# Numerical Investigation on Handling Stability of a Heavy Tractor Semi-Trailer under Crosswind
## 2 Numerical Method and Validation  
### Governing Equations  
Computational fluid dynamics (CFD), which solves the coupling of the Reynolds averaged Navier-Stokes (RANS) equations to a turbulence model, has proven to be a very powerful and practical tool [17; 18; 19; 20; 21]. Two-equation models, such as \(\kappa-\varepsilon\)[22], \(\kappa-\omega\)[23] and the shear stress transport (SST) model [24], are the favorable options for CFD calculations. In the current work, the Renormalization Group (RNG) \(\kappa-\varepsilon\) turbulence model [25] was selected to investigate the flow around the heavy truck, where the flow field induced was three-dimensional, turbulent and incompressible. It has been shown to better predict the separating flows and is widely used in engineering applications. For example, using two-equation models, previous scholars conducted research on street canyon flow [26], large tank fire [27; 28], internal combustion engine [29; 30] and heat exchange [31; 32]. The corresponding governing equations are as follows:  
Reynolds average continuity equation:  
\[\frac{\partial u_{i}}{\partial x_{i}}=0, \tag{1}\]  
RANS equations:  
\[\frac{\partial u_{i}}{\partial t}+u_{j}\frac{\partial u_{i}}{\partial x_{j}}= -\frac{1}{\rho}\frac{\partial p}{\partial x_{i}}+\frac{1}{\rho}\frac{\partial} {\partial x_{j}}\left(\mu\frac{\partial u_{i}}{\partial x_{j}}-\overline{u^{ \prime}_{i}u^{\prime}_{j}}\right) \tag{2}\]  
where \(u\) and \(u^{\prime}_{i}\) are mean and fluctuating velocities, respectively; subscripts \(i\), \(j\) = 1, 2, 3 stand for three different directions, x, y and z; \(\rho\) is the density, \(p\) represents the mean pressure and \(\mu\) denotes the dynamic viscosity.  
The last terms of the right side of Equation (2) represent the Reynolds stress using the approach of the eddy viscosity principle after Boussinesqq:  
\[-\overline{u^{\prime}_{i}u^{\prime}_{j}}=v_{t}\left(\frac{\partial u_{i}}{ \partial x_{j}}+\frac{\partial u_{j}}{\partial x_{i}}\right)-\frac{2}{3}k \delta_{ij}, \tag{3}\]  
where \(v_{t}\) is the turbulence eddy viscosity, \(k=\frac{1}{2}\overline{v^{\prime}_{i}v^{\prime}_{i}}\) is the turbulence kinetic energy and \(\delta_{ij}\) is the Kronecker delta. The turbulence eddy viscosity could be computed by the function of turbulence kinetic energy (\(k\)) and turbulence dissipation rate (\(\varepsilon\)):  
\[v_{t}=C_{\mu}\frac{k^{2}}{\varepsilon} \tag{4}\]  
\[\frac{\partial k}{\partial t}+u_{j}\frac{\partial k}{\partial x_{j}}=\frac{ \partial}{\partial x_{j}}\left[\left(v+\frac{v_{t}}{\varepsilon_{k}}\right) \frac{\partial k}{\partial x_{j}}\right]+G-\varepsilon \tag{5}\]\[\frac{\partial\varepsilon}{\partial t}+u_{j}\frac{\partial \varepsilon}{\partial x_{j}}=\frac{\partial}{\partial x_{j}}\bigg{[}\bigg{(}v+ \frac{v_{t}}{\sigma_{\varepsilon}}\bigg{)}\frac{\partial\varepsilon}{\partial x _{j}}\bigg{]}+C_{\varepsilon 1}\frac{\varepsilon}{k}G-C_{\varepsilon 2}\frac{ \varepsilon^{2}}{k}-\frac{C_{\mu}\eta^{3}(1-\eta/\eta_{0})}{1+\beta\eta^{3}} \frac{\varepsilon^{2}}{k} \tag{6}\] \[G =-\overline{u_{i}^{\prime}u_{j}^{\prime}}\frac{\partial u_{i}}{ \partial x_{j}}=v_{t}\bigg{(}\frac{\partial u_{i}}{\partial x_{j}}+\frac{ \partial u_{j}}{\partial x_{i}}\bigg{)}\frac{\partial u_{i}}{\partial x_{j}} \tag{7}\]  
where \(\eta=(k/\varepsilon)(G/v_{t})^{0.5}\), \(\sigma_{k}=\sigma_{\varepsilon}=0.7149\) is the turbulent Prandtl number in the \(k\) and the \(\varepsilon\) equation [33], \(C_{\mu}=0.0845\), \(C_{1\varepsilon}=1.42\), \(C_{2\varepsilon}=1.68\), \(\eta_{0}=4.38\), \(\beta=0.012\).  
The solutions of the governing equations were carried out in STAR-CCM+. The steady-state, implicit scheme was utilized in the incompressible flow solver. The governing equations were discretized by using a vertex centered unstructured finite volume method. The segregated solver was used in the computations. A second-order scheme was applied for the spatial discretization.