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Research Papers

# Numerical Evaluation of AASHTO Drag Force Coefficients of Water Flow Around Bridge PiersOPEN ACCESS

[+] Author and Article Information
Amin Almasri

Associate Professor
Civil Engineering Department,
Jordan University of Science and Technology,
Irbid 22110, Jordan
e-mail: ahalmasri@just.edu.jo

Assistant Professor
Civil Engineering Department,
The University of Jordan,
Amman 11942, Jordan
e-mail: s.moqbel@ju.edu.jo

1Corresponding author.

Contributed by the Materials Division of ASME for publication in the JOURNAL OF ENGINEERING MATERIALS AND TECHNOLOGY. Manuscript received April 13, 2016; final manuscript received August 31, 2016; published online February 1, 2017. Assoc. Editor: Xi Chen.

J. Eng. Mater. Technol 139(2), 021001 (Feb 01, 2017) (8 pages) Paper No: MATS-16-1112; doi: 10.1115/1.4035253 History: Received April 13, 2016; Revised August 31, 2016

## Abstract

Drag force is usually exerted on bridge piers due to running river water. This force is calculated empirically based on drag coefficients stated in design codes and specifications. Different values of drag coefficients have been reported in literature. For example, AASHTO LRFD Bridge Design Specifications uses a drag coefficient of 1.4 and 0.7 for square-ended and semicircular-nosed pier, respectively, while Coastal Construction Manual (FEMA P-55) recommends a value of two and 1.2 for square and round piles, respectively. In addition, many researchers have obtained other different values of drag coefficient under similar conditions (i.e., similar range of Reynolds number) reaching to 2.6 for square object. The present study investigates the drag coefficient of flow around square, semicircular-nosed, and 90 deg wedged-nosed and circular piers numerically using finite element method. Results showed that AASHTO values for drag force coefficient varied between very conservative to be under-reckoning. The study recommends that AASHTO drag coefficient values should be revised for different circumstances and under more severe conditions.

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## Introduction

The intensive use of vehicles in daily life and the race to reach destination in a short time and in the most economical way had resulted in laying roads in an unprecedented rate especially in the developing countries. Following this race to laying roads, bridges are being built over valleys and water paths. Bridge piers in these water paths can reach depths of 10's of meters. Designing these piers includes different parameters for various forces and stresses calculations.

