The Seventh International Colloquium on Bluff Body Aerodynamics and Applications (BBAA7)
Shanghai, China; September 2-6, 2012
Numerical simulation of hydrodynamic loading on
submerged rectangular bridge decks
Chia-Ren Chu1, Chih-Jung Huang1, Tso-Ren Wu2, Chung-Yue Wang1
1
2
Department of Civil Engineering, National Central University, Tao-Yuan, Taiwan, R.O.C.
Institute of Hydrological and Oceanic Sciences, National Central University, Taiwan, R.O.C.
ABSTRACT: This study used a Large Eddy Simulation (LES) model to investigate the hydrodynamic forces on fully submerged bridge decks. The numerical simulation was verified by comparing with the experimental results and was used to examine the influences of the deck Froude
number and blockage ratio on the drag and lift coefficients of the bridge deck. The results demonstrate that the force coefficients are dependent on the deck Froude number. The drag coefficient increases as the blockage ratio increases for low Froude number, and dramatically increases, due to the wave induced drag, for high Froude number. On the other hand, the lift
coefficient is a function of the submergence ratio of the deck. The downward velocity above the
bridge deck and the asymmetric pressure distribution on the upper and lower sides of the deck
generate a negative lift force on the bridge deck.
KEYWORDS: Bridge deck, Froude number, drag coefficient, lift coefficient, blockage ratio.
1 INTRODUCTION
The hydrodynamic loading on the bridge body is an essential parameter for evaluating bridge
safety, especially when a bridge deck is entirely submerged in the river flow during flood events.
The drag and lift forces can be calculated as:
1
FD CD UU o2 A
(1)
2
1
FL CL UU o2 A
(2)
2
where CD is the drag coefficient, CL is the lift coefficient, Uis the density of water, Uo is the freestream velocity of the undisturbed flow, and A is the frontal area of the bridge body. The drag
and lift coefficients are functions of the bridge shape, attack angle and the Reynolds number of
the flow (Hamill, 1999).
For fully submerged bridge decks, due to the presence of a free surface and the complicated
interaction between the bridge structure and river flow, the drag and lift coefficients are difficult
to determine. In previous studies, Naudascher & Medlarz (1983) measured the drag coefficient
of a girder bridge and found that the wave motion and vortex formation between the girders generated a peak loading on the bridge deck. Malavasi & Guadagnini (2003) used dynamometers to
measure the hydrodynamic forces on partially and fully submerged bridge decks with a rectangular cross-section. Their results revealed that the drag and lift coefficients were dependent on the
Reynolds number and submergence ratio of the bridge deck. The submergence ratio is defined
as:
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Ho h
(3)
D
where Ho is the water depth of undisturbed flow, D is the thickness of the bridge deck, and h is
the distance from the channel floor to the underside of the deck (see Fig. 1). They also point out
that the drag and lift coefficients significantly deviated from the corresponding values in an unbounded domain. In another paper, Malavasi & Guadagnini (2007) used a laboratory flume to
study the interaction between a free surface flow and a rectangular cylinder in the water. They
found that the rectangular cylinder created a blockage effect on the water flow, and the Reynolds
number and the blockage ratio together affected the hydrodynamic loading on the cylinder. The
blockage ratio is defined as the ratio of the frontal area of the bridge to the channel crosssectional area.
On the other hand, Malavasi & Trabucchi (2008) used a three-dimensional k-H turbulence
model to investigate the proximity effects of a solid wall on the wake flow of a rectangular cylinder (aspect ratio L/D = 3) placed above a solid wall with different gap ratios h/D (h is the distance of the lower side of the cylinder to the wall). Their simulation results showed that the drag
and lift coefficients increased as the gap ratio, S/D, decreased.
However, the above studies only covered a narrow range of flow conditions. Whether their
results could be applied to other flow conditions (for example, supercritical flow) still needed to
be examined. This study used a Large Eddy Simulation (LES) model to investigate the interactions between a free surface flow and fully submerged bridge decks. A series of numerical simulations were carried out to clarify the influences of the Froude number, the blockage ratio and the
submergence ratio on the force coefficients of the bridge deck.
h*
Uo
P=0
Free surface
Uo
L
9D
D
Ho
z
h
Channel floor
x
15D
60D
u = 0, w = 0
Figure 1. Schematic diagram of a fully submerged rectangular bridge deck.
