Effects of air relief openings on the mitigation of solitary wave forces on bridge decks *

Sheng-chao Xiao , An-xin Guo

1. Ministry-of-Education Key Laboratory of Structural Dynamic Behavior and Control, School of Civil Engineering, Harbin Institute of Technology, Harbin 150090, China

2. Department of Civil Engineering, The University of Queensland, Brisbane 4072, Australia

Abstract: For inverted T-type bridge decks, the air entrapped in the chambers between adjacent girders could increase the wave forces and lead to the destruction of the bridge decks. This paper studies the effects of the air relief openings (ARO) on the mitigation of the solitary wave-induced forces on the bridge decks. Hydrodynamic experiments are conducted for three inverted T-type decks with four, five, and six girders with different wave properties and deck clearances. The open source computational fluid dynamics toolbox OpenFOAM is adopted to conduct numerical simulations for the effects of the AROs. Since the numerical results correlate well with the measurements, the mechanism of the wave-structure interaction can be revealed by the numerical flow fields.Furthermore, the relationship between the shape and the volume of the ARO and the wave forces on the bridge decks, as well as the contribution of each ARO to the effect of the wave force mitigation, is also obtained from the numerical results of the OpenFOAM.Experimental and numerical results demonstrate that the AROs could effectively reduce the vertical wave forces on the bridge decks.The effects of the AROs increase with the increase of the volume of the ARO, while the shape of the ARO has no effect on the reduction of the wave forces.

Key words: Air entrapped, hydrodynamic experiment, solitary wave forces, bridge decks, OpenFOAM

Introduction

Tsunami-induced wave forces are known to be a significant threat to coastal bridges. It was reported that more than 80 bridge decks were totally washed away or seriously damaged in the 2004 Indonesian tsunami, and the number was more than 200 in the 2011 Japanese tsunami. The failure of a bridge deck is due to a combination of the vertical and horizontal wave forces, to raise the deck from the piers and then wash it away.

Since the two destructive tsunamis, the wave forces on bridge decks and jetties have been widely investigated. Valuable hydrodynamic experiments[1-3]were conducted to determine the magnitude of the wave forces because they are the most direct and convincing ways of analyzing the problem. Due to the limitations of scale in experiments, numerical models become an increasingly important approach to obtain a deeper insight into the failure mechanism of the bridge decks[4-6]. Large amounts of experimental and numerical investigations indicate that the air entrapped in the chambers of the girders greatly increases the magnitude of the wave forces. Therefore, more attention should be paid to the effects of the entrapped air on the wave forces on the bridge decks.

Previous studies of the wave-induced air entrapped between the chambers of the bridge girders were mainly focused on regular waves. Bradner demonstrated that the air entrapped between the girders leads to a sharp increase of wave forces[7].Mcpherson found that the air entrapment between the bridge girders increases the uplift buoyancy force significantly[8]. Through various experimental results,an empirical method was developed to take into consideration the forces induced by the air entrapment.Cuomo et al. performed a 3-D hydrodynamic experiment to investigate the wave-induced long-duration quasi-static force and the slamming force (highfrequency wave force) on a 1:10 bridge model. It was concluded that the slamming force could be up to three times larger than the quasi-static force[9]. Azadbakht and Yim[10-11]reported that the effect of the entrapped air could increase the vertical wave force, and an increasing trend of the effect can be found with a decreasing wave period.

Fig. 1 Schematic diagram of laboratory flume (m)

Experimental and numerical investigations of the solitary wave-induced air entrapment have received attention in recent years. The numerical results of Bozorgnia et al.[12]demonstrated that the vertical wave force can be mitigated by setting air relief openings (ARO) on the bridge deck. Hayatdavoodi et al.[13]experimentally and numerically investigated the effect of entrapped air on wave forces by comparing the results between bridge decks with and without the AROs. It was demonstrated that the entrapped air between the girders modified the wave surface,leading to a variation of the pressure field and an increase of the wave forces on the bridge deck. In addition, a numerical study revealed that the vertical wave force would be decreased significantly by installing the AROs on the bridge decks. Seiffert et al.[14]experimentally analyzed the relationship between the number of AROs and the wave-induced force by setting differently sized air openings in the sides of a bridge deck. Through analyzing the experimental results, it was found that a more apparent mitigated effect of the AROs could be found on the vertical force than on the horizontal force. Xu et al.[15]provided an engineering consultation for the installation of the AROs on the bridge decks based on the numerical study of the effects of the AROs on the solitary wave-induced forces. Bricker and Nakayama used the tsunami model Delf-3D and the CFD library OpenFOAM to show that the entrapped air would increase the vertical force and lead to the failure of the bridge deck[16].