These piers are susceptible to drag force resulting from water movement. Design codes have specific provisions on how to calculate the effect of this force on specific bridge pier shape depending on a constant drag coefficient (CD). However, this drag coefficient of any object is known to be function of fluid density and viscosity, flow speed and direction, and object position, shape, and size. Fluid velocity, fluid kinematic viscosity, and object size are incorporated into a dimensionless quantity called the Reynolds number. Therefore, drag coefficient is highly dependent on Reynolds number. Different values of drag coefficient can be found in literature for the same problem leaving designers in a dilemma on which drag factor is appropriate. For example, AASHTO LRFD Bridge Design Specifications [1] states a drag coefficient of 1.4 and 0.7 for square-ended and semicircular-nosed pier, respectively, while Coastal Construction Manual (FEMA P-55) [2] recommends a value of 2 and 1.2 for square and round piles, respectively. McConnell et al. [3] recommended for designing piers to use a drag coefficient of 0.65–0.9 for smooth circular piers, 1.05–1.5 for rough circular piers, 2 for square piers, and 1.6 for diamond-shape pier. Shore Protection Manual [4] presented the results of experiments and theories done by different researchers and showed that drag coefficient for circular pile is 1.2 for Reynolds number below 1 × 105 (called subcritical region) and drops to about 0.6–0.7 for Reynolds number above 4 × 105 (called supercritical region), with transition zone between the two regions. Similar to the Shore Protection Manual, Sumer and Fredsoe [5] compiled a plot of the drag coefficient for circular cylinder versus Reynolds number from the work of different researchers. In their plot, the drag coefficient value dropped from 60 at Reynolds number equal to 0.01 to 1.5 at Reynolds number equal to 1000; then, the drag coefficient was almost flat at 1.5 until 100,000 then dropped to 0.25 at Reynolds number equal to 4 × 105, and then, the drag coefficient increased as Reynolds number increased. Debus et al. [6] used computational fluid dynamics model to estimate drag coefficient and other flow characteristics for a square and circular pier. The study presented drag coefficient estimation at fixed Reynolds numbers. At Reynolds number of 1.15 × 105, drag coefficient for square pier fluctuated between 1.3 and 2.5 while for circular pier, it fluctuated between 0.6 and 1.2. At Reynolds number of 1.15 × 108, they showed results for circular pier only where it fluctuated between 0.6 and 0.81. Wenxiu et al. [7] conducted two-dimensional numerical simulation of flow around single square cylinder and two square cylinders in turbulent condition at a fixed Reynolds number (Re = 4 × 106). They found the drag coefficient for single square cylinder to be about 2.24. The drag coefficient value for two square cylinders was found to be depended on the dimension of the pier, spacing between piers and location of the piers relative to each other. Mallikarjuna et al. [8] investigated the drag force coefficient for bridge piers with semicircular-nose and lens-nose over a range of Reynolds number of 7 × 104–5 × 105. They found that drag force coefficients vary between 0.7 and 1.0 depending on flow contraction and pier length–width ratio. The observer of these studies and codes finds a wide variation in the value of the drag force coefficient for one directional flow. Moreover, these studies covered a limited range of Reynolds number or flow conditions. The studies which compiled a plot that covers the entire spectrum of Reynolds number were based on the work of researchers who used different approaches.

In literature, different values of the drag coefficient can be found that are not specifically found for piers but rather to any fluid flowing around cylinders. Sharma and Eswaran [9] studied the flow characteristics and heat transfer around an isolated square cylinder numerically. Both steady and unsteady laminar flows in the two-dimensional regime were considered at relatively low Reynolds number (1–160). Their results show a drag coefficient of about 1.5 at Reynolds number of 100. Lamura et al. [10] used a particle-based model to simulate steady and unsteady fluid flows around circular and square cylinders for Reynolds numbers ranging from 10 to 130. The approach they used is two-dimensional particle-based simulation method. The fluid is modeled by particles that propagate and collide, where their positions and velocities change continuously, while time is discretized. Their results showed that drag coefficient for square and circle cylinder reaches a value of about 1.4 and 1.3, respectively, at Reynolds number of about 100. Tsutsui [11] studied scour around circular pier caused by water flow in a wind tunnel. Reynolds number was fixed to 4.2 × 104. In Tsutsui [11] study, the pressure distributions on the pier surface were measured and the drag coefficient was obtained by integrating the surface pressure distribution on the pier. The study showed that drag coefficient of circular cylinder without a scour is about 1.3 and changes depending on scour geometry. Sato and Kobayashi [12] used abaqus/cfd to investigate the water flow around a circular cylinder and the occurrence of various phenomena associated with von Karman vortices. The study covered flow over Reynolds number range of 0.038–195. The obtained drag coefficient dropped from around 200 at low Reynolds to 1.8 at Reynolds number of 200. Wei-Bin and Bao-Chang [13] simulated the flow around a square cylinder inside a two-dimensional channel using nonuniform lattice Bhatnagar–Gross–Krook model over a Reynolds number range between 1 and 500. Wei-Bin and Bao-Chang found that drag coefficient reach has it is lowest value of 1.3 at Reynolds number of 170, and increases at other Reynolds numbers. Yoon et al. [14] studied the effect of Reynolds number and angle of incidence on flow around square cylinders numerically. The study calculated drag coefficient over Reynolds number range of 0–150 and angle of incidence 0–45 deg. At zero angle of incidence, the drag coefficient decreased from 4.4 at very low Reynolds to 1.38 at Re = 150. With variation of Reynolds number and angle of incidence, Yoon et al. presented a contour map for the drag coefficient. The minimum CD was 1.38 and the maximum CD was 1.76. Sohankar et al. [15] simulated the flow around square cylinder over Reynolds number range of 150–500 using two-dimensional and three-dimensional finite difference models. The drag coefficients obtained were between 1.45 and 1.65. In their simulation, the drag coefficient increased with increasing the Reynolds number. Dutta et al. [16] investigated the flow past a square cylinder over Reynolds number 250–700 and studied the flow with different incidence angles with cylinder at Reynolds number of 410. The study resulted in a minimum and a maximum drag force coefficient of 2.03 and 2.6, respectively. He et al. [17] studied the impact of corner cutoffs of a square cylinder on drag force at a fixed Reynolds number of 1035. In their study, the drag force coefficient dropped from 1.95 for square cylinder with no cutting for the corners to lower values depending on the minimum wake width and the cut-corners magnitude. These studies on drag coefficient for fluids around cylindrical shapes mainly estimated the drag coefficient at a Reynolds number range of as low as 0.038 and up to 500, which happens around bridge piers only if water is almost still with no significant flow velocity.

