**3. The numerical method**

The Lattice Boltzmann Method is used in this paper to model fluid flow past a high-speed train with two barriers. The velocity of the two-dimensional nine-speed (D2Q9) model in multiple directions can be defined as [23]:

$$e\_i = \begin{cases} (0,0) & (i=0) \\ \cos\left(\frac{i\pi}{2} - \frac{\pi}{2}\right), \sin\left(\frac{i\pi}{2} - \frac{\pi}{2}\right). \mathbf{c} & (i=1,...,4) \\ \sqrt{2}\left(\cos\left(\frac{i\pi}{2} - \frac{9\pi}{4}\right)\right), \sin\left(\frac{i\pi}{2} - \frac{9\pi}{4}\right). \mathbf{c} & (i=5,...,8) \end{cases} \tag{1}$$

where *c* ¼ Δ*x=*Δ*t* is the velocity of lattice, Δ*x* is the lattice space, Δ*t* is the time step and *i* is the different direction.

The governing equation in the lattice Boltzmann method is:

$$f\_i(\mathbf{x} + e\_i \Delta t, t + \Delta t) = f\_i(\mathbf{x}, t) + \frac{\Delta t}{\tau\_v} \left[ f\_i^{eq}(\mathbf{x}, t) - f\_i(\mathbf{x}, t) \right] \tag{2}$$

where *fi* is the distribution function and *f eq <sup>i</sup>* is the equilibrium distribution function, which can be calculated according to:

$$f\_i^{eq} = w\_i \rho \left[ \mathbf{1} + \frac{e\_i u}{c\_s^2} + \frac{\mathbf{1}}{2} \frac{(e\_i u)^2}{c\_s^4} - \frac{\mathbf{1}}{2} \frac{u^2}{c\_s^2} \right] \tag{3}$$

where *wi* are the weights, *w*<sup>0</sup> ¼ 4*=*9, *w*<sup>1</sup>�<sup>4</sup> ¼ 1*=*9, *w*<sup>5</sup>�<sup>8</sup> ¼ 1*=*36.

The distribution functions can be used to obtain the macroscopic variables,

$$
\rho = \sum f\_i \tag{4}
$$

$$
\rho u = \sum\_{i} \mathfrak{e}\_{i} \mathfrak{f}\_{i} \tag{5}
$$

With the multi-scaling expansion, the mass and the moment equations can be obtained:

*The Influence of Inclined Barriers on Airflow Over a High Speed Train under Crosswind… DOI: http://dx.doi.org/10.5772/intechopen.112751*

$$\frac{\partial \rho}{\partial t} + \nabla \bullet (\rho u) = \mathbf{0} \tag{6}$$

$$\frac{\partial \rho}{\partial t} + \nabla \cdot (\rho u u) = -\nabla P + \theta \left[\nabla^2 (\rho u) + \nabla (\nabla \cdot (\rho u))\right] \tag{7}$$

More information on the present numerical method can be found in the authors' previous paper [19].

In this study, Wall Modelled Large Eddy Simulation (WMLES) [24] approach was used to consider the turbulence. LES has been proved as a compliant numerical approach in computing and simulating unsteady turbulent flows. WMLES takes the wall models into account and its primary idea is that the near-wall turbulence length scales grow linearly with the wall distance, leading to the smaller and smaller eddies as the wall is approached.

The aerodynamic coefficients in terms of non-dimensional characteristics are described by EN 14067-1 as follows:

$$\text{Fores} : F = \frac{\rho U\_{\infty}^2}{2} \bullet A.\text{C} \tag{8}$$

$$\text{Momentums}: M = \frac{\rho U\_{\infty}^2}{2} \bullet A \bullet l.\text{C} \tag{9}$$

$$\text{Pressure} : P - P\_{\infty} = \frac{\rho U\_{\infty}^2}{2} \bullet C\_p \tag{10}$$

*U*<sup>∞</sup> is the free stream velocity, *P* is the local static pressure, and *P*<sup>∞</sup> is the free-stream static pressure.