2 NUMERICAL MODEL
This study used a Large Eddy Simulation (LES) model to simulate the flow field around the
bridge deck. The fluid motion was simulated by solving the continuity equation and the Navier-
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The Seventh International Colloquium on Bluff Body Aerodynamics and Applications (BBAA7)
Shanghai, China; September 2-6, 2012
Stokes equations. The free surface was tracked by the Volume of Fluid (VOF) algorithm. The
governing equations can be expressed as:
wu i
(4)
0
w xi
§ wu
w u j ·º
wP
w ª
«Peff ¨ i (5)
UgGi3 ¸»
¨ w x j w xi ¸»
w xi
wxj «
©
¹¼
¬
where the subscripts i, j = 1, 2 represent the x and z directions, respectively; t is the time, u and
P are the filtered velocity and pressure, U is the density of water, g is the gravitational acceleration, and Peff is the effective viscosity, defined as:
Peff = P + PSGS
(6)
where P is the dynamic viscosity of water, and PSGS is the viscosity of sub-grid scale turbulence,
defined as:
wU u i wUu i u j
wt
wxj
μ SGS
ȡ(Cs ǻ s ) 2 2Sij Sij
(7)
where Cs is the Smagorinsky parameter, Sij is the rate of strain, and 's is the characteristic length
of the spatial filter:
(8)
's 2('x'z)1/ 2
In this study, the value of the Smagorinsky parameter was set to be Cs = 0.30. In addition, the
projection method (DeLong, 1997) was used to solve the Poisson Pressure Equation (PPE) and
to decouple the velocity and pressure in the Navier-Stokes equations. The water surface was
simulated by the Volume of fluid (VOF) method (Hirt & Nichols, 1981).
sf m
(9)
+ ‹¸( f mu) = 0
st
The value of fm = 1 corresponds to a cell full of water; and fm = 0 represents the cell is full of air.
Further details of the numerical model can be found in Wu & Liu (2009).
3 RESULTS AND DISCUSSIONS
In order to demonstrate the accuracy of the numerical model, the present Large Eddy Simulation
(LES) model was compared with the experimental results of Malavasi & Guadagnini (2007). A
stationary, rectangular cross-sectional bridge deck (thickness D = 0.06 m, length L = 0.18 m, aspect ratio L/D = 3) was placed above the channel floor. The deck was fixed in the channel and
parallel to the channel floor. The distance from the channel floor to the underside of the deck
was h = 0.14 m (see Fig. 1). The width of the bridge deck was equal to the width of the channel,
and the flow direction was parallel to the length of the bridge deck. The bridge pier was not considered in this study. The water depth of undisturbed flow was in the range of Ho = 0.17 ~ 0.5 m,
upstream velocity Uo = 0.2 m/s, the submergence ratio h* = (Ho – h)/D = 0.5 ~ 5.0, the blockage
ratio Br = 0.12 ~ 0.35. The Reynolds number Re = UoD/Q = 1.2 × 104, and the Froude number Fr
= Uo/(gHo)1/2 was in the range of Fr = 0.09 ~ 0.15. In order to examine the influence of the flow
velocity on the force coefficients of the bridge deck, Malavasi & Guadagnini (2003) defined a
deck Froude number as:
Uo
(10)
FrD
gD
where D is the thickness of the bridge deck.
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The drag and lift coefficients were computed from the pressure distribution around the cylinder and compared with the experimental results of Malavasi & Guadagnini (2007). As shown in
Figure 2, the simulation results of the uniform grid (936 x 110) and the non-uniform grid (496 x
110) both are close to the measured drag and lift coefficients. The biggest difference for the
time-averaged drag coefficient was 1.9%. In order to reduce the computing time, the nonuniform grid was used for the rest of the simulation.
Figure 2(a) depicts that the drag coefficient CD increased as the submergence ratio, h*, increased in the range of 0 d h* < 1.0. This is because the bridge deck was partially submerged and
the frontal area of the deck under the water surface increased as h* increased. However, the value
of CD decreased as the submergence ratio, h*, increased when h* t 1.2. Malavasi & Guadagnini
(2007) explained this trend by asserting that the variations of h* induced the change in the blockage ratio, and the increasing blockage ratio resulted in the increase of the drag coefficient.