Other researchers work can also be found for higher Reynolds number but they mainly considered compressible fluid flow and covered limited range of Reynolds number. Lindsey [18] conducted an experimental study to estimate the drag coefficient for various simple shapes cylinders: circular, semitubular, elliptical, square, and triangular. In the experiment, different sizes of these shapes were tested in a wind tunnel at different Reynolds numbers. The Reynolds numbers were 840–8400 for small models and 31,000–310,000 for large models. The drag coefficient was found nearly 1.2 for circular cylinders, 2.3–2.4 for semitubular, 0.2–0.6 for elliptical, 1.7–2.6 for square cylinders, and 1–2.2 for triangular cylinders. In his experiment results, the drag coefficient tends to increase with increasing Reynolds number. Roshko [19] studied the flow of compressible fluid flow around bluff bodies and estimated the drag coefficient for circular and 90 deg wedge cylinders over Reynolds number range from 500 to 105. Roshko [19] estimated a drag coefficient range of 0.5–1.2 for circular cylinder and 0.65–1.6 for 90 deg wedge. Hoerner [20] recommended a value of 2.05 for square cylinders 1.55 for wedge shape cylinders and 1.17 for circular cylinders over a Reynolds number of 104–106.

The wide variation in drag force coefficient presented in the previous studies can affect the pier design significantly. Work of other researchers mostly considered a limited range of Reynolds number or flow conditions. Difference between these values and AASHTO drag coefficient value questions the suitability of AASHTO specification for drag force coefficient. Therefore, a drag force coefficient that is clearly set, safe to use, reliable, and systematically estimated for different flow conditions and different pier shapes is needed. In this study, the drag force coefficient for various pier shapes is estimated numerically using computational fluid dynamics within the finite element software abaqus/cfd. The values obtained cover conditions from still water to fast river flow with different inclination angles. Then, the drag coefficient values were compared to AASHTO design code and available literature values.

## Methodology

In this study, quantitative analysis was used for the evaluation of the drag force coefficient. The water flow around pier is simulated numerically using finite element analysis to obtain the drag force under different conditions. The analysis was focused on the interaction between the pier as a solid body and water as running fluid exerting pressure on the pier. The study concentrates on water flowing under natural conditions simulating river streams and lakes. Therefore, pier shapes were tested over a wide range of flow velocities covering conditions of nearly still water to flow velocity experienced in flash floods. In addition, the drag force was evaluated for flow toward the pier with different angle of inclinations. To achieve this goal, the finite element software abaqus/cfd was used with its computational fluid dynamics solving capabilities.