It can be seen from the above review, that the effect of an ARO on mitigating the wave forces on the inverted T-type bridge decks is a very typical problem,and is important for the survival of the coastal bridge decks during a tsunami. Although previous studies have demonstrated that the installation of the AROs can reduce the wave forces on the bridge decks, the relationship between the shape of an ARO and the mitigation of the wave forces has not been determined.Moreover, the importance of the AROs in each chamber of a bridge deck should be investigated to providethebestlocationsoftheAROsforcoastal-bridge designers.

This paper investigates the effect of the AROs on the mitigation of the solitary wave-induced forces on the bridge decks. Hydrodynamic experiments are conducted to determine quantitatively the relationship between the AROs and the wave forces on three inverted T-type decks with four, five, and six girders.In addition, the flow fields of the wave-structure interaction are studied by using the computational fluid dynamics (CFD) toolbox OpenFOAM. The relationship between the shape and the volume of an ARO and the mitigated effects of the wave forces is investigated by numerical analyses. Finally, the importance of the AROs in each bridge deck chamber is shown in the numerical results.

Fig. 2 (Color online) Image of the experimental configurations

1. Experiment design

Experiments are conducted at the Hydraulics Laboratory of the Harbin Engineering University in China. As shown in Fig. 1, the tank is 30 m long, 1 m wide, and 1 m deep. The water depth for the experiment is 0.20 m. A piston-type wave-maker is installed at a distance 1.51 m away from the upstream end of the wave tank, and the experimental model is placed 19.74 m from the upstream end. Three wave gauges are installed 0.74 m, 2.04 m, and 2.54 mawayfromthe upstream side of the bridge model. A wave absorber is installed at the downstream end of the wave tank to decrease the effect of the reflective waves.

Fig. 3 Positions and dimensional information of air relief openings (ARO) and pressure transducers for model I-III (m)

The experiment configuration is shown in Fig. 2.As can be seen in this figure, the model system includes the experimental model, a three-component load cell, four steel bolts, and a steel plate. Through the connection of the steel bolts and the plate, the wave forces on the experimental model can be fully transferred to the load cell so that the horizontal and vertical wave forces induced by the solitary waves can be measured. In the middle of the bridge model,eleven pressure transducers are installed to investigate the variation of the pressure field in the direction of the wave propagation. To reproduce the real boundary conditions for the bridge deck, two partial bridge decks are also installed on either side of the main bridge deck. The whole testing system is suspended in the portal frame by the steel tubes shown in Fig. 2.

The water surface elevation is measured using wave gauges (Fig. 1), and sampled at 200 Hz. The pressure on the bottom of the bridge deck is measured at 500 Hz by the pressure transducers. The horizontal and vertical wave forces, measured by a load cell, are sampled at 500 Hz. An NI PXI 6251 real-time data acquisition system records all data.

A bridge superstructure of a 1:30 scale of the I-10 Bridge over Escambia Bay is used to study the mitigating effect of the AROs. To investigate the effect of the AROs on the decks with different numbers of girders, the bridge decks with four and five girders are also used in the measurements. The AROs with the same diameter of 15 mm are installed in the middle of the chambers between the T-type girders on all three decks. Detailed information is shown in Fig. 3.

In this study, the method presented by Goring and Raichlen[17]is adopted to generate the first-order solitary-wave profile:

where η is the wave amplitude, y is the wave profile, h is the water depth, c is the wave velocity,x is the horizontal coordinate and t is the time. By matching the relative velocity between the wavemaker paddle and the water, the governing equation for determining the position of the push plate can be obtained as follows

where ξ( t) is the position of the paddle of the wavemaker, T is the period of the plate motion and S is the stroke of the piston. A detailed derivation of this equation can be found in Ref. [17].

Four ratios of η/h , 0.400, 0.325, 0.250 and 0.175 are used to examine the relationship between the initial wave amplitudes and the wave forces.Because the maximum wave force occurs in the case with the clearance at z=0 mm (z represents the height between the low chord of the bridge girders and the still water level), this clearance is selected for all measurements. To ensure repeatability, each experiment is repeated five times under the same conditions of the water depth, the clearance, and the wave amplitude. In the processing of this data, the maximum and minimum values are removed, and the remaining three values are used to obtain the mean value.

Fig. 4 Comparisons of solitary wave profiles between analytical solutions and measurements.

