are determined by a balance between these two. This is achieved in the software by solving equations for the two phenomena simultaneously. At both surfaces, convection coefficients are estimated using the Reynolds analogy for quasi-steady flows. The methods used to represent conduction within the tunnel and vehicle walls are different. Transient heat conduction equations are solved in the tunnel walls, assuming uniform wall properties, but a lumped thermal mass approximation is used to represent the vehicle. The principal reason for using different approaches is that the numerical grids in the air and the wall are fixed relative to the tunnel. Accordingly, no interpolation is required in communication between them. In contrast, this is not possible on the surfaces of vehicles that are moving along the tunnel. This is a decisive matter because the evaluation of information at fixed locations along a vehicle would necessarily involve interpolations. Errors due to this cause would interact with time-dependent quantities and, at best, make the solutions unreliable. It is far from certain that significantly better predictions would be obtained from typical 3-D software even though the use of sliding grids would be likely to reduce interpolation issues by moving the location of interpolations away from the surface. However, the detailed determination of convection coefficients depends on the choice of turbulence models and this challenge has not been fully mastered even for skin friction coefficients, let alone heat transfer coefficients (see [PERSON] et al. 2014 [15], for instance). Relatively few turbulence models make satisfactory allowance for rapidly accelerating flows and even fewer can do so when the same simulation also includes extensive regions of slowly varying flows. In practice, in the time scales of single journeys that are the focus herein, the expected rates of heat transfer will induce only small changes in surface temperatures of either the tunnel or the vehicles. This is because (i) the air density is small so the associated heat transfer coefficients are small and (ii) the thermal capacitances of the tunnel and vehicle are much larger than that of the air. Accordingly, it will normally be acceptable to regard the surface temperatures as constant. In contrast, however, the small density also implies small thermal capacitance and so the small heat transfers can potentially cause large changes in air temperature, albeit over larger time scales than those associated with the dominant pressure changes. This is illustrated in Figure 8, which compares the base case with alternative simulations in which at least one of the surfaces is treated as adiabatic. By inspection, this has a strong influence on the shapes of the temperature histories as well as on the maximum values. The differences increase with increasing time because (i) the time available for heat transfers to occur with the tunnel wall before the vehicle arrives increases with increasing time, (ii) the influence of heat transfers is cumulative, and (iii) the dominant direction of heat transfer is from the air to the surfaces. Figure 8: _Cont._  
The top row of the figure shows conditions when heat transfers are suppressed at both surfaces. In this idealised condition, the maximum temperature just ahead of the train is in excess of 100 \({}^{\circ}\)C and those further along the tunnel are even larger. In contrast, when heat transfers with the tunnel wall are modelled (Figure 8c,d), the corresponding values are much smaller.