The governing equation used for the fluid movement is the Navier–Stokes equation for the motion of fluid with incompressible viscous flow Display Formula

(1)$ρ(∂v∂t+v.∇v)=−∇p+μ∇2v+ρg$

where ρ is the water density, v is the flow velocity, p is the pressure on pier, μ is the dynamic viscosity of water, and g is the gravity acceleration constant.

The drag force is calculated in the software by integrating the pressure profile over the area. It can be calculated in the flow direction as well as the direction perpendicular to the flow. Generally, the drag coefficients can be calculated through the following relationships: Display Formula

(2)$Cd=Fx12V2Ad$
Display Formula
(3)$Ct=Fy12V2At$

where Cd is drag coefficient in longitudinal direction, Ct is drag coefficient in transverse (lateral) direction, Fx is drag force parallel to the flow direction, Fy is drag force perpendicular to the flow direction, V is flow design velocity, Ad is the area of the edge perpendicular to longitudinal axis of pier, and At is the area of the edge parallel to longitudinal axis of pier.

Reynolds number (Re) is obtained through the relation Display Formula

(4)$Re=ρvLμ$

where L is a characteristic length of the object, which will be the pier width in this case.

## The Numerical Model

The study is based on simulating three-dimensional system for free water flow around different pier shapes. The shapes include square pier, semicircular-nosed pier, and 90 deg wedged-nosed pier, as shown in Fig. 1. The model flow area has dimensions of 20 m long and 8 m width. The square pier has a 1 m side length. The semicircular-nosed pier has a diameter of 1 m and different tail lengths. The 90 deg wedged-nosed pier also has 1 m width and different tail lengths.

The model was discretized to create a mesh using several finite element types and sizes to ensure the convergence of the results. Element types included hexagonal element, tetrahedral element, and pentahedron (wedge) element. The hexagonal element was considered with two techniques of meshing: structured mesh and random mesh.

The water is flowing at a temperature of 293 K, with a density of 1000 kg/m3 and a dynamic viscosity of 0.001 N s/m2. The change in these properties has almost no effect on the results, and hence kept constant throughout the analysis. Water is considered uncompressible here. The flow is assumed to be two-dimensional flow; in X–Y plane, the velocity in Z-direction is assumed zero. No-slip was assumed at the surface of the pier, so friction between water and pier body is negligible.

## Results and Discussion

To ensure convergence of the numerical model, several meshing types and sizes have been tested. The flow around square pier was used as base for choosing appropriate meshing element for the simulation. The conducted test included a flow in the X-direction with a velocity of 0.1 m/s. The results for different mesh types and number of nodes based on element size are presented in Fig. 2. The figure shows that using fine mesh, the drag coefficient ranges from 1.4 to 1.7 depending on the element type. The wedge element obtains an average value of 1.5, and hence, the wedge element is used for the rest of the study with element side edge of approximately 0.1 m.

The drag coefficient is obtained for square pier at velocities of 0.001, 0.01, 0.1, 1, 3, and 5 m/s, representing flow velocities of almost still water to fast river flow. Figure 3 show the pressure contours at the typical velocities of 0.001–1 m/s, where the flow is perpendicular to pier surface. The pressure contour of flow at velocity 0.001 m/s is noticed to be the only case of symmetrical flow around the pier with no occurrence of vortices. The pressure contours at velocity 1 m/s show the formation of vortices behind the pier, which is why the pressure contour is not symmetric. They usually form in turbulent flow behind the piers. Vortex initiation results from the interaction of two streams that were separated from the original stream due to the obstacle (pier), after solving Navier–Stokes equations. The finite element software abaqus solver utilizes a hybrid discretization built on the integral conservation statements for an arbitrary deforming domain. It also uses an advanced second-order projection method with a node-centered finite element discretization for the pressure to handle time-dependent part. An edge-based implementation is used for all transport equations permitting a single implementation that spans a broad variety of element topologies ranging from simple tetrahedral and hexahedral elements to arbitrary polyhedral. This has the capability to simulate the initiation and formation of vortices behind the pier.

