*5.3.2. Discrete vortex method*

It should be noted that FE simulations are usually carried out for structural reasons, for example to assess the response to seismic or traffic loads, and therefore, its use in the assessment of the aerodynamic behaviour of the bridge [29, 47] is just another benefit to be collected from

Accompanying the advances in computing power, the dynamic analysis of bridges employs, nowadays, high-resolution three-dimensional FE modelling, encompassing thousands of finite elements. Among the various types of finite elements, the following should be highlighted: beam, shell and slab elements for deck and towers, and tension-only spar elements for cables. More information on the number and type of finite elements typically employed in FE analysis of bridges can be found in [46], where a number of case studies are presented.

Arbitrary flows of viscous fluids are described by the Navier-Stokes equations. The numerical solution of these nonlinear partial differential equations for turbulent flows is very difficult. The finite volume (FV) is the most common method used in the formulation of the numerical problem. The domain is divided into small volumes and the balance equations are solved for each, in an iterative procedure across the domain. However, the wide range of length scales imply the use of a very fine mesh up to a level that requires huge computational resources thus making direct numerical simulation (DNS) still infeasible for engineering problems. Another approach is to model the finer scales and to explicitly solve the larger turbulent scales. Yet, the resulting model, large eddy simulation (LES) [48], is still computationally intensive. For practical problems in engineering, less demanding approaches are used, in particular the one based on time-averaged equations (RANS) complemented with turbulence models. Popular turbulence models are the *(κ-ε*) (suitable for regions away from walls), the *κ-ω* (developed to tackle near wall effects), and the Reynolds stress model (which includes equations to deal with turbulent fluctuations in the three spatial directions). The single-equation Spalart-Allmaras model is also interesting for being much less computationally expensive and nevertheless

The problem can be further simplified by reducing the analysis to the two-dimensional space and/or by considering the flow to be steady. In two-dimensional, steady simulations, a crosssection of the deck is studied under the action of lateral wind to identify the existence of recir-

and how all of these features vary with angle of attack of the wind. Uniform wind or with

In turn, unsteady simulations allow studying the pattern of recirculation bubble formation and vortex shedding around the deck's contour and their effect on the variation in time of the

By allowing the structure's section to move in the domain in response to the aerodynamic forces acting over its contour and introducing mesh adaption techniques [49], it is possible to study fluid structure interaction, though without the three-dimensionality associated with the

, *CD*, and *CM*),

culation bubbles, their mean dimension, idea of the aerodynamic coefficients (*CL*

aerodynamic coefficients [42]. The effect of wind gusts can also be studied.

shear (simulating the atmospheric boundary layer) can be studied.

the effort put into carrying out the simulations.

useful in the study of airflow around bridge decks [16].

oscillation modes of the whole bridge.

**5.3. Modelling of the aerodynamics**

*5.3.1. Finite volume method*

98 Bridge Engineering

Based on the existing discrete vortex method, HonoréWalther and Larsen [50] developed a relatively fast numerical model well suited for the simulation of two-dimensional bluff-body flows such as around the cross-section of bridge decks. A great advantage of this numerical tool is that it is mesh free.

The method takes the vorticity transport equation, spit into an advection part and a diffusion part, with appropriate boundary conditions at the solid surface involving the surface vortex sheet concept. A Lagrangian approach is then used in which vortex particles introduced in the domain are followed. The diffusion part is tackled via random walks. A scheme for marching in time is obtained from a Euler integration.

The input to the numerical model is the deck cross-section discretised into vortex-panels, typically a few hundreds of them. The model outputs are time series of aerodynamic force and moment, maps of pressure distribution, and streamlines. Aerodynamic derivatives can also be obtained. Larsen and Walther have applied the model to bridge decks of long-span bridges for which wind tunnel data are available and found that the model made good to excellent predictions regarding drag coefficient, Strouhal number, critical velocity for the onset of torsional and coupled flutter, and onset of vortex-induced vibrations.

#### *5.3.3. Finite element method*

While in the past FEM would not be the method of choice for solving flow equations, because unlike the FVM it would not inherently satisfy the local mass and momentum conservation, more recent advances have overcome this [51]. The new finite element procedure for the solution of incompressible Navier-Stokes equations is referred to as flow condition–based interpolation (FCBI) finite element method.

Panneer Selvam et al. [52] have used FEM together with LES to model turbulence to simulate the flow around Great Belt East Bridge approach span. The high accuracy in approximating the convection term made it possible to use a relatively modest number of nodes. Long timesteps can be used if the grid can align with the flow. The results from the 2D simulations were in reasonable agreement with wind tunnel results.
