Simulation of 3D Flow and Bed Shear Stress
Fields around A Circular Pier
Wen-Yi Chang 1,
Jihn-Sung Lai 2 and Chin-lien Yen 3
1 Ph. D. Student, Department of
Civil Engineering, National Taiwan University (NTU),
Taipei, Taiwan, China
2 Assistant Research Fellow,
Hydrotech Research Institute, NTU
3 Professor, Department of Civil
Engineering; Senior Research Fellow, Hydrotech Research Institute, NTU
Address: 1, 2 and 3: 158 Chow Shan
Road, Taipei 106, Taiwan, China
Telephone: 1: (886-2) 23630231 ext. 3250 ext. 322; 2:
(886-2) 23644512; 3: (886-2) 23630489
Fax: 1 and 3: (886-2) 23639258; 2:
(886-2) 23644512
E-mail: 1: wychnag@hy.ntu.edu.tw;
2: jslai@hy.ntu.edu.tw; 3: clyen@ccms.ntu.edu.tw
Abstract: Water flowing around piers forms complex three-dimensional (3D)
flow patterns that induce local scour on the loose bed boundary. The large-eddy simulation (LES) approach
with Smagorinsky’s subgrid-scale turbulence model is employed for computing
the 3D flow and bed shear stress fields. The
simulated results are compared with experimental data. To investigate the scour mechanism
around the pier, the flow fields in various fixed scour holes with similar shape
are simulated, and the simulated results of bed shear stress and downflow are
discussed herein.
Keywords: pier, scour, 3D flow, downflow, bed shear stress
1 Introduction
Three-dimensional (3D) flow patterns around piers having local
scour hole are essentially complicated. In
recent years, several 3D numerical models have been developed for simulating the
flow field and bed scouring around piers. Dou
(1997) solved the 3D Navier-Stokes equations with the stochastic
turbulence-closure model to simulate the local scour at bridge crossing. Olsen and Kjellesvig (1998) computed the
development of scour hole by solving the 3D Navier-Stokes equations with
model for Reynolds stresses, and
the convection-diffusion equation for sediment transport. Richardson and Panchang (1998) used a
computational fluid dynamics model called Flow-3D® to compute the
flow field around a pier within a given fixed scour hole. In the present study, the LES approach
with Smagorinsky’s subgrid-scale turbulence model is employed for solving the
3D weakly compressible flow equations (Yen et al., 2001). The experimental data obtained by Ahmed
(1998) are adopted to validate the proposed 3D flow model. For further investigation on the scour
mechanism during scouring process, the flow fields within similar-shaped fixed
scour holes are simulated. Then,
the simulated results of bed shear stress and downflow are compared and
discussed.
2 Governing Equations
The mathematical expression of weakly
compressible flow equations and the LES approach with subgrid-scale turbulence
model are briefly stated herein. Further
details are given elsewhere (Yen et al., 2001).
When Mach number (M) of flow is small and, hence, the effect of density change is
negligibly small, the equation of state for fluid can be represented by the
following linear equation:
(1)
where p is the pressure; a is the speed of sound;
is the density of water; and the subscript “O” represents the reference value.
By substituting Equation (1) into the equations
of continuity and motion with
for weakly compressible flow, the
governing equations become
(2)
(3)
where
and
are the coordinates,
and
;
is the
-component of velocity;
is the
-component of velocity;
is the kinematic viscosity of
water; and t is
time. Equations (2) and (3) are defined as weakly compressible flow
equations, which are employed as the primary governing equations in this study
for hydraulic transient flow.
The concept of large eddy simulation is adopted for directly
computing the large-scale turbulence that can be resolved with the computational
mesh size and to model only the unresolvable small-scale turbulence. Based on the above concept and the
averaged quantity over a finite volume, the subgrid turbulence stress term (
) is produced on the right hand side of Equation (3). The subgrid turbulence stress modeled by
Smagorinsky (1963) is expressed as:
(4)
where
, is the subgrid eddy viscosity coefficient;
, is the resolvable rate of deformation tensor;
is the subgrid length scale
dependent on mesh size; and C is the Smagorinsky constant which is the only adjustable
parameter in the subgrid scale turbulence (SGS) model.
3 Validation of the
model
The experimental data of C2R case obtained by
Ahmed (1998) are employed to compare with simulated results. The basic information in this case: the
approaching mean velocity (
) is 0.2927 m/s; the flow depth (
) is 0.182 m; the approaching bed shear stress (
) is 0.4584
; the pier diameter (b) is 0.089 m; and the sediment size (
) is
m.