The change in drag coefficient with Reynolds number is plotted in Fig. 4. The current results are seen to be considerably less than most of the results obtained by other researchers [7,16,17,21] under similar circumstances (i.e., similar Reynolds number range, for water flow). Results in literature obtained for gas flow or at lower Reynolds number are believed not to be suitable for comparisons purposes here, and hence are not included. The current study results of drag coefficient are noticed to coincide with AASHTO results of 1.4 at Reynolds numbers ranging from 104 to 106, and deviate out of this range of Reynolds numbers. The drag coefficient is found to be about 1.8 at Reynolds number of 1000, which is at a flow velocity of 0.001 m/s. This can be neglected for pier design purposes since it represents still water and will not cause any significant force on the pier. However, when the Reynolds number is higher than 106, the drag coefficient is seen to be higher than the AASHTO value and could reach to 1.6 for a flow velocity of 5 m/s, which can occur in some rivers or flash floods. Therefore, the AASHTO value might not be safe for this higher range of flow velocities.

For flow attacking the pier at an angle, as shown in Fig. 5, AASHTO suggests using a lateral drag coefficient to calculate lateral force on the pier in addition to the longitudinal (in flow direction) force. The model for square pier is used to simulate the water flow occurring at angles θ = 0, 5, 10, 20, 30, and 45 deg, and to investigate the orientation effect on drag coefficient at a flow velocity of 1 m/s.

The pressure contours for different flow angles are obtained, where Fig. 6 illustrates the typical contours for angles θ = 10 deg and θ = 20 deg. The drag coefficient is plotted versus time for these flow angles in Fig. 7, where it shows longitudinal and lateral drag coefficients. If resultant of the two forces in Eqs. (2) and (3) is obtained, one can find a combined drag coefficient $Ccomb$ for the square pier for flow at specific angle θ

where A is the area of pier perpendicular to longitudinal flow component divided by cos(θ). The combined drag coefficient is also drawn in Fig. 7. At a flow angle of 5 deg, the longitudinal drag coefficient is oscillating around a value of 1.4 while the lateral one is oscillating around a value of 0.2, compared to values of 1.4 and 0.5 in AASHTO, respectively. For a flow angle of 10 deg, the drag coefficients are around 1.5 and 0.5 for longitudinal and lateral coefficients, while it is 1.4 and 0.7 in AASHTO, respectively. It is clear that AASHTO value for longitudinal drag coefficient is independent of flow angle, while only lateral drag coefficient changes with the flow angle with conservative values. The combined drag coefficient is found to be oscillating around 1.5, 1.55, 1.8, 1.5, and 1.3 for angles of 5, 10, 20, 30, and 45 deg, respectively. This could explain why drag coefficient is taken around two in some codes and manuals that do not include flow orientation effect. It is worth mentioning also that the oscillation of the force on the piers due to the vortices formation could have a negative effect on the pier in terms of cyclic loading and fatigue. The dynamic nature of this was totally neglected in codes, and hence should be investigated and included in design codes explicitly. This is in agreement with the findings of Kallaka and Wang [22], where they found that piers may fail due to resonant vibration from vortex shedding for a moderate flow velocity. Figure 8 shows how drag coefficient changes with flow orientation angle, with comparison to AASHTO values as well as some results obtained by other researchers [15,16,21]. It is clear that other researchers mostly overestimate the longitudinal drag coefficient at most of flow orientation angle, in addition to having different relation pattern between drag coefficient and flow angle.

The drag coefficient of semicircular-nosed pier is also investigated, and it is plotted in Fig. 9 for different tail lengths at a flow velocity of 1 m/s, which is a typical value for rivers water velocity. The figure shows that the drag coefficient reduces with increasing pier length, approaching a value of 0.8 for a considerably long tail pier compared to its width. The AASHTO value is 0.7 in this case. However, the drag coefficient drops to about 0.4 for a fully circular pier at this water velocity, which coincides well with other data from literature.

