# Impact of Different Nose Lengths on Flow-Field Structure around a High-Speed Train
## 2 Methodology
### Numerical Method  
At present, numerous scholars have used the IDDES (based on the \(k\)-\(\omega\) shear-stress transport turbulence model) method to study the unsteady aerodynamic performance around a train. For example, [PERSON] et al. [21] compared the accuracy of different turbulence models in predicting the slipstream. The results show that the slipstream velocity obtained with the IDDES method has good consistency with experimental data. [PERSON] et al. [22] used the IDDES method to study the influence of ground conditions on the slipstream velocity of a HST. [PERSON] et al. [23] studied the effects of ground configurations on the aerodynamic performance of the HST using the IDDES method. [PERSON] et al. [3] and [PERSON] et al. [12] studied the effects of train nose length on aerodynamic performance by the IDDES method. This paper also uses the IDDES (based on \(k\)-\(\omega\) shear-stress transport) method to study the flow field around a HST.  
The IDDES combines the advantage of delayed detached-eddy simulation (DDES) with an improved RANS (Reynolds-averaged Navier-Stokes)-LES hybrid model aimed at wall modelling in LES (WMLES) [24]. Because the IDDES combines DDES with WMLES, the length-scale must be redefined. The IDDES length-scale is defined as:  
\[l_{hyd}=\overline{f_{d}}(1+f_{e})l_{RANS}+(1-\overline{f_{d}})l_{LES}, \tag{1}\]  
where the RANS length-scale \(l_{\text{RANS}}=\kappa^{1/2}/(C_{\mu}\omega)\), the LES length-scale \(l_{\text{LES}}=C_{\text{DES}}\Psi\Delta\), the blending function, \(\overline{f_{d}}\), is defined as \(\overline{f_{d}}=\max\{(1-f_{dt}),f_{\text{B}}\}\), and \(f_{\text{dt}}\) is the delay function defined in [PERSON]'s literature [25]. \(f_{\text{B}}\) is an empirical blending-function, which varies from 0 to 1, and \(f_{e}\) is an elevating function, which is aimed at preventing the excessive reduction of the RANS Reynolds stresses.  
In addition, the new sub-grid length-scale is defined by:  
\[\Delta=\min\{\max[C_{w}d_{w},\ C_{w}h_{\max},\ h_{\text{wn}}],\ h_{\max}\} \tag{2}\]  
and  
\[h_{\max}=\max\{h_{x},\ h_{y},\ h_{z}\}, \tag{3}\]  
where \(h_{wn}\) is the grid step in the wall-normal direction and \(C_{w}\) is an empirical constant set as 0.15. \(h_{x}\), \(h_{y}\), and \(h_{z}\) are the local streamwise, wall-normal, and lateral cell sizes, respectively. For more information, see [PERSON] et al. [24].  
An implicit-unsteady segregated incompressible finite-volume solver was used in this study. The convection term used hybrid second-order upwind and bounded-central differencing (hybrid-BCD) and the temporal scheme was selected as second order for this numerical simulation. Because of the Mach number (\(Ma=U_{\infty}c\), where the upstream velocity \(U_{\infty}=60\) m/s and \(c\) is local velocity, which is equal to 340 m/s), \(Ma=0.176<0.3\) and the gas density is constant. In this study, the initial solver used steady RANS, which can make the residuals converge more easily and shorten the calculation time. For the unsteady numerical simulation calculation, the physical time step is \(0.062\ t^{*}\) (\(t^{*}=tU_{\infty}/H\)), and each time step is iterated 20 times and total number of time steps is 4000. The residual of each turbulence equation can be achieved \(10^{-4}\) in each time step. The STAR-CCM+ of the National Supercomputing Center (Wuxi, China) was used for calculation. There are altogether six cases in this calculation, with each case calculated by eight nodes.