2. Experimental result analysis

To guarantee the accuracy of the incident wave conditions, the wave profiles measured at the location of wave gauge 2 are selected as the governing wave conditions. The comparison of the wave profiles between the analytical solution and the measurement without the experimental model in the wave tank is shown in Fig. 4. It can be seen in this figure that the actual measured wave amplitudes in the cases of0.400, 0.325, 0.250, and 0.175 are 65 mm, 51 mm,39 mm and 25 mm, respectively, due to the attenuation property of the wave tank. Good correlation can be seen between the measured and analytical solutions of the wave profiles, with some fluctuation of the wave tails in the measurements. This uneven water surface is the result of the motion of the wave-maker, which should be moved to the upstream side of the stroke from the central position in order to generate solitary waves. Considering that the wave forces in the downward and upstream directions would be influenced by the fluctuations, the effects of the AROs on the wave forces are only investigated in the upward and downstream directions (in this paper,the wave forces in the upward and downstream directions are named the vertical and horizontal wave forces, respectively). Furthermore, it is assumed that the wave force on the bridge deck is the same across the whole span of the bridge model. Therefore, the wave force in unit length is considered in the following analysis, with the wave force unit N/m.

Figure 5 shows the comparison of the vertical force and the pressure for model I with and without the AROs for a wave amplitude of 65 mm. Comparisons of the vertical wave forces are presented in Fig.5(a). As can be seen in this figure, the vertical force decreases with the evacuation of the air from the relief openings. However, a stronger slamming force can be observed in the case with an ARO. In addition, the duration of the wave-structure interaction for the decks with and without venting holes is 0.5 s, 1.0 s,respectively, which indicates that the AROs can shorten the duration of the wave impingement. The same phenomenon can also be observed in Fig. 5(b),in the comparison of the pressure measured at the location of the pressure transducer six (Fig. 3). It is also shown in this figure that the presence of the AROs leads to a strong pulse-like pressure.

Fig. 5 Time history of vertical wave force and pressure for the 65 mm wave for model I

Figures 6(a)-6(c) show the maximum wave forces on the decks with and without the AROs for models I, II, and III, respectively. As can be seen in these figures, with the AROs, the wave forces decrease in both the vertical and horizontal directions for all bridge decks, and the effect decreases with the increase of the wave amplitude. The largest decrease of the wave forces occurs for Model II. For a wave amplitude of 65 mm, the AROs in model III have little influence on the mitigation of the total wave force in both the vertical and horizontal directions (Fig. 6(c)).

The reason is the presence of a large impulse pressure,since a significant decrease of the long duration wave force can be seen in Fig. 6(f). The corresponding comparisons of the long duration wave forces are presented in Figs. 6(d)-6(f). It is shown in these figures that with the AROs, the long duration wave forces can be significantly decreased in both the vertical and horizontal directions for all measurements.

Fig. 6 Total and long duration vertical (solid symbols) and horizontal(open symbols) wave forces for models I-III ((a)-(c)), models I-III ((d)-(f)), respectively

The pressures in the direction of the wave propagation for model I with and without the AROs are presented in Fig. 7. It can be seen in Fig. 7(a) that the pressure under the flat plate of the deck is zero for the deck with an ARO, while the pressure under the girders is similar to that for the deck without the AROs. A similar phenomenon can be observed in Fig.7(b), which shows the pressure on the bridge deck for a wave amplitude of 39 mm. As shown in this figure,compared to the pressure under the deck without the AROs the pressure under the flat plate of the deck with the AROs is significantly decreased. Moreover,the pressure under the girders is slightly modified by the presence of the ARO. However, for the wave amplitudes of 51 mm, 65 mm, the pressure under the bridge deck with the AROs is greater than that for the deck without the AROs, as shown in Figs. 7(c), 7(d),respectively. Although the existence of the air venting holes can amplify the pressure locally, the total wave forces on the bridge deck decrease (Fig. 6) because the strong pressure is of short duration (Fig. 5).

Fig. 7 Wave pressures measured at the pressure transducers installed in model I for wave amplitudes

3. Numerical computations

To investigate the effect of the AROs on the flow field with the wave-structure interaction and to explain the generation of a pulse-like pressure on the bridge deck, the wave-generation library, waves2Foam[18],based on an open source CFD toolbox OpenFOAM-2.4.0 (Open Field Operation and Manipulation), is used to model the numerical wave tank in this study.

3.1 Numerical wave tank

The library waves2Foam is a wave generation and absorption toolbox developed by Jacobsen et al.[18], and it was adopted to establish numerical wave tanks by Seiffert et al.[19], Hayatdavoodi et al.[13].

In the fluid-structure interaction problems, the effects of viscosity are generally considered to have little influence on the force on the structure. Based on this consideration, the following Euler's equations are used as the governing equations:

where ρ is the density of the water, =( u , v, w)u is the velocity vector of the fluid, x is the position vector, p*is the dynamic pressure and g is the gravitational acceleration.