The drag coefficient of wedged-nosed pier is also studied here and illustrated in Fig. 10 for different tail lengths at a flow velocity of 1 m/s. The figure shows that the drag coefficient also reduces with increasing pier tail length, approaching a value of 0.7 for a considerably long pier compared to its width. The AASHTO value is 0.8 in this case. However, the drag coefficient rises to about 1.3 for a fully wedged pier at this water velocity, which coincides well with the results of a square pier attacked by water at an angle of 45 deg.

The change in drag coefficient with Reynolds number is plotted in Fig. 11 for circular pier. The current results are seen to be equal to or less than most of the results documented by other researchers [11,12,2325] for Reynolds number ranging from 103 to 107, which covers the typical values of water flow pressuring circular bridge piers. It reaches a value of 0.45 at high Reynolds number, which is less than the semicircular-nosed pier value of about 0.7. Hence, using the AASHTO semicircular-nosed pier drag coefficient for fully circular piers will be very conservative. However, the current results of drag coefficient rise quickly for Reynolds number below 103 and diverge from other researchers values, but this range of Reynolds number is beyond the interest of this research. It is worth noting that experimental results on flow around circular cylindrical shapes usually show a dramatic drop in the drag coefficient between Reynolds number of 105 and 106 as a result of the change in the location of the separation point and reduction in the wake and negative pressure behind the pier. This phenomenon was translated into switching the flow regime from subcritical flow to supercritical flow [25]. In the model used for this study, this phenomenon was not captured.

The numerical analysis also showed clearly that the velocity of flow decreases directly behind the pier and increases on both sides behind the pier, as shown in Fig. 12, where the original flow velocity was 1 m/s and the increased flow velocity reaches to 1.4 m/s, which is 40% higher. If another pier exists in this path of faster flow, it would suffer higher drag force than the first pier, and this is not accounted for in the AASHTO code. Few researchers (such as Wenxiu et al. [7], Jempson [26], and Mussa et al. [27]) studied drag coefficients for two or more piers either parallel or directly behind each other, which is believed not to be the worst case. This is shown in Fig. 13 which shows the pressure contour for two tandem square piers, with the second pier shifted from being directly behind the original pier. The force exerted on the second pier causes the drag coefficient to oscillate about a value of 1.8, which is higher than that for the original pier value of about 1.35 coinciding with AASHTO value, as shown in Fig. 14.

## Conclusion and Recommendations

Wardhana and Hadipriono [28] found that flood and scour is responsible for about 53% of all bridge failures in U.S. between 1989 and 2000, which shows the high importance of right and accurate bridge and piers design for water flow and floods. Therefore, the present paper studies numerically the drag force and coefficient values of water flow for different pier shapes, and compares the results with AASHTO Bridge Design Code values as well as with values obtained by other researchers. The following conclusions can be drawn:

1. (1)The AASHTO longitudinal drag coefficient value of 1.4 for square piers might not be safe for high Reynolds numbers, which represent high river flow velocities. However, the AASHTO lateral drag coefficient values are in good agreement with the numerical values and are usually on the conservative side.
2. (2)The drag coefficient for the semicircular-nosed pier stated in AASHTO agrees well with the numerical results. However, this value can be very conservative if it is to be used for a perfectly circular pier, since AASHTO does not suggest a specific value for this case.
3. (3)The drag coefficient for wedged-nosed pier suggested in AASHTO agrees well with the numerically obtained one, only for a pier long enough compared to its width. For wedged-nosed piers with tail length-to-width ratio less than three, a value higher than that in the AASHTO should be used.
4. (4)It is recommended that the cyclic effect of the water flow force on bridge piers should be investigated thoroughly to ensure that fatigue problems are avoided in the piers.
5. (5)The effect of one pier on another should be studied and included in the design codes as it could lead to higher forces on the piers that are not accounted for by the current design methodologies. It could lead to piers failure due to the underestimation of these forces.

