In this chapter, a brief survey of the literature and work done by various researchers with regard to the modeling of flow around bridges is investigated and reported. The different numerical procedures and turbulence models that are used in the computational wind engineering are reviewed. The problems and issues in the fluid structure interaction modeling is also discussed in this chapter. Many researchers have worked on the Great Belt East Bridge section because there is extensive wind tunnel results against which their computer models could be validated.

The methodology to be used in the modeling of flow involves solving the structure and fluid equations simultaneously incorporating the fluid structure interaction. The solution is based upon a grid that is generated in the domain of the fluid flow around the structure. The generation of the grid is very important and critical in order to get the correct results. The grid generation involves several issues that need to be addressed. Hence those details are discussed in a separate chapter. The different types of methods that are adopted to solve the governing equations are the Finite Element Method (FEM), Finite Difference Method (FDM) or the Discrete Vortex Method (DVM). The turbulence in the flow is to be modeled using a turbulence model. The widely used turbulence models are the Reynolds averaged Navier-Stokes (RANS), Large Eddy Simulation (LES) and Vortex Method (VM).

1. Computational Methods

When compared with FEM, the FDM takes less computational time and storage space for the same number of grid points. These are some of the advantages of FDM over FEM. But at the same time FDM is geometrically restrictive whereas FEM is good for complex geometrical shapes and is flexible to impose any type of boundary conditions. Also, the accuracy of the FEM is much higher than that of FDM. (Hughes, 1993). FEM can approximate the convection term in the Navier-Stokes equations with more accuracy than FDM (Selvam, 1998).

The FEM and FDM approximates the unknowns into a set of simultaneous equations of the type AX=B. These equations can be solved by many procedures like Gauss-elimination, Gauss-Seidel and Preconditioned conjugate gradient (PCG) methods. Though the solving of pressure equation takes 80% of the time, the usage of Preconditioned conjugate gradient (PCG) solver makes the iterations faster and speeds up the convergence (Selvam, 1994). Larsen and Walther (1996a) have demonstrated reasonably good success with DVM, but it takes a lot of time and storage space for the 3D model. In this work, the FEM was used for a grid size of 14805 points and the 2D model was able to predict the vortex shedding vividly and the flutter velocity predicted was in good agreement with wind tunnel results.

2. Turbulence Models

In computational wind engineering, the phenomenon of turbulence can be modeled by Reynolds Averaged Navier-Stokes equations (RANS), Direct Numerical Simulation (DNS), and Large Eddy Simulation (LES) as reported by Selvam (1995 and 1998). The relative merits and demerits of these three methods are discussed below.

The RANS model is widely used in computational wind engineering to model turbulent flow. This model uses time - independent equations and solves for Reynolds averaged stresses, which represent the effect of turbulence on the model. Based on the type of solving Reynolds stresses, the RANS model represents different methods. These methods are Eddy Viscosity Models (EVM), Reynolds Stress Models (RSM), and the Algebraic Stress Models (ASM). Different EVM models are available including zero-equation models, one-equation models and two-equation models. The most commonly used form is the model, which is used by Selvam (1990 and 1992) to compute wind-induced pressures around buildings. These models are further explained in Selvam (1992a) and compared in Selvam (1995). To solve for Reynolds stresses, the RSM model requires six partial differential equations with an additional equation for dissipation. Hence, the RSM model requires more computer time and storage. Comparing to RSM, the Reynolds stresses are calculated using algebraic equations rather than solving partial differential equations as in the ASM model. The ASM model consumes less computer time in comparison with the RSM model.

The model is quite popular and can simulate a variety of flows. Its main practical limitation is the assumption of isotropic eddy viscosity. Another disadvantage of this method is that it cannot give time-dependent value like peak pressure.

In DNS, all eddies down to the dissipation scale must be simulated with accuracy. This drastically increases the number of grid points and so consumes much more computer time and storage. It is usually not economical to apply this method to wind engineering problems with available computer resources.

The LES model uses time dependent equations and has the advantage of generating a time dependent flow field. In the LES model, eddies which are larger than the grid size, are simulated. The smaller eddies, occurring below the limit of numerical resolution, are simulated using other methods, such as eddy viscosity model (Selvam, 1997). Eddies significantly larger than the grid size are calculated in detail so that their turbulent properties are modeled correctly. Selvam (1997) and Selvam and Peng (1997 and 1998) used this model to compute pressures on the Texas Tech University (TTU) building. The disadvantage of this model is that it consumes more computer time and storage than the RANS. Due to developments in computer techniques, it is possible to use LES for the prediction of wind problems. In this report, the LES turbulence model is used.

The FSI problem is intricate and complicated to solve because the structural equations are formulated in the Lagrangian co-ordinate system whereas the fluid equations are in the Eulerian co-ordinate system (Selvam and Govindaswamy, 2000). They also state that the FSI modeling needs the simultaneous solving of both the equations of structure and fluid. A moving grid at each time step for the fluid portion is necessary for the solution process. The different approaches in use at this time are Arbitrary Lagrangian-Eulerian (ALE) formulation (Nomura and Hughes, 1992, Selvam et al. 1998 and Tamura et al., 1995), co-rotational approach (Murakami and Mochida, 1995) and dynamic meshes (De Sampaio et al., 1993).

The co-rotational approach may be easier to implement by adding extra terms in the Navier-Stokes (NS) equations for movements in one direction. Therefore, it will be difficult to apply for general problems. In the dynamic mesh approach, for each time step a new mesh is formulated and hence needs a very sophisticated grid generator. In the ALE approach, grid can be moved as a whole in a rigid fashion with constant velocity for each node as reported by Tamura et. al. (1995) or with different velocity for each node in a flexible manner as reported by Selvam et. al. (1998) and Nomura and Hughes (1992). Moving the grid, as a whole is preferred for FSI problem since the structure has rigid body movement. If the structure is very flexible and each node on the structure is moving, then the latter grid moving procedure has to be used. In the ALE procedure, geometric conservation laws (GCL) are violated if the equations are solved as such (Thomas & Lombard, 1979 and Ferziger & Peric, 1999). In this work the GCL error is reduced by using the corrections similar to the one reported by Thomas & Lombard, 1979 and Ferziger & Peric, 1999. Moving the grid as a whole may be computationally easy to apply. In this work the bridge deck is assumed to be rigid and the rigid body moving technique is used in this research.

The critical flutter velocity can be calculated using free oscillation procedure or forced oscillation procedure as explained in chapter two. The free oscillation procedure was used by Frandsen & McRobie (1999), Enevoldsen et al. (1999), Nomura & Hughes (1992), Mendes & Branco (1995) and Selvam et al. (1998). Larsen and Walther (1997) and Enevoldsen et. al. (1999) used the forced oscillation procedure. In this work the free oscillation procedure is used and the critical flutter velocity is computed in a few computational runs.