**5. Numerical modelling**

#### **5.1. Introductory notes**

compared to sectional model tests, full three-dimensional tests involve higher cost and longer lead time; larger wind tunnels, which are costly to operate; and smaller scale of the model, which limits the level of detail of the flow that can be studied and possibly introduces *Re* dependency effects. Consequently, full aeroelastic model tests are often complemented with sectional model tests, as was the case in the design stage of the Trans-Tokyo Bay Bridge [38], to give an example. An interesting discussion of advantages and limitations of sectional models, taut-strip models, and full bridge models is given in [29]. Alternatively, instead of the full bridge, parts of the bridge, retaining some three-dimensional aspects, may be tested throughout the design phase; for example, at a certain

The test of sectional models has been widely understood as a good tool to predict critical flutter instability as well as critical vortex-induced vibration. This is particularly accurate for box-girder and plate girder decks, when sectional model testing appears to predict full scale vortex shedding excitation, and there is evidence that turbulence has only but a minor effect

To study in the wind tunnel the susceptibility of a bridge to aerodynamic instabilities, the model has to be free to oscillate in response to the action of the flow. This is a completely different situation of static testing. In what concerns sectional models, there are a few solutions, along with the associated instrumentation, to adequately support them in the wind tunnel. Typical suspension systems are based on helical springs. One configuration (used by e.g., [39]) is based on a single pair of suspension columns, which support the extremities of a shaft on the sectional model's axis of rotation. Other sets of helical springs, not contributing to the suspension though, can govern the torsional stiffness, and the setup also includes means of preventing rolling and horizontal displacements of the model. Another configuration employs four suspension columns to support the sectional model through two horizontal arms, one at each end of the model. The four columns tackle both torsional and flexural stiffness. Each suspension column includes two helical springs, one above and one below the suspension arm. Two horizontal wires limit the translation of the model in the horizontal plane. The entire assembly remains outside the test chamber. Systems of this type have been used,

Various types of sensors may be used with the suspension systems to monitor the windinduced motions of the deck model: laser reflexion sensors, piezoelectric accelerometers, or

The dynamics of the flow pattern around the model can be understood using diagnostic techniques such as laser sheet visualisation (LSV) [40]. Computational fluid dynamics (CFD), to be discussed in the next section, is becoming more often used also for this purpose, and the term CFD visualisation has been employed (e.g., [10, 42]). The classic techniques of smoke lines and

Another possibility is the simultaneous measurement of turbulent pressure fluctuations at multiple points on the surface of the model (e.g., [43]). By analysing the spectra correlation, the development and movement of recirculation bubbles can be understood. The interest in

point of the Messina Bridge Project, just the main span was tested at 1:250 scale.

for example, by Larsen and Wall [40], and by the present authors [41].

wool tufts are still valuable for expedite assessments.

on critical flutter speeds.

**4.4. Instrumentation**

96 Bridge Engineering

ring strain sensors [41].

At present, three-dimensional unsteady numerical simulations where all the relevant physics of the interaction of turbulent wind with a flexible structure are completely described still involve a considerable investment of time, despite the steady increase in computational power over time. The case becomes even more difficult if wind gusts, surrounding complex terrain, or coupling with neighbour bridges, are to be included. Therefore, numerical simulations are carried out to look into a certain aspect of the whole problem. Often, numerical simulations complement wind tunnel testing. The engineer has to integrate the several pieces of information to come up with a design that not only is robust from the wind action point of view but also complies with economic and structural constraints.

Generally speaking, structural aspects are studied with the finite element method (FEM), whereas aerodynamic aspects are studied with the finite volume method (FVM), or some other common method in computational fluid dynamics (CFD). It should be said that, recently, there have been advances towards using FEM for the study of the flow field.

In FEM, the structure of the bridge, or a part of it, is discretized into simple structural elements. Values of interest are then obtained at the ends (nodes) of the element, by solving the numerical model equations, and a weighting function is used within an element to obtain the values at intermediary positions on it. There is a wide variety of elements that may be used and a large choice of weighting functions ranging from simple to more elaborate and precise.

In CFD, the common approach is the Eulerian approach, in which the fluid around the deck, or pylons, or even the whole bridge, up to a long distance from the structure, is discretized into small control volumes for which transport equations (mass, momentum, turbulence, and scalars) are solved. Another approach is to employ meshless methods.

The challenge has been to couple these two tools to describe the coupling that exists in the real world between structure and wind, in order to be able to study aeroelastic phenomena.

#### **5.2. Modelling of the structural dynamics**

Finite element (FE) is the method of choice to model the structural dynamics of a bridge and can provide insight into its dynamic response, such as the main modes of oscillation and associated natural frequencies. This is especially important for bridges with low structural damping, as is the case of cable-stayed or suspension bridges. Among many studies, a few published examples on suspension bridges are The John A. Roebling Bridge in Kentucky, USA [44]; Tamar Bridge in Plymouth, UK [45]; and the Bosphorus Bridge and the Fatih Sultan Mehmet Bridge, in Istanbul, Turkey [46].

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 the effort put into carrying out the simulations.

Three-dimensional simulations are much more computationally intensive. An example of three-dimensional simulation is that of [43], in which LES has been used to obtain flutter-

Wind Action Phenomena Associated with Large-Span Bridges

http://dx.doi.org/10.5772/intechopen.73061

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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

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

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 tor-

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 inter-

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

To begin with, it should be noted that the aerodynamic study of a deck can only be initiated after the overall geometry and main dimensions of its section have been established. Thus, at that time, considerable effort has already been put in the bridge structural design. Therefore, to avoid delays and costs, the corrective modifications to be proposed in order to diminish the possibility of wind exciting a vibration mode, as inferred from wind tunnel tests, should

derivatives of a deck sectional model.

in time is obtained from a Euler integration.

sional and coupled flutter, and onset of vortex-induced vibrations.

*5.3.2. Discrete vortex method*

tool is that it is mesh free.

*5.3.3. Finite element method*

polation (FCBI) finite element method.

in reasonable agreement with wind tunnel results.

**6. Attenuation of adverse aerodynamic phenomena**

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.

## **5.3. Modelling of the aerodynamics**

#### *5.3.1. Finite volume method*

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

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 recirculation bubbles, their mean dimension, idea of the aerodynamic coefficients (*CL* , *CD*, and *CM*), and how all of these features vary with angle of attack of the wind. Uniform wind or with shear (simulating the atmospheric boundary layer) can be studied.

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

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 oscillation modes of the whole bridge.

Three-dimensional simulations are much more computationally intensive. An example of three-dimensional simulation is that of [43], in which LES has been used to obtain flutterderivatives of a deck sectional model.