(a)
5
Malavasi & Guadagnini (2007)
non-uniform grid (496 x 110)
uniform grid (936 x 110)
4
CD
3
2
1
0
0
1
2
3
h*
4
5
6
(b)
3
0
CL
-3
-6
-9
Malavasi & Guadagnini (2007)
non-uniform grid (496 x 110)
uniform grid (936 x 110)
-12
-15
0
1
2
3
h*
4
5
6
Figure 2. Comparison of model predictions and experimental results of Malavasi & Guadagnini (2007): (a) drag coefficient; (b) lift coefficient.
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The Seventh International Colloquium on Bluff Body Aerodynamics and Applications (BBAA7)
Shanghai, China; September 2-6, 2012
However, the submergence ratio and the blockage ratio were related to each other in the experimental conditions of Malavasi & Guadagnini (2007). Discriminating the influences of the
blockage ratio and the submergence ratio on the drag coefficient can be difficult.
3.0
(a)
2.5
CD
2.0
1.5
Malavasi & Guadagnini (2007), D = 0.06 m
Present study, D = 0.06 m
Present study, D = 0.6 m
1.0
0.0
0.4
0.8
1.2
1.6
2.0
FrD
0
(b)
-2
CL
-4
-6
-8
-10
-12
0.0
Malavasi & Guadagnini (2007), D = 0.06 m
Present study, D = 0.06 m
Present study, D = 0.6 m
0.4
0.8
1.2
1.6
2.0
FrD
Figure 3. Relationship between force coefficients and deck Froude number for submergence ratio h* = 2.0, and
blockage ratio Br = 0.23: (a) drag coefficient; (b) lift coefficient.
On the other hand, Figure 2(b) shows that the lift coefficient, CL, was negative and decreased as the submergence ratio, h*, increased in the range of 0ʳ d h* d 1.0. This is due to the
negative pressure (suction) produced by the separation bubble only occurred on the lower side of
the deck when h* < 1.0. When the deck was fully submerged (h* t 1.2), the lift coefficient, CL,
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increased until it approached zero (h* t 3.5). It was easy to understand this phenomenon because
the pressure coefficients, Cp, became symmetric on the upper and lower sides when the water
surface no longer influenced the separation flow on the upper side of the deck.
The drag and lift coefficients of the present study were plotted against the deck Froude number in Figure 3. The submergence ratio was h* = 2.0, the aspect ratio was L/D = 3, and the blockage ratio was Br = 0.23. The Reynolds number was Re = 1.2 × 104 ~ 6.54 × 104, and the Froude
number based on water depth Ho, Fr = Uo/(gHo)1/2, was in the range of Fr = 0.13 ~ 0.88. Namely,
the channel flows were all sub-critical flows. However, the flow condition covered sub-critical
flow to super-critical flow while using the deck Froude number (FrD = 0.26 ~ 1.82).
Figure 3(a) shows that when FrD < 0.50, the computed drag coefficient, CD = 2.25, was close
to the value CD = 2.0~2.2 suggested by Hamill (1999) for bridge deck design. But the value of
CD increased slightly as the deck Froude number increased, and became a constant CD = 2.6
when FrD > 0.80. In addition, the lift coefficient suddenly decreased from CL = -1.5 to CL = -5.0
when the deck Froude number was FrD > 0.80 (see Fig. 3(b)). This implied that there was a flow
transition when the deck Froude number increased from 0.50 to 0.80.
The time-averaged water surface curves of different deck Froude numbers are shown in Figure 4. The gray area represents the location of the bridge deck. The location of the water surface
was determined by setting the volume fraction fm = 0.5. The water surface was smooth and undisturbed for low Froude number (FrD d 0.52). However, due to the back-water effect, the water
surface was levitated in front of the bridge deck and suddenly dropped behind the deck when the
Froude number was FrD t 0.78. This was the reason that the drag and lift coefficients went
through a transition as the deck Froude number increased. The flow condition transformed from
a sub-critical flow to a trans-critical flow as the deck Froude number increased from FrD d 0.52
to FrD t 0.78. The trans-critical flow covered the near-critical flow (0.8 < FrD < 1.0), critical
flow (FrD = 1.0) and super-critical flow (FrD > 1.0).
