Hydrodynamic Forces on Inundated Bridge Decks

508 Captions

Figure 14. Image. Comparison of flow fields for the 2-D and 3-D models for the STAR-CD® simulations. Two velocity maps generated from the STAR-CD® are shown. The top map depicts the 2-D model results, and the bottom, the 3-D results. The maps are centered on the six-girder bridge in a cross section, which is shown in white. The velocity mapped is only the horizontal vector and is positive moving from right to left in the direction of flow. Differences in velocity are shown by variations in color. The velocity scale goes from red (highest velocity) to orange, yellow, green, and blue (zero potentially negative velocity). Red corresponds to a velocity of 0.40 m/s, yellow to 0.20 m/s, light blue to 0 m/s, and dark blue to slightly negative velocity. For the 2-D results map, the region above the bridge is nearly completely red. Just left of the bridge, a mostly green zone begins somewhat wide and is pinched down by the yellow and red to the top of the railing and the bottom of the leftmost girder. The zone in between the two railings and the spaces in between all show blue zero velocity zones. A band of green traces the top of the railings and bottom of the girders. The green bands widen downstream of the bridge and constrict the blue zone downstream of the bridge. These green bands transition to thin, nearly horizontal bands of yellow above and below the bridge. Below the yellow band, velocity increases to red again. The 3-D figure has a very similar map except the green zones do not constrict the blue zone downstream of the bridge and instead expand away from the bridge. The yellow bands are therefore pitched upward above the bridge and slightly downward below the bridge.

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Figure 19. Image. Velocity profile from the 2-D Fluent® model. This image shows the velocity profile in the vicinity of the six-girder bridge cross section by color coding flow according to velocity. The scale ranges from dark blue, which corresponds to 0 or slightly negative velocity, through light blue and green, which correspond to low velocity (0.10 m/s to 0.40 m/s), up to yellow, orange, and red, which represent velocities up to 0.6 m/s. At the top of the map, the water surface is shown by a clear distinction among solid blue, yellows, and oranges. The bridge deck is located about two-thirds of the way down in the water. Below the bridge deck, the spaces in between the girders are shown as dark blue, but the color rapidly transitions from blue to red just slightly below the bottom of the girders. Upstream of the left edge bridge deck, the flow is characterized by the color green which then turns to blue very close to the bridge. However, the color shifts to yellow above and below the bridge's elevation. On the top half of the bridge, the railing blocks flow and have a blue zone to about halfway across the bridge deck. Above the blue zone, there is a rapid transition to green and then red. On the right half of the top of the bridge deck, a large "bubble" of green is present, raised to roughly twice the height of the railing. Above the green area, the velocity reaches only to the orange color. Downstream of the bridge, yellow zones spread away from the top of the right railing and the bottom of the right most girder in a slight V pattern, though much more pronounced on the top. In between, the flow is green and then blue. Above this downstream zone, the flow is mostly orange and below, mostly red.

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Figure 20. Image. Velocity profile from the 3-D Fluent® model. This image shows a velocity profile similar to that of figure 19 but with the flow data from the 3-D Fluent® model. At the top of the profile, the transition from air to water is not as sharp as in figure 19 and has defined bands of light blue, green, yellow, and eventually red. Upstream of the bridge, the flow is mostly yellow-orange but shifts toward green at the bridge's elevation and the very bottom of the profile. Directly above and below the bridge and continuing downstream, the flow is predominantly red. At the leading (left) edge of the bridge, thin bands of yellow extend slightly above the top of the railing and slightly below the bottom but remain parallel going downstream. On the bottom half of the bridge, the space between the girders is dark blue. Below the girder, a very thin band of light blue and a wider band of green are visible before the yellow band. On the top half of the bridge, there is a dark blue triangle between the top of the left railing, the bottom of the railing, and the midway point of the road deck. Above this triangle, there is rapid transition from blue to green to yellow. The downstream half of the road surface has horizontal bands of light blue, green, and then yellow. Downstream of the bridge, there is a blue zone which is rapidly constricted by expanding zones of green. These green zones expand to the mostly horizontal yellow bands above and below the bridge.

