# Effects of Tunnel and Its Ventilation Modes on the Aerodynamic Drag of a Subway Train
## 3 Measurement Results
### Measured \(\tau_{w}\)  
Since Equation (2) will be used to evaluate \(\tau_{w}\), it is necessary to confirm that the hotwire was submerged within the viscous sublayer. Since \(C_{f}\) ranges from 0.002-0.004 for most passenger trains [10], the \(C_{f}\)was chosen to be 0.003 for present estimation of wall shear stress. The wall shear stress at \(V=11.5\) m/s may be estimated by \(\tau_{w}=C_{f}\cdot 0.5\rho V^{2}\approx 0.23\) (N/m\({}^{2}\)). Substitute \(\tau_{w}\) into following equations,  
\[u_{\tau}=\sqrt{\tau_{w}/\rho}, \tag{5}\]  
\[\lambda=\frac{\
u}{u_{\tau}}, \tag{6}\]  
where \(u_{\tau}\) is the wall fiction velocity, \(\
u\) is kinematic viscosity of air, and \(\lambda\) is the wall unit. We may obtain \(\lambda\approx 36.11\times 10^{-3}\) mm. Since the boundary layer over the test train is fully turbulent, the viscous sublayer thickness is about 15\(\lambda\)[25], i.e., 0.54 mm. Therefore, the hotwire, mounted 0.41 mm away from the train roof surface (Section 2.1), was submerged within the viscous sublayer, thus providing a validation for the use of Equation (2) to estimate \(\tau_{w}\). Note that since the thickness of the viscous sublayer depends on the train speed, \(\tau_{w}\) may be underestimated based on Equation (2), if this thickness is smaller than 0.41 mm. However, this underestimate is expected to be about the same for different ventilation modes or inside and outside the tunnel, and therefore should not affect the qualitative comparison for \(\tau_{w}\) between different circumstances. Figure 6 presents the instantaneous wall shear stress on the first carriage, \(\tau_{wf}\), calculated from Equation (2), for the recirculation mode. The dependence on \(V\) of the instantaneous wall stress on the last carriage, \(\tau_{wl}\), was qualitatively similar to that of \(\tau_{wf}\), thus not presented. The second order polynomial curve fit (solid curve) for the data obtained using the least square method is also presented to facilitate data interpretation. Apparently, \(\tau_{wf}\) increases rapidly with \(V\), and its trend, as indicated by the fitting curve, is consistent with the fact that the aerodynamic drag is approximately proportional to \(V^{2}\)[2]. The fitting curves of \(\tau_{wf}\) and \(\tau_{wl}\), along with their average \(\tau_{wf}\) (\(=\frac{\tau_{wf}+\tau_{wl}}{2}\)), are presented in Figure 7 for the recirculation mode. The measured \(\tau_{wl}\) is slightly higher than \(\tau_{wf}\). For example, at \(V=11.5\) m/s, \(\tau_{wl}\) exceeds \(\tau_{wf}\) by approximately 13%. A similar observation is made for the free-cooling mode, as will be shown later. It is well known that \(\tau_{w}\) on a flat plate immersed in a turbulent boundary layer decreases with increasing distance from  
Figure 6: Instantaneous \(\tau_{wf}\) measured on the first carriage and the second order polynomial curve fit in the recirculation mode. the leading edge [26] because of the growing thickness of the boundary layer or viscous sublayer. The present observation may be ascribed to the tunnel blockage and the gap, about 0.9 m, between two adjacent carriages, which altered the boundary layer development along the train. There are seven gaps between the first and last carriages, which interrupt the growth of the boundary layer and viscous sublayer. As a matter of fact, a gap generates additional turbulence in flow surrounding the surfaces of the train. As such, the sublayer thickness on the last carriage may shrink [27; 28], resulting in larger \(\tau_{val}\) than \(\tau_{val}\). The result suggests that the boundary layer over the presently tested train surface differs from that over a continuous flat plate. The results of \(\tau_{val}\), \(\tau_{val}\) and \(\tau_{val}\) for the free-cooling mode are presented in Figure 8. The difference between \(\tau_{val}\) and \(\tau_{val}\) is reduced, compared with the recirculation mode; for example, \(\tau_{val}\) exceeds \(\tau_{val}\) by only approximately 8% at \(V=11.5\) m/s.