1.4
1.2
H/Ho
1.0
0.8
FrD = 0.26
FrD = 0.98
FrD = 1.56
0.6
0.4
-10
0
10
x/D
FrD = 0.78
FrD = 1.30
FrD = 1.69
20
30
Figure 4. Water surface variations of different deck Froude numbers for h* = 2. The gray area represents the location
of the bridge deck, x is the distance from the leading edge of the deck.
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The Seventh International Colloquium on Bluff Body Aerodynamics and Applications (BBAA7)
Shanghai, China; September 2-6, 2012
It should be noted that, for trans-critical flows, the height of the water surface above the deck
and the amplitude of the surface disturbance increased as the deck Froude number increased. The
surface disturbance behind the deck gradually decreased as the water flowed downstream. This is
similar to the ship wave generated by a moving object near the water surface (Bal, 2008). The
increase of the drag coefficient from CD = 2.2 to CD = 2.6, as the flow condition changed from
sub-critical flow to trans-critical flow, was due to the wave-induced drag.
Figure 5 compares the distribution of the time-averaged pressure coefficients on the upper and
lower sides of the deck for FrD = 0.26, 0.78 and 1.30, h* = 2.0. The spatial average pressure coefficients were Cp3 = -2.01, 0.66, 0.92 on the upper sides and Cp4 = -3.21, -4.32, -4.17 on the lower
sides, respectively. As can be seen in Figure 5, the pressure coefficients Cp4 on the lower side of
the deck were all negative and were close to each other for different Froude numbers. But the
pressure coefficients Cp3 < 0 on the upper side of the deck for the sub-critical flow (FrD = 0.26)
and Cp3 > 0 for the trans-critical flows (FrD t 0.78). The positive pressure coefficient, Cp3, was
caused by the downward flow above the bridge deck. This positive pressure on the upper side
together with the negative pressure on the lower side engendered a large downward force on the
deck (negative lift coefficient CL).
8
6
4
FrD = 0.26, upper side,
FrD = 0.78, upper side,
FrD = 1.30, upper side,
FrD = 0.26, lower side
FrD = 0.78, lower side
FrD = 1.30, lower side
Cp
2
0
-2
-4
-6
-8
0.0
0.5
1.0
1.5
x/D
2.0
2.5
3.0
Figure 5. Distribution of time-averaged pressure coefficients on the upper and lower sides of the deck for FrD = 0.26,
0.78 and 1.30. The submergence ratio h* = 2.
Besides the Froude number effect, the force coefficients were influenced by the blockage ratio. This section focuses on the blockage effect on the force coefficients. The deck thickness D =
0.06 m, length L = 0.18 m (aspect ratio L/D = 3), and the free-stream velocity Uo = 0.20 m s-1
were kept constant throughout the simulation. The Reynolds number was Re = 1.20 x 104, and
the deck Froude number was FrD = 0.26. The variation of the blockage ratio included two different simulation series. The first simulation series kept the distance from the channel floor to the
underside of the deck constant (h = 0.14 m), while changing the water depth Ho = 0.23 ~ 0.50 m.
This was similar to the experimental setup of Malavasi & Guadagnini (2007). The submergence
ratio was in the range of h* = 1.0 ~ 6.0, and the blockage ratio was Br = 0.12 ~ 0.26. The second
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simulation series changed the water depth, Ho and the height above the channel floor, h, while
keeping the submergence ratio (h* = 2.0) constant. The blockage ratio was in the range of Br =
0.05 ~ 0.20.
(a)
3.6
Malavasi & Guadagnini (2007)
Present study, h* = 1.0 ~ 6.0
Present study, h* = 2.0
3.2
CD
2.8
2.4
2.0
1.6
1.2
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.24
0.28
0.35
Br
2
(b)
0
-2
CL
-4
-6
-8
-10
Malavasi & Guadagnini (2007)
Present study, h* = 1.0 ~ 6.0
Present study, h* = 2.0
-12
-14
0.00
0.04
0.08
0.12
0.16
0.20
0.32
Br
Figure 6. Relationship between force coefficients and blockage ratio for FrD = 0.26: (a) drag coefficient; (b) lift coefficient. The line is the prediction of Eq. (11).