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Figure 28. Graph. Drag coefficient versus inundation ratio for the six-girder bridge. This graph displays the inundation ratio on the x-axis and the drag coefficient on the y-axis. The x-axis scale goes from 0 to 3.5 and the y-axis from 0 to 2.5. This graph, as do figures 29-36, displays experimental results for four Froude numbers, an upper and lower fitting equation, and CFD results. The experimental results for the 0.16, 0.22, 0.28, and 0.32 Froude numbers are denoted by a white square with black outline, white triangle with black outline, black square and black triangle, respectively. The fitting equations are denoted by blue lines. The STAR-CD® results are denoted by pink circles, Fluent® LES results by red diamonds, and Fluent® k-epsilon results by green squares. The upper fitting equation starts on the left at 2.15 on the y-axis, follows a rough, parabolic shape down to a minimum of 1.2 at h (star) equals 0.8, and then smoothly returns to 2.15 when h (star) is 2.5. It continues as a flat line to the end of the plot. The lower fitting equation curve is the same shape, only 0.4 units lower. The Froude number equals 0.32 results roughly trace the upper curve at intervals of about 0.125 on the x-axis. The Froude number equals 0.16 results roughly follow the lower curve, and the other two experimental results fall in between. The CFD results are shown for illustration purposes only and are described in the main text.

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Figure 30. Graph. Moment coefficient versus inundation ratio for the six-girder bridge. This graph displays the inundation ratio on the x-axis and the moment coefficient on the y-axis. The x-axis scale goes from 0 to 3.5 and the y-axis from -0.15 to 0.35. The curve shapes appear to be an inversion of the shape in figure 28. The upper fitting curve starts at 0, with a coefficient value around 0.03. The curve increases in a rough parabola to a maximum of about 0.29 at roughly h (star) equals 0.85 and descends at roughly the same rate to become a flat line after h (star) equals 2.5 at 0.03 again. The lower curve is the same shape but shifted downward by roughly 0.1. The top of the curve is somewhat flattened with a maximum of about 0.11. The Froude number equals 0.16 and the Froude number equals 0.22 data series are very close to the upper curve. The Froude number equals 0.32 follows the lower curve, and the Froude number equals 0.28 data fall in the middle. The CFD results are described in the text, but all three fall mainly between the curves from h (star) equals 1.2 onward.

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Figure 31. Graph. Drag coefficient versus inundation ratio for the three-girder bridge. This graph displays the inundation ratio on the x-axis and the drag coefficient on the y-axis. The x-axis scale goes from 0 to 3 and the y-axis from 0 to 2.5. This graph has all the same elements as figure 28, except for no Fluent® LES CFD results. (Figures 32-36 do not have Fluent® LES results either.) The shape of the fitting equation curves is similar to those in figure 28. The y-intercept of the upper curve is 1.75, the minimum value is roughly 1.2, and the curve reaches a plateau of 1.95 when h (star) reaches 2.25 or so. The lower fitting equation envelope curve is virtually identical but is shifted down roughly 0.35. The experimental data are generally well contained by the curves, but the high Froude numbers (0.28 and 0.32) are erratic and frequently overlap. At h (star) greater than 2, the 0.28 results follow the upper curve with the 0.32 results below. The Froude number equals 0.16 and 0.22 results remain lower and generally follow the lower envelope curve. The STAR-CD® results (four data points) fall within the upper portion of the envelope when h (star) is greater than 1.5. The Fluent® k-epsilon results (five data points) fall in the middle of the envelope range when the drag coefficient is at its minimum value but rise higher than the upper fitting equation when h (star) is greater than 1.5 and stabilize around 2.3 when h (star) exceeds 2.