The interface between the wave and the air is captured by the volume of fluid (VOF) method in the waves2Foam. The volume fraction function α( x , t)is used to identify the property of the fluids in a cell,as shown in the following equation

where α is the volume fraction function related to the density ρ of the fluid ( α=1 is a water cell,is an air cell, and 0 ＜ α＜1 represents an intermediate cell).

In the waves2Foam, a series of waves, including the regular waves, the irregular waves, and the solitary waves can be generated in the inlet region by prescribing the velocity, the surface elevation, and the pressure. The Dirichlet boundary condition and the Neumann boundary are used to initialize the velocity and the pressure, respectively. In the waves2Foam, an extension of the relaxation technique is introduced to remove the contaminants and discontinuities. The detailed information about the method can be found in Refs. [18, 20].

To reproduce a wave tank similar to the experimental one in the waves2foam, the configuration of the experimental wave tank is used to provide the physical background of the boundary conditions. In the numerical wave tank, the inlet and outlet regions are used to represent the wave generation and absorption ends of the experimental wave tank. The Korteweg-de Vrise solitary wave solution is used in the inlet zone to generate the initial wave according to the experiment. The boundary conditions of the bottom of the numerical wave tank and the bridge model are defined as the “wall”. In the OpenFOAM,the “empty” in the front and back directions means a 2-D numerical simulation, while others mean a 3-D numerical simulation. The boundary condition“atmosphere” is used to represent the air in the experimental wave tank. The setup of the boundary conditions of the wave tank and the bridge deck are shown in Table 1. A detailed explanation of boundary conditions can be found in the user guide of the OpenFOAM-2.4.0[21].

The lengths of the inlet and outlet regions of the numerical tank are set to 5 m. Three wave gauges are installed in the wave tank at the same positions as in the physical tank. It was reported by Seiffert et al.[19]that a sudden drop of the amplitude of the wave appears when the solitary wave exits the inlet region.However, this phenomenon disappears when the wave propagates out of the generated region. In the numerical cases the distance between the wave-maker and the model is 18.23 m, which is long enough to eliminate the influence of the instability.

A grid size of 5 mm is used for the area near the bridge model, while a 20 mm grid is used for other areas. In the waves2Foam, an adjustable function can be used to change the value of the time step if the Courant number exceeds the maximum value given in the computations. In the numerical simulations, the adjustable time step function is applied and the maximum Courant number is set as 0.25 in all cases.The wave forces on the bridge model are obtained by integrating the pressure in the vertical and horizontal directions (the library FORCE in the OpenFOAM).

Fig. 8 (Color online) Schematic diagram

3.2 Wave forces

In the numerical model of this study, the round ARO (15 mm in diameter according to the test model)as shown in Fig. 8(a), is used in the 3-D numerical model. In view of the symmetries of the structure,only a portion of the test model, as shown in Fig. 8(b),from the whole structure, is used for establishing the 3-D numerical model in order to improve the computational efficiency. In this study, a simplified 2-D numerical model is also established for comparative purposes. Because the round ARO cannot be directly employed in the 2-D numerical model, an equivalent cuboid ARO with the same volume as the round ARO is used. Because the cuboid ARO are set along the span wise direction of the bridge deck, the numerical model can be simplified as the 2-D model,as shown in Fig. 8(c), due to the symmetry. Using this method, the efficiency and the accuracy of the 2-D modelcanbevalidatedfromthecomparisonofthesimulation results. Furthermore, the 2-D numerical model can also be used to investigate the effect of the ARO's shape on the wave forces. All 2-D, 3-D numerical models are generated by the CAD, and then the mesh-generating software “ICEM” is used to obtain the numerical grid.

Table 1 Boundary conditions used in the numerical wave tank

Figure 9 shows the comparison of the wave profile, and the vertical and horizontal wave forces between the measurements and the simulation results of the 2-D, 3-D numerical models. From Fig. 9(a), it can be seen that the wave profiles between the numerical results and the measurements correlate well,which shows the identical initial wave conditions in the numerical simulations and the experiments. In the vertical direction, the long duration and the slamming wave forces can be clearly observed from the measured and simulated results in Fig. 9(b). From the comparison, it is seen that the 2-D, 3-D simulated results agree well with the experiment results. However, because the slamming force is strongly affected by the boundary conditions, the high-frequency slamming force is somewhat different from the testing results. As for the horizontal wave force shown in Fig.9(c), both numerical models overestimate the wave forces. Furthermore, Figs. 9(b), 9(c) show that the simplified 2-D model can be used effectively for the numerical simulation and that the shape of the ARO has little influence on the wave forces.