The force coefficients were plotted against the blockage ratio in Figure 6. The flow condition
was sub-critical flow (FrD = 0.26). The experimental results of Malavasi & Guadagnini (2007)
were also plotted in the figure for comparison. Note that in Figure 6(a), the relationship between
the drag coefficient and the blockage ratio was independent of the submergence ratio h* for sub-
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The Seventh International Colloquium on Bluff Body Aerodynamics and Applications (BBAA7)
Shanghai, China; September 2-6, 2012
critical flow. The drag coefficient was a constant CD = 1.6 when the blockage ratio was Br <
0.14. When Br > 0.14, the value of CD increased from 1.6 to 2.8 as the blockage ratio Br increased. This is because the pressure difference between the frontal and rear faces of the deck increased by the blockage effect, and higher pressure drop resulted in a greater drag on the deck. In
other words, the drag coefficient CD = 2.0~2.2 suggested by Hamill (1999) may not be applicable
for bridge decks with a large blockage ratio (Br > 0.23) or in trans-critical flows (FrD t 0.78). An
empirical equation between the drag coefficient, CD, and blockage ratio Br can be found by least
square regression:
CD
1.0 D ˜ Br n
(11)
CDo
The parameters D =128.4, n = 4.0 and CDo = 1.58 is the drag coefficient of rectangular cylinders of aspect ratio L/D = 3 in an unbounded domain. The coefficient of determination was R2 =
0.938. This equation could be used to compute the drag coefficients of fully submerged bridge
decks (h* > 1) with a rectangular cross-section in sub-critical flows.
On the other hand, Figure 6(b) shows that the lift coefficient, CL, decreased as the blockage ratio increased. The lift coefficient was CL > -2.0 when the blockage ratio Br < 0.25. This is because the water surface restricted the development of separation flow on the upper side of the
deck, and resulted in the asymmetric pressure distribution on the upper and lower sides of the
deck. However, the negative lift coefficient became fairly large (CL < -3.0) when the blockage
ratio Br > 0.25. When the blockage ratio Br > 0.25, even in sub-critical flows, the pressure distribution is similar to trans-critical flow shown in Figure 5. The spatial average pressure coefficients, Cp3, on the upper side of the deck were positive, together with the negative pressure on
the lower side generated a large downward force on the deck.
4 CONCLUSIONS
This study used a Large Eddy Simulation (LES) model and the Volume of fluid (VOF) method
to investigate the interactions between the free surface flows and fully submerged bridge decks.
The model prediction was verified by comparing it with the experimental data of Malavasi &
Guadagnini (2007). The numerical model then was used to calculate the hydrodynamic forces on
bridge decks of rectangular cross-section. The results indicated that the drag and lift coefficients
were dependent on the deck Froude number, FrD = Uo/(gD)1/2. The water surface was flat and
undisturbed for sub-critical flow (FrD d 0.52). When the deck Froude number was FrD t 0.78, the
water surface rose up in front of the submerged deck and dropped behind the deck, created a
large surface disturbance.
In addition, the drag coefficient remained a constant CD = 1.6 when Br < 0.14, and the value
of CD increased as the blockage ratio, Br, increased for sub-critical flows (FrD d 0.52). For transcritical flows (FrD t 0.78), due to the wave-induced drag, the drag coefficient was larger than
that for sub-critical flows with the same blockage ratio. Furthermore, the lift coefficient was a
function of the deck Froude number, FrD, and the submergence ratio, h*. This is because the
separation shear flow on the upper side of the deck was constrained by the water surface and resulted in an asymmetric pressure distribution on the upper and lower sides of the deck. For transcritical flows and submergence ratio h* = 2.0, the downward flow above the deck produced positive pressures on the upper side of the deck and subsequently generated a large negative lift force
on the deck.
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5 ACKNOWLEDGEMENTS
The financial support from the China Engineering Consultants Inc. (CECI) of Taiwan, R.O.C.
under grants no. 99932 is gratefully appreciated.
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1
2
3
4
5
6
7
8
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