## 2 Background
### CFD Simulations  
Computational Fluid Dynamics (CFD) is a branch of fluid mechanics that utilises the computational power of modern computers to model fluid flow (Figure 3). The essence of the method is to numerically solve partial differential equations that govern the conservation of mass, momentum and energy of fluids. In the study of bridge aerodynamics, the air is often assumed to be incompressible, Newtonian and isothermal. Therefore, the air is governed by the momentum equation, Equation (1), and the continuity equation, Equation (2),  
\[\frac{\partial\mathbf{u}}{\partial t}+\mathbf{u}\cdot\
abla\mathbf{u}=-\
abla \mathbf{p}+\
u\
abla^{2}\mathbf{u}+\mathbf{g} \tag{1}\]  
\[\
abla\cdot\mathbf{u}=0 \tag{2}\]  
in which, \(\mathbf{u}\) is the velocity vector, \(\mathbf{p}\) is the kinematic pressure, \(\
u\) is the kinematic viscosity and \(\mathbf{g}\) is the gravity as a source term. Equations (1) and (2) are known as the Navier-Stokes equations. By solving these two equations, the fluid velocity and fluid pressure at points of interest in a flow can be determined. Simulations based on Equations (1) and (2) directly are called direct numerical simulations (DNSs). Although DNSs are very accurate in solving simple flow problems, they are inefficient for practical problems due to the significant requirement for computational power. In practice, the Reynolds-Averaged Navier-Stokes (RANS) formulation is often used as a replacement of the original Navier-Stokes equations,  
\[\frac{\partial\mathbf{u}}{\partial t}+\overline{\mathbf{u}}\cdot\
abla \overline{\mathbf{u}}=-\
abla\overline{\mathbf{p}}+v\
abla^{2}\overline{ \mathbf{u}}+\mathbf{g}+\
abla\cdot(-\overline{\mathbf{u}\mathbf{u}}) \tag{3}\]  
Figure 3: Domain of a Computational Fluid Dynamics (CFD) simulation of a bridge. \[\
abla\cdot\overline{\mathbf{u}}=0 \tag{4}\]  
where \(\overline{\mathbf{u}}\) is the time-averaged velocity, \(\overline{\mathbf{p}}\) is the mean kinematic pressure and \(\mathbf{u}^{{}^{\prime}}\) is the fluctuation velocity. Compared to Equation (1), Equation (3) has an extra term \(\
abla\cdot(-\overline{\mathbf{u}}\overline{\mathbf{u}}\overline{\mathbf{u}})\), in which \((-\overline{\mathbf{u}}\overline{\mathbf{u}}\overline{\mathbf{u}})\) is known as the Reynolds stress. To close the equations, the Reynolds stress must be approximated using so-called turbulence models. The most commonly used turbulence models in bridge aerodynamics are eddy-viscosity models, in which the Boussinesq hypothesis is applied so that the Reynolds stress can be calculated by the mean velocity field and the turbulence viscosity. These models normally contain one or two transport equations, and so the computational cost is relatively low. Table 1 lists a few frequently used eddy-viscosity models. Among the models listed in Table 1, the k-\(\varepsilon\) models are particularly favoured in applications related to civil and structural engineering. In order to validate the use of k-\(\varepsilon\) models in the built environment, [PERSON] and [PERSON] (2018) conducted a pioneering study to examine the accuracy of results derived from RANS simulations with the standard k-\(\varepsilon\) model. Three-dimensional steady-state simulations with various mesh schemes and boundary conditions were performed on a cubic geometry. By comparing results from these numerical simulations to wind tunnel results, it is found that RANS simulations using the standard k-\(\varepsilon\) model can faithfully reproduce the velocity field and pressure field measured in wind tunnel tests if configured with sufficiently fine mesh and compatible boundary conditions. This study emphasised the significant effect of the mesh and boundary conditions on numerical results, and also showed the enormous potential of CFD simulations. However, a study by [PERSON] et al.