Verification of the Wave Forces on a Bridge Deck

The capability of the developed wave model to predict the wave forces is ensured by the verification of the wave forces on a bridge deck with girders and side railings conducted by McPherson (2008) in a wave basin. The bridge model with the Biloxi Bay Bridge as the prototype one was built with a 1:20 scale and tested in the Haynes Coastal Engineering Laboratory 3D shallow water wave basin at the Texas A&M University. Perforated railings were considered on the seaward side of the bridge model. Two dummy bridge sections, supported by the bridge pilings, were placed on each side in

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order to eliminate the three dimensional bridge deck end (3D) effects and to simulate in field scenarios.

Fig. 5.3 Comparisons of the free surface profiles for solitary waves at two different simulation times. (a) 𝜀 = 0.24; (b) 𝜀 = 0.30; and (c) 𝜀 = 0.36.

In this verification process, the bridge model geometry is shown in Fig. 5.4 for the 2D simulations, where the perforated railings cannot be fully realized owing to the limitation of the 2D computational domain. To accommodate the accuracy of the bridge model, a railing height of 3 cm is considered above the bridge deck with a 2 cm clearance. Only one elevation of the bridge model is considered with the bottom of the girders located 0.41 m above the bed. Four water depths, 0.39 m, 0.41 m, 0.48 m and 0.54 m, are considered. Only one wave height, 0.14 m, is used with the ε values falling in the range from 0.24 to 0.36 that are already verified with the analytical wave profile.

Fig. 5.4 Schematic diagram for the bridge model adopted by McPherson (2008)

To accurately capture the near-wall features (the velocity field and pressure field) and hence the predicted wave forces, the wall boundaries of the structure model should be paid great attention. Based on the log-law for the “law-of-the-wall” used for identifying the viscous layer, blending layer, and the fully turbulent layer, very fine

Bottom d H=0.14m 3.8cm 7.6cm 68.58cm 41 cm 13 cm 1c m 7c m SWL

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meshes should be adopted near these wall boundaries. To take full advantage of the SST

k-ω model, the wall-coordinate (dimensionless) y+ should be less than 2, where y+ is

used to calculate the height of the first grid cell along the walls of the structure model in the turbulent flow. While it is very difficult to satisfy this requirement, the height of the first grid should be in the logarithmic layer and valid results can still be guaranteed. In the literature, y+ is desirable to be set in a range from 11.6 to 300 in order to achieve an acceptable accuracy in bridge engineering (Bredberg 2000; Xiong et al. 2014). In the current study, the range of y+ is from 30 to 50 in order to ensure that reliable pressure field and velocity field around the near wall regions can be obtained and to avoid the extensive computation.

For this verification, the computational domain is 13 m in length and 0.9 m in height. The grid resolutions are: 𝑑𝑦=0.02 m, 0.0025 m and 0.005 m for the air zone, the near water zone, and the deep water zone, respectively; 𝑑𝑥=0.005 m, 0.0025 m, and 0.02 m for the near velocity inlet zone, main computational zone, and far field from the main computational zone, respectively. A structured mesh method is employed and the grid mesh in the computational domain is shown in Fig. 5.5. The simulation time is 6 s with the time step of 𝑑𝑡=0.0025 s.

Fig. 5.5 Grid mesh for the bridge model adopted by McPherson (2008)

The total force component along the specified force vector 𝑎⃗ (horizontal force or vertical force) on the wall zones of the bridge deck model is computed by summing the dot product of the pressure and viscous forces on each face with the specified force vector. The terms in this summation represent the pressure and viscous force components in the direction of the vector a⃗⃗ as:

𝐹𝑎 = 𝑎⃗ ∙ 𝐹⃗𝑝+ 𝑎⃗ ∙ 𝐹⃗𝜈 (5.4)

where 𝑎⃗ is the specified force vector, 𝐹⃗_𝑝 is the pressure force vector, and 𝐹⃗_𝜈 is the viscous force vector. A reference pressure, 𝑝_𝑟𝑒𝑓, (the operating pressure, 101, 325 Pa) is used to normalize the cell pressure for computation of the pressure force to reduce the round-off error:

𝐹⃗_𝑝 = ∑𝑛 (𝑝 − 𝑝_𝑟𝑒𝑓)𝐴𝑛̂

𝑖=1 (5.5)

where 𝑛 is the number of faces, A is the area of the face, and 𝑛̂ is the unit normal to the face. The associated force coefficients (cl and cd, uplift and drag force coefficients, respectively) are computed for the selected wall zones and are recorded since the simulation begins in the monitor setups. The force coefficient is defined as the force

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divided by 1₂𝜌𝑣2𝐴, where 𝜌, 𝑣 and 𝐴 are the referenced density, velocity, and area, respectively, and these values are set in the “Reference Values” in Fluent.

Comparisons between the numerical results and the experimental measurements by McPherson (2008) are shown in Figs. 5.6 and 5.7. Fig. 5.6 shows a small difference between the maximum horizontal forces when d =0.54 m, which may be due to that the simplified 2D railing has shortcomings when compared with the 3D perforated railings in the laboratory experiments and this was expected. Similarly, Fig. 5.7 shows some small differences between the maximum vertical forces when d =0.39 m and 0.41 m. This is probably due to the effects of the entrapped air since the entrapped air cannot escape in a timely manner in 2D simulations. In general, good agreements are obtained, indicating that the current wave simulation and force prediction procedure can be further employed in the bridge deck-wave interaction problem.

Fig. 5.6 Comparisons of the horizontal forces

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