## References

AASHTO, 2013, “ AASHTO LRFD Bridge Design Specifications,” 6th ed., American Association of State Highway and Transportation Officials, Washington, DC.
FEMA, 2011, “ Coastal Construction Manual: Principles and Practices of Planning, Siting, Designing, Constructing, and Maintaining Residential Buildings in Coastal Areas,” 4th ed., Federal Emergency Management Agency, Washington, DC, Standard No. FEMA P-55.
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## References

AASHTO, 2013, “ AASHTO LRFD Bridge Design Specifications,” 6th ed., American Association of State Highway and Transportation Officials, Washington, DC.
FEMA, 2011, “ Coastal Construction Manual: Principles and Practices of Planning, Siting, Designing, Constructing, and Maintaining Residential Buildings in Coastal Areas,” 4th ed., Federal Emergency Management Agency, Washington, DC, Standard No. FEMA P-55.
McConnell, K. , Allsop, W. , and Cruickshank, I. , 2004, Piers, Jetties and Related Structures Exposed to Waves: Guidelines for Hydraulic Loadings, Thomas Telford Publishing, London.
U.S. Army Corp of Engineers, 1984, Shore Protection Manual, 4th ed., Department of the Army, Waterways Experiment Station, Corps of Engineers Coastal Engineering Research Center, Vicksburg, MS.
Sumer, B. , and Fredsoe, J. , 2006, Hydrodynamics Around Cylindrical Structures (Advanced Series on Ocean Engineering, Vol. 26), Revised ed., World Scientific Publishing, Ithaca, NY.
Debus, K. , Berkoe, J. , Rosendall, B. , and Shakib, F. , 2003, “ Computational Fluid Dynamics Model for Tacoma Narrows Bridge Upgrade Project,” ASME Paper No. FEDSM2003-45514.
Wenxiu, L. , Bing, Z. , and Xulin, H. , 2011, “ Two-Dimensional Numerical Simulation of Drag Coefficients on Two Square Cylinders at High Reynolds Number,” International Conference on Opto-Electronics Engineering and Information Science, Xi'an, China, Dec. 23–25, Vol. 3, pp. 2283–2286.
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## Figures

Fig. 1

Vertical and horizontal dimensions of the models (dimensions in meters): (a) square pier, (b) semicircular-nosed pier, and (c) wedged-nosed pier

Fig. 2

Convergence of drag coefficient value for different types of finite elements

Fig. 3

Pressure contours at (a) velocity of 0.001 m/s and (b) velocity of 1 m/s for square pier

Fig. 4

Longitudinal drag coefficient at different Reynolds numbers

Fig. 5

Inclined flow direction

Fig. 6

Pressure contours at flow angle of (a) θ = 10 deg and (b) θ = 20 deg, for square pier at flow velocity of 1 m/s

Fig. 7

Drag coefficient versus time at different flow angles for a square pier at 1 m/s flow velocity: (a) V = 1 m/s, angle = 5 deg; (b) V = 1 m/s, angle = 10 deg; (c) V = 1 m/s, angle = 20 deg; (d) V = 1 m/s, angle = 30 deg; and (e) V = 1 m/s, angle = 45 deg

Fig. 8

Drag coefficient for a square pier for different flow orientation angles

Fig. 9

Longitudinal drag coefficient for semicircular-nosed pier

Fig. 10

Longitudinal drag coefficient for wedged-nosed pier

Fig. 11

Drag coefficient versus Reynolds number for circular pier

Fig. 12

Velocity contour around square pier at flow velocity of 1 m/s

Fig. 13

Velocity contour around two tandem square piers at flow velocity of 1 m/s

Fig. 14

Drag coefficient of two tandem square piers at flow velocity of 1 m/s

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