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Figure 32. Graph. Lift coefficient versus inundation ratio for the three-girder bridge. This graph displays the inundation ratio on the x-axis and the lift coefficient on the y-axis. The x-axis scale goes from 0 to 3 and the y-axis from -2 to 0.5. The shape of the two envelope curves is similar to those in figure 29. The upper curves has a y-intercept of 0, descends quickly to minimum value of about -1.1 at h (star) equals 0.83, and gradually increases at decreasing rate to approximately -0.05 at h (star) equals 3. The lower curve starts at the same intercept but drops more steeply to minimum of -1.85 and rises only to -0.2 at the right end of the plot. Unlike figure 29, the experimental data are generally well contained. The experimental results are erratic and overlapping for partially inundated cases. There is a near convergence of all four series at h (star) equals 1 near the lower envelope curve. Beyond h (star) equals 1, the results separate and appear in order with the Froude number equals 0.16 near the upper curve and the Froude number equals 0.32 results near the lower curve. At h (star) values above 1, both CFD results are within the envelope, with the STAR-CD® results being well centered, and the Fluent® k-epsilon results being somewhat high.

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Figure 33. Graph. Moment coefficient versus inundation ratio for the three-girder bridge. This graph displays the inundation ratio on the x-axis and the moment coefficient on the y-axis. The x-axis scale goes from 0 to 3 and the y-axis from -0.2 to 0.3. The shape of the two envelope curves is similar to those in figure 30. The upper curve has y-intercept of 0, reaches a maximum of 0.245 around h (star) equals 0.8, and returns back to 0 where h (star) is greater than 2.2. The lower curve is shifted down by about 0.11, and the bump in the curve is somewhat shallower, reaching a maximum of around 0.8. The experimental data are fairly well contained by the envelope curves, but they are highly erratic within the curves and cross each other numerous times. The STAR-CD® results, starting at h (star) equals 1.2, stay within the lower portion of the envelope but have a linear decreasing pattern instead of flattening. The Fluent® k-epsilon starts below the lower curve at h (star) equals 0.75 and remains mostly flat until h (star) equals 1.6. The results are within the envelope and then rapidly decrease at an increasing rate until the last point (at h (star) equals 2.55) is below the lower envelope curve by 0.02.

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Figure 35. Graph. Lift coefficient versus inundation ratio for the streamlined bridge. This graph displays the inundation ratio on the x-axis and the lift coefficient on the y-axis. The x-axis scale goes from 0 to 4 and the y-axis from -1.4 to 0.4. The envelope curves are actually similar in shape to those in figure 34. The upper curves start at h (star) equals 0.25 from a value of 0.3 and is already at rapid downward slope. The slope begins to curve upward and bottoms out at roughly h (star) equals 1.3 with a value of -0.81. The curve continues upward to roughly -0.4 at h (star) equals 2.5, and then it transitions to a linear path, ending at h (star) equals 4 with a value of roughly -0.25. The lower curve is shifted downward by about 0.4, with the minimum point at about -1.21. The experimental results are well contained within the curves. The Froude number equals 0.16 dataset follows the upper envelope curve, but the other three sets of experimental results all cluster near the lower curve. The Fluent® k-epsilon data are above the upper curve and show little variation with h (star) remaining at a steady -0.2 to -0.25.

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Figure 36. Graph. Moment coefficient versus inundation ratio for the streamlined bridge. This graph displays the inundation ratio on the x-axis and the moment coefficient on the y-axis. The x-axis scale goes from 0 to 4 and the y-axis from -0.05 to 0.25. The envelope curves share a similar shape to those in the other moment coefficient plots, figure 30 and figure 33. The upper curve starts at left with a value 0.05, reaches a maximum around 0.19 around h (star) equals 1.3, and then descends smoothly to transition to a flat line value 0.05 again when h (star) exceeds 3. The lower curve is merely shifted downward by 0.05. The experimental results are not perfectly bounded by the curves. The higher Froude number results are above the highest curve at h (star) of about 1.1, where they are just to the left of the high point in the curve and between 2.5 and 3, where the coefficient values descend more slowly than the curve. The lower Froude number results mostly follow the lower curve but fall just slightly below it between h (star) equals 1.25 and 2.25. The Fluent® k-epsilon results do not match well with the experimental results and remain at 0 or slightly less for their entire range of h (star) equals 1.5 to 4.

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