# Aerodynamic Analysis of Simple Grider Bridges under Construction Phase
## 2 Materials and Methods  
### Simulation Setup  
The beams' height for simple girder bridges with spans around 40 m ranged from two to three meters, so the values of \(h=\) [2, 2.5, 3] \(m\) were tested. The distance between beams is usually equal to or larger than the section's height, so the values were taken depending on it, leaving \(d=h+\Delta\), where \(\Delta=\) [0, 0.5, 1] \(m\). The joint procedure usually determines the flanges' width with the concrete slab, assuming a constant value of \(b=0.9\) m. The bridge's height was assumed to be \(z_{0}=\) [3,5,10] \(m\), the usual heights for this type of bridge. Moreover, an additional height of \(z_{0}=120\) m was considered, representing the free flow. Finally, the thickness of the flanges and the web were set to \(t_{f}=4\) cm and \(t_{w}=2\) cm, respectively. The Reynolds number falls between \(6.8\times 10^{6}\) and \(3.1\times 10^{6}\), depending on the case parameters. Since the tested shapes have sharp corners and the separation of the boundary layer is expected to be always in the same location, the vortex shedding around this type of section can be considered constant for this small range of Reynolds numbers [27]. To reduce computational costs, all CFD simulations in this study were conducted in two-dimensional settings. This is a valid assumption in this particular case study, as the modeled bridge has a constant cross-section area all along the bridge. In addition, two-dimensional models showed an excellent approximation to reality when representing the vortex shedding phenomenon [28]. To achieve statistically significant results, each case study was tested with five different flow speeds within the interval where vortex shedding vibrations are more likely to occur [25]: \(U=\) [5, 7.5, 10, 12.5, 15] m/s. Therefore, 420 simulations were made with 84 different geometries. In order to consider the turbulence effect, the Re-Normalization Group (RNG) k-\(\epsilon\) model [29] combined with enhanced wall treatment was used. RNG model was developed using Re-Normalization group theory and can be written as follows:  
\[\frac{\partial(\rho k)}{\partial t}+\frac{\partial(\rho ku_{i})}{\partial x_{i }}=\frac{\partial}{\partial x_{j}}\Bigg{[}\bigg{(}\mu+\frac{\mu_{t}}{\sigma_{ k}}\bigg{)}\frac{\partial k}{\partial x_{j}}\Bigg{]}+P_{k}-\rho\epsilon \tag{1}\]  
\[\frac{\partial(\rho\epsilon)}{\partial t}+\frac{\partial(\rho\epsilon u_{i})} {\partial x_{i}}=\frac{\partial}{\partial x_{j}}\Bigg{[}\bigg{(}\mu+\frac{ \mu_{t}}{\sigma_{t}}\bigg{)}\frac{\partial\epsilon}{\partial x_{j}}\Bigg{]}+C _{1\epsilon}\frac{\epsilon}{k}P_{k}-C_{2\epsilon}^{*}\frac{\epsilon^{2}}{k} \tag{2}\]  
where \(C_{2\epsilon}^{*}=C_{2\epsilon}+\frac{C_{2\epsilon}\eta^{3}(1-\eta/\eta_{0})}{ 1+\beta\eta^{3}}\) with \(\eta=(Sk/\epsilon)\) and \(S=\sqrt{2S_{ij}S_{ij}}\), \(k\) is turbulence kinetic energy per unit mass and \(\epsilon\) is turbulent dissipation rate. More details and all the constants are given by [PERSON] et al. [29]. Since the wind speeds tested imply a Mach number below 0.3 (\(Ma=0.015\)-\(0.044\)), an incompressible flow could be assumed [30]. The air density and dynamic viscosity were assigned to be \(\rho_{a}=1.225\) kg/m\({}^{3}\) and \(\mu=1.7894\times 10^{-5}\) kg/ms, respectively. The second-order time domain numerical solver was used to improve accuracy, and the maximum number of iterations per time step was set at 200 to ensure convergence. The time step size and the simulation duration for each case depend on the wind speed tested, ensuring that there are around 200 time-steps per vortex shedding cycle (Table 1). A steady-state simulation with sufficient iterations to ensure a satisfactory convergence was used as initial data to start the transient simulation. Since the case study is considered to be adiabatic, the buoyancy effect and gravitational acceleration were ignored. Wall boundary conditions with zero roughness were set for all rigid limits such as the beams and the valley.