Fig. 9 Comparative results between 2-D and 3-D numerical results and the measurements

The snapshots of the pressure and the flow fields obtained by the 3-D numerical model for the time instants corresponding to the five peaks of the vertical force (Fig. 9(b)) are shown in Fig. 10. With the evolution of the solitary wave, high pressure areas are sequentially observed in the chambers between the T-type girders in the direction of the wave propagation (Figs. 10(a)-10(e)). The corresponding flow fields (Figs. 10(f)-10(j)) demonstrate that the highpressure areas appear in the chambers when the wave particles just reach the bottom of the deck plate. After that, the pressure in the chambers is decreased due to the ejection of the wave particles from the AROs installed in the plate.

Fig. 10 (Color online) Pressure fields (a-c) and flow fields (f-j)for the 3D numerical model at the five peak-pressure times

Fig. 11 Vertical and horizontal wave forces obtained by measurements,2-D numerical model and 3-D numerical model

The comparisons of the wave forces between the measurements, and the 2-D, 3-D numerical results for the four measured wave amplitudes are presented in Fig. 11. It can be seen in this figure that the experimental forces correlate well with both the 2-D, 3-D numerical results in the vertical direction, which demonstrates that the shape of the ARO plays a negligible role in mitigating the effects of the vertical wave forces. In the horizontal direction, both the numerical models overestimate the wave forces for the wave amplitudes of 39 mm, 51 mm and 65 mm.

To investigate the effect of the AROs in each chamber of the bridge model on the mitigation of the wave forces, five 2-D numerical cases, No. 1, No. 2,No. 3, No. 4 and No. 5 (without the first, second, third,fourth, or fifth ARO (Fig. 8(c)), respectively), are selected for the comparisons. Numerical results (Fig.12) indicate that the fifth ARO plays an important role in the mitigation of the vertical wave forces for the wave amplitudes of 39 mm, 51 mm and 65 mm.

Fig. 12 (Color online) Vertical forces (red bars) and horizontal forces (cyan bars) obtained by the 2-D numerical model for the decks with 5 AROs (5 holes) and 4 AROs(where the number means without the ARO of that number (Fig. 8(c)), respectively) for wave amplitudes

Fig. 13 Vertical- (solid symbol) and horizontal-wave (open symbol) forces obtained by 2-D numerical model for ARO widths 1.8 mm, 2.7 mm, 3.6 mm and 4.5 mm

To investigate the relationship between the volume of the AROs and the wave forces, numerical wave forces on bridge decks with widths of 1.8 mm,2.7 mm, 3.6 mm and 4.5 mm are presented in Fig. 13.As can be seen in Fig. 13, the same vertical forces are obtained for all ARO widths for a wave amplitude of 25 mm. With the increase of the wave amplitude,large ARO widths result in small vertical wave forces on the bridge decks, except for the widths of 2.7 mm,3.6 mm, in which the same vertical wave forces are obtained. In the horizontal direction, the same wave forces in all numerical cases indicate that the wave forces are independent of the width of the AROs.

4. Conclusions

In this paper, the ARO are used to reduce the solitary wave-induced forces on the inverted T-type bridge decks. Hydrodynamic experiments are conducted for three types of bridge decks, including the inverted T-type decks with four, five, and six girders.The open source toolbox OpenFOAM is used to analyze the mechanism of the wave-structure interaction. Because of the good correlation between the numerical results and the measurements, the relationship between the shape and the volume of the ARO and the wave forces are numerically investigated using the numerical model. Based on the measurements and the numerical results, the following conclusions can be drawn:

(1) For all three types of bridge decks, experimental results indicate that the AROs could reduce both the amplitudes and the duration of the wave forces in the vertical and horizontal directions. The best mitigation effect on the wave forces is observed for the inverted T-type deck with five girders.However, the presence of the AROs results in strong local pulse-like pressures in the chambers of the bridge decks for large wave amplitudes.

(2) Numerical results obtained by the OpenFOAM demonstrate that high-pressure fields can be sequentially observed in the chambers between the T-type girders from the upstream to the downstream sides. The pulse-like high-pressure fields appear when the wave particles touch the bottom of the plate.

(3) Comparisons between 2-D and 3-D numerical results indicate that the mitigation of the wave forces is independent of the shape of the ARO. For the inverted T-type decks, the ARO installed in the chamber of the downstream side is more important than other AROs in the mitigation of the wave forces.With the increase of the ARO volume, the mitigation of the wave forces is increased.