The simulated domain is
,
, and
. The upstream boundary condition is imposed by the measured velocity profile
along z-direction. The downstream
boundary condition is given by Neumann condition (
). The condition on the sides
(i.e.
) is given by fully slip condition (
and
). The boundary condition on the
water surface is also given by the fully slip condition. The solid boundary condition is given by
the partial slip condition (wall function method) to avoid too much
computational mesh constructed near the bed and pier wall. In addition, the Smagorinsky constant is
calibrated to be 0.15.
According to the wall function method, the bed
shear stress can be calculated by the following relation:
(5)
where
is the bed shear stress;
is the velocity near the bed;
is von Karman constant;
is the shortest distance from the
bed to the position of
;
for
;
is the shear velocity;
is the roughness.
Figure 1 presents the comparison of simulated
and measured velocity profiles along the centerline in front of the pier, and
the simulated results agree with the measured quite well. Figure 2 presents the simulated bed shear stresses compared
with measured data along the centerline in front of the pier and the 90°-line which is normal to the direction of incoming flow. In general, the simulated bed shear stress distributions are
in good agreement with the experiment except the area near the maximum bed shear
stress.
4 Simulation
In order to study the variation of bed shear
stress and the maximum downflow velocity in the scouring process, several
simulation runs have been conducted with the same flow condition as the
experiment C2R by Ahmed (1998). The
simulated flow fields within the fixed scour holes with the various maximum
scour depths are shown in Figure 3.
As can be seen in Figure 4, the maximum bed
shear stress in front of the pier increases with increasing depth of scour hole.
The variation of the maximum downflow velocity, as shown in Figure 5, exhibits a
similar trend. This seems to indicate that the development of downflow in the
scouring process further strengthens the capability of the flow to scour in
front of the pier. Figure 6
presents the variation of bed shear stress along the 90°-line as the depth of
scour hole increases. The simulated results show that the maximum bed shear
stress along the 90°-line decreases with increasing depth of
scour hole. This phenomenon can be directly explained by the continuity of flow.
Furthermore, the variation of bed shear stress in front of the pier is quite
different from that along the 90°-line.
5 Conclusion
The LES approach with Smagorinsky’s subgrid-scale
turbulence model is employed for computing the velocity and bed shear stress
fields in 3D flow simulation. The simulated results are in reasonably good
agreement with measured data. Furthermore,
the development of downflow seems to strength the capability of the flow to
scour in front of the pier in the scouring process. The bed shear stress in
front of the pier increases with increasing depth of scour hole, while the bed
shear stress along the 90°-line decreases.
Acknowledgements
The research facilities provided for the study
reported herein by the Hydrotech Research Institute, National Taiwan
University, is hereby gratefully acknowledged. Financial support by the National
Science Council, under grant NSC 89-2211-E-002-139 is highly appreciated.
References
[1] Ahmed, F. and Rajaratnam,
N. (1998)“Flow
around bridge piers.”J. of Hydr. Eng. ASCE,
124(3), 288-300.
[2] Dou, X. (1997)“Numerical simulation
of three-dimensional flow field and local scour at bridge crossings. ”Ph.D. Dissertation. University of Mississippi, Oxford, MS, U.S.A.
[3] Olsen, N.R.B. and
Kjellesvig, H.M. (1998).“Three-dimensional numerical flow modeling for estimation of maximum
local scour depth.”J. of Hydr. Res., IAHR, 36(4), 579-590.
[4] Richardson, J.E. and
Panchang, V.G. (1998).“Three-dimensional
simulation of scour-inducing flow at bridge piers.” J. of Hydr. Eng. ASCE, 124(5), 530-540.
[5] Smagorinsky, J (1963). “General circulation
experiments with primitive equation.”
Monthly Weather Review, 91(3), 99-154.
[6] Yen, C.L., Lai, J.S., and
Chang, W.Y. (2001). “Modeling of 3D flow and scouring around piers.” Proc. of the National Science Council, Part A: Phys. Sci. and
Eng., 25(1).
Fig. 1 Comparison of simulated and measured
velocity profiles along the centerline in front of the pier
Fig. 2
Comparison of simulated and measured bed shear stress along (a) centerline in
front of the pier, and (b) 90o-line
(a)
= -0.2 (b)
= -0.4
(c)
= -0.6
Fig. 3 Given fixed scour holes in various
sizes for flow simulation
(
= maximum scour depth)
Fig. 4 Variations of (a) bed shear stress along the centerline
in front of the pier, and (b) its maximum value with depth of scour hole
Fig.
5 Variation of the maximum downflow velocity with depth of
scour hole
(a)
(b)
Fig. 6 Riations of (a) bed shear stress along 90o-line, and (b) its maximum value with depth of scour hole
